%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU661^1 : 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 : n005.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 09:36:01 AM UTC 2026
% Result : Theorem 3.02s 1.82s
% Output : Refutation 3.02s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 3
% Syntax : Number of formulae : 11 ( 2 unt; 0 typ; 0 def)
% Number of atoms : 2742 ( 734 equ; 0 cnn)
% Maximal formula atoms : 180 ( 249 avg)
% Number of connectives : 1360 ( 11 ~; 0 |; 543 &; 88 @)
% ( 0 <=>; 718 =>; 0 <=; 0 <~>)
% Maximal formula depth : 187 ( 118 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 219 ( 216 usr; 183 con; 0-3 aty)
% Number of variables : 30 ( 0 ^; 16 !; 14 ?; 30 :)
% 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_231,type,
sK5: $i > $i ).
thf(func_def_232,type,
sK6: $i > $i > $i ).
thf(func_def_233,type,
sK7: $i > $i > $i ).
thf(func_def_234,type,
sK8: $i > $i > $i ).
thf(func_def_235,type,
sK9: $o > $i > $i > $i ).
thf(func_def_236,type,
sK10: $i > ( $i > $o ) > $i ).
thf(f1264,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f723]) ).
thf(f723,plain,
( ( kpair @ sK2 @ sK3 )
!= ( kpair @ sK2 @ sK3 ) ),
inference(cnf_transformation,[status(thm)],[f547]) ).
thf(f547,plain,
( ( emptyset__Cong = $true )
& ( setadjoinSub2 = $true )
& ( setunionE2 = $true )
& ( quantDeMorgan1 = $true )
& ( ubforcartprodlem1 = $true )
& ( nonemptyI1 = $true )
& ( dsetconstrER = $true )
& ( upairequniteq = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( foundationAx = $true )
& ( subsetE = $true )
& ( in__Cong = $true )
& ( singletonsuniq = $true )
& ( setadjoinOr = $true )
& ( setunionsingleton = $true )
& ( upairsetE = $true )
& ( eqimpsubset2 = $true )
& ( cartprodmempair1 = $true )
& ( upairsubunion = $true )
& ( setukpairinjR = $true )
& ( notdallE = $true )
& ( setadjoinAx = $true )
& ( binintersectSubset2 = $true )
& ( setminusEL = $true )
& ( upairinpowunion = $true )
& ( inPowerset = $true )
& ( setukpairinjL = $true )
& ( exu__Cong = $true )
& ( eqimpsubset1 = $true )
& ( descr__Cong = $true )
& ( symdiffI2 = $true )
& ( omegaSAx = $true )
& ( setunionAx = $true )
& ( omegaIndAx = $true )
& ( binintersectI = $true )
& ( exuE3e = $true )
& ( emptyinPowerset = $true )
& ( setukpairinjL1 = $true )
& ( emptysetimpfalse = $true )
& ( omega0Ax = $true )
& ( emptysetsubset = $true )
& ( dsetconstrEL = $true )
& ( cartprodfstin = $true )
& ( powersetI = $true )
& ( wellorderingAx = $true )
& ( exuEu = $true )
& ( binintersectSubset1 = $true )
& ( ksndsingleton = $true )
& ( emptyE1 = $true )
& ( ( kpair @ sK2 @ sK3 )
!= ( kpair @ sK2 @ sK3 ) )
& ( ( in @ sK3 @ sK1 )
= $true )
& ( ( in @ sK2 @ sK0 )
= $true )
& ( setadjoinE = $true )
& ( emptyinunitempty = $true )
& ( dsetconstr__Cong = $true )
& ( setadjoinIL = $true )
& ( setukpairinjR11 = $true )
& ( setunionE = $true )
& ( exuI3 = $true )
& ( notdexE = $true )
& ( eqinunit = $true )
& ( exuI2 = $true )
& ( powersetAx = $true )
& ( dsetconstrI = $true )
& ( setunionsingleton2 = $true )
& ( setukpairIL = $true )
& ( setextsub = $true )
& ( setminusI = $true )
& ( binunionLsub = $true )
& ( setukpairinjR12 = $true )
& ( subset2powerset = $true )
& ( powersetE = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan2 = $true )
& ( binintersectLsub = $true )
& ( powersetE1 = $true )
& ( binintersectSubset4 = $true )
& ( omega__Cong = $true )
& ( bs114d = $true )
& ( prop2set2propI = $true )
& ( subsetI2 = $true )
& ( setunionI = $true )
& ( quantDeMorgan3 = $true )
& ( upairset2E = $true )
& ( notequalI2 = $true )
& ( symdiffE = $true )
& ( binunionEcases = $true )
& ( upairsetIL = $true )
& ( ex1I = $true )
& ( binintersectRsub = $true )
& ( kfstpairEq = $true )
& ( exuI1 = $true )
& ( kpairp = $true )
& ( singletoninpowerset = $true )
& ( descrp = $true )
& ( setminusELneg = $true )
& ( setadjoinSub = $true )
& ( notinemptyset = $true )
& ( ex1I2 = $true )
& ( cartprodpairmemER = $true )
& ( symdiffIneg2 = $true )
& ( subsetRefl = $true )
& ( exuE1 = $true )
& ( binintersectER = $true )
& ( powersetsubset = $true )
& ( disjointsetsI1 = $true )
& ( binintersectSubset5 = $true )
& ( nonemptyI = $true )
& ( setukpairinjL2 = $true )
& ( prop2setI = $true )
& ( nonemptyImpWitness = $true )
& ( setminusSubset2 = $true )
& ( setminusERneg = $true )
& ( singletonsswitch = $true )
& ( binunionRsub = $true )
& ( singletonsubset = $true )
& ( exuE2 = $true )
& ( setunionsingleton1 = $true )
& ( setadjoinIR = $true )
& ( subsetTrans = $true )
& ( cartprodpairin = $true )
& ( setminusSubset1 = $true )
& ( cartprodpairmemEL = $true )
& ( binunionE = $true )
& ( subsetE2 = $true )
& ( ksndpairEq = $true )
& ( binunionIR = $true )
& ( notsubsetI = $true )
& ( subsetI1 = $true )
& ( binintersectEL = $true )
& ( secondinupair = $true )
& ( setminusIRneg = $true )
& ( setextAx = $true )
& ( upairset2IR = $true )
& ( emptysetE = $true )
& ( replAx = $true )
& ( setadjoin__Cong = $true )
& ( ex1E1 = $true )
& ( symdiffIneg1 = $true )
& ( notequalI1 = $true )
& ( setminusER = $true )
& ( sepInPowerset = $true )
& ( setunion__Cong = $true )
& ( setbeta = $true )
& ( setukpairIR = $true )
& ( nonemptyE1 = $true )
& ( subPowSU = $true )
& ( kfstsingleton = $true )
& ( kpairiskpair = $true )
& ( noeltsimpempty = $true )
& ( ubforcartprodlem2 = $true )
& ( uniqinunit = $true )
& ( prop2setE = $true )
& ( setukpairinjR1 = $true )
& ( upairsetIR = $true )
& ( setminusILneg = $true )
& ( setoftrueEq = $true )
& ( notinsingleton = $true )
& ( sepSubset = $true )
& ( exuE3u = $true )
& ( emptyInPowerset = $true )
& ( powersetI1 = $true )
& ( inCongP = $true )
& ( ubforcartprodlem3 = $true )
& ( kpairsurjEq = $true )
& ( subsetemptysetimpeq = $true )
& ( binintersectSubset3 = $true )
& ( singletoninpowunion = $true )
& ( powerset__Cong = $true )
& ( emptysetAx = $true )
& ( vacuousDall = $true )
& ( cartprodmempair = $true )
& ( cartprodsndin = $true )
& ( symdiffI1 = $true )
& ( theprop = $true )
& ( setukpairinjR2 = $true )
& ( setminusLsub = $true )
& ( binunionIL = $true )
& ( singletonprop = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3])],[f544,f546,f545]) ).
