%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU666^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n014.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 09:36:03 AM UTC 2026
% Result : Theorem 2.55s 1.64s
% Output : Refutation 2.55s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 4
% Syntax : Number of formulae : 27 ( 19 unt; 0 typ; 0 def)
% Number of atoms : 2861 ( 758 equ; 0 cnn)
% Maximal formula atoms : 183 ( 105 avg)
% Number of connectives : 1580 ( 11 ~; 0 |; 564 &; 258 @)
% ( 0 <=>; 729 =>; 0 <=; 0 <~>)
% Maximal formula depth : 194 ( 50 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 41 ( 41 >; 0 *; 0 +; 0 <<)
% Number of symbols : 276 ( 271 usr; 188 con; 0-4 aty)
% ( 10 !!; 8 ??; 0 @@+; 0 @@-)
% Number of variables : 103 ( 68 ^; 22 !; 13 ?; 103 :)
% 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_233,type,
sK0: $i > $i > $o ).
thf(func_def_237,type,
sK4: ( $i > $o ) > $i > $i ).
thf(func_def_238,type,
sK5: ( $i > $o ) > $i > $i ).
thf(func_def_239,type,
sK6: ( $i > $o ) > $i ).
thf(func_def_240,type,
sK7: $i > $i ).
thf(func_def_241,type,
sK8: $i > ( $i > $o ) > $i ).
thf(func_def_242,type,
sK9: $i > ( $i > $o ) > $i ).
thf(func_def_243,type,
sK10: ( $i > $o ) > $i > $i ).
thf(func_def_244,type,
sK11: $i > $i > $i ).
thf(func_def_245,type,
sK12: $i > $i ).
thf(func_def_246,type,
sK13: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_247,type,
sK14: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_248,type,
sK15: ( $i > $o ) > $i ).
thf(func_def_249,type,
sK16: ( $i > $o ) > $i ).
thf(func_def_250,type,
sK17: $i > $i ).
thf(func_def_251,type,
sK18: ( $i > $o ) > $i > $i ).
thf(func_def_252,type,
sK19: $i > $i > $i ).
thf(func_def_253,type,
sK20: $i > $i > $i ).
thf(func_def_254,type,
sK21: $i > $i > ( $i > $o ) > $i ).
thf(func_def_255,type,
sK22: $i > ( $i > $o ) > $i ).
thf(func_def_256,type,
sK23: $i > $i > $i ).
thf(func_def_257,type,
sK24: $i > ( $i > $o ) > $i ).
thf(func_def_258,type,
sK25: $i > $i ).
thf(func_def_259,type,
sK26: $i > $i > $i ).
thf(func_def_260,type,
sK27: $i > $i > $i ).
thf(func_def_261,type,
sK28: $i > $i > $i ).
thf(func_def_262,type,
sK29: $i > $i > $i > $i ).
thf(func_def_263,type,
sK30: $i > $i ).
thf(func_def_264,type,
sK31: ( $i > $i > $o ) > $i > $i ).
thf(func_def_265,type,
sK32: ( $i > $i > $o ) > $i > $i ).
thf(func_def_266,type,
sK33: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_267,type,
sK34: ( $i > $i > $o ) > $i > $i ).
thf(func_def_268,type,
sK35: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_269,type,
sK36: $i > $i > $i ).
thf(func_def_270,type,
sK37: ( $i > $o ) > $i ).
thf(func_def_271,type,
sK38: $i > $i > $i ).
thf(func_def_272,type,
sK39: $i > ( $i > $o ) > $i ).
thf(func_def_273,type,
sK40: ( $i > $o ) > $i ).
thf(func_def_274,type,
sK41: ( $i > $o ) > $i > ( $i > $o ) > $i > $i ).
thf(func_def_275,type,
sK42: $i > $i > ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_276,type,
sK43: $i > $i > $i > $i ).
thf(func_def_277,type,
sK44: $i > $i > $i > $i ).
thf(func_def_278,type,
sK45: $i > $i > $i ).
thf(func_def_279,type,
sK46: $i > $i ).
thf(func_def_280,type,
sK47: $i > $i > $i ).
thf(func_def_281,type,
sK48: $i > $i > $i ).
thf(func_def_282,type,
sK49: $i > $i > $i ).
thf(func_def_283,type,
sK50: $i > $i > $i ).
thf(func_def_286,type,
sK51: $i > ( $i > $o ) > $i ).
thf(func_def_287,type,
sK52: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_288,type,
sK53: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_289,type,
sK54: ( $i > $o ) > $i > $i ).
thf(func_def_290,type,
sK55: $i > $o > $i > $i ).
thf(func_def_291,type,
sK56: ( $i > $o ) > $i > $i ).
thf(func_def_292,type,
sK57: ( $i > $o ) > $i > $i ).
thf(func_def_293,type,
sK58: $i > $i ).
thf(f2814,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1807,f2076]) ).
thf(f2076,plain,
! [X2: $i > $o,X1: $i] :
( $true
= ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 ) ),
inference(beta_eta_normalization,[status(thm)],[f2075]) ).
thf(f2075,plain,
! [X2: $i > $o,X1: $i] :
( ( ^ [Y0: $i > $o] : ( subset @ ( dsetconstr @ X1 @ Y0 ) @ X1 )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f2074]) ).
thf(f2074,plain,
! [X1: $i] :
( $true
= ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] : ( subset @ ( dsetconstr @ X1 @ Y0 ) @ X1 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f2073]) ).
thf(f2073,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
( !! @ ( $i > $o )
@ ^ [Y1: $i > $o] : ( subset @ ( dsetconstr @ Y0 @ Y1 ) @ Y0 ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f2072]) ).
thf(f2072,plain,
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ ( $i > $o )
@ ^ [Y1: $i > $o] : ( subset @ ( dsetconstr @ Y0 @ Y1 ) @ Y0 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1065]) ).
thf(f1065,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ ( $i > $o )
@ ^ [Y1: $i > $o] :
( subset
@ ( dsetconstr @ Y0
@ ^ [Y2: $i] : ( Y1 @ Y2 ) )
@ Y0 ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f880,f701]) ).
thf(f701,plain,
sepSubset = $true,
inference(cnf_transformation,[status(thm)],[f567]) ).
thf(f567,plain,
( ( cartprodfstin = $true )
& ( noeltsimpempty = $true )
& ( symdiffI2 = $true )
& ( kfstpairEq = $true )
& ( setukpairinjL1 = $true )
& ( inPowerset = $true )
& ( setunionE = $true )
& ( subsetRefl = $true )
& ( setextsub = $true )
& ( eqimpsubset2 = $true )
& ( singletonprop = $true )
& ( descr__Cong = $true )
& ( ksndpairEq = $true )
& ( upairsetE = $true )
& ( setukpairinjR = $true )
& ( setukpairinjL = $true )
& ( upairsetIR = $true )
& ( exuE3u = $true )
& ( descrp = $true )
& ( notequalI2 = $true )
& ( dpsetconstrI = $true )
& ( upairequniteq = $true )
& ( setminusELneg = $true )
& ( prop2set2propI = $true )
& ( binintersectSubset2 = $true )
& ( upairsetIL = $true )
& ( kpairsurjEq = $true )
& ( replAx = $true )
& ( ubforcartprodlem3 = $true )
& ( setadjoinE = $true )
& ( bs114d = $true )
& ( emptyinPowerset = $true )
& ( binunionE = $true )
& ( nonemptyImpWitness = $true )
& ( subset2powerset = $true )
& ( setminusI = $true )
& ( powersetsubset = $true )
& ( exuI1 = $true )
& ( setext = $true )
& ( singletonsubset = $true )
& ( omegaSAx = $true )
& ( cartprodpairmemER = $true )
& ( dsetconstrEL = $true )
& ( setukpairIL = $true )
& ( notinemptyset = $true )
& ( setminusLsub = $true )
& ( powersetI = $true )
& ( setbeta = $true )
& ( emptyE1 = $true )
& ( cartprodmempaircEq = $true )
& ( binintersectEL = $true )
& ( exuE1 = $true )
& ( subsetI2 = $true )
& ( quantDeMorgan3 = $true )
& ( setadjoinSub = $true )
& ( prop2setI = $true )
& ( sepSubset = $true )
& ( omegaIndAx = $true )
& ( upairset2IR = $true )
& ( subsetTrans = $true )
& ( symdiffIneg1 = $true )
& ( singletonsswitch = $true )
& ( secondinupair = $true )
& ( cartprodpairin = $true )
& ( setminusSubset2 = $true )
& ( setoftrueEq = $true )
& ( notequalI1 = $true )
& ( powersetE1 = $true )
& ( in__Cong = $true )
& ( setadjoinIR = $true )
& ( setukpairinjR2 = $true )
& ( exu__Cong = $true )
& ( emptyset__Cong = $true )
& ( kpairiskpair = $true )
& ( binintersectSubset1 = $true )
& ( notdallE = $true )
& ( emptyI = $true )
& ( quantDeMorgan4 = $true )
& ( uniqinunit = $true )
& ( kfstsingleton = $true )
& ( emptyInPowerset = $true )
& ( notdexE = $true )
& ( eqimpsubset1 = $true )
& ( prop2setE = $true )
& ( subsetE = $true )
& ( disjointsetsI1 = $true )
& ( dsetconstrI = $true )
& ( ksndsingleton = $true )
& ( setukpairIR = $true )
& ( binunionEcases = $true )
& ( setukpairinjR12 = $true )
& ( exuI2 = $true )
& ( exuE3e = $true )
& ( ubforcartprodlem1 = $true )
& ( cartprodfstpairEq = $true )
& ( $true
!= ( subset
@ ( dpsetconstr @ sK1 @ sK2
@ ^ [Y0: $i,Y1: $i] : ( sK0 @ Y0 @ Y1 ) )
@ ( cartprod @ sK1 @ sK2 ) ) )
& ( cartprodmempair1 = $true )
& ( subsetI1 = $true )
& ( eqinunit = $true )
& ( ex1E1 = $true )
& ( emptysetsubset = $true )
& ( kpairp = $true )
& ( setadjoinSub2 = $true )
& ( cartprodsndpairEq = $true )
& ( powersetAx = $true )
& ( setunionsingleton2 = $true )
& ( singletoninpowerset = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( setunionE2 = $true )
& ( dsetconstrER = $true )
& ( binunionIR = $true )
& ( quantDeMorgan2 = $true )
& ( notsubsetI = $true )
& ( binintersectRsub = $true )
& ( emptysetimpfalse = $true )
& ( setadjoinIL = $true )
& ( exuI3 = $true )
& ( singletoninpowunion = $true )
& ( setextAx = $true )
& ( vacuousDall = $true )
& ( setunionAx = $true )
& ( binintersectI = $true )
& ( setminusILneg = $true )
& ( nonemptyI1 = $true )
& ( subPowSU = $true )
& ( setadjoinAx = $true )
& ( symdiffI1 = $true )
& ( emptysetE = $true )
& ( foundationAx = $true )
& ( nonemptyE1 = $true )
& ( cartprodpairsurjEq = $true )
& ( setminusERneg = $true )
& ( sepInPowerset = $true )
& ( exuEu = $true )
& ( singletonsuniq = $true )
& ( cartprodmempair = $true )
& ( powersetE = $true )
& ( setunionsingleton = $true )
& ( ubforcartprodlem2 = $true )
& ( setminusIRneg = $true )
& ( setukpairinjL2 = $true )
& ( symdiffIneg2 = $true )
& ( wellorderingAx = $true )
& ( binintersectSubset5 = $true )
& ( symdiffE = $true )
& ( quantDeMorgan1 = $true )
& ( upairsubunion = $true )
& ( theprop = $true )
& ( binunionRsub = $true )
& ( upairset2E = $true )
& ( setunionI = $true )
& ( upairinpowunion = $true )
& ( setadjoinOr = $true )
& ( exuE2 = $true )
& ( setunion__Cong = $true )
& ( emptyinunitempty = $true )
& ( binintersectER = $true )
& ( binintersectLsub = $true )
& ( subsetE2 = $true )
& ( dsetconstr__Cong = $true )
& ( cartprodpairmemEL = $true )
& ( subsetemptysetimpeq = $true )
& ( binunionIL = $true )
& ( cartprodsndin = $true )
& ( inCongP = $true )
& ( setminusSubset1 = $true )
& ( ex1I2 = $true )
& ( setunionsingleton1 = $true )
& ( powersetI1 = $true )
& ( omega__Cong = $true )
& ( setminusEL = $true )
& ( binunionLsub = $true )
& ( binintersectSubset3 = $true )
& ( nonemptyI = $true )
& ( emptysetAx = $true )
& ( omega0Ax = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjR1 = $true )
& ( notinsingleton = $true )
& ( ex1I = $true )
& ( setminusER = $true )
& ( binintersectSubset4 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f565,f566]) ).
thf(f566,plain,
( ? [X0: $i > $i > $o,X1: $i,X2: $i] :
( $true
!= ( subset
@ ( dpsetconstr @ X1 @ X2
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) )
@ ( cartprod @ X1 @ X2 ) ) )
=> ( $true
!= ( subset
@ ( dpsetconstr @ sK1 @ sK2
@ ^ [Y0: $i,Y1: $i] : ( sK0 @ Y0 @ Y1 ) )
@ ( cartprod @ sK1 @ sK2 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f565,plain,
( ( cartprodfstin = $true )
& ( noeltsimpempty = $true )
& ( symdiffI2 = $true )
& ( kfstpairEq = $true )
& ( setukpairinjL1 = $true )
& ( inPowerset = $true )
& ( setunionE = $true )
& ( subsetRefl = $true )
& ( setextsub = $true )
& ( eqimpsubset2 = $true )
& ( singletonprop = $true )
& ( descr__Cong = $true )
& ( ksndpairEq = $true )
& ( upairsetE = $true )
& ( setukpairinjR = $true )
& ( setukpairinjL = $true )
& ( upairsetIR = $true )
& ( exuE3u = $true )
& ( descrp = $true )
& ( notequalI2 = $true )
& ( dpsetconstrI = $true )
& ( upairequniteq = $true )
& ( setminusELneg = $true )
& ( prop2set2propI = $true )
& ( binintersectSubset2 = $true )
& ( upairsetIL = $true )
& ( kpairsurjEq = $true )
& ( replAx = $true )
& ( ubforcartprodlem3 = $true )
& ( setadjoinE = $true )
& ( bs114d = $true )
& ( emptyinPowerset = $true )
& ( binunionE = $true )
& ( nonemptyImpWitness = $true )
& ( subset2powerset = $true )
& ( setminusI = $true )
& ( powersetsubset = $true )
& ( exuI1 = $true )
& ( setext = $true )
& ( singletonsubset = $true )
& ( omegaSAx = $true )
& ( cartprodpairmemER = $true )
& ( dsetconstrEL = $true )
& ( setukpairIL = $true )
& ( notinemptyset = $true )
& ( setminusLsub = $true )
& ( powersetI = $true )
& ( setbeta = $true )
& ( emptyE1 = $true )
& ( cartprodmempaircEq = $true )
& ( binintersectEL = $true )
& ( exuE1 = $true )
& ( subsetI2 = $true )
& ( quantDeMorgan3 = $true )
& ( setadjoinSub = $true )
& ( prop2setI = $true )
& ( sepSubset = $true )
& ( omegaIndAx = $true )
& ( upairset2IR = $true )
& ( subsetTrans = $true )
& ( symdiffIneg1 = $true )
& ( singletonsswitch = $true )
& ( secondinupair = $true )
& ( cartprodpairin = $true )
& ( setminusSubset2 = $true )
& ( setoftrueEq = $true )
& ( notequalI1 = $true )
& ( powersetE1 = $true )
& ( in__Cong = $true )
& ( setadjoinIR = $true )
& ( setukpairinjR2 = $true )
& ( exu__Cong = $true )
& ( emptyset__Cong = $true )
& ( kpairiskpair = $true )
& ( binintersectSubset1 = $true )
& ( notdallE = $true )
& ( emptyI = $true )
& ( quantDeMorgan4 = $true )
& ( uniqinunit = $true )
& ( kfstsingleton = $true )
& ( emptyInPowerset = $true )
& ( notdexE = $true )
& ( eqimpsubset1 = $true )
& ( prop2setE = $true )
& ( subsetE = $true )
& ( disjointsetsI1 = $true )
& ( dsetconstrI = $true )
& ( ksndsingleton = $true )
& ( setukpairIR = $true )
& ( binunionEcases = $true )
& ( setukpairinjR12 = $true )
& ( exuI2 = $true )
& ( exuE3e = $true )
& ( ubforcartprodlem1 = $true )
& ( cartprodfstpairEq = $true )
& ? [X0: $i > $i > $o,X1: $i,X2: $i] :
( $true
!= ( subset
@ ( dpsetconstr @ X1 @ X2
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) )
@ ( cartprod @ X1 @ X2 ) ) )
& ( cartprodmempair1 = $true )
& ( subsetI1 = $true )
& ( eqinunit = $true )
& ( ex1E1 = $true )
& ( emptysetsubset = $true )
& ( kpairp = $true )
& ( setadjoinSub2 = $true )
& ( cartprodsndpairEq = $true )
& ( powersetAx = $true )
& ( setunionsingleton2 = $true )
& ( singletoninpowerset = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( setunionE2 = $true )
& ( dsetconstrER = $true )
& ( binunionIR = $true )
& ( quantDeMorgan2 = $true )
& ( notsubsetI = $true )
& ( binintersectRsub = $true )
& ( emptysetimpfalse = $true )
& ( setadjoinIL = $true )
& ( exuI3 = $true )
& ( singletoninpowunion = $true )
& ( setextAx = $true )
& ( vacuousDall = $true )
& ( setunionAx = $true )
& ( binintersectI = $true )
& ( setminusILneg = $true )
& ( nonemptyI1 = $true )
& ( subPowSU = $true )
& ( setadjoinAx = $true )
& ( symdiffI1 = $true )
& ( emptysetE = $true )
& ( foundationAx = $true )
& ( nonemptyE1 = $true )
& ( cartprodpairsurjEq = $true )
& ( setminusERneg = $true )
& ( sepInPowerset = $true )
& ( exuEu = $true )
& ( singletonsuniq = $true )
& ( cartprodmempair = $true )
& ( powersetE = $true )
& ( setunionsingleton = $true )
& ( ubforcartprodlem2 = $true )
& ( setminusIRneg = $true )
& ( setukpairinjL2 = $true )
& ( symdiffIneg2 = $true )
& ( wellorderingAx = $true )
& ( binintersectSubset5 = $true )
& ( symdiffE = $true )
& ( quantDeMorgan1 = $true )
& ( upairsubunion = $true )
& ( theprop = $true )
& ( binunionRsub = $true )
& ( upairset2E = $true )
& ( setunionI = $true )
& ( upairinpowunion = $true )
& ( setadjoinOr = $true )
& ( exuE2 = $true )
& ( setunion__Cong = $true )
& ( emptyinunitempty = $true )
& ( binintersectER = $true )
& ( binintersectLsub = $true )
& ( subsetE2 = $true )
& ( dsetconstr__Cong = $true )
& ( cartprodpairmemEL = $true )
& ( subsetemptysetimpeq = $true )
& ( binunionIL = $true )
& ( cartprodsndin = $true )
& ( inCongP = $true )
& ( setminusSubset1 = $true )
& ( ex1I2 = $true )
& ( setunionsingleton1 = $true )
& ( powersetI1 = $true )
& ( omega__Cong = $true )
& ( setminusEL = $true )
& ( binunionLsub = $true )
& ( binintersectSubset3 = $true )
& ( nonemptyI = $true )
& ( emptysetAx = $true )
& ( omega0Ax = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjR1 = $true )
& ( notinsingleton = $true )
& ( ex1I = $true )
& ( setminusER = $true )
& ( binintersectSubset4 = $true ) ),
inference(flattening,[status(thm)],[f564]) ).
thf(f564,plain,
( ? [X0: $i > $i > $o,X1: $i,X2: $i] :
( $true
!= ( subset
@ ( dpsetconstr @ X1 @ X2
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) )
@ ( cartprod @ X1 @ X2 ) ) )
& ( 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)],[f388]) ).
thf(f388,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 )
=> ! [X0: $i > $i > $o,X1: $i,X2: $i] :
( $true
= ( subset
@ ( dpsetconstr @ X1 @ X2
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) )
@ ( cartprod @ X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f387]) ).
thf(f387,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
=> ! [X0: $i > $i > $o,X1: $i,X2: $i] :
( subset
@ ( dpsetconstr @ X1 @ X2
@ ^ [X3: $i,X4: $i] : ( X0 @ X3 @ X4 ) )
@ ( cartprod @ X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f190]) ).
thf(f190,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
=> ! [X0: $i > $i > $o,X3: $i,X4: $i] :
( subset
@ ( dpsetconstr @ X3 @ X4
@ ^ [X1: $i,X2: $i] : ( X0 @ X1 @ X2 ) )
@ ( cartprod @ X3 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f189]) ).
thf(f189,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
=> ! [X0: $i > $i > $o,X3: $i,X4: $i] :
( subset
@ ( dpsetconstr @ X3 @ X4
@ ^ [X1: $i,X2: $i] : ( X0 @ X1 @ X2 ) )
@ ( cartprod @ X3 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.frxx7lxlXX/DTF2THF_31262.p',dpsetconstrSub) ).
thf(f880,plain,
( sepSubset
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ ( $i > $o )
@ ^ [Y1: $i > $o] :
( subset
@ ( dsetconstr @ Y0
@ ^ [Y2: $i] : ( Y1 @ Y2 ) )
@ Y0 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f561]) ).
thf(f561,plain,
( sepSubset
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ ( $i > $o )
@ ^ [Y1: $i > $o] :
( subset
@ ( dsetconstr @ Y0
@ ^ [Y2: $i] : ( Y1 @ Y2 ) )
@ Y0 ) ) ) ),
inference(fool_elimination,[status(thm)],[f560]) ).
thf(f560,plain,
! [X0: $i > $o,X1: $i] :
( ( subset
@ ( dsetconstr @ X1
@ ^ [X2: $i] : ( X0 @ X2 ) )
@ X1 )
= sepSubset ),
inference(rectify,[status(thm)],[f100]) ).
thf(f100,axiom,
! [X0: $i > $o,X3: $i] :
( ( subset
@ ( dsetconstr @ X3
@ ^ [X1: $i] : ( X0 @ X1 ) )
@ X3 )
= sepSubset ),
file('/export/starexec/sandbox/tmp/tmp.frxx7lxlXX/DTF2THF_31262.p',sepSubset) ).
thf(f1807,plain,
( ( subset
@ ( dsetconstr @ ( cartprod @ sK1 @ sK2 )
@ ^ [Y0: $i] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ sK1 )
& ( ?? @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ sK2 )
& ( ( kpair @ Y1 @ Y2 )
= Y0 )
& ( sK0 @ Y1 @ Y2 ) ) ) ) ) )
@ ( cartprod @ sK1 @ sK2 ) )
!= $true ),
inference(beta_eta_normalization,[status(thm)],[f937]) ).
thf(f937,plain,
( $true
!= ( subset
@ ( ^ [Y0: $i,Y1: $i,Y2: $i > $i > $o] :
( dsetconstr @ ( cartprod @ Y0 @ Y1 )
@ ^ [Y3: $i] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
& ( ?? @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
& ( ( kpair @ Y4 @ Y5 )
= Y3 )
& ( Y2 @ Y4 @ Y5 ) ) ) ) ) )
@ sK1
@ sK2
@ ^ [Y0: $i,Y1: $i] : ( sK0 @ Y0 @ Y1 ) )
@ ( cartprod @ sK1 @ sK2 ) ) ),
inference(definition_unfolding,[status(thm)],[f662,f895]) ).
thf(f895,plain,
( dpsetconstr
= ( ^ [Y0: $i,Y1: $i,Y2: $i > $i > $o] :
( dsetconstr @ ( cartprod @ Y0 @ Y1 )
@ ^ [Y3: $i] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
& ( ?? @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
& ( ( kpair @ Y4 @ Y5 )
= Y3 )
& ( Y2 @ Y4 @ Y5 ) ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f558]) ).
thf(f558,plain,
( dpsetconstr
= ( ^ [Y0: $i,Y1: $i,Y2: $i > $i > $o] :
( dsetconstr @ ( cartprod @ Y0 @ Y1 )
@ ^ [Y3: $i] :
( ?? @ $i
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
& ( ?? @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
& ( ( kpair @ Y4 @ Y5 )
= Y3 )
& ( Y2 @ Y4 @ Y5 ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f557]) ).
thf(f557,plain,
( dpsetconstr
= ( ^ [X0: $i,X1: $i,X2: $i > $i > $o] :
( dsetconstr @ ( cartprod @ X0 @ X1 )
@ ^ [X3: $i] :
? [X4: $i] :
( ? [X5: $i] :
( ( X2 @ X4 @ X5 )
& ( ( kpair @ X4 @ X5 )
= X3 )
& ( in @ X5 @ X1 ) )
& ( in @ X4 @ X0 ) ) ) ) ),
inference(rectify,[status(thm)],[f187]) ).
thf(f187,axiom,
( dpsetconstr
= ( ^ [X3: $i,X4: $i,X0: $i > $i > $o] :
( dsetconstr @ ( cartprod @ X3 @ X4 )
@ ^ [X10: $i] :
? [X1: $i] :
( ? [X2: $i] :
( ( X0 @ X1 @ X2 )
& ( ( kpair @ X1 @ X2 )
= X10 )
& ( in @ X2 @ X4 ) )
& ( in @ X1 @ X3 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.frxx7lxlXX/DTF2THF_31262.p',dpsetconstr) ).
thf(f662,plain,
( $true
!= ( subset
@ ( dpsetconstr @ sK1 @ sK2
@ ^ [Y0: $i,Y1: $i] : ( sK0 @ Y0 @ Y1 ) )
@ ( cartprod @ sK1 @ sK2 ) ) ),
inference(cnf_transformation,[status(thm)],[f567]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU666^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.33 % Computer : n014.cluster.edu
% 0.17/0.33 % Model : x86_64 x86_64
% 0.17/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.33 % Memory : 8042.1875MB
% 0.17/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.33 % CPULimit : 300
% 0.17/0.33 % WCLimit : 300
% 0.17/0.33 % DateTime : Tue Feb 24 12:30:27 EST 2026
% 0.17/0.33 % CPUTime :
% 0.17/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.45 ---- Original DTF file ---
% 0.19/0.45 thf(spec,logic,$$dhol).
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 % File : SEU666^1 : TPTP v9.2.1. Released v3.7.0.
% 0.19/0.45 % Domain : Set Theory
% 0.19/0.45 % Problem : Ordered Pairs - Sets of Pairs
% 0.19/0.45 % Version : Especial.
% 0.19/0.45 % English : (! A:i.! B:i.! phi:i>(i>o).subset (dpsetconstr A B
% 0.19/0.45 % (^ x,y:i.phi x y)) (cartprod A B))
% 0.19/0.45
% 0.19/0.45 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.19/0.45 % Source : [Bro08]
% 0.19/0.45 % Names : ZFC168g [Bro08]
% 0.19/0.45
% 0.19/0.45 % Status : Theorem
% 0.19/0.45 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.40 v8.2.0, 0.54 v8.1.0, 0.55 v7.5.0, 0.29 v7.4.0, 0.33 v7.2.0, 0.25 v7.1.0, 0.50 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.57 v5.5.0, 0.50 v5.4.0, 0.60 v4.1.0, 1.00 v3.7.0
% 0.19/0.45 % Syntax : Number of formulae : 404 ( 188 unt; 215 typ; 188 def)
% 0.19/0.45 % Number of atoms : 996 ( 282 equ; 0 cnn)
% 0.19/0.45 % Maximal formula atoms : 183 ( 5 avg)
% 0.19/0.45 % Number of connectives : 1766 ( 47 ~; 7 |; 38 &;1222 @)
% 0.19/0.45 % ( 14 <=>; 438 =>; 0 <=; 0 <~>)
% 0.19/0.45 % Maximal formula depth : 193 ( 2 avg)
% 0.19/0.45 % Number of types : 2 ( 0 usr)
% 0.19/0.45 % Number of type conns : 98 ( 98 >; 0 *; 0 +; 0 <<)
% 0.19/0.45 % Number of symbols : 218 ( 215 usr; 186 con; 0-3 aty)
% 0.19/0.45 % Number of variables : 562 ( 51 ^; 473 !; 38 ?; 562 :)
% 0.19/0.45 % SPC : TH0_THM_EQU_NAR
% 0.19/0.45
% 0.19/0.45 % Comments : http://mathgate.info/detsetitem.php?id=230
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 thf(in_type,type,
% 0.19/0.45 in: $i > $i > $o ).
