%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU667^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n009.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 09:36:04 AM UTC 2026
% Result : Theorem 2.25s 1.38s
% Output : Refutation 2.25s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 5
% Syntax : Number of formulae : 28 ( 13 unt; 0 typ; 0 def)
% Number of atoms : 2898 ( 770 equ; 0 cnn)
% Maximal formula atoms : 184 ( 103 avg)
% Number of connectives : 1564 ( 21 ~; 8 |; 552 &; 248 @)
% ( 1 <=>; 734 =>; 0 <=; 0 <~>)
% Maximal formula depth : 194 ( 50 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 40 ( 40 >; 0 *; 0 +; 0 <<)
% Number of symbols : 335 ( 332 usr; 189 con; 0-4 aty)
% Number of variables : 125 ( 65 ^; 42 !; 18 ?; 125 :)
% 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_234,type,
sP3: $o > $i > $i > $i > $o ).
thf(func_def_235,type,
sP4: $i > $i > $o ).
thf(func_def_236,type,
sP5: $i > $o ).
thf(func_def_237,type,
sP6: $i > $i > $o ).
thf(func_def_238,type,
sP7: $i > $i > $o ).
thf(func_def_242,type,
sK11: $i > $i > $i ).
thf(func_def_243,type,
sK12: $i > $i > $i ).
thf(func_def_248,type,
sK17: $i > $o ).
thf(func_def_251,type,
sK20: $i > ( $i > $o ) > $i ).
thf(func_def_252,type,
sK21: $i > $o ).
thf(func_def_259,type,
sK28: $i > $i > $i ).
thf(func_def_263,type,
sK32: $i > $o ).
thf(func_def_265,type,
sK34: ( $i > $o ) > $i > $i ).
thf(func_def_266,type,
sK35: ( $i > $o ) > $i > $i ).
thf(func_def_267,type,
sK36: $i > $i > $o ).
thf(func_def_278,type,
sK47: $i > $o ).
thf(func_def_279,type,
sK48: ( $i > $o ) > $i ).
thf(func_def_289,type,
sK58: $i > $o ).
thf(func_def_292,type,
sK61: $i > ( $i > $o ) > $i ).
thf(func_def_293,type,
sK62: $i > ( $i > $o ) > $i ).
thf(func_def_294,type,
sK63: $i > $o ).
thf(func_def_296,type,
sK65: ( $i > $o ) > $i ).
thf(func_def_297,type,
sK66: $i > $o ).
thf(func_def_298,type,
sK67: $i > $i ).
thf(func_def_302,type,
sK71: $i > $o ).
thf(func_def_304,type,
sK73: ( $i > $o ) > $i ).
thf(func_def_305,type,
sK74: ( $i > $o ) > $i ).
thf(func_def_309,type,
sK78: $i > $o ).
thf(func_def_312,type,
sK81: $i > ( $i > $o ) > $i ).
thf(func_def_313,type,
sK82: $i > ( $i > $o ) > $i ).
thf(func_def_322,type,
sK91: $i > $i > $i ).
thf(func_def_326,type,
sK95: $i > ( $i > $o ) > $i > $i ).
thf(func_def_328,type,
sK97: $i > $o ).
thf(func_def_334,type,
sK103: $i > $o ).
thf(func_def_345,type,
sK114: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_346,type,
sK115: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_347,type,
sK116: $i > $o ).
thf(func_def_348,type,
sK117: $i > $o ).
thf(func_def_359,type,
sK128: $i > $o ).
thf(func_def_366,type,
sK135: $i > $i ).
thf(func_def_367,type,
sK136: $i > $i ).
thf(func_def_389,type,
sK158: $i > $i > $o ).
thf(func_def_390,type,
sK159: $i > $i ).
thf(func_def_391,type,
sK160: $i > $i ).
thf(func_def_392,type,
sK161: ( $i > $i > $o ) > $i > $i ).
thf(func_def_393,type,
sK162: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_394,type,
sK163: ( $i > $i > $o ) > $i > $i ).
thf(func_def_397,type,
sK166: $i > $o ).
thf(func_def_400,type,
sK169: $i > $o ).
thf(func_def_415,type,
sK184: $i > $o ).
thf(func_def_419,type,
sK188: $i > $i > $i ).
thf(func_def_427,type,
sK196: $i > $i > $i > $i ).
thf(func_def_428,type,
sK197: $i > $i > $i > $i ).
thf(func_def_434,type,
sK203: $i > $o ).
thf(func_def_439,type,
sK208: $i > $o ).
thf(func_def_451,type,
sK220: $i > $i > $i ).
thf(func_def_452,type,
sK221: $i > ( $i > $o ) > $i ).
thf(func_def_453,type,
sK222: $i > $o ).
thf(func_def_458,type,
sK227: $i > ( $i > $o ) > $i ).
thf(func_def_459,type,
sK228: $i > $o ).
thf(func_def_473,type,
sK242: ( $i > $o ) > $i > $i ).
thf(func_def_475,type,
sK244: $i > $o ).
thf(func_def_479,type,
sK248: $i > $o ).
thf(func_def_483,type,
sK252: $i > $i ).
thf(func_def_485,type,
sK254: $i > $i ).
thf(func_def_498,type,
sK267: $i > $i > $i ).
thf(func_def_500,type,
sK269: $i > $i ).
thf(func_def_519,type,
sK288: ( $i > $o ) > $i ).
thf(func_def_520,type,
sK289: $i > $o ).
thf(func_def_521,type,
sK290: $i > $i ).
thf(func_def_532,type,
sK301: $i > $o ).
thf(func_def_538,type,
sK307: $i > ( $i > $o ) > $i ).
thf(func_def_539,type,
sK308: $i > $o ).
thf(func_def_542,type,
sK311: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_543,type,
sK312: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_544,type,
sK313: $i > $o ).
thf(func_def_545,type,
sK314: $i > $o ).
thf(func_def_548,type,
sK317: $i > $o ).
thf(func_def_549,type,
sK318: $i > $o ).
thf(func_def_550,type,
sK319: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_551,type,
sK320: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_555,type,
sK324: $i > $i > $o ).
thf(func_def_563,type,
sK332: $i > $i > $i ).
thf(func_def_571,type,
sK340: $i > $i > $i ).
thf(func_def_610,type,
sK379: $i > $i > $i ).
thf(func_def_611,type,
sK380: $i > $i > $i ).
thf(func_def_612,type,
sK381: $i > $i > $i ).
thf(func_def_613,type,
sK382: $i > $i > $i ).
thf(func_def_614,type,
sK383: $i > $i > $i ).
thf(func_def_615,type,
sK384: $i > $i > $i > $i ).
thf(func_def_616,type,
sK385: $i > $i ).
thf(func_def_617,type,
sK386: $i > $i ).
thf(func_def_618,type,
sK387: $i > $i ).
thf(func_def_619,type,
sK388: $i > $i ).
thf(func_def_620,type,
sK389: $i > $i > $i ).
thf(func_def_621,type,
sK390: $i > $i > $i > $i ).
thf(func_def_622,type,
sK391: $i > $i > $i > $i ).
thf(func_def_623,type,
sK392: $i > $i > $i ).
thf(func_def_624,type,
sK393: $i > $i > $i ).
thf(func_def_625,type,
sK394: $i > $i ).
thf(func_def_627,type,
sK396: $i > $i ).
thf(func_def_628,type,
sK397: $i > $i ).
thf(func_def_630,type,
sK399: $i > $i > $o ).
thf(func_def_633,type,
sK402: $i > $i ).
thf(func_def_645,type,
sK414: $i > $o ).
thf(func_def_646,type,
sK415: $i > ( $i > $o ) > $i ).
thf(func_def_647,type,
sK416: $i > $o ).
thf(func_def_683,type,
sK452: $i > $i > $i ).
thf(func_def_698,type,
sK467: $i > $o ).
thf(func_def_706,type,
sK475: $i > $i ).
thf(func_def_718,type,
sK487: $i > $o ).
thf(func_def_720,type,
sK489: ( $i > $o ) > $i > $i ).
thf(func_def_727,type,
sK496: $o > $i > $i > $i ).
thf(func_def_751,type,
ph520:
!>[X0: $tType] : X0 ).
thf(f3481,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f3355,f3420]) ).
thf(f3420,plain,
! [X2: $i,X0: $i > $i > $o,X1: $i] :
( $true
= ( subset @ ( dpsetconstr @ X1 @ X2 @ X0 ) @ ( cartprod @ X1 @ X2 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f3419]) ).
thf(f3419,plain,
! [X2: $i,X0: $i > $i > $o,X1: $i] :
( $true
= ( subset
@ ( dpsetconstr @ X1 @ X2
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) )
@ ( cartprod @ X1 @ X2 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2917]) ).
thf(f2917,plain,
! [X2: $i,X0: $i > $i > $o,X1: $i] :
( ( $true
= ( subset
@ ( dpsetconstr @ X1 @ X2
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) )
@ ( cartprod @ X1 @ X2 ) ) )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f2068,f2265]) ).
thf(f2265,plain,
dpsetconstrSub = $true,
inference(cnf_transformation,[status(thm)],[f1434]) ).
thf(f1434,plain,
( ( omega__Cong = $true )
& ( subset2powerset = $true )
& ( setukpairIL = $true )
& ( setext = $true )
& ( uniqinunit = $true )
& ( binintersectSubset5 = $true )
& ( setukpairinjR = $true )
& ( exuE1 = $true )
& ( emptyset__Cong = $true )
& ( sepSubset = $true )
& ( exuEu = $true )
& ( ksndpairEq = $true )
& ( setadjoinE = $true )
& ( prop2set2propI = $true )
& ( emptyInPowerset = $true )
& ( powersetE1 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjR12 = $true )
& ( notinemptyset = $true )
& ( cartprodpairin = $true )
& ( setminusELneg = $true )
& ( emptyinPowerset = $true )
& ( setadjoinSub = $true )
& ( setukpairinjL1 = $true )
& ( cartprodpairmemEL = $true )
& ( powersetsubset = $true )
& ( setminusERneg = $true )
& ( symdiffIneg2 = $true )
& ( setminusIRneg = $true )
& ( in__Cong = $true )
& ( dsetconstrER = $true )
& ( sepInPowerset = $true )
& ( setbeta = $true )
& ( setoftrueEq = $true )
& ( singletonsswitch = $true )
& ( notinsingleton = $true )
& ( descr__Cong = $true )
& ( powersetE = $true )
& ( exuI3 = $true )
& ( kpairiskpair = $true )
& ( binunionRsub = $true )
& ( binunionE = $true )
& ( emptyE1 = $true )
& ( binunionLsub = $true )
& ( subsetI1 = $true )
& ( setukpairinjL2 = $true )
& ( singletonsubset = $true )
& ( bs114d = $true )
& ( kpairsurjEq = $true )
& ( eqinunit = $true )
& ( setukpairinjR2 = $true )
& ( exuI2 = $true )
& ( dsetconstrI = $true )
& ( upairset2IR = $true )
& ( singletoninpowunion = $true )
& ( setextsub = $true )
& ( upairset2E = $true )
& ( upairsetE = $true )
& ( upairinpowunion = $true )
& ( subPowSU = $true )
& ( setadjoin__Cong = $true )
& ( subsetI2 = $true )
& ( ex1E1 = $true )
& ( wellorderingAx = $true )
& ( setunionsingleton = $true )
& ( emptysetimpfalse = $true )
& ( ex1I2 = $true )
& ( setunionsingleton1 = $true )
& ( singletoninpowerset = $true )
& ( setunionAx = $true )
& ( omega0Ax = $true )
& ( quantDeMorgan3 = $true )
& ( notsubsetI = $true )
& ( binintersectLsub = $true )
& ( setukpairIR = $true )
& ( symdiffIneg1 = $true )
& ( setminusEL = $true )
& ( nonemptyI1 = $true )
& ( upairsetIR = $true )
& ( disjointsetsI1 = $true )
& ( quantDeMorgan1 = $true )
& ( setadjoinIR = $true )
& ( setunionE = $true )
& ( kfstsingleton = $true )
& ( eqimpsubset1 = $true )
& ( nonemptyE1 = $true )
& ( binintersectI = $true )
& ( subsetE = $true )
& ( cartprodsndpairEq = $true )
& ( quantDeMorgan2 = $true )
& ( notdallE = $true )
& ( cartprodpairsurjEq = $true )
& ( setminusILneg = $true )
& ( setminusSubset2 = $true )
& ( kfstpairEq = $true )
& ( noeltsimpempty = $true )
& ( binintersectSubset4 = $true )
& ( dpsetconstrSub = $true )
& ( exuE3e = $true )
& ( binunionEcases = $true )
& ( cartprodfstin = $true )
& ( secondinupair = $true )
& ( ubforcartprodlem2 = $true )
& ( binintersectER = $true )
& ( singletonsuniq = $true )
& ( setunionI = $true )
& ( vacuousDall = $true )
& ( cartprodsndin = $true )
& ( setminusLsub = $true )
& ( upairsetIL = $true )
& ( setadjoinIL = $true )
& ( cartprodmempair1 = $true )
& ( binintersectEL = $true )
& ( prop2setI = $true )
& ( replAx = $true )
& ( ksndsingleton = $true )
& ( cartprodfstpairEq = $true )
& ( setunionE2 = $true )
& ( symdiffI1 = $true )
& ( cartprodmempair = $true )
& ( subsetemptysetimpeq = $true )
& ( inCongP = $true )
& ( emptysetAx = $true )
& ( ex1I = $true )
& ( setextAx = $true )
& ( quantDeMorgan4 = $true )
& ( emptyI = $true )
& ( symdiffI2 = $true )
& ( dsetconstrEL = $true )
& ( upairsubunion = $true )
& ( subsetRefl = $true )
& ( prop2setE = $true )
& ( ubforcartprodlem3 = $true )
& ( setadjoinAx = $true )
& ( binintersectSubset2 = $true )
& ( notdexE = $true )
& ( setminusI = $true )
& ( exuI1 = $true )
& ( powersetAx = $true )
& ( dpsetconstrI = $true )
& ( symdiffE = $true )
& ( nonemptyImpWitness = $true )
& ( foundationAx = $true )
& ( setminusSubset1 = $true )
& ( emptysetE = $true )
& ( setadjoinOr = $true )
& ( binintersectRsub = $true )
& ( notequalI1 = $true )
& ( inPowerset = $true )
& ( setunionsingleton2 = $true )
& ( setunion__Cong = $true )
& ( exu__Cong = $true )
& ( notequalI2 = $true )
& ( nonemptyI = $true )
& ( ubforcartprodlem1 = $true )
& ( emptyinunitempty = $true )
& ( subsetTrans = $true )
& ( emptysetsubset = $true )
& ( setukpairinjR1 = $true )
& ( theprop = $true )
& ( binintersectSubset3 = $true )
& ( omegaSAx = $true )
& ( exuE2 = $true )
& ( subsetE2 = $true )
& ( kpairp = $true )
& ( setukpairinjL = $true )
& ( descrp = $true )
& ( setminusER = $true )
& ( powersetI1 = $true )
& ( singletonprop = $true )
& ( dsetconstr__Cong = $true )
& ( binintersectSubset1 = $true )
& ( powersetI = $true )
& ( setadjoinSub2 = $true )
& ( eqimpsubset2 = $true )
& ( omegaIndAx = $true )
& ( binunionIR = $true )
& ( powerset__Cong = $true )
& ( cartprodpairmemER = $true )
& ( upairequniteq = $true )
& ( $true
!= ( breln @ sK400 @ sK398
@ ( dpsetconstr @ sK400 @ sK398
@ ^ [Y0: $i,Y1: $i] : ( sK399 @ Y0 @ Y1 ) ) ) )
& ( cartprodmempaircEq = $true )
& ( binunionIL = $true )
& ( exuE3u = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK398,sK399,sK400])],[f773,f1433]) ).
thf(f1433,plain,
( ? [X0: $i,X1: $i > $i > $o,X2: $i] :
( $true
!= ( breln @ X2 @ X0
@ ( dpsetconstr @ X2 @ X0
@ ^ [Y0: $i,Y1: $i] : ( X1 @ Y0 @ Y1 ) ) ) )
=> ( $true
!= ( breln @ sK400 @ sK398
@ ( dpsetconstr @ sK400 @ sK398
@ ^ [Y0: $i,Y1: $i] : ( sK399 @ Y0 @ Y1 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f773,plain,
( ( omega__Cong = $true )
& ( subset2powerset = $true )
& ( setukpairIL = $true )
& ( setext = $true )
& ( uniqinunit = $true )
& ( binintersectSubset5 = $true )
& ( setukpairinjR = $true )
& ( exuE1 = $true )
& ( emptyset__Cong = $true )
& ( sepSubset = $true )
& ( exuEu = $true )
& ( ksndpairEq = $true )
& ( setadjoinE = $true )
& ( prop2set2propI = $true )
& ( emptyInPowerset = $true )
& ( powersetE1 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjR12 = $true )
& ( notinemptyset = $true )
& ( cartprodpairin = $true )
& ( setminusELneg = $true )
& ( emptyinPowerset = $true )
& ( setadjoinSub = $true )
& ( setukpairinjL1 = $true )
& ( cartprodpairmemEL = $true )
& ( powersetsubset = $true )
& ( setminusERneg = $true )
& ( symdiffIneg2 = $true )
& ( setminusIRneg = $true )
& ( in__Cong = $true )
& ( dsetconstrER = $true )
& ( sepInPowerset = $true )
& ( setbeta = $true )
& ( setoftrueEq = $true )
& ( singletonsswitch = $true )
& ( notinsingleton = $true )
& ( descr__Cong = $true )
& ( powersetE = $true )
& ( exuI3 = $true )
& ( kpairiskpair = $true )
& ( binunionRsub = $true )
& ( binunionE = $true )
& ( emptyE1 = $true )
& ( binunionLsub = $true )
& ( subsetI1 = $true )
& ( setukpairinjL2 = $true )
& ( singletonsubset = $true )
& ( bs114d = $true )
& ( kpairsurjEq = $true )
& ( eqinunit = $true )
& ( setukpairinjR2 = $true )
& ( exuI2 = $true )
& ( dsetconstrI = $true )
& ( upairset2IR = $true )
& ( singletoninpowunion = $true )
& ( setextsub = $true )
& ( upairset2E = $true )
& ( upairsetE = $true )
& ( upairinpowunion = $true )
& ( subPowSU = $true )
& ( setadjoin__Cong = $true )
& ( subsetI2 = $true )
& ( ex1E1 = $true )
& ( wellorderingAx = $true )
& ( setunionsingleton = $true )
& ( emptysetimpfalse = $true )
& ( ex1I2 = $true )
& ( setunionsingleton1 = $true )
& ( singletoninpowerset = $true )
& ( setunionAx = $true )
& ( omega0Ax = $true )
& ( quantDeMorgan3 = $true )
& ( notsubsetI = $true )
& ( binintersectLsub = $true )
& ( setukpairIR = $true )
& ( symdiffIneg1 = $true )
& ( setminusEL = $true )
& ( nonemptyI1 = $true )
& ( upairsetIR = $true )
& ( disjointsetsI1 = $true )
& ( quantDeMorgan1 = $true )
& ( setadjoinIR = $true )
& ( setunionE = $true )
& ( kfstsingleton = $true )
& ( eqimpsubset1 = $true )
& ( nonemptyE1 = $true )
& ( binintersectI = $true )
& ( subsetE = $true )
& ( cartprodsndpairEq = $true )
& ( quantDeMorgan2 = $true )
& ( notdallE = $true )
& ( cartprodpairsurjEq = $true )
& ( setminusILneg = $true )
& ( setminusSubset2 = $true )
& ( kfstpairEq = $true )
& ( noeltsimpempty = $true )
& ( binintersectSubset4 = $true )
& ( dpsetconstrSub = $true )
& ( exuE3e = $true )
& ( binunionEcases = $true )
& ( cartprodfstin = $true )
& ( secondinupair = $true )
& ( ubforcartprodlem2 = $true )
& ( binintersectER = $true )
& ( singletonsuniq = $true )
& ( setunionI = $true )
& ( vacuousDall = $true )
& ( cartprodsndin = $true )
& ( setminusLsub = $true )
& ( upairsetIL = $true )
& ( setadjoinIL = $true )
& ( cartprodmempair1 = $true )
& ( binintersectEL = $true )
& ( prop2setI = $true )
& ( replAx = $true )
& ( ksndsingleton = $true )
& ( cartprodfstpairEq = $true )
& ( setunionE2 = $true )
& ( symdiffI1 = $true )
& ( cartprodmempair = $true )
& ( subsetemptysetimpeq = $true )
& ( inCongP = $true )
& ( emptysetAx = $true )
& ( ex1I = $true )
& ( setextAx = $true )
& ( quantDeMorgan4 = $true )
& ( emptyI = $true )
& ( symdiffI2 = $true )
& ( dsetconstrEL = $true )
& ( upairsubunion = $true )
& ( subsetRefl = $true )
& ( prop2setE = $true )
& ( ubforcartprodlem3 = $true )
& ( setadjoinAx = $true )
& ( binintersectSubset2 = $true )
& ( notdexE = $true )
& ( setminusI = $true )
& ( exuI1 = $true )
& ( powersetAx = $true )
& ( dpsetconstrI = $true )
& ( symdiffE = $true )
& ( nonemptyImpWitness = $true )
& ( foundationAx = $true )
& ( setminusSubset1 = $true )
& ( emptysetE = $true )
& ( setadjoinOr = $true )
& ( binintersectRsub = $true )
& ( notequalI1 = $true )
& ( inPowerset = $true )
& ( setunionsingleton2 = $true )
& ( setunion__Cong = $true )
& ( exu__Cong = $true )
& ( notequalI2 = $true )
& ( nonemptyI = $true )
& ( ubforcartprodlem1 = $true )
& ( emptyinunitempty = $true )
& ( subsetTrans = $true )
& ( emptysetsubset = $true )
& ( setukpairinjR1 = $true )
& ( theprop = $true )
& ( binintersectSubset3 = $true )
& ( omegaSAx = $true )
& ( exuE2 = $true )
& ( subsetE2 = $true )
& ( kpairp = $true )
& ( setukpairinjL = $true )
& ( descrp = $true )
& ( setminusER = $true )
& ( powersetI1 = $true )
& ( singletonprop = $true )
& ( dsetconstr__Cong = $true )
& ( binintersectSubset1 = $true )
& ( powersetI = $true )
& ( setadjoinSub2 = $true )
& ( eqimpsubset2 = $true )
& ( omegaIndAx = $true )
& ( binunionIR = $true )
& ( powerset__Cong = $true )
& ( cartprodpairmemER = $true )
& ( upairequniteq = $true )
& ? [X0: $i,X1: $i > $i > $o,X2: $i] :
( $true
!= ( breln @ X2 @ X0
@ ( dpsetconstr @ X2 @ X0
@ ^ [Y0: $i,Y1: $i] : ( X1 @ Y0 @ Y1 ) ) ) )
& ( cartprodmempaircEq = $true )
& ( binunionIL = $true )
& ( exuE3u = $true ) ),
inference(flattening,[status(thm)],[f772]) ).
thf(f772,plain,
( ? [X0: $i,X1: $i > $i > $o,X2: $i] :
( $true
!= ( breln @ X2 @ X0
@ ( dpsetconstr @ X2 @ X0
@ ^ [Y0: $i,Y1: $i] : ( X1 @ Y0 @ Y1 ) ) ) )
& ( 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)],[f534]) ).
thf(f534,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 )
=> ! [X0: $i,X1: $i > $i > $o,X2: $i] :
( $true
= ( breln @ X2 @ X0
@ ( dpsetconstr @ X2 @ X0
@ ^ [Y0: $i,Y1: $i] : ( X1 @ Y0 @ Y1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f533]) ).
thf(f533,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
=> ! [X0: $i,X1: $i > $i > $o,X2: $i] :
( breln @ X2 @ X0
@ ( dpsetconstr @ X2 @ X0
@ ^ [X3: $i,X4: $i] : ( X1 @ X3 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f191]) ).
thf(f191,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
=> ! [X4: $i,X0: $i > $i > $o,X3: $i] :
( breln @ X3 @ X4
@ ( dpsetconstr @ X3 @ X4
@ ^ [X1: $i,X2: $i] : ( X0 @ X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f190]) ).
thf(f190,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
=> ! [X4: $i,X0: $i > $i > $o,X3: $i] :
( breln @ X3 @ X4
@ ( dpsetconstr @ X3 @ X4
@ ^ [X1: $i,X2: $i] : ( X0 @ X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.wdvij0GjC6/DTF2THF_20708.p',setOfPairsIsBReln) ).
thf(f2068,plain,
! [X2: $i,X0: $i > $i > $o,X1: $i] :
( ( $true
= ( subset
@ ( dpsetconstr @ X1 @ X2
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) )
@ ( cartprod @ X1 @ X2 ) ) )
| ( dpsetconstrSub != $true ) ),
inference(cnf_transformation,[status(thm)],[f1304]) ).
thf(f1304,plain,
( ( ! [X0: $i > $i > $o,X1: $i,X2: $i] :
( $true
= ( subset
@ ( dpsetconstr @ X1 @ X2
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) )
@ ( cartprod @ X1 @ X2 ) ) )
| ( dpsetconstrSub != $true ) )
& ( ( dpsetconstrSub = $true )
| ( $true
!= ( subset
@ ( dpsetconstr @ sK325 @ sK326
@ ^ [Y0: $i,Y1: $i] : ( sK324 @ Y0 @ Y1 ) )
@ ( cartprod @ sK325 @ sK326 ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK324,sK325,sK326])],[f1302,f1303]) ).
thf(f1303,plain,
( ? [X3: $i > $i > $o,X4: $i,X5: $i] :
( ( subset
@ ( dpsetconstr @ X4 @ X5
@ ^ [Y0: $i,Y1: $i] : ( X3 @ Y0 @ Y1 ) )
@ ( cartprod @ X4 @ X5 ) )
!= $true )
=> ( $true
!= ( subset
@ ( dpsetconstr @ sK325 @ sK326
@ ^ [Y0: $i,Y1: $i] : ( sK324 @ Y0 @ Y1 ) )
@ ( cartprod @ sK325 @ sK326 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1302,plain,
( ( ! [X0: $i > $i > $o,X1: $i,X2: $i] :
( $true
= ( subset
@ ( dpsetconstr @ X1 @ X2
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) )
@ ( cartprod @ X1 @ X2 ) ) )
| ( dpsetconstrSub != $true ) )
& ( ( dpsetconstrSub = $true )
| ? [X3: $i > $i > $o,X4: $i,X5: $i] :
( ( subset
@ ( dpsetconstr @ X4 @ X5
@ ^ [Y0: $i,Y1: $i] : ( X3 @ Y0 @ Y1 ) )
@ ( cartprod @ X4 @ X5 ) )
!= $true ) ) ),
inference(rectify,[status(thm)],[f1301]) ).
thf(f1301,plain,
( ( ! [X0: $i > $i > $o,X1: $i,X2: $i] :
( $true
= ( subset
@ ( dpsetconstr @ X1 @ X2
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) )
@ ( cartprod @ X1 @ X2 ) ) )
| ( dpsetconstrSub != $true ) )
& ( ( dpsetconstrSub = $true )
| ? [X0: $i > $i > $o,X1: $i,X2: $i] :
( $true
!= ( subset
@ ( dpsetconstr @ X1 @ X2
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) )
@ ( cartprod @ X1 @ X2 ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f248]) ).
thf(f248,plain,
( ! [X0: $i > $i > $o,X1: $i,X2: $i] :
( $true
= ( subset
@ ( dpsetconstr @ X1 @ X2
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) )
@ ( cartprod @ X1 @ X2 ) ) )
<=> ( dpsetconstrSub = $true ) ),
inference(fool_elimination,[status(thm)],[f247]) ).
thf(f247,plain,
! [X0: $i > $i > $o,X1: $i,X2: $i] :
( ( subset
@ ( dpsetconstr @ X1 @ X2
@ ^ [X3: $i,X4: $i] : ( X0 @ X3 @ X4 ) )
@ ( cartprod @ X1 @ X2 ) )
= dpsetconstrSub ),
inference(rectify,[status(thm)],[f189]) ).
thf(f189,axiom,
! [X0: $i > $i > $o,X3: $i,X4: $i] :
( ( subset
@ ( dpsetconstr @ X3 @ X4
@ ^ [X1: $i,X2: $i] : ( X0 @ X1 @ X2 ) )
@ ( cartprod @ X3 @ X4 ) )
= dpsetconstrSub ),
file('/export/starexec/sandbox2/tmp/tmp.wdvij0GjC6/DTF2THF_20708.p',dpsetconstrSub) ).
thf(f3355,plain,
( $true
!= ( subset @ ( dpsetconstr @ sK400 @ sK398 @ sK399 ) @ ( cartprod @ sK400 @ sK398 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f3001]) ).
thf(f3001,plain,
( $true
!= ( ^ [Y0: $i,Y1: $i,Y2: $i] : ( subset @ Y2 @ ( cartprod @ Y0 @ Y1 ) )
@ sK400
@ sK398
@ ( dpsetconstr @ sK400 @ sK398
@ ^ [Y0: $i,Y1: $i] : ( sK399 @ Y0 @ Y1 ) ) ) ),
inference(definition_unfolding,[status(thm)],[f2182,f2061]) ).
thf(f2061,plain,
( breln
= ( ^ [Y0: $i,Y1: $i,Y2: $i] : ( subset @ Y2 @ ( cartprod @ Y0 @ Y1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f372]) ).
thf(f372,plain,
( breln
= ( ^ [Y0: $i,Y1: $i,Y2: $i] : ( subset @ Y2 @ ( cartprod @ Y0 @ Y1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f371]) ).
thf(f371,plain,
( breln
= ( ^ [X0: $i,X1: $i,X2: $i] : ( subset @ X2 @ ( cartprod @ X0 @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f186]) ).
thf(f186,axiom,
( breln
= ( ^ [X3: $i,X4: $i,X5: $i] : ( subset @ X5 @ ( cartprod @ X3 @ X4 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.wdvij0GjC6/DTF2THF_20708.p',breln) ).
thf(f2182,plain,
( $true
!= ( breln @ sK400 @ sK398
@ ( dpsetconstr @ sK400 @ sK398
@ ^ [Y0: $i,Y1: $i] : ( sK399 @ Y0 @ Y1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f1434]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU667^1 : TPTP v9.2.1. Released v3.7.0.