thf(f545,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( kpair @ X2 @ X3 )
!= ( kpair @ X2 @ X3 ) )
& ( ( in @ X3 @ X1 )
= $true ) )
& ( $true
= ( in @ X2 @ X0 ) ) )
=> ( ? [X3: $i] :
( ( ( kpair @ sK2 @ X3 )
!= ( kpair @ sK2 @ X3 ) )
& ( ( in @ X3 @ sK1 )
= $true ) )
& ( ( in @ sK2 @ sK0 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f546,plain,
( ? [X3: $i] :
( ( ( kpair @ sK2 @ X3 )
!= ( kpair @ sK2 @ X3 ) )
& ( ( in @ X3 @ sK1 )
= $true ) )
=> ( ( ( kpair @ sK2 @ sK3 )
!= ( kpair @ sK2 @ sK3 ) )
& ( ( in @ sK3 @ sK1 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f544,plain,
( ( emptyset__Cong = $true )
& ( setadjoinSub2 = $true )
& ( setunionE2 = $true )
& ( quantDeMorgan1 = $true )
& ( ubforcartprodlem1 = $true )
& ( nonemptyI1 = $true )
& ( dsetconstrER = $true )
& ( upairequniteq = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( foundationAx = $true )
& ( subsetE = $true )
& ( in__Cong = $true )
& ( singletonsuniq = $true )
& ( setadjoinOr = $true )
& ( setunionsingleton = $true )
& ( upairsetE = $true )
& ( eqimpsubset2 = $true )
& ( cartprodmempair1 = $true )
& ( upairsubunion = $true )
& ( setukpairinjR = $true )
& ( notdallE = $true )
& ( setadjoinAx = $true )
& ( binintersectSubset2 = $true )
& ( setminusEL = $true )
& ( upairinpowunion = $true )
& ( inPowerset = $true )
& ( setukpairinjL = $true )
& ( exu__Cong = $true )
& ( eqimpsubset1 = $true )
& ( descr__Cong = $true )
& ( symdiffI2 = $true )
& ( omegaSAx = $true )
& ( setunionAx = $true )
& ( omegaIndAx = $true )
& ( binintersectI = $true )
& ( exuE3e = $true )
& ( emptyinPowerset = $true )
& ( setukpairinjL1 = $true )
& ( emptysetimpfalse = $true )
& ( omega0Ax = $true )
& ( emptysetsubset = $true )
& ( dsetconstrEL = $true )
& ( cartprodfstin = $true )
& ( powersetI = $true )
& ( wellorderingAx = $true )
& ( exuEu = $true )
& ( binintersectSubset1 = $true )
& ( ksndsingleton = $true )
& ( emptyE1 = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( kpair @ X2 @ X3 )
!= ( kpair @ X2 @ X3 ) )
& ( ( in @ X3 @ X1 )
= $true ) )
& ( $true
= ( in @ X2 @ X0 ) ) )
& ( setadjoinE = $true )
& ( emptyinunitempty = $true )
& ( dsetconstr__Cong = $true )
& ( setadjoinIL = $true )
& ( setukpairinjR11 = $true )
& ( setunionE = $true )
& ( exuI3 = $true )
& ( notdexE = $true )
& ( eqinunit = $true )
& ( exuI2 = $true )
& ( powersetAx = $true )
& ( dsetconstrI = $true )
& ( setunionsingleton2 = $true )
& ( setukpairIL = $true )
& ( setextsub = $true )
& ( setminusI = $true )
& ( binunionLsub = $true )
& ( setukpairinjR12 = $true )
& ( subset2powerset = $true )
& ( powersetE = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan2 = $true )
& ( binintersectLsub = $true )
& ( powersetE1 = $true )
& ( binintersectSubset4 = $true )
& ( omega__Cong = $true )
& ( bs114d = $true )
& ( prop2set2propI = $true )
& ( subsetI2 = $true )
& ( setunionI = $true )
& ( quantDeMorgan3 = $true )
& ( upairset2E = $true )
& ( notequalI2 = $true )
& ( symdiffE = $true )
& ( binunionEcases = $true )
& ( upairsetIL = $true )
& ( ex1I = $true )
& ( binintersectRsub = $true )
& ( kfstpairEq = $true )
& ( exuI1 = $true )
& ( kpairp = $true )
& ( singletoninpowerset = $true )
& ( descrp = $true )
& ( setminusELneg = $true )
& ( setadjoinSub = $true )
& ( notinemptyset = $true )
& ( ex1I2 = $true )
& ( cartprodpairmemER = $true )
& ( symdiffIneg2 = $true )
& ( subsetRefl = $true )
& ( exuE1 = $true )
& ( binintersectER = $true )
& ( powersetsubset = $true )
& ( disjointsetsI1 = $true )
& ( binintersectSubset5 = $true )
& ( nonemptyI = $true )
& ( setukpairinjL2 = $true )
& ( prop2setI = $true )
& ( nonemptyImpWitness = $true )
& ( setminusSubset2 = $true )
& ( setminusERneg = $true )
& ( singletonsswitch = $true )
& ( binunionRsub = $true )
& ( singletonsubset = $true )
& ( exuE2 = $true )
& ( setunionsingleton1 = $true )
& ( setadjoinIR = $true )
& ( subsetTrans = $true )
& ( cartprodpairin = $true )
& ( setminusSubset1 = $true )
& ( cartprodpairmemEL = $true )
& ( binunionE = $true )
& ( subsetE2 = $true )
& ( ksndpairEq = $true )
& ( binunionIR = $true )
& ( notsubsetI = $true )
& ( subsetI1 = $true )
& ( binintersectEL = $true )
& ( secondinupair = $true )
& ( setminusIRneg = $true )
& ( setextAx = $true )
& ( upairset2IR = $true )
& ( emptysetE = $true )
& ( replAx = $true )
& ( setadjoin__Cong = $true )
& ( ex1E1 = $true )
& ( symdiffIneg1 = $true )
& ( notequalI1 = $true )
& ( setminusER = $true )
& ( sepInPowerset = $true )
& ( setunion__Cong = $true )
& ( setbeta = $true )
& ( setukpairIR = $true )
& ( nonemptyE1 = $true )
& ( subPowSU = $true )
& ( kfstsingleton = $true )
& ( kpairiskpair = $true )
& ( noeltsimpempty = $true )
& ( ubforcartprodlem2 = $true )
& ( uniqinunit = $true )
& ( prop2setE = $true )
& ( setukpairinjR1 = $true )
& ( upairsetIR = $true )
& ( setminusILneg = $true )
& ( setoftrueEq = $true )
& ( notinsingleton = $true )
& ( sepSubset = $true )
& ( exuE3u = $true )
& ( emptyInPowerset = $true )
& ( powersetI1 = $true )
& ( inCongP = $true )
& ( ubforcartprodlem3 = $true )
& ( kpairsurjEq = $true )
& ( subsetemptysetimpeq = $true )
& ( binintersectSubset3 = $true )
& ( singletoninpowunion = $true )
& ( powerset__Cong = $true )
& ( emptysetAx = $true )
& ( vacuousDall = $true )
& ( cartprodmempair = $true )
& ( cartprodsndin = $true )
& ( symdiffI1 = $true )
& ( theprop = $true )
& ( setukpairinjR2 = $true )
& ( setminusLsub = $true )
& ( binunionIL = $true )
& ( singletonprop = $true ) ),
inference(flattening,[status(thm)],[f543]) ).
thf(f543,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( kpair @ X2 @ X3 )
!= ( kpair @ X2 @ X3 ) )
& ( ( in @ X3 @ X1 )
= $true ) )
& ( $true
= ( in @ X2 @ X0 ) ) )
& ( 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)],[f290]) ).
thf(f290,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 )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
=> ! [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
=> ( ( kpair @ X2 @ X3 )
= ( kpair @ X2 @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f289]) ).
thf(f289,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
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ X0 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( kpair @ X2 @ X3 )
= ( kpair @ X2 @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f183]) ).
thf(f183,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
=> ! [X3: $i,X4: $i,X1: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( ( kpair @ X1 @ X2 )
= ( kpair @ X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f182]) ).
thf(f182,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
=> ! [X3: $i,X4: $i,X1: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( ( kpair @ X1 @ X2 )
= ( kpair @ X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.CrCPL8bmZ3/DTF2THF_9448.p',cartprodmempaircEq) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU661^1 : 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
% 0.15/0.34 % Computer : n005.cluster.edu
% 0.15/0.34 % Model : x86_64 x86_64
% 0.15/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.34 % Memory : 8042.1875MB
% 0.15/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34 % CPULimit : 300
% 0.15/0.34 % WCLimit : 300
% 0.15/0.34 % DateTime : Tue Feb 24 12:29:44 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.18/0.47 ---- Original DTF file ---
% 0.18/0.47 thf(spec,logic,$$dhol).
% 0.18/0.47 %------------------------------------------------------------------------------
% 0.18/0.47 % File : SEU661^1 : TPTP v9.2.1. Released v3.7.0.
% 0.18/0.47 % Domain : Set Theory
% 0.18/0.47 % Problem : Ordered Pairs - Properties of Pairs
% 0.18/0.47 % Version : Especial.
% 0.18/0.47 % English : (! A:i.! B:i.! x:i.in x A -> (! y:i.in y B ->
% 0.18/0.47 % kpair x y = kpair x y))
% 0.18/0.47
% 0.18/0.47 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.18/0.47 % Source : [Bro08]
% 0.18/0.47 % Names : ZFC163g [Bro08]
% 0.18/0.47
% 0.18/0.47 % Status : Theorem
% 0.18/0.47 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.15 v8.1.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.12 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.67 v4.0.0, 0.33 v3.7.0
% 0.18/0.47 % Syntax : Number of formulae : 390 ( 181 unt; 208 typ; 181 def)
% 0.18/0.47 % Number of atoms : 962 ( 271 equ; 0 cnn)
% 0.18/0.47 % Maximal formula atoms : 180 ( 5 avg)
% 0.18/0.47 % Number of connectives : 1688 ( 47 ~; 7 |; 35 &;1160 @)
% 0.18/0.47 % ( 14 <=>; 425 =>; 0 <=; 0 <~>)
% 0.18/0.47 % Maximal formula depth : 186 ( 2 avg)
% 0.18/0.47 % Number of types : 2 ( 0 usr)
% 0.18/0.47 % Number of type conns : 84 ( 84 >; 0 *; 0 +; 0 <<)
% 0.18/0.47 % Number of symbols : 211 ( 208 usr; 181 con; 0-2 aty)
% 0.18/0.47 % Number of variables : 530 ( 40 ^; 454 !; 36 ?; 530 :)
% 0.18/0.47 % SPC : TH0_THM_EQU_NAR
% 0.18/0.47
% 0.18/0.47 % Comments : http://mathgate.info/detsetitem.php?id=215
% 0.18/0.47 %------------------------------------------------------------------------------
% 0.18/0.47 thf(in_type,type,
% 0.18/0.47 in: $i > $i > $o ).
% 0.18/0.47
% 0.18/0.47 thf(exu_type,type,
% 0.18/0.47 exu: ( $i > $o ) > $o ).
% 0.18/0.47
% 0.18/0.47 thf(exu,definition,
% 0.18/0.47 ( exu
% 0.18/0.47 = ( ^ [Xphi: $i > $o] :
% 0.18/0.47 ? [Xx: $i] :
% 0.18/0.47 ( ( Xphi @ Xx )
% 0.18/0.47 & ! [Xy: $i] :
% 0.18/0.47 ( ( Xphi @ Xy )
% 0.18/0.47 => ( Xx = Xy ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(setextAx_type,type,
% 0.18/0.47 setextAx: $o ).
% 0.18/0.47
% 0.18/0.47 thf(setextAx,definition,
% 0.18/0.47 ( setextAx
% 0.18/0.47 = ( ! [A: $i,B: $i] :
% 0.18/0.47 ( ! [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ A )
% 0.18/0.47 <=> ( in @ Xx @ B ) )
% 0.18/0.47 => ( A = B ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(emptyset_type,type,
% 0.18/0.47 emptyset: $i ).
% 0.18/0.47
% 0.18/0.47 thf(emptysetAx_type,type,
% 0.18/0.47 emptysetAx: $o ).
% 0.18/0.47
% 0.18/0.47 thf(emptysetAx,definition,
% 0.18/0.47 ( emptysetAx
% 0.18/0.47 = ( ! [Xx: $i] :
% 0.18/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(setadjoin_type,type,
% 0.18/0.47 setadjoin: $i > $i > $i ).
% 0.18/0.47
% 0.18/0.47 thf(setadjoinAx_type,type,
% 0.18/0.47 setadjoinAx: $o ).
% 0.18/0.47
% 0.18/0.47 thf(setadjoinAx,definition,
% 0.18/0.47 ( setadjoinAx
% 0.18/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.18/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.18/0.47 <=> ( ( Xy = Xx )
% 0.18/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(powerset_type,type,
% 0.18/0.47 powerset: $i > $i ).
% 0.18/0.47
% 0.18/0.47 thf(powersetAx_type,type,
% 0.18/0.47 powersetAx: $o ).
% 0.18/0.47
% 0.18/0.47 thf(powersetAx,definition,
% 0.18/0.47 ( powersetAx
% 0.18/0.47 = ( ! [A: $i,B: $i] :
% 0.18/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.18/0.47 <=> ! [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ B )
% 0.18/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(setunion_type,type,
% 0.18/0.47 setunion: $i > $i ).
% 0.18/0.47
% 0.18/0.47 thf(setunionAx_type,type,
% 0.18/0.47 setunionAx: $o ).
% 0.18/0.47
% 0.18/0.47 thf(setunionAx,definition,
% 0.18/0.47 ( setunionAx
% 0.18/0.47 = ( ! [A: $i,Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.18/0.47 <=> ? [B: $i] :
% 0.18/0.47 ( ( in @ Xx @ B )
% 0.18/0.47 & ( in @ B @ A ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(omega_type,type,
% 0.18/0.47 omega: $i ).
% 0.18/0.47
% 0.18/0.47 thf(omega0Ax_type,type,
% 0.18/0.47 omega0Ax: $o ).
% 0.18/0.47
% 0.18/0.47 thf(omega0Ax,definition,
% 0.18/0.47 ( omega0Ax
% 0.18/0.47 = ( in @ emptyset @ omega ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(omegaSAx_type,type,
% 0.18/0.47 omegaSAx: $o ).
% 0.18/0.47
% 0.18/0.47 thf(omegaSAx,definition,
% 0.18/0.47 ( omegaSAx
% 0.18/0.47 = ( ! [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ omega )
% 0.18/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(omegaIndAx_type,type,
% 0.18/0.47 omegaIndAx: $o ).