% 0.19/0.45
% 0.19/0.45 thf(exu_type,type,
% 0.19/0.45 exu: ( $i > $o ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(exu,definition,
% 0.19/0.45 ( exu
% 0.19/0.45 = ( ^ [Xphi: $i > $o] :
% 0.19/0.45 ? [Xx: $i] :
% 0.19/0.45 ( ( Xphi @ Xx )
% 0.19/0.45 & ! [Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xy )
% 0.19/0.45 => ( Xx = Xy ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setextAx_type,type,
% 0.19/0.45 setextAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setextAx,definition,
% 0.19/0.45 ( setextAx
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 <=> ( in @ Xx @ B ) )
% 0.19/0.45 => ( A = B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyset_type,type,
% 0.19/0.45 emptyset: $i ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetAx_type,type,
% 0.19/0.45 emptysetAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetAx,definition,
% 0.19/0.45 ( emptysetAx
% 0.19/0.45 = ( ! [Xx: $i] :
% 0.19/0.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoin_type,type,
% 0.19/0.45 setadjoin: $i > $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinAx_type,type,
% 0.19/0.45 setadjoinAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinAx,definition,
% 0.19/0.45 ( setadjoinAx
% 0.19/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.45 <=> ( ( Xy = Xx )
% 0.19/0.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(powerset_type,type,
% 0.19/0.45 powerset: $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(powersetAx_type,type,
% 0.19/0.45 powersetAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(powersetAx,definition,
% 0.19/0.45 ( powersetAx
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.45 <=> ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setunion_type,type,
% 0.19/0.45 setunion: $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(setunionAx_type,type,
% 0.19/0.45 setunionAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setunionAx,definition,
% 0.19/0.45 ( setunionAx
% 0.19/0.45 = ( ! [A: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.45 <=> ? [B: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 & ( in @ B @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(omega_type,type,
% 0.19/0.45 omega: $i ).
% 0.19/0.45
% 0.19/0.45 thf(omega0Ax_type,type,
% 0.19/0.45 omega0Ax: $o ).
% 0.19/0.45
% 0.19/0.45 thf(omega0Ax,definition,
% 0.19/0.45 ( omega0Ax
% 0.19/0.45 = ( in @ emptyset @ omega ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(omegaSAx_type,type,
% 0.19/0.45 omegaSAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(omegaSAx,definition,
% 0.19/0.45 ( omegaSAx
% 0.19/0.45 = ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ omega )
% 0.19/0.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(omegaIndAx_type,type,
% 0.19/0.45 omegaIndAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(omegaIndAx,definition,
% 0.19/0.45 ( omegaIndAx
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ( ( in @ emptyset @ A )
% 0.19/0.45 & ! [Xx: $i] :
% 0.19/0.45 ( ( ( in @ Xx @ omega )
% 0.19/0.45 & ( in @ Xx @ A ) )
% 0.19/0.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.19/0.45 => ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ omega )
% 0.19/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(replAx_type,type,
% 0.19/0.45 replAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(replAx,definition,
% 0.19/0.45 ( replAx
% 0.19/0.45 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( exu
% 0.19/0.45 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.19/0.45 => ? [B: $i] :
% 0.19/0.45 ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 <=> ? [Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ A )
% 0.19/0.45 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(foundationAx_type,type,
% 0.19/0.45 foundationAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(foundationAx,definition,
% 0.19/0.45 ( foundationAx
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.45 => ? [B: $i] :
% 0.19/0.45 ( ( in @ B @ A )
% 0.19/0.45 & ~ ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(wellorderingAx_type,type,
% 0.19/0.45 wellorderingAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(wellorderingAx,definition,
% 0.19/0.45 ( wellorderingAx
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ? [B: $i] :
% 0.19/0.45 ( ! [C: $i] :
% 0.19/0.45 ( ( in @ C @ B )
% 0.19/0.45 => ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ C )
% 0.19/0.45 => ( in @ Xx @ A ) ) )
% 0.19/0.45 & ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( ( in @ Xx @ A )
% 0.19/0.45 & ( in @ Xy @ A ) )
% 0.19/0.45 => ( ! [C: $i] :
% 0.19/0.45 ( ( in @ C @ B )
% 0.19/0.45 => ( ( in @ Xx @ C )
% 0.19/0.45 <=> ( in @ Xy @ C ) ) )
% 0.19/0.45 => ( Xx = Xy ) ) )
% 0.19/0.45 & ! [C: $i,D: $i] :
% 0.19/0.45 ( ( ( in @ C @ B )
% 0.19/0.45 & ( in @ D @ B ) )
% 0.19/0.45 => ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ C )
% 0.19/0.45 => ( in @ Xx @ D ) )
% 0.19/0.45 | ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ D )
% 0.19/0.45 => ( in @ Xx @ C ) ) ) )
% 0.19/0.45 & ! [C: $i] :
% 0.19/0.45 ( ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ C )
% 0.19/0.45 => ( in @ Xx @ A ) )
% 0.19/0.45 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.19/0.45 => ? [D: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ D @ B )
% 0.19/0.45 & ( in @ Xx @ C )
% 0.19/0.45 & ~ ? [Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ D )
% 0.19/0.45 & ( in @ Xy @ C ) )
% 0.19/0.45 & ! [E: $i] :
% 0.19/0.45 ( ( in @ E @ B )
% 0.19/0.45 => ( ! [Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ E )
% 0.19/0.45 => ( in @ Xy @ D ) )
% 0.19/0.45 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(descr_type,type,
% 0.19/0.45 descr: ( $i > $o ) > $i ).
% 0.19/0.45
% 0.19/0.45 thf(descrp_type,type,
% 0.19/0.45 descrp: $o ).
% 0.19/0.45
% 0.19/0.45 thf(descrp,definition,
% 0.19/0.45 ( descrp
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 => ( Xphi
% 0.19/0.45 @ ( descr
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstr_type,type,
% 0.19/0.45 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrI_type,type,
% 0.19/0.45 dsetconstrI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrI,definition,
% 0.19/0.45 ( dsetconstrI
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ( Xphi @ Xx )
% 0.19/0.45 => ( in @ Xx
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrEL_type,type,
% 0.19/0.45 dsetconstrEL: $o ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrEL,definition,
% 0.19/0.45 ( dsetconstrEL
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrER_type,type,
% 0.19/0.45 dsetconstrER: $o ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrER,definition,
% 0.19/0.45 ( dsetconstrER
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.45 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuE1_type,type,
% 0.19/0.45 exuE1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuE1,definition,
% 0.19/0.45 ( exuE1
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 => ? [Xx: $i] :
% 0.19/0.45 ( ( Xphi @ Xx )
% 0.19/0.45 & ! [Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xy )
% 0.19/0.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(prop2set_type,type,
% 0.19/0.45 prop2set: $o > $i ).
% 0.19/0.45
% 0.19/0.45 thf(prop2setE_type,type,
% 0.19/0.45 prop2setE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(prop2setE,definition,
% 0.19/0.45 ( prop2setE
% 0.19/0.45 = ( ! [Xphi: $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.19/0.45 => Xphi ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetE_type,type,
% 0.19/0.45 emptysetE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetE,definition,
% 0.19/0.45 ( emptysetE
% 0.19/0.45 = ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ emptyset )
% 0.19/0.45 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetimpfalse_type,type,
% 0.19/0.45 emptysetimpfalse: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetimpfalse,definition,
% 0.19/0.45 ( emptysetimpfalse
% 0.19/0.45 = ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ emptyset )
% 0.19/0.45 => $false ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(notinemptyset_type,type,
% 0.19/0.45 notinemptyset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(notinemptyset,definition,
% 0.19/0.45 ( notinemptyset
% 0.19/0.45 = ( ! [Xx: $i] :
% 0.19/0.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuE3e_type,type,
% 0.19/0.45 exuE3e: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuE3e,definition,
% 0.19/0.45 ( exuE3e
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setext_type,type,
% 0.19/0.45 setext: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setext,definition,
% 0.19/0.45 ( setext
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( in @ Xx @ B ) )
% 0.19/0.45 => ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ A ) )
% 0.19/0.45 => ( A = B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyI_type,type,
% 0.19/0.45 emptyI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptyI,definition,
% 0.19/0.45 ( emptyI
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ~ ( in @ Xx @ A )
% 0.19/0.45 => ( A = emptyset ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(noeltsimpempty_type,type,
% 0.19/0.45 noeltsimpempty: $o ).
% 0.19/0.45
% 0.19/0.45 thf(noeltsimpempty,definition,
% 0.19/0.45 ( noeltsimpempty
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ~ ( in @ Xx @ A )
% 0.19/0.45 => ( A = emptyset ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setbeta_type,type,
% 0.19/0.45 setbeta: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setbeta,definition,
% 0.19/0.45 ( setbeta
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ( in @ Xx
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.45 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(nonempty_type,type,
% 0.19/0.45 nonempty: $i > $o ).
% 0.19/0.45
% 0.19/0.45 thf(nonempty,definition,
% 0.19/0.45 ( nonempty
% 0.19/0.45 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyE1_type,type,
% 0.19/0.45 nonemptyE1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyE1,definition,
% 0.19/0.45 ( nonemptyE1
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ( nonempty @ A )
% 0.19/0.45 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyI_type,type,
% 0.19/0.45 nonemptyI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyI,definition,
% 0.19/0.45 ( nonemptyI
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ( Xphi @ Xx )
% 0.19/0.45 => ( nonempty
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyI1_type,type,
% 0.19/0.45 nonemptyI1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyI1,definition,
% 0.19/0.45 ( nonemptyI1
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.45 => ( nonempty @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinIL_type,type,
% 0.19/0.45 setadjoinIL: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinIL,definition,
% 0.19/0.45 ( setadjoinIL
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyinunitempty_type,type,
% 0.19/0.45 emptyinunitempty: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptyinunitempty,definition,
% 0.19/0.45 ( emptyinunitempty
% 0.19/0.45 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinIR_type,type,
% 0.19/0.45 setadjoinIR: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinIR,definition,
% 0.19/0.45 ( setadjoinIR
% 0.19/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ A )
% 0.19/0.45 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinE_type,type,
% 0.19/0.45 setadjoinE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinE,definition,
% 0.19/0.45 ( setadjoinE
% 0.19/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.45 => ! [Xphi: $o] :
% 0.19/0.45 ( ( ( Xy = Xx )
% 0.19/0.45 => Xphi )
% 0.19/0.45 => ( ( ( in @ Xy @ A )
% 0.19/0.45 => Xphi )
% 0.19/0.45 => Xphi ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinOr_type,type,
% 0.19/0.45 setadjoinOr: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinOr,definition,
% 0.19/0.45 ( setadjoinOr
% 0.19/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.45 => ( ( Xy = Xx )
% 0.19/0.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setoftrueEq_type,type,
% 0.19/0.45 setoftrueEq: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setoftrueEq,definition,
% 0.19/0.45 ( setoftrueEq
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xx: $i] : $true )
% 0.19/0.45 = A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(powersetI_type,type,
% 0.19/0.45 powersetI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(powersetI,definition,
% 0.19/0.45 ( powersetI
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ A ) )
% 0.19/0.45 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyinPowerset_type,type,
% 0.19/0.45 emptyinPowerset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptyinPowerset,definition,
% 0.19/0.45 ( emptyinPowerset
% 0.19/0.45 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyInPowerset_type,type,
% 0.19/0.45 emptyInPowerset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptyInPowerset,definition,
% 0.19/0.45 ( emptyInPowerset
% 0.19/0.45 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(powersetE_type,type,
% 0.19/0.45 powersetE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(powersetE,definition,
% 0.19/0.45 ( powersetE
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.45 => ( ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setunionI_type,type,
% 0.19/0.45 setunionI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setunionI,definition,
% 0.19/0.45 ( setunionI
% 0.19/0.45 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ( ( in @ B @ A )
% 0.19/0.45 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setunionE_type,type,
% 0.19/0.45 setunionE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setunionE,definition,
% 0.19/0.45 ( setunionE
% 0.19/0.45 = ( ! [A: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.45 => ! [Xphi: $o] :
% 0.19/0.45 ( ! [B: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ( ( in @ B @ A )
% 0.19/0.45 => Xphi ) )
% 0.19/0.45 => Xphi ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(subPowSU_type,type,
% 0.19/0.45 subPowSU: $o ).
% 0.19/0.45
% 0.19/0.45 thf(subPowSU,definition,
% 0.19/0.45 ( subPowSU
% 0.19/0.45 = ( ! [A: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuE2_type,type,
% 0.19/0.45 exuE2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuE2,definition,
% 0.19/0.45 ( exuE2
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 => ? [Xx: $i] :
% 0.19/0.45 ! [Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xy )
% 0.19/0.45 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyImpWitness_type,type,
% 0.19/0.45 nonemptyImpWitness: $o ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyImpWitness,definition,
% 0.19/0.45 ( nonemptyImpWitness
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ( nonempty @ A )
% 0.19/0.45 => ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & $true ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(uniqinunit_type,type,
% 0.19/0.45 uniqinunit: $o ).
% 0.19/0.45
% 0.19/0.45 thf(uniqinunit,definition,
% 0.19/0.45 ( uniqinunit
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.45 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(notinsingleton_type,type,
% 0.19/0.45 notinsingleton: $o ).
% 0.19/0.45
% 0.19/0.45 thf(notinsingleton,definition,
% 0.19/0.45 ( notinsingleton
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xx != Xy )
% 0.19/0.45 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(eqinunit_type,type,
% 0.19/0.45 eqinunit: $o ).
% 0.19/0.45
% 0.19/0.45 thf(eqinunit,definition,
% 0.19/0.45 ( eqinunit
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xx = Xy )
% 0.19/0.45 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(singletonsswitch_type,type,
% 0.19/0.45 singletonsswitch: $o ).
% 0.19/0.45
% 0.19/0.45 thf(singletonsswitch,definition,
% 0.19/0.45 ( singletonsswitch
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.45 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(upairsetE_type,type,
% 0.19/0.45 upairsetE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(upairsetE,definition,
% 0.19/0.45 ( upairsetE
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.45 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.19/0.45 => ( ( Xz = Xx )
% 0.19/0.45 | ( Xz = Xy ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(upairsetIL_type,type,
% 0.19/0.45 upairsetIL: $o ).
% 0.19/0.45
% 0.19/0.45 thf(upairsetIL,definition,
% 0.19/0.45 ( upairsetIL
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(upairsetIR_type,type,
% 0.19/0.45 upairsetIR: $o ).
% 0.19/0.45
% 0.19/0.45 thf(upairsetIR,definition,
% 0.19/0.45 ( upairsetIR
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyE1_type,type,
% 0.19/0.45 emptyE1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptyE1,definition,
% 0.19/0.45 ( emptyE1
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ( Xphi @ Xx ) )
% 0.19/0.45 => ( ( ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 = emptyset )
% 0.19/0.45 => $false ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(vacuousDall_type,type,
% 0.19/0.45 vacuousDall: $o ).
% 0.19/0.45
% 0.19/0.45 thf(vacuousDall,definition,
% 0.19/0.45 ( vacuousDall
% 0.19/0.45 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ emptyset )
% 0.19/0.45 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan1_type,type,
% 0.19/0.45 quantDeMorgan1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan1,definition,
% 0.19/0.45 ( quantDeMorgan1
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ~ ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( Xphi @ Xx ) )
% 0.19/0.45 => ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan2_type,type,
% 0.19/0.45 quantDeMorgan2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan2,definition,
% 0.19/0.45 ( quantDeMorgan2
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ~ ( Xphi @ Xx ) )
% 0.19/0.45 => ~ ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan3_type,type,
% 0.19/0.45 quantDeMorgan3: $o ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan3,definition,
% 0.19/0.45 ( quantDeMorgan3
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ~ ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ( Xphi @ Xx ) )
% 0.19/0.45 => ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan4_type,type,
% 0.19/0.45 quantDeMorgan4: $o ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan4,definition,
% 0.19/0.45 ( quantDeMorgan4
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ~ ( Xphi @ Xx ) )
% 0.19/0.45 => ~ ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(prop2setI_type,type,
% 0.19/0.45 prop2setI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(prop2setI,definition,
% 0.19/0.45 ( prop2setI
% 0.19/0.45 = ( ! [Xphi: $o] :
% 0.19/0.45 ( Xphi
% 0.19/0.45 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(set2prop_type,type,
% 0.19/0.45 set2prop: $i > $o ).
% 0.19/0.45
% 0.19/0.45 thf(prop2set2propI_type,type,
% 0.19/0.45 prop2set2propI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(prop2set2propI,definition,
% 0.19/0.45 ( prop2set2propI
% 0.19/0.45 = ( ! [Xphi: $o] :
% 0.19/0.45 ( Xphi
% 0.19/0.45 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(notdexE_type,type,
% 0.19/0.45 notdexE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(notdexE,definition,
% 0.19/0.45 ( notdexE
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ~ ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ( Xphi @ Xx ) )
% 0.19/0.45 => ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(notdallE_type,type,
% 0.19/0.45 notdallE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(notdallE,definition,
% 0.19/0.45 ( notdallE
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ~ ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( Xphi @ Xx ) )
% 0.19/0.45 => ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuI1_type,type,
% 0.19/0.45 exuI1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuI1,definition,
% 0.19/0.45 ( exuI1
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ? [Xx: $i] :
% 0.19/0.45 ( ( Xphi @ Xx )
% 0.19/0.45 & ! [Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xy )
% 0.19/0.45 => ( Xx = Xy ) ) )
% 0.19/0.45 => ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuI3_type,type,
% 0.19/0.45 exuI3: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuI3,definition,
% 0.19/0.45 ( exuI3
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.19/0.45 => ( ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xx )
% 0.19/0.45 => ( ( Xphi @ Xy )
% 0.19/0.45 => ( Xx = Xy ) ) )
% 0.19/0.45 => ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuI2_type,type,
% 0.19/0.45 exuI2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuI2,definition,
% 0.19/0.45 ( exuI2
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ? [Xx: $i] :
% 0.19/0.45 ! [Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xy )
% 0.19/0.45 <=> ( Xy = Xx ) )
% 0.19/0.45 => ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(inCongP_type,type,
% 0.19/0.45 inCongP: $o ).
% 0.19/0.45
% 0.19/0.45 thf(inCongP,definition,
% 0.19/0.45 ( inCongP
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( A = B )
% 0.19/0.45 => ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xx = Xy )
% 0.19/0.45 => ( ( in @ Xx @ A )
% 0.19/0.45 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(in__Cong_type,type,
% 0.19/0.45 in__Cong: $o ).
% 0.19/0.45
% 0.19/0.45 thf(in__Cong,definition,
% 0.19/0.45 ( in__Cong
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( A = B )
% 0.19/0.45 => ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xx = Xy )
% 0.19/0.45 => ( ( in @ Xx @ A )
% 0.19/0.45 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuE3u_type,type,
% 0.19/0.45 exuE3u: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuE3u,definition,
% 0.19/0.45 ( exuE3u
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 => ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xx )
% 0.19/0.45 => ( ( Xphi @ Xy )
% 0.19/0.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exu__Cong_type,type,
% 0.19/0.45 exu__Cong: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exu__Cong,definition,
% 0.19/0.45 ( exu__Cong
% 0.19/0.45 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.45 ( ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xx = Xy )
% 0.19/0.45 => ( ( Xphi @ Xx )
% 0.19/0.45 <=> ( Xpsi @ Xy ) ) )
% 0.19/0.45 => ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 <=> ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyset__Cong_type,type,
% 0.19/0.45 emptyset__Cong: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptyset__Cong,definition,
% 0.19/0.45 ( emptyset__Cong
% 0.19/0.45 = ( emptyset = emptyset ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoin__Cong_type,type,
% 0.19/0.45 setadjoin__Cong: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoin__Cong,definition,
% 0.19/0.45 ( setadjoin__Cong
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xx = Xy )
% 0.19/0.45 => ! [Xz: $i,Xu: $i] :
% 0.19/0.45 ( ( Xz = Xu )
% 0.19/0.45 => ( ( setadjoin @ Xx @ Xz )
% 0.19/0.45 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(powerset__Cong_type,type,
% 0.19/0.45 powerset__Cong: $o ).
% 0.19/0.45
% 0.19/0.45 thf(powerset__Cong,definition,
% 0.19/0.45 ( powerset__Cong
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( A = B )
% 0.19/0.45 => ( ( powerset @ A )
% 0.19/0.45 = ( powerset @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setunion__Cong_type,type,
% 0.19/0.45 setunion__Cong: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setunion__Cong,definition,
% 0.19/0.45 ( setunion__Cong
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( A = B )
% 0.19/0.45 => ( ( setunion @ A )
% 0.19/0.45 = ( setunion @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(omega__Cong_type,type,
% 0.19/0.45 omega__Cong: $o ).
% 0.19/0.45
% 0.19/0.45 thf(omega__Cong,definition,
% 0.19/0.45 ( omega__Cong
% 0.19/0.45 = ( omega = omega ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuEu_type,type,
% 0.19/0.45 exuEu: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuEu,definition,
% 0.19/0.45 ( exuEu
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 => ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xx )
% 0.19/0.45 => ( ( Xphi @ Xy )
% 0.19/0.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(descr__Cong_type,type,
% 0.19/0.45 descr__Cong: $o ).
% 0.19/0.45
% 0.19/0.45 thf(descr__Cong,definition,
% 0.19/0.45 ( descr__Cong
% 0.19/0.45 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.45 ( ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xx = Xy )
% 0.19/0.45 => ( ( Xphi @ Xx )
% 0.19/0.45 <=> ( Xpsi @ Xy ) ) )
% 0.19/0.45 => ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 => ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.19/0.45 => ( ( descr
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 = ( descr
% 0.19/0.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstr__Cong_type,type,
% 0.19/0.45 dsetconstr__Cong: $o ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstr__Cong,definition,
% 0.19/0.45 ( dsetconstr__Cong
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( A = B )
% 0.19/0.45 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ! [Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ B )
% 0.19/0.45 => ( ( Xx = Xy )
% 0.19/0.45 => ( ( Xphi @ Xx )
% 0.19/0.45 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.19/0.45 => ( ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 = ( dsetconstr @ B
% 0.19/0.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(subset_type,type,
% 0.19/0.45 subset: $i > $i > $o ).
% 0.19/0.45
% 0.19/0.45 thf(disjoint_type,type,
% 0.19/0.45 disjoint: $i > $i > $o ).
% 0.19/0.45
% 0.19/0.45 thf(setsmeet_type,type,
% 0.19/0.45 setsmeet: $i > $i > $o ).
% 0.19/0.45
% 0.19/0.45 thf(subsetI1_type,type,
% 0.19/0.45 subsetI1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(subsetI1,definition,
% 0.19/0.45 ( subsetI1
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( in @ Xx @ B ) )
% 0.19/0.45 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(eqimpsubset2_type,type,
% 0.19/0.45 eqimpsubset2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(eqimpsubset2,definition,
% 0.19/0.45 ( eqimpsubset2
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( A = B )
% 0.19/0.45 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(eqimpsubset1_type,type,
% 0.19/0.45 eqimpsubset1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(eqimpsubset1,definition,
% 0.19/0.45 ( eqimpsubset1
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( A = B )
% 0.19/0.45 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(subsetI2_type,type,
% 0.19/0.45 subsetI2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(subsetI2,definition,
% 0.19/0.45 ( subsetI2
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( in @ Xx @ B ) )
% 0.19/0.45 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetsubset_type,type,
% 0.19/0.45 emptysetsubset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetsubset,definition,
% 0.19/0.45 ( emptysetsubset
% 0.19/0.45 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(subsetE_type,type,
% 0.19/0.45 subsetE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(subsetE,definition,
% 0.19/0.45 ( subsetE
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( subset @ A @ B )
% 0.19/0.45 => ( ( in @ Xx @ A )
% 0.19/0.45 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(subsetE2_type,type,
% 0.19/0.45 subsetE2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(subsetE2,definition,
% 0.19/0.45 ( subsetE2
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( subset @ A @ B )
% 0.19/0.45 => ( ~ ( in @ Xx @ B )
% 0.19/0.45 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(notsubsetI_type,type,
% 0.19/0.45 notsubsetI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(notsubsetI,definition,
% 0.19/0.45 ( notsubsetI
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ~ ( in @ Xx @ B )
% 0.19/0.45 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(notequalI1_type,type,
% 0.19/0.45 notequalI1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(notequalI1,definition,
% 0.19/0.45 ( notequalI1
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ~ ( subset @ A @ B )
% 0.19/0.45 => ( A != B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(notequalI2_type,type,
% 0.19/0.45 notequalI2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(notequalI2,definition,
% 0.19/0.45 ( notequalI2
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ~ ( in @ Xx @ B )
% 0.19/0.45 => ( A != B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(subsetRefl_type,type,
% 0.19/0.45 subsetRefl: $o ).
% 0.19/0.45
% 0.19/0.45 thf(subsetRefl,definition,
% 0.19/0.45 ( subsetRefl
% 0.19/0.45 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(subsetTrans_type,type,
% 0.19/0.45 subsetTrans: $o ).
% 0.19/0.45
% 0.19/0.45 thf(subsetTrans,definition,
% 0.19/0.45 ( subsetTrans
% 0.19/0.45 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.45 ( ( subset @ A @ B )
% 0.19/0.45 => ( ( subset @ B @ C )
% 0.19/0.45 => ( subset @ A @ C ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinSub_type,type,
% 0.19/0.45 setadjoinSub: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinSub,definition,
% 0.19/0.45 ( setadjoinSub
% 0.19/0.45 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinSub2_type,type,
% 0.19/0.45 setadjoinSub2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinSub2,definition,
% 0.19/0.45 ( setadjoinSub2
% 0.19/0.45 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.19/0.45 ( ( subset @ A @ B )
% 0.19/0.45 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(subset2powerset_type,type,
% 0.19/0.45 subset2powerset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(subset2powerset,definition,
% 0.19/0.45 ( subset2powerset
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( subset @ A @ B )
% 0.19/0.45 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setextsub_type,type,
% 0.19/0.45 setextsub: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setextsub,definition,
% 0.19/0.45 ( setextsub
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( subset @ A @ B )
% 0.19/0.45 => ( ( subset @ B @ A )
% 0.19/0.45 => ( A = B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(subsetemptysetimpeq_type,type,
% 0.19/0.45 subsetemptysetimpeq: $o ).