% 0.10/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.13/0.33 % Computer : n009.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33 % CPULimit : 300
% 0.13/0.33 % WCLimit : 300
% 0.13/0.33 % DateTime : Tue Feb 24 12:30:10 EST 2026
% 0.13/0.34 % CPUTime :
% 0.13/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.43 ---- Original DTF file ---
% 0.16/0.43 thf(spec,logic,$$dhol).
% 0.16/0.43 %------------------------------------------------------------------------------
% 0.16/0.43 % File : SEU667^1 : TPTP v9.2.1. Released v3.7.0.
% 0.16/0.43 % Domain : Set Theory
% 0.16/0.43 % Problem : Ordered Pairs - Sets of Pairs
% 0.16/0.43 % Version : Especial.
% 0.16/0.43 % English : (! A:i.! B:i.! phi:i>(i>o).breln A B (dpsetconstr A B
% 0.16/0.43 % (^ x,y:i.phi x y)))
% 0.16/0.43
% 0.16/0.43 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.16/0.43 % Source : [Bro08]
% 0.16/0.43 % Names : ZFC169g [Bro08]
% 0.16/0.43
% 0.16/0.43 % Status : Theorem
% 0.16/0.43 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.38 v8.1.0, 0.27 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.25 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v4.1.0, 0.67 v4.0.1, 1.00 v4.0.0, 0.67 v3.7.0
% 0.16/0.43 % Syntax : Number of formulae : 406 ( 189 unt; 216 typ; 189 def)
% 0.16/0.43 % Number of atoms : 1000 ( 283 equ; 0 cnn)
% 0.16/0.43 % Maximal formula atoms : 184 ( 5 avg)
% 0.16/0.43 % Number of connectives : 1775 ( 47 ~; 7 |; 38 &;1230 @)
% 0.16/0.43 % ( 14 <=>; 439 =>; 0 <=; 0 <~>)
% 0.16/0.43 % Maximal formula depth : 193 ( 2 avg)
% 0.16/0.43 % Number of types : 2 ( 0 usr)
% 0.16/0.43 % Number of type conns : 100 ( 100 >; 0 *; 0 +; 0 <<)
% 0.16/0.43 % Number of symbols : 219 ( 216 usr; 187 con; 0-3 aty)
% 0.16/0.43 % Number of variables : 567 ( 53 ^; 476 !; 38 ?; 567 :)
% 0.16/0.43 % SPC : TH0_THM_EQU_NAR
% 0.16/0.43
% 0.16/0.43 % Comments : http://mathgate.info/detsetitem.php?id=232
% 0.16/0.43 %------------------------------------------------------------------------------
% 0.16/0.43 thf(in_type,type,
% 0.16/0.43 in: $i > $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(exu_type,type,
% 0.16/0.43 exu: ( $i > $o ) > $o ).
% 0.16/0.43
% 0.16/0.43 thf(exu,definition,
% 0.16/0.43 ( exu
% 0.16/0.43 = ( ^ [Xphi: $i > $o] :
% 0.16/0.43 ? [Xx: $i] :
% 0.16/0.43 ( ( Xphi @ Xx )
% 0.16/0.43 & ! [Xy: $i] :
% 0.16/0.43 ( ( Xphi @ Xy )
% 0.16/0.43 => ( Xx = Xy ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setextAx_type,type,
% 0.16/0.43 setextAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setextAx,definition,
% 0.16/0.43 ( setextAx
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 <=> ( in @ Xx @ B ) )
% 0.16/0.43 => ( A = B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptyset_type,type,
% 0.16/0.43 emptyset: $i ).
% 0.16/0.43
% 0.16/0.43 thf(emptysetAx_type,type,
% 0.16/0.43 emptysetAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptysetAx,definition,
% 0.16/0.43 ( emptysetAx
% 0.16/0.43 = ( ! [Xx: $i] :
% 0.16/0.43 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoin_type,type,
% 0.16/0.43 setadjoin: $i > $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinAx_type,type,
% 0.16/0.43 setadjoinAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinAx,definition,
% 0.16/0.43 ( setadjoinAx
% 0.16/0.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.43 <=> ( ( Xy = Xx )
% 0.16/0.43 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(powerset_type,type,
% 0.16/0.43 powerset: $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(powersetAx_type,type,
% 0.16/0.43 powersetAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(powersetAx,definition,
% 0.16/0.43 ( powersetAx
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.43 <=> ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setunion_type,type,
% 0.16/0.43 setunion: $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(setunionAx_type,type,
% 0.16/0.43 setunionAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setunionAx,definition,
% 0.16/0.43 ( setunionAx
% 0.16/0.43 = ( ! [A: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.43 <=> ? [B: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 & ( in @ B @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(omega_type,type,
% 0.16/0.43 omega: $i ).
% 0.16/0.43
% 0.16/0.43 thf(omega0Ax_type,type,
% 0.16/0.43 omega0Ax: $o ).
% 0.16/0.43
% 0.16/0.43 thf(omega0Ax,definition,
% 0.16/0.43 ( omega0Ax
% 0.16/0.43 = ( in @ emptyset @ omega ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(omegaSAx_type,type,
% 0.16/0.43 omegaSAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(omegaSAx,definition,
% 0.16/0.43 ( omegaSAx
% 0.16/0.43 = ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ omega )
% 0.16/0.43 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(omegaIndAx_type,type,
% 0.16/0.43 omegaIndAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(omegaIndAx,definition,
% 0.16/0.43 ( omegaIndAx
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ( ( in @ emptyset @ A )
% 0.16/0.43 & ! [Xx: $i] :
% 0.16/0.43 ( ( ( in @ Xx @ omega )
% 0.16/0.43 & ( in @ Xx @ A ) )
% 0.16/0.43 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ omega )
% 0.16/0.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(replAx_type,type,
% 0.16/0.43 replAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(replAx,definition,
% 0.16/0.43 ( replAx
% 0.16/0.43 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( exu
% 0.16/0.43 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.16/0.43 => ? [B: $i] :
% 0.16/0.43 ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 <=> ? [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ A )
% 0.16/0.43 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(foundationAx_type,type,
% 0.16/0.43 foundationAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(foundationAx,definition,
% 0.16/0.43 ( foundationAx
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.43 => ? [B: $i] :
% 0.16/0.43 ( ( in @ B @ A )
% 0.16/0.43 & ~ ? [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(wellorderingAx_type,type,
% 0.16/0.43 wellorderingAx: $o ).
% 0.16/0.43
% 0.16/0.43 thf(wellorderingAx,definition,
% 0.16/0.43 ( wellorderingAx
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ? [B: $i] :
% 0.16/0.43 ( ! [C: $i] :
% 0.16/0.43 ( ( in @ C @ B )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ C )
% 0.16/0.43 => ( in @ Xx @ A ) ) )
% 0.16/0.43 & ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( ( in @ Xx @ A )
% 0.16/0.43 & ( in @ Xy @ A ) )
% 0.16/0.43 => ( ! [C: $i] :
% 0.16/0.43 ( ( in @ C @ B )
% 0.16/0.43 => ( ( in @ Xx @ C )
% 0.16/0.43 <=> ( in @ Xy @ C ) ) )
% 0.16/0.43 => ( Xx = Xy ) ) )
% 0.16/0.43 & ! [C: $i,D: $i] :
% 0.16/0.43 ( ( ( in @ C @ B )
% 0.16/0.43 & ( in @ D @ B ) )
% 0.16/0.43 => ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ C )
% 0.16/0.43 => ( in @ Xx @ D ) )
% 0.16/0.43 | ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ D )
% 0.16/0.43 => ( in @ Xx @ C ) ) ) )
% 0.16/0.43 & ! [C: $i] :
% 0.16/0.43 ( ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ C )
% 0.16/0.43 => ( in @ Xx @ A ) )
% 0.16/0.43 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.16/0.43 => ? [D: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ D @ B )
% 0.16/0.43 & ( in @ Xx @ C )
% 0.16/0.43 & ~ ? [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ D )
% 0.16/0.43 & ( in @ Xy @ C ) )
% 0.16/0.43 & ! [E: $i] :
% 0.16/0.43 ( ( in @ E @ B )
% 0.16/0.43 => ( ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ E )
% 0.16/0.43 => ( in @ Xy @ D ) )
% 0.16/0.43 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(descr_type,type,
% 0.16/0.43 descr: ( $i > $o ) > $i ).
% 0.16/0.43
% 0.16/0.43 thf(descrp_type,type,
% 0.16/0.43 descrp: $o ).
% 0.16/0.43
% 0.16/0.43 thf(descrp,definition,
% 0.16/0.43 ( descrp
% 0.16/0.43 = ( ! [Xphi: $i > $o] :
% 0.16/0.43 ( ( exu
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 => ( Xphi
% 0.16/0.43 @ ( descr
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(dsetconstr_type,type,
% 0.16/0.43 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.16/0.43
% 0.16/0.43 thf(dsetconstrI_type,type,
% 0.16/0.43 dsetconstrI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(dsetconstrI,definition,
% 0.16/0.43 ( dsetconstrI
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( Xphi @ Xx )
% 0.16/0.43 => ( in @ Xx
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(dsetconstrEL_type,type,
% 0.16/0.43 dsetconstrEL: $o ).
% 0.16/0.43
% 0.16/0.43 thf(dsetconstrEL,definition,
% 0.16/0.43 ( dsetconstrEL
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.43 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(dsetconstrER_type,type,
% 0.16/0.43 dsetconstrER: $o ).
% 0.16/0.43
% 0.16/0.43 thf(dsetconstrER,definition,
% 0.16/0.43 ( dsetconstrER
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.43 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(exuE1_type,type,
% 0.16/0.43 exuE1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(exuE1,definition,
% 0.16/0.43 ( exuE1
% 0.16/0.43 = ( ! [Xphi: $i > $o] :
% 0.16/0.43 ( ( exu
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 => ? [Xx: $i] :
% 0.16/0.43 ( ( Xphi @ Xx )
% 0.16/0.43 & ! [Xy: $i] :
% 0.16/0.43 ( ( Xphi @ Xy )
% 0.16/0.43 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(prop2set_type,type,
% 0.16/0.43 prop2set: $o > $i ).
% 0.16/0.43
% 0.16/0.43 thf(prop2setE_type,type,
% 0.16/0.43 prop2setE: $o ).
% 0.16/0.43
% 0.16/0.43 thf(prop2setE,definition,
% 0.16/0.43 ( prop2setE
% 0.16/0.43 = ( ! [Xphi: $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.16/0.43 => Xphi ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptysetE_type,type,
% 0.16/0.43 emptysetE: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptysetE,definition,
% 0.16/0.43 ( emptysetE
% 0.16/0.43 = ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ emptyset )
% 0.16/0.43 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptysetimpfalse_type,type,
% 0.16/0.43 emptysetimpfalse: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptysetimpfalse,definition,
% 0.16/0.43 ( emptysetimpfalse
% 0.16/0.43 = ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ emptyset )
% 0.16/0.43 => $false ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(notinemptyset_type,type,
% 0.16/0.43 notinemptyset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(notinemptyset,definition,
% 0.16/0.43 ( notinemptyset
% 0.16/0.43 = ( ! [Xx: $i] :
% 0.16/0.43 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(exuE3e_type,type,
% 0.16/0.43 exuE3e: $o ).
% 0.16/0.43
% 0.16/0.43 thf(exuE3e,definition,
% 0.16/0.43 ( exuE3e
% 0.16/0.43 = ( ! [Xphi: $i > $o] :
% 0.16/0.43 ( ( exu
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setext_type,type,
% 0.16/0.43 setext: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setext,definition,
% 0.16/0.43 ( setext
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ Xx @ B ) )
% 0.16/0.43 => ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 => ( in @ Xx @ A ) )
% 0.16/0.43 => ( A = B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptyI_type,type,
% 0.16/0.43 emptyI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptyI,definition,
% 0.16/0.43 ( emptyI
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ~ ( in @ Xx @ A )
% 0.16/0.43 => ( A = emptyset ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(noeltsimpempty_type,type,
% 0.16/0.43 noeltsimpempty: $o ).
% 0.16/0.43
% 0.16/0.43 thf(noeltsimpempty,definition,
% 0.16/0.43 ( noeltsimpempty
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ~ ( in @ Xx @ A )
% 0.16/0.43 => ( A = emptyset ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setbeta_type,type,
% 0.16/0.43 setbeta: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setbeta,definition,
% 0.16/0.43 ( setbeta
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( in @ Xx
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.43 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(nonempty_type,type,
% 0.16/0.43 nonempty: $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(nonempty,definition,
% 0.16/0.43 ( nonempty
% 0.16/0.43 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(nonemptyE1_type,type,
% 0.16/0.43 nonemptyE1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(nonemptyE1,definition,
% 0.16/0.43 ( nonemptyE1
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ( nonempty @ A )
% 0.16/0.43 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(nonemptyI_type,type,
% 0.16/0.43 nonemptyI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(nonemptyI,definition,
% 0.16/0.43 ( nonemptyI
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( Xphi @ Xx )
% 0.16/0.43 => ( nonempty
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(nonemptyI1_type,type,
% 0.16/0.43 nonemptyI1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(nonemptyI1,definition,
% 0.16/0.43 ( nonemptyI1
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.43 => ( nonempty @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinIL_type,type,
% 0.16/0.43 setadjoinIL: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinIL,definition,
% 0.16/0.43 ( setadjoinIL
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptyinunitempty_type,type,
% 0.16/0.43 emptyinunitempty: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptyinunitempty,definition,
% 0.16/0.43 ( emptyinunitempty
% 0.16/0.43 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinIR_type,type,
% 0.16/0.43 setadjoinIR: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinIR,definition,
% 0.16/0.43 ( setadjoinIR
% 0.16/0.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ A )
% 0.16/0.43 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinE_type,type,
% 0.16/0.43 setadjoinE: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinE,definition,
% 0.16/0.43 ( setadjoinE
% 0.16/0.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.43 => ! [Xphi: $o] :
% 0.16/0.43 ( ( ( Xy = Xx )
% 0.16/0.43 => Xphi )
% 0.16/0.43 => ( ( ( in @ Xy @ A )
% 0.16/0.43 => Xphi )
% 0.16/0.43 => Xphi ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinOr_type,type,
% 0.16/0.43 setadjoinOr: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinOr,definition,
% 0.16/0.43 ( setadjoinOr
% 0.16/0.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.43 => ( ( Xy = Xx )
% 0.16/0.43 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setoftrueEq_type,type,
% 0.16/0.43 setoftrueEq: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setoftrueEq,definition,
% 0.16/0.43 ( setoftrueEq
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xx: $i] : $true )
% 0.16/0.43 = A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(powersetI_type,type,
% 0.16/0.43 powersetI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(powersetI,definition,
% 0.16/0.43 ( powersetI
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 => ( in @ Xx @ A ) )
% 0.16/0.43 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptyinPowerset_type,type,
% 0.16/0.43 emptyinPowerset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptyinPowerset,definition,
% 0.16/0.43 ( emptyinPowerset
% 0.16/0.43 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptyInPowerset_type,type,
% 0.16/0.43 emptyInPowerset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptyInPowerset,definition,
% 0.16/0.43 ( emptyInPowerset
% 0.16/0.43 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(powersetE_type,type,
% 0.16/0.43 powersetE: $o ).
% 0.16/0.43
% 0.16/0.43 thf(powersetE,definition,
% 0.16/0.43 ( powersetE
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.43 => ( ( in @ Xx @ B )
% 0.16/0.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setunionI_type,type,
% 0.16/0.43 setunionI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setunionI,definition,
% 0.16/0.43 ( setunionI
% 0.16/0.43 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 => ( ( in @ B @ A )
% 0.16/0.43 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setunionE_type,type,
% 0.16/0.43 setunionE: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setunionE,definition,
% 0.16/0.43 ( setunionE
% 0.16/0.43 = ( ! [A: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.43 => ! [Xphi: $o] :
% 0.16/0.43 ( ! [B: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 => ( ( in @ B @ A )
% 0.16/0.43 => Xphi ) )
% 0.16/0.43 => Xphi ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(subPowSU_type,type,
% 0.16/0.43 subPowSU: $o ).
% 0.16/0.43
% 0.16/0.43 thf(subPowSU,definition,
% 0.16/0.43 ( subPowSU
% 0.16/0.43 = ( ! [A: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(exuE2_type,type,
% 0.16/0.43 exuE2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(exuE2,definition,
% 0.16/0.43 ( exuE2
% 0.16/0.43 = ( ! [Xphi: $i > $o] :
% 0.16/0.43 ( ( exu
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 => ? [Xx: $i] :
% 0.16/0.43 ! [Xy: $i] :
% 0.16/0.43 ( ( Xphi @ Xy )
% 0.16/0.43 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(nonemptyImpWitness_type,type,
% 0.16/0.43 nonemptyImpWitness: $o ).
% 0.16/0.43
% 0.16/0.43 thf(nonemptyImpWitness,definition,
% 0.16/0.43 ( nonemptyImpWitness
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ( nonempty @ A )
% 0.16/0.43 => ? [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 & $true ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(uniqinunit_type,type,
% 0.16/0.43 uniqinunit: $o ).
% 0.16/0.43
% 0.16/0.43 thf(uniqinunit,definition,
% 0.16/0.43 ( uniqinunit
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.43 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(notinsingleton_type,type,
% 0.16/0.43 notinsingleton: $o ).
% 0.16/0.43
% 0.16/0.43 thf(notinsingleton,definition,
% 0.16/0.43 ( notinsingleton
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( Xx != Xy )
% 0.16/0.43 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(eqinunit_type,type,
% 0.16/0.43 eqinunit: $o ).
% 0.16/0.43
% 0.16/0.43 thf(eqinunit,definition,
% 0.16/0.43 ( eqinunit
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( Xx = Xy )
% 0.16/0.43 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(singletonsswitch_type,type,
% 0.16/0.43 singletonsswitch: $o ).
% 0.16/0.43
% 0.16/0.43 thf(singletonsswitch,definition,
% 0.16/0.43 ( singletonsswitch
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.43 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(upairsetE_type,type,
% 0.16/0.43 upairsetE: $o ).
% 0.16/0.43
% 0.16/0.43 thf(upairsetE,definition,
% 0.16/0.43 ( upairsetE
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.43 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.16/0.43 => ( ( Xz = Xx )
% 0.16/0.43 | ( Xz = Xy ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(upairsetIL_type,type,
% 0.16/0.43 upairsetIL: $o ).
% 0.16/0.43
% 0.16/0.43 thf(upairsetIL,definition,
% 0.16/0.43 ( upairsetIL
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(upairsetIR_type,type,
% 0.16/0.43 upairsetIR: $o ).
% 0.16/0.43
% 0.16/0.43 thf(upairsetIR,definition,
% 0.16/0.43 ( upairsetIR
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptyE1_type,type,
% 0.16/0.43 emptyE1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptyE1,definition,
% 0.16/0.43 ( emptyE1
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.43 ( ? [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 & ( Xphi @ Xx ) )
% 0.16/0.43 => ( ( ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 = emptyset )
% 0.16/0.43 => $false ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(vacuousDall_type,type,
% 0.16/0.43 vacuousDall: $o ).
% 0.16/0.43
% 0.16/0.43 thf(vacuousDall,definition,
% 0.16/0.43 ( vacuousDall
% 0.16/0.43 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ emptyset )
% 0.16/0.43 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(quantDeMorgan1_type,type,
% 0.16/0.43 quantDeMorgan1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(quantDeMorgan1,definition,
% 0.16/0.43 ( quantDeMorgan1
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.43 ( ~ ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( Xphi @ Xx ) )
% 0.16/0.43 => ? [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(quantDeMorgan2_type,type,
% 0.16/0.43 quantDeMorgan2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(quantDeMorgan2,definition,
% 0.16/0.43 ( quantDeMorgan2
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ~ ( Xphi @ Xx ) )
% 0.16/0.43 => ~ ? [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(quantDeMorgan3_type,type,
% 0.16/0.43 quantDeMorgan3: $o ).
% 0.16/0.43
% 0.16/0.43 thf(quantDeMorgan3,definition,
% 0.16/0.43 ( quantDeMorgan3
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.43 ( ~ ? [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 & ( Xphi @ Xx ) )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(quantDeMorgan4_type,type,
% 0.16/0.43 quantDeMorgan4: $o ).
% 0.16/0.43
% 0.16/0.43 thf(quantDeMorgan4,definition,
% 0.16/0.43 ( quantDeMorgan4
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.43 ( ? [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 & ~ ( Xphi @ Xx ) )
% 0.16/0.43 => ~ ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(prop2setI_type,type,
% 0.16/0.43 prop2setI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(prop2setI,definition,
% 0.16/0.43 ( prop2setI
% 0.16/0.43 = ( ! [Xphi: $o] :
% 0.16/0.43 ( Xphi
% 0.16/0.43 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(set2prop_type,type,
% 0.16/0.43 set2prop: $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(prop2set2propI_type,type,
% 0.16/0.43 prop2set2propI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(prop2set2propI,definition,
% 0.16/0.43 ( prop2set2propI
% 0.16/0.43 = ( ! [Xphi: $o] :
% 0.16/0.43 ( Xphi
% 0.16/0.43 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(notdexE_type,type,
% 0.16/0.43 notdexE: $o ).
% 0.16/0.43
% 0.16/0.43 thf(notdexE,definition,
% 0.16/0.43 ( notdexE
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.43 ( ~ ? [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 & ( Xphi @ Xx ) )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(notdallE_type,type,
% 0.16/0.43 notdallE: $o ).
% 0.16/0.43
% 0.16/0.43 thf(notdallE,definition,
% 0.16/0.43 ( notdallE
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.43 ( ~ ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( Xphi @ Xx ) )
% 0.16/0.43 => ? [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(exuI1_type,type,
% 0.16/0.43 exuI1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(exuI1,definition,
% 0.16/0.43 ( exuI1
% 0.16/0.43 = ( ! [Xphi: $i > $o] :
% 0.16/0.43 ( ? [Xx: $i] :
% 0.16/0.43 ( ( Xphi @ Xx )
% 0.16/0.43 & ! [Xy: $i] :
% 0.16/0.43 ( ( Xphi @ Xy )
% 0.16/0.43 => ( Xx = Xy ) ) )
% 0.16/0.43 => ( exu
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(exuI3_type,type,
% 0.16/0.43 exuI3: $o ).
% 0.16/0.43
% 0.16/0.43 thf(exuI3,definition,
% 0.16/0.43 ( exuI3
% 0.16/0.43 = ( ! [Xphi: $i > $o] :
% 0.16/0.43 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.16/0.43 => ( ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( Xphi @ Xx )
% 0.16/0.43 => ( ( Xphi @ Xy )
% 0.16/0.43 => ( Xx = Xy ) ) )
% 0.16/0.43 => ( exu
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(exuI2_type,type,
% 0.16/0.43 exuI2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(exuI2,definition,
% 0.16/0.43 ( exuI2
% 0.16/0.43 = ( ! [Xphi: $i > $o] :
% 0.16/0.43 ( ? [Xx: $i] :
% 0.16/0.43 ! [Xy: $i] :
% 0.16/0.43 ( ( Xphi @ Xy )
% 0.16/0.43 <=> ( Xy = Xx ) )
% 0.16/0.43 => ( exu
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(inCongP_type,type,
% 0.16/0.43 inCongP: $o ).
% 0.16/0.43
% 0.16/0.43 thf(inCongP,definition,
% 0.16/0.43 ( inCongP
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( A = B )
% 0.16/0.43 => ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( Xx = Xy )
% 0.16/0.43 => ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(in__Cong_type,type,
% 0.16/0.43 in__Cong: $o ).
% 0.16/0.43
% 0.16/0.43 thf(in__Cong,definition,
% 0.16/0.43 ( in__Cong
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( A = B )
% 0.16/0.43 => ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( Xx = Xy )
% 0.16/0.43 => ( ( in @ Xx @ A )
% 0.16/0.43 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(exuE3u_type,type,
% 0.16/0.43 exuE3u: $o ).
% 0.16/0.43
% 0.16/0.43 thf(exuE3u,definition,
% 0.16/0.43 ( exuE3u
% 0.16/0.43 = ( ! [Xphi: $i > $o] :
% 0.16/0.43 ( ( exu
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 => ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( Xphi @ Xx )
% 0.16/0.43 => ( ( Xphi @ Xy )
% 0.16/0.43 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(exu__Cong_type,type,
% 0.16/0.43 exu__Cong: $o ).
% 0.16/0.43
% 0.16/0.43 thf(exu__Cong,definition,
% 0.16/0.43 ( exu__Cong
% 0.16/0.43 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.43 ( ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( Xx = Xy )
% 0.16/0.43 => ( ( Xphi @ Xx )
% 0.16/0.43 <=> ( Xpsi @ Xy ) ) )
% 0.16/0.43 => ( ( exu
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 <=> ( exu
% 0.16/0.43 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptyset__Cong_type,type,
% 0.16/0.43 emptyset__Cong: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptyset__Cong,definition,
% 0.16/0.43 ( emptyset__Cong
% 0.16/0.43 = ( emptyset = emptyset ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoin__Cong_type,type,
% 0.16/0.43 setadjoin__Cong: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoin__Cong,definition,
% 0.16/0.43 ( setadjoin__Cong
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( Xx = Xy )
% 0.16/0.43 => ! [Xz: $i,Xu: $i] :
% 0.16/0.43 ( ( Xz = Xu )
% 0.16/0.43 => ( ( setadjoin @ Xx @ Xz )
% 0.16/0.43 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(powerset__Cong_type,type,
% 0.16/0.43 powerset__Cong: $o ).
% 0.16/0.43
% 0.16/0.43 thf(powerset__Cong,definition,
% 0.16/0.43 ( powerset__Cong
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( A = B )
% 0.16/0.43 => ( ( powerset @ A )
% 0.16/0.43 = ( powerset @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setunion__Cong_type,type,
% 0.16/0.43 setunion__Cong: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setunion__Cong,definition,
% 0.16/0.43 ( setunion__Cong
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( A = B )
% 0.16/0.43 => ( ( setunion @ A )
% 0.16/0.43 = ( setunion @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(omega__Cong_type,type,
% 0.16/0.43 omega__Cong: $o ).