% 0.18/0.47
% 0.18/0.47 thf(omegaIndAx,definition,
% 0.18/0.47 ( omegaIndAx
% 0.18/0.47 = ( ! [A: $i] :
% 0.18/0.47 ( ( ( in @ emptyset @ A )
% 0.18/0.47 & ! [Xx: $i] :
% 0.18/0.47 ( ( ( in @ Xx @ omega )
% 0.18/0.47 & ( in @ Xx @ A ) )
% 0.18/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.18/0.47 => ! [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ omega )
% 0.18/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(replAx_type,type,
% 0.18/0.47 replAx: $o ).
% 0.18/0.47
% 0.18/0.47 thf(replAx,definition,
% 0.18/0.47 ( replAx
% 0.18/0.47 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.18/0.47 ( ! [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ A )
% 0.18/0.47 => ( exu
% 0.18/0.47 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.18/0.47 => ? [B: $i] :
% 0.18/0.47 ! [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ B )
% 0.18/0.47 <=> ? [Xy: $i] :
% 0.18/0.47 ( ( in @ Xy @ A )
% 0.18/0.47 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(foundationAx_type,type,
% 0.18/0.47 foundationAx: $o ).
% 0.18/0.47
% 0.18/0.47 thf(foundationAx,definition,
% 0.18/0.47 ( foundationAx
% 0.18/0.47 = ( ! [A: $i] :
% 0.18/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.18/0.47 => ? [B: $i] :
% 0.18/0.47 ( ( in @ B @ A )
% 0.18/0.47 & ~ ? [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ B )
% 0.18/0.47 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(wellorderingAx_type,type,
% 0.18/0.47 wellorderingAx: $o ).
% 0.18/0.47
% 0.18/0.47 thf(wellorderingAx,definition,
% 0.18/0.47 ( wellorderingAx
% 0.18/0.47 = ( ! [A: $i] :
% 0.18/0.47 ? [B: $i] :
% 0.18/0.47 ( ! [C: $i] :
% 0.18/0.47 ( ( in @ C @ B )
% 0.18/0.47 => ! [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ C )
% 0.18/0.47 => ( in @ Xx @ A ) ) )
% 0.18/0.47 & ! [Xx: $i,Xy: $i] :
% 0.18/0.47 ( ( ( in @ Xx @ A )
% 0.18/0.47 & ( in @ Xy @ A ) )
% 0.18/0.47 => ( ! [C: $i] :
% 0.18/0.47 ( ( in @ C @ B )
% 0.18/0.47 => ( ( in @ Xx @ C )
% 0.18/0.47 <=> ( in @ Xy @ C ) ) )
% 0.18/0.47 => ( Xx = Xy ) ) )
% 0.18/0.47 & ! [C: $i,D: $i] :
% 0.18/0.47 ( ( ( in @ C @ B )
% 0.18/0.47 & ( in @ D @ B ) )
% 0.18/0.47 => ( ! [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ C )
% 0.18/0.47 => ( in @ Xx @ D ) )
% 0.18/0.47 | ! [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ D )
% 0.18/0.47 => ( in @ Xx @ C ) ) ) )
% 0.18/0.47 & ! [C: $i] :
% 0.18/0.47 ( ( ! [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ C )
% 0.18/0.47 => ( in @ Xx @ A ) )
% 0.18/0.47 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.18/0.47 => ? [D: $i,Xx: $i] :
% 0.18/0.47 ( ( in @ D @ B )
% 0.18/0.47 & ( in @ Xx @ C )
% 0.18/0.47 & ~ ? [Xy: $i] :
% 0.18/0.47 ( ( in @ Xy @ D )
% 0.18/0.47 & ( in @ Xy @ C ) )
% 0.18/0.47 & ! [E: $i] :
% 0.18/0.47 ( ( in @ E @ B )
% 0.18/0.47 => ( ! [Xy: $i] :
% 0.18/0.47 ( ( in @ Xy @ E )
% 0.18/0.47 => ( in @ Xy @ D ) )
% 0.18/0.47 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(descr_type,type,
% 0.18/0.47 descr: ( $i > $o ) > $i ).
% 0.18/0.47
% 0.18/0.47 thf(descrp_type,type,
% 0.18/0.47 descrp: $o ).
% 0.18/0.47
% 0.18/0.47 thf(descrp,definition,
% 0.18/0.47 ( descrp
% 0.18/0.47 = ( ! [Xphi: $i > $o] :
% 0.18/0.47 ( ( exu
% 0.18/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.47 => ( Xphi
% 0.18/0.47 @ ( descr
% 0.18/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(dsetconstr_type,type,
% 0.18/0.47 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.18/0.47
% 0.18/0.47 thf(dsetconstrI_type,type,
% 0.18/0.47 dsetconstrI: $o ).
% 0.18/0.47
% 0.18/0.47 thf(dsetconstrI,definition,
% 0.18/0.47 ( dsetconstrI
% 0.18/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ A )
% 0.18/0.47 => ( ( Xphi @ Xx )
% 0.18/0.47 => ( in @ Xx
% 0.18/0.47 @ ( dsetconstr @ A
% 0.18/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(dsetconstrEL_type,type,
% 0.18/0.47 dsetconstrEL: $o ).
% 0.18/0.47
% 0.18/0.47 thf(dsetconstrEL,definition,
% 0.18/0.47 ( dsetconstrEL
% 0.18/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.47 ( ( in @ Xx
% 0.18/0.47 @ ( dsetconstr @ A
% 0.18/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.18/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(dsetconstrER_type,type,
% 0.18/0.47 dsetconstrER: $o ).
% 0.18/0.47
% 0.18/0.47 thf(dsetconstrER,definition,
% 0.18/0.47 ( dsetconstrER
% 0.18/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.47 ( ( in @ Xx
% 0.18/0.47 @ ( dsetconstr @ A
% 0.18/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.18/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(exuE1_type,type,
% 0.18/0.47 exuE1: $o ).
% 0.18/0.47
% 0.18/0.47 thf(exuE1,definition,
% 0.18/0.47 ( exuE1
% 0.18/0.47 = ( ! [Xphi: $i > $o] :
% 0.18/0.47 ( ( exu
% 0.18/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.47 => ? [Xx: $i] :
% 0.18/0.47 ( ( Xphi @ Xx )
% 0.18/0.47 & ! [Xy: $i] :
% 0.18/0.47 ( ( Xphi @ Xy )
% 0.18/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(prop2set_type,type,
% 0.18/0.47 prop2set: $o > $i ).
% 0.18/0.47
% 0.18/0.47 thf(prop2setE_type,type,
% 0.18/0.47 prop2setE: $o ).
% 0.18/0.47
% 0.18/0.47 thf(prop2setE,definition,
% 0.18/0.47 ( prop2setE
% 0.18/0.47 = ( ! [Xphi: $o,Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.18/0.47 => Xphi ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(emptysetE_type,type,
% 0.18/0.47 emptysetE: $o ).
% 0.18/0.47
% 0.18/0.47 thf(emptysetE,definition,
% 0.18/0.47 ( emptysetE
% 0.18/0.47 = ( ! [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ emptyset )
% 0.18/0.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(emptysetimpfalse_type,type,
% 0.18/0.47 emptysetimpfalse: $o ).
% 0.18/0.47
% 0.18/0.47 thf(emptysetimpfalse,definition,
% 0.18/0.47 ( emptysetimpfalse
% 0.18/0.47 = ( ! [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ emptyset )
% 0.18/0.47 => $false ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(notinemptyset_type,type,
% 0.18/0.47 notinemptyset: $o ).
% 0.18/0.47
% 0.18/0.47 thf(notinemptyset,definition,
% 0.18/0.47 ( notinemptyset
% 0.18/0.47 = ( ! [Xx: $i] :
% 0.18/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(exuE3e_type,type,
% 0.18/0.47 exuE3e: $o ).
% 0.18/0.47
% 0.18/0.47 thf(exuE3e,definition,
% 0.18/0.47 ( exuE3e
% 0.18/0.47 = ( ! [Xphi: $i > $o] :
% 0.18/0.47 ( ( exu
% 0.18/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.47 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(setext_type,type,
% 0.18/0.47 setext: $o ).
% 0.18/0.47
% 0.18/0.47 thf(setext,definition,
% 0.18/0.47 ( setext
% 0.18/0.47 = ( ! [A: $i,B: $i] :
% 0.18/0.47 ( ! [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ A )
% 0.18/0.47 => ( in @ Xx @ B ) )
% 0.18/0.47 => ( ! [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ B )
% 0.18/0.47 => ( in @ Xx @ A ) )
% 0.18/0.47 => ( A = B ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(emptyI_type,type,
% 0.18/0.47 emptyI: $o ).
% 0.18/0.47
% 0.18/0.47 thf(emptyI,definition,
% 0.18/0.47 ( emptyI
% 0.18/0.47 = ( ! [A: $i] :
% 0.18/0.47 ( ! [Xx: $i] :
% 0.18/0.47 ~ ( in @ Xx @ A )
% 0.18/0.47 => ( A = emptyset ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(noeltsimpempty_type,type,
% 0.18/0.47 noeltsimpempty: $o ).
% 0.18/0.47
% 0.18/0.47 thf(noeltsimpempty,definition,
% 0.18/0.47 ( noeltsimpempty
% 0.18/0.47 = ( ! [A: $i] :
% 0.18/0.47 ( ! [Xx: $i] :
% 0.18/0.47 ~ ( in @ Xx @ A )
% 0.18/0.47 => ( A = emptyset ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(setbeta_type,type,
% 0.18/0.47 setbeta: $o ).
% 0.18/0.47
% 0.18/0.47 thf(setbeta,definition,
% 0.18/0.47 ( setbeta
% 0.18/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ A )
% 0.18/0.47 => ( ( in @ Xx
% 0.18/0.47 @ ( dsetconstr @ A
% 0.18/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.18/0.47 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(nonempty_type,type,
% 0.18/0.47 nonempty: $i > $o ).
% 0.18/0.47
% 0.18/0.47 thf(nonempty,definition,
% 0.18/0.47 ( nonempty
% 0.18/0.47 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(nonemptyE1_type,type,
% 0.18/0.47 nonemptyE1: $o ).
% 0.18/0.47
% 0.18/0.47 thf(nonemptyE1,definition,
% 0.18/0.47 ( nonemptyE1
% 0.18/0.47 = ( ! [A: $i] :
% 0.18/0.47 ( ( nonempty @ A )
% 0.18/0.47 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(nonemptyI_type,type,
% 0.18/0.47 nonemptyI: $o ).
% 0.18/0.47
% 0.18/0.47 thf(nonemptyI,definition,
% 0.18/0.47 ( nonemptyI
% 0.18/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ A )
% 0.18/0.47 => ( ( Xphi @ Xx )
% 0.18/0.47 => ( nonempty
% 0.18/0.47 @ ( dsetconstr @ A
% 0.18/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(nonemptyI1_type,type,
% 0.18/0.47 nonemptyI1: $o ).