% 0.19/0.45
% 0.19/0.45 thf(subsetemptysetimpeq,definition,
% 0.19/0.45 ( subsetemptysetimpeq
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ( subset @ A @ emptyset )
% 0.19/0.45 => ( A = emptyset ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(powersetI1_type,type,
% 0.19/0.45 powersetI1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(powersetI1,definition,
% 0.19/0.45 ( powersetI1
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( subset @ B @ A )
% 0.19/0.45 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(powersetE1_type,type,
% 0.19/0.45 powersetE1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(powersetE1,definition,
% 0.19/0.45 ( powersetE1
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.45 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(inPowerset_type,type,
% 0.19/0.45 inPowerset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(inPowerset,definition,
% 0.19/0.45 ( inPowerset
% 0.19/0.45 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(powersetsubset_type,type,
% 0.19/0.45 powersetsubset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(powersetsubset,definition,
% 0.19/0.45 ( powersetsubset
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( subset @ A @ B )
% 0.19/0.45 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sepInPowerset_type,type,
% 0.19/0.45 sepInPowerset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(sepInPowerset,definition,
% 0.19/0.45 ( sepInPowerset
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( in
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 @ ( powerset @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sepSubset_type,type,
% 0.19/0.45 sepSubset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(sepSubset,definition,
% 0.19/0.45 ( sepSubset
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( subset
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binunion_type,type,
% 0.19/0.45 binunion: $i > $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(binunionIL_type,type,
% 0.19/0.45 binunionIL: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binunionIL,definition,
% 0.19/0.45 ( binunionIL
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(upairset2IR_type,type,
% 0.19/0.45 upairset2IR: $o ).
% 0.19/0.45
% 0.19/0.45 thf(upairset2IR,definition,
% 0.19/0.45 ( upairset2IR
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binunionIR_type,type,
% 0.19/0.45 binunionIR: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binunionIR,definition,
% 0.19/0.45 ( binunionIR
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binunionEcases_type,type,
% 0.19/0.45 binunionEcases: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binunionEcases,definition,
% 0.19/0.45 ( binunionEcases
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.19/0.45 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.19/0.45 => ( ( ( in @ Xx @ A )
% 0.19/0.45 => Xphi )
% 0.19/0.45 => ( ( ( in @ Xx @ B )
% 0.19/0.45 => Xphi )
% 0.19/0.45 => Xphi ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binunionE_type,type,
% 0.19/0.45 binunionE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binunionE,definition,
% 0.19/0.45 ( binunionE
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.19/0.45 => ( ( in @ Xx @ A )
% 0.19/0.45 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binunionLsub_type,type,
% 0.19/0.45 binunionLsub: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binunionLsub,definition,
% 0.19/0.45 ( binunionLsub
% 0.19/0.45 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binunionRsub_type,type,
% 0.19/0.45 binunionRsub: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binunionRsub,definition,
% 0.19/0.45 ( binunionRsub
% 0.19/0.45 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersect_type,type,
% 0.19/0.45 binintersect: $i > $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectI_type,type,
% 0.19/0.45 binintersectI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectI,definition,
% 0.19/0.45 ( binintersectI
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset5_type,type,
% 0.19/0.45 binintersectSubset5: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset5,definition,
% 0.19/0.45 ( binintersectSubset5
% 0.19/0.45 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.45 ( ( subset @ C @ A )
% 0.19/0.45 => ( ( subset @ C @ B )
% 0.19/0.45 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectEL_type,type,
% 0.19/0.45 binintersectEL: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectEL,definition,
% 0.19/0.45 ( binintersectEL
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.19/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectLsub_type,type,
% 0.19/0.45 binintersectLsub: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectLsub,definition,
% 0.19/0.45 ( binintersectLsub
% 0.19/0.45 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset2_type,type,
% 0.19/0.45 binintersectSubset2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset2,definition,
% 0.19/0.45 ( binintersectSubset2
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( subset @ A @ B )
% 0.19/0.45 => ( ( binintersect @ A @ B )
% 0.19/0.45 = A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset3_type,type,
% 0.19/0.45 binintersectSubset3: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset3,definition,
% 0.19/0.45 ( binintersectSubset3
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( ( binintersect @ A @ B )
% 0.19/0.45 = B )
% 0.19/0.45 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectER_type,type,
% 0.19/0.45 binintersectER: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectER,definition,
% 0.19/0.45 ( binintersectER
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.19/0.45 => ( in @ Xx @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(disjointsetsI1_type,type,
% 0.19/0.45 disjointsetsI1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(disjointsetsI1,definition,
% 0.19/0.45 ( disjointsetsI1
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ~ ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ( in @ Xx @ B ) )
% 0.19/0.45 => ( ( binintersect @ A @ B )
% 0.19/0.45 = emptyset ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectRsub_type,type,
% 0.19/0.45 binintersectRsub: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectRsub,definition,
% 0.19/0.45 ( binintersectRsub
% 0.19/0.45 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset4_type,type,
% 0.19/0.45 binintersectSubset4: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset4,definition,
% 0.19/0.45 ( binintersectSubset4
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( subset @ B @ A )
% 0.19/0.45 => ( ( binintersect @ A @ B )
% 0.19/0.45 = B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset1_type,type,
% 0.19/0.45 binintersectSubset1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset1,definition,
% 0.19/0.45 ( binintersectSubset1
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( ( binintersect @ A @ B )
% 0.19/0.45 = A )
% 0.19/0.45 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(bs114d_type,type,
% 0.19/0.45 bs114d: $o ).
% 0.19/0.45
% 0.19/0.45 thf(bs114d,definition,
% 0.19/0.45 ( bs114d
% 0.19/0.45 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.45 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.19/0.45 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(regular_type,type,
% 0.19/0.45 regular: $i > $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminus_type,type,
% 0.19/0.45 setminus: $i > $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(setminusI_type,type,
% 0.19/0.45 setminusI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusI,definition,
% 0.19/0.45 ( setminusI
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ~ ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusEL_type,type,
% 0.19/0.45 setminusEL: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusEL,definition,
% 0.19/0.45 ( setminusEL
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusER_type,type,
% 0.19/0.45 setminusER: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusER,definition,
% 0.19/0.45 ( setminusER
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.45 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusSubset2_type,type,
% 0.19/0.45 setminusSubset2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusSubset2,definition,
% 0.19/0.45 ( setminusSubset2
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( subset @ A @ B )
% 0.19/0.45 => ( ( setminus @ A @ B )
% 0.19/0.45 = emptyset ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusERneg_type,type,
% 0.19/0.45 setminusERneg: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusERneg,definition,
% 0.19/0.45 ( setminusERneg
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.45 => ( ( in @ Xx @ A )
% 0.19/0.45 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusELneg_type,type,
% 0.19/0.45 setminusELneg: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusELneg,definition,
% 0.19/0.45 ( setminusELneg
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.45 => ( ~ ( in @ Xx @ B )
% 0.19/0.45 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusILneg_type,type,
% 0.19/0.45 setminusILneg: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusILneg,definition,
% 0.19/0.45 ( setminusILneg
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ~ ( in @ Xx @ A )
% 0.19/0.45 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusIRneg_type,type,
% 0.19/0.45 setminusIRneg: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusIRneg,definition,
% 0.19/0.45 ( setminusIRneg
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusLsub_type,type,
% 0.19/0.45 setminusLsub: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusLsub,definition,
% 0.19/0.45 ( setminusLsub
% 0.19/0.45 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusSubset1_type,type,
% 0.19/0.45 setminusSubset1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusSubset1,definition,
% 0.19/0.45 ( setminusSubset1
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( ( setminus @ A @ B )
% 0.19/0.45 = emptyset )
% 0.19/0.45 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(symdiff_type,type,
% 0.19/0.45 symdiff: $i > $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffE_type,type,
% 0.19/0.45 symdiffE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffE,definition,
% 0.19/0.45 ( symdiffE
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.19/0.45 => ! [Xphi: $o] :
% 0.19/0.45 ( ( ( in @ Xx @ A )
% 0.19/0.45 => ( ~ ( in @ Xx @ B )
% 0.19/0.45 => Xphi ) )
% 0.19/0.45 => ( ( ~ ( in @ Xx @ A )
% 0.19/0.45 => ( ( in @ Xx @ B )
% 0.19/0.45 => Xphi ) )
% 0.19/0.45 => Xphi ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffI1_type,type,
% 0.19/0.45 symdiffI1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffI1,definition,
% 0.19/0.45 ( symdiffI1
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ~ ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffI2_type,type,
% 0.19/0.45 symdiffI2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffI2,definition,
% 0.19/0.45 ( symdiffI2
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ~ ( in @ Xx @ A )
% 0.19/0.45 => ( ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffIneg1_type,type,
% 0.19/0.45 symdiffIneg1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffIneg1,definition,
% 0.19/0.45 ( symdiffIneg1
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ( in @ Xx @ B )
% 0.19/0.45 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffIneg2_type,type,
% 0.19/0.45 symdiffIneg2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffIneg2,definition,
% 0.19/0.45 ( symdiffIneg2
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ~ ( in @ Xx @ A )
% 0.19/0.45 => ( ~ ( in @ Xx @ B )
% 0.19/0.45 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(iskpair_type,type,
% 0.19/0.45 iskpair: $i > $o ).
% 0.19/0.45
% 0.19/0.45 thf(secondinupair_type,type,
% 0.19/0.45 secondinupair: $o ).
% 0.19/0.45
% 0.19/0.45 thf(secondinupair,definition,
% 0.19/0.45 ( secondinupair
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setukpairIL_type,type,
% 0.19/0.45 setukpairIL: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setukpairIL,definition,
% 0.19/0.45 ( setukpairIL
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setukpairIR_type,type,
% 0.19/0.45 setukpairIR: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setukpairIR,definition,
% 0.30/0.45 ( setukpairIR
% 0.30/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(kpairiskpair_type,type,
% 0.30/0.45 kpairiskpair: $o ).
% 0.30/0.45
% 0.30/0.45 thf(kpairiskpair,definition,
% 0.30/0.45 ( kpairiskpair
% 0.30/0.45 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(kpair_type,type,
% 0.30/0.45 kpair: $i > $i > $i ).
% 0.30/0.45
% 0.30/0.45 thf(kpairp_type,type,
% 0.30/0.45 kpairp: $o ).
% 0.30/0.45
% 0.30/0.45 thf(kpairp,definition,
% 0.30/0.45 ( kpairp
% 0.30/0.45 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(cartprod_type,type,
% 0.30/0.45 cartprod: $i > $i > $i ).
% 0.30/0.45
% 0.30/0.45 thf(singletonsubset_type,type,
% 0.30/0.45 singletonsubset: $o ).
% 0.30/0.45
% 0.30/0.45 thf(singletonsubset,definition,
% 0.30/0.45 ( singletonsubset
% 0.30/0.45 = ( ! [A: $i,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(singletoninpowerset_type,type,
% 0.30/0.45 singletoninpowerset: $o ).
% 0.30/0.45
% 0.30/0.45 thf(singletoninpowerset,definition,
% 0.30/0.45 ( singletoninpowerset
% 0.30/0.45 = ( ! [A: $i,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(singletoninpowunion_type,type,
% 0.30/0.45 singletoninpowunion: $o ).
% 0.30/0.45
% 0.30/0.45 thf(singletoninpowunion,definition,
% 0.30/0.45 ( singletoninpowunion
% 0.30/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(upairset2E_type,type,
% 0.30/0.45 upairset2E: $o ).
% 0.30/0.45
% 0.30/0.45 thf(upairset2E,definition,
% 0.30/0.45 ( upairset2E
% 0.30/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.45 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.30/0.45 => ( ( Xz = Xx )
% 0.30/0.45 | ( Xz = Xy ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(upairsubunion_type,type,
% 0.30/0.45 upairsubunion: $o ).
% 0.30/0.45
% 0.30/0.45 thf(upairsubunion,definition,
% 0.30/0.45 ( upairsubunion
% 0.30/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ! [Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ B )
% 0.30/0.45 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(upairinpowunion_type,type,
% 0.30/0.45 upairinpowunion: $o ).
% 0.30/0.45
% 0.30/0.45 thf(upairinpowunion,definition,
% 0.30/0.45 ( upairinpowunion
% 0.30/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ! [Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ B )
% 0.30/0.45 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(ubforcartprodlem1_type,type,
% 0.30/0.45 ubforcartprodlem1: $o ).
% 0.30/0.45
% 0.30/0.45 thf(ubforcartprodlem1,definition,
% 0.30/0.45 ( ubforcartprodlem1
% 0.30/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ! [Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ B )
% 0.30/0.45 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(ubforcartprodlem2_type,type,
% 0.30/0.45 ubforcartprodlem2: $o ).
% 0.30/0.45
% 0.30/0.45 thf(ubforcartprodlem2,definition,
% 0.30/0.45 ( ubforcartprodlem2
% 0.30/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ! [Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ B )
% 0.30/0.45 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(ubforcartprodlem3_type,type,
% 0.30/0.45 ubforcartprodlem3: $o ).
% 0.30/0.45
% 0.30/0.45 thf(ubforcartprodlem3,definition,
% 0.30/0.45 ( ubforcartprodlem3
% 0.30/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ! [Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ B )
% 0.30/0.45 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodpairin_type,type,
% 0.30/0.45 cartprodpairin: $o ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodpairin,definition,
% 0.30/0.45 ( cartprodpairin
% 0.30/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ! [Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ B )
% 0.30/0.45 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodmempair1_type,type,
% 0.30/0.45 cartprodmempair1: $o ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodmempair1,definition,
% 0.30/0.45 ( cartprodmempair1
% 0.30/0.45 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.45 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.45 => ? [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 & ? [Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ B )
% 0.30/0.45 & ( Xu
% 0.30/0.45 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodmempair_type,type,
% 0.30/0.45 cartprodmempair: $o ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodmempair,definition,
% 0.30/0.45 ( cartprodmempair
% 0.30/0.45 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.45 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.45 => ( iskpair @ Xu ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setunionE2_type,type,
% 0.30/0.45 setunionE2: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setunionE2,definition,
% 0.30/0.45 ( setunionE2
% 0.30/0.45 = ( ! [A: $i,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.45 => ? [X: $i] :
% 0.30/0.45 ( ( in @ X @ A )
% 0.30/0.45 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setunionsingleton1_type,type,
% 0.30/0.45 setunionsingleton1: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setunionsingleton1,definition,
% 0.30/0.45 ( setunionsingleton1
% 0.30/0.45 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setunionsingleton2_type,type,
% 0.30/0.45 setunionsingleton2: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setunionsingleton2,definition,
% 0.30/0.45 ( setunionsingleton2
% 0.30/0.45 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setunionsingleton_type,type,
% 0.30/0.45 setunionsingleton: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setunionsingleton,definition,
% 0.30/0.45 ( setunionsingleton
% 0.30/0.45 = ( ! [Xx: $i] :
% 0.30/0.45 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.30/0.45 = Xx ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(singleton_type,type,
% 0.30/0.45 singleton: $i > $o ).
% 0.30/0.45
% 0.30/0.45 thf(singleton,definition,
% 0.30/0.45 ( singleton
% 0.30/0.45 = ( ^ [A: $i] :
% 0.30/0.45 ? [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 & ( A
% 0.30/0.45 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(singletonprop_type,type,
% 0.30/0.45 singletonprop: $o ).
% 0.30/0.45
% 0.30/0.45 thf(singletonprop,definition,
% 0.30/0.45 ( singletonprop
% 0.30/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.45 ( ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ! [Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ A )
% 0.30/0.45 => ( ( Xphi @ Xx )
% 0.30/0.45 => ( ( Xphi @ Xy )
% 0.30/0.45 => ( Xx = Xy ) ) ) ) )
% 0.30/0.45 => ( ? [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 & ( Xphi @ Xx ) )
% 0.30/0.45 => ( singleton
% 0.30/0.45 @ ( dsetconstr @ A
% 0.30/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(ex1_type,type,
% 0.30/0.45 ex1: $i > ( $i > $o ) > $o ).
% 0.30/0.45
% 0.30/0.45 thf(ex1,definition,
% 0.30/0.45 ( ex1
% 0.30/0.45 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.30/0.45 ( singleton
% 0.30/0.45 @ ( dsetconstr @ A
% 0.30/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(ex1E1_type,type,
% 0.30/0.45 ex1E1: $o ).
% 0.30/0.45
% 0.30/0.45 thf(ex1E1,definition,
% 0.30/0.45 ( ex1E1
% 0.30/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.45 ( ( ex1 @ A
% 0.30/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.45 => ? [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(ex1I_type,type,
% 0.30/0.45 ex1I: $o ).
% 0.30/0.45
% 0.30/0.45 thf(ex1I,definition,
% 0.30/0.45 ( ex1I
% 0.30/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ( ( Xphi @ Xx )
% 0.30/0.45 => ( ! [Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ A )
% 0.30/0.45 => ( ( Xphi @ Xy )
% 0.30/0.45 => ( Xy = Xx ) ) )
% 0.30/0.45 => ( ex1 @ A
% 0.30/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(ex1I2_type,type,
% 0.30/0.45 ex1I2: $o ).
% 0.30/0.45
% 0.30/0.45 thf(ex1I2,definition,
% 0.30/0.45 ( ex1I2
% 0.30/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.45 ( ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ! [Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ A )
% 0.30/0.45 => ( ( Xphi @ Xx )
% 0.30/0.45 => ( ( Xphi @ Xy )
% 0.30/0.45 => ( Xx = Xy ) ) ) ) )
% 0.30/0.45 => ( ? [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 & ( Xphi @ Xx ) )
% 0.30/0.45 => ( ex1 @ A
% 0.30/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(singletonsuniq_type,type,
% 0.30/0.45 singletonsuniq: $o ).
% 0.30/0.45
% 0.30/0.45 thf(singletonsuniq,definition,
% 0.30/0.45 ( singletonsuniq
% 0.30/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.45 ( ( ( setadjoin @ Xx @ emptyset )
% 0.30/0.45 = ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.45 => ( Xx = Xy ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(atmost1p_type,type,
% 0.30/0.45 atmost1p: $i > $o ).
% 0.30/0.45
% 0.30/0.45 thf(atleast2p_type,type,
% 0.30/0.45 atleast2p: $i > $o ).
% 0.30/0.45
% 0.30/0.45 thf(atmost2p_type,type,
% 0.30/0.45 atmost2p: $i > $o ).
% 0.30/0.45
% 0.30/0.45 thf(upairsetp_type,type,
% 0.30/0.45 upairsetp: $i > $o ).
% 0.30/0.45
% 0.30/0.45 thf(setukpairinjL1_type,type,
% 0.30/0.45 setukpairinjL1: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setukpairinjL1,definition,
% 0.30/0.45 ( setukpairinjL1
% 0.30/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.45 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.45 => ( Xx = Xz ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(kfstsingleton_type,type,
% 0.30/0.45 kfstsingleton: $o ).
% 0.30/0.45
% 0.30/0.45 thf(kfstsingleton,definition,
% 0.30/0.45 ( kfstsingleton
% 0.30/0.45 = ( ! [Xu: $i] :
% 0.30/0.45 ( ( iskpair @ Xu )
% 0.30/0.45 => ( singleton
% 0.30/0.45 @ ( dsetconstr @ ( setunion @ Xu )
% 0.30/0.45 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(theprop_type,type,
% 0.30/0.45 theprop: $o ).
% 0.30/0.45
% 0.30/0.45 thf(theprop,definition,
% 0.30/0.45 ( theprop
% 0.30/0.45 = ( ! [X: $i] :
% 0.30/0.45 ( ( singleton @ X )
% 0.30/0.45 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(kfst_type,type,
% 0.30/0.45 kfst: $i > $i ).
% 0.30/0.45
% 0.30/0.45 thf(kfstpairEq_type,type,
% 0.30/0.45 kfstpairEq: $o ).
% 0.30/0.45
% 0.30/0.45 thf(kfstpairEq,definition,
% 0.30/0.45 ( kfstpairEq
% 0.30/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.45 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.30/0.45 = Xx ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodfstin_type,type,
% 0.30/0.45 cartprodfstin: $o ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodfstin,definition,
% 0.30/0.45 ( cartprodfstin
% 0.30/0.45 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.45 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.45 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setukpairinjL2_type,type,
% 0.30/0.45 setukpairinjL2: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setukpairinjL2,definition,
% 0.30/0.45 ( setukpairinjL2
% 0.30/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.45 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.45 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.45 => ( Xx = Xz ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setukpairinjL_type,type,
% 0.30/0.45 setukpairinjL: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setukpairinjL,definition,
% 0.30/0.45 ( setukpairinjL
% 0.30/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.45 ( ( ( kpair @ Xx @ Xy )
% 0.30/0.45 = ( kpair @ Xz @ Xu ) )
% 0.30/0.45 => ( Xx = Xz ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setukpairinjR11_type,type,
% 0.30/0.45 setukpairinjR11: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setukpairinjR11,definition,
% 0.30/0.45 ( setukpairinjR11
% 0.30/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.45 ( ( Xx = Xy )
% 0.30/0.45 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.45 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setukpairinjR12_type,type,
% 0.30/0.45 setukpairinjR12: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setukpairinjR12,definition,
% 0.30/0.45 ( setukpairinjR12
% 0.30/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.45 ( ( Xx = Xy )
% 0.30/0.45 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.45 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setukpairinjR1_type,type,
% 0.30/0.45 setukpairinjR1: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setukpairinjR1,definition,
% 0.30/0.45 ( setukpairinjR1
% 0.30/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.45 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.45 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.45 => ( ( Xz = Xu )
% 0.30/0.45 => ( Xy = Xu ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(upairequniteq_type,type,
% 0.30/0.45 upairequniteq: $o ).
% 0.30/0.45
% 0.30/0.45 thf(upairequniteq,definition,
% 0.30/0.45 ( upairequniteq
% 0.30/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.45 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.45 = ( setadjoin @ Xz @ emptyset ) )
% 0.30/0.45 => ( Xx = Xy ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setukpairinjR2_type,type,
% 0.30/0.45 setukpairinjR2: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setukpairinjR2,definition,
% 0.30/0.45 ( setukpairinjR2
% 0.30/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.45 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.45 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.45 => ( Xy = Xu ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setukpairinjR_type,type,
% 0.30/0.45 setukpairinjR: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setukpairinjR,definition,
% 0.30/0.45 ( setukpairinjR
% 0.30/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.45 ( ( ( kpair @ Xx @ Xy )
% 0.30/0.45 = ( kpair @ Xz @ Xu ) )
% 0.30/0.45 => ( Xy = Xu ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(ksndsingleton_type,type,
% 0.30/0.45 ksndsingleton: $o ).
% 0.30/0.45
% 0.30/0.45 thf(ksndsingleton,definition,
% 0.30/0.45 ( ksndsingleton
% 0.30/0.45 = ( ! [Xu: $i] :
% 0.30/0.45 ( ( iskpair @ Xu )
% 0.30/0.45 => ( singleton
% 0.30/0.45 @ ( dsetconstr @ ( setunion @ Xu )
% 0.30/0.45 @ ^ [Xx: $i] :
% 0.30/0.45 ( Xu
% 0.30/0.45 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(ksnd_type,type,
% 0.30/0.45 ksnd: $i > $i ).
% 0.30/0.45
% 0.30/0.45 thf(ksndpairEq_type,type,
% 0.30/0.45 ksndpairEq: $o ).
% 0.30/0.45
% 0.30/0.45 thf(ksndpairEq,definition,
% 0.30/0.45 ( ksndpairEq
% 0.30/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.45 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.30/0.45 = Xy ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(kpairsurjEq_type,type,
% 0.30/0.45 kpairsurjEq: $o ).
% 0.30/0.45
% 0.30/0.45 thf(kpairsurjEq,definition,
% 0.30/0.45 ( kpairsurjEq
% 0.30/0.45 = ( ! [Xu: $i] :
% 0.30/0.45 ( ( iskpair @ Xu )
% 0.30/0.45 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.30/0.45 = Xu ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodsndin_type,type,
% 0.30/0.45 cartprodsndin: $o ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodsndin,definition,
% 0.30/0.45 ( cartprodsndin
% 0.30/0.45 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.45 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.45 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodpairmemEL_type,type,
% 0.30/0.45 cartprodpairmemEL: $o ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodpairmemEL,definition,
% 0.30/0.45 ( cartprodpairmemEL
% 0.30/0.45 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.30/0.45 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.30/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodpairmemER_type,type,
% 0.30/0.45 cartprodpairmemER: $o ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodpairmemER,definition,
% 0.30/0.45 ( cartprodpairmemER
% 0.30/0.45 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.30/0.45 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.30/0.45 => ( in @ Xy @ B ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodmempaircEq_type,type,
% 0.30/0.45 cartprodmempaircEq: $o ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodmempaircEq,definition,
% 0.30/0.45 ( cartprodmempaircEq
% 0.30/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ! [Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ B )
% 0.30/0.45 => ( ( kpair @ Xx @ Xy )
% 0.30/0.45 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodfstpairEq_type,type,
% 0.30/0.45 cartprodfstpairEq: $o ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodfstpairEq,definition,
% 0.30/0.45 ( cartprodfstpairEq
% 0.30/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ! [Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ B )
% 0.30/0.45 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.30/0.45 = Xx ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodsndpairEq_type,type,
% 0.30/0.45 cartprodsndpairEq: $o ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodsndpairEq,definition,
% 0.30/0.45 ( cartprodsndpairEq
% 0.30/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ! [Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ B )
% 0.30/0.45 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.30/0.45 = Xy ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodpairsurjEq_type,type,
% 0.30/0.45 cartprodpairsurjEq: $o ).
% 0.30/0.45
% 0.30/0.45 thf(cartprodpairsurjEq,definition,
% 0.30/0.45 ( cartprodpairsurjEq
% 0.30/0.45 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.45 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.45 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.30/0.45 = Xu ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(breln_type,type,
% 0.30/0.45 breln: $i > $i > $i > $o ).
% 0.30/0.45
% 0.30/0.45 thf(breln,definition,
% 0.30/0.45 ( breln
% 0.30/0.45 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(dpsetconstr_type,type,
% 0.30/0.45 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.30/0.45
% 0.30/0.45 thf(dpsetconstr,definition,
% 0.30/0.45 ( dpsetconstr
% 0.30/0.45 = ( ^ [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.30/0.45 ( dsetconstr @ ( cartprod @ A @ B )
% 0.30/0.45 @ ^ [Xu: $i] :
% 0.30/0.45 ? [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 & ? [Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ B )
% 0.30/0.45 & ( Xphi @ Xx @ Xy )
% 0.30/0.45 & ( Xu
% 0.30/0.45 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(dpsetconstrI_type,type,
% 0.30/0.45 dpsetconstrI: $o ).