% 0.16/0.43
% 0.16/0.43 thf(omega__Cong,definition,
% 0.16/0.43 ( omega__Cong
% 0.16/0.43 = ( omega = omega ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(exuEu_type,type,
% 0.16/0.43 exuEu: $o ).
% 0.16/0.43
% 0.16/0.43 thf(exuEu,definition,
% 0.16/0.43 ( exuEu
% 0.16/0.43 = ( ! [Xphi: $i > $o] :
% 0.16/0.43 ( ( exu
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 => ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( Xphi @ Xx )
% 0.16/0.43 => ( ( Xphi @ Xy )
% 0.16/0.43 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(descr__Cong_type,type,
% 0.16/0.43 descr__Cong: $o ).
% 0.16/0.43
% 0.16/0.43 thf(descr__Cong,definition,
% 0.16/0.43 ( descr__Cong
% 0.16/0.43 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.43 ( ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( Xx = Xy )
% 0.16/0.43 => ( ( Xphi @ Xx )
% 0.16/0.43 <=> ( Xpsi @ Xy ) ) )
% 0.16/0.43 => ( ( exu
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 => ( ( exu
% 0.16/0.43 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.16/0.43 => ( ( descr
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 = ( descr
% 0.16/0.43 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(dsetconstr__Cong_type,type,
% 0.16/0.43 dsetconstr__Cong: $o ).
% 0.16/0.43
% 0.16/0.43 thf(dsetconstr__Cong,definition,
% 0.16/0.43 ( dsetconstr__Cong
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( A = B )
% 0.16/0.43 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( ( Xx = Xy )
% 0.16/0.43 => ( ( Xphi @ Xx )
% 0.16/0.43 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.16/0.43 => ( ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 = ( dsetconstr @ B
% 0.16/0.43 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(subset_type,type,
% 0.16/0.43 subset: $i > $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(disjoint_type,type,
% 0.16/0.43 disjoint: $i > $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(setsmeet_type,type,
% 0.16/0.43 setsmeet: $i > $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(subsetI1_type,type,
% 0.16/0.43 subsetI1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(subsetI1,definition,
% 0.16/0.43 ( subsetI1
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ Xx @ B ) )
% 0.16/0.43 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(eqimpsubset2_type,type,
% 0.16/0.43 eqimpsubset2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(eqimpsubset2,definition,
% 0.16/0.43 ( eqimpsubset2
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( A = B )
% 0.16/0.43 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(eqimpsubset1_type,type,
% 0.16/0.43 eqimpsubset1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(eqimpsubset1,definition,
% 0.16/0.43 ( eqimpsubset1
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( A = B )
% 0.16/0.43 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(subsetI2_type,type,
% 0.16/0.43 subsetI2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(subsetI2,definition,
% 0.16/0.43 ( subsetI2
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ Xx @ B ) )
% 0.16/0.43 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptysetsubset_type,type,
% 0.16/0.43 emptysetsubset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptysetsubset,definition,
% 0.16/0.43 ( emptysetsubset
% 0.16/0.43 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(subsetE_type,type,
% 0.16/0.43 subsetE: $o ).
% 0.16/0.43
% 0.16/0.43 thf(subsetE,definition,
% 0.16/0.43 ( subsetE
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( subset @ A @ B )
% 0.16/0.43 => ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(subsetE2_type,type,
% 0.16/0.43 subsetE2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(subsetE2,definition,
% 0.16/0.43 ( subsetE2
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( subset @ A @ B )
% 0.16/0.43 => ( ~ ( in @ Xx @ B )
% 0.16/0.43 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(notsubsetI_type,type,
% 0.16/0.43 notsubsetI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(notsubsetI,definition,
% 0.16/0.43 ( notsubsetI
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ~ ( in @ Xx @ B )
% 0.16/0.43 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(notequalI1_type,type,
% 0.16/0.43 notequalI1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(notequalI1,definition,
% 0.16/0.43 ( notequalI1
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ~ ( subset @ A @ B )
% 0.16/0.43 => ( A != B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(notequalI2_type,type,
% 0.16/0.43 notequalI2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(notequalI2,definition,
% 0.16/0.43 ( notequalI2
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ~ ( in @ Xx @ B )
% 0.16/0.43 => ( A != B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(subsetRefl_type,type,
% 0.16/0.43 subsetRefl: $o ).
% 0.16/0.43
% 0.16/0.43 thf(subsetRefl,definition,
% 0.16/0.43 ( subsetRefl
% 0.16/0.43 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(subsetTrans_type,type,
% 0.16/0.43 subsetTrans: $o ).
% 0.16/0.43
% 0.16/0.43 thf(subsetTrans,definition,
% 0.16/0.43 ( subsetTrans
% 0.16/0.43 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.43 ( ( subset @ A @ B )
% 0.16/0.43 => ( ( subset @ B @ C )
% 0.16/0.43 => ( subset @ A @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinSub_type,type,
% 0.16/0.43 setadjoinSub: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinSub,definition,
% 0.16/0.43 ( setadjoinSub
% 0.16/0.43 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinSub2_type,type,
% 0.16/0.43 setadjoinSub2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinSub2,definition,
% 0.16/0.43 ( setadjoinSub2
% 0.16/0.43 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.16/0.43 ( ( subset @ A @ B )
% 0.16/0.43 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(subset2powerset_type,type,
% 0.16/0.43 subset2powerset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(subset2powerset,definition,
% 0.16/0.43 ( subset2powerset
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( subset @ A @ B )
% 0.16/0.43 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setextsub_type,type,
% 0.16/0.43 setextsub: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setextsub,definition,
% 0.16/0.43 ( setextsub
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( subset @ A @ B )
% 0.16/0.43 => ( ( subset @ B @ A )
% 0.16/0.43 => ( A = B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(subsetemptysetimpeq_type,type,
% 0.16/0.43 subsetemptysetimpeq: $o ).
% 0.16/0.43
% 0.16/0.43 thf(subsetemptysetimpeq,definition,
% 0.16/0.43 ( subsetemptysetimpeq
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ( subset @ A @ emptyset )
% 0.16/0.43 => ( A = emptyset ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(powersetI1_type,type,
% 0.16/0.43 powersetI1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(powersetI1,definition,
% 0.16/0.43 ( powersetI1
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( subset @ B @ A )
% 0.16/0.43 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(powersetE1_type,type,
% 0.16/0.43 powersetE1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(powersetE1,definition,
% 0.16/0.43 ( powersetE1
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.43 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(inPowerset_type,type,
% 0.16/0.43 inPowerset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(inPowerset,definition,
% 0.16/0.43 ( inPowerset
% 0.16/0.43 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(powersetsubset_type,type,
% 0.16/0.43 powersetsubset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(powersetsubset,definition,
% 0.16/0.43 ( powersetsubset
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( subset @ A @ B )
% 0.16/0.43 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(sepInPowerset_type,type,
% 0.16/0.43 sepInPowerset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(sepInPowerset,definition,
% 0.16/0.43 ( sepInPowerset
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.43 ( in
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 @ ( powerset @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(sepSubset_type,type,
% 0.16/0.43 sepSubset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(sepSubset,definition,
% 0.16/0.43 ( sepSubset
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.43 ( subset
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binunion_type,type,
% 0.16/0.43 binunion: $i > $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(binunionIL_type,type,
% 0.16/0.43 binunionIL: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binunionIL,definition,
% 0.16/0.43 ( binunionIL
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(upairset2IR_type,type,
% 0.16/0.43 upairset2IR: $o ).
% 0.16/0.43
% 0.16/0.43 thf(upairset2IR,definition,
% 0.16/0.43 ( upairset2IR
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binunionIR_type,type,
% 0.16/0.43 binunionIR: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binunionIR,definition,
% 0.16/0.43 ( binunionIR
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binunionEcases_type,type,
% 0.16/0.43 binunionEcases: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binunionEcases,definition,
% 0.16/0.43 ( binunionEcases
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.16/0.43 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.16/0.43 => ( ( ( in @ Xx @ A )
% 0.16/0.43 => Xphi )
% 0.16/0.43 => ( ( ( in @ Xx @ B )
% 0.16/0.43 => Xphi )
% 0.16/0.43 => Xphi ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binunionE_type,type,
% 0.16/0.43 binunionE: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binunionE,definition,
% 0.16/0.43 ( binunionE
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.16/0.43 => ( ( in @ Xx @ A )
% 0.16/0.43 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binunionLsub_type,type,
% 0.16/0.43 binunionLsub: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binunionLsub,definition,
% 0.16/0.43 ( binunionLsub
% 0.16/0.43 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binunionRsub_type,type,
% 0.16/0.43 binunionRsub: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binunionRsub,definition,
% 0.16/0.43 ( binunionRsub
% 0.16/0.43 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binintersect_type,type,
% 0.16/0.43 binintersect: $i > $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectI_type,type,
% 0.16/0.43 binintersectI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectI,definition,
% 0.16/0.43 ( binintersectI
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( in @ Xx @ B )
% 0.16/0.43 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectSubset5_type,type,
% 0.16/0.43 binintersectSubset5: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectSubset5,definition,
% 0.16/0.43 ( binintersectSubset5
% 0.16/0.43 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.43 ( ( subset @ C @ A )
% 0.16/0.43 => ( ( subset @ C @ B )
% 0.16/0.43 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectEL_type,type,
% 0.16/0.43 binintersectEL: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectEL,definition,
% 0.16/0.43 ( binintersectEL
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.16/0.43 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectLsub_type,type,
% 0.16/0.43 binintersectLsub: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectLsub,definition,
% 0.16/0.43 ( binintersectLsub
% 0.16/0.43 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectSubset2_type,type,
% 0.16/0.43 binintersectSubset2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectSubset2,definition,
% 0.16/0.43 ( binintersectSubset2
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( subset @ A @ B )
% 0.16/0.43 => ( ( binintersect @ A @ B )
% 0.16/0.43 = A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectSubset3_type,type,
% 0.16/0.43 binintersectSubset3: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectSubset3,definition,
% 0.16/0.43 ( binintersectSubset3
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( ( binintersect @ A @ B )
% 0.16/0.43 = B )
% 0.16/0.43 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectER_type,type,
% 0.16/0.43 binintersectER: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectER,definition,
% 0.16/0.43 ( binintersectER
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.16/0.43 => ( in @ Xx @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(disjointsetsI1_type,type,
% 0.16/0.43 disjointsetsI1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(disjointsetsI1,definition,
% 0.16/0.43 ( disjointsetsI1
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ~ ? [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 & ( in @ Xx @ B ) )
% 0.16/0.43 => ( ( binintersect @ A @ B )
% 0.16/0.43 = emptyset ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectRsub_type,type,
% 0.16/0.43 binintersectRsub: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectRsub,definition,
% 0.16/0.43 ( binintersectRsub
% 0.16/0.43 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectSubset4_type,type,
% 0.16/0.43 binintersectSubset4: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectSubset4,definition,
% 0.16/0.43 ( binintersectSubset4
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( subset @ B @ A )
% 0.16/0.43 => ( ( binintersect @ A @ B )
% 0.16/0.43 = B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectSubset1_type,type,
% 0.16/0.43 binintersectSubset1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectSubset1,definition,
% 0.16/0.43 ( binintersectSubset1
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( ( binintersect @ A @ B )
% 0.16/0.43 = A )
% 0.16/0.43 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(bs114d_type,type,
% 0.16/0.43 bs114d: $o ).
% 0.16/0.43
% 0.16/0.43 thf(bs114d,definition,
% 0.16/0.43 ( bs114d
% 0.16/0.43 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.43 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.16/0.43 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(regular_type,type,
% 0.16/0.43 regular: $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(setminus_type,type,
% 0.16/0.43 setminus: $i > $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(setminusI_type,type,
% 0.16/0.43 setminusI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setminusI,definition,
% 0.16/0.43 ( setminusI
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ~ ( in @ Xx @ B )
% 0.16/0.43 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setminusEL_type,type,
% 0.16/0.43 setminusEL: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setminusEL,definition,
% 0.16/0.43 ( setminusEL
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.43 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setminusER_type,type,
% 0.16/0.43 setminusER: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setminusER,definition,
% 0.16/0.43 ( setminusER
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.43 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setminusSubset2_type,type,
% 0.16/0.43 setminusSubset2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setminusSubset2,definition,
% 0.16/0.43 ( setminusSubset2
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( subset @ A @ B )
% 0.16/0.43 => ( ( setminus @ A @ B )
% 0.16/0.43 = emptyset ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setminusERneg_type,type,
% 0.16/0.43 setminusERneg: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setminusERneg,definition,
% 0.16/0.43 ( setminusERneg
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.43 => ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setminusELneg_type,type,
% 0.16/0.43 setminusELneg: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setminusELneg,definition,
% 0.16/0.43 ( setminusELneg
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.43 => ( ~ ( in @ Xx @ B )
% 0.16/0.43 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setminusILneg_type,type,
% 0.16/0.43 setminusILneg: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setminusILneg,definition,
% 0.16/0.43 ( setminusILneg
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ~ ( in @ Xx @ A )
% 0.16/0.43 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setminusIRneg_type,type,
% 0.16/0.43 setminusIRneg: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setminusIRneg,definition,
% 0.16/0.43 ( setminusIRneg
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setminusLsub_type,type,
% 0.16/0.43 setminusLsub: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setminusLsub,definition,
% 0.16/0.43 ( setminusLsub
% 0.16/0.43 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setminusSubset1_type,type,
% 0.16/0.43 setminusSubset1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setminusSubset1,definition,
% 0.16/0.43 ( setminusSubset1
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( ( setminus @ A @ B )
% 0.16/0.43 = emptyset )
% 0.16/0.43 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(symdiff_type,type,
% 0.16/0.43 symdiff: $i > $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(symdiffE_type,type,
% 0.16/0.43 symdiffE: $o ).
% 0.16/0.43
% 0.16/0.43 thf(symdiffE,definition,
% 0.16/0.43 ( symdiffE
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.16/0.43 => ! [Xphi: $o] :
% 0.16/0.43 ( ( ( in @ Xx @ A )
% 0.16/0.43 => ( ~ ( in @ Xx @ B )
% 0.16/0.43 => Xphi ) )
% 0.16/0.43 => ( ( ~ ( in @ Xx @ A )
% 0.16/0.43 => ( ( in @ Xx @ B )
% 0.16/0.43 => Xphi ) )
% 0.16/0.43 => Xphi ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(symdiffI1_type,type,
% 0.16/0.43 symdiffI1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(symdiffI1,definition,
% 0.16/0.43 ( symdiffI1
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ~ ( in @ Xx @ B )
% 0.16/0.43 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(symdiffI2_type,type,
% 0.16/0.43 symdiffI2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(symdiffI2,definition,
% 0.16/0.43 ( symdiffI2
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ~ ( in @ Xx @ A )
% 0.16/0.43 => ( ( in @ Xx @ B )
% 0.16/0.43 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(symdiffIneg1_type,type,
% 0.16/0.43 symdiffIneg1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(symdiffIneg1,definition,
% 0.16/0.43 ( symdiffIneg1
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( in @ Xx @ B )
% 0.16/0.43 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(symdiffIneg2_type,type,
% 0.16/0.43 symdiffIneg2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(symdiffIneg2,definition,
% 0.16/0.43 ( symdiffIneg2
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ~ ( in @ Xx @ A )
% 0.16/0.43 => ( ~ ( in @ Xx @ B )
% 0.16/0.43 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(iskpair_type,type,
% 0.16/0.43 iskpair: $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(secondinupair_type,type,
% 0.16/0.43 secondinupair: $o ).
% 0.16/0.43
% 0.16/0.43 thf(secondinupair,definition,
% 0.16/0.43 ( secondinupair
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairIL_type,type,
% 0.16/0.43 setukpairIL: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairIL,definition,
% 0.16/0.43 ( setukpairIL
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairIR_type,type,
% 0.16/0.43 setukpairIR: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairIR,definition,
% 0.16/0.43 ( setukpairIR
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(kpairiskpair_type,type,
% 0.16/0.43 kpairiskpair: $o ).
% 0.16/0.43
% 0.16/0.43 thf(kpairiskpair,definition,
% 0.16/0.43 ( kpairiskpair
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(kpair_type,type,
% 0.16/0.43 kpair: $i > $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(kpairp_type,type,
% 0.16/0.43 kpairp: $o ).
% 0.16/0.43
% 0.16/0.43 thf(kpairp,definition,
% 0.16/0.43 ( kpairp
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(cartprod_type,type,
% 0.16/0.43 cartprod: $i > $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(singletonsubset_type,type,
% 0.16/0.43 singletonsubset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(singletonsubset,definition,
% 0.16/0.43 ( singletonsubset
% 0.16/0.43 = ( ! [A: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(singletoninpowerset_type,type,
% 0.16/0.43 singletoninpowerset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(singletoninpowerset,definition,
% 0.16/0.43 ( singletoninpowerset
% 0.16/0.43 = ( ! [A: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(singletoninpowunion_type,type,
% 0.16/0.43 singletoninpowunion: $o ).
% 0.16/0.43
% 0.16/0.43 thf(singletoninpowunion,definition,
% 0.16/0.43 ( singletoninpowunion
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(upairset2E_type,type,
% 0.16/0.43 upairset2E: $o ).
% 0.16/0.43
% 0.16/0.43 thf(upairset2E,definition,
% 0.16/0.43 ( upairset2E
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.43 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.16/0.43 => ( ( Xz = Xx )
% 0.16/0.43 | ( Xz = Xy ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(upairsubunion_type,type,
% 0.16/0.43 upairsubunion: $o ).
% 0.16/0.43
% 0.16/0.43 thf(upairsubunion,definition,
% 0.16/0.43 ( upairsubunion
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(upairinpowunion_type,type,
% 0.16/0.43 upairinpowunion: $o ).
% 0.16/0.43
% 0.16/0.43 thf(upairinpowunion,definition,
% 0.16/0.43 ( upairinpowunion
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(ubforcartprodlem1_type,type,
% 0.16/0.43 ubforcartprodlem1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(ubforcartprodlem1,definition,
% 0.16/0.43 ( ubforcartprodlem1
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(ubforcartprodlem2_type,type,
% 0.16/0.43 ubforcartprodlem2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(ubforcartprodlem2,definition,
% 0.16/0.43 ( ubforcartprodlem2
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(ubforcartprodlem3_type,type,
% 0.16/0.43 ubforcartprodlem3: $o ).
% 0.16/0.43
% 0.16/0.43 thf(ubforcartprodlem3,definition,
% 0.16/0.43 ( ubforcartprodlem3
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodpairin_type,type,
% 0.16/0.43 cartprodpairin: $o ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodpairin,definition,
% 0.16/0.43 ( cartprodpairin
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodmempair1_type,type,
% 0.16/0.43 cartprodmempair1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodmempair1,definition,
% 0.16/0.43 ( cartprodmempair1
% 0.16/0.43 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.43 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.43 => ? [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 & ? [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 & ( Xu
% 0.16/0.43 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodmempair_type,type,
% 0.16/0.43 cartprodmempair: $o ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodmempair,definition,
% 0.16/0.43 ( cartprodmempair
% 0.16/0.43 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.43 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.43 => ( iskpair @ Xu ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setunionE2_type,type,
% 0.16/0.43 setunionE2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setunionE2,definition,
% 0.16/0.43 ( setunionE2
% 0.16/0.43 = ( ! [A: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.43 => ? [X: $i] :
% 0.16/0.43 ( ( in @ X @ A )
% 0.16/0.43 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setunionsingleton1_type,type,
% 0.16/0.43 setunionsingleton1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setunionsingleton1,definition,
% 0.16/0.43 ( setunionsingleton1
% 0.16/0.43 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setunionsingleton2_type,type,
% 0.16/0.43 setunionsingleton2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setunionsingleton2,definition,
% 0.16/0.43 ( setunionsingleton2
% 0.16/0.43 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setunionsingleton_type,type,
% 0.16/0.43 setunionsingleton: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setunionsingleton,definition,
% 0.16/0.43 ( setunionsingleton
% 0.16/0.43 = ( ! [Xx: $i] :
% 0.16/0.43 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.16/0.43 = Xx ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(singleton_type,type,
% 0.16/0.43 singleton: $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(singleton,definition,
% 0.16/0.43 ( singleton
% 0.16/0.43 = ( ^ [A: $i] :
% 0.16/0.43 ? [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 & ( A
% 0.16/0.43 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(singletonprop_type,type,
% 0.16/0.43 singletonprop: $o ).
% 0.16/0.43
% 0.16/0.43 thf(singletonprop,definition,
% 0.16/0.43 ( singletonprop
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ A )
% 0.16/0.43 => ( ( Xphi @ Xx )
% 0.16/0.43 => ( ( Xphi @ Xy )
% 0.16/0.43 => ( Xx = Xy ) ) ) ) )
% 0.16/0.43 => ( ? [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 & ( Xphi @ Xx ) )
% 0.16/0.43 => ( singleton
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(ex1_type,type,
% 0.16/0.43 ex1: $i > ( $i > $o ) > $o ).
% 0.16/0.43
% 0.16/0.43 thf(ex1,definition,
% 0.16/0.43 ( ex1
% 0.16/0.43 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.16/0.43 ( singleton
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(ex1E1_type,type,
% 0.16/0.43 ex1E1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(ex1E1,definition,
% 0.16/0.43 ( ex1E1
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.43 ( ( ex1 @ A
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 => ? [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(ex1I_type,type,
% 0.16/0.43 ex1I: $o ).
% 0.16/0.43
% 0.16/0.43 thf(ex1I,definition,
% 0.16/0.43 ( ex1I
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( Xphi @ Xx )
% 0.16/0.43 => ( ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ A )
% 0.16/0.43 => ( ( Xphi @ Xy )
% 0.16/0.43 => ( Xy = Xx ) ) )
% 0.16/0.43 => ( ex1 @ A
% 0.16/0.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(ex1I2_type,type,
% 0.16/0.43 ex1I2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(ex1I2,definition,
% 0.16/0.43 ( ex1I2
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ A )
% 0.16/0.43 => ( ( Xphi @ Xx )
% 0.16/0.43 => ( ( Xphi @ Xy )
% 0.16/0.43 => ( Xx = Xy ) ) ) ) )
% 0.16/0.43 => ( ? [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 & ( Xphi @ Xx ) )
% 0.16/0.43 => ( ex1 @ A
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(singletonsuniq_type,type,
% 0.16/0.43 singletonsuniq: $o ).
% 0.16/0.43
% 0.16/0.43 thf(singletonsuniq,definition,
% 0.16/0.43 ( singletonsuniq
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( ( setadjoin @ Xx @ emptyset )
% 0.16/0.43 = ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.43 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(atmost1p_type,type,
% 0.16/0.43 atmost1p: $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(atleast2p_type,type,
% 0.16/0.43 atleast2p: $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(atmost2p_type,type,
% 0.16/0.43 atmost2p: $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(upairsetp_type,type,
% 0.16/0.43 upairsetp: $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjL1_type,type,
% 0.16/0.43 setukpairinjL1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjL1,definition,
% 0.16/0.43 ( setukpairinjL1
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.43 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.43 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(kfstsingleton_type,type,
% 0.16/0.43 kfstsingleton: $o ).
% 0.16/0.43
% 0.16/0.43 thf(kfstsingleton,definition,
% 0.16/0.43 ( kfstsingleton
% 0.16/0.43 = ( ! [Xu: $i] :
% 0.16/0.43 ( ( iskpair @ Xu )
% 0.16/0.43 => ( singleton
% 0.16/0.43 @ ( dsetconstr @ ( setunion @ Xu )
% 0.16/0.43 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(theprop_type,type,
% 0.16/0.43 theprop: $o ).
% 0.16/0.43
% 0.16/0.43 thf(theprop,definition,
% 0.16/0.43 ( theprop
% 0.16/0.43 = ( ! [X: $i] :
% 0.16/0.43 ( ( singleton @ X )
% 0.16/0.43 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(kfst_type,type,
% 0.16/0.43 kfst: $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(kfstpairEq_type,type,
% 0.16/0.43 kfstpairEq: $o ).
% 0.16/0.43
% 0.16/0.43 thf(kfstpairEq,definition,
% 0.16/0.43 ( kfstpairEq
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.16/0.43 = Xx ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodfstin_type,type,
% 0.16/0.43 cartprodfstin: $o ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodfstin,definition,
% 0.16/0.43 ( cartprodfstin
% 0.16/0.43 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.43 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.43 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjL2_type,type,
% 0.16/0.43 setukpairinjL2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjL2,definition,
% 0.16/0.43 ( setukpairinjL2
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.43 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.43 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.43 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjL_type,type,
% 0.16/0.43 setukpairinjL: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjL,definition,
% 0.16/0.43 ( setukpairinjL
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.43 ( ( ( kpair @ Xx @ Xy )
% 0.16/0.43 = ( kpair @ Xz @ Xu ) )
% 0.16/0.43 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjR11_type,type,
% 0.16/0.43 setukpairinjR11: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjR11,definition,
% 0.16/0.43 ( setukpairinjR11
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( Xx = Xy )
% 0.16/0.43 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.43 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjR12_type,type,
% 0.16/0.43 setukpairinjR12: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjR12,definition,
% 0.16/0.43 ( setukpairinjR12
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( Xx = Xy )
% 0.16/0.43 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.43 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjR1_type,type,
% 0.16/0.43 setukpairinjR1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjR1,definition,
% 0.16/0.43 ( setukpairinjR1
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.43 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.43 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.43 => ( ( Xz = Xu )
% 0.16/0.43 => ( Xy = Xu ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(upairequniteq_type,type,
% 0.16/0.43 upairequniteq: $o ).
% 0.16/0.43
% 0.16/0.43 thf(upairequniteq,definition,
% 0.16/0.43 ( upairequniteq
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.43 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.43 = ( setadjoin @ Xz @ emptyset ) )
% 0.16/0.43 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjR2_type,type,
% 0.16/0.43 setukpairinjR2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjR2,definition,
% 0.16/0.43 ( setukpairinjR2
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.43 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.43 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.43 => ( Xy = Xu ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjR_type,type,
% 0.16/0.43 setukpairinjR: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjR,definition,
% 0.16/0.43 ( setukpairinjR
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.43 ( ( ( kpair @ Xx @ Xy )
% 0.16/0.43 = ( kpair @ Xz @ Xu ) )
% 0.16/0.43 => ( Xy = Xu ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(ksndsingleton_type,type,
% 0.16/0.43 ksndsingleton: $o ).
% 0.16/0.43
% 0.16/0.43 thf(ksndsingleton,definition,
% 0.16/0.43 ( ksndsingleton
% 0.16/0.43 = ( ! [Xu: $i] :
% 0.16/0.43 ( ( iskpair @ Xu )
% 0.16/0.43 => ( singleton
% 0.16/0.43 @ ( dsetconstr @ ( setunion @ Xu )
% 0.16/0.43 @ ^ [Xx: $i] :
% 0.16/0.43 ( Xu
% 0.16/0.43 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(ksnd_type,type,
% 0.16/0.43 ksnd: $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(ksndpairEq_type,type,
% 0.16/0.43 ksndpairEq: $o ).
% 0.16/0.43
% 0.16/0.43 thf(ksndpairEq,definition,
% 0.16/0.43 ( ksndpairEq
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.16/0.43 = Xy ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(kpairsurjEq_type,type,
% 0.16/0.43 kpairsurjEq: $o ).