% 0.18/0.47
% 0.18/0.47 thf(nonemptyI1,definition,
% 0.18/0.47 ( nonemptyI1
% 0.18/0.47 = ( ! [A: $i] :
% 0.18/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.18/0.47 => ( nonempty @ A ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(setadjoinIL_type,type,
% 0.18/0.47 setadjoinIL: $o ).
% 0.18/0.47
% 0.18/0.47 thf(setadjoinIL,definition,
% 0.18/0.47 ( setadjoinIL
% 0.18/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(emptyinunitempty_type,type,
% 0.18/0.47 emptyinunitempty: $o ).
% 0.18/0.47
% 0.18/0.47 thf(emptyinunitempty,definition,
% 0.18/0.47 ( emptyinunitempty
% 0.18/0.47 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(setadjoinIR_type,type,
% 0.18/0.47 setadjoinIR: $o ).
% 0.18/0.47
% 0.18/0.47 thf(setadjoinIR,definition,
% 0.18/0.47 ( setadjoinIR
% 0.18/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.18/0.47 ( ( in @ Xy @ A )
% 0.18/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(setadjoinE_type,type,
% 0.18/0.47 setadjoinE: $o ).
% 0.18/0.47
% 0.18/0.47 thf(setadjoinE,definition,
% 0.18/0.47 ( setadjoinE
% 0.18/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.18/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.18/0.47 => ! [Xphi: $o] :
% 0.18/0.47 ( ( ( Xy = Xx )
% 0.18/0.47 => Xphi )
% 0.18/0.47 => ( ( ( in @ Xy @ A )
% 0.18/0.47 => Xphi )
% 0.18/0.47 => Xphi ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(setadjoinOr_type,type,
% 0.18/0.47 setadjoinOr: $o ).
% 0.18/0.47
% 0.18/0.47 thf(setadjoinOr,definition,
% 0.18/0.47 ( setadjoinOr
% 0.18/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.18/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.18/0.47 => ( ( Xy = Xx )
% 0.18/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(setoftrueEq_type,type,
% 0.18/0.47 setoftrueEq: $o ).
% 0.18/0.47
% 0.18/0.47 thf(setoftrueEq,definition,
% 0.18/0.47 ( setoftrueEq
% 0.18/0.47 = ( ! [A: $i] :
% 0.18/0.47 ( ( dsetconstr @ A
% 0.18/0.47 @ ^ [Xx: $i] : $true )
% 0.18/0.47 = A ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(powersetI_type,type,
% 0.18/0.47 powersetI: $o ).
% 0.18/0.47
% 0.18/0.47 thf(powersetI,definition,
% 0.18/0.47 ( powersetI
% 0.18/0.47 = ( ! [A: $i,B: $i] :
% 0.18/0.47 ( ! [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ B )
% 0.18/0.47 => ( in @ Xx @ A ) )
% 0.18/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(emptyinPowerset_type,type,
% 0.18/0.47 emptyinPowerset: $o ).
% 0.18/0.47
% 0.18/0.47 thf(emptyinPowerset,definition,
% 0.18/0.47 ( emptyinPowerset
% 0.18/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(emptyInPowerset_type,type,
% 0.18/0.47 emptyInPowerset: $o ).
% 0.18/0.47
% 0.18/0.47 thf(emptyInPowerset,definition,
% 0.18/0.47 ( emptyInPowerset
% 0.18/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(powersetE_type,type,
% 0.18/0.47 powersetE: $o ).
% 0.18/0.47
% 0.18/0.47 thf(powersetE,definition,
% 0.18/0.47 ( powersetE
% 0.18/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.18/0.47 => ( ( in @ Xx @ B )
% 0.18/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(setunionI_type,type,
% 0.18/0.47 setunionI: $o ).
% 0.18/0.47
% 0.18/0.47 thf(setunionI,definition,
% 0.18/0.47 ( setunionI
% 0.18/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.18/0.47 ( ( in @ Xx @ B )
% 0.18/0.47 => ( ( in @ B @ A )
% 0.18/0.47 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(setunionE_type,type,
% 0.18/0.47 setunionE: $o ).
% 0.18/0.47
% 0.18/0.47 thf(setunionE,definition,
% 0.18/0.47 ( setunionE
% 0.18/0.47 = ( ! [A: $i,Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.18/0.47 => ! [Xphi: $o] :
% 0.18/0.47 ( ! [B: $i] :
% 0.18/0.47 ( ( in @ Xx @ B )
% 0.18/0.47 => ( ( in @ B @ A )
% 0.18/0.47 => Xphi ) )
% 0.18/0.47 => Xphi ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(subPowSU_type,type,
% 0.18/0.47 subPowSU: $o ).
% 0.18/0.47
% 0.18/0.47 thf(subPowSU,definition,
% 0.18/0.47 ( subPowSU
% 0.18/0.47 = ( ! [A: $i,Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ A )
% 0.18/0.47 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(exuE2_type,type,
% 0.18/0.47 exuE2: $o ).
% 0.18/0.47
% 0.18/0.47 thf(exuE2,definition,
% 0.18/0.47 ( exuE2
% 0.18/0.47 = ( ! [Xphi: $i > $o] :
% 0.18/0.47 ( ( exu
% 0.18/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.47 => ? [Xx: $i] :
% 0.18/0.47 ! [Xy: $i] :
% 0.18/0.47 ( ( Xphi @ Xy )
% 0.18/0.47 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(nonemptyImpWitness_type,type,
% 0.18/0.47 nonemptyImpWitness: $o ).
% 0.18/0.47
% 0.18/0.47 thf(nonemptyImpWitness,definition,
% 0.18/0.47 ( nonemptyImpWitness
% 0.18/0.47 = ( ! [A: $i] :
% 0.18/0.47 ( ( nonempty @ A )
% 0.18/0.47 => ? [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ A )
% 0.18/0.47 & $true ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(uniqinunit_type,type,
% 0.18/0.47 uniqinunit: $o ).
% 0.18/0.47
% 0.18/0.47 thf(uniqinunit,definition,
% 0.18/0.47 ( uniqinunit
% 0.18/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.18/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(notinsingleton_type,type,
% 0.18/0.47 notinsingleton: $o ).
% 0.18/0.47
% 0.18/0.47 thf(notinsingleton,definition,
% 0.18/0.47 ( notinsingleton
% 0.18/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.47 ( ( Xx != Xy )
% 0.18/0.47 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(eqinunit_type,type,
% 0.18/0.47 eqinunit: $o ).
% 0.18/0.47
% 0.18/0.47 thf(eqinunit,definition,
% 0.18/0.47 ( eqinunit
% 0.18/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.47 ( ( Xx = Xy )
% 0.18/0.47 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(singletonsswitch_type,type,
% 0.18/0.47 singletonsswitch: $o ).
% 0.18/0.47
% 0.18/0.47 thf(singletonsswitch,definition,
% 0.18/0.47 ( singletonsswitch
% 0.18/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.18/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(upairsetE_type,type,
% 0.18/0.47 upairsetE: $o ).
% 0.18/0.47
% 0.18/0.47 thf(upairsetE,definition,
% 0.18/0.47 ( upairsetE
% 0.18/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.18/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.18/0.47 => ( ( Xz = Xx )
% 0.18/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(upairsetIL_type,type,
% 0.18/0.47 upairsetIL: $o ).
% 0.18/0.47
% 0.18/0.47 thf(upairsetIL,definition,
% 0.18/0.47 ( upairsetIL
% 0.18/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(upairsetIR_type,type,
% 0.18/0.47 upairsetIR: $o ).
% 0.18/0.47
% 0.18/0.47 thf(upairsetIR,definition,
% 0.18/0.47 ( upairsetIR
% 0.18/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(emptyE1_type,type,
% 0.18/0.47 emptyE1: $o ).
% 0.18/0.47
% 0.18/0.47 thf(emptyE1,definition,
% 0.18/0.47 ( emptyE1
% 0.18/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.47 ( ? [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ A )
% 0.18/0.47 & ( Xphi @ Xx ) )
% 0.18/0.47 => ( ( ( dsetconstr @ A
% 0.18/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.47 = emptyset )
% 0.18/0.47 => $false ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(vacuousDall_type,type,
% 0.18/0.47 vacuousDall: $o ).
% 0.18/0.47
% 0.18/0.47 thf(vacuousDall,definition,
% 0.18/0.47 ( vacuousDall
% 0.18/0.47 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ emptyset )
% 0.18/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(quantDeMorgan1_type,type,
% 0.18/0.47 quantDeMorgan1: $o ).
% 0.18/0.47
% 0.18/0.47 thf(quantDeMorgan1,definition,
% 0.18/0.47 ( quantDeMorgan1
% 0.18/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.47 ( ~ ! [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ A )
% 0.18/0.47 => ( Xphi @ Xx ) )
% 0.18/0.47 => ? [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ A )
% 0.18/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(quantDeMorgan2_type,type,
% 0.18/0.47 quantDeMorgan2: $o ).
% 0.18/0.47
% 0.18/0.47 thf(quantDeMorgan2,definition,
% 0.18/0.47 ( quantDeMorgan2
% 0.18/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.47 ( ! [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ A )
% 0.18/0.47 => ~ ( Xphi @ Xx ) )
% 0.18/0.47 => ~ ? [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ A )
% 0.18/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(quantDeMorgan3_type,type,
% 0.18/0.47 quantDeMorgan3: $o ).
% 0.18/0.47
% 0.18/0.47 thf(quantDeMorgan3,definition,
% 0.18/0.47 ( quantDeMorgan3
% 0.18/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.47 ( ~ ? [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ A )
% 0.18/0.47 & ( Xphi @ Xx ) )
% 0.18/0.47 => ! [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ A )
% 0.18/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(quantDeMorgan4_type,type,
% 0.18/0.47 quantDeMorgan4: $o ).
% 0.18/0.47
% 0.18/0.47 thf(quantDeMorgan4,definition,
% 0.18/0.47 ( quantDeMorgan4
% 0.18/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.47 ( ? [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ A )
% 0.18/0.47 & ~ ( Xphi @ Xx ) )
% 0.18/0.47 => ~ ! [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ A )
% 0.18/0.47 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(prop2setI_type,type,
% 0.18/0.47 prop2setI: $o ).
% 0.18/0.47
% 0.18/0.47 thf(prop2setI,definition,
% 0.18/0.47 ( prop2setI
% 0.18/0.47 = ( ! [Xphi: $o] :
% 0.18/0.47 ( Xphi
% 0.18/0.47 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(set2prop_type,type,
% 0.18/0.47 set2prop: $i > $o ).
% 0.18/0.47
% 0.18/0.47 thf(prop2set2propI_type,type,
% 0.18/0.47 prop2set2propI: $o ).
% 0.18/0.47
% 0.18/0.47 thf(prop2set2propI,definition,
% 0.18/0.47 ( prop2set2propI
% 0.18/0.47 = ( ! [Xphi: $o] :
% 0.18/0.47 ( Xphi
% 0.18/0.47 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(notdexE_type,type,
% 0.18/0.47 notdexE: $o ).
% 0.18/0.47
% 0.18/0.47 thf(notdexE,definition,
% 0.18/0.47 ( notdexE
% 0.18/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.47 ( ~ ? [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ A )
% 0.18/0.47 & ( Xphi @ Xx ) )
% 0.18/0.47 => ! [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ A )
% 0.18/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(notdallE_type,type,
% 0.18/0.47 notdallE: $o ).