% 0.30/0.45
% 0.30/0.45 thf(dpsetconstrI,definition,
% 0.30/0.45 ( dpsetconstrI
% 0.30/0.45 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ! [Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ B )
% 0.30/0.45 => ( ( Xphi @ Xx @ Xy )
% 0.30/0.45 => ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.45 @ ( dpsetconstr @ A @ B
% 0.30/0.45 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(dpsetconstrSub,conjecture,
% 0.30/0.45 ( setextAx
% 0.30/0.45 => ( emptysetAx
% 0.30/0.45 => ( setadjoinAx
% 0.30/0.45 => ( powersetAx
% 0.30/0.45 => ( setunionAx
% 0.30/0.45 => ( omega0Ax
% 0.30/0.45 => ( omegaSAx
% 0.30/0.45 => ( omegaIndAx
% 0.30/0.45 => ( replAx
% 0.30/0.45 => ( foundationAx
% 0.30/0.45 => ( wellorderingAx
% 0.30/0.45 => ( descrp
% 0.30/0.45 => ( dsetconstrI
% 0.30/0.45 => ( dsetconstrEL
% 0.30/0.45 => ( dsetconstrER
% 0.30/0.45 => ( exuE1
% 0.30/0.45 => ( prop2setE
% 0.30/0.45 => ( emptysetE
% 0.30/0.45 => ( emptysetimpfalse
% 0.30/0.45 => ( notinemptyset
% 0.30/0.45 => ( exuE3e
% 0.30/0.45 => ( setext
% 0.30/0.45 => ( emptyI
% 0.30/0.45 => ( noeltsimpempty
% 0.30/0.46 => ( setbeta
% 0.30/0.46 => ( nonemptyE1
% 0.30/0.46 => ( nonemptyI
% 0.30/0.46 => ( nonemptyI1
% 0.30/0.46 => ( setadjoinIL
% 0.30/0.46 => ( emptyinunitempty
% 0.30/0.46 => ( setadjoinIR
% 0.30/0.46 => ( setadjoinE
% 0.30/0.46 => ( setadjoinOr
% 0.30/0.46 => ( setoftrueEq
% 0.30/0.46 => ( powersetI
% 0.30/0.46 => ( emptyinPowerset
% 0.30/0.46 => ( emptyInPowerset
% 0.30/0.46 => ( powersetE
% 0.30/0.46 => ( setunionI
% 0.30/0.46 => ( setunionE
% 0.30/0.46 => ( subPowSU
% 0.30/0.46 => ( exuE2
% 0.30/0.46 => ( nonemptyImpWitness
% 0.30/0.46 => ( uniqinunit
% 0.30/0.46 => ( notinsingleton
% 0.30/0.46 => ( eqinunit
% 0.30/0.46 => ( singletonsswitch
% 0.30/0.46 => ( upairsetE
% 0.30/0.46 => ( upairsetIL
% 0.30/0.46 => ( upairsetIR
% 0.30/0.46 => ( emptyE1
% 0.30/0.46 => ( vacuousDall
% 0.30/0.46 => ( quantDeMorgan1
% 0.30/0.46 => ( quantDeMorgan2
% 0.30/0.46 => ( quantDeMorgan3
% 0.30/0.46 => ( quantDeMorgan4
% 0.30/0.46 => ( prop2setI
% 0.30/0.46 => ( prop2set2propI
% 0.30/0.46 => ( notdexE
% 0.30/0.46 => ( notdallE
% 0.30/0.46 => ( exuI1
% 0.30/0.46 => ( exuI3
% 0.30/0.46 => ( exuI2
% 0.30/0.46 => ( inCongP
% 0.30/0.46 => ( in__Cong
% 0.30/0.46 => ( exuE3u
% 0.30/0.46 => ( exu__Cong
% 0.30/0.46 => ( emptyset__Cong
% 0.30/0.46 => ( setadjoin__Cong
% 0.30/0.46 => ( powerset__Cong
% 0.30/0.46 => ( setunion__Cong
% 0.30/0.46 => ( omega__Cong
% 0.30/0.46 => ( exuEu
% 0.30/0.46 => ( descr__Cong
% 0.30/0.46 => ( dsetconstr__Cong
% 0.30/0.46 => ( subsetI1
% 0.30/0.46 => ( eqimpsubset2
% 0.30/0.46 => ( eqimpsubset1
% 0.30/0.46 => ( subsetI2
% 0.30/0.46 => ( emptysetsubset
% 0.30/0.46 => ( subsetE
% 0.30/0.46 => ( subsetE2
% 0.30/0.46 => ( notsubsetI
% 0.30/0.46 => ( notequalI1
% 0.30/0.46 => ( notequalI2
% 0.30/0.46 => ( subsetRefl
% 0.30/0.46 => ( subsetTrans
% 0.30/0.46 => ( setadjoinSub
% 0.30/0.46 => ( setadjoinSub2
% 0.30/0.46 => ( subset2powerset
% 0.30/0.46 => ( setextsub
% 0.30/0.46 => ( subsetemptysetimpeq
% 0.30/0.46 => ( powersetI1
% 0.30/0.46 => ( powersetE1
% 0.30/0.46 => ( inPowerset
% 0.30/0.46 => ( powersetsubset
% 0.30/0.46 => ( sepInPowerset
% 0.30/0.46 => ( sepSubset
% 0.30/0.46 => ( binunionIL
% 0.30/0.46 => ( upairset2IR
% 0.30/0.46 => ( binunionIR
% 0.30/0.46 => ( binunionEcases
% 0.30/0.46 => ( binunionE
% 0.30/0.46 => ( binunionLsub
% 0.30/0.46 => ( binunionRsub
% 0.30/0.46 => ( binintersectI
% 0.30/0.46 => ( binintersectSubset5
% 0.30/0.46 => ( binintersectEL
% 0.30/0.46 => ( binintersectLsub
% 0.30/0.46 => ( binintersectSubset2
% 0.30/0.46 => ( binintersectSubset3
% 0.30/0.46 => ( binintersectER
% 0.30/0.46 => ( disjointsetsI1
% 0.30/0.46 => ( binintersectRsub
% 0.30/0.46 => ( binintersectSubset4
% 0.30/0.46 => ( binintersectSubset1
% 0.30/0.46 => ( bs114d
% 0.30/0.46 => ( setminusI
% 0.30/0.46 => ( setminusEL
% 0.30/0.46 => ( setminusER
% 0.30/0.46 => ( setminusSubset2
% 0.30/0.46 => ( setminusERneg
% 0.30/0.46 => ( setminusELneg
% 0.30/0.46 => ( setminusILneg
% 0.30/0.46 => ( setminusIRneg
% 0.30/0.46 => ( setminusLsub
% 0.30/0.46 => ( setminusSubset1
% 0.30/0.46 => ( symdiffE
% 0.30/0.46 => ( symdiffI1
% 0.30/0.46 => ( symdiffI2
% 0.30/0.46 => ( symdiffIneg1
% 0.30/0.46 => ( symdiffIneg2
% 0.30/0.46 => ( secondinupair
% 0.30/0.46 => ( setukpairIL
% 0.30/0.46 => ( setukpairIR
% 0.30/0.46 => ( kpairiskpair
% 0.30/0.46 => ( kpairp
% 0.30/0.46 => ( singletonsubset
% 0.30/0.46 => ( singletoninpowerset
% 0.30/0.46 => ( singletoninpowunion
% 0.30/0.46 => ( upairset2E
% 0.30/0.46 => ( upairsubunion
% 0.30/0.46 => ( upairinpowunion
% 0.30/0.46 => ( ubforcartprodlem1
% 0.30/0.46 => ( ubforcartprodlem2
% 0.30/0.46 => ( ubforcartprodlem3
% 0.30/0.46 => ( cartprodpairin
% 0.30/0.46 => ( cartprodmempair1
% 0.30/0.46 => ( cartprodmempair
% 0.30/0.46 => ( setunionE2
% 0.30/0.46 => ( setunionsingleton1
% 0.30/0.46 => ( setunionsingleton2
% 0.30/0.46 => ( setunionsingleton
% 0.30/0.46 => ( singletonprop
% 0.30/0.46 => ( ex1E1
% 0.30/0.46 => ( ex1I
% 0.30/0.46 => ( ex1I2
% 0.30/0.46 => ( singletonsuniq
% 0.30/0.46 => ( setukpairinjL1
% 0.30/0.46 => ( kfstsingleton
% 0.30/0.46 => ( theprop
% 0.30/0.46 => ( kfstpairEq
% 0.30/0.46 => ( cartprodfstin
% 0.30/0.46 => ( setukpairinjL2
% 0.30/0.46 => ( setukpairinjL
% 0.30/0.46 => ( setukpairinjR11
% 0.30/0.46 => ( setukpairinjR12
% 0.30/0.46 => ( setukpairinjR1
% 0.30/0.46 => ( upairequniteq
% 0.30/0.46 => ( setukpairinjR2
% 0.30/0.46 => ( setukpairinjR
% 0.30/0.46 => ( ksndsingleton
% 0.30/0.46 => ( ksndpairEq
% 0.30/0.46 => ( kpairsurjEq
% 0.30/0.46 => ( cartprodsndin
% 0.30/0.46 => ( cartprodpairmemEL
% 0.30/0.46 => ( cartprodpairmemER
% 0.30/0.46 => ( cartprodmempaircEq
% 0.30/0.46 => ( cartprodfstpairEq
% 0.30/0.46 => ( cartprodsndpairEq
% 0.30/0.46 => ( cartprodpairsurjEq
% 0.30/0.46 => ( dpsetconstrI
% 0.30/0.46 => ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.30/0.46 ( subset
% 0.30/0.46 @ ( dpsetconstr @ A @ B
% 0.30/0.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.30/0.46 @ ( cartprod @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 ------------------------
% 0.30/1.40 ---- Embedded in THF ---
% 0.30/1.41 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.30/1.41 %%% Generated on Tue Feb 24 12:30:28 EST 2026
% 0.30/1.41 %%% using '$$dhol' embedding, version 1.3.0.
% 0.30/1.41 %%% Logic specification used:
% 0.30/1.41 %%% thf(spec, logic, $$dhol).
% 0.30/1.41
% 0.30/1.41 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.frxx7lxlXX/DTF2DTF31262.p
% See solution above
% 0.30/1.41 ------------------------
% 0.30/1.43 ---- Cleaned THF ---
% 0.30/1.43 thf(in_type,type,
% 0.30/1.43 in: $i > $i > $o ).
% 0.30/1.43
% 0.30/1.43 thf(exu_type,type,
% 0.30/1.43 exu: ( $i > $o ) > $o ).
% 0.30/1.43
% 0.30/1.43 thf(exu,definition,
% 0.30/1.43 ( exu
% 0.30/1.43 = ( ^ [Xphi: $i > $o] :
% 0.30/1.43 ? [Xx: $i] :
% 0.30/1.43 ( ( Xphi @ Xx )
% 0.30/1.43 & ! [Xy: $i] :
% 0.30/1.43 ( ( Xphi @ Xy )
% 0.30/1.43 => ( Xx = Xy ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setextAx_type,type,
% 0.30/1.43 setextAx: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setextAx,definition,
% 0.30/1.43 ( setextAx
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ! [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 <=> ( in @ Xx @ B ) )
% 0.30/1.43 => ( A = B ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(emptyset_type,type,
% 0.30/1.43 emptyset: $i ).
% 0.30/1.43
% 0.30/1.43 thf(emptysetAx_type,type,
% 0.30/1.43 emptysetAx: $o ).
% 0.30/1.43
% 0.30/1.43 thf(emptysetAx,definition,
% 0.30/1.43 ( emptysetAx
% 0.30/1.43 = ( ! [Xx: $i] :
% 0.30/1.43 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setadjoin_type,type,
% 0.30/1.43 setadjoin: $i > $i > $i ).
% 0.30/1.43
% 0.30/1.43 thf(setadjoinAx_type,type,
% 0.30/1.43 setadjoinAx: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setadjoinAx,definition,
% 0.30/1.43 ( setadjoinAx
% 0.30/1.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/1.43 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/1.43 <=> ( ( Xy = Xx )
% 0.30/1.43 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(powerset_type,type,
% 0.30/1.43 powerset: $i > $i ).
% 0.30/1.43
% 0.30/1.43 thf(powersetAx_type,type,
% 0.30/1.43 powersetAx: $o ).
% 0.30/1.43
% 0.30/1.43 thf(powersetAx,definition,
% 0.30/1.43 ( powersetAx
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ( in @ B @ ( powerset @ A ) )
% 0.30/1.43 <=> ! [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ B )
% 0.30/1.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setunion_type,type,
% 0.30/1.43 setunion: $i > $i ).
% 0.30/1.43
% 0.30/1.43 thf(setunionAx_type,type,
% 0.30/1.43 setunionAx: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setunionAx,definition,
% 0.30/1.43 ( setunionAx
% 0.30/1.43 = ( ! [A: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/1.43 <=> ? [B: $i] :
% 0.30/1.43 ( ( in @ Xx @ B )
% 0.30/1.43 & ( in @ B @ A ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(omega_type,type,
% 0.30/1.43 omega: $i ).
% 0.30/1.43
% 0.30/1.43 thf(omega0Ax_type,type,
% 0.30/1.43 omega0Ax: $o ).
% 0.30/1.43
% 0.30/1.43 thf(omega0Ax,definition,
% 0.30/1.43 ( omega0Ax
% 0.30/1.43 = ( in @ emptyset @ omega ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(omegaSAx_type,type,
% 0.30/1.43 omegaSAx: $o ).
% 0.30/1.43
% 0.30/1.43 thf(omegaSAx,definition,
% 0.30/1.43 ( omegaSAx
% 0.30/1.43 = ( ! [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ omega )
% 0.30/1.43 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(omegaIndAx_type,type,
% 0.30/1.43 omegaIndAx: $o ).
% 0.30/1.43
% 0.30/1.43 thf(omegaIndAx,definition,
% 0.30/1.43 ( omegaIndAx
% 0.30/1.43 = ( ! [A: $i] :
% 0.30/1.43 ( ( ( in @ emptyset @ A )
% 0.30/1.43 & ! [Xx: $i] :
% 0.30/1.43 ( ( ( in @ Xx @ omega )
% 0.30/1.43 & ( in @ Xx @ A ) )
% 0.30/1.43 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.30/1.43 => ! [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ omega )
% 0.30/1.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(replAx_type,type,
% 0.30/1.43 replAx: $o ).
% 0.30/1.43
% 0.30/1.43 thf(replAx,definition,
% 0.30/1.43 ( replAx
% 0.30/1.43 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.30/1.43 ( ! [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ( exu
% 0.30/1.43 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.30/1.43 => ? [B: $i] :
% 0.30/1.43 ! [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ B )
% 0.30/1.43 <=> ? [Xy: $i] :
% 0.30/1.43 ( ( in @ Xy @ A )
% 0.30/1.43 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(foundationAx_type,type,
% 0.30/1.43 foundationAx: $o ).
% 0.30/1.43
% 0.30/1.43 thf(foundationAx,definition,
% 0.30/1.43 ( foundationAx
% 0.30/1.43 = ( ! [A: $i] :
% 0.30/1.43 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/1.43 => ? [B: $i] :
% 0.30/1.43 ( ( in @ B @ A )
% 0.30/1.43 & ~ ? [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ B )
% 0.30/1.43 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(wellorderingAx_type,type,
% 0.30/1.43 wellorderingAx: $o ).
% 0.30/1.43
% 0.30/1.43 thf(wellorderingAx,definition,
% 0.30/1.43 ( wellorderingAx
% 0.30/1.43 = ( ! [A: $i] :
% 0.30/1.43 ? [B: $i] :
% 0.30/1.43 ( ! [C: $i] :
% 0.30/1.43 ( ( in @ C @ B )
% 0.30/1.43 => ! [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ C )
% 0.30/1.43 => ( in @ Xx @ A ) ) )
% 0.30/1.43 & ! [Xx: $i,Xy: $i] :
% 0.30/1.43 ( ( ( in @ Xx @ A )
% 0.30/1.43 & ( in @ Xy @ A ) )
% 0.30/1.43 => ( ! [C: $i] :
% 0.30/1.43 ( ( in @ C @ B )
% 0.30/1.43 => ( ( in @ Xx @ C )
% 0.30/1.43 <=> ( in @ Xy @ C ) ) )
% 0.30/1.43 => ( Xx = Xy ) ) )
% 0.30/1.43 & ! [C: $i,D: $i] :
% 0.30/1.43 ( ( ( in @ C @ B )
% 0.30/1.43 & ( in @ D @ B ) )
% 0.30/1.43 => ( ! [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ C )
% 0.30/1.43 => ( in @ Xx @ D ) )
% 0.30/1.43 | ! [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ D )
% 0.30/1.43 => ( in @ Xx @ C ) ) ) )
% 0.30/1.43 & ! [C: $i] :
% 0.30/1.43 ( ( ! [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ C )
% 0.30/1.43 => ( in @ Xx @ A ) )
% 0.30/1.43 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.30/1.43 => ? [D: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ D @ B )
% 0.30/1.43 & ( in @ Xx @ C )
% 0.30/1.43 & ~ ? [Xy: $i] :
% 0.30/1.43 ( ( in @ Xy @ D )
% 0.30/1.43 & ( in @ Xy @ C ) )
% 0.30/1.43 & ! [E: $i] :
% 0.30/1.43 ( ( in @ E @ B )
% 0.30/1.43 => ( ! [Xy: $i] :
% 0.30/1.43 ( ( in @ Xy @ E )
% 0.30/1.43 => ( in @ Xy @ D ) )
% 0.30/1.43 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(descr_type,type,
% 0.30/1.43 descr: ( $i > $o ) > $i ).
% 0.30/1.43
% 0.30/1.43 thf(descrp_type,type,
% 0.30/1.43 descrp: $o ).
% 0.30/1.43
% 0.30/1.43 thf(descrp,definition,
% 0.30/1.43 ( descrp
% 0.30/1.43 = ( ! [Xphi: $i > $o] :
% 0.30/1.43 ( ( exu
% 0.30/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.43 => ( Xphi
% 0.30/1.43 @ ( descr
% 0.30/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(dsetconstr_type,type,
% 0.30/1.43 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.30/1.43
% 0.30/1.43 thf(dsetconstrI_type,type,
% 0.30/1.43 dsetconstrI: $o ).
% 0.30/1.43
% 0.30/1.43 thf(dsetconstrI,definition,
% 0.30/1.43 ( dsetconstrI
% 0.30/1.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ( ( Xphi @ Xx )
% 0.30/1.43 => ( in @ Xx
% 0.30/1.43 @ ( dsetconstr @ A
% 0.30/1.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(dsetconstrEL_type,type,
% 0.30/1.43 dsetconstrEL: $o ).
% 0.30/1.43
% 0.30/1.43 thf(dsetconstrEL,definition,
% 0.30/1.43 ( dsetconstrEL
% 0.30/1.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx
% 0.30/1.43 @ ( dsetconstr @ A
% 0.30/1.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/1.43 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(dsetconstrER_type,type,
% 0.30/1.43 dsetconstrER: $o ).
% 0.30/1.43
% 0.30/1.43 thf(dsetconstrER,definition,
% 0.30/1.43 ( dsetconstrER
% 0.30/1.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx
% 0.30/1.43 @ ( dsetconstr @ A
% 0.30/1.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/1.43 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(exuE1_type,type,
% 0.30/1.43 exuE1: $o ).
% 0.30/1.43
% 0.30/1.43 thf(exuE1,definition,
% 0.30/1.43 ( exuE1
% 0.30/1.43 = ( ! [Xphi: $i > $o] :
% 0.30/1.43 ( ( exu
% 0.30/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.43 => ? [Xx: $i] :
% 0.30/1.43 ( ( Xphi @ Xx )
% 0.30/1.43 & ! [Xy: $i] :
% 0.30/1.43 ( ( Xphi @ Xy )
% 0.30/1.43 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(prop2set_type,type,
% 0.30/1.43 prop2set: $o > $i ).
% 0.30/1.43
% 0.30/1.43 thf(prop2setE_type,type,
% 0.30/1.43 prop2setE: $o ).
% 0.30/1.43
% 0.30/1.43 thf(prop2setE,definition,
% 0.30/1.43 ( prop2setE
% 0.30/1.43 = ( ! [Xphi: $o,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.30/1.43 => Xphi ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(emptysetE_type,type,
% 0.30/1.43 emptysetE: $o ).
% 0.30/1.43
% 0.30/1.43 thf(emptysetE,definition,
% 0.30/1.43 ( emptysetE
% 0.30/1.43 = ( ! [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ emptyset )
% 0.30/1.43 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(emptysetimpfalse_type,type,
% 0.30/1.43 emptysetimpfalse: $o ).
% 0.30/1.43
% 0.30/1.43 thf(emptysetimpfalse,definition,
% 0.30/1.43 ( emptysetimpfalse
% 0.30/1.43 = ( ! [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ emptyset )
% 0.30/1.43 => $false ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(notinemptyset_type,type,
% 0.30/1.43 notinemptyset: $o ).
% 0.30/1.43
% 0.30/1.43 thf(notinemptyset,definition,
% 0.30/1.43 ( notinemptyset
% 0.30/1.43 = ( ! [Xx: $i] :
% 0.30/1.43 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(exuE3e_type,type,
% 0.30/1.43 exuE3e: $o ).
% 0.30/1.43
% 0.30/1.43 thf(exuE3e,definition,
% 0.30/1.43 ( exuE3e
% 0.30/1.43 = ( ! [Xphi: $i > $o] :
% 0.30/1.43 ( ( exu
% 0.30/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.43 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setext_type,type,
% 0.30/1.43 setext: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setext,definition,
% 0.30/1.43 ( setext
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ! [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ( in @ Xx @ B ) )
% 0.30/1.43 => ( ! [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ B )
% 0.30/1.43 => ( in @ Xx @ A ) )
% 0.30/1.43 => ( A = B ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(emptyI_type,type,
% 0.30/1.43 emptyI: $o ).
% 0.30/1.43
% 0.30/1.43 thf(emptyI,definition,
% 0.30/1.43 ( emptyI
% 0.30/1.43 = ( ! [A: $i] :
% 0.30/1.43 ( ! [Xx: $i] :
% 0.30/1.43 ~ ( in @ Xx @ A )
% 0.30/1.43 => ( A = emptyset ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(noeltsimpempty_type,type,
% 0.30/1.43 noeltsimpempty: $o ).
% 0.30/1.43
% 0.30/1.43 thf(noeltsimpempty,definition,
% 0.30/1.43 ( noeltsimpempty
% 0.30/1.43 = ( ! [A: $i] :
% 0.30/1.43 ( ! [Xx: $i] :
% 0.30/1.43 ~ ( in @ Xx @ A )
% 0.30/1.43 => ( A = emptyset ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setbeta_type,type,
% 0.30/1.43 setbeta: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setbeta,definition,
% 0.30/1.43 ( setbeta
% 0.30/1.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ( ( in @ Xx
% 0.30/1.43 @ ( dsetconstr @ A
% 0.30/1.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/1.43 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(nonempty_type,type,
% 0.30/1.43 nonempty: $i > $o ).
% 0.30/1.43
% 0.30/1.43 thf(nonempty,definition,
% 0.30/1.43 ( nonempty
% 0.30/1.43 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(nonemptyE1_type,type,
% 0.30/1.43 nonemptyE1: $o ).
% 0.30/1.43
% 0.30/1.43 thf(nonemptyE1,definition,
% 0.30/1.43 ( nonemptyE1
% 0.30/1.43 = ( ! [A: $i] :
% 0.30/1.43 ( ( nonempty @ A )
% 0.30/1.43 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(nonemptyI_type,type,
% 0.30/1.43 nonemptyI: $o ).
% 0.30/1.43
% 0.30/1.43 thf(nonemptyI,definition,
% 0.30/1.43 ( nonemptyI
% 0.30/1.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ( ( Xphi @ Xx )
% 0.30/1.43 => ( nonempty
% 0.30/1.43 @ ( dsetconstr @ A
% 0.30/1.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(nonemptyI1_type,type,
% 0.30/1.43 nonemptyI1: $o ).
% 0.30/1.43
% 0.30/1.43 thf(nonemptyI1,definition,
% 0.30/1.43 ( nonemptyI1
% 0.30/1.43 = ( ! [A: $i] :
% 0.30/1.43 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/1.43 => ( nonempty @ A ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setadjoinIL_type,type,
% 0.30/1.43 setadjoinIL: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setadjoinIL,definition,
% 0.30/1.43 ( setadjoinIL
% 0.30/1.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(emptyinunitempty_type,type,
% 0.30/1.43 emptyinunitempty: $o ).
% 0.30/1.43
% 0.30/1.43 thf(emptyinunitempty,definition,
% 0.30/1.43 ( emptyinunitempty
% 0.30/1.43 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setadjoinIR_type,type,
% 0.30/1.43 setadjoinIR: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setadjoinIR,definition,
% 0.30/1.43 ( setadjoinIR
% 0.30/1.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/1.43 ( ( in @ Xy @ A )
% 0.30/1.43 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setadjoinE_type,type,
% 0.30/1.43 setadjoinE: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setadjoinE,definition,
% 0.30/1.43 ( setadjoinE
% 0.30/1.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/1.43 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/1.43 => ! [Xphi: $o] :
% 0.30/1.43 ( ( ( Xy = Xx )
% 0.30/1.43 => Xphi )
% 0.30/1.43 => ( ( ( in @ Xy @ A )
% 0.30/1.43 => Xphi )
% 0.30/1.43 => Xphi ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setadjoinOr_type,type,
% 0.30/1.43 setadjoinOr: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setadjoinOr,definition,
% 0.30/1.43 ( setadjoinOr
% 0.30/1.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/1.43 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/1.43 => ( ( Xy = Xx )
% 0.30/1.43 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setoftrueEq_type,type,
% 0.30/1.43 setoftrueEq: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setoftrueEq,definition,
% 0.30/1.43 ( setoftrueEq
% 0.30/1.43 = ( ! [A: $i] :
% 0.30/1.43 ( ( dsetconstr @ A
% 0.30/1.43 @ ^ [Xx: $i] : $true )
% 0.30/1.43 = A ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(powersetI_type,type,
% 0.30/1.43 powersetI: $o ).
% 0.30/1.43
% 0.30/1.43 thf(powersetI,definition,
% 0.30/1.43 ( powersetI
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ! [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ B )
% 0.30/1.43 => ( in @ Xx @ A ) )
% 0.30/1.43 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(emptyinPowerset_type,type,
% 0.30/1.43 emptyinPowerset: $o ).
% 0.30/1.43
% 0.30/1.43 thf(emptyinPowerset,definition,
% 0.30/1.43 ( emptyinPowerset
% 0.30/1.43 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(emptyInPowerset_type,type,
% 0.30/1.43 emptyInPowerset: $o ).
% 0.30/1.43
% 0.30/1.43 thf(emptyInPowerset,definition,
% 0.30/1.43 ( emptyInPowerset
% 0.30/1.43 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(powersetE_type,type,
% 0.30/1.43 powersetE: $o ).
% 0.30/1.43
% 0.30/1.43 thf(powersetE,definition,
% 0.30/1.43 ( powersetE
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ B @ ( powerset @ A ) )
% 0.30/1.43 => ( ( in @ Xx @ B )
% 0.30/1.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setunionI_type,type,
% 0.30/1.43 setunionI: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setunionI,definition,
% 0.30/1.43 ( setunionI
% 0.30/1.43 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/1.43 ( ( in @ Xx @ B )
% 0.30/1.43 => ( ( in @ B @ A )
% 0.30/1.43 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setunionE_type,type,
% 0.30/1.43 setunionE: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setunionE,definition,
% 0.30/1.43 ( setunionE
% 0.30/1.43 = ( ! [A: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/1.43 => ! [Xphi: $o] :
% 0.30/1.43 ( ! [B: $i] :
% 0.30/1.43 ( ( in @ Xx @ B )
% 0.30/1.43 => ( ( in @ B @ A )
% 0.30/1.43 => Xphi ) )
% 0.30/1.43 => Xphi ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(subPowSU_type,type,
% 0.30/1.43 subPowSU: $o ).
% 0.30/1.43
% 0.30/1.43 thf(subPowSU,definition,
% 0.30/1.43 ( subPowSU
% 0.30/1.43 = ( ! [A: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(exuE2_type,type,
% 0.30/1.43 exuE2: $o ).
% 0.30/1.43
% 0.30/1.43 thf(exuE2,definition,
% 0.30/1.43 ( exuE2
% 0.30/1.43 = ( ! [Xphi: $i > $o] :
% 0.30/1.43 ( ( exu
% 0.30/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.43 => ? [Xx: $i] :
% 0.30/1.43 ! [Xy: $i] :
% 0.30/1.43 ( ( Xphi @ Xy )
% 0.30/1.43 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(nonemptyImpWitness_type,type,
% 0.30/1.43 nonemptyImpWitness: $o ).
% 0.30/1.43
% 0.30/1.43 thf(nonemptyImpWitness,definition,
% 0.30/1.43 ( nonemptyImpWitness
% 0.30/1.43 = ( ! [A: $i] :
% 0.30/1.43 ( ( nonempty @ A )
% 0.30/1.43 => ? [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 & $true ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(uniqinunit_type,type,
% 0.30/1.43 uniqinunit: $o ).