% 0.16/0.43
% 0.16/0.43 thf(kpairsurjEq,definition,
% 0.16/0.43 ( kpairsurjEq
% 0.16/0.43 = ( ! [Xu: $i] :
% 0.16/0.43 ( ( iskpair @ Xu )
% 0.16/0.43 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.16/0.43 = Xu ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodsndin_type,type,
% 0.16/0.43 cartprodsndin: $o ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodsndin,definition,
% 0.16/0.43 ( cartprodsndin
% 0.16/0.43 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.43 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.43 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodpairmemEL_type,type,
% 0.16/0.43 cartprodpairmemEL: $o ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodpairmemEL,definition,
% 0.16/0.43 ( cartprodpairmemEL
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.16/0.43 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodpairmemER_type,type,
% 0.16/0.43 cartprodpairmemER: $o ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodpairmemER,definition,
% 0.16/0.43 ( cartprodpairmemER
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.16/0.43 => ( in @ Xy @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodmempaircEq_type,type,
% 0.16/0.43 cartprodmempaircEq: $o ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodmempaircEq,definition,
% 0.16/0.43 ( cartprodmempaircEq
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( ( kpair @ Xx @ Xy )
% 0.16/0.43 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodfstpairEq_type,type,
% 0.16/0.43 cartprodfstpairEq: $o ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodfstpairEq,definition,
% 0.16/0.43 ( cartprodfstpairEq
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.16/0.43 = Xx ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodsndpairEq_type,type,
% 0.16/0.43 cartprodsndpairEq: $o ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodsndpairEq,definition,
% 0.16/0.43 ( cartprodsndpairEq
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.16/0.43 = Xy ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodpairsurjEq_type,type,
% 0.16/0.43 cartprodpairsurjEq: $o ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodpairsurjEq,definition,
% 0.16/0.43 ( cartprodpairsurjEq
% 0.16/0.43 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.43 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.43 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.16/0.43 = Xu ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(breln_type,type,
% 0.16/0.43 breln: $i > $i > $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(breln,definition,
% 0.16/0.43 ( breln
% 0.16/0.43 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(dpsetconstr_type,type,
% 0.16/0.43 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.16/0.43
% 0.16/0.43 thf(dpsetconstr,definition,
% 0.16/0.43 ( dpsetconstr
% 0.16/0.43 = ( ^ [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.16/0.43 ( dsetconstr @ ( cartprod @ A @ B )
% 0.16/0.43 @ ^ [Xu: $i] :
% 0.16/0.43 ? [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 & ? [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 & ( Xphi @ Xx @ Xy )
% 0.16/0.43 & ( Xu
% 0.16/0.43 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(dpsetconstrI_type,type,
% 0.16/0.43 dpsetconstrI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(dpsetconstrI,definition,
% 0.16/0.43 ( dpsetconstrI
% 0.16/0.43 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( ( Xphi @ Xx @ Xy )
% 0.16/0.43 => ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.43 @ ( dpsetconstr @ A @ B
% 0.16/0.43 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(dpsetconstrSub_type,type,
% 0.16/0.43 dpsetconstrSub: $o ).
% 0.16/0.43
% 0.16/0.43 thf(dpsetconstrSub,definition,
% 0.16/0.43 ( dpsetconstrSub
% 0.16/0.43 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.16/0.43 ( subset
% 0.16/0.43 @ ( dpsetconstr @ A @ B
% 0.16/0.43 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.16/0.43 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setOfPairsIsBReln,conjecture,
% 0.16/0.43 ( setextAx
% 0.16/0.43 => ( emptysetAx
% 0.16/0.43 => ( setadjoinAx
% 0.16/0.43 => ( powersetAx
% 0.16/0.43 => ( setunionAx
% 0.16/0.43 => ( omega0Ax
% 0.16/0.43 => ( omegaSAx
% 0.16/0.43 => ( omegaIndAx
% 0.16/0.43 => ( replAx
% 0.16/0.43 => ( foundationAx
% 0.16/0.43 => ( wellorderingAx
% 0.16/0.43 => ( descrp
% 0.16/0.43 => ( dsetconstrI
% 0.16/0.43 => ( dsetconstrEL
% 0.16/0.43 => ( dsetconstrER
% 0.16/0.43 => ( exuE1
% 0.16/0.43 => ( prop2setE
% 0.16/0.43 => ( emptysetE
% 0.16/0.43 => ( emptysetimpfalse
% 0.16/0.43 => ( notinemptyset
% 0.16/0.43 => ( exuE3e
% 0.16/0.43 => ( setext
% 0.16/0.43 => ( emptyI
% 0.16/0.43 => ( noeltsimpempty
% 0.16/0.43 => ( setbeta
% 0.16/0.43 => ( nonemptyE1
% 0.16/0.43 => ( nonemptyI
% 0.16/0.43 => ( nonemptyI1
% 0.16/0.43 => ( setadjoinIL
% 0.16/0.43 => ( emptyinunitempty
% 0.16/0.43 => ( setadjoinIR
% 0.16/0.43 => ( setadjoinE
% 0.16/0.43 => ( setadjoinOr
% 0.16/0.43 => ( setoftrueEq
% 0.16/0.43 => ( powersetI
% 0.16/0.43 => ( emptyinPowerset
% 0.16/0.43 => ( emptyInPowerset
% 0.16/0.43 => ( powersetE
% 0.16/0.43 => ( setunionI
% 0.16/0.43 => ( setunionE
% 0.16/0.43 => ( subPowSU
% 0.16/0.43 => ( exuE2
% 0.16/0.43 => ( nonemptyImpWitness
% 0.16/0.43 => ( uniqinunit
% 0.16/0.43 => ( notinsingleton
% 0.16/0.43 => ( eqinunit
% 0.16/0.43 => ( singletonsswitch
% 0.16/0.43 => ( upairsetE
% 0.16/0.43 => ( upairsetIL
% 0.16/0.43 => ( upairsetIR
% 0.16/0.43 => ( emptyE1
% 0.16/0.43 => ( vacuousDall
% 0.16/0.43 => ( quantDeMorgan1
% 0.16/0.43 => ( quantDeMorgan2
% 0.16/0.43 => ( quantDeMorgan3
% 0.16/0.43 => ( quantDeMorgan4
% 0.16/0.43 => ( prop2setI
% 0.16/0.43 => ( prop2set2propI
% 0.16/0.43 => ( notdexE
% 0.16/0.43 => ( notdallE
% 0.16/0.43 => ( exuI1
% 0.16/0.43 => ( exuI3
% 0.16/0.43 => ( exuI2
% 0.16/0.43 => ( inCongP
% 0.16/0.43 => ( in__Cong
% 0.16/0.43 => ( exuE3u
% 0.16/0.43 => ( exu__Cong
% 0.16/0.43 => ( emptyset__Cong
% 0.16/0.43 => ( setadjoin__Cong
% 0.16/0.43 => ( powerset__Cong
% 0.16/0.43 => ( setunion__Cong
% 0.16/0.43 => ( omega__Cong
% 0.16/0.43 => ( exuEu
% 0.16/0.43 => ( descr__Cong
% 0.16/0.43 => ( dsetconstr__Cong
% 0.16/0.43 => ( subsetI1
% 0.16/0.43 => ( eqimpsubset2
% 0.16/0.43 => ( eqimpsubset1
% 0.16/0.43 => ( subsetI2
% 0.16/0.43 => ( emptysetsubset
% 0.16/0.43 => ( subsetE
% 0.16/0.43 => ( subsetE2
% 0.16/0.43 => ( notsubsetI
% 0.16/0.43 => ( notequalI1
% 0.16/0.43 => ( notequalI2
% 0.16/0.43 => ( subsetRefl
% 0.16/0.43 => ( subsetTrans
% 0.16/0.43 => ( setadjoinSub
% 0.16/0.43 => ( setadjoinSub2
% 0.16/0.43 => ( subset2powerset
% 0.16/0.43 => ( setextsub
% 0.16/0.43 => ( subsetemptysetimpeq
% 0.16/0.43 => ( powersetI1
% 0.16/0.43 => ( powersetE1
% 0.16/0.43 => ( inPowerset
% 0.16/0.43 => ( powersetsubset
% 0.16/0.43 => ( sepInPowerset
% 0.16/0.43 => ( sepSubset
% 0.16/0.43 => ( binunionIL
% 0.16/0.43 => ( upairset2IR
% 0.16/0.43 => ( binunionIR
% 0.16/0.43 => ( binunionEcases
% 0.16/0.43 => ( binunionE
% 0.16/0.43 => ( binunionLsub
% 0.16/0.43 => ( binunionRsub
% 0.16/0.43 => ( binintersectI
% 0.16/0.43 => ( binintersectSubset5
% 0.16/0.43 => ( binintersectEL
% 0.16/0.43 => ( binintersectLsub
% 0.16/0.43 => ( binintersectSubset2
% 0.16/0.43 => ( binintersectSubset3
% 0.16/0.43 => ( binintersectER
% 0.16/0.43 => ( disjointsetsI1
% 0.16/0.43 => ( binintersectRsub
% 0.16/0.43 => ( binintersectSubset4
% 0.16/0.43 => ( binintersectSubset1
% 0.16/0.43 => ( bs114d
% 0.16/0.43 => ( setminusI
% 0.16/0.43 => ( setminusEL
% 0.16/0.43 => ( setminusER
% 0.16/0.43 => ( setminusSubset2
% 0.16/0.43 => ( setminusERneg
% 0.16/0.43 => ( setminusELneg
% 0.16/0.43 => ( setminusILneg
% 0.16/0.43 => ( setminusIRneg
% 0.16/0.43 => ( setminusLsub
% 0.16/0.43 => ( setminusSubset1
% 0.16/0.43 => ( symdiffE
% 0.16/0.43 => ( symdiffI1
% 0.16/0.43 => ( symdiffI2
% 0.16/0.43 => ( symdiffIneg1
% 0.16/0.43 => ( symdiffIneg2
% 0.16/0.43 => ( secondinupair
% 0.16/0.43 => ( setukpairIL
% 0.16/0.43 => ( setukpairIR
% 0.16/0.43 => ( kpairiskpair
% 0.16/0.43 => ( kpairp
% 0.16/0.43 => ( singletonsubset
% 0.16/0.43 => ( singletoninpowerset
% 0.16/0.43 => ( singletoninpowunion
% 0.16/0.43 => ( upairset2E
% 0.16/0.43 => ( upairsubunion
% 0.16/0.43 => ( upairinpowunion
% 0.16/0.43 => ( ubforcartprodlem1
% 0.16/0.43 => ( ubforcartprodlem2
% 0.16/0.43 => ( ubforcartprodlem3
% 0.16/0.43 => ( cartprodpairin
% 0.16/0.43 => ( cartprodmempair1
% 0.16/0.43 => ( cartprodmempair
% 0.16/0.43 => ( setunionE2
% 0.16/0.43 => ( setunionsingleton1
% 0.16/0.43 => ( setunionsingleton2
% 0.16/0.43 => ( setunionsingleton
% 0.16/0.43 => ( singletonprop
% 0.16/0.43 => ( ex1E1
% 0.16/0.43 => ( ex1I
% 0.16/0.43 => ( ex1I2
% 0.16/0.43 => ( singletonsuniq
% 0.16/0.43 => ( setukpairinjL1
% 0.16/0.43 => ( kfstsingleton
% 0.16/0.43 => ( theprop
% 0.16/0.43 => ( kfstpairEq
% 0.16/0.43 => ( cartprodfstin
% 0.16/0.43 => ( setukpairinjL2
% 0.16/0.43 => ( setukpairinjL
% 0.16/0.43 => ( setukpairinjR11
% 0.16/0.43 => ( setukpairinjR12
% 0.16/0.43 => ( setukpairinjR1
% 0.16/0.43 => ( upairequniteq
% 0.16/0.43 => ( setukpairinjR2
% 0.16/0.43 => ( setukpairinjR
% 0.16/0.43 => ( ksndsingleton
% 0.16/0.43 => ( ksndpairEq
% 0.16/0.43 => ( kpairsurjEq
% 0.16/0.43 => ( cartprodsndin
% 0.16/0.43 => ( cartprodpairmemEL
% 0.16/0.43 => ( cartprodpairmemER
% 0.16/0.43 => ( cartprodmempaircEq
% 0.16/0.43 => ( cartprodfstpairEq
% 0.16/0.43 => ( cartprodsndpairEq
% 0.16/0.43 => ( cartprodpairsurjEq
% 0.16/0.43 => ( dpsetconstrI
% 0.16/0.43 => ( dpsetconstrSub
% 0.16/0.43 => ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.16/0.43 ( breln @ A @ B
% 0.16/0.43 @ ( dpsetconstr @ A @ B
% 0.16/0.43 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 %------------------------------------------------------------------------------
% 0.16/0.43 ------------------------
% 1.98/1.29 ---- Embedded in THF ---
% 1.98/1.29 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.98/1.29 %%% Generated on Tue Feb 24 12:30:11 EST 2026
% 1.98/1.29 %%% using '$$dhol' embedding, version 1.3.0.
% 1.98/1.29 %%% Logic specification used:
% 1.98/1.29 %%% thf(spec, logic, $$dhol).
% 1.98/1.29
% 1.98/1.29 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.wdvij0GjC6/DTF2DTF20708.p
% See solution above
% 1.98/1.29 ------------------------
% 2.25/1.31 ---- Cleaned THF ---
% 2.25/1.31 thf(in_type,type,
% 2.25/1.31 in: $i > $i > $o ).
% 2.25/1.31
% 2.25/1.31 thf(exu_type,type,
% 2.25/1.31 exu: ( $i > $o ) > $o ).
% 2.25/1.31
% 2.25/1.31 thf(exu,definition,
% 2.25/1.31 ( exu
% 2.25/1.31 = ( ^ [Xphi: $i > $o] :
% 2.25/1.31 ? [Xx: $i] :
% 2.25/1.31 ( ( Xphi @ Xx )
% 2.25/1.31 & ! [Xy: $i] :
% 2.25/1.31 ( ( Xphi @ Xy )
% 2.25/1.31 => ( Xx = Xy ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setextAx_type,type,
% 2.25/1.31 setextAx: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setextAx,definition,
% 2.25/1.31 ( setextAx
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 <=> ( in @ Xx @ B ) )
% 2.25/1.31 => ( A = B ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(emptyset_type,type,
% 2.25/1.31 emptyset: $i ).
% 2.25/1.31
% 2.25/1.31 thf(emptysetAx_type,type,
% 2.25/1.31 emptysetAx: $o ).
% 2.25/1.31
% 2.25/1.31 thf(emptysetAx,definition,
% 2.25/1.31 ( emptysetAx
% 2.25/1.31 = ( ! [Xx: $i] :
% 2.25/1.31 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setadjoin_type,type,
% 2.25/1.31 setadjoin: $i > $i > $i ).
% 2.25/1.31
% 2.25/1.31 thf(setadjoinAx_type,type,
% 2.25/1.31 setadjoinAx: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setadjoinAx,definition,
% 2.25/1.31 ( setadjoinAx
% 2.25/1.31 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.25/1.31 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.25/1.31 <=> ( ( Xy = Xx )
% 2.25/1.31 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(powerset_type,type,
% 2.25/1.31 powerset: $i > $i ).
% 2.25/1.31
% 2.25/1.31 thf(powersetAx_type,type,
% 2.25/1.31 powersetAx: $o ).
% 2.25/1.31
% 2.25/1.31 thf(powersetAx,definition,
% 2.25/1.31 ( powersetAx
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ( in @ B @ ( powerset @ A ) )
% 2.25/1.31 <=> ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ B )
% 2.25/1.31 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setunion_type,type,
% 2.25/1.31 setunion: $i > $i ).
% 2.25/1.31
% 2.25/1.31 thf(setunionAx_type,type,
% 2.25/1.31 setunionAx: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setunionAx,definition,
% 2.25/1.31 ( setunionAx
% 2.25/1.31 = ( ! [A: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ ( setunion @ A ) )
% 2.25/1.31 <=> ? [B: $i] :
% 2.25/1.31 ( ( in @ Xx @ B )
% 2.25/1.31 & ( in @ B @ A ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(omega_type,type,
% 2.25/1.31 omega: $i ).
% 2.25/1.31
% 2.25/1.31 thf(omega0Ax_type,type,
% 2.25/1.31 omega0Ax: $o ).
% 2.25/1.31
% 2.25/1.31 thf(omega0Ax,definition,
% 2.25/1.31 ( omega0Ax
% 2.25/1.31 = ( in @ emptyset @ omega ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(omegaSAx_type,type,
% 2.25/1.31 omegaSAx: $o ).
% 2.25/1.31
% 2.25/1.31 thf(omegaSAx,definition,
% 2.25/1.31 ( omegaSAx
% 2.25/1.31 = ( ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ omega )
% 2.25/1.31 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(omegaIndAx_type,type,
% 2.25/1.31 omegaIndAx: $o ).
% 2.25/1.31
% 2.25/1.31 thf(omegaIndAx,definition,
% 2.25/1.31 ( omegaIndAx
% 2.25/1.31 = ( ! [A: $i] :
% 2.25/1.31 ( ( ( in @ emptyset @ A )
% 2.25/1.31 & ! [Xx: $i] :
% 2.25/1.31 ( ( ( in @ Xx @ omega )
% 2.25/1.31 & ( in @ Xx @ A ) )
% 2.25/1.31 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.25/1.31 => ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ omega )
% 2.25/1.31 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(replAx_type,type,
% 2.25/1.31 replAx: $o ).
% 2.25/1.31
% 2.25/1.31 thf(replAx,definition,
% 2.25/1.31 ( replAx
% 2.25/1.31 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.25/1.31 ( ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ( exu
% 2.25/1.31 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.25/1.31 => ? [B: $i] :
% 2.25/1.31 ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ B )
% 2.25/1.31 <=> ? [Xy: $i] :
% 2.25/1.31 ( ( in @ Xy @ A )
% 2.25/1.31 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(foundationAx_type,type,
% 2.25/1.31 foundationAx: $o ).
% 2.25/1.31
% 2.25/1.31 thf(foundationAx,definition,
% 2.25/1.31 ( foundationAx
% 2.25/1.31 = ( ! [A: $i] :
% 2.25/1.31 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.25/1.31 => ? [B: $i] :
% 2.25/1.31 ( ( in @ B @ A )
% 2.25/1.31 & ~ ? [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ B )
% 2.25/1.31 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(wellorderingAx_type,type,
% 2.25/1.31 wellorderingAx: $o ).
% 2.25/1.31
% 2.25/1.31 thf(wellorderingAx,definition,
% 2.25/1.31 ( wellorderingAx
% 2.25/1.31 = ( ! [A: $i] :
% 2.25/1.31 ? [B: $i] :
% 2.25/1.31 ( ! [C: $i] :
% 2.25/1.31 ( ( in @ C @ B )
% 2.25/1.31 => ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ C )
% 2.25/1.31 => ( in @ Xx @ A ) ) )
% 2.25/1.31 & ! [Xx: $i,Xy: $i] :
% 2.25/1.31 ( ( ( in @ Xx @ A )
% 2.25/1.31 & ( in @ Xy @ A ) )
% 2.25/1.31 => ( ! [C: $i] :
% 2.25/1.31 ( ( in @ C @ B )
% 2.25/1.31 => ( ( in @ Xx @ C )
% 2.25/1.31 <=> ( in @ Xy @ C ) ) )
% 2.25/1.31 => ( Xx = Xy ) ) )
% 2.25/1.31 & ! [C: $i,D: $i] :
% 2.25/1.31 ( ( ( in @ C @ B )
% 2.25/1.31 & ( in @ D @ B ) )
% 2.25/1.31 => ( ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ C )
% 2.25/1.31 => ( in @ Xx @ D ) )
% 2.25/1.31 | ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ D )
% 2.25/1.31 => ( in @ Xx @ C ) ) ) )
% 2.25/1.31 & ! [C: $i] :
% 2.25/1.31 ( ( ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ C )
% 2.25/1.31 => ( in @ Xx @ A ) )
% 2.25/1.31 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.25/1.31 => ? [D: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ D @ B )
% 2.25/1.31 & ( in @ Xx @ C )
% 2.25/1.31 & ~ ? [Xy: $i] :
% 2.25/1.31 ( ( in @ Xy @ D )
% 2.25/1.31 & ( in @ Xy @ C ) )
% 2.25/1.31 & ! [E: $i] :
% 2.25/1.31 ( ( in @ E @ B )
% 2.25/1.31 => ( ! [Xy: $i] :
% 2.25/1.31 ( ( in @ Xy @ E )
% 2.25/1.31 => ( in @ Xy @ D ) )
% 2.25/1.31 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(descr_type,type,
% 2.25/1.31 descr: ( $i > $o ) > $i ).
% 2.25/1.31
% 2.25/1.31 thf(descrp_type,type,
% 2.25/1.31 descrp: $o ).
% 2.25/1.31
% 2.25/1.31 thf(descrp,definition,
% 2.25/1.31 ( descrp
% 2.25/1.31 = ( ! [Xphi: $i > $o] :
% 2.25/1.31 ( ( exu
% 2.25/1.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.25/1.31 => ( Xphi
% 2.25/1.31 @ ( descr
% 2.25/1.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(dsetconstr_type,type,
% 2.25/1.31 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.25/1.31
% 2.25/1.31 thf(dsetconstrI_type,type,
% 2.25/1.31 dsetconstrI: $o ).
% 2.25/1.31
% 2.25/1.31 thf(dsetconstrI,definition,
% 2.25/1.31 ( dsetconstrI
% 2.25/1.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ( ( Xphi @ Xx )
% 2.25/1.31 => ( in @ Xx
% 2.25/1.31 @ ( dsetconstr @ A
% 2.25/1.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(dsetconstrEL_type,type,
% 2.25/1.31 dsetconstrEL: $o ).
% 2.25/1.31
% 2.25/1.31 thf(dsetconstrEL,definition,
% 2.25/1.31 ( dsetconstrEL
% 2.25/1.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx
% 2.25/1.31 @ ( dsetconstr @ A
% 2.25/1.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.25/1.31 => ( in @ Xx @ A ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(dsetconstrER_type,type,
% 2.25/1.31 dsetconstrER: $o ).
% 2.25/1.31
% 2.25/1.31 thf(dsetconstrER,definition,
% 2.25/1.31 ( dsetconstrER
% 2.25/1.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx
% 2.25/1.31 @ ( dsetconstr @ A
% 2.25/1.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.25/1.31 => ( Xphi @ Xx ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(exuE1_type,type,
% 2.25/1.31 exuE1: $o ).
% 2.25/1.31
% 2.25/1.31 thf(exuE1,definition,
% 2.25/1.31 ( exuE1
% 2.25/1.31 = ( ! [Xphi: $i > $o] :
% 2.25/1.31 ( ( exu
% 2.25/1.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.25/1.31 => ? [Xx: $i] :
% 2.25/1.31 ( ( Xphi @ Xx )
% 2.25/1.31 & ! [Xy: $i] :
% 2.25/1.31 ( ( Xphi @ Xy )
% 2.25/1.31 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(prop2set_type,type,
% 2.25/1.31 prop2set: $o > $i ).
% 2.25/1.31
% 2.25/1.31 thf(prop2setE_type,type,
% 2.25/1.31 prop2setE: $o ).
% 2.25/1.31
% 2.25/1.31 thf(prop2setE,definition,
% 2.25/1.31 ( prop2setE
% 2.25/1.31 = ( ! [Xphi: $o,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.25/1.31 => Xphi ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(emptysetE_type,type,
% 2.25/1.31 emptysetE: $o ).
% 2.25/1.31
% 2.25/1.31 thf(emptysetE,definition,
% 2.25/1.31 ( emptysetE
% 2.25/1.31 = ( ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ emptyset )
% 2.25/1.31 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(emptysetimpfalse_type,type,
% 2.25/1.31 emptysetimpfalse: $o ).
% 2.25/1.31
% 2.25/1.31 thf(emptysetimpfalse,definition,
% 2.25/1.31 ( emptysetimpfalse
% 2.25/1.31 = ( ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ emptyset )
% 2.25/1.31 => $false ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(notinemptyset_type,type,
% 2.25/1.31 notinemptyset: $o ).
% 2.25/1.31
% 2.25/1.31 thf(notinemptyset,definition,
% 2.25/1.31 ( notinemptyset
% 2.25/1.31 = ( ! [Xx: $i] :
% 2.25/1.31 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(exuE3e_type,type,
% 2.25/1.31 exuE3e: $o ).
% 2.25/1.31
% 2.25/1.31 thf(exuE3e,definition,
% 2.25/1.31 ( exuE3e
% 2.25/1.31 = ( ! [Xphi: $i > $o] :
% 2.25/1.31 ( ( exu
% 2.25/1.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.25/1.31 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setext_type,type,
% 2.25/1.31 setext: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setext,definition,
% 2.25/1.31 ( setext
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ( in @ Xx @ B ) )
% 2.25/1.31 => ( ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ B )
% 2.25/1.31 => ( in @ Xx @ A ) )
% 2.25/1.31 => ( A = B ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(emptyI_type,type,
% 2.25/1.31 emptyI: $o ).
% 2.25/1.31
% 2.25/1.31 thf(emptyI,definition,
% 2.25/1.31 ( emptyI
% 2.25/1.31 = ( ! [A: $i] :
% 2.25/1.31 ( ! [Xx: $i] :
% 2.25/1.31 ~ ( in @ Xx @ A )
% 2.25/1.31 => ( A = emptyset ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(noeltsimpempty_type,type,
% 2.25/1.31 noeltsimpempty: $o ).
% 2.25/1.31
% 2.25/1.31 thf(noeltsimpempty,definition,
% 2.25/1.31 ( noeltsimpempty
% 2.25/1.31 = ( ! [A: $i] :
% 2.25/1.31 ( ! [Xx: $i] :
% 2.25/1.31 ~ ( in @ Xx @ A )
% 2.25/1.31 => ( A = emptyset ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setbeta_type,type,
% 2.25/1.31 setbeta: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setbeta,definition,
% 2.25/1.31 ( setbeta
% 2.25/1.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ( ( in @ Xx
% 2.25/1.31 @ ( dsetconstr @ A
% 2.25/1.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.25/1.31 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(nonempty_type,type,
% 2.25/1.31 nonempty: $i > $o ).
% 2.25/1.31
% 2.25/1.31 thf(nonempty,definition,
% 2.25/1.31 ( nonempty
% 2.25/1.31 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(nonemptyE1_type,type,
% 2.25/1.31 nonemptyE1: $o ).
% 2.25/1.31
% 2.25/1.31 thf(nonemptyE1,definition,
% 2.25/1.31 ( nonemptyE1
% 2.25/1.31 = ( ! [A: $i] :
% 2.25/1.31 ( ( nonempty @ A )
% 2.25/1.31 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(nonemptyI_type,type,
% 2.25/1.31 nonemptyI: $o ).
% 2.25/1.31
% 2.25/1.31 thf(nonemptyI,definition,
% 2.25/1.31 ( nonemptyI
% 2.25/1.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ( ( Xphi @ Xx )
% 2.25/1.31 => ( nonempty
% 2.25/1.31 @ ( dsetconstr @ A
% 2.25/1.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(nonemptyI1_type,type,
% 2.25/1.31 nonemptyI1: $o ).
% 2.25/1.31
% 2.25/1.31 thf(nonemptyI1,definition,
% 2.25/1.31 ( nonemptyI1
% 2.25/1.31 = ( ! [A: $i] :
% 2.25/1.31 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.25/1.31 => ( nonempty @ A ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setadjoinIL_type,type,
% 2.25/1.31 setadjoinIL: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setadjoinIL,definition,
% 2.25/1.31 ( setadjoinIL
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(emptyinunitempty_type,type,
% 2.25/1.31 emptyinunitempty: $o ).
% 2.25/1.31
% 2.25/1.31 thf(emptyinunitempty,definition,
% 2.25/1.31 ( emptyinunitempty
% 2.25/1.31 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setadjoinIR_type,type,
% 2.25/1.31 setadjoinIR: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setadjoinIR,definition,
% 2.25/1.31 ( setadjoinIR
% 2.25/1.31 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.25/1.31 ( ( in @ Xy @ A )
% 2.25/1.31 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setadjoinE_type,type,
% 2.25/1.31 setadjoinE: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setadjoinE,definition,
% 2.25/1.31 ( setadjoinE
% 2.25/1.31 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.25/1.31 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.25/1.31 => ! [Xphi: $o] :
% 2.25/1.31 ( ( ( Xy = Xx )
% 2.25/1.31 => Xphi )
% 2.25/1.31 => ( ( ( in @ Xy @ A )
% 2.25/1.31 => Xphi )
% 2.25/1.31 => Xphi ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setadjoinOr_type,type,
% 2.25/1.31 setadjoinOr: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setadjoinOr,definition,
% 2.25/1.31 ( setadjoinOr
% 2.25/1.31 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.25/1.31 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.25/1.31 => ( ( Xy = Xx )
% 2.25/1.31 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setoftrueEq_type,type,
% 2.25/1.31 setoftrueEq: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setoftrueEq,definition,
% 2.25/1.31 ( setoftrueEq
% 2.25/1.31 = ( ! [A: $i] :
% 2.25/1.31 ( ( dsetconstr @ A
% 2.25/1.31 @ ^ [Xx: $i] : $true )
% 2.25/1.31 = A ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(powersetI_type,type,
% 2.25/1.31 powersetI: $o ).