% 0.18/0.47
% 0.18/0.47 thf(notdallE,definition,
% 0.18/0.47 ( notdallE
% 0.18/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.47 ( ~ ! [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ A )
% 0.18/0.47 => ( Xphi @ Xx ) )
% 0.18/0.47 => ? [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ A )
% 0.18/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(exuI1_type,type,
% 0.18/0.47 exuI1: $o ).
% 0.18/0.47
% 0.18/0.47 thf(exuI1,definition,
% 0.18/0.47 ( exuI1
% 0.18/0.47 = ( ! [Xphi: $i > $o] :
% 0.18/0.47 ( ? [Xx: $i] :
% 0.18/0.47 ( ( Xphi @ Xx )
% 0.18/0.47 & ! [Xy: $i] :
% 0.18/0.47 ( ( Xphi @ Xy )
% 0.18/0.47 => ( Xx = Xy ) ) )
% 0.18/0.47 => ( exu
% 0.18/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(exuI3_type,type,
% 0.18/0.47 exuI3: $o ).
% 0.18/0.47
% 0.18/0.47 thf(exuI3,definition,
% 0.18/0.47 ( exuI3
% 0.18/0.47 = ( ! [Xphi: $i > $o] :
% 0.18/0.47 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.18/0.47 => ( ! [Xx: $i,Xy: $i] :
% 0.18/0.47 ( ( Xphi @ Xx )
% 0.18/0.47 => ( ( Xphi @ Xy )
% 0.18/0.47 => ( Xx = Xy ) ) )
% 0.18/0.47 => ( exu
% 0.18/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(exuI2_type,type,
% 0.18/0.47 exuI2: $o ).
% 0.18/0.47
% 0.18/0.47 thf(exuI2,definition,
% 0.18/0.47 ( exuI2
% 0.18/0.47 = ( ! [Xphi: $i > $o] :
% 0.18/0.47 ( ? [Xx: $i] :
% 0.18/0.47 ! [Xy: $i] :
% 0.18/0.47 ( ( Xphi @ Xy )
% 0.18/0.47 <=> ( Xy = Xx ) )
% 0.18/0.47 => ( exu
% 0.18/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(inCongP_type,type,
% 0.18/0.47 inCongP: $o ).
% 0.18/0.47
% 0.18/0.47 thf(inCongP,definition,
% 0.18/0.47 ( inCongP
% 0.18/0.47 = ( ! [A: $i,B: $i] :
% 0.18/0.47 ( ( A = B )
% 0.18/0.47 => ! [Xx: $i,Xy: $i] :
% 0.18/0.47 ( ( Xx = Xy )
% 0.18/0.47 => ( ( in @ Xx @ A )
% 0.18/0.47 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(in__Cong_type,type,
% 0.18/0.47 in__Cong: $o ).
% 0.18/0.47
% 0.18/0.47 thf(in__Cong,definition,
% 0.18/0.47 ( in__Cong
% 0.18/0.47 = ( ! [A: $i,B: $i] :
% 0.18/0.47 ( ( A = B )
% 0.18/0.47 => ! [Xx: $i,Xy: $i] :
% 0.18/0.47 ( ( Xx = Xy )
% 0.18/0.47 => ( ( in @ Xx @ A )
% 0.18/0.47 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(exuE3u_type,type,
% 0.18/0.47 exuE3u: $o ).
% 0.18/0.47
% 0.18/0.47 thf(exuE3u,definition,
% 0.18/0.47 ( exuE3u
% 0.18/0.47 = ( ! [Xphi: $i > $o] :
% 0.18/0.47 ( ( exu
% 0.18/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.47 => ! [Xx: $i,Xy: $i] :
% 0.18/0.47 ( ( Xphi @ Xx )
% 0.18/0.47 => ( ( Xphi @ Xy )
% 0.18/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(exu__Cong_type,type,
% 0.18/0.47 exu__Cong: $o ).
% 0.18/0.47
% 0.18/0.47 thf(exu__Cong,definition,
% 0.18/0.47 ( exu__Cong
% 0.18/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.18/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.18/0.47 ( ( Xx = Xy )
% 0.18/0.47 => ( ( Xphi @ Xx )
% 0.18/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.18/0.47 => ( ( exu
% 0.18/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.47 <=> ( exu
% 0.18/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(emptyset__Cong_type,type,
% 0.18/0.47 emptyset__Cong: $o ).
% 0.18/0.47
% 0.18/0.47 thf(emptyset__Cong,definition,
% 0.18/0.47 ( emptyset__Cong
% 0.18/0.47 = ( emptyset = emptyset ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(setadjoin__Cong_type,type,
% 0.18/0.47 setadjoin__Cong: $o ).
% 0.18/0.47
% 0.18/0.47 thf(setadjoin__Cong,definition,
% 0.18/0.47 ( setadjoin__Cong
% 0.18/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.47 ( ( Xx = Xy )
% 0.18/0.47 => ! [Xz: $i,Xu: $i] :
% 0.18/0.47 ( ( Xz = Xu )
% 0.18/0.47 => ( ( setadjoin @ Xx @ Xz )
% 0.18/0.47 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(powerset__Cong_type,type,
% 0.18/0.47 powerset__Cong: $o ).
% 0.18/0.47
% 0.18/0.47 thf(powerset__Cong,definition,
% 0.18/0.47 ( powerset__Cong
% 0.18/0.47 = ( ! [A: $i,B: $i] :
% 0.18/0.47 ( ( A = B )
% 0.18/0.47 => ( ( powerset @ A )
% 0.18/0.47 = ( powerset @ B ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(setunion__Cong_type,type,
% 0.18/0.47 setunion__Cong: $o ).
% 0.18/0.47
% 0.18/0.47 thf(setunion__Cong,definition,
% 0.18/0.47 ( setunion__Cong
% 0.18/0.47 = ( ! [A: $i,B: $i] :
% 0.18/0.47 ( ( A = B )
% 0.18/0.47 => ( ( setunion @ A )
% 0.18/0.47 = ( setunion @ B ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(omega__Cong_type,type,
% 0.18/0.47 omega__Cong: $o ).
% 0.18/0.47
% 0.18/0.47 thf(omega__Cong,definition,
% 0.18/0.47 ( omega__Cong
% 0.18/0.47 = ( omega = omega ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(exuEu_type,type,
% 0.18/0.47 exuEu: $o ).
% 0.18/0.47
% 0.18/0.47 thf(exuEu,definition,
% 0.18/0.47 ( exuEu
% 0.18/0.47 = ( ! [Xphi: $i > $o] :
% 0.18/0.47 ( ( exu
% 0.18/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.47 => ! [Xx: $i,Xy: $i] :
% 0.18/0.47 ( ( Xphi @ Xx )
% 0.18/0.47 => ( ( Xphi @ Xy )
% 0.18/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(descr__Cong_type,type,
% 0.18/0.47 descr__Cong: $o ).
% 0.18/0.47
% 0.18/0.47 thf(descr__Cong,definition,
% 0.18/0.47 ( descr__Cong
% 0.18/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.18/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.18/0.47 ( ( Xx = Xy )
% 0.18/0.47 => ( ( Xphi @ Xx )
% 0.18/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.18/0.47 => ( ( exu
% 0.18/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.47 => ( ( exu
% 0.18/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.18/0.47 => ( ( descr
% 0.18/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.47 = ( descr
% 0.18/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(dsetconstr__Cong_type,type,
% 0.18/0.47 dsetconstr__Cong: $o ).
% 0.18/0.47
% 0.18/0.47 thf(dsetconstr__Cong,definition,
% 0.18/0.47 ( dsetconstr__Cong
% 0.18/0.47 = ( ! [A: $i,B: $i] :
% 0.18/0.47 ( ( A = B )
% 0.18/0.47 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.18/0.47 ( ! [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ A )
% 0.18/0.47 => ! [Xy: $i] :
% 0.18/0.47 ( ( in @ Xy @ B )
% 0.18/0.47 => ( ( Xx = Xy )
% 0.18/0.47 => ( ( Xphi @ Xx )
% 0.18/0.47 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.18/0.47 => ( ( dsetconstr @ A
% 0.18/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.47 = ( dsetconstr @ B
% 0.18/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(subset_type,type,
% 0.18/0.47 subset: $i > $i > $o ).
% 0.18/0.47
% 0.18/0.47 thf(disjoint_type,type,
% 0.18/0.47 disjoint: $i > $i > $o ).
% 0.18/0.47
% 0.18/0.47 thf(setsmeet_type,type,
% 0.18/0.47 setsmeet: $i > $i > $o ).
% 0.18/0.47
% 0.18/0.47 thf(subsetI1_type,type,
% 0.18/0.47 subsetI1: $o ).
% 0.18/0.47
% 0.18/0.47 thf(subsetI1,definition,
% 0.18/0.47 ( subsetI1
% 0.18/0.47 = ( ! [A: $i,B: $i] :
% 0.18/0.47 ( ! [Xx: $i] :
% 0.18/0.47 ( ( in @ Xx @ A )
% 0.18/0.47 => ( in @ Xx @ B ) )
% 0.18/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(eqimpsubset2_type,type,
% 0.18/0.47 eqimpsubset2: $o ).
% 0.18/0.47
% 0.18/0.47 thf(eqimpsubset2,definition,
% 0.18/0.47 ( eqimpsubset2
% 0.18/0.47 = ( ! [A: $i,B: $i] :
% 0.18/0.47 ( ( A = B )
% 0.18/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(eqimpsubset1_type,type,
% 0.18/0.47 eqimpsubset1: $o ).
% 0.18/0.47
% 0.18/0.47 thf(eqimpsubset1,definition,
% 0.18/0.47 ( eqimpsubset1
% 0.18/0.47 = ( ! [A: $i,B: $i] :
% 0.18/0.47 ( ( A = B )
% 0.18/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.18/0.47
% 0.18/0.47 thf(subsetI2_type,type,
% 0.18/0.48 subsetI2: $o ).
% 0.18/0.48
% 0.18/0.48 thf(subsetI2,definition,
% 0.18/0.48 ( subsetI2
% 0.18/0.48 = ( ! [A: $i,B: $i] :
% 0.18/0.48 ( ! [Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 => ( in @ Xx @ B ) )
% 0.18/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(emptysetsubset_type,type,
% 0.18/0.48 emptysetsubset: $o ).
% 0.18/0.48
% 0.18/0.48 thf(emptysetsubset,definition,
% 0.18/0.48 ( emptysetsubset
% 0.18/0.48 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(subsetE_type,type,
% 0.18/0.48 subsetE: $o ).
% 0.18/0.48
% 0.18/0.48 thf(subsetE,definition,
% 0.18/0.48 ( subsetE
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ( subset @ A @ B )
% 0.18/0.48 => ( ( in @ Xx @ A )
% 0.18/0.48 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(subsetE2_type,type,
% 0.18/0.48 subsetE2: $o ).
% 0.18/0.48
% 0.18/0.48 thf(subsetE2,definition,
% 0.18/0.48 ( subsetE2
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ( subset @ A @ B )
% 0.18/0.48 => ( ~ ( in @ Xx @ B )
% 0.18/0.48 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(notsubsetI_type,type,
% 0.18/0.48 notsubsetI: $o ).