% 0.30/1.43
% 0.30/1.43 thf(uniqinunit,definition,
% 0.30/1.43 ( uniqinunit
% 0.30/1.43 = ( ! [Xx: $i,Xy: $i] :
% 0.30/1.43 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/1.43 => ( Xx = Xy ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(notinsingleton_type,type,
% 0.30/1.43 notinsingleton: $o ).
% 0.30/1.43
% 0.30/1.43 thf(notinsingleton,definition,
% 0.30/1.43 ( notinsingleton
% 0.30/1.43 = ( ! [Xx: $i,Xy: $i] :
% 0.30/1.43 ( ( Xx != Xy )
% 0.30/1.43 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(eqinunit_type,type,
% 0.30/1.43 eqinunit: $o ).
% 0.30/1.43
% 0.30/1.43 thf(eqinunit,definition,
% 0.30/1.43 ( eqinunit
% 0.30/1.43 = ( ! [Xx: $i,Xy: $i] :
% 0.30/1.43 ( ( Xx = Xy )
% 0.30/1.43 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(singletonsswitch_type,type,
% 0.30/1.43 singletonsswitch: $o ).
% 0.30/1.43
% 0.30/1.43 thf(singletonsswitch,definition,
% 0.30/1.43 ( singletonsswitch
% 0.30/1.43 = ( ! [Xx: $i,Xy: $i] :
% 0.30/1.43 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/1.43 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(upairsetE_type,type,
% 0.30/1.43 upairsetE: $o ).
% 0.30/1.43
% 0.30/1.43 thf(upairsetE,definition,
% 0.30/1.43 ( upairsetE
% 0.30/1.43 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/1.43 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.30/1.43 => ( ( Xz = Xx )
% 0.30/1.43 | ( Xz = Xy ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(upairsetIL_type,type,
% 0.30/1.43 upairsetIL: $o ).
% 0.30/1.43
% 0.30/1.43 thf(upairsetIL,definition,
% 0.30/1.43 ( upairsetIL
% 0.30/1.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(upairsetIR_type,type,
% 0.30/1.43 upairsetIR: $o ).
% 0.30/1.43
% 0.30/1.43 thf(upairsetIR,definition,
% 0.30/1.43 ( upairsetIR
% 0.30/1.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(emptyE1_type,type,
% 0.30/1.43 emptyE1: $o ).
% 0.30/1.43
% 0.30/1.43 thf(emptyE1,definition,
% 0.30/1.43 ( emptyE1
% 0.30/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.43 ( ? [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 & ( Xphi @ Xx ) )
% 0.30/1.43 => ( ( ( dsetconstr @ A
% 0.30/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.43 = emptyset )
% 0.30/1.43 => $false ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(vacuousDall_type,type,
% 0.30/1.43 vacuousDall: $o ).
% 0.30/1.43
% 0.30/1.43 thf(vacuousDall,definition,
% 0.30/1.43 ( vacuousDall
% 0.30/1.43 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ emptyset )
% 0.30/1.43 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(quantDeMorgan1_type,type,
% 0.30/1.43 quantDeMorgan1: $o ).
% 0.30/1.43
% 0.30/1.43 thf(quantDeMorgan1,definition,
% 0.30/1.43 ( quantDeMorgan1
% 0.30/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.43 ( ~ ! [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ( Xphi @ Xx ) )
% 0.30/1.43 => ? [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(quantDeMorgan2_type,type,
% 0.30/1.43 quantDeMorgan2: $o ).
% 0.30/1.43
% 0.30/1.43 thf(quantDeMorgan2,definition,
% 0.30/1.43 ( quantDeMorgan2
% 0.30/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.43 ( ! [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ~ ( Xphi @ Xx ) )
% 0.30/1.43 => ~ ? [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(quantDeMorgan3_type,type,
% 0.30/1.43 quantDeMorgan3: $o ).
% 0.30/1.43
% 0.30/1.43 thf(quantDeMorgan3,definition,
% 0.30/1.43 ( quantDeMorgan3
% 0.30/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.43 ( ~ ? [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 & ( Xphi @ Xx ) )
% 0.30/1.43 => ! [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(quantDeMorgan4_type,type,
% 0.30/1.43 quantDeMorgan4: $o ).
% 0.30/1.43
% 0.30/1.43 thf(quantDeMorgan4,definition,
% 0.30/1.43 ( quantDeMorgan4
% 0.30/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.43 ( ? [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 & ~ ( Xphi @ Xx ) )
% 0.30/1.43 => ~ ! [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(prop2setI_type,type,
% 0.30/1.43 prop2setI: $o ).
% 0.30/1.43
% 0.30/1.43 thf(prop2setI,definition,
% 0.30/1.43 ( prop2setI
% 0.30/1.43 = ( ! [Xphi: $o] :
% 0.30/1.43 ( Xphi
% 0.30/1.43 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(set2prop_type,type,
% 0.30/1.43 set2prop: $i > $o ).
% 0.30/1.43
% 0.30/1.43 thf(prop2set2propI_type,type,
% 0.30/1.43 prop2set2propI: $o ).
% 0.30/1.43
% 0.30/1.43 thf(prop2set2propI,definition,
% 0.30/1.43 ( prop2set2propI
% 0.30/1.43 = ( ! [Xphi: $o] :
% 0.30/1.43 ( Xphi
% 0.30/1.43 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(notdexE_type,type,
% 0.30/1.43 notdexE: $o ).
% 0.30/1.43
% 0.30/1.43 thf(notdexE,definition,
% 0.30/1.43 ( notdexE
% 0.30/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.43 ( ~ ? [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 & ( Xphi @ Xx ) )
% 0.30/1.43 => ! [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(notdallE_type,type,
% 0.30/1.43 notdallE: $o ).
% 0.30/1.43
% 0.30/1.43 thf(notdallE,definition,
% 0.30/1.43 ( notdallE
% 0.30/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.43 ( ~ ! [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ( Xphi @ Xx ) )
% 0.30/1.43 => ? [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(exuI1_type,type,
% 0.30/1.43 exuI1: $o ).
% 0.30/1.43
% 0.30/1.43 thf(exuI1,definition,
% 0.30/1.43 ( exuI1
% 0.30/1.43 = ( ! [Xphi: $i > $o] :
% 0.30/1.43 ( ? [Xx: $i] :
% 0.30/1.43 ( ( Xphi @ Xx )
% 0.30/1.43 & ! [Xy: $i] :
% 0.30/1.43 ( ( Xphi @ Xy )
% 0.30/1.43 => ( Xx = Xy ) ) )
% 0.30/1.43 => ( exu
% 0.30/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(exuI3_type,type,
% 0.30/1.43 exuI3: $o ).
% 0.30/1.43
% 0.30/1.43 thf(exuI3,definition,
% 0.30/1.43 ( exuI3
% 0.30/1.43 = ( ! [Xphi: $i > $o] :
% 0.30/1.43 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.30/1.43 => ( ! [Xx: $i,Xy: $i] :
% 0.30/1.43 ( ( Xphi @ Xx )
% 0.30/1.43 => ( ( Xphi @ Xy )
% 0.30/1.43 => ( Xx = Xy ) ) )
% 0.30/1.43 => ( exu
% 0.30/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(exuI2_type,type,
% 0.30/1.43 exuI2: $o ).
% 0.30/1.43
% 0.30/1.43 thf(exuI2,definition,
% 0.30/1.43 ( exuI2
% 0.30/1.43 = ( ! [Xphi: $i > $o] :
% 0.30/1.43 ( ? [Xx: $i] :
% 0.30/1.43 ! [Xy: $i] :
% 0.30/1.43 ( ( Xphi @ Xy )
% 0.30/1.43 <=> ( Xy = Xx ) )
% 0.30/1.43 => ( exu
% 0.30/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(inCongP_type,type,
% 0.30/1.43 inCongP: $o ).
% 0.30/1.43
% 0.30/1.43 thf(inCongP,definition,
% 0.30/1.43 ( inCongP
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ( A = B )
% 0.30/1.43 => ! [Xx: $i,Xy: $i] :
% 0.30/1.43 ( ( Xx = Xy )
% 0.30/1.43 => ( ( in @ Xx @ A )
% 0.30/1.43 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(in__Cong_type,type,
% 0.30/1.43 in__Cong: $o ).
% 0.30/1.43
% 0.30/1.43 thf(in__Cong,definition,
% 0.30/1.43 ( in__Cong
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ( A = B )
% 0.30/1.43 => ! [Xx: $i,Xy: $i] :
% 0.30/1.43 ( ( Xx = Xy )
% 0.30/1.43 => ( ( in @ Xx @ A )
% 0.30/1.43 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(exuE3u_type,type,
% 0.30/1.43 exuE3u: $o ).
% 0.30/1.43
% 0.30/1.43 thf(exuE3u,definition,
% 0.30/1.43 ( exuE3u
% 0.30/1.43 = ( ! [Xphi: $i > $o] :
% 0.30/1.43 ( ( exu
% 0.30/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.43 => ! [Xx: $i,Xy: $i] :
% 0.30/1.43 ( ( Xphi @ Xx )
% 0.30/1.43 => ( ( Xphi @ Xy )
% 0.30/1.43 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(exu__Cong_type,type,
% 0.30/1.43 exu__Cong: $o ).
% 0.30/1.43
% 0.30/1.43 thf(exu__Cong,definition,
% 0.30/1.43 ( exu__Cong
% 0.30/1.43 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/1.43 ( ! [Xx: $i,Xy: $i] :
% 0.30/1.43 ( ( Xx = Xy )
% 0.30/1.43 => ( ( Xphi @ Xx )
% 0.30/1.43 <=> ( Xpsi @ Xy ) ) )
% 0.30/1.43 => ( ( exu
% 0.30/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.43 <=> ( exu
% 0.30/1.43 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(emptyset__Cong_type,type,
% 0.30/1.43 emptyset__Cong: $o ).
% 0.30/1.43
% 0.30/1.43 thf(emptyset__Cong,definition,
% 0.30/1.43 ( emptyset__Cong
% 0.30/1.43 = ( emptyset = emptyset ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setadjoin__Cong_type,type,
% 0.30/1.43 setadjoin__Cong: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setadjoin__Cong,definition,
% 0.30/1.43 ( setadjoin__Cong
% 0.30/1.43 = ( ! [Xx: $i,Xy: $i] :
% 0.30/1.43 ( ( Xx = Xy )
% 0.30/1.43 => ! [Xz: $i,Xu: $i] :
% 0.30/1.43 ( ( Xz = Xu )
% 0.30/1.43 => ( ( setadjoin @ Xx @ Xz )
% 0.30/1.43 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(powerset__Cong_type,type,
% 0.30/1.43 powerset__Cong: $o ).
% 0.30/1.43
% 0.30/1.43 thf(powerset__Cong,definition,
% 0.30/1.43 ( powerset__Cong
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ( A = B )
% 0.30/1.43 => ( ( powerset @ A )
% 0.30/1.43 = ( powerset @ B ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setunion__Cong_type,type,
% 0.30/1.43 setunion__Cong: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setunion__Cong,definition,
% 0.30/1.43 ( setunion__Cong
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ( A = B )
% 0.30/1.43 => ( ( setunion @ A )
% 0.30/1.43 = ( setunion @ B ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(omega__Cong_type,type,
% 0.30/1.43 omega__Cong: $o ).
% 0.30/1.43
% 0.30/1.43 thf(omega__Cong,definition,
% 0.30/1.43 ( omega__Cong
% 0.30/1.43 = ( omega = omega ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(exuEu_type,type,
% 0.30/1.43 exuEu: $o ).
% 0.30/1.43
% 0.30/1.43 thf(exuEu,definition,
% 0.30/1.43 ( exuEu
% 0.30/1.43 = ( ! [Xphi: $i > $o] :
% 0.30/1.43 ( ( exu
% 0.30/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.43 => ! [Xx: $i,Xy: $i] :
% 0.30/1.43 ( ( Xphi @ Xx )
% 0.30/1.43 => ( ( Xphi @ Xy )
% 0.30/1.43 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(descr__Cong_type,type,
% 0.30/1.43 descr__Cong: $o ).
% 0.30/1.43
% 0.30/1.43 thf(descr__Cong,definition,
% 0.30/1.43 ( descr__Cong
% 0.30/1.43 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/1.43 ( ! [Xx: $i,Xy: $i] :
% 0.30/1.43 ( ( Xx = Xy )
% 0.30/1.43 => ( ( Xphi @ Xx )
% 0.30/1.43 <=> ( Xpsi @ Xy ) ) )
% 0.30/1.43 => ( ( exu
% 0.30/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.43 => ( ( exu
% 0.30/1.43 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.30/1.43 => ( ( descr
% 0.30/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.43 = ( descr
% 0.30/1.43 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(dsetconstr__Cong_type,type,
% 0.30/1.43 dsetconstr__Cong: $o ).
% 0.30/1.43
% 0.30/1.43 thf(dsetconstr__Cong,definition,
% 0.30/1.43 ( dsetconstr__Cong
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ( A = B )
% 0.30/1.43 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/1.43 ( ! [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ! [Xy: $i] :
% 0.30/1.43 ( ( in @ Xy @ B )
% 0.30/1.43 => ( ( Xx = Xy )
% 0.30/1.43 => ( ( Xphi @ Xx )
% 0.30/1.43 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.30/1.43 => ( ( dsetconstr @ A
% 0.30/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.43 = ( dsetconstr @ B
% 0.30/1.43 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(subset_type,type,
% 0.30/1.43 subset: $i > $i > $o ).
% 0.30/1.43
% 0.30/1.43 thf(disjoint_type,type,
% 0.30/1.43 disjoint: $i > $i > $o ).
% 0.30/1.43
% 0.30/1.43 thf(setsmeet_type,type,
% 0.30/1.43 setsmeet: $i > $i > $o ).
% 0.30/1.43
% 0.30/1.43 thf(subsetI1_type,type,
% 0.30/1.43 subsetI1: $o ).
% 0.30/1.43
% 0.30/1.43 thf(subsetI1,definition,
% 0.30/1.43 ( subsetI1
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ! [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ( in @ Xx @ B ) )
% 0.30/1.43 => ( subset @ A @ B ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(eqimpsubset2_type,type,
% 0.30/1.43 eqimpsubset2: $o ).
% 0.30/1.43
% 0.30/1.43 thf(eqimpsubset2,definition,
% 0.30/1.43 ( eqimpsubset2
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ( A = B )
% 0.30/1.43 => ( subset @ B @ A ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(eqimpsubset1_type,type,
% 0.30/1.43 eqimpsubset1: $o ).
% 0.30/1.43
% 0.30/1.43 thf(eqimpsubset1,definition,
% 0.30/1.43 ( eqimpsubset1
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ( A = B )
% 0.30/1.43 => ( subset @ A @ B ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(subsetI2_type,type,
% 0.30/1.43 subsetI2: $o ).
% 0.30/1.43
% 0.30/1.43 thf(subsetI2,definition,
% 0.30/1.43 ( subsetI2
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ! [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ( in @ Xx @ B ) )
% 0.30/1.43 => ( subset @ A @ B ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(emptysetsubset_type,type,
% 0.30/1.43 emptysetsubset: $o ).
% 0.30/1.43
% 0.30/1.43 thf(emptysetsubset,definition,
% 0.30/1.43 ( emptysetsubset
% 0.30/1.43 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(subsetE_type,type,
% 0.30/1.43 subsetE: $o ).
% 0.30/1.43
% 0.30/1.43 thf(subsetE,definition,
% 0.30/1.43 ( subsetE
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ( subset @ A @ B )
% 0.30/1.43 => ( ( in @ Xx @ A )
% 0.30/1.43 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(subsetE2_type,type,
% 0.30/1.43 subsetE2: $o ).
% 0.30/1.43
% 0.30/1.43 thf(subsetE2,definition,
% 0.30/1.43 ( subsetE2
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ( subset @ A @ B )
% 0.30/1.43 => ( ~ ( in @ Xx @ B )
% 0.30/1.43 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(notsubsetI_type,type,
% 0.30/1.43 notsubsetI: $o ).
% 0.30/1.43
% 0.30/1.43 thf(notsubsetI,definition,
% 0.30/1.43 ( notsubsetI
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ( ~ ( in @ Xx @ B )
% 0.30/1.43 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(notequalI1_type,type,
% 0.30/1.43 notequalI1: $o ).
% 0.30/1.43
% 0.30/1.43 thf(notequalI1,definition,
% 0.30/1.43 ( notequalI1
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ~ ( subset @ A @ B )
% 0.30/1.43 => ( A != B ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(notequalI2_type,type,
% 0.30/1.43 notequalI2: $o ).
% 0.30/1.43
% 0.30/1.43 thf(notequalI2,definition,
% 0.30/1.43 ( notequalI2
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ( ~ ( in @ Xx @ B )
% 0.30/1.43 => ( A != B ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(subsetRefl_type,type,
% 0.30/1.43 subsetRefl: $o ).
% 0.30/1.43
% 0.30/1.43 thf(subsetRefl,definition,
% 0.30/1.43 ( subsetRefl
% 0.30/1.43 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(subsetTrans_type,type,
% 0.30/1.43 subsetTrans: $o ).
% 0.30/1.43
% 0.30/1.43 thf(subsetTrans,definition,
% 0.30/1.43 ( subsetTrans
% 0.30/1.43 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/1.43 ( ( subset @ A @ B )
% 0.30/1.43 => ( ( subset @ B @ C )
% 0.30/1.43 => ( subset @ A @ C ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setadjoinSub_type,type,
% 0.30/1.43 setadjoinSub: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setadjoinSub,definition,
% 0.30/1.43 ( setadjoinSub
% 0.30/1.43 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setadjoinSub2_type,type,
% 0.30/1.43 setadjoinSub2: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setadjoinSub2,definition,
% 0.30/1.43 ( setadjoinSub2
% 0.30/1.43 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/1.43 ( ( subset @ A @ B )
% 0.30/1.43 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(subset2powerset_type,type,
% 0.30/1.43 subset2powerset: $o ).
% 0.30/1.43
% 0.30/1.43 thf(subset2powerset,definition,
% 0.30/1.43 ( subset2powerset
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ( subset @ A @ B )
% 0.30/1.43 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setextsub_type,type,
% 0.30/1.43 setextsub: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setextsub,definition,
% 0.30/1.43 ( setextsub
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ( subset @ A @ B )
% 0.30/1.43 => ( ( subset @ B @ A )
% 0.30/1.43 => ( A = B ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(subsetemptysetimpeq_type,type,
% 0.30/1.43 subsetemptysetimpeq: $o ).
% 0.30/1.43
% 0.30/1.43 thf(subsetemptysetimpeq,definition,
% 0.30/1.43 ( subsetemptysetimpeq
% 0.30/1.43 = ( ! [A: $i] :
% 0.30/1.43 ( ( subset @ A @ emptyset )
% 0.30/1.43 => ( A = emptyset ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(powersetI1_type,type,
% 0.30/1.43 powersetI1: $o ).
% 0.30/1.43
% 0.30/1.43 thf(powersetI1,definition,
% 0.30/1.43 ( powersetI1
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ( subset @ B @ A )
% 0.30/1.43 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(powersetE1_type,type,
% 0.30/1.43 powersetE1: $o ).
% 0.30/1.43
% 0.30/1.43 thf(powersetE1,definition,
% 0.30/1.43 ( powersetE1
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ( in @ B @ ( powerset @ A ) )
% 0.30/1.43 => ( subset @ B @ A ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(inPowerset_type,type,
% 0.30/1.43 inPowerset: $o ).
% 0.30/1.43
% 0.30/1.43 thf(inPowerset,definition,
% 0.30/1.43 ( inPowerset
% 0.30/1.43 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(powersetsubset_type,type,
% 0.30/1.43 powersetsubset: $o ).
% 0.30/1.43
% 0.30/1.43 thf(powersetsubset,definition,
% 0.30/1.43 ( powersetsubset
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ( subset @ A @ B )
% 0.30/1.43 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(sepInPowerset_type,type,
% 0.30/1.43 sepInPowerset: $o ).
% 0.30/1.43
% 0.30/1.43 thf(sepInPowerset,definition,
% 0.30/1.43 ( sepInPowerset
% 0.30/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.43 ( in
% 0.30/1.43 @ ( dsetconstr @ A
% 0.30/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.43 @ ( powerset @ A ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(sepSubset_type,type,
% 0.30/1.43 sepSubset: $o ).
% 0.30/1.43
% 0.30/1.43 thf(sepSubset,definition,
% 0.30/1.43 ( sepSubset
% 0.30/1.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.43 ( subset
% 0.30/1.43 @ ( dsetconstr @ A
% 0.30/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.43 @ A ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(binunion_type,type,
% 0.30/1.43 binunion: $i > $i > $i ).
% 0.30/1.43
% 0.30/1.43 thf(binunionIL_type,type,
% 0.30/1.43 binunionIL: $o ).
% 0.30/1.43
% 0.30/1.43 thf(binunionIL,definition,
% 0.30/1.43 ( binunionIL
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(upairset2IR_type,type,
% 0.30/1.43 upairset2IR: $o ).
% 0.30/1.43
% 0.30/1.43 thf(upairset2IR,definition,
% 0.30/1.43 ( upairset2IR
% 0.30/1.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(binunionIR_type,type,
% 0.30/1.43 binunionIR: $o ).
% 0.30/1.43
% 0.30/1.43 thf(binunionIR,definition,
% 0.30/1.43 ( binunionIR
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ B )
% 0.30/1.43 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(binunionEcases_type,type,
% 0.30/1.43 binunionEcases: $o ).
% 0.30/1.43
% 0.30/1.43 thf(binunionEcases,definition,
% 0.30/1.43 ( binunionEcases
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.30/1.43 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.30/1.43 => ( ( ( in @ Xx @ A )
% 0.30/1.43 => Xphi )
% 0.30/1.43 => ( ( ( in @ Xx @ B )
% 0.30/1.43 => Xphi )
% 0.30/1.43 => Xphi ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(binunionE_type,type,
% 0.30/1.43 binunionE: $o ).
% 0.30/1.43
% 0.30/1.43 thf(binunionE,definition,
% 0.30/1.43 ( binunionE
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.30/1.43 => ( ( in @ Xx @ A )
% 0.30/1.43 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(binunionLsub_type,type,
% 0.30/1.43 binunionLsub: $o ).
% 0.30/1.43
% 0.30/1.43 thf(binunionLsub,definition,
% 0.30/1.43 ( binunionLsub
% 0.30/1.43 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(binunionRsub_type,type,
% 0.30/1.43 binunionRsub: $o ).
% 0.30/1.43
% 0.30/1.43 thf(binunionRsub,definition,
% 0.30/1.43 ( binunionRsub
% 0.30/1.43 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(binintersect_type,type,
% 0.30/1.43 binintersect: $i > $i > $i ).
% 0.30/1.43
% 0.30/1.43 thf(binintersectI_type,type,
% 0.30/1.43 binintersectI: $o ).
% 0.30/1.43
% 0.30/1.43 thf(binintersectI,definition,
% 0.30/1.43 ( binintersectI
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ( ( in @ Xx @ B )
% 0.30/1.43 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(binintersectSubset5_type,type,
% 0.30/1.43 binintersectSubset5: $o ).
% 0.30/1.43
% 0.30/1.43 thf(binintersectSubset5,definition,
% 0.30/1.43 ( binintersectSubset5
% 0.30/1.43 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/1.43 ( ( subset @ C @ A )
% 0.30/1.43 => ( ( subset @ C @ B )
% 0.30/1.43 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(binintersectEL_type,type,
% 0.30/1.43 binintersectEL: $o ).
% 0.30/1.43
% 0.30/1.43 thf(binintersectEL,definition,
% 0.30/1.43 ( binintersectEL
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.30/1.43 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(binintersectLsub_type,type,
% 0.30/1.43 binintersectLsub: $o ).
% 0.30/1.43
% 0.30/1.43 thf(binintersectLsub,definition,
% 0.30/1.43 ( binintersectLsub
% 0.30/1.43 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(binintersectSubset2_type,type,
% 0.30/1.43 binintersectSubset2: $o ).
% 0.30/1.43
% 0.30/1.43 thf(binintersectSubset2,definition,
% 0.30/1.43 ( binintersectSubset2
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ( subset @ A @ B )
% 0.30/1.43 => ( ( binintersect @ A @ B )
% 0.30/1.43 = A ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(binintersectSubset3_type,type,
% 0.30/1.43 binintersectSubset3: $o ).
% 0.30/1.43
% 0.30/1.43 thf(binintersectSubset3,definition,
% 0.30/1.43 ( binintersectSubset3
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ( ( binintersect @ A @ B )
% 0.30/1.43 = B )
% 0.30/1.43 => ( subset @ B @ A ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(binintersectER_type,type,
% 0.30/1.43 binintersectER: $o ).
% 0.30/1.43
% 0.30/1.43 thf(binintersectER,definition,
% 0.30/1.43 ( binintersectER
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.30/1.43 => ( in @ Xx @ B ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(disjointsetsI1_type,type,
% 0.30/1.43 disjointsetsI1: $o ).
% 0.30/1.43
% 0.30/1.43 thf(disjointsetsI1,definition,
% 0.30/1.43 ( disjointsetsI1
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ~ ? [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 & ( in @ Xx @ B ) )
% 0.30/1.43 => ( ( binintersect @ A @ B )
% 0.30/1.43 = emptyset ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(binintersectRsub_type,type,
% 0.30/1.43 binintersectRsub: $o ).
% 0.30/1.43
% 0.30/1.43 thf(binintersectRsub,definition,
% 0.30/1.43 ( binintersectRsub
% 0.30/1.43 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(binintersectSubset4_type,type,
% 0.30/1.43 binintersectSubset4: $o ).
% 0.30/1.43
% 0.30/1.43 thf(binintersectSubset4,definition,
% 0.30/1.43 ( binintersectSubset4
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ( subset @ B @ A )
% 0.30/1.43 => ( ( binintersect @ A @ B )
% 0.30/1.43 = B ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(binintersectSubset1_type,type,
% 0.30/1.43 binintersectSubset1: $o ).
% 0.30/1.43
% 0.30/1.43 thf(binintersectSubset1,definition,
% 0.30/1.43 ( binintersectSubset1
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ( ( binintersect @ A @ B )
% 0.30/1.43 = A )
% 0.30/1.43 => ( subset @ A @ B ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(bs114d_type,type,
% 0.30/1.43 bs114d: $o ).
% 0.30/1.43
% 0.30/1.43 thf(bs114d,definition,
% 0.30/1.43 ( bs114d
% 0.30/1.43 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/1.43 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.30/1.43 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(regular_type,type,
% 0.30/1.43 regular: $i > $o ).
% 0.30/1.43
% 0.30/1.43 thf(setminus_type,type,
% 0.30/1.43 setminus: $i > $i > $i ).