% 2.25/1.31
% 2.25/1.31 thf(powersetI,definition,
% 2.25/1.31 ( powersetI
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ B )
% 2.25/1.31 => ( in @ Xx @ A ) )
% 2.25/1.31 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(emptyinPowerset_type,type,
% 2.25/1.31 emptyinPowerset: $o ).
% 2.25/1.31
% 2.25/1.31 thf(emptyinPowerset,definition,
% 2.25/1.31 ( emptyinPowerset
% 2.25/1.31 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(emptyInPowerset_type,type,
% 2.25/1.31 emptyInPowerset: $o ).
% 2.25/1.31
% 2.25/1.31 thf(emptyInPowerset,definition,
% 2.25/1.31 ( emptyInPowerset
% 2.25/1.31 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(powersetE_type,type,
% 2.25/1.31 powersetE: $o ).
% 2.25/1.31
% 2.25/1.31 thf(powersetE,definition,
% 2.25/1.31 ( powersetE
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ B @ ( powerset @ A ) )
% 2.25/1.31 => ( ( in @ Xx @ B )
% 2.25/1.31 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setunionI_type,type,
% 2.25/1.31 setunionI: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setunionI,definition,
% 2.25/1.31 ( setunionI
% 2.25/1.31 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.25/1.31 ( ( in @ Xx @ B )
% 2.25/1.31 => ( ( in @ B @ A )
% 2.25/1.31 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setunionE_type,type,
% 2.25/1.31 setunionE: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setunionE,definition,
% 2.25/1.31 ( setunionE
% 2.25/1.31 = ( ! [A: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ ( setunion @ A ) )
% 2.25/1.31 => ! [Xphi: $o] :
% 2.25/1.31 ( ! [B: $i] :
% 2.25/1.31 ( ( in @ Xx @ B )
% 2.25/1.31 => ( ( in @ B @ A )
% 2.25/1.31 => Xphi ) )
% 2.25/1.31 => Xphi ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(subPowSU_type,type,
% 2.25/1.31 subPowSU: $o ).
% 2.25/1.31
% 2.25/1.31 thf(subPowSU,definition,
% 2.25/1.31 ( subPowSU
% 2.25/1.31 = ( ! [A: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(exuE2_type,type,
% 2.25/1.31 exuE2: $o ).
% 2.25/1.31
% 2.25/1.31 thf(exuE2,definition,
% 2.25/1.31 ( exuE2
% 2.25/1.31 = ( ! [Xphi: $i > $o] :
% 2.25/1.31 ( ( exu
% 2.25/1.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.25/1.31 => ? [Xx: $i] :
% 2.25/1.31 ! [Xy: $i] :
% 2.25/1.31 ( ( Xphi @ Xy )
% 2.25/1.31 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(nonemptyImpWitness_type,type,
% 2.25/1.31 nonemptyImpWitness: $o ).
% 2.25/1.31
% 2.25/1.31 thf(nonemptyImpWitness,definition,
% 2.25/1.31 ( nonemptyImpWitness
% 2.25/1.31 = ( ! [A: $i] :
% 2.25/1.31 ( ( nonempty @ A )
% 2.25/1.31 => ? [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 & $true ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(uniqinunit_type,type,
% 2.25/1.31 uniqinunit: $o ).
% 2.25/1.31
% 2.25/1.31 thf(uniqinunit,definition,
% 2.25/1.31 ( uniqinunit
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i] :
% 2.25/1.31 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.25/1.31 => ( Xx = Xy ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(notinsingleton_type,type,
% 2.25/1.31 notinsingleton: $o ).
% 2.25/1.31
% 2.25/1.31 thf(notinsingleton,definition,
% 2.25/1.31 ( notinsingleton
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i] :
% 2.25/1.31 ( ( Xx != Xy )
% 2.25/1.31 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(eqinunit_type,type,
% 2.25/1.31 eqinunit: $o ).
% 2.25/1.31
% 2.25/1.31 thf(eqinunit,definition,
% 2.25/1.31 ( eqinunit
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i] :
% 2.25/1.31 ( ( Xx = Xy )
% 2.25/1.31 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(singletonsswitch_type,type,
% 2.25/1.31 singletonsswitch: $o ).
% 2.25/1.31
% 2.25/1.31 thf(singletonsswitch,definition,
% 2.25/1.31 ( singletonsswitch
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i] :
% 2.25/1.31 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.25/1.31 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(upairsetE_type,type,
% 2.25/1.31 upairsetE: $o ).
% 2.25/1.31
% 2.25/1.31 thf(upairsetE,definition,
% 2.25/1.31 ( upairsetE
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.25/1.31 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.25/1.31 => ( ( Xz = Xx )
% 2.25/1.31 | ( Xz = Xy ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(upairsetIL_type,type,
% 2.25/1.31 upairsetIL: $o ).
% 2.25/1.31
% 2.25/1.31 thf(upairsetIL,definition,
% 2.25/1.31 ( upairsetIL
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(upairsetIR_type,type,
% 2.25/1.31 upairsetIR: $o ).
% 2.25/1.31
% 2.25/1.31 thf(upairsetIR,definition,
% 2.25/1.31 ( upairsetIR
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(emptyE1_type,type,
% 2.25/1.31 emptyE1: $o ).
% 2.25/1.31
% 2.25/1.31 thf(emptyE1,definition,
% 2.25/1.31 ( emptyE1
% 2.25/1.31 = ( ! [A: $i,Xphi: $i > $o] :
% 2.25/1.31 ( ? [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 & ( Xphi @ Xx ) )
% 2.25/1.31 => ( ( ( dsetconstr @ A
% 2.25/1.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.25/1.31 = emptyset )
% 2.25/1.31 => $false ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(vacuousDall_type,type,
% 2.25/1.31 vacuousDall: $o ).
% 2.25/1.31
% 2.25/1.31 thf(vacuousDall,definition,
% 2.25/1.31 ( vacuousDall
% 2.25/1.31 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ emptyset )
% 2.25/1.31 => ( Xphi @ Xx ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(quantDeMorgan1_type,type,
% 2.25/1.31 quantDeMorgan1: $o ).
% 2.25/1.31
% 2.25/1.31 thf(quantDeMorgan1,definition,
% 2.25/1.31 ( quantDeMorgan1
% 2.25/1.31 = ( ! [A: $i,Xphi: $i > $o] :
% 2.25/1.31 ( ~ ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ( Xphi @ Xx ) )
% 2.25/1.31 => ? [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(quantDeMorgan2_type,type,
% 2.25/1.31 quantDeMorgan2: $o ).
% 2.25/1.31
% 2.25/1.31 thf(quantDeMorgan2,definition,
% 2.25/1.31 ( quantDeMorgan2
% 2.25/1.31 = ( ! [A: $i,Xphi: $i > $o] :
% 2.25/1.31 ( ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ~ ( Xphi @ Xx ) )
% 2.25/1.31 => ~ ? [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(quantDeMorgan3_type,type,
% 2.25/1.31 quantDeMorgan3: $o ).
% 2.25/1.31
% 2.25/1.31 thf(quantDeMorgan3,definition,
% 2.25/1.31 ( quantDeMorgan3
% 2.25/1.31 = ( ! [A: $i,Xphi: $i > $o] :
% 2.25/1.31 ( ~ ? [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 & ( Xphi @ Xx ) )
% 2.25/1.31 => ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(quantDeMorgan4_type,type,
% 2.25/1.31 quantDeMorgan4: $o ).
% 2.25/1.31
% 2.25/1.31 thf(quantDeMorgan4,definition,
% 2.25/1.31 ( quantDeMorgan4
% 2.25/1.31 = ( ! [A: $i,Xphi: $i > $o] :
% 2.25/1.31 ( ? [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 & ~ ( Xphi @ Xx ) )
% 2.25/1.31 => ~ ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(prop2setI_type,type,
% 2.25/1.31 prop2setI: $o ).
% 2.25/1.31
% 2.25/1.31 thf(prop2setI,definition,
% 2.25/1.31 ( prop2setI
% 2.25/1.31 = ( ! [Xphi: $o] :
% 2.25/1.31 ( Xphi
% 2.25/1.31 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(set2prop_type,type,
% 2.25/1.31 set2prop: $i > $o ).
% 2.25/1.31
% 2.25/1.31 thf(prop2set2propI_type,type,
% 2.25/1.31 prop2set2propI: $o ).
% 2.25/1.31
% 2.25/1.31 thf(prop2set2propI,definition,
% 2.25/1.31 ( prop2set2propI
% 2.25/1.31 = ( ! [Xphi: $o] :
% 2.25/1.31 ( Xphi
% 2.25/1.31 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(notdexE_type,type,
% 2.25/1.31 notdexE: $o ).
% 2.25/1.31
% 2.25/1.31 thf(notdexE,definition,
% 2.25/1.31 ( notdexE
% 2.25/1.31 = ( ! [A: $i,Xphi: $i > $o] :
% 2.25/1.31 ( ~ ? [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 & ( Xphi @ Xx ) )
% 2.25/1.31 => ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(notdallE_type,type,
% 2.25/1.31 notdallE: $o ).
% 2.25/1.31
% 2.25/1.31 thf(notdallE,definition,
% 2.25/1.31 ( notdallE
% 2.25/1.31 = ( ! [A: $i,Xphi: $i > $o] :
% 2.25/1.31 ( ~ ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ( Xphi @ Xx ) )
% 2.25/1.31 => ? [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(exuI1_type,type,
% 2.25/1.31 exuI1: $o ).
% 2.25/1.31
% 2.25/1.31 thf(exuI1,definition,
% 2.25/1.31 ( exuI1
% 2.25/1.31 = ( ! [Xphi: $i > $o] :
% 2.25/1.31 ( ? [Xx: $i] :
% 2.25/1.31 ( ( Xphi @ Xx )
% 2.25/1.31 & ! [Xy: $i] :
% 2.25/1.31 ( ( Xphi @ Xy )
% 2.25/1.31 => ( Xx = Xy ) ) )
% 2.25/1.31 => ( exu
% 2.25/1.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(exuI3_type,type,
% 2.25/1.31 exuI3: $o ).
% 2.25/1.31
% 2.25/1.31 thf(exuI3,definition,
% 2.25/1.31 ( exuI3
% 2.25/1.31 = ( ! [Xphi: $i > $o] :
% 2.25/1.31 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.25/1.31 => ( ! [Xx: $i,Xy: $i] :
% 2.25/1.31 ( ( Xphi @ Xx )
% 2.25/1.31 => ( ( Xphi @ Xy )
% 2.25/1.31 => ( Xx = Xy ) ) )
% 2.25/1.31 => ( exu
% 2.25/1.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(exuI2_type,type,
% 2.25/1.31 exuI2: $o ).
% 2.25/1.31
% 2.25/1.31 thf(exuI2,definition,
% 2.25/1.31 ( exuI2
% 2.25/1.31 = ( ! [Xphi: $i > $o] :
% 2.25/1.31 ( ? [Xx: $i] :
% 2.25/1.31 ! [Xy: $i] :
% 2.25/1.31 ( ( Xphi @ Xy )
% 2.25/1.31 <=> ( Xy = Xx ) )
% 2.25/1.31 => ( exu
% 2.25/1.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(inCongP_type,type,
% 2.25/1.31 inCongP: $o ).
% 2.25/1.31
% 2.25/1.31 thf(inCongP,definition,
% 2.25/1.31 ( inCongP
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ( A = B )
% 2.25/1.31 => ! [Xx: $i,Xy: $i] :
% 2.25/1.31 ( ( Xx = Xy )
% 2.25/1.31 => ( ( in @ Xx @ A )
% 2.25/1.31 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(in__Cong_type,type,
% 2.25/1.31 in__Cong: $o ).
% 2.25/1.31
% 2.25/1.31 thf(in__Cong,definition,
% 2.25/1.31 ( in__Cong
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ( A = B )
% 2.25/1.31 => ! [Xx: $i,Xy: $i] :
% 2.25/1.31 ( ( Xx = Xy )
% 2.25/1.31 => ( ( in @ Xx @ A )
% 2.25/1.31 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(exuE3u_type,type,
% 2.25/1.31 exuE3u: $o ).
% 2.25/1.31
% 2.25/1.31 thf(exuE3u,definition,
% 2.25/1.31 ( exuE3u
% 2.25/1.31 = ( ! [Xphi: $i > $o] :
% 2.25/1.31 ( ( exu
% 2.25/1.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.25/1.31 => ! [Xx: $i,Xy: $i] :
% 2.25/1.31 ( ( Xphi @ Xx )
% 2.25/1.31 => ( ( Xphi @ Xy )
% 2.25/1.31 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(exu__Cong_type,type,
% 2.25/1.31 exu__Cong: $o ).
% 2.25/1.31
% 2.25/1.31 thf(exu__Cong,definition,
% 2.25/1.31 ( exu__Cong
% 2.25/1.31 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.25/1.31 ( ! [Xx: $i,Xy: $i] :
% 2.25/1.31 ( ( Xx = Xy )
% 2.25/1.31 => ( ( Xphi @ Xx )
% 2.25/1.31 <=> ( Xpsi @ Xy ) ) )
% 2.25/1.31 => ( ( exu
% 2.25/1.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.25/1.31 <=> ( exu
% 2.25/1.31 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(emptyset__Cong_type,type,
% 2.25/1.31 emptyset__Cong: $o ).
% 2.25/1.31
% 2.25/1.31 thf(emptyset__Cong,definition,
% 2.25/1.31 ( emptyset__Cong
% 2.25/1.31 = ( emptyset = emptyset ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setadjoin__Cong_type,type,
% 2.25/1.31 setadjoin__Cong: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setadjoin__Cong,definition,
% 2.25/1.31 ( setadjoin__Cong
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i] :
% 2.25/1.31 ( ( Xx = Xy )
% 2.25/1.31 => ! [Xz: $i,Xu: $i] :
% 2.25/1.31 ( ( Xz = Xu )
% 2.25/1.31 => ( ( setadjoin @ Xx @ Xz )
% 2.25/1.31 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(powerset__Cong_type,type,
% 2.25/1.31 powerset__Cong: $o ).
% 2.25/1.31
% 2.25/1.31 thf(powerset__Cong,definition,
% 2.25/1.31 ( powerset__Cong
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ( A = B )
% 2.25/1.31 => ( ( powerset @ A )
% 2.25/1.31 = ( powerset @ B ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setunion__Cong_type,type,
% 2.25/1.31 setunion__Cong: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setunion__Cong,definition,
% 2.25/1.31 ( setunion__Cong
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ( A = B )
% 2.25/1.31 => ( ( setunion @ A )
% 2.25/1.31 = ( setunion @ B ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(omega__Cong_type,type,
% 2.25/1.31 omega__Cong: $o ).
% 2.25/1.31
% 2.25/1.31 thf(omega__Cong,definition,
% 2.25/1.31 ( omega__Cong
% 2.25/1.31 = ( omega = omega ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(exuEu_type,type,
% 2.25/1.31 exuEu: $o ).
% 2.25/1.31
% 2.25/1.31 thf(exuEu,definition,
% 2.25/1.31 ( exuEu
% 2.25/1.31 = ( ! [Xphi: $i > $o] :
% 2.25/1.31 ( ( exu
% 2.25/1.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.25/1.31 => ! [Xx: $i,Xy: $i] :
% 2.25/1.31 ( ( Xphi @ Xx )
% 2.25/1.31 => ( ( Xphi @ Xy )
% 2.25/1.31 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(descr__Cong_type,type,
% 2.25/1.31 descr__Cong: $o ).
% 2.25/1.31
% 2.25/1.31 thf(descr__Cong,definition,
% 2.25/1.31 ( descr__Cong
% 2.25/1.31 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.25/1.31 ( ! [Xx: $i,Xy: $i] :
% 2.25/1.31 ( ( Xx = Xy )
% 2.25/1.31 => ( ( Xphi @ Xx )
% 2.25/1.31 <=> ( Xpsi @ Xy ) ) )
% 2.25/1.31 => ( ( exu
% 2.25/1.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.25/1.31 => ( ( exu
% 2.25/1.31 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.25/1.31 => ( ( descr
% 2.25/1.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.25/1.31 = ( descr
% 2.25/1.31 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(dsetconstr__Cong_type,type,
% 2.25/1.31 dsetconstr__Cong: $o ).
% 2.25/1.31
% 2.25/1.31 thf(dsetconstr__Cong,definition,
% 2.25/1.31 ( dsetconstr__Cong
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ( A = B )
% 2.25/1.31 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.25/1.31 ( ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ! [Xy: $i] :
% 2.25/1.31 ( ( in @ Xy @ B )
% 2.25/1.31 => ( ( Xx = Xy )
% 2.25/1.31 => ( ( Xphi @ Xx )
% 2.25/1.31 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.25/1.31 => ( ( dsetconstr @ A
% 2.25/1.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.25/1.31 = ( dsetconstr @ B
% 2.25/1.31 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(subset_type,type,
% 2.25/1.31 subset: $i > $i > $o ).
% 2.25/1.31
% 2.25/1.31 thf(disjoint_type,type,
% 2.25/1.31 disjoint: $i > $i > $o ).
% 2.25/1.31
% 2.25/1.31 thf(setsmeet_type,type,
% 2.25/1.31 setsmeet: $i > $i > $o ).
% 2.25/1.31
% 2.25/1.31 thf(subsetI1_type,type,
% 2.25/1.31 subsetI1: $o ).
% 2.25/1.31
% 2.25/1.31 thf(subsetI1,definition,
% 2.25/1.31 ( subsetI1
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ( in @ Xx @ B ) )
% 2.25/1.31 => ( subset @ A @ B ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(eqimpsubset2_type,type,
% 2.25/1.31 eqimpsubset2: $o ).
% 2.25/1.31
% 2.25/1.31 thf(eqimpsubset2,definition,
% 2.25/1.31 ( eqimpsubset2
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ( A = B )
% 2.25/1.31 => ( subset @ B @ A ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(eqimpsubset1_type,type,
% 2.25/1.31 eqimpsubset1: $o ).
% 2.25/1.31
% 2.25/1.31 thf(eqimpsubset1,definition,
% 2.25/1.31 ( eqimpsubset1
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ( A = B )
% 2.25/1.31 => ( subset @ A @ B ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(subsetI2_type,type,
% 2.25/1.31 subsetI2: $o ).
% 2.25/1.31
% 2.25/1.31 thf(subsetI2,definition,
% 2.25/1.31 ( subsetI2
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ( in @ Xx @ B ) )
% 2.25/1.31 => ( subset @ A @ B ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(emptysetsubset_type,type,
% 2.25/1.31 emptysetsubset: $o ).
% 2.25/1.31
% 2.25/1.31 thf(emptysetsubset,definition,
% 2.25/1.31 ( emptysetsubset
% 2.25/1.31 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(subsetE_type,type,
% 2.25/1.31 subsetE: $o ).
% 2.25/1.31
% 2.25/1.31 thf(subsetE,definition,
% 2.25/1.31 ( subsetE
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( subset @ A @ B )
% 2.25/1.31 => ( ( in @ Xx @ A )
% 2.25/1.31 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(subsetE2_type,type,
% 2.25/1.31 subsetE2: $o ).
% 2.25/1.31
% 2.25/1.31 thf(subsetE2,definition,
% 2.25/1.31 ( subsetE2
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( subset @ A @ B )
% 2.25/1.31 => ( ~ ( in @ Xx @ B )
% 2.25/1.31 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(notsubsetI_type,type,
% 2.25/1.31 notsubsetI: $o ).
% 2.25/1.31
% 2.25/1.31 thf(notsubsetI,definition,
% 2.25/1.31 ( notsubsetI
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ( ~ ( in @ Xx @ B )
% 2.25/1.31 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(notequalI1_type,type,
% 2.25/1.31 notequalI1: $o ).
% 2.25/1.31
% 2.25/1.31 thf(notequalI1,definition,
% 2.25/1.31 ( notequalI1
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ~ ( subset @ A @ B )
% 2.25/1.31 => ( A != B ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(notequalI2_type,type,
% 2.25/1.31 notequalI2: $o ).
% 2.25/1.31
% 2.25/1.31 thf(notequalI2,definition,
% 2.25/1.31 ( notequalI2
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ( ~ ( in @ Xx @ B )
% 2.25/1.31 => ( A != B ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(subsetRefl_type,type,
% 2.25/1.31 subsetRefl: $o ).
% 2.25/1.31
% 2.25/1.31 thf(subsetRefl,definition,
% 2.25/1.31 ( subsetRefl
% 2.25/1.31 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(subsetTrans_type,type,
% 2.25/1.31 subsetTrans: $o ).
% 2.25/1.31
% 2.25/1.31 thf(subsetTrans,definition,
% 2.25/1.31 ( subsetTrans
% 2.25/1.31 = ( ! [A: $i,B: $i,C: $i] :
% 2.25/1.31 ( ( subset @ A @ B )
% 2.25/1.31 => ( ( subset @ B @ C )
% 2.25/1.31 => ( subset @ A @ C ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setadjoinSub_type,type,
% 2.25/1.31 setadjoinSub: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setadjoinSub,definition,
% 2.25/1.31 ( setadjoinSub
% 2.25/1.31 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setadjoinSub2_type,type,
% 2.25/1.31 setadjoinSub2: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setadjoinSub2,definition,
% 2.25/1.31 ( setadjoinSub2
% 2.25/1.31 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.25/1.31 ( ( subset @ A @ B )
% 2.25/1.31 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(subset2powerset_type,type,
% 2.25/1.31 subset2powerset: $o ).
% 2.25/1.31
% 2.25/1.31 thf(subset2powerset,definition,
% 2.25/1.31 ( subset2powerset
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ( subset @ A @ B )
% 2.25/1.31 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setextsub_type,type,
% 2.25/1.31 setextsub: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setextsub,definition,
% 2.25/1.31 ( setextsub
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ( subset @ A @ B )
% 2.25/1.31 => ( ( subset @ B @ A )
% 2.25/1.31 => ( A = B ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(subsetemptysetimpeq_type,type,
% 2.25/1.31 subsetemptysetimpeq: $o ).
% 2.25/1.31
% 2.25/1.31 thf(subsetemptysetimpeq,definition,
% 2.25/1.31 ( subsetemptysetimpeq
% 2.25/1.31 = ( ! [A: $i] :
% 2.25/1.31 ( ( subset @ A @ emptyset )
% 2.25/1.31 => ( A = emptyset ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(powersetI1_type,type,
% 2.25/1.31 powersetI1: $o ).
% 2.25/1.31
% 2.25/1.31 thf(powersetI1,definition,
% 2.25/1.31 ( powersetI1
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ( subset @ B @ A )
% 2.25/1.31 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(powersetE1_type,type,
% 2.25/1.31 powersetE1: $o ).
% 2.25/1.31
% 2.25/1.31 thf(powersetE1,definition,
% 2.25/1.31 ( powersetE1
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ( in @ B @ ( powerset @ A ) )
% 2.25/1.31 => ( subset @ B @ A ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(inPowerset_type,type,
% 2.25/1.31 inPowerset: $o ).
% 2.25/1.31
% 2.25/1.31 thf(inPowerset,definition,
% 2.25/1.31 ( inPowerset
% 2.25/1.31 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(powersetsubset_type,type,
% 2.25/1.31 powersetsubset: $o ).
% 2.25/1.31
% 2.25/1.31 thf(powersetsubset,definition,
% 2.25/1.31 ( powersetsubset
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ( subset @ A @ B )
% 2.25/1.31 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(sepInPowerset_type,type,
% 2.25/1.31 sepInPowerset: $o ).
% 2.25/1.31
% 2.25/1.31 thf(sepInPowerset,definition,
% 2.25/1.31 ( sepInPowerset
% 2.25/1.31 = ( ! [A: $i,Xphi: $i > $o] :
% 2.25/1.31 ( in
% 2.25/1.31 @ ( dsetconstr @ A
% 2.25/1.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.25/1.31 @ ( powerset @ A ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(sepSubset_type,type,
% 2.25/1.31 sepSubset: $o ).
% 2.25/1.31
% 2.25/1.31 thf(sepSubset,definition,
% 2.25/1.31 ( sepSubset
% 2.25/1.31 = ( ! [A: $i,Xphi: $i > $o] :
% 2.25/1.31 ( subset
% 2.25/1.31 @ ( dsetconstr @ A
% 2.25/1.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.25/1.31 @ A ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(binunion_type,type,
% 2.25/1.31 binunion: $i > $i > $i ).
% 2.25/1.31
% 2.25/1.31 thf(binunionIL_type,type,
% 2.25/1.31 binunionIL: $o ).
% 2.25/1.31
% 2.25/1.31 thf(binunionIL,definition,
% 2.25/1.31 ( binunionIL
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(upairset2IR_type,type,
% 2.25/1.31 upairset2IR: $o ).
% 2.25/1.31
% 2.25/1.31 thf(upairset2IR,definition,
% 2.25/1.31 ( upairset2IR
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(binunionIR_type,type,
% 2.25/1.31 binunionIR: $o ).
% 2.25/1.31
% 2.25/1.31 thf(binunionIR,definition,
% 2.25/1.31 ( binunionIR
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ B )
% 2.25/1.31 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(binunionEcases_type,type,
% 2.25/1.31 binunionEcases: $o ).
% 2.25/1.31
% 2.25/1.31 thf(binunionEcases,definition,
% 2.25/1.31 ( binunionEcases
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.25/1.31 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.25/1.31 => ( ( ( in @ Xx @ A )
% 2.25/1.31 => Xphi )
% 2.25/1.31 => ( ( ( in @ Xx @ B )
% 2.25/1.31 => Xphi )
% 2.25/1.31 => Xphi ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(binunionE_type,type,
% 2.25/1.31 binunionE: $o ).
% 2.25/1.31
% 2.25/1.31 thf(binunionE,definition,
% 2.25/1.31 ( binunionE
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.25/1.31 => ( ( in @ Xx @ A )
% 2.25/1.31 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(binunionLsub_type,type,
% 2.25/1.31 binunionLsub: $o ).
% 2.25/1.31
% 2.25/1.31 thf(binunionLsub,definition,
% 2.25/1.31 ( binunionLsub
% 2.25/1.31 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(binunionRsub_type,type,
% 2.25/1.31 binunionRsub: $o ).
% 2.25/1.31
% 2.25/1.31 thf(binunionRsub,definition,
% 2.25/1.31 ( binunionRsub
% 2.25/1.31 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(binintersect_type,type,
% 2.25/1.31 binintersect: $i > $i > $i ).
% 2.25/1.31
% 2.25/1.31 thf(binintersectI_type,type,
% 2.25/1.31 binintersectI: $o ).
% 2.25/1.31
% 2.25/1.31 thf(binintersectI,definition,
% 2.25/1.31 ( binintersectI
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ( ( in @ Xx @ B )
% 2.25/1.31 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(binintersectSubset5_type,type,
% 2.25/1.31 binintersectSubset5: $o ).
% 2.25/1.31
% 2.25/1.31 thf(binintersectSubset5,definition,
% 2.25/1.31 ( binintersectSubset5
% 2.25/1.31 = ( ! [A: $i,B: $i,C: $i] :
% 2.25/1.31 ( ( subset @ C @ A )
% 2.25/1.31 => ( ( subset @ C @ B )
% 2.25/1.31 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(binintersectEL_type,type,
% 2.25/1.31 binintersectEL: $o ).
% 2.25/1.31
% 2.25/1.31 thf(binintersectEL,definition,
% 2.25/1.31 ( binintersectEL
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.25/1.31 => ( in @ Xx @ A ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(binintersectLsub_type,type,
% 2.25/1.31 binintersectLsub: $o ).
% 2.25/1.31
% 2.25/1.31 thf(binintersectLsub,definition,
% 2.25/1.31 ( binintersectLsub
% 2.25/1.31 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(binintersectSubset2_type,type,
% 2.25/1.31 binintersectSubset2: $o ).