% 0.18/0.48
% 0.18/0.48 thf(notsubsetI,definition,
% 0.18/0.48 ( notsubsetI
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 => ( ~ ( in @ Xx @ B )
% 0.18/0.48 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(notequalI1_type,type,
% 0.18/0.48 notequalI1: $o ).
% 0.18/0.48
% 0.18/0.48 thf(notequalI1,definition,
% 0.18/0.48 ( notequalI1
% 0.18/0.48 = ( ! [A: $i,B: $i] :
% 0.18/0.48 ( ~ ( subset @ A @ B )
% 0.18/0.48 => ( A != B ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(notequalI2_type,type,
% 0.18/0.48 notequalI2: $o ).
% 0.18/0.48
% 0.18/0.48 thf(notequalI2,definition,
% 0.18/0.48 ( notequalI2
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 => ( ~ ( in @ Xx @ B )
% 0.18/0.48 => ( A != B ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(subsetRefl_type,type,
% 0.18/0.48 subsetRefl: $o ).
% 0.18/0.48
% 0.18/0.48 thf(subsetRefl,definition,
% 0.18/0.48 ( subsetRefl
% 0.18/0.48 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(subsetTrans_type,type,
% 0.18/0.48 subsetTrans: $o ).
% 0.18/0.48
% 0.18/0.48 thf(subsetTrans,definition,
% 0.18/0.48 ( subsetTrans
% 0.18/0.48 = ( ! [A: $i,B: $i,C: $i] :
% 0.18/0.48 ( ( subset @ A @ B )
% 0.18/0.48 => ( ( subset @ B @ C )
% 0.18/0.48 => ( subset @ A @ C ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(setadjoinSub_type,type,
% 0.18/0.48 setadjoinSub: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setadjoinSub,definition,
% 0.18/0.48 ( setadjoinSub
% 0.18/0.48 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(setadjoinSub2_type,type,
% 0.18/0.48 setadjoinSub2: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setadjoinSub2,definition,
% 0.18/0.48 ( setadjoinSub2
% 0.18/0.48 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.18/0.48 ( ( subset @ A @ B )
% 0.18/0.48 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(subset2powerset_type,type,
% 0.18/0.48 subset2powerset: $o ).
% 0.18/0.48
% 0.18/0.48 thf(subset2powerset,definition,
% 0.18/0.48 ( subset2powerset
% 0.18/0.48 = ( ! [A: $i,B: $i] :
% 0.18/0.48 ( ( subset @ A @ B )
% 0.18/0.48 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(setextsub_type,type,
% 0.18/0.48 setextsub: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setextsub,definition,
% 0.18/0.48 ( setextsub
% 0.18/0.48 = ( ! [A: $i,B: $i] :
% 0.18/0.48 ( ( subset @ A @ B )
% 0.18/0.48 => ( ( subset @ B @ A )
% 0.18/0.48 => ( A = B ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(subsetemptysetimpeq_type,type,
% 0.18/0.48 subsetemptysetimpeq: $o ).
% 0.18/0.48
% 0.18/0.48 thf(subsetemptysetimpeq,definition,
% 0.18/0.48 ( subsetemptysetimpeq
% 0.18/0.48 = ( ! [A: $i] :
% 0.18/0.48 ( ( subset @ A @ emptyset )
% 0.18/0.48 => ( A = emptyset ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(powersetI1_type,type,
% 0.18/0.48 powersetI1: $o ).
% 0.18/0.48
% 0.18/0.48 thf(powersetI1,definition,
% 0.18/0.48 ( powersetI1
% 0.18/0.48 = ( ! [A: $i,B: $i] :
% 0.18/0.48 ( ( subset @ B @ A )
% 0.18/0.48 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(powersetE1_type,type,
% 0.18/0.48 powersetE1: $o ).
% 0.18/0.48
% 0.18/0.48 thf(powersetE1,definition,
% 0.18/0.48 ( powersetE1
% 0.18/0.48 = ( ! [A: $i,B: $i] :
% 0.18/0.48 ( ( in @ B @ ( powerset @ A ) )
% 0.18/0.48 => ( subset @ B @ A ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(inPowerset_type,type,
% 0.18/0.48 inPowerset: $o ).
% 0.18/0.48
% 0.18/0.48 thf(inPowerset,definition,
% 0.18/0.48 ( inPowerset
% 0.18/0.48 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(powersetsubset_type,type,
% 0.18/0.48 powersetsubset: $o ).
% 0.18/0.48
% 0.18/0.48 thf(powersetsubset,definition,
% 0.18/0.48 ( powersetsubset
% 0.18/0.48 = ( ! [A: $i,B: $i] :
% 0.18/0.48 ( ( subset @ A @ B )
% 0.18/0.48 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(sepInPowerset_type,type,
% 0.18/0.48 sepInPowerset: $o ).
% 0.18/0.48
% 0.18/0.48 thf(sepInPowerset,definition,
% 0.18/0.48 ( sepInPowerset
% 0.18/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.48 ( in
% 0.18/0.48 @ ( dsetconstr @ A
% 0.18/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.48 @ ( powerset @ A ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(sepSubset_type,type,
% 0.18/0.48 sepSubset: $o ).
% 0.18/0.48
% 0.18/0.48 thf(sepSubset,definition,
% 0.18/0.48 ( sepSubset
% 0.18/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.48 ( subset
% 0.18/0.48 @ ( dsetconstr @ A
% 0.18/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.48 @ A ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(binunion_type,type,
% 0.18/0.48 binunion: $i > $i > $i ).
% 0.18/0.48
% 0.18/0.48 thf(binunionIL_type,type,
% 0.18/0.48 binunionIL: $o ).
% 0.18/0.48
% 0.18/0.48 thf(binunionIL,definition,
% 0.18/0.48 ( binunionIL
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(upairset2IR_type,type,
% 0.18/0.48 upairset2IR: $o ).
% 0.18/0.48
% 0.18/0.48 thf(upairset2IR,definition,
% 0.18/0.48 ( upairset2IR
% 0.18/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(binunionIR_type,type,
% 0.18/0.48 binunionIR: $o ).
% 0.18/0.48
% 0.18/0.48 thf(binunionIR,definition,
% 0.18/0.48 ( binunionIR
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ B )
% 0.18/0.48 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(binunionEcases_type,type,
% 0.18/0.48 binunionEcases: $o ).
% 0.18/0.48
% 0.18/0.48 thf(binunionEcases,definition,
% 0.18/0.48 ( binunionEcases
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.18/0.48 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.18/0.48 => ( ( ( in @ Xx @ A )
% 0.18/0.48 => Xphi )
% 0.18/0.48 => ( ( ( in @ Xx @ B )
% 0.18/0.48 => Xphi )
% 0.18/0.48 => Xphi ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(binunionE_type,type,
% 0.18/0.48 binunionE: $o ).
% 0.18/0.48
% 0.18/0.48 thf(binunionE,definition,
% 0.18/0.48 ( binunionE
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.18/0.48 => ( ( in @ Xx @ A )
% 0.18/0.48 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(binunionLsub_type,type,
% 0.18/0.48 binunionLsub: $o ).
% 0.18/0.48
% 0.18/0.48 thf(binunionLsub,definition,
% 0.18/0.48 ( binunionLsub
% 0.18/0.48 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(binunionRsub_type,type,
% 0.18/0.48 binunionRsub: $o ).
% 0.18/0.48
% 0.18/0.48 thf(binunionRsub,definition,
% 0.18/0.48 ( binunionRsub
% 0.18/0.48 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(binintersect_type,type,
% 0.18/0.48 binintersect: $i > $i > $i ).
% 0.18/0.48
% 0.18/0.48 thf(binintersectI_type,type,
% 0.18/0.48 binintersectI: $o ).
% 0.18/0.48
% 0.18/0.48 thf(binintersectI,definition,
% 0.18/0.48 ( binintersectI
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 => ( ( in @ Xx @ B )
% 0.18/0.48 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(binintersectSubset5_type,type,
% 0.18/0.48 binintersectSubset5: $o ).
% 0.18/0.48
% 0.18/0.48 thf(binintersectSubset5,definition,
% 0.18/0.48 ( binintersectSubset5
% 0.18/0.48 = ( ! [A: $i,B: $i,C: $i] :
% 0.18/0.48 ( ( subset @ C @ A )
% 0.18/0.48 => ( ( subset @ C @ B )
% 0.18/0.48 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(binintersectEL_type,type,
% 0.18/0.48 binintersectEL: $o ).
% 0.18/0.48
% 0.18/0.48 thf(binintersectEL,definition,
% 0.18/0.48 ( binintersectEL
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.18/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(binintersectLsub_type,type,
% 0.18/0.48 binintersectLsub: $o ).
% 0.18/0.48
% 0.18/0.48 thf(binintersectLsub,definition,
% 0.18/0.48 ( binintersectLsub
% 0.18/0.48 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(binintersectSubset2_type,type,
% 0.18/0.48 binintersectSubset2: $o ).
% 0.18/0.48
% 0.18/0.48 thf(binintersectSubset2,definition,
% 0.18/0.48 ( binintersectSubset2
% 0.18/0.48 = ( ! [A: $i,B: $i] :
% 0.18/0.48 ( ( subset @ A @ B )
% 0.18/0.48 => ( ( binintersect @ A @ B )
% 0.18/0.48 = A ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(binintersectSubset3_type,type,
% 0.18/0.48 binintersectSubset3: $o ).
% 0.18/0.48
% 0.18/0.48 thf(binintersectSubset3,definition,
% 0.18/0.48 ( binintersectSubset3
% 0.18/0.48 = ( ! [A: $i,B: $i] :
% 0.18/0.48 ( ( ( binintersect @ A @ B )
% 0.18/0.48 = B )
% 0.18/0.48 => ( subset @ B @ A ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(binintersectER_type,type,
% 0.18/0.48 binintersectER: $o ).
% 0.18/0.48
% 0.18/0.48 thf(binintersectER,definition,
% 0.18/0.48 ( binintersectER
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.18/0.48 => ( in @ Xx @ B ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(disjointsetsI1_type,type,
% 0.18/0.48 disjointsetsI1: $o ).
% 0.18/0.48
% 0.18/0.48 thf(disjointsetsI1,definition,
% 0.18/0.48 ( disjointsetsI1
% 0.18/0.48 = ( ! [A: $i,B: $i] :
% 0.18/0.48 ( ~ ? [Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 & ( in @ Xx @ B ) )
% 0.18/0.48 => ( ( binintersect @ A @ B )
% 0.18/0.48 = emptyset ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(binintersectRsub_type,type,
% 0.18/0.48 binintersectRsub: $o ).
% 0.18/0.48
% 0.18/0.48 thf(binintersectRsub,definition,
% 0.18/0.48 ( binintersectRsub
% 0.18/0.48 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(binintersectSubset4_type,type,
% 0.18/0.48 binintersectSubset4: $o ).
% 0.18/0.48
% 0.18/0.48 thf(binintersectSubset4,definition,
% 0.18/0.48 ( binintersectSubset4
% 0.18/0.48 = ( ! [A: $i,B: $i] :
% 0.18/0.48 ( ( subset @ B @ A )
% 0.18/0.48 => ( ( binintersect @ A @ B )
% 0.18/0.48 = B ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(binintersectSubset1_type,type,
% 0.18/0.48 binintersectSubset1: $o ).