% 0.30/1.43
% 0.30/1.43 thf(setminusI_type,type,
% 0.30/1.43 setminusI: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setminusI,definition,
% 0.30/1.43 ( setminusI
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ( ~ ( in @ Xx @ B )
% 0.30/1.43 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setminusEL_type,type,
% 0.30/1.43 setminusEL: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setminusEL,definition,
% 0.30/1.43 ( setminusEL
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/1.43 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setminusER_type,type,
% 0.30/1.43 setminusER: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setminusER,definition,
% 0.30/1.43 ( setminusER
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/1.43 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setminusSubset2_type,type,
% 0.30/1.43 setminusSubset2: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setminusSubset2,definition,
% 0.30/1.43 ( setminusSubset2
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ( subset @ A @ B )
% 0.30/1.43 => ( ( setminus @ A @ B )
% 0.30/1.43 = emptyset ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setminusERneg_type,type,
% 0.30/1.43 setminusERneg: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setminusERneg,definition,
% 0.30/1.43 ( setminusERneg
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/1.43 => ( ( in @ Xx @ A )
% 0.30/1.43 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setminusELneg_type,type,
% 0.30/1.43 setminusELneg: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setminusELneg,definition,
% 0.30/1.43 ( setminusELneg
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/1.43 => ( ~ ( in @ Xx @ B )
% 0.30/1.43 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setminusILneg_type,type,
% 0.30/1.43 setminusILneg: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setminusILneg,definition,
% 0.30/1.43 ( setminusILneg
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ~ ( in @ Xx @ A )
% 0.30/1.43 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setminusIRneg_type,type,
% 0.30/1.43 setminusIRneg: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setminusIRneg,definition,
% 0.30/1.43 ( setminusIRneg
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ B )
% 0.30/1.43 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setminusLsub_type,type,
% 0.30/1.43 setminusLsub: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setminusLsub,definition,
% 0.30/1.43 ( setminusLsub
% 0.30/1.43 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setminusSubset1_type,type,
% 0.30/1.43 setminusSubset1: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setminusSubset1,definition,
% 0.30/1.43 ( setminusSubset1
% 0.30/1.43 = ( ! [A: $i,B: $i] :
% 0.30/1.43 ( ( ( setminus @ A @ B )
% 0.30/1.43 = emptyset )
% 0.30/1.43 => ( subset @ A @ B ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(symdiff_type,type,
% 0.30/1.43 symdiff: $i > $i > $i ).
% 0.30/1.43
% 0.30/1.43 thf(symdiffE_type,type,
% 0.30/1.43 symdiffE: $o ).
% 0.30/1.43
% 0.30/1.43 thf(symdiffE,definition,
% 0.30/1.43 ( symdiffE
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.30/1.43 => ! [Xphi: $o] :
% 0.30/1.43 ( ( ( in @ Xx @ A )
% 0.30/1.43 => ( ~ ( in @ Xx @ B )
% 0.30/1.43 => Xphi ) )
% 0.30/1.43 => ( ( ~ ( in @ Xx @ A )
% 0.30/1.43 => ( ( in @ Xx @ B )
% 0.30/1.43 => Xphi ) )
% 0.30/1.43 => Xphi ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(symdiffI1_type,type,
% 0.30/1.43 symdiffI1: $o ).
% 0.30/1.43
% 0.30/1.43 thf(symdiffI1,definition,
% 0.30/1.43 ( symdiffI1
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ( ~ ( in @ Xx @ B )
% 0.30/1.43 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(symdiffI2_type,type,
% 0.30/1.43 symdiffI2: $o ).
% 0.30/1.43
% 0.30/1.43 thf(symdiffI2,definition,
% 0.30/1.43 ( symdiffI2
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ~ ( in @ Xx @ A )
% 0.30/1.43 => ( ( in @ Xx @ B )
% 0.30/1.43 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(symdiffIneg1_type,type,
% 0.30/1.43 symdiffIneg1: $o ).
% 0.30/1.43
% 0.30/1.43 thf(symdiffIneg1,definition,
% 0.30/1.43 ( symdiffIneg1
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ( ( in @ Xx @ B )
% 0.30/1.43 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(symdiffIneg2_type,type,
% 0.30/1.43 symdiffIneg2: $o ).
% 0.30/1.43
% 0.30/1.43 thf(symdiffIneg2,definition,
% 0.30/1.43 ( symdiffIneg2
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ~ ( in @ Xx @ A )
% 0.30/1.43 => ( ~ ( in @ Xx @ B )
% 0.30/1.43 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(iskpair_type,type,
% 0.30/1.43 iskpair: $i > $o ).
% 0.30/1.43
% 0.30/1.43 thf(secondinupair_type,type,
% 0.30/1.43 secondinupair: $o ).
% 0.30/1.43
% 0.30/1.43 thf(secondinupair,definition,
% 0.30/1.43 ( secondinupair
% 0.30/1.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setukpairIL_type,type,
% 0.30/1.43 setukpairIL: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setukpairIL,definition,
% 0.30/1.43 ( setukpairIL
% 0.30/1.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setukpairIR_type,type,
% 0.30/1.43 setukpairIR: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setukpairIR,definition,
% 0.30/1.43 ( setukpairIR
% 0.30/1.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(kpairiskpair_type,type,
% 0.30/1.43 kpairiskpair: $o ).
% 0.30/1.43
% 0.30/1.43 thf(kpairiskpair,definition,
% 0.30/1.43 ( kpairiskpair
% 0.30/1.43 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(kpair_type,type,
% 0.30/1.43 kpair: $i > $i > $i ).
% 0.30/1.43
% 0.30/1.43 thf(kpairp_type,type,
% 0.30/1.43 kpairp: $o ).
% 0.30/1.43
% 0.30/1.43 thf(kpairp,definition,
% 0.30/1.43 ( kpairp
% 0.30/1.43 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(cartprod_type,type,
% 0.30/1.43 cartprod: $i > $i > $i ).
% 0.30/1.43
% 0.30/1.43 thf(singletonsubset_type,type,
% 0.30/1.43 singletonsubset: $o ).
% 0.30/1.43
% 0.30/1.43 thf(singletonsubset,definition,
% 0.30/1.43 ( singletonsubset
% 0.30/1.43 = ( ! [A: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(singletoninpowerset_type,type,
% 0.30/1.43 singletoninpowerset: $o ).
% 0.30/1.43
% 0.30/1.43 thf(singletoninpowerset,definition,
% 0.30/1.43 ( singletoninpowerset
% 0.30/1.43 = ( ! [A: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(singletoninpowunion_type,type,
% 0.30/1.43 singletoninpowunion: $o ).
% 0.30/1.43
% 0.30/1.43 thf(singletoninpowunion,definition,
% 0.30/1.43 ( singletoninpowunion
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(upairset2E_type,type,
% 0.30/1.43 upairset2E: $o ).
% 0.30/1.43
% 0.30/1.43 thf(upairset2E,definition,
% 0.30/1.43 ( upairset2E
% 0.30/1.43 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/1.43 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.30/1.43 => ( ( Xz = Xx )
% 0.30/1.43 | ( Xz = Xy ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(upairsubunion_type,type,
% 0.30/1.43 upairsubunion: $o ).
% 0.30/1.43
% 0.30/1.43 thf(upairsubunion,definition,
% 0.30/1.43 ( upairsubunion
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ! [Xy: $i] :
% 0.30/1.43 ( ( in @ Xy @ B )
% 0.30/1.43 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(upairinpowunion_type,type,
% 0.30/1.43 upairinpowunion: $o ).
% 0.30/1.43
% 0.30/1.43 thf(upairinpowunion,definition,
% 0.30/1.43 ( upairinpowunion
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ! [Xy: $i] :
% 0.30/1.43 ( ( in @ Xy @ B )
% 0.30/1.43 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(ubforcartprodlem1_type,type,
% 0.30/1.43 ubforcartprodlem1: $o ).
% 0.30/1.43
% 0.30/1.43 thf(ubforcartprodlem1,definition,
% 0.30/1.43 ( ubforcartprodlem1
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ! [Xy: $i] :
% 0.30/1.43 ( ( in @ Xy @ B )
% 0.30/1.43 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(ubforcartprodlem2_type,type,
% 0.30/1.43 ubforcartprodlem2: $o ).
% 0.30/1.43
% 0.30/1.43 thf(ubforcartprodlem2,definition,
% 0.30/1.43 ( ubforcartprodlem2
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ! [Xy: $i] :
% 0.30/1.43 ( ( in @ Xy @ B )
% 0.30/1.43 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(ubforcartprodlem3_type,type,
% 0.30/1.43 ubforcartprodlem3: $o ).
% 0.30/1.43
% 0.30/1.43 thf(ubforcartprodlem3,definition,
% 0.30/1.43 ( ubforcartprodlem3
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ! [Xy: $i] :
% 0.30/1.43 ( ( in @ Xy @ B )
% 0.30/1.43 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(cartprodpairin_type,type,
% 0.30/1.43 cartprodpairin: $o ).
% 0.30/1.43
% 0.30/1.43 thf(cartprodpairin,definition,
% 0.30/1.43 ( cartprodpairin
% 0.30/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 => ! [Xy: $i] :
% 0.30/1.43 ( ( in @ Xy @ B )
% 0.30/1.43 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(cartprodmempair1_type,type,
% 0.30/1.43 cartprodmempair1: $o ).
% 0.30/1.43
% 0.30/1.43 thf(cartprodmempair1,definition,
% 0.30/1.43 ( cartprodmempair1
% 0.30/1.43 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/1.43 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/1.43 => ? [Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ A )
% 0.30/1.43 & ? [Xy: $i] :
% 0.30/1.43 ( ( in @ Xy @ B )
% 0.30/1.43 & ( Xu
% 0.30/1.43 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(cartprodmempair_type,type,
% 0.30/1.43 cartprodmempair: $o ).
% 0.30/1.43
% 0.30/1.43 thf(cartprodmempair,definition,
% 0.30/1.43 ( cartprodmempair
% 0.30/1.43 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/1.43 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/1.43 => ( iskpair @ Xu ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setunionE2_type,type,
% 0.30/1.43 setunionE2: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setunionE2,definition,
% 0.30/1.43 ( setunionE2
% 0.30/1.43 = ( ! [A: $i,Xx: $i] :
% 0.30/1.43 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/1.43 => ? [X: $i] :
% 0.30/1.43 ( ( in @ X @ A )
% 0.30/1.43 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setunionsingleton1_type,type,
% 0.30/1.43 setunionsingleton1: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setunionsingleton1,definition,
% 0.30/1.43 ( setunionsingleton1
% 0.30/1.43 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setunionsingleton2_type,type,
% 0.30/1.43 setunionsingleton2: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setunionsingleton2,definition,
% 0.30/1.43 ( setunionsingleton2
% 0.30/1.43 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(setunionsingleton_type,type,
% 0.30/1.43 setunionsingleton: $o ).
% 0.30/1.43
% 0.30/1.43 thf(setunionsingleton,definition,
% 0.30/1.43 ( setunionsingleton
% 0.30/1.43 = ( ! [Xx: $i] :
% 0.30/1.43 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.30/1.43 = Xx ) ) ) ).
% 0.30/1.43
% 0.30/1.43 thf(singleton_type,type,
% 0.30/1.43 singleton: $i > $o ).
% 0.30/1.44
% 0.30/1.44 thf(singleton,definition,
% 0.30/1.44 ( singleton
% 0.30/1.44 = ( ^ [A: $i] :
% 0.30/1.44 ? [Xx: $i] :
% 0.30/1.44 ( ( in @ Xx @ A )
% 0.30/1.44 & ( A
% 0.30/1.44 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(singletonprop_type,type,
% 0.30/1.44 singletonprop: $o ).
% 0.30/1.44
% 0.30/1.44 thf(singletonprop,definition,
% 0.30/1.44 ( singletonprop
% 0.30/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.44 ( ! [Xx: $i] :
% 0.30/1.44 ( ( in @ Xx @ A )
% 0.30/1.44 => ! [Xy: $i] :
% 0.30/1.44 ( ( in @ Xy @ A )
% 0.30/1.44 => ( ( Xphi @ Xx )
% 0.30/1.44 => ( ( Xphi @ Xy )
% 0.30/1.44 => ( Xx = Xy ) ) ) ) )
% 0.30/1.44 => ( ? [Xx: $i] :
% 0.30/1.44 ( ( in @ Xx @ A )
% 0.30/1.44 & ( Xphi @ Xx ) )
% 0.30/1.44 => ( singleton
% 0.30/1.44 @ ( dsetconstr @ A
% 0.30/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(ex1_type,type,
% 0.30/1.44 ex1: $i > ( $i > $o ) > $o ).
% 0.30/1.44
% 0.30/1.44 thf(ex1,definition,
% 0.30/1.44 ( ex1
% 0.30/1.44 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.30/1.44 ( singleton
% 0.30/1.44 @ ( dsetconstr @ A
% 0.30/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(ex1E1_type,type,
% 0.30/1.44 ex1E1: $o ).
% 0.30/1.44
% 0.30/1.44 thf(ex1E1,definition,
% 0.30/1.44 ( ex1E1
% 0.30/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.44 ( ( ex1 @ A
% 0.30/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.44 => ? [Xx: $i] :
% 0.30/1.44 ( ( in @ Xx @ A )
% 0.30/1.44 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(ex1I_type,type,
% 0.30/1.44 ex1I: $o ).
% 0.30/1.44
% 0.30/1.44 thf(ex1I,definition,
% 0.30/1.44 ( ex1I
% 0.30/1.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/1.44 ( ( in @ Xx @ A )
% 0.30/1.44 => ( ( Xphi @ Xx )
% 0.30/1.44 => ( ! [Xy: $i] :
% 0.30/1.44 ( ( in @ Xy @ A )
% 0.30/1.44 => ( ( Xphi @ Xy )
% 0.30/1.44 => ( Xy = Xx ) ) )
% 0.30/1.44 => ( ex1 @ A
% 0.30/1.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(ex1I2_type,type,
% 0.30/1.44 ex1I2: $o ).
% 0.30/1.44
% 0.30/1.44 thf(ex1I2,definition,
% 0.30/1.44 ( ex1I2
% 0.30/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.44 ( ! [Xx: $i] :
% 0.30/1.44 ( ( in @ Xx @ A )
% 0.30/1.44 => ! [Xy: $i] :
% 0.30/1.44 ( ( in @ Xy @ A )
% 0.30/1.44 => ( ( Xphi @ Xx )
% 0.30/1.44 => ( ( Xphi @ Xy )
% 0.30/1.44 => ( Xx = Xy ) ) ) ) )
% 0.30/1.44 => ( ? [Xx: $i] :
% 0.30/1.44 ( ( in @ Xx @ A )
% 0.30/1.44 & ( Xphi @ Xx ) )
% 0.30/1.44 => ( ex1 @ A
% 0.30/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(singletonsuniq_type,type,
% 0.30/1.44 singletonsuniq: $o ).
% 0.30/1.44
% 0.30/1.44 thf(singletonsuniq,definition,
% 0.30/1.44 ( singletonsuniq
% 0.30/1.44 = ( ! [Xx: $i,Xy: $i] :
% 0.30/1.44 ( ( ( setadjoin @ Xx @ emptyset )
% 0.30/1.44 = ( setadjoin @ Xy @ emptyset ) )
% 0.30/1.44 => ( Xx = Xy ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(atmost1p_type,type,
% 0.30/1.44 atmost1p: $i > $o ).
% 0.30/1.44
% 0.30/1.44 thf(atleast2p_type,type,
% 0.30/1.44 atleast2p: $i > $o ).
% 0.30/1.44
% 0.30/1.44 thf(atmost2p_type,type,
% 0.30/1.44 atmost2p: $i > $o ).
% 0.30/1.44
% 0.30/1.44 thf(upairsetp_type,type,
% 0.30/1.44 upairsetp: $i > $o ).
% 0.30/1.44
% 0.30/1.44 thf(setukpairinjL1_type,type,
% 0.30/1.44 setukpairinjL1: $o ).
% 0.30/1.44
% 0.30/1.44 thf(setukpairinjL1,definition,
% 0.30/1.44 ( setukpairinjL1
% 0.30/1.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/1.44 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.30/1.44 => ( Xx = Xz ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(kfstsingleton_type,type,
% 0.30/1.44 kfstsingleton: $o ).
% 0.30/1.44
% 0.30/1.44 thf(kfstsingleton,definition,
% 0.30/1.44 ( kfstsingleton
% 0.30/1.44 = ( ! [Xu: $i] :
% 0.30/1.44 ( ( iskpair @ Xu )
% 0.30/1.44 => ( singleton
% 0.30/1.44 @ ( dsetconstr @ ( setunion @ Xu )
% 0.30/1.44 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(theprop_type,type,
% 0.30/1.44 theprop: $o ).
% 0.30/1.44
% 0.30/1.44 thf(theprop,definition,
% 0.30/1.44 ( theprop
% 0.30/1.44 = ( ! [X: $i] :
% 0.30/1.44 ( ( singleton @ X )
% 0.30/1.44 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(kfst_type,type,
% 0.30/1.44 kfst: $i > $i ).
% 0.30/1.44
% 0.30/1.44 thf(kfstpairEq_type,type,
% 0.30/1.44 kfstpairEq: $o ).
% 0.30/1.44
% 0.30/1.44 thf(kfstpairEq,definition,
% 0.30/1.44 ( kfstpairEq
% 0.30/1.44 = ( ! [Xx: $i,Xy: $i] :
% 0.30/1.44 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.30/1.44 = Xx ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(cartprodfstin_type,type,
% 0.30/1.44 cartprodfstin: $o ).
% 0.30/1.44
% 0.30/1.44 thf(cartprodfstin,definition,
% 0.30/1.44 ( cartprodfstin
% 0.30/1.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/1.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/1.44 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(setukpairinjL2_type,type,
% 0.30/1.44 setukpairinjL2: $o ).
% 0.30/1.44
% 0.30/1.44 thf(setukpairinjL2,definition,
% 0.30/1.44 ( setukpairinjL2
% 0.30/1.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/1.44 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/1.44 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/1.44 => ( Xx = Xz ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(setukpairinjL_type,type,
% 0.30/1.44 setukpairinjL: $o ).
% 0.30/1.44
% 0.30/1.44 thf(setukpairinjL,definition,
% 0.30/1.44 ( setukpairinjL
% 0.30/1.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/1.44 ( ( ( kpair @ Xx @ Xy )
% 0.30/1.44 = ( kpair @ Xz @ Xu ) )
% 0.30/1.44 => ( Xx = Xz ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(setukpairinjR11_type,type,
% 0.30/1.44 setukpairinjR11: $o ).
% 0.30/1.44
% 0.30/1.44 thf(setukpairinjR11,definition,
% 0.30/1.44 ( setukpairinjR11
% 0.30/1.44 = ( ! [Xx: $i,Xy: $i] :
% 0.30/1.44 ( ( Xx = Xy )
% 0.30/1.44 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/1.44 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(setukpairinjR12_type,type,
% 0.30/1.44 setukpairinjR12: $o ).
% 0.30/1.44
% 0.30/1.44 thf(setukpairinjR12,definition,
% 0.30/1.44 ( setukpairinjR12
% 0.30/1.44 = ( ! [Xx: $i,Xy: $i] :
% 0.30/1.44 ( ( Xx = Xy )
% 0.30/1.44 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/1.44 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(setukpairinjR1_type,type,
% 0.30/1.44 setukpairinjR1: $o ).
% 0.30/1.44
% 0.30/1.44 thf(setukpairinjR1,definition,
% 0.30/1.44 ( setukpairinjR1
% 0.30/1.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/1.44 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/1.44 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/1.44 => ( ( Xz = Xu )
% 0.30/1.44 => ( Xy = Xu ) ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(upairequniteq_type,type,
% 0.30/1.44 upairequniteq: $o ).
% 0.30/1.44
% 0.30/1.44 thf(upairequniteq,definition,
% 0.30/1.44 ( upairequniteq
% 0.30/1.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/1.44 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/1.44 = ( setadjoin @ Xz @ emptyset ) )
% 0.30/1.44 => ( Xx = Xy ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(setukpairinjR2_type,type,
% 0.30/1.44 setukpairinjR2: $o ).
% 0.30/1.44
% 0.30/1.44 thf(setukpairinjR2,definition,
% 0.30/1.44 ( setukpairinjR2
% 0.30/1.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/1.44 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/1.44 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/1.44 => ( Xy = Xu ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(setukpairinjR_type,type,
% 0.30/1.44 setukpairinjR: $o ).
% 0.30/1.44
% 0.30/1.44 thf(setukpairinjR,definition,
% 0.30/1.44 ( setukpairinjR
% 0.30/1.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/1.44 ( ( ( kpair @ Xx @ Xy )
% 0.30/1.44 = ( kpair @ Xz @ Xu ) )
% 0.30/1.44 => ( Xy = Xu ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(ksndsingleton_type,type,
% 0.30/1.44 ksndsingleton: $o ).
% 0.30/1.44
% 0.30/1.44 thf(ksndsingleton,definition,
% 0.30/1.44 ( ksndsingleton
% 0.30/1.44 = ( ! [Xu: $i] :
% 0.30/1.44 ( ( iskpair @ Xu )
% 0.30/1.44 => ( singleton
% 0.30/1.44 @ ( dsetconstr @ ( setunion @ Xu )
% 0.30/1.44 @ ^ [Xx: $i] :
% 0.30/1.44 ( Xu
% 0.30/1.44 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(ksnd_type,type,
% 0.30/1.44 ksnd: $i > $i ).
% 0.30/1.44
% 0.30/1.44 thf(ksndpairEq_type,type,
% 0.30/1.44 ksndpairEq: $o ).
% 0.30/1.44
% 0.30/1.44 thf(ksndpairEq,definition,
% 0.30/1.44 ( ksndpairEq
% 0.30/1.44 = ( ! [Xx: $i,Xy: $i] :
% 0.30/1.44 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.30/1.44 = Xy ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(kpairsurjEq_type,type,
% 0.30/1.44 kpairsurjEq: $o ).
% 0.30/1.44
% 0.30/1.44 thf(kpairsurjEq,definition,
% 0.30/1.44 ( kpairsurjEq
% 0.30/1.44 = ( ! [Xu: $i] :
% 0.30/1.44 ( ( iskpair @ Xu )
% 0.30/1.44 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.30/1.44 = Xu ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(cartprodsndin_type,type,
% 0.30/1.44 cartprodsndin: $o ).
% 0.30/1.44
% 0.30/1.44 thf(cartprodsndin,definition,
% 0.30/1.44 ( cartprodsndin
% 0.30/1.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/1.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/1.44 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(cartprodpairmemEL_type,type,
% 0.30/1.44 cartprodpairmemEL: $o ).
% 0.30/1.44
% 0.30/1.44 thf(cartprodpairmemEL,definition,
% 0.30/1.44 ( cartprodpairmemEL
% 0.30/1.44 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.30/1.44 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.30/1.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(cartprodpairmemER_type,type,
% 0.30/1.44 cartprodpairmemER: $o ).
% 0.30/1.44
% 0.30/1.44 thf(cartprodpairmemER,definition,
% 0.30/1.44 ( cartprodpairmemER
% 0.30/1.44 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.30/1.44 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.30/1.44 => ( in @ Xy @ B ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(cartprodmempaircEq_type,type,
% 0.30/1.44 cartprodmempaircEq: $o ).
% 0.30/1.44
% 0.30/1.44 thf(cartprodmempaircEq,definition,
% 0.30/1.44 ( cartprodmempaircEq
% 0.30/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.44 ( ( in @ Xx @ A )
% 0.30/1.44 => ! [Xy: $i] :
% 0.30/1.44 ( ( in @ Xy @ B )
% 0.30/1.44 => ( ( kpair @ Xx @ Xy )
% 0.30/1.44 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(cartprodfstpairEq_type,type,
% 0.30/1.44 cartprodfstpairEq: $o ).
% 0.30/1.44
% 0.30/1.44 thf(cartprodfstpairEq,definition,
% 0.30/1.44 ( cartprodfstpairEq
% 0.30/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.44 ( ( in @ Xx @ A )
% 0.30/1.44 => ! [Xy: $i] :
% 0.30/1.44 ( ( in @ Xy @ B )
% 0.30/1.44 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.30/1.44 = Xx ) ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(cartprodsndpairEq_type,type,
% 0.30/1.44 cartprodsndpairEq: $o ).
% 0.30/1.44
% 0.30/1.44 thf(cartprodsndpairEq,definition,
% 0.30/1.44 ( cartprodsndpairEq
% 0.30/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.44 ( ( in @ Xx @ A )
% 0.30/1.44 => ! [Xy: $i] :
% 0.30/1.44 ( ( in @ Xy @ B )
% 0.30/1.44 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.30/1.44 = Xy ) ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(cartprodpairsurjEq_type,type,
% 0.30/1.44 cartprodpairsurjEq: $o ).
% 0.30/1.44
% 0.30/1.44 thf(cartprodpairsurjEq,definition,
% 0.30/1.44 ( cartprodpairsurjEq
% 0.30/1.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/1.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/1.44 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.30/1.44 = Xu ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(breln_type,type,
% 0.30/1.44 breln: $i > $i > $i > $o ).
% 0.30/1.44
% 0.30/1.44 thf(breln,definition,
% 0.30/1.44 ( breln
% 0.30/1.44 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(dpsetconstr_type,type,
% 0.30/1.44 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.30/1.44
% 0.30/1.44 thf(dpsetconstr,definition,
% 0.30/1.44 ( dpsetconstr
% 0.30/1.44 = ( ^ [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.30/1.44 ( dsetconstr @ ( cartprod @ A @ B )
% 0.30/1.44 @ ^ [Xu: $i] :
% 0.30/1.44 ? [Xx: $i] :
% 0.30/1.44 ( ( in @ Xx @ A )
% 0.30/1.44 & ? [Xy: $i] :
% 0.30/1.44 ( ( in @ Xy @ B )
% 0.30/1.44 & ( Xphi @ Xx @ Xy )
% 0.30/1.44 & ( Xu
% 0.30/1.44 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(dpsetconstrI_type,type,
% 0.30/1.44 dpsetconstrI: $o ).