% 2.25/1.31
% 2.25/1.31 thf(binintersectSubset2,definition,
% 2.25/1.31 ( binintersectSubset2
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ( subset @ A @ B )
% 2.25/1.31 => ( ( binintersect @ A @ B )
% 2.25/1.31 = A ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(binintersectSubset3_type,type,
% 2.25/1.31 binintersectSubset3: $o ).
% 2.25/1.31
% 2.25/1.31 thf(binintersectSubset3,definition,
% 2.25/1.31 ( binintersectSubset3
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ( ( binintersect @ A @ B )
% 2.25/1.31 = B )
% 2.25/1.31 => ( subset @ B @ A ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(binintersectER_type,type,
% 2.25/1.31 binintersectER: $o ).
% 2.25/1.31
% 2.25/1.31 thf(binintersectER,definition,
% 2.25/1.31 ( binintersectER
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.25/1.31 => ( in @ Xx @ B ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(disjointsetsI1_type,type,
% 2.25/1.31 disjointsetsI1: $o ).
% 2.25/1.31
% 2.25/1.31 thf(disjointsetsI1,definition,
% 2.25/1.31 ( disjointsetsI1
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ~ ? [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 & ( in @ Xx @ B ) )
% 2.25/1.31 => ( ( binintersect @ A @ B )
% 2.25/1.31 = emptyset ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(binintersectRsub_type,type,
% 2.25/1.31 binintersectRsub: $o ).
% 2.25/1.31
% 2.25/1.31 thf(binintersectRsub,definition,
% 2.25/1.31 ( binintersectRsub
% 2.25/1.31 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(binintersectSubset4_type,type,
% 2.25/1.31 binintersectSubset4: $o ).
% 2.25/1.31
% 2.25/1.31 thf(binintersectSubset4,definition,
% 2.25/1.31 ( binintersectSubset4
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ( subset @ B @ A )
% 2.25/1.31 => ( ( binintersect @ A @ B )
% 2.25/1.31 = B ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(binintersectSubset1_type,type,
% 2.25/1.31 binintersectSubset1: $o ).
% 2.25/1.31
% 2.25/1.31 thf(binintersectSubset1,definition,
% 2.25/1.31 ( binintersectSubset1
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ( ( binintersect @ A @ B )
% 2.25/1.31 = A )
% 2.25/1.31 => ( subset @ A @ B ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(bs114d_type,type,
% 2.25/1.31 bs114d: $o ).
% 2.25/1.31
% 2.25/1.31 thf(bs114d,definition,
% 2.25/1.31 ( bs114d
% 2.25/1.31 = ( ! [A: $i,B: $i,C: $i] :
% 2.25/1.31 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.25/1.31 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(regular_type,type,
% 2.25/1.31 regular: $i > $o ).
% 2.25/1.31
% 2.25/1.31 thf(setminus_type,type,
% 2.25/1.31 setminus: $i > $i > $i ).
% 2.25/1.31
% 2.25/1.31 thf(setminusI_type,type,
% 2.25/1.31 setminusI: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setminusI,definition,
% 2.25/1.31 ( setminusI
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ( ~ ( in @ Xx @ B )
% 2.25/1.31 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setminusEL_type,type,
% 2.25/1.31 setminusEL: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setminusEL,definition,
% 2.25/1.31 ( setminusEL
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.25/1.31 => ( in @ Xx @ A ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setminusER_type,type,
% 2.25/1.31 setminusER: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setminusER,definition,
% 2.25/1.31 ( setminusER
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.25/1.31 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setminusSubset2_type,type,
% 2.25/1.31 setminusSubset2: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setminusSubset2,definition,
% 2.25/1.31 ( setminusSubset2
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ( subset @ A @ B )
% 2.25/1.31 => ( ( setminus @ A @ B )
% 2.25/1.31 = emptyset ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setminusERneg_type,type,
% 2.25/1.31 setminusERneg: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setminusERneg,definition,
% 2.25/1.31 ( setminusERneg
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.25/1.31 => ( ( in @ Xx @ A )
% 2.25/1.31 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setminusELneg_type,type,
% 2.25/1.31 setminusELneg: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setminusELneg,definition,
% 2.25/1.31 ( setminusELneg
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.25/1.31 => ( ~ ( in @ Xx @ B )
% 2.25/1.31 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setminusILneg_type,type,
% 2.25/1.31 setminusILneg: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setminusILneg,definition,
% 2.25/1.31 ( setminusILneg
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ~ ( in @ Xx @ A )
% 2.25/1.31 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setminusIRneg_type,type,
% 2.25/1.31 setminusIRneg: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setminusIRneg,definition,
% 2.25/1.31 ( setminusIRneg
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ B )
% 2.25/1.31 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setminusLsub_type,type,
% 2.25/1.31 setminusLsub: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setminusLsub,definition,
% 2.25/1.31 ( setminusLsub
% 2.25/1.31 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setminusSubset1_type,type,
% 2.25/1.31 setminusSubset1: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setminusSubset1,definition,
% 2.25/1.31 ( setminusSubset1
% 2.25/1.31 = ( ! [A: $i,B: $i] :
% 2.25/1.31 ( ( ( setminus @ A @ B )
% 2.25/1.31 = emptyset )
% 2.25/1.31 => ( subset @ A @ B ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(symdiff_type,type,
% 2.25/1.31 symdiff: $i > $i > $i ).
% 2.25/1.31
% 2.25/1.31 thf(symdiffE_type,type,
% 2.25/1.31 symdiffE: $o ).
% 2.25/1.31
% 2.25/1.31 thf(symdiffE,definition,
% 2.25/1.31 ( symdiffE
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.25/1.31 => ! [Xphi: $o] :
% 2.25/1.31 ( ( ( in @ Xx @ A )
% 2.25/1.31 => ( ~ ( in @ Xx @ B )
% 2.25/1.31 => Xphi ) )
% 2.25/1.31 => ( ( ~ ( in @ Xx @ A )
% 2.25/1.31 => ( ( in @ Xx @ B )
% 2.25/1.31 => Xphi ) )
% 2.25/1.31 => Xphi ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(symdiffI1_type,type,
% 2.25/1.31 symdiffI1: $o ).
% 2.25/1.31
% 2.25/1.31 thf(symdiffI1,definition,
% 2.25/1.31 ( symdiffI1
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ( ~ ( in @ Xx @ B )
% 2.25/1.31 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(symdiffI2_type,type,
% 2.25/1.31 symdiffI2: $o ).
% 2.25/1.31
% 2.25/1.31 thf(symdiffI2,definition,
% 2.25/1.31 ( symdiffI2
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ~ ( in @ Xx @ A )
% 2.25/1.31 => ( ( in @ Xx @ B )
% 2.25/1.31 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(symdiffIneg1_type,type,
% 2.25/1.31 symdiffIneg1: $o ).
% 2.25/1.31
% 2.25/1.31 thf(symdiffIneg1,definition,
% 2.25/1.31 ( symdiffIneg1
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ( ( in @ Xx @ B )
% 2.25/1.31 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(symdiffIneg2_type,type,
% 2.25/1.31 symdiffIneg2: $o ).
% 2.25/1.31
% 2.25/1.31 thf(symdiffIneg2,definition,
% 2.25/1.31 ( symdiffIneg2
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ~ ( in @ Xx @ A )
% 2.25/1.31 => ( ~ ( in @ Xx @ B )
% 2.25/1.31 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(iskpair_type,type,
% 2.25/1.31 iskpair: $i > $o ).
% 2.25/1.31
% 2.25/1.31 thf(secondinupair_type,type,
% 2.25/1.31 secondinupair: $o ).
% 2.25/1.31
% 2.25/1.31 thf(secondinupair,definition,
% 2.25/1.31 ( secondinupair
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setukpairIL_type,type,
% 2.25/1.31 setukpairIL: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setukpairIL,definition,
% 2.25/1.31 ( setukpairIL
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setukpairIR_type,type,
% 2.25/1.31 setukpairIR: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setukpairIR,definition,
% 2.25/1.31 ( setukpairIR
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(kpairiskpair_type,type,
% 2.25/1.31 kpairiskpair: $o ).
% 2.25/1.31
% 2.25/1.31 thf(kpairiskpair,definition,
% 2.25/1.31 ( kpairiskpair
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(kpair_type,type,
% 2.25/1.31 kpair: $i > $i > $i ).
% 2.25/1.31
% 2.25/1.31 thf(kpairp_type,type,
% 2.25/1.31 kpairp: $o ).
% 2.25/1.31
% 2.25/1.31 thf(kpairp,definition,
% 2.25/1.31 ( kpairp
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(cartprod_type,type,
% 2.25/1.31 cartprod: $i > $i > $i ).
% 2.25/1.31
% 2.25/1.31 thf(singletonsubset_type,type,
% 2.25/1.31 singletonsubset: $o ).
% 2.25/1.31
% 2.25/1.31 thf(singletonsubset,definition,
% 2.25/1.31 ( singletonsubset
% 2.25/1.31 = ( ! [A: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(singletoninpowerset_type,type,
% 2.25/1.31 singletoninpowerset: $o ).
% 2.25/1.31
% 2.25/1.31 thf(singletoninpowerset,definition,
% 2.25/1.31 ( singletoninpowerset
% 2.25/1.31 = ( ! [A: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(singletoninpowunion_type,type,
% 2.25/1.31 singletoninpowunion: $o ).
% 2.25/1.31
% 2.25/1.31 thf(singletoninpowunion,definition,
% 2.25/1.31 ( singletoninpowunion
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(upairset2E_type,type,
% 2.25/1.31 upairset2E: $o ).
% 2.25/1.31
% 2.25/1.31 thf(upairset2E,definition,
% 2.25/1.31 ( upairset2E
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.25/1.31 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.25/1.31 => ( ( Xz = Xx )
% 2.25/1.31 | ( Xz = Xy ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(upairsubunion_type,type,
% 2.25/1.31 upairsubunion: $o ).
% 2.25/1.31
% 2.25/1.31 thf(upairsubunion,definition,
% 2.25/1.31 ( upairsubunion
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ! [Xy: $i] :
% 2.25/1.31 ( ( in @ Xy @ B )
% 2.25/1.31 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(upairinpowunion_type,type,
% 2.25/1.31 upairinpowunion: $o ).
% 2.25/1.31
% 2.25/1.31 thf(upairinpowunion,definition,
% 2.25/1.31 ( upairinpowunion
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ! [Xy: $i] :
% 2.25/1.31 ( ( in @ Xy @ B )
% 2.25/1.31 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(ubforcartprodlem1_type,type,
% 2.25/1.31 ubforcartprodlem1: $o ).
% 2.25/1.31
% 2.25/1.31 thf(ubforcartprodlem1,definition,
% 2.25/1.31 ( ubforcartprodlem1
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ! [Xy: $i] :
% 2.25/1.31 ( ( in @ Xy @ B )
% 2.25/1.31 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(ubforcartprodlem2_type,type,
% 2.25/1.31 ubforcartprodlem2: $o ).
% 2.25/1.31
% 2.25/1.31 thf(ubforcartprodlem2,definition,
% 2.25/1.31 ( ubforcartprodlem2
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ! [Xy: $i] :
% 2.25/1.31 ( ( in @ Xy @ B )
% 2.25/1.31 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(ubforcartprodlem3_type,type,
% 2.25/1.31 ubforcartprodlem3: $o ).
% 2.25/1.31
% 2.25/1.31 thf(ubforcartprodlem3,definition,
% 2.25/1.31 ( ubforcartprodlem3
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ! [Xy: $i] :
% 2.25/1.31 ( ( in @ Xy @ B )
% 2.25/1.31 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(cartprodpairin_type,type,
% 2.25/1.31 cartprodpairin: $o ).
% 2.25/1.31
% 2.25/1.31 thf(cartprodpairin,definition,
% 2.25/1.31 ( cartprodpairin
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ! [Xy: $i] :
% 2.25/1.31 ( ( in @ Xy @ B )
% 2.25/1.31 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(cartprodmempair1_type,type,
% 2.25/1.31 cartprodmempair1: $o ).
% 2.25/1.31
% 2.25/1.31 thf(cartprodmempair1,definition,
% 2.25/1.31 ( cartprodmempair1
% 2.25/1.31 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.25/1.31 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.25/1.31 => ? [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 & ? [Xy: $i] :
% 2.25/1.31 ( ( in @ Xy @ B )
% 2.25/1.31 & ( Xu
% 2.25/1.31 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(cartprodmempair_type,type,
% 2.25/1.31 cartprodmempair: $o ).
% 2.25/1.31
% 2.25/1.31 thf(cartprodmempair,definition,
% 2.25/1.31 ( cartprodmempair
% 2.25/1.31 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.25/1.31 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.25/1.31 => ( iskpair @ Xu ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setunionE2_type,type,
% 2.25/1.31 setunionE2: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setunionE2,definition,
% 2.25/1.31 ( setunionE2
% 2.25/1.31 = ( ! [A: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ ( setunion @ A ) )
% 2.25/1.31 => ? [X: $i] :
% 2.25/1.31 ( ( in @ X @ A )
% 2.25/1.31 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setunionsingleton1_type,type,
% 2.25/1.31 setunionsingleton1: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setunionsingleton1,definition,
% 2.25/1.31 ( setunionsingleton1
% 2.25/1.31 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setunionsingleton2_type,type,
% 2.25/1.31 setunionsingleton2: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setunionsingleton2,definition,
% 2.25/1.31 ( setunionsingleton2
% 2.25/1.31 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setunionsingleton_type,type,
% 2.25/1.31 setunionsingleton: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setunionsingleton,definition,
% 2.25/1.31 ( setunionsingleton
% 2.25/1.31 = ( ! [Xx: $i] :
% 2.25/1.31 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.25/1.31 = Xx ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(singleton_type,type,
% 2.25/1.31 singleton: $i > $o ).
% 2.25/1.31
% 2.25/1.31 thf(singleton,definition,
% 2.25/1.31 ( singleton
% 2.25/1.31 = ( ^ [A: $i] :
% 2.25/1.31 ? [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 & ( A
% 2.25/1.31 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(singletonprop_type,type,
% 2.25/1.31 singletonprop: $o ).
% 2.25/1.31
% 2.25/1.31 thf(singletonprop,definition,
% 2.25/1.31 ( singletonprop
% 2.25/1.31 = ( ! [A: $i,Xphi: $i > $o] :
% 2.25/1.31 ( ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ! [Xy: $i] :
% 2.25/1.31 ( ( in @ Xy @ A )
% 2.25/1.31 => ( ( Xphi @ Xx )
% 2.25/1.31 => ( ( Xphi @ Xy )
% 2.25/1.31 => ( Xx = Xy ) ) ) ) )
% 2.25/1.31 => ( ? [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 & ( Xphi @ Xx ) )
% 2.25/1.31 => ( singleton
% 2.25/1.31 @ ( dsetconstr @ A
% 2.25/1.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(ex1_type,type,
% 2.25/1.31 ex1: $i > ( $i > $o ) > $o ).
% 2.25/1.31
% 2.25/1.31 thf(ex1,definition,
% 2.25/1.31 ( ex1
% 2.25/1.31 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.25/1.31 ( singleton
% 2.25/1.31 @ ( dsetconstr @ A
% 2.25/1.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(ex1E1_type,type,
% 2.25/1.31 ex1E1: $o ).
% 2.25/1.31
% 2.25/1.31 thf(ex1E1,definition,
% 2.25/1.31 ( ex1E1
% 2.25/1.31 = ( ! [A: $i,Xphi: $i > $o] :
% 2.25/1.31 ( ( ex1 @ A
% 2.25/1.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.25/1.31 => ? [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(ex1I_type,type,
% 2.25/1.31 ex1I: $o ).
% 2.25/1.31
% 2.25/1.31 thf(ex1I,definition,
% 2.25/1.31 ( ex1I
% 2.25/1.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ( ( Xphi @ Xx )
% 2.25/1.31 => ( ! [Xy: $i] :
% 2.25/1.31 ( ( in @ Xy @ A )
% 2.25/1.31 => ( ( Xphi @ Xy )
% 2.25/1.31 => ( Xy = Xx ) ) )
% 2.25/1.31 => ( ex1 @ A
% 2.25/1.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(ex1I2_type,type,
% 2.25/1.31 ex1I2: $o ).
% 2.25/1.31
% 2.25/1.31 thf(ex1I2,definition,
% 2.25/1.31 ( ex1I2
% 2.25/1.31 = ( ! [A: $i,Xphi: $i > $o] :
% 2.25/1.31 ( ! [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ! [Xy: $i] :
% 2.25/1.31 ( ( in @ Xy @ A )
% 2.25/1.31 => ( ( Xphi @ Xx )
% 2.25/1.31 => ( ( Xphi @ Xy )
% 2.25/1.31 => ( Xx = Xy ) ) ) ) )
% 2.25/1.31 => ( ? [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 & ( Xphi @ Xx ) )
% 2.25/1.31 => ( ex1 @ A
% 2.25/1.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(singletonsuniq_type,type,
% 2.25/1.31 singletonsuniq: $o ).
% 2.25/1.31
% 2.25/1.31 thf(singletonsuniq,definition,
% 2.25/1.31 ( singletonsuniq
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i] :
% 2.25/1.31 ( ( ( setadjoin @ Xx @ emptyset )
% 2.25/1.31 = ( setadjoin @ Xy @ emptyset ) )
% 2.25/1.31 => ( Xx = Xy ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(atmost1p_type,type,
% 2.25/1.31 atmost1p: $i > $o ).
% 2.25/1.31
% 2.25/1.31 thf(atleast2p_type,type,
% 2.25/1.31 atleast2p: $i > $o ).
% 2.25/1.31
% 2.25/1.31 thf(atmost2p_type,type,
% 2.25/1.31 atmost2p: $i > $o ).
% 2.25/1.31
% 2.25/1.31 thf(upairsetp_type,type,
% 2.25/1.31 upairsetp: $i > $o ).
% 2.25/1.31
% 2.25/1.31 thf(setukpairinjL1_type,type,
% 2.25/1.31 setukpairinjL1: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setukpairinjL1,definition,
% 2.25/1.31 ( setukpairinjL1
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.25/1.31 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.25/1.31 => ( Xx = Xz ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(kfstsingleton_type,type,
% 2.25/1.31 kfstsingleton: $o ).
% 2.25/1.31
% 2.25/1.31 thf(kfstsingleton,definition,
% 2.25/1.31 ( kfstsingleton
% 2.25/1.31 = ( ! [Xu: $i] :
% 2.25/1.31 ( ( iskpair @ Xu )
% 2.25/1.31 => ( singleton
% 2.25/1.31 @ ( dsetconstr @ ( setunion @ Xu )
% 2.25/1.31 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(theprop_type,type,
% 2.25/1.31 theprop: $o ).
% 2.25/1.31
% 2.25/1.31 thf(theprop,definition,
% 2.25/1.31 ( theprop
% 2.25/1.31 = ( ! [X: $i] :
% 2.25/1.31 ( ( singleton @ X )
% 2.25/1.31 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(kfst_type,type,
% 2.25/1.31 kfst: $i > $i ).
% 2.25/1.31
% 2.25/1.31 thf(kfstpairEq_type,type,
% 2.25/1.31 kfstpairEq: $o ).
% 2.25/1.31
% 2.25/1.31 thf(kfstpairEq,definition,
% 2.25/1.31 ( kfstpairEq
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i] :
% 2.25/1.31 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.25/1.31 = Xx ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(cartprodfstin_type,type,
% 2.25/1.31 cartprodfstin: $o ).
% 2.25/1.31
% 2.25/1.31 thf(cartprodfstin,definition,
% 2.25/1.31 ( cartprodfstin
% 2.25/1.31 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.25/1.31 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.25/1.31 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setukpairinjL2_type,type,
% 2.25/1.31 setukpairinjL2: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setukpairinjL2,definition,
% 2.25/1.31 ( setukpairinjL2
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.25/1.31 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.25/1.31 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.25/1.31 => ( Xx = Xz ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setukpairinjL_type,type,
% 2.25/1.31 setukpairinjL: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setukpairinjL,definition,
% 2.25/1.31 ( setukpairinjL
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.25/1.31 ( ( ( kpair @ Xx @ Xy )
% 2.25/1.31 = ( kpair @ Xz @ Xu ) )
% 2.25/1.31 => ( Xx = Xz ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setukpairinjR11_type,type,
% 2.25/1.31 setukpairinjR11: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setukpairinjR11,definition,
% 2.25/1.31 ( setukpairinjR11
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i] :
% 2.25/1.31 ( ( Xx = Xy )
% 2.25/1.31 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.25/1.31 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setukpairinjR12_type,type,
% 2.25/1.31 setukpairinjR12: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setukpairinjR12,definition,
% 2.25/1.31 ( setukpairinjR12
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i] :
% 2.25/1.31 ( ( Xx = Xy )
% 2.25/1.31 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.25/1.31 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setukpairinjR1_type,type,
% 2.25/1.31 setukpairinjR1: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setukpairinjR1,definition,
% 2.25/1.31 ( setukpairinjR1
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.25/1.31 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.25/1.31 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.25/1.31 => ( ( Xz = Xu )
% 2.25/1.31 => ( Xy = Xu ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(upairequniteq_type,type,
% 2.25/1.31 upairequniteq: $o ).
% 2.25/1.31
% 2.25/1.31 thf(upairequniteq,definition,
% 2.25/1.31 ( upairequniteq
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.25/1.31 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.25/1.31 = ( setadjoin @ Xz @ emptyset ) )
% 2.25/1.31 => ( Xx = Xy ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setukpairinjR2_type,type,
% 2.25/1.31 setukpairinjR2: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setukpairinjR2,definition,
% 2.25/1.31 ( setukpairinjR2
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.25/1.31 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.25/1.31 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.25/1.31 => ( Xy = Xu ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setukpairinjR_type,type,
% 2.25/1.31 setukpairinjR: $o ).
% 2.25/1.31
% 2.25/1.31 thf(setukpairinjR,definition,
% 2.25/1.31 ( setukpairinjR
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.25/1.31 ( ( ( kpair @ Xx @ Xy )
% 2.25/1.31 = ( kpair @ Xz @ Xu ) )
% 2.25/1.31 => ( Xy = Xu ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(ksndsingleton_type,type,
% 2.25/1.31 ksndsingleton: $o ).
% 2.25/1.31
% 2.25/1.31 thf(ksndsingleton,definition,
% 2.25/1.31 ( ksndsingleton
% 2.25/1.31 = ( ! [Xu: $i] :
% 2.25/1.31 ( ( iskpair @ Xu )
% 2.25/1.31 => ( singleton
% 2.25/1.31 @ ( dsetconstr @ ( setunion @ Xu )
% 2.25/1.31 @ ^ [Xx: $i] :
% 2.25/1.31 ( Xu
% 2.25/1.31 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(ksnd_type,type,
% 2.25/1.31 ksnd: $i > $i ).
% 2.25/1.31
% 2.25/1.31 thf(ksndpairEq_type,type,
% 2.25/1.31 ksndpairEq: $o ).
% 2.25/1.31
% 2.25/1.31 thf(ksndpairEq,definition,
% 2.25/1.31 ( ksndpairEq
% 2.25/1.31 = ( ! [Xx: $i,Xy: $i] :
% 2.25/1.31 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.25/1.31 = Xy ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(kpairsurjEq_type,type,
% 2.25/1.31 kpairsurjEq: $o ).
% 2.25/1.31
% 2.25/1.31 thf(kpairsurjEq,definition,
% 2.25/1.31 ( kpairsurjEq
% 2.25/1.31 = ( ! [Xu: $i] :
% 2.25/1.31 ( ( iskpair @ Xu )
% 2.25/1.31 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.25/1.31 = Xu ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(cartprodsndin_type,type,
% 2.25/1.31 cartprodsndin: $o ).
% 2.25/1.31
% 2.25/1.31 thf(cartprodsndin,definition,
% 2.25/1.31 ( cartprodsndin
% 2.25/1.31 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.25/1.31 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.25/1.31 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(cartprodpairmemEL_type,type,
% 2.25/1.31 cartprodpairmemEL: $o ).
% 2.25/1.31
% 2.25/1.31 thf(cartprodpairmemEL,definition,
% 2.25/1.31 ( cartprodpairmemEL
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.25/1.31 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.25/1.31 => ( in @ Xx @ A ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(cartprodpairmemER_type,type,
% 2.25/1.31 cartprodpairmemER: $o ).
% 2.25/1.31
% 2.25/1.31 thf(cartprodpairmemER,definition,
% 2.25/1.31 ( cartprodpairmemER
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.25/1.31 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.25/1.31 => ( in @ Xy @ B ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(cartprodmempaircEq_type,type,
% 2.25/1.31 cartprodmempaircEq: $o ).
% 2.25/1.31
% 2.25/1.31 thf(cartprodmempaircEq,definition,
% 2.25/1.31 ( cartprodmempaircEq
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ! [Xy: $i] :
% 2.25/1.31 ( ( in @ Xy @ B )
% 2.25/1.31 => ( ( kpair @ Xx @ Xy )
% 2.25/1.31 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(cartprodfstpairEq_type,type,
% 2.25/1.31 cartprodfstpairEq: $o ).
% 2.25/1.31
% 2.25/1.31 thf(cartprodfstpairEq,definition,
% 2.25/1.31 ( cartprodfstpairEq
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ! [Xy: $i] :
% 2.25/1.31 ( ( in @ Xy @ B )
% 2.25/1.31 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.25/1.31 = Xx ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(cartprodsndpairEq_type,type,
% 2.25/1.31 cartprodsndpairEq: $o ).
% 2.25/1.31
% 2.25/1.31 thf(cartprodsndpairEq,definition,
% 2.25/1.31 ( cartprodsndpairEq
% 2.25/1.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ! [Xy: $i] :
% 2.25/1.31 ( ( in @ Xy @ B )
% 2.25/1.31 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.25/1.31 = Xy ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(cartprodpairsurjEq_type,type,
% 2.25/1.31 cartprodpairsurjEq: $o ).
% 2.25/1.31
% 2.25/1.31 thf(cartprodpairsurjEq,definition,
% 2.25/1.31 ( cartprodpairsurjEq
% 2.25/1.31 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.25/1.31 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.25/1.31 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.25/1.31 = Xu ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(breln_type,type,
% 2.25/1.31 breln: $i > $i > $i > $o ).
% 2.25/1.31
% 2.25/1.31 thf(breln,definition,
% 2.25/1.31 ( breln
% 2.25/1.31 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(dpsetconstr_type,type,
% 2.25/1.31 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.25/1.31
% 2.25/1.31 thf(dpsetconstr,definition,
% 2.25/1.31 ( dpsetconstr
% 2.25/1.31 = ( ^ [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.25/1.31 ( dsetconstr @ ( cartprod @ A @ B )
% 2.25/1.31 @ ^ [Xu: $i] :
% 2.25/1.31 ? [Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 & ? [Xy: $i] :
% 2.25/1.31 ( ( in @ Xy @ B )
% 2.25/1.31 & ( Xphi @ Xx @ Xy )
% 2.25/1.31 & ( Xu
% 2.25/1.31 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(dpsetconstrI_type,type,
% 2.25/1.31 dpsetconstrI: $o ).
% 2.25/1.31
% 2.25/1.31 thf(dpsetconstrI,definition,
% 2.25/1.31 ( dpsetconstrI
% 2.25/1.31 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.25/1.31 ( ( in @ Xx @ A )
% 2.25/1.31 => ! [Xy: $i] :
% 2.25/1.31 ( ( in @ Xy @ B )
% 2.25/1.31 => ( ( Xphi @ Xx @ Xy )
% 2.25/1.31 => ( in @ ( kpair @ Xx @ Xy )
% 2.25/1.31 @ ( dpsetconstr @ A @ B
% 2.25/1.31 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(dpsetconstrSub_type,type,
% 2.25/1.31 dpsetconstrSub: $o ).