% 0.18/0.48
% 0.18/0.48 thf(binintersectSubset1,definition,
% 0.18/0.48 ( binintersectSubset1
% 0.18/0.48 = ( ! [A: $i,B: $i] :
% 0.18/0.48 ( ( ( binintersect @ A @ B )
% 0.18/0.48 = A )
% 0.18/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(bs114d_type,type,
% 0.18/0.48 bs114d: $o ).
% 0.18/0.48
% 0.18/0.48 thf(bs114d,definition,
% 0.18/0.48 ( bs114d
% 0.18/0.48 = ( ! [A: $i,B: $i,C: $i] :
% 0.18/0.48 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.18/0.48 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(regular_type,type,
% 0.18/0.48 regular: $i > $o ).
% 0.18/0.48
% 0.18/0.48 thf(setminus_type,type,
% 0.18/0.48 setminus: $i > $i > $i ).
% 0.18/0.48
% 0.18/0.48 thf(setminusI_type,type,
% 0.18/0.48 setminusI: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setminusI,definition,
% 0.18/0.48 ( setminusI
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 => ( ~ ( in @ Xx @ B )
% 0.18/0.48 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(setminusEL_type,type,
% 0.18/0.48 setminusEL: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setminusEL,definition,
% 0.18/0.48 ( setminusEL
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.18/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(setminusER_type,type,
% 0.18/0.48 setminusER: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setminusER,definition,
% 0.18/0.48 ( setminusER
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.18/0.48 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(setminusSubset2_type,type,
% 0.18/0.48 setminusSubset2: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setminusSubset2,definition,
% 0.18/0.48 ( setminusSubset2
% 0.18/0.48 = ( ! [A: $i,B: $i] :
% 0.18/0.48 ( ( subset @ A @ B )
% 0.18/0.48 => ( ( setminus @ A @ B )
% 0.18/0.48 = emptyset ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(setminusERneg_type,type,
% 0.18/0.48 setminusERneg: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setminusERneg,definition,
% 0.18/0.48 ( setminusERneg
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.18/0.48 => ( ( in @ Xx @ A )
% 0.18/0.48 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(setminusELneg_type,type,
% 0.18/0.48 setminusELneg: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setminusELneg,definition,
% 0.18/0.48 ( setminusELneg
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.18/0.48 => ( ~ ( in @ Xx @ B )
% 0.18/0.48 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(setminusILneg_type,type,
% 0.18/0.48 setminusILneg: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setminusILneg,definition,
% 0.18/0.48 ( setminusILneg
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ~ ( in @ Xx @ A )
% 0.18/0.48 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(setminusIRneg_type,type,
% 0.18/0.48 setminusIRneg: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setminusIRneg,definition,
% 0.18/0.48 ( setminusIRneg
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ B )
% 0.18/0.48 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(setminusLsub_type,type,
% 0.18/0.48 setminusLsub: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setminusLsub,definition,
% 0.18/0.48 ( setminusLsub
% 0.18/0.48 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(setminusSubset1_type,type,
% 0.18/0.48 setminusSubset1: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setminusSubset1,definition,
% 0.18/0.48 ( setminusSubset1
% 0.18/0.48 = ( ! [A: $i,B: $i] :
% 0.18/0.48 ( ( ( setminus @ A @ B )
% 0.18/0.48 = emptyset )
% 0.18/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(symdiff_type,type,
% 0.18/0.48 symdiff: $i > $i > $i ).
% 0.18/0.48
% 0.18/0.48 thf(symdiffE_type,type,
% 0.18/0.48 symdiffE: $o ).
% 0.18/0.48
% 0.18/0.48 thf(symdiffE,definition,
% 0.18/0.48 ( symdiffE
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.18/0.48 => ! [Xphi: $o] :
% 0.18/0.48 ( ( ( in @ Xx @ A )
% 0.18/0.48 => ( ~ ( in @ Xx @ B )
% 0.18/0.48 => Xphi ) )
% 0.18/0.48 => ( ( ~ ( in @ Xx @ A )
% 0.18/0.48 => ( ( in @ Xx @ B )
% 0.18/0.48 => Xphi ) )
% 0.18/0.48 => Xphi ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(symdiffI1_type,type,
% 0.18/0.48 symdiffI1: $o ).
% 0.18/0.48
% 0.18/0.48 thf(symdiffI1,definition,
% 0.18/0.48 ( symdiffI1
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 => ( ~ ( in @ Xx @ B )
% 0.18/0.48 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(symdiffI2_type,type,
% 0.18/0.48 symdiffI2: $o ).
% 0.18/0.48
% 0.18/0.48 thf(symdiffI2,definition,
% 0.18/0.48 ( symdiffI2
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ~ ( in @ Xx @ A )
% 0.18/0.48 => ( ( in @ Xx @ B )
% 0.18/0.48 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(symdiffIneg1_type,type,
% 0.18/0.48 symdiffIneg1: $o ).
% 0.18/0.48
% 0.18/0.48 thf(symdiffIneg1,definition,
% 0.18/0.48 ( symdiffIneg1
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 => ( ( in @ Xx @ B )
% 0.18/0.48 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(symdiffIneg2_type,type,
% 0.18/0.48 symdiffIneg2: $o ).
% 0.18/0.48
% 0.18/0.48 thf(symdiffIneg2,definition,
% 0.18/0.48 ( symdiffIneg2
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ~ ( in @ Xx @ A )
% 0.18/0.48 => ( ~ ( in @ Xx @ B )
% 0.18/0.48 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(iskpair_type,type,
% 0.18/0.48 iskpair: $i > $o ).
% 0.18/0.48
% 0.18/0.48 thf(secondinupair_type,type,
% 0.18/0.48 secondinupair: $o ).
% 0.18/0.48
% 0.18/0.48 thf(secondinupair,definition,
% 0.18/0.48 ( secondinupair
% 0.18/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(setukpairIL_type,type,
% 0.18/0.48 setukpairIL: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setukpairIL,definition,
% 0.18/0.48 ( setukpairIL
% 0.18/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(setukpairIR_type,type,
% 0.18/0.48 setukpairIR: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setukpairIR,definition,
% 0.18/0.48 ( setukpairIR
% 0.18/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(kpairiskpair_type,type,
% 0.18/0.48 kpairiskpair: $o ).
% 0.18/0.48
% 0.18/0.48 thf(kpairiskpair,definition,
% 0.18/0.48 ( kpairiskpair
% 0.18/0.48 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(kpair_type,type,
% 0.18/0.48 kpair: $i > $i > $i ).
% 0.18/0.48
% 0.18/0.48 thf(kpairp_type,type,
% 0.18/0.48 kpairp: $o ).
% 0.18/0.48
% 0.18/0.48 thf(kpairp,definition,
% 0.18/0.48 ( kpairp
% 0.18/0.48 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(cartprod_type,type,
% 0.18/0.48 cartprod: $i > $i > $i ).
% 0.18/0.48
% 0.18/0.48 thf(singletonsubset_type,type,
% 0.18/0.48 singletonsubset: $o ).
% 0.18/0.48
% 0.18/0.48 thf(singletonsubset,definition,
% 0.18/0.48 ( singletonsubset
% 0.18/0.48 = ( ! [A: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(singletoninpowerset_type,type,
% 0.18/0.48 singletoninpowerset: $o ).
% 0.18/0.48
% 0.18/0.48 thf(singletoninpowerset,definition,
% 0.18/0.48 ( singletoninpowerset
% 0.18/0.48 = ( ! [A: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(singletoninpowunion_type,type,
% 0.18/0.48 singletoninpowunion: $o ).
% 0.18/0.48
% 0.18/0.48 thf(singletoninpowunion,definition,
% 0.18/0.48 ( singletoninpowunion
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(upairset2E_type,type,
% 0.18/0.48 upairset2E: $o ).
% 0.18/0.48
% 0.18/0.48 thf(upairset2E,definition,
% 0.18/0.48 ( upairset2E
% 0.18/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.18/0.48 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.18/0.48 => ( ( Xz = Xx )
% 0.18/0.48 | ( Xz = Xy ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(upairsubunion_type,type,
% 0.18/0.48 upairsubunion: $o ).
% 0.18/0.48
% 0.18/0.48 thf(upairsubunion,definition,
% 0.18/0.48 ( upairsubunion
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 => ! [Xy: $i] :
% 0.18/0.48 ( ( in @ Xy @ B )
% 0.18/0.48 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(upairinpowunion_type,type,
% 0.18/0.48 upairinpowunion: $o ).
% 0.18/0.48
% 0.18/0.48 thf(upairinpowunion,definition,
% 0.18/0.48 ( upairinpowunion
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 => ! [Xy: $i] :
% 0.18/0.48 ( ( in @ Xy @ B )
% 0.18/0.48 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(ubforcartprodlem1_type,type,
% 0.18/0.48 ubforcartprodlem1: $o ).
% 0.18/0.48
% 0.18/0.48 thf(ubforcartprodlem1,definition,
% 0.18/0.48 ( ubforcartprodlem1
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 => ! [Xy: $i] :
% 0.18/0.48 ( ( in @ Xy @ B )
% 0.18/0.48 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(ubforcartprodlem2_type,type,
% 0.18/0.48 ubforcartprodlem2: $o ).
% 0.18/0.48
% 0.18/0.48 thf(ubforcartprodlem2,definition,
% 0.18/0.48 ( ubforcartprodlem2
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 => ! [Xy: $i] :
% 0.18/0.48 ( ( in @ Xy @ B )
% 0.18/0.48 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(ubforcartprodlem3_type,type,
% 0.18/0.48 ubforcartprodlem3: $o ).
% 0.18/0.48
% 0.18/0.48 thf(ubforcartprodlem3,definition,
% 0.18/0.48 ( ubforcartprodlem3
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 => ! [Xy: $i] :
% 0.18/0.48 ( ( in @ Xy @ B )
% 0.18/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(cartprodpairin_type,type,
% 0.18/0.48 cartprodpairin: $o ).
% 0.18/0.48
% 0.18/0.48 thf(cartprodpairin,definition,
% 0.18/0.48 ( cartprodpairin
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 => ! [Xy: $i] :
% 0.18/0.48 ( ( in @ Xy @ B )
% 0.18/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(cartprodmempair1_type,type,
% 0.18/0.48 cartprodmempair1: $o ).
% 0.18/0.48
% 0.18/0.48 thf(cartprodmempair1,definition,
% 0.18/0.48 ( cartprodmempair1
% 0.18/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.18/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.18/0.48 => ? [Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 & ? [Xy: $i] :
% 0.18/0.48 ( ( in @ Xy @ B )
% 0.18/0.48 & ( Xu
% 0.18/0.48 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(cartprodmempair_type,type,
% 0.18/0.48 cartprodmempair: $o ).