% 0.30/1.44
% 0.30/1.44 thf(dpsetconstrI,definition,
% 0.30/1.44 ( dpsetconstrI
% 0.30/1.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.30/1.44 ( ( in @ Xx @ A )
% 0.30/1.44 => ! [Xy: $i] :
% 0.30/1.44 ( ( in @ Xy @ B )
% 0.30/1.44 => ( ( Xphi @ Xx @ Xy )
% 0.30/1.44 => ( in @ ( kpair @ Xx @ Xy )
% 0.30/1.44 @ ( dpsetconstr @ A @ B
% 0.30/1.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.30/1.44
% 0.30/1.44 thf(dpsetconstrSub,conjecture,
% 0.30/1.44 ( setextAx
% 0.30/1.44 => ( emptysetAx
% 0.30/1.44 => ( setadjoinAx
% 0.30/1.44 => ( powersetAx
% 0.30/1.44 => ( setunionAx
% 0.30/1.44 => ( omega0Ax
% 0.30/1.44 => ( omegaSAx
% 0.30/1.44 => ( omegaIndAx
% 0.30/1.44 => ( replAx
% 0.30/1.44 => ( foundationAx
% 0.30/1.44 => ( wellorderingAx
% 0.30/1.44 => ( descrp
% 0.30/1.44 => ( dsetconstrI
% 0.30/1.44 => ( dsetconstrEL
% 0.30/1.44 => ( dsetconstrER
% 0.30/1.44 => ( exuE1
% 0.30/1.44 => ( prop2setE
% 0.30/1.44 => ( emptysetE
% 0.30/1.44 => ( emptysetimpfalse
% 0.30/1.44 => ( notinemptyset
% 0.30/1.44 => ( exuE3e
% 0.30/1.44 => ( setext
% 0.30/1.44 => ( emptyI
% 0.30/1.44 => ( noeltsimpempty
% 0.30/1.44 => ( setbeta
% 0.30/1.44 => ( nonemptyE1
% 0.30/1.44 => ( nonemptyI
% 0.30/1.44 => ( nonemptyI1
% 0.30/1.44 => ( setadjoinIL
% 0.30/1.44 => ( emptyinunitempty
% 0.30/1.44 => ( setadjoinIR
% 0.30/1.44 => ( setadjoinE
% 0.30/1.44 => ( setadjoinOr
% 0.30/1.44 => ( setoftrueEq
% 0.30/1.44 => ( powersetI
% 0.30/1.44 => ( emptyinPowerset
% 0.30/1.44 => ( emptyInPowerset
% 0.30/1.44 => ( powersetE
% 0.30/1.44 => ( setunionI
% 0.30/1.44 => ( setunionE
% 0.30/1.44 => ( subPowSU
% 0.30/1.44 => ( exuE2
% 0.30/1.44 => ( nonemptyImpWitness
% 0.30/1.44 => ( uniqinunit
% 0.30/1.44 => ( notinsingleton
% 0.30/1.44 => ( eqinunit
% 0.30/1.44 => ( singletonsswitch
% 0.30/1.44 => ( upairsetE
% 0.30/1.44 => ( upairsetIL
% 0.30/1.44 => ( upairsetIR
% 0.30/1.44 => ( emptyE1
% 0.30/1.44 => ( vacuousDall
% 0.30/1.44 => ( quantDeMorgan1
% 0.30/1.44 => ( quantDeMorgan2
% 0.30/1.44 => ( quantDeMorgan3
% 0.30/1.44 => ( quantDeMorgan4
% 0.30/1.44 => ( prop2setI
% 0.30/1.44 => ( prop2set2propI
% 0.30/1.44 => ( notdexE
% 0.30/1.44 => ( notdallE
% 0.30/1.44 => ( exuI1
% 0.30/1.44 => ( exuI3
% 0.30/1.44 => ( exuI2
% 0.30/1.44 => ( inCongP
% 0.30/1.44 => ( in__Cong
% 0.30/1.44 => ( exuE3u
% 0.30/1.44 => ( exu__Cong
% 0.30/1.44 => ( emptyset__Cong
% 0.30/1.44 => ( setadjoin__Cong
% 0.30/1.44 => ( powerset__Cong
% 0.30/1.44 => ( setunion__Cong
% 0.30/1.44 => ( omega__Cong
% 0.30/1.44 => ( exuEu
% 0.30/1.44 => ( descr__Cong
% 0.30/1.44 => ( dsetconstr__Cong
% 0.30/1.44 => ( subsetI1
% 0.30/1.44 => ( eqimpsubset2
% 0.30/1.44 => ( eqimpsubset1
% 0.30/1.44 => ( subsetI2
% 0.30/1.44 => ( emptysetsubset
% 0.30/1.44 => ( subsetE
% 0.30/1.44 => ( subsetE2
% 0.30/1.44 => ( notsubsetI
% 0.30/1.44 => ( notequalI1
% 0.30/1.44 => ( notequalI2
% 0.30/1.44 => ( subsetRefl
% 0.30/1.44 => ( subsetTrans
% 0.30/1.44 => ( setadjoinSub
% 0.30/1.44 => ( setadjoinSub2
% 0.30/1.44 => ( subset2powerset
% 0.30/1.44 => ( setextsub
% 0.30/1.44 => ( subsetemptysetimpeq
% 0.30/1.44 => ( powersetI1
% 0.30/1.44 => ( powersetE1
% 0.30/1.44 => ( inPowerset
% 0.30/1.44 => ( powersetsubset
% 0.30/1.44 => ( sepInPowerset
% 0.30/1.44 => ( sepSubset
% 0.30/1.44 => ( binunionIL
% 0.30/1.44 => ( upairset2IR
% 0.30/1.44 => ( binunionIR
% 0.30/1.44 => ( binunionEcases
% 0.30/1.44 => ( binunionE
% 0.30/1.44 => ( binunionLsub
% 0.30/1.44 => ( binunionRsub
% 0.30/1.44 => ( binintersectI
% 0.30/1.44 => ( binintersectSubset5
% 0.30/1.44 => ( binintersectEL
% 0.30/1.44 => ( binintersectLsub
% 0.30/1.44 => ( binintersectSubset2
% 0.30/1.44 => ( binintersectSubset3
% 0.30/1.44 => ( binintersectER
% 0.30/1.44 => ( disjointsetsI1
% 0.30/1.44 => ( binintersectRsub
% 0.30/1.44 => ( binintersectSubset4
% 0.30/1.44 => ( binintersectSubset1
% 0.30/1.44 => ( bs114d
% 0.30/1.44 => ( setminusI
% 0.30/1.44 => ( setminusEL
% 0.30/1.44 => ( setminusER
% 0.30/1.44 => ( setminusSubset2
% 0.30/1.44 => ( setminusERneg
% 0.30/1.44 => ( setminusELneg
% 0.30/1.44 => ( setminusILneg
% 0.30/1.44 => ( setminusIRneg
% 0.30/1.44 => ( setminusLsub
% 0.30/1.44 => ( setminusSubset1
% 0.30/1.44 => ( symdiffE
% 0.30/1.44 => ( symdiffI1
% 0.30/1.44 => ( symdiffI2
% 0.30/1.44 => ( symdiffIneg1
% 0.30/1.44 => ( symdiffIneg2
% 0.30/1.44 => ( secondinupair
% 0.30/1.44 => ( setukpairIL
% 0.30/1.44 => ( setukpairIR
% 0.30/1.44 => ( kpairiskpair
% 0.30/1.44 => ( kpairp
% 0.30/1.44 => ( singletonsubset
% 0.30/1.44 => ( singletoninpowerset
% 0.30/1.44 => ( singletoninpowunion
% 0.30/1.44 => ( upairset2E
% 0.30/1.44 => ( upairsubunion
% 0.30/1.44 => ( upairinpowunion
% 0.30/1.44 => ( ubforcartprodlem1
% 0.30/1.44 => ( ubforcartprodlem2
% 0.30/1.44 => ( ubforcartprodlem3
% 0.30/1.44 => ( cartprodpairin
% 0.30/1.44 => ( cartprodmempair1
% 0.30/1.44 => ( cartprodmempair
% 0.30/1.44 => ( setunionE2
% 0.30/1.44 => ( setunionsingleton1
% 0.30/1.44 => ( setunionsingleton2
% 0.30/1.44 => ( setunionsingleton
% 0.30/1.44 => ( singletonprop
% 0.30/1.44 => ( ex1E1
% 0.30/1.44 => ( ex1I
% 0.30/1.44 => ( ex1I2
% 0.30/1.44 => ( singletonsuniq
% 0.30/1.44 => ( setukpairinjL1
% 0.30/1.44 => ( kfstsingleton
% 0.30/1.44 => ( theprop
% 0.30/1.44 => ( kfstpairEq
% 0.30/1.44 => ( cartprodfstin
% 0.30/1.44 => ( setukpairinjL2
% 0.30/1.44 => ( setukpairinjL
% 0.30/1.44 => ( setukpairinjR11
% 0.30/1.44 => ( setukpairinjR12
% 0.30/1.44 => ( setukpairinjR1
% 0.30/1.44 => ( upairequniteq
% 0.30/1.44 => ( setukpairinjR2
% 0.30/1.44 => ( setukpairinjR
% 0.30/1.44 => ( ksndsingleton
% 0.30/1.44 => ( ksndpairEq
% 0.30/1.44 => ( kpairsurjEq
% 0.30/1.44 => ( cartprodsndin
% 0.30/1.44 => ( cartprodpairmemEL
% 0.30/1.44 => ( cartprodpairmemER
% 0.30/1.44 => ( cartprodmempaircEq
% 0.30/1.44 => ( cartprodfstpairEq
% 0.30/1.44 => ( cartprodsndpairEq
% 0.30/1.44 => ( cartprodpairsurjEq
% 0.30/1.44 => ( dpsetconstrI
% 0.30/1.44 => ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.30/1.44 ( subset
% 0.30/1.44 @ ( dpsetconstr @ A @ B
% 0.30/1.44 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.30/1.44 @ ( cartprod @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/1.44 ------------------------
% 0.30/1.46 % (31361)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_31262 for (2999ds/183Mi)
% 0.30/1.46 % (31362)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_31262 for (2999ds/4Mi)
% 0.30/1.46 % (31363)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_31262 for (2999ds/27Mi)
% 0.30/1.46 % (31362)Instruction limit reached!
% 0.30/1.46 % (31362)------------------------------
% 0.30/1.46 % (31362)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.46 % (31362)Termination reason: Unknown
% 0.30/1.46 % (31362)Termination phase: shuffling
% 0.30/1.46
% 0.30/1.46 % (31362)Memory used [KB]: 1407
% 0.30/1.46 % (31362)Time elapsed: 0.003 s
% 0.30/1.46 % (31362)Instructions burned: 7 (million)
% 0.30/1.46 % (31362)------------------------------
% 0.30/1.46 % (31362)------------------------------
% 0.30/1.47 % (31364)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_31262 for (2999ds/2Mi)
% 0.30/1.47 % (31366)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_31262 for (2999ds/2Mi)
% 0.30/1.47 % (31367)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_31262 for (2999ds/275Mi)
% 0.30/1.47 % (31368)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_31262 for (2999ds/18Mi)
% 0.30/1.47 % (31363)Instruction limit reached!
% 0.30/1.47 % (31363)------------------------------
% 0.30/1.47 % (31363)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.47 % (31363)Termination reason: Unknown
% 0.30/1.47 % (31363)Termination phase: shuffling
% 0.30/1.47
% 0.30/1.47 % (31363)Memory used [KB]: 1918
% 0.30/1.47 % (31363)Time elapsed: 0.009 s
% 0.30/1.47 % (31363)Instructions burned: 27 (million)
% 0.30/1.47 % (31363)------------------------------
% 0.30/1.47 % (31363)------------------------------
% 0.30/1.47 % (31364)Instruction limit reached!
% 0.30/1.47 % (31364)------------------------------
% 0.30/1.47 % (31364)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.47 % (31364)Termination reason: Unknown
% 0.30/1.47 % (31364)Termination phase: shuffling
% 0.30/1.47 % (31366)Instruction limit reached!
% 0.30/1.47 % (31366)------------------------------
% 0.30/1.47 % (31366)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.47 % (31366)Termination reason: Unknown
% 0.30/1.47 % (31366)Termination phase: shuffling
% 0.30/1.47
% 0.30/1.47 % (31366)Memory used [KB]: 1279
% 0.30/1.47 % (31366)Time elapsed: 0.004 s
% 0.30/1.47 % (31366)Instructions burned: 3 (million)
% 0.30/1.47 % (31366)------------------------------
% 0.30/1.47 % (31366)------------------------------
% 0.30/1.47
% 0.30/1.47 % (31364)Memory used [KB]: 1279
% 0.30/1.47 % (31364)Time elapsed: 0.004 s
% 0.30/1.47 % (31364)Instructions burned: 3 (million)
% 0.30/1.47 % (31364)------------------------------
% 0.30/1.47 % (31364)------------------------------
% 0.30/1.48 % (31369)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_31262 for (2999ds/3Mi)
% 0.30/1.48 % (31369)Instruction limit reached!
% 0.30/1.48 % (31369)------------------------------
% 0.30/1.48 % (31369)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.48 % (31369)Termination reason: Unknown
% 0.30/1.48 % (31369)Termination phase: shuffling
% 0.30/1.48
% 0.30/1.48 % (31369)Memory used [KB]: 1407
% 0.30/1.48 % (31369)Time elapsed: 0.003 s
% 0.30/1.48 % (31369)Instructions burned: 4 (million)
% 0.30/1.48 % (31369)------------------------------
% 0.30/1.48 % (31369)------------------------------
% 0.30/1.48 % (31370)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_31262 for (2999ds/37Mi)
% 0.30/1.48 % (31368)Instruction limit reached!
% 0.30/1.48 % (31368)------------------------------
% 0.30/1.48 % (31368)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.48 % (31368)Termination reason: Unknown
% 0.30/1.48 % (31368)Termination phase: Property scanning
% 0.30/1.48
% 0.30/1.48 % (31368)Memory used [KB]: 1663
% 0.30/1.48 % (31368)Time elapsed: 0.013 s
% 0.30/1.48 % (31368)Instructions burned: 18 (million)
% 0.30/1.48 % (31368)------------------------------
% 0.30/1.48 % (31368)------------------------------
% 0.30/1.49 % (31373)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_31262 for (2999ds/1041Mi)
% 0.30/1.49 % (31372)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_31262 for (2999ds/3Mi)
% 0.30/1.49 % (31372)Instruction limit reached!
% 0.30/1.49 % (31372)------------------------------
% 0.30/1.49 % (31372)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.49 % (31372)Termination reason: Unknown
% 0.30/1.49 % (31372)Termination phase: shuffling
% 0.30/1.49
% 0.30/1.49 % (31372)Memory used [KB]: 1279
% 0.30/1.49 % (31372)Time elapsed: 0.003 s
% 0.30/1.49 % (31372)Instructions burned: 4 (million)
% 0.30/1.49 % (31372)------------------------------
% 0.30/1.49 % (31372)------------------------------
% 0.30/1.49 % (31371)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_31262 for (2999ds/15Mi)
% 0.30/1.49 % (31370)Instruction limit reached!
% 0.30/1.49 % (31370)------------------------------
% 0.30/1.49 % (31370)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.49 % (31370)Termination reason: Unknown
% 0.30/1.49 % (31370)Termination phase: Property scanning
% 0.30/1.49
% 0.30/1.49 % (31370)Memory used [KB]: 1791
% 0.30/1.49 % (31370)Time elapsed: 0.012 s
% 0.30/1.49 % (31370)Instructions burned: 37 (million)
% 0.30/1.49 % (31370)------------------------------
% 0.30/1.49 % (31370)------------------------------
% 2.55/1.50 % (31375)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_31262 for (2999ds/16Mi)
% 2.55/1.50 % (31371)Instruction limit reached!
% 2.55/1.50 % (31371)------------------------------
% 2.55/1.50 % (31371)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.50 % (31371)Termination reason: Unknown
% 2.55/1.50 % (31371)Termination phase: shuffling
% 2.55/1.50
% 2.55/1.50 % (31371)Memory used [KB]: 1663
% 2.55/1.50 % (31371)Time elapsed: 0.011 s
% 2.55/1.50 % (31371)Instructions burned: 16 (million)
% 2.55/1.50 % (31371)------------------------------
% 2.55/1.50 % (31371)------------------------------
% 2.55/1.50 % (31374)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_31262 for (2999ds/7Mi)
% 2.55/1.50 % (31376)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_31262 for (2999ds/3Mi)
% 2.55/1.50 % (31376)Instruction limit reached!
% 2.55/1.50 % (31376)------------------------------
% 2.55/1.50 % (31376)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.50 % (31376)Termination reason: Unknown
% 2.55/1.50 % (31376)Termination phase: shuffling
% 2.55/1.50
% 2.55/1.50 % (31376)Memory used [KB]: 1279
% 2.55/1.50 % (31376)Time elapsed: 0.002 s
% 2.55/1.50 % (31376)Instructions burned: 3 (million)
% 2.55/1.50 % (31376)------------------------------
% 2.55/1.50 % (31376)------------------------------
% 2.55/1.50 % (31374)Instruction limit reached!
% 2.55/1.50 % (31374)------------------------------
% 2.55/1.50 % (31374)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.50 % (31374)Termination reason: Unknown
% 2.55/1.50 % (31374)Termination phase: shuffling
% 2.55/1.50
% 2.55/1.50 % (31374)Memory used [KB]: 1407
% 2.55/1.50 % (31374)Time elapsed: 0.005 s
% 2.55/1.50 % (31374)Instructions burned: 7 (million)
% 2.55/1.50 % (31374)------------------------------
% 2.55/1.50 % (31374)------------------------------
% 2.55/1.50 % (31375)Instruction limit reached!
% 2.55/1.50 % (31375)------------------------------
% 2.55/1.50 % (31375)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.50 % (31375)Termination reason: Unknown
% 2.55/1.50 % (31375)Termination phase: shuffling
% 2.55/1.50
% 2.55/1.50 % (31375)Memory used [KB]: 1663
% 2.55/1.50 % (31375)Time elapsed: 0.006 s
% 2.55/1.50 % (31375)Instructions burned: 16 (million)
% 2.55/1.50 % (31375)------------------------------
% 2.55/1.50 % (31375)------------------------------
% 2.55/1.51 % (31361)Instruction limit reached!
% 2.55/1.51 % (31361)------------------------------
% 2.55/1.51 % (31361)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.51 % (31361)Termination reason: Unknown
% 2.55/1.51 % (31361)Termination phase: Saturation
% 2.55/1.51
% 2.55/1.51 % (31361)Memory used [KB]: 7419
% 2.55/1.51 % (31361)Time elapsed: 0.050 s
% 2.55/1.51 % (31361)Instructions burned: 186 (million)
% 2.55/1.51 % (31361)------------------------------
% 2.55/1.51 % (31361)------------------------------
% 2.55/1.51 % (31378)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_31262 for (2999ds/7Mi)
% 2.55/1.51 % (31378)Instruction limit reached!
% 2.55/1.51 % (31378)------------------------------
% 2.55/1.51 % (31378)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.51 % (31378)Termination reason: Unknown
% 2.55/1.51 % (31378)Termination phase: shuffling
% 2.55/1.51
% 2.55/1.51 % (31378)Memory used [KB]: 1407
% 2.55/1.51 % (31378)Time elapsed: 0.003 s
% 2.55/1.51 % (31378)Instructions burned: 8 (million)
% 2.55/1.51 % (31378)------------------------------
% 2.55/1.51 % (31378)------------------------------
% 2.55/1.51 % (31377)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_31262 for (2999ds/3Mi)
% 2.55/1.51 % (31380)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_31262 for (2999ds/4Mi)
% 2.55/1.51 % (31380)Instruction limit reached!
% 2.55/1.51 % (31380)------------------------------
% 2.55/1.51 % (31380)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.51 % (31380)Termination reason: Unknown
% 2.55/1.51 % (31380)Termination phase: shuffling
% 2.55/1.51
% 2.55/1.51 % (31380)Memory used [KB]: 1407
% 2.55/1.51 % (31380)Time elapsed: 0.002 s
% 2.55/1.51 % (31380)Instructions burned: 4 (million)
% 2.55/1.51 % (31380)------------------------------
% 2.55/1.51 % (31380)------------------------------
% 2.55/1.52 % (31377)Instruction limit reached!
% 2.55/1.52 % (31377)------------------------------
% 2.55/1.52 % (31377)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.52 % (31377)Termination reason: Unknown
% 2.55/1.52 % (31377)Termination phase: shuffling
% 2.55/1.52
% 2.55/1.52 % (31377)Memory used [KB]: 1407
% 2.55/1.52 % (31377)Time elapsed: 0.003 s
% 2.55/1.52 % (31377)Instructions burned: 5 (million)
% 2.55/1.52 % (31377)------------------------------
% 2.55/1.52 % (31377)------------------------------
% 2.55/1.52 % (31379)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_31262 for (2999ds/3Mi)
% 2.55/1.52 % (31379)Instruction limit reached!
% 2.55/1.52 % (31379)------------------------------
% 2.55/1.52 % (31379)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.52 % (31379)Termination reason: Unknown
% 2.55/1.52 % (31379)Termination phase: shuffling
% 2.55/1.52
% 2.55/1.52 % (31379)Memory used [KB]: 1279
% 2.55/1.52 % (31379)Time elapsed: 0.002 s
% 2.55/1.52 % (31379)Instructions burned: 4 (million)
% 2.55/1.52 % (31379)------------------------------
% 2.55/1.52 % (31379)------------------------------
% 2.55/1.52 % (31381)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_31262 for (2999ds/18Mi)
% 2.55/1.52 % (31382)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_31262 for (2999ds/710Mi)
% 2.55/1.52 % (31383)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_31262 for (2999ds/6Mi)
% 2.55/1.53 % (31381)Instruction limit reached!
% 2.55/1.53 % (31381)------------------------------
% 2.55/1.53 % (31381)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.53 % (31381)Termination reason: Unknown
% 2.55/1.53 % (31381)Termination phase: shuffling
% 2.55/1.53
% 2.55/1.53 % (31381)Memory used [KB]: 1663
% 2.55/1.53 % (31381)Time elapsed: 0.006 s
% 2.55/1.53 % (31381)Instructions burned: 19 (million)
% 2.55/1.53 % (31381)------------------------------
% 2.55/1.53 % (31381)------------------------------
% 2.55/1.53 % (31384)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_31262 for (2999ds/902Mi)
% 2.55/1.53 % (31383)Instruction limit reached!
% 2.55/1.53 % (31383)------------------------------
% 2.55/1.53 % (31383)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.53 % (31383)Termination reason: Unknown
% 2.55/1.53 % (31383)Termination phase: shuffling
% 2.55/1.53
% 2.55/1.53 % (31383)Memory used [KB]: 1407
% 2.55/1.53 % (31383)Time elapsed: 0.004 s
% 2.55/1.53 % (31383)Instructions burned: 8 (million)
% 2.55/1.53 % (31383)------------------------------
% 2.55/1.53 % (31383)------------------------------
% 2.55/1.53 % (31385)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_31262 for (2999ds/21Mi)
% 2.55/1.53 % (31386)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_31262 for (2999ds/5Mi)
% 2.55/1.54 % (31385)Instruction limit reached!
% 2.55/1.54 % (31385)------------------------------
% 2.55/1.54 % (31385)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.54 % (31385)Termination reason: Unknown
% 2.55/1.54 % (31385)Termination phase: shuffling
% 2.55/1.54
% 2.55/1.54 % (31385)Memory used [KB]: 1791
% 2.55/1.54 % (31385)Time elapsed: 0.008 s
% 2.55/1.54 % (31385)Instructions burned: 22 (million)
% 2.55/1.54 % (31385)------------------------------
% 2.55/1.54 % (31385)------------------------------
% 2.55/1.54 % (31386)Instruction limit reached!
% 2.55/1.54 % (31386)------------------------------
% 2.55/1.54 % (31386)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.54 % (31386)Termination reason: Unknown
% 2.55/1.54 % (31386)Termination phase: shuffling
% 2.55/1.54
% 2.55/1.54 % (31386)Memory used [KB]: 1407
% 2.55/1.54 % (31386)Time elapsed: 0.003 s
% 2.55/1.54 % (31386)Instructions burned: 8 (million)
% 2.55/1.54 % (31386)------------------------------
% 2.55/1.54 % (31386)------------------------------
% 2.55/1.54 % (31387)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_31262 for (2999ds/6Mi)
% 2.55/1.54 % (31387)Instruction limit reached!
% 2.55/1.54 % (31387)------------------------------
% 2.55/1.54 % (31387)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.54 % (31387)Termination reason: Unknown
% 2.55/1.54 % (31387)Termination phase: shuffling
% 2.55/1.54
% 2.55/1.54 % (31387)Memory used [KB]: 1407
% 2.55/1.54 % (31387)Time elapsed: 0.003 s
% 2.55/1.54 % (31387)Instructions burned: 8 (million)
% 2.55/1.54 % (31387)------------------------------
% 2.55/1.54 % (31387)------------------------------
% 2.55/1.55 % (31389)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_31262 for (2999ds/779Mi)
% 2.55/1.55 % (31388)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_31262 for (2999ds/377Mi)
% 2.55/1.55 % (31390)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_31262 for (2998ds/19Mi)
% 2.55/1.56 % (31390)Instruction limit reached!
% 2.55/1.56 % (31390)------------------------------
% 2.55/1.56 % (31390)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.56 % (31390)Termination reason: Unknown
% 2.55/1.56 % (31390)Termination phase: shuffling
% 2.55/1.56
% 2.55/1.56 % (31390)Memory used [KB]: 1663
% 2.55/1.56 % (31390)Time elapsed: 0.008 s
% 2.55/1.56 % (31390)Instructions burned: 21 (million)
% 2.55/1.56 % (31390)------------------------------
% 2.55/1.56 % (31390)------------------------------
% 2.55/1.57 % (31391)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_31262 for (2998ds/879Mi)
% 2.55/1.61 % (31367)Instruction limit reached!
% 2.55/1.61 % (31367)------------------------------
% 2.55/1.61 % (31367)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.61 % (31367)Termination reason: Unknown
% 2.55/1.61 % (31367)Termination phase: Saturation
% 2.55/1.61
% 2.55/1.61 % (31367)Memory used [KB]: 8443
% 2.55/1.61 % (31367)Time elapsed: 0.139 s
% 2.55/1.61 % (31367)Instructions burned: 277 (million)
% 2.55/1.61 % (31367)------------------------------
% 2.55/1.61 % (31367)------------------------------
% 2.55/1.62 % (31392)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_31262 for (2998ds/17Mi)
% 2.55/1.62 % (31392)Instruction limit reached!
% 2.55/1.62 % (31392)------------------------------
% 2.55/1.62 % (31392)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.62 % (31392)Termination reason: Unknown
% 2.55/1.62 % (31392)Termination phase: shuffling
% 2.55/1.62
% 2.55/1.62 % (31392)Memory used [KB]: 1791
% 2.55/1.62 % (31392)Time elapsed: 0.007 s
% 2.55/1.62 % (31392)Instructions burned: 20 (million)
% 2.55/1.62 % (31392)------------------------------
% 2.55/1.62 % (31392)------------------------------
% 2.55/1.63 % (31393)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_31262 for (2998ds/3Mi)
% 2.55/1.64 % (31393)Instruction limit reached!
% 2.55/1.64 % (31393)------------------------------
% 2.55/1.64 % (31393)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.64 % (31393)Termination reason: Unknown
% 2.55/1.64 % (31393)Termination phase: shuffling
% 2.55/1.64
% 2.55/1.64 % (31393)Memory used [KB]: 1279
% 2.55/1.64 % (31393)Time elapsed: 0.003 s
% 2.55/1.64 % (31393)Instructions burned: 4 (million)
% 2.55/1.64 % (31393)------------------------------
% 2.55/1.64 % (31393)------------------------------
% 2.55/1.64 % (31391)First to succeed.
% 2.55/1.64 % (31391)Refutation found. Thanks to Tanya!
% 2.55/1.64 % SZS status Theorem for DTF2THF_31262
% 2.55/1.64 % SZS output start Proof for DTF2THF_31262
% 2.55/1.64 thf(func_def_0, type, in: $i > $i > $o).
% 2.55/1.64 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.55/1.64 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.55/1.64 thf(func_def_8, type, powerset: $i > $i).
% 2.55/1.64 thf(func_def_10, type, setunion: $i > $i).
% 2.55/1.64 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.55/1.64 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.55/1.64 thf(func_def_26, type, prop2set: $o > $i).
% 2.55/1.64 thf(func_def_36, type, nonempty: $i > $o).
% 2.55/1.64 thf(func_def_69, type, set2prop: $i > $o).
% 2.55/1.64 thf(func_def_88, type, subset: $i > $i > $o).
% 2.55/1.64 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.55/1.64 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.55/1.64 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.55/1.64 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.55/1.64 thf(func_def_135, type, regular: $i > $o).
% 2.55/1.64 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.55/1.64 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.55/1.64 thf(func_def_153, type, iskpair: $i > $o).
% 2.55/1.64 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.55/1.64 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.55/1.64 thf(func_def_177, type, singleton: $i > $o).
% 2.55/1.64 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.55/1.64 thf(func_def_184, type, atmost1p: $i > $o).
% 2.55/1.64 thf(func_def_185, type, atleast2p: $i > $o).
% 2.55/1.64 thf(func_def_186, type, atmost2p: $i > $o).
% 2.55/1.64 thf(func_def_187, type, upairsetp: $i > $o).
% 2.55/1.64 thf(func_def_191, type, kfst: $i > $i).
% 2.55/1.64 thf(func_def_203, type, ksnd: $i > $i).
% 2.55/1.64 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.55/1.64 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.55/1.64 thf(func_def_233, type, sK0: $i > $i > $o).
% 2.55/1.64 thf(func_def_237, type, sK4: ($i > $o) > $i > $i).
% 2.55/1.64 thf(func_def_238, type, sK5: ($i > $o) > $i > $i).
% 2.55/1.64 thf(func_def_239, type, sK6: ($i > $o) > $i).
% 2.55/1.64 thf(func_def_240, type, sK7: $i > $i).
% 2.55/1.64 thf(func_def_241, type, sK8: $i > ($i > $o) > $i).
% 2.55/1.64 thf(func_def_242, type, sK9: $i > ($i > $o) > $i).