% 2.25/1.31
% 2.25/1.31 thf(dpsetconstrSub,definition,
% 2.25/1.31 ( dpsetconstrSub
% 2.25/1.31 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.25/1.31 ( subset
% 2.25/1.31 @ ( dpsetconstr @ A @ B
% 2.25/1.31 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.25/1.31 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.25/1.31
% 2.25/1.31 thf(setOfPairsIsBReln,conjecture,
% 2.25/1.31 ( setextAx
% 2.25/1.31 => ( emptysetAx
% 2.25/1.31 => ( setadjoinAx
% 2.25/1.31 => ( powersetAx
% 2.25/1.31 => ( setunionAx
% 2.25/1.31 => ( omega0Ax
% 2.25/1.31 => ( omegaSAx
% 2.25/1.31 => ( omegaIndAx
% 2.25/1.31 => ( replAx
% 2.25/1.31 => ( foundationAx
% 2.25/1.31 => ( wellorderingAx
% 2.25/1.31 => ( descrp
% 2.25/1.31 => ( dsetconstrI
% 2.25/1.31 => ( dsetconstrEL
% 2.25/1.31 => ( dsetconstrER
% 2.25/1.31 => ( exuE1
% 2.25/1.31 => ( prop2setE
% 2.25/1.31 => ( emptysetE
% 2.25/1.31 => ( emptysetimpfalse
% 2.25/1.31 => ( notinemptyset
% 2.25/1.31 => ( exuE3e
% 2.25/1.31 => ( setext
% 2.25/1.31 => ( emptyI
% 2.25/1.31 => ( noeltsimpempty
% 2.25/1.31 => ( setbeta
% 2.25/1.31 => ( nonemptyE1
% 2.25/1.31 => ( nonemptyI
% 2.25/1.31 => ( nonemptyI1
% 2.25/1.31 => ( setadjoinIL
% 2.25/1.31 => ( emptyinunitempty
% 2.25/1.31 => ( setadjoinIR
% 2.25/1.31 => ( setadjoinE
% 2.25/1.31 => ( setadjoinOr
% 2.25/1.31 => ( setoftrueEq
% 2.25/1.31 => ( powersetI
% 2.25/1.31 => ( emptyinPowerset
% 2.25/1.31 => ( emptyInPowerset
% 2.25/1.31 => ( powersetE
% 2.25/1.31 => ( setunionI
% 2.25/1.31 => ( setunionE
% 2.25/1.31 => ( subPowSU
% 2.25/1.31 => ( exuE2
% 2.25/1.31 => ( nonemptyImpWitness
% 2.25/1.31 => ( uniqinunit
% 2.25/1.31 => ( notinsingleton
% 2.25/1.31 => ( eqinunit
% 2.25/1.31 => ( singletonsswitch
% 2.25/1.31 => ( upairsetE
% 2.25/1.31 => ( upairsetIL
% 2.25/1.31 => ( upairsetIR
% 2.25/1.31 => ( emptyE1
% 2.25/1.31 => ( vacuousDall
% 2.25/1.31 => ( quantDeMorgan1
% 2.25/1.31 => ( quantDeMorgan2
% 2.25/1.31 => ( quantDeMorgan3
% 2.25/1.31 => ( quantDeMorgan4
% 2.25/1.31 => ( prop2setI
% 2.25/1.31 => ( prop2set2propI
% 2.25/1.31 => ( notdexE
% 2.25/1.31 => ( notdallE
% 2.25/1.31 => ( exuI1
% 2.25/1.31 => ( exuI3
% 2.25/1.31 => ( exuI2
% 2.25/1.31 => ( inCongP
% 2.25/1.31 => ( in__Cong
% 2.25/1.31 => ( exuE3u
% 2.25/1.31 => ( exu__Cong
% 2.25/1.31 => ( emptyset__Cong
% 2.25/1.31 => ( setadjoin__Cong
% 2.25/1.31 => ( powerset__Cong
% 2.25/1.31 => ( setunion__Cong
% 2.25/1.31 => ( omega__Cong
% 2.25/1.31 => ( exuEu
% 2.25/1.31 => ( descr__Cong
% 2.25/1.31 => ( dsetconstr__Cong
% 2.25/1.31 => ( subsetI1
% 2.25/1.31 => ( eqimpsubset2
% 2.25/1.31 => ( eqimpsubset1
% 2.25/1.31 => ( subsetI2
% 2.25/1.31 => ( emptysetsubset
% 2.25/1.31 => ( subsetE
% 2.25/1.31 => ( subsetE2
% 2.25/1.31 => ( notsubsetI
% 2.25/1.31 => ( notequalI1
% 2.25/1.31 => ( notequalI2
% 2.25/1.31 => ( subsetRefl
% 2.25/1.31 => ( subsetTrans
% 2.25/1.31 => ( setadjoinSub
% 2.25/1.31 => ( setadjoinSub2
% 2.25/1.31 => ( subset2powerset
% 2.25/1.31 => ( setextsub
% 2.25/1.31 => ( subsetemptysetimpeq
% 2.25/1.31 => ( powersetI1
% 2.25/1.31 => ( powersetE1
% 2.25/1.31 => ( inPowerset
% 2.25/1.31 => ( powersetsubset
% 2.25/1.31 => ( sepInPowerset
% 2.25/1.31 => ( sepSubset
% 2.25/1.31 => ( binunionIL
% 2.25/1.31 => ( upairset2IR
% 2.25/1.31 => ( binunionIR
% 2.25/1.31 => ( binunionEcases
% 2.25/1.31 => ( binunionE
% 2.25/1.31 => ( binunionLsub
% 2.25/1.31 => ( binunionRsub
% 2.25/1.31 => ( binintersectI
% 2.25/1.31 => ( binintersectSubset5
% 2.25/1.31 => ( binintersectEL
% 2.25/1.31 => ( binintersectLsub
% 2.25/1.31 => ( binintersectSubset2
% 2.25/1.31 => ( binintersectSubset3
% 2.25/1.31 => ( binintersectER
% 2.25/1.31 => ( disjointsetsI1
% 2.25/1.31 => ( binintersectRsub
% 2.25/1.31 => ( binintersectSubset4
% 2.25/1.31 => ( binintersectSubset1
% 2.25/1.31 => ( bs114d
% 2.25/1.31 => ( setminusI
% 2.25/1.31 => ( setminusEL
% 2.25/1.31 => ( setminusER
% 2.25/1.31 => ( setminusSubset2
% 2.25/1.31 => ( setminusERneg
% 2.25/1.31 => ( setminusELneg
% 2.25/1.31 => ( setminusILneg
% 2.25/1.31 => ( setminusIRneg
% 2.25/1.31 => ( setminusLsub
% 2.25/1.31 => ( setminusSubset1
% 2.25/1.31 => ( symdiffE
% 2.25/1.31 => ( symdiffI1
% 2.25/1.31 => ( symdiffI2
% 2.25/1.31 => ( symdiffIneg1
% 2.25/1.31 => ( symdiffIneg2
% 2.25/1.31 => ( secondinupair
% 2.25/1.31 => ( setukpairIL
% 2.25/1.31 => ( setukpairIR
% 2.25/1.31 => ( kpairiskpair
% 2.25/1.31 => ( kpairp
% 2.25/1.31 => ( singletonsubset
% 2.25/1.31 => ( singletoninpowerset
% 2.25/1.31 => ( singletoninpowunion
% 2.25/1.31 => ( upairset2E
% 2.25/1.31 => ( upairsubunion
% 2.25/1.31 => ( upairinpowunion
% 2.25/1.31 => ( ubforcartprodlem1
% 2.25/1.31 => ( ubforcartprodlem2
% 2.25/1.31 => ( ubforcartprodlem3
% 2.25/1.31 => ( cartprodpairin
% 2.25/1.31 => ( cartprodmempair1
% 2.25/1.31 => ( cartprodmempair
% 2.25/1.31 => ( setunionE2
% 2.25/1.31 => ( setunionsingleton1
% 2.25/1.31 => ( setunionsingleton2
% 2.25/1.31 => ( setunionsingleton
% 2.25/1.31 => ( singletonprop
% 2.25/1.31 => ( ex1E1
% 2.25/1.31 => ( ex1I
% 2.25/1.31 => ( ex1I2
% 2.25/1.31 => ( singletonsuniq
% 2.25/1.31 => ( setukpairinjL1
% 2.25/1.31 => ( kfstsingleton
% 2.25/1.31 => ( theprop
% 2.25/1.31 => ( kfstpairEq
% 2.25/1.31 => ( cartprodfstin
% 2.25/1.31 => ( setukpairinjL2
% 2.25/1.31 => ( setukpairinjL
% 2.25/1.31 => ( setukpairinjR11
% 2.25/1.31 => ( setukpairinjR12
% 2.25/1.31 => ( setukpairinjR1
% 2.25/1.31 => ( upairequniteq
% 2.25/1.31 => ( setukpairinjR2
% 2.25/1.31 => ( setukpairinjR
% 2.25/1.31 => ( ksndsingleton
% 2.25/1.31 => ( ksndpairEq
% 2.25/1.31 => ( kpairsurjEq
% 2.25/1.31 => ( cartprodsndin
% 2.25/1.31 => ( cartprodpairmemEL
% 2.25/1.31 => ( cartprodpairmemER
% 2.25/1.31 => ( cartprodmempaircEq
% 2.25/1.31 => ( cartprodfstpairEq
% 2.25/1.31 => ( cartprodsndpairEq
% 2.25/1.31 => ( cartprodpairsurjEq
% 2.25/1.31 => ( dpsetconstrI
% 2.25/1.31 => ( dpsetconstrSub
% 2.25/1.31 => ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.25/1.32 ( breln @ A @ B
% 2.25/1.32 @ ( dpsetconstr @ A @ B
% 2.25/1.32 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.25/1.32 ------------------------
% 2.25/1.33 % (20878)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_20708 for (2999ds/275Mi)
% 2.25/1.33 % (20879)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_20708 for (2999ds/18Mi)
% 2.25/1.33 % (20874)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_20708 for (2999ds/4Mi)
% 2.25/1.33 % (20877)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_20708 for (2999ds/2Mi)
% 2.25/1.33 % (20876)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_20708 for (2999ds/2Mi)
% 2.25/1.33 % (20875)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_20708 for (2999ds/27Mi)
% 2.25/1.33 % (20877)Instruction limit reached!
% 2.25/1.33 % (20877)------------------------------
% 2.25/1.33 % (20877)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.33 % (20877)Termination reason: Unknown
% 2.25/1.33 % (20877)Termination phase: shuffling
% 2.25/1.33
% 2.25/1.33 % (20877)Memory used [KB]: 1279
% 2.25/1.33 % (20877)Time elapsed: 0.002 s
% 2.25/1.33 % (20877)Instructions burned: 2 (million)
% 2.25/1.33 % (20877)------------------------------
% 2.25/1.33 % (20877)------------------------------
% 2.25/1.33 % (20874)Instruction limit reached!
% 2.25/1.33 % (20874)------------------------------
% 2.25/1.33 % (20874)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.33 % (20874)Termination reason: Unknown
% 2.25/1.33 % (20874)Termination phase: shuffling
% 2.25/1.33
% 2.25/1.33 % (20874)Memory used [KB]: 1407
% 2.25/1.33 % (20874)Time elapsed: 0.003 s
% 2.25/1.33 % (20874)Instructions burned: 5 (million)
% 2.25/1.33 % (20874)------------------------------
% 2.25/1.33 % (20874)------------------------------
% 2.25/1.33 % (20876)Instruction limit reached!
% 2.25/1.33 % (20876)------------------------------
% 2.25/1.33 % (20876)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.33 % (20876)Termination reason: Unknown
% 2.25/1.33 % (20876)Termination phase: shuffling
% 2.25/1.33
% 2.25/1.33 % (20876)Memory used [KB]: 1407
% 2.25/1.33 % (20876)Time elapsed: 0.003 s
% 2.25/1.33 % (20876)Instructions burned: 5 (million)
% 2.25/1.33 % (20876)------------------------------
% 2.25/1.33 % (20876)------------------------------
% 2.25/1.33 % (20873)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_20708 for (2999ds/183Mi)
% 2.25/1.34 % (20879)Instruction limit reached!
% 2.25/1.34 % (20879)------------------------------
% 2.25/1.34 % (20879)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.34 % (20879)Termination reason: Unknown
% 2.25/1.34 % (20879)Termination phase: Property scanning
% 2.25/1.34
% 2.25/1.34 % (20879)Memory used [KB]: 1663
% 2.25/1.34 % (20879)Time elapsed: 0.008 s
% 2.25/1.34 % (20879)Instructions burned: 21 (million)
% 2.25/1.34 % (20879)------------------------------
% 2.25/1.34 % (20879)------------------------------
% 2.25/1.34 % (20875)Instruction limit reached!
% 2.25/1.34 % (20875)------------------------------
% 2.25/1.34 % (20875)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.34 % (20875)Termination reason: Unknown
% 2.25/1.34 % (20875)Termination phase: shuffling
% 2.25/1.34
% 2.25/1.34 % (20875)Memory used [KB]: 1918
% 2.25/1.34 % (20875)Time elapsed: 0.009 s
% 2.25/1.34 % (20875)Instructions burned: 27 (million)
% 2.25/1.34 % (20875)------------------------------
% 2.25/1.34 % (20875)------------------------------
% 2.25/1.34 % (20886)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_20708 for (2999ds/3Mi)
% 2.25/1.34 % (20886)Instruction limit reached!
% 2.25/1.34 % (20886)------------------------------
% 2.25/1.34 % (20886)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.34 % (20886)Termination reason: Unknown
% 2.25/1.34 % (20886)Termination phase: shuffling
% 2.25/1.34
% 2.25/1.34 % (20886)Memory used [KB]: 1279
% 2.25/1.34 % (20886)Time elapsed: 0.002 s
% 2.25/1.34 % (20886)Instructions burned: 3 (million)
% 2.25/1.34 % (20886)------------------------------
% 2.25/1.34 % (20886)------------------------------
% 2.25/1.34 % (20888)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_20708 for (2999ds/37Mi)
% 2.25/1.34 % (20889)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_20708 for (2999ds/15Mi)
% 2.25/1.35 % (20889)Instruction limit reached!
% 2.25/1.35 % (20889)------------------------------
% 2.25/1.35 % (20889)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.35 % (20889)Termination reason: Unknown
% 2.25/1.35 % (20889)Termination phase: shuffling
% 2.25/1.35
% 2.25/1.35 % (20889)Memory used [KB]: 1663
% 2.25/1.35 % (20889)Time elapsed: 0.006 s
% 2.25/1.35 % (20889)Instructions burned: 15 (million)
% 2.25/1.35 % (20889)------------------------------
% 2.25/1.35 % (20889)------------------------------
% 2.25/1.35 % (20891)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_20708 for (2999ds/3Mi)
% 2.25/1.35 % (20891)Instruction limit reached!
% 2.25/1.35 % (20891)------------------------------
% 2.25/1.35 % (20891)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.35 % (20891)Termination reason: Unknown
% 2.25/1.35 % (20891)Termination phase: shuffling
% 2.25/1.35
% 2.25/1.35 % (20891)Memory used [KB]: 1279
% 2.25/1.35 % (20891)Time elapsed: 0.002 s
% 2.25/1.35 % (20891)Instructions burned: 3 (million)
% 2.25/1.35 % (20891)------------------------------
% 2.25/1.35 % (20891)------------------------------
% 2.25/1.35 % (20892)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_20708 for (2999ds/1041Mi)
% 2.25/1.35 % (20893)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_20708 for (2999ds/7Mi)
% 2.25/1.35 % (20888)Instruction limit reached!
% 2.25/1.35 % (20888)------------------------------
% 2.25/1.35 % (20888)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.35 % (20888)Termination reason: Unknown
% 2.25/1.35 % (20888)Termination phase: Property scanning
% 2.25/1.35
% 2.25/1.35 % (20888)Memory used [KB]: 1918
% 2.25/1.35 % (20888)Time elapsed: 0.012 s
% 2.25/1.35 % (20888)Instructions burned: 40 (million)
% 2.25/1.35 % (20888)------------------------------
% 2.25/1.35 % (20888)------------------------------
% 2.25/1.35 % (20893)Instruction limit reached!
% 2.25/1.35 % (20893)------------------------------
% 2.25/1.35 % (20893)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.35 % (20893)Termination reason: Unknown
% 2.25/1.35 % (20893)Termination phase: shuffling
% 2.25/1.35
% 2.25/1.35 % (20893)Memory used [KB]: 1407
% 2.25/1.35 % (20893)Time elapsed: 0.004 s
% 2.25/1.35 % (20893)Instructions burned: 9 (million)
% 2.25/1.35 % (20893)------------------------------
% 2.25/1.35 % (20893)------------------------------
% 2.25/1.36 % (20894)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_20708 for (2999ds/16Mi)
% 2.25/1.36 % (20895)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_20708 for (2999ds/3Mi)
% 2.25/1.36 % (20895)Instruction limit reached!
% 2.25/1.36 % (20895)------------------------------
% 2.25/1.36 % (20895)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.36 % (20895)Termination reason: Unknown
% 2.25/1.36 % (20895)Termination phase: shuffling
% 2.25/1.36
% 2.25/1.36 % (20895)Memory used [KB]: 1279
% 2.25/1.36 % (20895)Time elapsed: 0.002 s
% 2.25/1.36 % (20895)Instructions burned: 3 (million)
% 2.25/1.36 % (20895)------------------------------
% 2.25/1.36 % (20895)------------------------------
% 2.25/1.36 % (20894)Instruction limit reached!
% 2.25/1.36 % (20894)------------------------------
% 2.25/1.36 % (20894)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.36 % (20894)Termination reason: Unknown
% 2.25/1.36 % (20894)Termination phase: shuffling
% 2.25/1.36
% 2.25/1.36 % (20894)Memory used [KB]: 1791
% 2.25/1.36 % (20894)Time elapsed: 0.006 s
% 2.25/1.36 % (20894)Instructions burned: 18 (million)
% 2.25/1.36 % (20894)------------------------------
% 2.25/1.36 % (20894)------------------------------
% 2.25/1.36 % (20898)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_20708 for (2999ds/3Mi)
% 2.25/1.36 % (20898)Instruction limit reached!
% 2.25/1.36 % (20898)------------------------------
% 2.25/1.36 % (20898)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.36 % (20898)Termination reason: Unknown
% 2.25/1.36 % (20898)Termination phase: shuffling
% 2.25/1.36
% 2.25/1.36 % (20898)Memory used [KB]: 1279
% 2.25/1.36 % (20898)Time elapsed: 0.002 s
% 2.25/1.36 % (20898)Instructions burned: 3 (million)
% 2.25/1.36 % (20898)------------------------------
% 2.25/1.36 % (20898)------------------------------
% 2.25/1.36 % (20902)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_20708 for (2999ds/7Mi)
% 2.25/1.37 % (20902)Instruction limit reached!
% 2.25/1.37 % (20902)------------------------------
% 2.25/1.37 % (20902)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.37 % (20902)Termination reason: Unknown
% 2.25/1.37 % (20902)Termination phase: shuffling
% 2.25/1.37
% 2.25/1.37 % (20902)Memory used [KB]: 1407
% 2.25/1.37 % (20902)Time elapsed: 0.003 s
% 2.25/1.37 % (20902)Instructions burned: 7 (million)
% 2.25/1.37 % (20902)------------------------------
% 2.25/1.37 % (20902)------------------------------
% 2.25/1.37 % (20908)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_20708 for (2999ds/3Mi)
% 2.25/1.37 % (20908)Instruction limit reached!
% 2.25/1.37 % (20908)------------------------------
% 2.25/1.37 % (20908)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.37 % (20908)Termination reason: Unknown
% 2.25/1.37 % (20908)Termination phase: shuffling
% 2.25/1.37
% 2.25/1.37 % (20908)Memory used [KB]: 1279
% 2.25/1.37 % (20908)Time elapsed: 0.002 s
% 2.25/1.37 % (20908)Instructions burned: 3 (million)
% 2.25/1.37 % (20908)------------------------------
% 2.25/1.37 % (20908)------------------------------
% 2.25/1.37 % (20910)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_20708 for (2999ds/4Mi)
% 2.25/1.37 % (20910)Instruction limit reached!
% 2.25/1.37 % (20910)------------------------------
% 2.25/1.37 % (20910)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.37 % (20910)Termination reason: Unknown
% 2.25/1.37 % (20910)Termination phase: shuffling
% 2.25/1.37
% 2.25/1.37 % (20910)Memory used [KB]: 1407
% 2.25/1.37 % (20910)Time elapsed: 0.003 s
% 2.25/1.37 % (20910)Instructions burned: 7 (million)
% 2.25/1.37 % (20910)------------------------------
% 2.25/1.37 % (20910)------------------------------
% 2.25/1.37 % (20911)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_20708 for (2999ds/18Mi)
% 2.25/1.38 % (20913)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_20708 for (2999ds/710Mi)
% 2.25/1.38 % (20878)First to succeed.
% 2.25/1.38 % (20916)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_20708 for (2999ds/6Mi)
% 2.25/1.38 % (20911)Instruction limit reached!
% 2.25/1.38 % (20911)------------------------------
% 2.25/1.38 % (20911)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.38 % (20911)Termination reason: Unknown
% 2.25/1.38 % (20911)Termination phase: shuffling
% 2.25/1.38
% 2.25/1.38 % (20911)Memory used [KB]: 1791
% 2.25/1.38 % (20911)Time elapsed: 0.007 s
% 2.25/1.38 % (20911)Instructions burned: 21 (million)
% 2.25/1.38 % (20911)------------------------------
% 2.25/1.38 % (20911)------------------------------
% 2.25/1.38 % (20916)Instruction limit reached!
% 2.25/1.38 % (20916)------------------------------
% 2.25/1.38 % (20916)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.38 % (20916)Termination reason: Unknown
% 2.25/1.38 % (20916)Termination phase: shuffling
% 2.25/1.38
% 2.25/1.38 % (20916)Memory used [KB]: 1407
% 2.25/1.38 % (20916)Time elapsed: 0.003 s
% 2.25/1.38 % (20916)Instructions burned: 7 (million)
% 2.25/1.38 % (20916)------------------------------
% 2.25/1.38 % (20916)------------------------------
% 2.25/1.38 % (20878)Refutation found. Thanks to Tanya!
% 2.25/1.38 % SZS status Theorem for DTF2THF_20708
% 2.25/1.38 % SZS output start Proof for DTF2THF_20708
% 2.25/1.38 thf(func_def_0, type, in: $i > $i > $o).
% 2.25/1.38 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.25/1.38 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.25/1.38 thf(func_def_8, type, powerset: $i > $i).
% 2.25/1.38 thf(func_def_10, type, setunion: $i > $i).
% 2.25/1.38 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.25/1.38 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.25/1.38 thf(func_def_26, type, prop2set: $o > $i).
% 2.25/1.38 thf(func_def_36, type, nonempty: $i > $o).
% 2.25/1.38 thf(func_def_69, type, set2prop: $i > $o).
% 2.25/1.38 thf(func_def_88, type, subset: $i > $i > $o).
% 2.25/1.38 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.25/1.38 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.25/1.38 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.25/1.38 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.25/1.38 thf(func_def_135, type, regular: $i > $o).
% 2.25/1.38 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.25/1.38 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.25/1.38 thf(func_def_153, type, iskpair: $i > $o).
% 2.25/1.38 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.25/1.38 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.25/1.38 thf(func_def_177, type, singleton: $i > $o).
% 2.25/1.38 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.25/1.38 thf(func_def_184, type, atmost1p: $i > $o).
% 2.25/1.38 thf(func_def_185, type, atleast2p: $i > $o).
% 2.25/1.38 thf(func_def_186, type, atmost2p: $i > $o).
% 2.25/1.38 thf(func_def_187, type, upairsetp: $i > $o).
% 2.25/1.38 thf(func_def_191, type, kfst: $i > $i).
% 2.25/1.38 thf(func_def_203, type, ksnd: $i > $i).
% 2.25/1.38 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.25/1.38 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.25/1.38 thf(func_def_234, type, sP3: $o > $i > $i > $i > $o).
% 2.25/1.38 thf(func_def_235, type, sP4: $i > $i > $o).
% 2.25/1.38 thf(func_def_236, type, sP5: $i > $o).
% 2.25/1.38 thf(func_def_237, type, sP6: $i > $i > $o).
% 2.25/1.38 thf(func_def_238, type, sP7: $i > $i > $o).
% 2.25/1.38 thf(func_def_242, type, sK11: $i > $i > $i).
% 2.25/1.38 thf(func_def_243, type, sK12: $i > $i > $i).
% 2.25/1.38 thf(func_def_248, type, sK17: $i > $o).
% 2.25/1.38 thf(func_def_251, type, sK20: $i > ($i > $o) > $i).
% 2.25/1.38 thf(func_def_252, type, sK21: $i > $o).
% 2.25/1.38 thf(func_def_259, type, sK28: $i > $i > $i).
% 2.25/1.38 thf(func_def_263, type, sK32: $i > $o).
% 2.25/1.38 thf(func_def_265, type, sK34: ($i > $o) > $i > $i).
% 2.25/1.38 thf(func_def_266, type, sK35: ($i > $o) > $i > $i).
% 2.25/1.38 thf(func_def_267, type, sK36: $i > $i > $o).
% 2.25/1.38 thf(func_def_278, type, sK47: $i > $o).
% 2.25/1.38 thf(func_def_279, type, sK48: ($i > $o) > $i).
% 2.25/1.38 thf(func_def_289, type, sK58: $i > $o).
% 2.25/1.38 thf(func_def_292, type, sK61: $i > ($i > $o) > $i).
% 2.25/1.38 thf(func_def_293, type, sK62: $i > ($i > $o) > $i).
% 2.25/1.38 thf(func_def_294, type, sK63: $i > $o).
% 2.25/1.38 thf(func_def_296, type, sK65: ($i > $o) > $i).
% 2.25/1.38 thf(func_def_297, type, sK66: $i > $o).
% 2.25/1.38 thf(func_def_298, type, sK67: $i > $i).
% 2.25/1.38 thf(func_def_302, type, sK71: $i > $o).
% 2.25/1.38 thf(func_def_304, type, sK73: ($i > $o) > $i).
% 2.25/1.38 thf(func_def_305, type, sK74: ($i > $o) > $i).
% 2.25/1.38 thf(func_def_309, type, sK78: $i > $o).
% 2.25/1.38 thf(func_def_312, type, sK81: $i > ($i > $o) > $i).
% 2.25/1.38 thf(func_def_313, type, sK82: $i > ($i > $o) > $i).
% 2.25/1.38 thf(func_def_322, type, sK91: $i > $i > $i).
% 2.25/1.38 thf(func_def_326, type, sK95: $i > ($i > $o) > $i > $i).
% 2.25/1.38 thf(func_def_328, type, sK97: $i > $o).
% 2.25/1.38 thf(func_def_334, type, sK103: $i > $o).
% 2.25/1.38 thf(func_def_345, type, sK114: ($i > $o) > ($i > $o) > $i).
% 2.25/1.38 thf(func_def_346, type, sK115: ($i > $o) > ($i > $o) > $i).
% 2.25/1.38 thf(func_def_347, type, sK116: $i > $o).
% 2.25/1.38 thf(func_def_348, type, sK117: $i > $o).
% 2.25/1.38 thf(func_def_359, type, sK128: $i > $o).
% 2.25/1.38 thf(func_def_366, type, sK135: $i > $i).
% 2.25/1.38 thf(func_def_367, type, sK136: $i > $i).
% 2.25/1.38 thf(func_def_389, type, sK158: $i > $i > $o).
% 2.25/1.38 thf(func_def_390, type, sK159: $i > $i).
% 2.25/1.38 thf(func_def_391, type, sK160: $i > $i).
% 2.25/1.38 thf(func_def_392, type, sK161: ($i > $i > $o) > $i > $i).
% 2.25/1.38 thf(func_def_393, type, sK162: $i > ($i > $i > $o) > $i > $i).
% 2.25/1.38 thf(func_def_394, type, sK163: ($i > $i > $o) > $i > $i).
% 2.25/1.38 thf(func_def_397, type, sK166: $i > $o).
% 2.25/1.38 thf(func_def_400, type, sK169: $i > $o).
% 2.25/1.38 thf(func_def_415, type, sK184: $i > $o).
% 2.25/1.38 thf(func_def_419, type, sK188: $i > $i > $i).
% 2.25/1.38 thf(func_def_427, type, sK196: $i > $i > $i > $i).
% 2.25/1.38 thf(func_def_428, type, sK197: $i > $i > $i > $i).