% 2.55/1.64 thf(func_def_243, type, sK10: ($i > $o) > $i > $i).
% 2.55/1.64 thf(func_def_244, type, sK11: $i > $i > $i).
% 2.55/1.64 thf(func_def_245, type, sK12: $i > $i).
% 2.55/1.64 thf(func_def_246, type, sK13: ($i > $o) > ($i > $o) > $i).
% 2.55/1.64 thf(func_def_247, type, sK14: ($i > $o) > ($i > $o) > $i).
% 2.55/1.64 thf(func_def_248, type, sK15: ($i > $o) > $i).
% 2.55/1.64 thf(func_def_249, type, sK16: ($i > $o) > $i).
% 2.55/1.64 thf(func_def_250, type, sK17: $i > $i).
% 2.55/1.64 thf(func_def_251, type, sK18: ($i > $o) > $i > $i).
% 2.55/1.64 thf(func_def_252, type, sK19: $i > $i > $i).
% 2.55/1.64 thf(func_def_253, type, sK20: $i > $i > $i).
% 2.55/1.64 thf(func_def_254, type, sK21: $i > $i > ($i > $o) > $i).
% 2.55/1.64 thf(func_def_255, type, sK22: $i > ($i > $o) > $i).
% 2.55/1.64 thf(func_def_256, type, sK23: $i > $i > $i).
% 2.55/1.64 thf(func_def_257, type, sK24: $i > ($i > $o) > $i).
% 2.55/1.64 thf(func_def_258, type, sK25: $i > $i).
% 2.55/1.64 thf(func_def_259, type, sK26: $i > $i > $i).
% 2.55/1.64 thf(func_def_260, type, sK27: $i > $i > $i).
% 2.55/1.64 thf(func_def_261, type, sK28: $i > $i > $i).
% 2.55/1.64 thf(func_def_262, type, sK29: $i > $i > $i > $i).
% 2.55/1.64 thf(func_def_263, type, sK30: $i > $i).
% 2.55/1.64 thf(func_def_264, type, sK31: ($i > $i > $o) > $i > $i).
% 2.55/1.64 thf(func_def_265, type, sK32: ($i > $i > $o) > $i > $i).
% 2.55/1.64 thf(func_def_266, type, sK33: $i > ($i > $i > $o) > $i > $i).
% 2.55/1.64 thf(func_def_267, type, sK34: ($i > $i > $o) > $i > $i).
% 2.55/1.64 thf(func_def_268, type, sK35: $i > ($i > $i > $o) > $i > $i).
% 2.55/1.64 thf(func_def_269, type, sK36: $i > $i > $i).
% 2.55/1.64 thf(func_def_270, type, sK37: ($i > $o) > $i).
% 2.55/1.64 thf(func_def_271, type, sK38: $i > $i > $i).
% 2.55/1.64 thf(func_def_272, type, sK39: $i > ($i > $o) > $i).
% 2.55/1.64 thf(func_def_273, type, sK40: ($i > $o) > $i).
% 2.55/1.64 thf(func_def_274, type, sK41: ($i > $o) > $i > ($i > $o) > $i > $i).
% 2.55/1.64 thf(func_def_275, type, sK42: $i > $i > ($i > $o) > ($i > $o) > $i).
% 2.55/1.64 thf(func_def_276, type, sK43: $i > $i > $i > $i).
% 2.55/1.64 thf(func_def_277, type, sK44: $i > $i > $i > $i).
% 2.55/1.64 thf(func_def_278, type, sK45: $i > $i > $i).
% 2.55/1.64 thf(func_def_279, type, sK46: $i > $i).
% 2.55/1.64 thf(func_def_280, type, sK47: $i > $i > $i).
% 2.55/1.64 thf(func_def_281, type, sK48: $i > $i > $i).
% 2.55/1.64 thf(func_def_282, type, sK49: $i > $i > $i).
% 2.55/1.64 thf(func_def_283, type, sK50: $i > $i > $i).
% 2.55/1.64 thf(func_def_286, type, sK51: $i > ($i > $o) > $i).
% 2.55/1.64 thf(func_def_287, type, sK52: ($i > $o) > ($i > $o) > $i).
% 2.55/1.64 thf(func_def_288, type, sK53: ($i > $o) > ($i > $o) > $i).
% 2.55/1.64 thf(func_def_289, type, sK54: ($i > $o) > $i > $i).
% 2.55/1.64 thf(func_def_290, type, sK55: $i > $o > $i > $i).
% 2.55/1.64 thf(func_def_291, type, sK56: ($i > $o) > $i > $i).
% 2.55/1.64 thf(func_def_292, type, sK57: ($i > $o) > $i > $i).
% 2.55/1.64 thf(func_def_293, type, sK58: $i > $i).
% 2.55/1.64 thf(f2814,plain,(
% 2.55/1.64 $false),
% 2.55/1.64 inference(subsumption_resolution,[status(thm)],[f1807,f2076])).
% 2.55/1.64 thf(f2076,plain,(
% 2.55/1.64 ( ! [X2 : $i > $o,X1 : $i] : (($true = (subset @ (dsetconstr @ X1 @ X2) @ X1))) )),
% 2.55/1.64 inference(beta_eta_normalization,[status(thm)],[f2075])).
% 2.55/1.64 thf(f2075,plain,(
% 2.55/1.64 ( ! [X2 : $i > $o,X1 : $i] : ((((^[Y0 : $i > $o]: (subset @ (dsetconstr @ X1 @ Y0) @ X1)) @ X2) = $true)) )),
% 2.55/1.64 inference(pi_clausification,[status(thm)],[f2074])).
% 2.55/1.64 thf(f2074,plain,(
% 2.55/1.64 ( ! [X1 : $i] : (($true = (!! @ ($i > $o) @ (^[Y0 : $i > $o]: (subset @ (dsetconstr @ X1 @ Y0) @ X1))))) )),
% 2.55/1.64 inference(beta_eta_normalization,[status(thm)],[f2073])).
% 2.55/1.64 thf(f2073,plain,(
% 2.55/1.64 ( ! [X1 : $i] : ((((^[Y0 : $i]: (!! @ ($i > $o) @ (^[Y1 : $i > $o]: (subset @ (dsetconstr @ Y0 @ Y1) @ Y0)))) @ X1) = $true)) )),
% 2.55/1.64 inference(pi_clausification,[status(thm)],[f2072])).
% 2.55/1.64 thf(f2072,plain,(
% 2.55/1.64 ($true = (!! @ $i @ (^[Y0 : $i]: (!! @ ($i > $o) @ (^[Y1 : $i > $o]: (subset @ (dsetconstr @ Y0 @ Y1) @ Y0))))))),
% 2.55/1.64 inference(beta_eta_normalization,[status(thm)],[f1065])).
% 2.55/1.64 thf(f1065,plain,(
% 2.55/1.64 ((!! @ $i @ (^[Y0 : $i]: (!! @ ($i > $o) @ (^[Y1 : $i > $o]: (subset @ (dsetconstr @ Y0 @ (^[Y2 : $i]: (Y1 @ Y2))) @ Y0))))) = $true)),
% 2.55/1.64 inference(definition_unfolding,[status(thm)],[f880,f701])).
% 2.55/1.64 thf(f701,plain,(
% 2.55/1.64 (sepSubset = $true)),
% 2.55/1.64 inference(cnf_transformation,[status(thm)],[f567])).
% 2.55/1.64 thf(f567,plain,(
% 2.55/1.64 (cartprodfstin = $true) & (noeltsimpempty = $true) & (symdiffI2 = $true) & (kfstpairEq = $true) & (setukpairinjL1 = $true) & (inPowerset = $true) & (setunionE = $true) & (subsetRefl = $true) & (setextsub = $true) & (eqimpsubset2 = $true) & (singletonprop = $true) & (descr__Cong = $true) & (ksndpairEq = $true) & (upairsetE = $true) & (setukpairinjR = $true) & (setukpairinjL = $true) & (upairsetIR = $true) & (exuE3u = $true) & (descrp = $true) & (notequalI2 = $true) & (dpsetconstrI = $true) & (upairequniteq = $true) & (setminusELneg = $true) & (prop2set2propI = $true) & (binintersectSubset2 = $true) & (upairsetIL = $true) & (kpairsurjEq = $true) & (replAx = $true) & (ubforcartprodlem3 = $true) & (setadjoinE = $true) & (bs114d = $true) & (emptyinPowerset = $true) & (binunionE = $true) & (nonemptyImpWitness = $true) & (subset2powerset = $true) & (setminusI = $true) & (powersetsubset = $true) & (exuI1 = $true) & (setext = $true) & (singletonsubset = $true) & (omegaSAx = $true) & (cartprodpairmemER = $true) & (dsetconstrEL = $true) & (setukpairIL = $true) & (notinemptyset = $true) & (setminusLsub = $true) & (powersetI = $true) & (setbeta = $true) & (emptyE1 = $true) & (cartprodmempaircEq = $true) & (binintersectEL = $true) & (exuE1 = $true) & (subsetI2 = $true) & (quantDeMorgan3 = $true) & (setadjoinSub = $true) & (prop2setI = $true) & (sepSubset = $true) & (omegaIndAx = $true) & (upairset2IR = $true) & (subsetTrans = $true) & (symdiffIneg1 = $true) & (singletonsswitch = $true) & (secondinupair = $true) & (cartprodpairin = $true) & (setminusSubset2 = $true) & (setoftrueEq = $true) & (notequalI1 = $true) & (powersetE1 = $true) & (in__Cong = $true) & (setadjoinIR = $true) & (setukpairinjR2 = $true) & (exu__Cong = $true) & (emptyset__Cong = $true) & (kpairiskpair = $true) & (binintersectSubset1 = $true) & (notdallE = $true) & (emptyI = $true) & (quantDeMorgan4 = $true) & (uniqinunit = $true) & (kfstsingleton = $true) & (emptyInPowerset = $true) & (notdexE = $true) & (eqimpsubset1 = $true) & (prop2setE = $true) & (subsetE = $true) & (disjointsetsI1 = $true) & (dsetconstrI = $true) & (ksndsingleton = $true) & (setukpairIR = $true) & (binunionEcases = $true) & (setukpairinjR12 = $true) & (exuI2 = $true) & (exuE3e = $true) & (ubforcartprodlem1 = $true) & (cartprodfstpairEq = $true) & ($true != (subset @ (dpsetconstr @ sK1 @ sK2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK0 @ Y0 @ Y1))))) @ (cartprod @ sK1 @ sK2))) & (cartprodmempair1 = $true) & (subsetI1 = $true) & (eqinunit = $true) & (ex1E1 = $true) & (emptysetsubset = $true) & (kpairp = $true) & (setadjoinSub2 = $true) & (cartprodsndpairEq = $true) & (powersetAx = $true) & (setunionsingleton2 = $true) & (singletoninpowerset = $true) & (powerset__Cong = $true) & (setadjoin__Cong = $true) & (setunionE2 = $true) & (dsetconstrER = $true) & (binunionIR = $true) & (quantDeMorgan2 = $true) & (notsubsetI = $true) & (binintersectRsub = $true) & (emptysetimpfalse = $true) & (setadjoinIL = $true) & (exuI3 = $true) & (singletoninpowunion = $true) & (setextAx = $true) & (vacuousDall = $true) & (setunionAx = $true) & (binintersectI = $true) & (setminusILneg = $true) & (nonemptyI1 = $true) & (subPowSU = $true) & (setadjoinAx = $true) & (symdiffI1 = $true) & (emptysetE = $true) & (foundationAx = $true) & (nonemptyE1 = $true) & (cartprodpairsurjEq = $true) & (setminusERneg = $true) & (sepInPowerset = $true) & (exuEu = $true) & (singletonsuniq = $true) & (cartprodmempair = $true) & (powersetE = $true) & (setunionsingleton = $true) & (ubforcartprodlem2 = $true) & (setminusIRneg = $true) & (setukpairinjL2 = $true) & (symdiffIneg2 = $true) & (wellorderingAx = $true) & (binintersectSubset5 = $true) & (symdiffE = $true) & (quantDeMorgan1 = $true) & (upairsubunion = $true) & (theprop = $true) & (binunionRsub = $true) & (upairset2E = $true) & (setunionI = $true) & (upairinpowunion = $true) & (setadjoinOr = $true) & (exuE2 = $true) & (setunion__Cong = $true) & (emptyinunitempty = $true) & (binintersectER = $true) & (binintersectLsub = $true) & (subsetE2 = $true) & (dsetconstr__Cong = $true) & (cartprodpairmemEL = $true) & (subsetemptysetimpeq = $true) & (binunionIL = $true) & (cartprodsndin = $true) & (inCongP = $true) & (setminusSubset1 = $true) & (ex1I2 = $true) & (setunionsingleton1 = $true) & (powersetI1 = $true) & (omega__Cong = $true) & (setminusEL = $true) & (binunionLsub = $true) & (binintersectSubset3 = $true) & (nonemptyI = $true) & (emptysetAx = $true) & (omega0Ax = $true) & (setukpairinjR11 = $true) & (setukpairinjR1 = $true) & (notinsingleton = $true) & (ex1I = $true) & (setminusER = $true) & (binintersectSubset4 = $true)),
% 2.55/1.64 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f565,f566])).
% 2.55/1.64 thf(f566,plain,(
% 2.55/1.64 ? [X0 : $i > $i > $o,X1 : $i,X2 : $i] : ($true != (subset @ (dpsetconstr @ X1 @ X2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1))))) @ (cartprod @ X1 @ X2))) => ($true != (subset @ (dpsetconstr @ sK1 @ sK2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK0 @ Y0 @ Y1))))) @ (cartprod @ sK1 @ sK2)))),
% 2.55/1.64 introduced(definition,[],[choice_axiom])).
% 2.55/1.64 thf(f565,plain,(
% 2.55/1.64 (cartprodfstin = $true) & (noeltsimpempty = $true) & (symdiffI2 = $true) & (kfstpairEq = $true) & (setukpairinjL1 = $true) & (inPowerset = $true) & (setunionE = $true) & (subsetRefl = $true) & (setextsub = $true) & (eqimpsubset2 = $true) & (singletonprop = $true) & (descr__Cong = $true) & (ksndpairEq = $true) & (upairsetE = $true) & (setukpairinjR = $true) & (setukpairinjL = $true) & (upairsetIR = $true) & (exuE3u = $true) & (descrp = $true) & (notequalI2 = $true) & (dpsetconstrI = $true) & (upairequniteq = $true) & (setminusELneg = $true) & (prop2set2propI = $true) & (binintersectSubset2 = $true) & (upairsetIL = $true) & (kpairsurjEq = $true) & (replAx = $true) & (ubforcartprodlem3 = $true) & (setadjoinE = $true) & (bs114d = $true) & (emptyinPowerset = $true) & (binunionE = $true) & (nonemptyImpWitness = $true) & (subset2powerset = $true) & (setminusI = $true) & (powersetsubset = $true) & (exuI1 = $true) & (setext = $true) & (singletonsubset = $true) & (omegaSAx = $true) & (cartprodpairmemER = $true) & (dsetconstrEL = $true) & (setukpairIL = $true) & (notinemptyset = $true) & (setminusLsub = $true) & (powersetI = $true) & (setbeta = $true) & (emptyE1 = $true) & (cartprodmempaircEq = $true) & (binintersectEL = $true) & (exuE1 = $true) & (subsetI2 = $true) & (quantDeMorgan3 = $true) & (setadjoinSub = $true) & (prop2setI = $true) & (sepSubset = $true) & (omegaIndAx = $true) & (upairset2IR = $true) & (subsetTrans = $true) & (symdiffIneg1 = $true) & (singletonsswitch = $true) & (secondinupair = $true) & (cartprodpairin = $true) & (setminusSubset2 = $true) & (setoftrueEq = $true) & (notequalI1 = $true) & (powersetE1 = $true) & (in__Cong = $true) & (setadjoinIR = $true) & (setukpairinjR2 = $true) & (exu__Cong = $true) & (emptyset__Cong = $true) & (kpairiskpair = $true) & (binintersectSubset1 = $true) & (notdallE = $true) & (emptyI = $true) & (quantDeMorgan4 = $true) & (uniqinunit = $true) & (kfstsingleton = $true) & (emptyInPowerset = $true) & (notdexE = $true) & (eqimpsubset1 = $true) & (prop2setE = $true) & (subsetE = $true) & (disjointsetsI1 = $true) & (dsetconstrI = $true) & (ksndsingleton = $true) & (setukpairIR = $true) & (binunionEcases = $true) & (setukpairinjR12 = $true) & (exuI2 = $true) & (exuE3e = $true) & (ubforcartprodlem1 = $true) & (cartprodfstpairEq = $true) & ? [X0 : $i > $i > $o,X1 : $i,X2 : $i] : ($true != (subset @ (dpsetconstr @ X1 @ X2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1))))) @ (cartprod @ X1 @ X2))) & (cartprodmempair1 = $true) & (subsetI1 = $true) & (eqinunit = $true) & (ex1E1 = $true) & (emptysetsubset = $true) & (kpairp = $true) & (setadjoinSub2 = $true) & (cartprodsndpairEq = $true) & (powersetAx = $true) & (setunionsingleton2 = $true) & (singletoninpowerset = $true) & (powerset__Cong = $true) & (setadjoin__Cong = $true) & (setunionE2 = $true) & (dsetconstrER = $true) & (binunionIR = $true) & (quantDeMorgan2 = $true) & (notsubsetI = $true) & (binintersectRsub = $true) & (emptysetimpfalse = $true) & (setadjoinIL = $true) & (exuI3 = $true) & (singletoninpowunion = $true) & (setextAx = $true) & (vacuousDall = $true) & (setunionAx = $true) & (binintersectI = $true) & (setminusILneg = $true) & (nonemptyI1 = $true) & (subPowSU = $true) & (setadjoinAx = $true) & (symdiffI1 = $true) & (emptysetE = $true) & (foundationAx = $true) & (nonemptyE1 = $true) & (cartprodpairsurjEq = $true) & (setminusERneg = $true) & (sepInPowerset = $true) & (exuEu = $true) & (singletonsuniq = $true) & (cartprodmempair = $true) & (powersetE = $true) & (setunionsingleton = $true) & (ubforcartprodlem2 = $true) & (setminusIRneg = $true) & (setukpairinjL2 = $true) & (symdiffIneg2 = $true) & (wellorderingAx = $true) & (binintersectSubset5 = $true) & (symdiffE = $true) & (quantDeMorgan1 = $true) & (upairsubunion = $true) & (theprop = $true) & (binunionRsub = $true) & (upairset2E = $true) & (setunionI = $true) & (upairinpowunion = $true) & (setadjoinOr = $true) & (exuE2 = $true) & (setunion__Cong = $true) & (emptyinunitempty = $true) & (binintersectER = $true) & (binintersectLsub = $true) & (subsetE2 = $true) & (dsetconstr__Cong = $true) & (cartprodpairmemEL = $true) & (subsetemptysetimpeq = $true) & (binunionIL = $true) & (cartprodsndin = $true) & (inCongP = $true) & (setminusSubset1 = $true) & (ex1I2 = $true) & (setunionsingleton1 = $true) & (powersetI1 = $true) & (omega__Cong = $true) & (setminusEL = $true) & (binunionLsub = $true) & (binintersectSubset3 = $true) & (nonemptyI = $true) & (emptysetAx = $true) & (omega0Ax = $true) & (setukpairinjR11 = $true) & (setukpairinjR1 = $true) & (notinsingleton = $true) & (ex1I = $true) & (setminusER = $true) & (binintersectSubset4 = $true)),
% 2.55/1.64 inference(flattening,[status(thm)],[f564])).
% 2.55/1.64 thf(f564,plain,(
% 2.55/1.64 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i > $i > $o,X1 : $i,X2 : $i] : ($true != (subset @ (dpsetconstr @ X1 @ X2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1))))) @ (cartprod @ X1 @ X2))) & (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)),
% 2.55/1.64 inference(ennf_transformation,[status(thm)],[f388])).
% 2.55/1.64 thf(f388,plain,(
% 2.55/1.64 ~((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) => ! [X0 : $i > $i > $o,X1 : $i,X2 : $i] : ($true = (subset @ (dpsetconstr @ X1 @ X2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1))))) @ (cartprod @ X1 @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.55/1.64 inference(fool_elimination,[status(thm)],[f387])).
% 2.55/1.64 thf(f387,plain,(
% 2.55/1.64 ~(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 => ! [X0 : $i > $i > $o,X1 : $i,X2 : $i] : (subset @ (dpsetconstr @ X1 @ X2 @ (^[X3 : $i, X4 : $i] : (X0 @ X3 @ X4))) @ (cartprod @ X1 @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.55/1.64 inference(rectify,[status(thm)],[f190])).
% 2.55/1.64 thf(f190,negated_conjecture,(
% 2.55/1.64 ~(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 => ! [X0 : $i > $i > $o,X3 : $i,X4 : $i] : (subset @ (dpsetconstr @ X3 @ X4 @ (^[X1 : $i, X2 : $i] : (X0 @ X1 @ X2))) @ (cartprod @ X3 @ X4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.55/1.64 inference(negated_conjecture,[status(cth)],[f189])).
% 2.55/1.64 thf(f189,conjecture,(
% 2.55/1.64 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 => ! [X0 : $i > $i > $o,X3 : $i,X4 : $i] : (subset @ (dpsetconstr @ X3 @ X4 @ (^[X1 : $i, X2 : $i] : (X0 @ X1 @ X2))) @ (cartprod @ X3 @ X4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.55/1.64 file('/export/starexec/sandbox/tmp/tmp.frxx7lxlXX/DTF2THF_31262.p',dpsetconstrSub)).
% 2.55/1.64 thf(f880,plain,(
% 2.55/1.64 (sepSubset = (!! @ $i @ (^[Y0 : $i]: (!! @ ($i > $o) @ (^[Y1 : $i > $o]: (subset @ (dsetconstr @ Y0 @ (^[Y2 : $i]: (Y1 @ Y2))) @ Y0))))))),
% 2.55/1.64 inference(cnf_transformation,[status(thm)],[f561])).
% 2.55/1.64 thf(f561,plain,(
% 2.55/1.64 (sepSubset = (!! @ $i @ (^[Y0 : $i]: (!! @ ($i > $o) @ (^[Y1 : $i > $o]: (subset @ (dsetconstr @ Y0 @ (^[Y2 : $i]: (Y1 @ Y2))) @ Y0))))))),
% 2.55/1.64 inference(fool_elimination,[status(thm)],[f560])).
% 2.55/1.64 thf(f560,plain,(
% 2.55/1.64 (! [X0 : $i > $o,X1 : $i] : (subset @ (dsetconstr @ X1 @ (^[X2 : $i] : (X0 @ X2))) @ X1) = sepSubset)),
% 2.55/1.64 inference(rectify,[status(thm)],[f100])).
% 2.55/1.64 thf(f100,axiom,(
% 2.55/1.64 (! [X0 : $i > $o,X3 : $i] : (subset @ (dsetconstr @ X3 @ (^[X1 : $i] : (X0 @ X1))) @ X3) = sepSubset)),
% 2.55/1.64 file('/export/starexec/sandbox/tmp/tmp.frxx7lxlXX/DTF2THF_31262.p',sepSubset)).
% 2.55/1.64 thf(f1807,plain,(
% 2.55/1.64 ((subset @ (dsetconstr @ (cartprod @ sK1 @ sK2) @ (^[Y0 : $i]: (?? @ $i @ (^[Y1 : $i]: ((in @ Y1 @ sK1) & (?? @ $i @ (^[Y2 : $i]: (((in @ Y2 @ sK2) & ((kpair @ Y1 @ Y2) = Y0)) & (sK0 @ Y1 @ Y2))))))))) @ (cartprod @ sK1 @ sK2)) != $true)),
% 2.55/1.64 inference(beta_eta_normalization,[status(thm)],[f937])).
% 2.55/1.64 thf(f937,plain,(
% 2.55/1.64 ($true != (subset @ ((^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i > $i > $o]: (dsetconstr @ (cartprod @ Y0 @ Y1) @ (^[Y3 : $i]: (?? @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) & (?? @ $i @ (^[Y5 : $i]: (((in @ Y5 @ Y1) & ((kpair @ Y4 @ Y5) = Y3)) & (Y2 @ Y4 @ Y5)))))))))))))) @ sK1 @ sK2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK0 @ Y0 @ Y1))))) @ (cartprod @ sK1 @ sK2)))),
% 2.55/1.64 inference(definition_unfolding,[status(thm)],[f662,f895])).
% 2.55/1.64 thf(f895,plain,(
% 2.55/1.64 (dpsetconstr = (^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i > $i > $o]: (dsetconstr @ (cartprod @ Y0 @ Y1) @ (^[Y3 : $i]: (?? @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) & (?? @ $i @ (^[Y5 : $i]: (((in @ Y5 @ Y1) & ((kpair @ Y4 @ Y5) = Y3)) & (Y2 @ Y4 @ Y5)))))))))))))))),
% 2.55/1.64 inference(cnf_transformation,[status(thm)],[f558])).
% 2.55/1.64 thf(f558,plain,(
% 2.55/1.64 (dpsetconstr = (^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i > $i > $o]: (dsetconstr @ (cartprod @ Y0 @ Y1) @ (^[Y3 : $i]: (?? @ $i @ (^[Y4 : $i]: ((in @ Y4 @ Y0) & (?? @ $i @ (^[Y5 : $i]: (((in @ Y5 @ Y1) & ((kpair @ Y4 @ Y5) = Y3)) & (Y2 @ Y4 @ Y5)))))))))))))))),
% 2.55/1.64 inference(fool_elimination,[status(thm)],[f557])).
% 2.55/1.64 thf(f557,plain,(
% 2.55/1.64 (dpsetconstr = (^[X0 : $i, X1 : $i, X2 : $i > $i > $o] : (dsetconstr @ (cartprod @ X0 @ X1) @ (^[X3 : $i] : (? [X4 : $i] : (? [X5 : $i] : ((X2 @ X4 @ X5) & ((kpair @ X4 @ X5) = X3) & (in @ X5 @ X1)) & (in @ X4 @ X0)))))))),
% 2.55/1.64 inference(rectify,[status(thm)],[f187])).
% 2.55/1.64 thf(f187,axiom,(
% 2.55/1.64 (dpsetconstr = (^[X3 : $i, X4 : $i, X0 : $i > $i > $o] : (dsetconstr @ (cartprod @ X3 @ X4) @ (^[X10 : $i] : (? [X1 : $i] : (? [X2 : $i] : ((X0 @ X1 @ X2) & ((kpair @ X1 @ X2) = X10) & (in @ X2 @ X4)) & (in @ X1 @ X3)))))))),
% 2.55/1.64 file('/export/starexec/sandbox/tmp/tmp.frxx7lxlXX/DTF2THF_31262.p',dpsetconstr)).
% 2.55/1.64 thf(f662,plain,(
% 2.55/1.64 ($true != (subset @ (dpsetconstr @ sK1 @ sK2 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK0 @ Y0 @ Y1))))) @ (cartprod @ sK1 @ sK2)))),
% 2.55/1.64 inference(cnf_transformation,[status(thm)],[f567])).
% 2.55/1.64 % SZS output end Proof for DTF2THF_31262
% 2.55/1.64 % (31391)------------------------------
% 2.55/1.64 % (31391)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.55/1.64 % (31391)Termination reason: Refutation
% 2.55/1.64
% 2.55/1.64 % (31391)Memory used [KB]: 7931
% 2.55/1.64 % (31391)Time elapsed: 0.076 s
% 2.55/1.64 % (31391)Instructions burned: 239 (million)
% 2.55/1.64 % (31391)------------------------------
% 2.55/1.64 % (31391)------------------------------
% 2.55/1.64 % (31360)Success in time 0.2 s
%------------------------------------------------------------------------------