% 2.25/1.38 thf(func_def_434, type, sK203: $i > $o).
% 2.25/1.38 thf(func_def_439, type, sK208: $i > $o).
% 2.25/1.38 thf(func_def_451, type, sK220: $i > $i > $i).
% 2.25/1.38 thf(func_def_452, type, sK221: $i > ($i > $o) > $i).
% 2.25/1.38 thf(func_def_453, type, sK222: $i > $o).
% 2.25/1.38 thf(func_def_458, type, sK227: $i > ($i > $o) > $i).
% 2.25/1.38 thf(func_def_459, type, sK228: $i > $o).
% 2.25/1.38 thf(func_def_473, type, sK242: ($i > $o) > $i > $i).
% 2.25/1.38 thf(func_def_475, type, sK244: $i > $o).
% 2.25/1.38 thf(func_def_479, type, sK248: $i > $o).
% 2.25/1.38 thf(func_def_483, type, sK252: $i > $i).
% 2.25/1.38 thf(func_def_485, type, sK254: $i > $i).
% 2.25/1.38 thf(func_def_498, type, sK267: $i > $i > $i).
% 2.25/1.38 thf(func_def_500, type, sK269: $i > $i).
% 2.25/1.38 thf(func_def_519, type, sK288: ($i > $o) > $i).
% 2.25/1.38 thf(func_def_520, type, sK289: $i > $o).
% 2.25/1.38 thf(func_def_521, type, sK290: $i > $i).
% 2.25/1.38 thf(func_def_532, type, sK301: $i > $o).
% 2.25/1.38 thf(func_def_538, type, sK307: $i > ($i > $o) > $i).
% 2.25/1.38 thf(func_def_539, type, sK308: $i > $o).
% 2.25/1.38 thf(func_def_542, type, sK311: ($i > $o) > ($i > $o) > $i).
% 2.25/1.38 thf(func_def_543, type, sK312: ($i > $o) > ($i > $o) > $i).
% 2.25/1.38 thf(func_def_544, type, sK313: $i > $o).
% 2.25/1.38 thf(func_def_545, type, sK314: $i > $o).
% 2.25/1.38 thf(func_def_548, type, sK317: $i > $o).
% 2.25/1.38 thf(func_def_549, type, sK318: $i > $o).
% 2.25/1.38 thf(func_def_550, type, sK319: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.25/1.38 thf(func_def_551, type, sK320: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.25/1.38 thf(func_def_555, type, sK324: $i > $i > $o).
% 2.25/1.38 thf(func_def_563, type, sK332: $i > $i > $i).
% 2.25/1.38 thf(func_def_571, type, sK340: $i > $i > $i).
% 2.25/1.38 thf(func_def_610, type, sK379: $i > $i > $i).
% 2.25/1.38 thf(func_def_611, type, sK380: $i > $i > $i).
% 2.25/1.38 thf(func_def_612, type, sK381: $i > $i > $i).
% 2.25/1.38 thf(func_def_613, type, sK382: $i > $i > $i).
% 2.25/1.38 thf(func_def_614, type, sK383: $i > $i > $i).
% 2.25/1.38 thf(func_def_615, type, sK384: $i > $i > $i > $i).
% 2.25/1.38 thf(func_def_616, type, sK385: $i > $i).
% 2.25/1.38 thf(func_def_617, type, sK386: $i > $i).
% 2.25/1.38 thf(func_def_618, type, sK387: $i > $i).
% 2.25/1.38 thf(func_def_619, type, sK388: $i > $i).
% 2.25/1.38 thf(func_def_620, type, sK389: $i > $i > $i).
% 2.25/1.38 thf(func_def_621, type, sK390: $i > $i > $i > $i).
% 2.25/1.38 thf(func_def_622, type, sK391: $i > $i > $i > $i).
% 2.25/1.38 thf(func_def_623, type, sK392: $i > $i > $i).
% 2.25/1.38 thf(func_def_624, type, sK393: $i > $i > $i).
% 2.25/1.38 thf(func_def_625, type, sK394: $i > $i).
% 2.25/1.38 thf(func_def_627, type, sK396: $i > $i).
% 2.25/1.38 thf(func_def_628, type, sK397: $i > $i).
% 2.25/1.38 thf(func_def_630, type, sK399: $i > $i > $o).
% 2.25/1.38 thf(func_def_633, type, sK402: $i > $i).
% 2.25/1.38 thf(func_def_645, type, sK414: $i > $o).
% 2.25/1.38 thf(func_def_646, type, sK415: $i > ($i > $o) > $i).
% 2.25/1.38 thf(func_def_647, type, sK416: $i > $o).
% 2.25/1.38 thf(func_def_683, type, sK452: $i > $i > $i).
% 2.25/1.38 thf(func_def_698, type, sK467: $i > $o).
% 2.25/1.38 thf(func_def_706, type, sK475: $i > $i).
% 2.25/1.38 thf(func_def_718, type, sK487: $i > $o).
% 2.25/1.38 thf(func_def_720, type, sK489: ($i > $o) > $i > $i).
% 2.25/1.38 thf(func_def_727, type, sK496: $o > $i > $i > $i).
% 2.25/1.38 thf(func_def_751, type, ph520: !>[X0: $tType]:(X0)).
% 2.25/1.38 thf(f3481,plain,(
% 2.25/1.38 $false),
% 2.25/1.38 inference(subsumption_resolution,[status(thm)],[f3355,f3420])).
% 2.25/1.38 thf(f3420,plain,(
% 2.25/1.38 ( ! [X2 : $i,X0 : $i > $i > $o,X1 : $i] : (($true = (subset @ (dpsetconstr @ X1 @ X2 @ X0) @ (cartprod @ X1 @ X2)))) )),
% 2.25/1.38 inference(beta_eta_normalization,[status(thm)],[f3419])).
% 2.25/1.38 thf(f3419,plain,(
% 2.25/1.38 ( ! [X2 : $i,X0 : $i > $i > $o,X1 : $i] : (($true = (subset @ (dpsetconstr @ X1 @ X2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1))))) @ (cartprod @ X1 @ X2)))) )),
% 2.25/1.38 inference(trivial_inequality_removal,[status(thm)],[f2917])).
% 2.25/1.38 thf(f2917,plain,(
% 2.25/1.38 ( ! [X2 : $i,X0 : $i > $i > $o,X1 : $i] : (($true = (subset @ (dpsetconstr @ X1 @ X2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1))))) @ (cartprod @ X1 @ X2))) | ($true != $true)) )),
% 2.25/1.38 inference(definition_unfolding,[status(thm)],[f2068,f2265])).
% 2.25/1.38 thf(f2265,plain,(
% 2.25/1.38 (dpsetconstrSub = $true)),
% 2.25/1.38 inference(cnf_transformation,[status(thm)],[f1434])).
% 2.25/1.38 thf(f1434,plain,(
% 2.25/1.38 (omega__Cong = $true) & (subset2powerset = $true) & (setukpairIL = $true) & (setext = $true) & (uniqinunit = $true) & (binintersectSubset5 = $true) & (setukpairinjR = $true) & (exuE1 = $true) & (emptyset__Cong = $true) & (sepSubset = $true) & (exuEu = $true) & (ksndpairEq = $true) & (setadjoinE = $true) & (prop2set2propI = $true) & (emptyInPowerset = $true) & (powersetE1 = $true) & (setukpairinjR11 = $true) & (setukpairinjR12 = $true) & (notinemptyset = $true) & (cartprodpairin = $true) & (setminusELneg = $true) & (emptyinPowerset = $true) & (setadjoinSub = $true) & (setukpairinjL1 = $true) & (cartprodpairmemEL = $true) & (powersetsubset = $true) & (setminusERneg = $true) & (symdiffIneg2 = $true) & (setminusIRneg = $true) & (in__Cong = $true) & (dsetconstrER = $true) & (sepInPowerset = $true) & (setbeta = $true) & (setoftrueEq = $true) & (singletonsswitch = $true) & (notinsingleton = $true) & (descr__Cong = $true) & (powersetE = $true) & (exuI3 = $true) & (kpairiskpair = $true) & (binunionRsub = $true) & (binunionE = $true) & (emptyE1 = $true) & (binunionLsub = $true) & (subsetI1 = $true) & (setukpairinjL2 = $true) & (singletonsubset = $true) & (bs114d = $true) & (kpairsurjEq = $true) & (eqinunit = $true) & (setukpairinjR2 = $true) & (exuI2 = $true) & (dsetconstrI = $true) & (upairset2IR = $true) & (singletoninpowunion = $true) & (setextsub = $true) & (upairset2E = $true) & (upairsetE = $true) & (upairinpowunion = $true) & (subPowSU = $true) & (setadjoin__Cong = $true) & (subsetI2 = $true) & (ex1E1 = $true) & (wellorderingAx = $true) & (setunionsingleton = $true) & (emptysetimpfalse = $true) & (ex1I2 = $true) & (setunionsingleton1 = $true) & (singletoninpowerset = $true) & (setunionAx = $true) & (omega0Ax = $true) & (quantDeMorgan3 = $true) & (notsubsetI = $true) & (binintersectLsub = $true) & (setukpairIR = $true) & (symdiffIneg1 = $true) & (setminusEL = $true) & (nonemptyI1 = $true) & (upairsetIR = $true) & (disjointsetsI1 = $true) & (quantDeMorgan1 = $true) & (setadjoinIR = $true) & (setunionE = $true) & (kfstsingleton = $true) & (eqimpsubset1 = $true) & (nonemptyE1 = $true) & (binintersectI = $true) & (subsetE = $true) & (cartprodsndpairEq = $true) & (quantDeMorgan2 = $true) & (notdallE = $true) & (cartprodpairsurjEq = $true) & (setminusILneg = $true) & (setminusSubset2 = $true) & (kfstpairEq = $true) & (noeltsimpempty = $true) & (binintersectSubset4 = $true) & (dpsetconstrSub = $true) & (exuE3e = $true) & (binunionEcases = $true) & (cartprodfstin = $true) & (secondinupair = $true) & (ubforcartprodlem2 = $true) & (binintersectER = $true) & (singletonsuniq = $true) & (setunionI = $true) & (vacuousDall = $true) & (cartprodsndin = $true) & (setminusLsub = $true) & (upairsetIL = $true) & (setadjoinIL = $true) & (cartprodmempair1 = $true) & (binintersectEL = $true) & (prop2setI = $true) & (replAx = $true) & (ksndsingleton = $true) & (cartprodfstpairEq = $true) & (setunionE2 = $true) & (symdiffI1 = $true) & (cartprodmempair = $true) & (subsetemptysetimpeq = $true) & (inCongP = $true) & (emptysetAx = $true) & (ex1I = $true) & (setextAx = $true) & (quantDeMorgan4 = $true) & (emptyI = $true) & (symdiffI2 = $true) & (dsetconstrEL = $true) & (upairsubunion = $true) & (subsetRefl = $true) & (prop2setE = $true) & (ubforcartprodlem3 = $true) & (setadjoinAx = $true) & (binintersectSubset2 = $true) & (notdexE = $true) & (setminusI = $true) & (exuI1 = $true) & (powersetAx = $true) & (dpsetconstrI = $true) & (symdiffE = $true) & (nonemptyImpWitness = $true) & (foundationAx = $true) & (setminusSubset1 = $true) & (emptysetE = $true) & (setadjoinOr = $true) & (binintersectRsub = $true) & (notequalI1 = $true) & (inPowerset = $true) & (setunionsingleton2 = $true) & (setunion__Cong = $true) & (exu__Cong = $true) & (notequalI2 = $true) & (nonemptyI = $true) & (ubforcartprodlem1 = $true) & (emptyinunitempty = $true) & (subsetTrans = $true) & (emptysetsubset = $true) & (setukpairinjR1 = $true) & (theprop = $true) & (binintersectSubset3 = $true) & (omegaSAx = $true) & (exuE2 = $true) & (subsetE2 = $true) & (kpairp = $true) & (setukpairinjL = $true) & (descrp = $true) & (setminusER = $true) & (powersetI1 = $true) & (singletonprop = $true) & (dsetconstr__Cong = $true) & (binintersectSubset1 = $true) & (powersetI = $true) & (setadjoinSub2 = $true) & (eqimpsubset2 = $true) & (omegaIndAx = $true) & (binunionIR = $true) & (powerset__Cong = $true) & (cartprodpairmemER = $true) & (upairequniteq = $true) & ($true != (breln @ sK400 @ sK398 @ (dpsetconstr @ sK400 @ sK398 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK399 @ Y0 @ Y1))))))) & (cartprodmempaircEq = $true) & (binunionIL = $true) & (exuE3u = $true)),
% 2.25/1.38 inference(skolemisation,[status(esa),new_symbols(skolem,[sK398,sK399,sK400])],[f773,f1433])).
% 2.25/1.38 thf(f1433,plain,(
% 2.25/1.38 ? [X0 : $i,X1 : $i > $i > $o,X2 : $i] : ($true != (breln @ X2 @ X0 @ (dpsetconstr @ X2 @ X0 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X1 @ Y0 @ Y1))))))) => ($true != (breln @ sK400 @ sK398 @ (dpsetconstr @ sK400 @ sK398 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK399 @ Y0 @ Y1)))))))),
% 2.25/1.38 introduced(definition,[],[choice_axiom])).
% 2.25/1.38 thf(f773,plain,(
% 2.25/1.38 (omega__Cong = $true) & (subset2powerset = $true) & (setukpairIL = $true) & (setext = $true) & (uniqinunit = $true) & (binintersectSubset5 = $true) & (setukpairinjR = $true) & (exuE1 = $true) & (emptyset__Cong = $true) & (sepSubset = $true) & (exuEu = $true) & (ksndpairEq = $true) & (setadjoinE = $true) & (prop2set2propI = $true) & (emptyInPowerset = $true) & (powersetE1 = $true) & (setukpairinjR11 = $true) & (setukpairinjR12 = $true) & (notinemptyset = $true) & (cartprodpairin = $true) & (setminusELneg = $true) & (emptyinPowerset = $true) & (setadjoinSub = $true) & (setukpairinjL1 = $true) & (cartprodpairmemEL = $true) & (powersetsubset = $true) & (setminusERneg = $true) & (symdiffIneg2 = $true) & (setminusIRneg = $true) & (in__Cong = $true) & (dsetconstrER = $true) & (sepInPowerset = $true) & (setbeta = $true) & (setoftrueEq = $true) & (singletonsswitch = $true) & (notinsingleton = $true) & (descr__Cong = $true) & (powersetE = $true) & (exuI3 = $true) & (kpairiskpair = $true) & (binunionRsub = $true) & (binunionE = $true) & (emptyE1 = $true) & (binunionLsub = $true) & (subsetI1 = $true) & (setukpairinjL2 = $true) & (singletonsubset = $true) & (bs114d = $true) & (kpairsurjEq = $true) & (eqinunit = $true) & (setukpairinjR2 = $true) & (exuI2 = $true) & (dsetconstrI = $true) & (upairset2IR = $true) & (singletoninpowunion = $true) & (setextsub = $true) & (upairset2E = $true) & (upairsetE = $true) & (upairinpowunion = $true) & (subPowSU = $true) & (setadjoin__Cong = $true) & (subsetI2 = $true) & (ex1E1 = $true) & (wellorderingAx = $true) & (setunionsingleton = $true) & (emptysetimpfalse = $true) & (ex1I2 = $true) & (setunionsingleton1 = $true) & (singletoninpowerset = $true) & (setunionAx = $true) & (omega0Ax = $true) & (quantDeMorgan3 = $true) & (notsubsetI = $true) & (binintersectLsub = $true) & (setukpairIR = $true) & (symdiffIneg1 = $true) & (setminusEL = $true) & (nonemptyI1 = $true) & (upairsetIR = $true) & (disjointsetsI1 = $true) & (quantDeMorgan1 = $true) & (setadjoinIR = $true) & (setunionE = $true) & (kfstsingleton = $true) & (eqimpsubset1 = $true) & (nonemptyE1 = $true) & (binintersectI = $true) & (subsetE = $true) & (cartprodsndpairEq = $true) & (quantDeMorgan2 = $true) & (notdallE = $true) & (cartprodpairsurjEq = $true) & (setminusILneg = $true) & (setminusSubset2 = $true) & (kfstpairEq = $true) & (noeltsimpempty = $true) & (binintersectSubset4 = $true) & (dpsetconstrSub = $true) & (exuE3e = $true) & (binunionEcases = $true) & (cartprodfstin = $true) & (secondinupair = $true) & (ubforcartprodlem2 = $true) & (binintersectER = $true) & (singletonsuniq = $true) & (setunionI = $true) & (vacuousDall = $true) & (cartprodsndin = $true) & (setminusLsub = $true) & (upairsetIL = $true) & (setadjoinIL = $true) & (cartprodmempair1 = $true) & (binintersectEL = $true) & (prop2setI = $true) & (replAx = $true) & (ksndsingleton = $true) & (cartprodfstpairEq = $true) & (setunionE2 = $true) & (symdiffI1 = $true) & (cartprodmempair = $true) & (subsetemptysetimpeq = $true) & (inCongP = $true) & (emptysetAx = $true) & (ex1I = $true) & (setextAx = $true) & (quantDeMorgan4 = $true) & (emptyI = $true) & (symdiffI2 = $true) & (dsetconstrEL = $true) & (upairsubunion = $true) & (subsetRefl = $true) & (prop2setE = $true) & (ubforcartprodlem3 = $true) & (setadjoinAx = $true) & (binintersectSubset2 = $true) & (notdexE = $true) & (setminusI = $true) & (exuI1 = $true) & (powersetAx = $true) & (dpsetconstrI = $true) & (symdiffE = $true) & (nonemptyImpWitness = $true) & (foundationAx = $true) & (setminusSubset1 = $true) & (emptysetE = $true) & (setadjoinOr = $true) & (binintersectRsub = $true) & (notequalI1 = $true) & (inPowerset = $true) & (setunionsingleton2 = $true) & (setunion__Cong = $true) & (exu__Cong = $true) & (notequalI2 = $true) & (nonemptyI = $true) & (ubforcartprodlem1 = $true) & (emptyinunitempty = $true) & (subsetTrans = $true) & (emptysetsubset = $true) & (setukpairinjR1 = $true) & (theprop = $true) & (binintersectSubset3 = $true) & (omegaSAx = $true) & (exuE2 = $true) & (subsetE2 = $true) & (kpairp = $true) & (setukpairinjL = $true) & (descrp = $true) & (setminusER = $true) & (powersetI1 = $true) & (singletonprop = $true) & (dsetconstr__Cong = $true) & (binintersectSubset1 = $true) & (powersetI = $true) & (setadjoinSub2 = $true) & (eqimpsubset2 = $true) & (omegaIndAx = $true) & (binunionIR = $true) & (powerset__Cong = $true) & (cartprodpairmemER = $true) & (upairequniteq = $true) & ? [X0 : $i,X1 : $i > $i > $o,X2 : $i] : ($true != (breln @ X2 @ X0 @ (dpsetconstr @ X2 @ X0 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X1 @ Y0 @ Y1))))))) & (cartprodmempaircEq = $true) & (binunionIL = $true) & (exuE3u = $true)),
% 2.25/1.38 inference(flattening,[status(thm)],[f772])).
% 2.25/1.38 thf(f772,plain,(
% 2.25/1.38 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i > $i > $o,X2 : $i] : ($true != (breln @ X2 @ X0 @ (dpsetconstr @ X2 @ X0 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X1 @ Y0 @ Y1))))))) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.25/1.38 inference(ennf_transformation,[status(thm)],[f534])).
% 2.25/1.38 thf(f534,plain,(
% 2.25/1.38 ~((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) => ! [X0 : $i,X1 : $i > $i > $o,X2 : $i] : ($true = (breln @ X2 @ X0 @ (dpsetconstr @ X2 @ X0 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X1 @ Y0 @ Y1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.25/1.38 inference(fool_elimination,[status(thm)],[f533])).
% 2.25/1.38 thf(f533,plain,(
% 2.25/1.38 ~(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 => ! [X0 : $i,X1 : $i > $i > $o,X2 : $i] : (breln @ X2 @ X0 @ (dpsetconstr @ X2 @ X0 @ (^[X3 : $i, X4 : $i] : (X1 @ X3 @ X4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.25/1.38 inference(rectify,[status(thm)],[f191])).
% 2.25/1.38 thf(f191,negated_conjecture,(
% 2.25/1.38 ~(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 => ! [X4 : $i,X0 : $i > $i > $o,X3 : $i] : (breln @ X3 @ X4 @ (dpsetconstr @ X3 @ X4 @ (^[X1 : $i, X2 : $i] : (X0 @ X1 @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.25/1.38 inference(negated_conjecture,[status(cth)],[f190])).
% 2.25/1.38 thf(f190,conjecture,(
% 2.25/1.38 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 => ! [X4 : $i,X0 : $i > $i > $o,X3 : $i] : (breln @ X3 @ X4 @ (dpsetconstr @ X3 @ X4 @ (^[X1 : $i, X2 : $i] : (X0 @ X1 @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.25/1.38 file('/export/starexec/sandbox2/tmp/tmp.wdvij0GjC6/DTF2THF_20708.p',setOfPairsIsBReln)).
% 2.25/1.38 thf(f2068,plain,(
% 2.25/1.38 ( ! [X2 : $i,X0 : $i > $i > $o,X1 : $i] : (($true = (subset @ (dpsetconstr @ X1 @ X2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1))))) @ (cartprod @ X1 @ X2))) | (dpsetconstrSub != $true)) )),
% 2.25/1.38 inference(cnf_transformation,[status(thm)],[f1304])).
% 2.25/1.38 thf(f1304,plain,(
% 2.25/1.38 (! [X0 : $i > $i > $o,X1 : $i,X2 : $i] : ($true = (subset @ (dpsetconstr @ X1 @ X2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1))))) @ (cartprod @ X1 @ X2))) | (dpsetconstrSub != $true)) & ((dpsetconstrSub = $true) | ($true != (subset @ (dpsetconstr @ sK325 @ sK326 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK324 @ Y0 @ Y1))))) @ (cartprod @ sK325 @ sK326))))),
% 2.25/1.38 inference(skolemisation,[status(esa),new_symbols(skolem,[sK324,sK325,sK326])],[f1302,f1303])).
% 2.25/1.38 thf(f1303,plain,(
% 2.25/1.38 ? [X3 : $i > $i > $o,X4 : $i,X5 : $i] : ((subset @ (dpsetconstr @ X4 @ X5 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X3 @ Y0 @ Y1))))) @ (cartprod @ X4 @ X5)) != $true) => ($true != (subset @ (dpsetconstr @ sK325 @ sK326 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK324 @ Y0 @ Y1))))) @ (cartprod @ sK325 @ sK326)))),
% 2.25/1.38 introduced(definition,[],[choice_axiom])).
% 2.25/1.38 thf(f1302,plain,(
% 2.25/1.38 (! [X0 : $i > $i > $o,X1 : $i,X2 : $i] : ($true = (subset @ (dpsetconstr @ X1 @ X2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1))))) @ (cartprod @ X1 @ X2))) | (dpsetconstrSub != $true)) & ((dpsetconstrSub = $true) | ? [X3 : $i > $i > $o,X4 : $i,X5 : $i] : ((subset @ (dpsetconstr @ X4 @ X5 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X3 @ Y0 @ Y1))))) @ (cartprod @ X4 @ X5)) != $true))),
% 2.25/1.38 inference(rectify,[status(thm)],[f1301])).
% 2.25/1.38 thf(f1301,plain,(
% 2.25/1.38 (! [X0 : $i > $i > $o,X1 : $i,X2 : $i] : ($true = (subset @ (dpsetconstr @ X1 @ X2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1))))) @ (cartprod @ X1 @ X2))) | (dpsetconstrSub != $true)) & ((dpsetconstrSub = $true) | ? [X0 : $i > $i > $o,X1 : $i,X2 : $i] : ($true != (subset @ (dpsetconstr @ X1 @ X2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1))))) @ (cartprod @ X1 @ X2))))),
% 2.25/1.38 inference(nnf_transformation,[status(thm)],[f248])).
% 2.25/1.38 thf(f248,plain,(
% 2.25/1.38 ! [X0 : $i > $i > $o,X1 : $i,X2 : $i] : ($true = (subset @ (dpsetconstr @ X1 @ X2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1))))) @ (cartprod @ X1 @ X2))) <=> (dpsetconstrSub = $true)),
% 2.25/1.38 inference(fool_elimination,[status(thm)],[f247])).
% 2.25/1.38 thf(f247,plain,(
% 2.25/1.38 (! [X0 : $i > $i > $o,X1 : $i,X2 : $i] : (subset @ (dpsetconstr @ X1 @ X2 @ (^[X3 : $i, X4 : $i] : (X0 @ X3 @ X4))) @ (cartprod @ X1 @ X2)) = dpsetconstrSub)),
% 2.25/1.38 inference(rectify,[status(thm)],[f189])).
% 2.25/1.38 thf(f189,axiom,(
% 2.25/1.38 (! [X0 : $i > $i > $o,X3 : $i,X4 : $i] : (subset @ (dpsetconstr @ X3 @ X4 @ (^[X1 : $i, X2 : $i] : (X0 @ X1 @ X2))) @ (cartprod @ X3 @ X4)) = dpsetconstrSub)),
% 2.25/1.38 file('/export/starexec/sandbox2/tmp/tmp.wdvij0GjC6/DTF2THF_20708.p',dpsetconstrSub)).
% 2.25/1.38 thf(f3355,plain,(
% 2.25/1.38 ($true != (subset @ (dpsetconstr @ sK400 @ sK398 @ sK399) @ (cartprod @ sK400 @ sK398)))),
% 2.25/1.38 inference(beta_eta_normalization,[status(thm)],[f3001])).
% 2.25/1.38 thf(f3001,plain,(
% 2.25/1.38 ($true != ((^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i]: (subset @ Y2 @ (cartprod @ Y0 @ Y1))))))) @ sK400 @ sK398 @ (dpsetconstr @ sK400 @ sK398 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK399 @ Y0 @ Y1)))))))),
% 2.25/1.38 inference(definition_unfolding,[status(thm)],[f2182,f2061])).
% 2.25/1.38 thf(f2061,plain,(
% 2.25/1.38 (breln = (^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i]: (subset @ Y2 @ (cartprod @ Y0 @ Y1))))))))),
% 2.25/1.38 inference(cnf_transformation,[status(thm)],[f372])).
% 2.25/1.38 thf(f372,plain,(
% 2.25/1.38 (breln = (^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i]: (subset @ Y2 @ (cartprod @ Y0 @ Y1))))))))),
% 2.25/1.38 inference(fool_elimination,[status(thm)],[f371])).
% 2.25/1.38 thf(f371,plain,(
% 2.25/1.38 (breln = (^[X0 : $i, X1 : $i, X2 : $i] : (subset @ X2 @ (cartprod @ X0 @ X1))))),
% 2.25/1.38 inference(rectify,[status(thm)],[f186])).
% 2.25/1.38 thf(f186,axiom,(
% 2.25/1.38 (breln = (^[X3 : $i, X4 : $i, X5 : $i] : (subset @ X5 @ (cartprod @ X3 @ X4))))),
% 2.25/1.38 file('/export/starexec/sandbox2/tmp/tmp.wdvij0GjC6/DTF2THF_20708.p',breln)).
% 2.25/1.38 thf(f2182,plain,(
% 2.25/1.38 ($true != (breln @ sK400 @ sK398 @ (dpsetconstr @ sK400 @ sK398 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK399 @ Y0 @ Y1)))))))),
% 2.25/1.38 inference(cnf_transformation,[status(thm)],[f1434])).
% 2.25/1.38 % SZS output end Proof for DTF2THF_20708
% 2.25/1.38 % (20878)------------------------------
% 2.25/1.38 % (20878)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.38 % (20878)Termination reason: Refutation
% 2.25/1.38
% 2.25/1.38 % (20878)Memory used [KB]: 8187
% 2.25/1.38 % (20878)Time elapsed: 0.050 s
% 2.25/1.38 % (20878)Instructions burned: 160 (million)
% 2.25/1.38 % (20878)------------------------------
% 2.25/1.38 % (20878)------------------------------
% 2.25/1.38 % (20871)Success in time 0.055 s
%------------------------------------------------------------------------------