% 0.18/0.48
% 0.18/0.48 thf(cartprodmempair,definition,
% 0.18/0.48 ( cartprodmempair
% 0.18/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.18/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.18/0.48 => ( iskpair @ Xu ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(setunionE2_type,type,
% 0.18/0.48 setunionE2: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setunionE2,definition,
% 0.18/0.48 ( setunionE2
% 0.18/0.48 = ( ! [A: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ ( setunion @ A ) )
% 0.18/0.48 => ? [X: $i] :
% 0.18/0.48 ( ( in @ X @ A )
% 0.18/0.48 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(setunionsingleton1_type,type,
% 0.18/0.48 setunionsingleton1: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setunionsingleton1,definition,
% 0.18/0.48 ( setunionsingleton1
% 0.18/0.48 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(setunionsingleton2_type,type,
% 0.18/0.48 setunionsingleton2: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setunionsingleton2,definition,
% 0.18/0.48 ( setunionsingleton2
% 0.18/0.48 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(setunionsingleton_type,type,
% 0.18/0.48 setunionsingleton: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setunionsingleton,definition,
% 0.18/0.48 ( setunionsingleton
% 0.18/0.48 = ( ! [Xx: $i] :
% 0.18/0.48 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.18/0.48 = Xx ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(singleton_type,type,
% 0.18/0.48 singleton: $i > $o ).
% 0.18/0.48
% 0.18/0.48 thf(singleton,definition,
% 0.18/0.48 ( singleton
% 0.18/0.48 = ( ^ [A: $i] :
% 0.18/0.48 ? [Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 & ( A
% 0.18/0.48 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(singletonprop_type,type,
% 0.18/0.48 singletonprop: $o ).
% 0.18/0.48
% 0.18/0.48 thf(singletonprop,definition,
% 0.18/0.48 ( singletonprop
% 0.18/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.48 ( ! [Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 => ! [Xy: $i] :
% 0.18/0.48 ( ( in @ Xy @ A )
% 0.18/0.48 => ( ( Xphi @ Xx )
% 0.18/0.48 => ( ( Xphi @ Xy )
% 0.18/0.48 => ( Xx = Xy ) ) ) ) )
% 0.18/0.48 => ( ? [Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 & ( Xphi @ Xx ) )
% 0.18/0.48 => ( singleton
% 0.18/0.48 @ ( dsetconstr @ A
% 0.18/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(ex1_type,type,
% 0.18/0.48 ex1: $i > ( $i > $o ) > $o ).
% 0.18/0.48
% 0.18/0.48 thf(ex1,definition,
% 0.18/0.48 ( ex1
% 0.18/0.48 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.18/0.48 ( singleton
% 0.18/0.48 @ ( dsetconstr @ A
% 0.18/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(ex1E1_type,type,
% 0.18/0.48 ex1E1: $o ).
% 0.18/0.48
% 0.18/0.48 thf(ex1E1,definition,
% 0.18/0.48 ( ex1E1
% 0.18/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.48 ( ( ex1 @ A
% 0.18/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.48 => ? [Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(ex1I_type,type,
% 0.18/0.48 ex1I: $o ).
% 0.18/0.48
% 0.18/0.48 thf(ex1I,definition,
% 0.18/0.48 ( ex1I
% 0.18/0.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 => ( ( Xphi @ Xx )
% 0.18/0.48 => ( ! [Xy: $i] :
% 0.18/0.48 ( ( in @ Xy @ A )
% 0.18/0.48 => ( ( Xphi @ Xy )
% 0.18/0.48 => ( Xy = Xx ) ) )
% 0.18/0.48 => ( ex1 @ A
% 0.18/0.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(ex1I2_type,type,
% 0.18/0.48 ex1I2: $o ).
% 0.18/0.48
% 0.18/0.48 thf(ex1I2,definition,
% 0.18/0.48 ( ex1I2
% 0.18/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.48 ( ! [Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 => ! [Xy: $i] :
% 0.18/0.48 ( ( in @ Xy @ A )
% 0.18/0.48 => ( ( Xphi @ Xx )
% 0.18/0.48 => ( ( Xphi @ Xy )
% 0.18/0.48 => ( Xx = Xy ) ) ) ) )
% 0.18/0.48 => ( ? [Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 & ( Xphi @ Xx ) )
% 0.18/0.48 => ( ex1 @ A
% 0.18/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(singletonsuniq_type,type,
% 0.18/0.48 singletonsuniq: $o ).
% 0.18/0.48
% 0.18/0.48 thf(singletonsuniq,definition,
% 0.18/0.48 ( singletonsuniq
% 0.18/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.48 ( ( ( setadjoin @ Xx @ emptyset )
% 0.18/0.48 = ( setadjoin @ Xy @ emptyset ) )
% 0.18/0.48 => ( Xx = Xy ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(atmost1p_type,type,
% 0.18/0.48 atmost1p: $i > $o ).
% 0.18/0.48
% 0.18/0.48 thf(atleast2p_type,type,
% 0.18/0.48 atleast2p: $i > $o ).
% 0.18/0.48
% 0.18/0.48 thf(atmost2p_type,type,
% 0.18/0.48 atmost2p: $i > $o ).
% 0.18/0.48
% 0.18/0.48 thf(upairsetp_type,type,
% 0.18/0.48 upairsetp: $i > $o ).
% 0.18/0.48
% 0.18/0.48 thf(setukpairinjL1_type,type,
% 0.18/0.48 setukpairinjL1: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setukpairinjL1,definition,
% 0.18/0.48 ( setukpairinjL1
% 0.18/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.18/0.48 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.18/0.48 => ( Xx = Xz ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(kfstsingleton_type,type,
% 0.18/0.48 kfstsingleton: $o ).
% 0.18/0.48
% 0.18/0.48 thf(kfstsingleton,definition,
% 0.18/0.48 ( kfstsingleton
% 0.18/0.48 = ( ! [Xu: $i] :
% 0.18/0.48 ( ( iskpair @ Xu )
% 0.18/0.48 => ( singleton
% 0.18/0.48 @ ( dsetconstr @ ( setunion @ Xu )
% 0.18/0.48 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(theprop_type,type,
% 0.18/0.48 theprop: $o ).
% 0.18/0.48
% 0.18/0.48 thf(theprop,definition,
% 0.18/0.48 ( theprop
% 0.18/0.48 = ( ! [X: $i] :
% 0.18/0.48 ( ( singleton @ X )
% 0.18/0.48 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(kfst_type,type,
% 0.18/0.48 kfst: $i > $i ).
% 0.18/0.48
% 0.18/0.48 thf(kfstpairEq_type,type,
% 0.18/0.48 kfstpairEq: $o ).
% 0.18/0.48
% 0.18/0.48 thf(kfstpairEq,definition,
% 0.18/0.48 ( kfstpairEq
% 0.18/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.48 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.18/0.48 = Xx ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(cartprodfstin_type,type,
% 0.18/0.48 cartprodfstin: $o ).
% 0.18/0.48
% 0.18/0.48 thf(cartprodfstin,definition,
% 0.18/0.48 ( cartprodfstin
% 0.18/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.18/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.18/0.48 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(setukpairinjL2_type,type,
% 0.18/0.48 setukpairinjL2: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setukpairinjL2,definition,
% 0.18/0.48 ( setukpairinjL2
% 0.18/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.18/0.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.18/0.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.18/0.48 => ( Xx = Xz ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(setukpairinjL_type,type,
% 0.18/0.48 setukpairinjL: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setukpairinjL,definition,
% 0.18/0.48 ( setukpairinjL
% 0.18/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.18/0.48 ( ( ( kpair @ Xx @ Xy )
% 0.18/0.48 = ( kpair @ Xz @ Xu ) )
% 0.18/0.48 => ( Xx = Xz ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(setukpairinjR11_type,type,
% 0.18/0.48 setukpairinjR11: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setukpairinjR11,definition,
% 0.18/0.48 ( setukpairinjR11
% 0.18/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.48 ( ( Xx = Xy )
% 0.18/0.48 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.18/0.48 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(setukpairinjR12_type,type,
% 0.18/0.48 setukpairinjR12: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setukpairinjR12,definition,
% 0.18/0.48 ( setukpairinjR12
% 0.18/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.48 ( ( Xx = Xy )
% 0.18/0.48 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.18/0.48 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(setukpairinjR1_type,type,
% 0.18/0.48 setukpairinjR1: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setukpairinjR1,definition,
% 0.18/0.48 ( setukpairinjR1
% 0.18/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.18/0.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.18/0.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.18/0.48 => ( ( Xz = Xu )
% 0.18/0.48 => ( Xy = Xu ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(upairequniteq_type,type,
% 0.18/0.48 upairequniteq: $o ).
% 0.18/0.48
% 0.18/0.48 thf(upairequniteq,definition,
% 0.18/0.48 ( upairequniteq
% 0.18/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.18/0.48 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.18/0.48 = ( setadjoin @ Xz @ emptyset ) )
% 0.18/0.48 => ( Xx = Xy ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(setukpairinjR2_type,type,
% 0.18/0.48 setukpairinjR2: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setukpairinjR2,definition,
% 0.18/0.48 ( setukpairinjR2
% 0.18/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.18/0.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.18/0.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.18/0.48 => ( Xy = Xu ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(setukpairinjR_type,type,
% 0.18/0.48 setukpairinjR: $o ).
% 0.18/0.48
% 0.18/0.48 thf(setukpairinjR,definition,
% 0.18/0.48 ( setukpairinjR
% 0.18/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.18/0.48 ( ( ( kpair @ Xx @ Xy )
% 0.18/0.48 = ( kpair @ Xz @ Xu ) )
% 0.18/0.48 => ( Xy = Xu ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(ksndsingleton_type,type,
% 0.18/0.48 ksndsingleton: $o ).
% 0.18/0.48
% 0.18/0.48 thf(ksndsingleton,definition,
% 0.18/0.48 ( ksndsingleton
% 0.18/0.48 = ( ! [Xu: $i] :
% 0.18/0.48 ( ( iskpair @ Xu )
% 0.18/0.48 => ( singleton
% 0.18/0.48 @ ( dsetconstr @ ( setunion @ Xu )
% 0.18/0.48 @ ^ [Xx: $i] :
% 0.18/0.48 ( Xu
% 0.18/0.48 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(ksnd_type,type,
% 0.18/0.48 ksnd: $i > $i ).
% 0.18/0.48
% 0.18/0.48 thf(ksndpairEq_type,type,
% 0.18/0.48 ksndpairEq: $o ).
% 0.18/0.48
% 0.18/0.48 thf(ksndpairEq,definition,
% 0.18/0.48 ( ksndpairEq
% 0.18/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.48 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.18/0.48 = Xy ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(kpairsurjEq_type,type,
% 0.18/0.48 kpairsurjEq: $o ).
% 0.18/0.48
% 0.18/0.48 thf(kpairsurjEq,definition,
% 0.18/0.48 ( kpairsurjEq
% 0.18/0.48 = ( ! [Xu: $i] :
% 0.18/0.48 ( ( iskpair @ Xu )
% 0.18/0.48 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.18/0.48 = Xu ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(cartprodsndin_type,type,
% 0.18/0.48 cartprodsndin: $o ).
% 0.18/0.48
% 0.18/0.48 thf(cartprodsndin,definition,
% 0.18/0.48 ( cartprodsndin
% 0.18/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.18/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.18/0.48 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(cartprodpairmemEL_type,type,
% 0.18/0.48 cartprodpairmemEL: $o ).
% 0.18/0.48
% 0.18/0.48 thf(cartprodpairmemEL,definition,
% 0.18/0.48 ( cartprodpairmemEL
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.18/0.48 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.18/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(cartprodpairmemER_type,type,
% 0.18/0.48 cartprodpairmemER: $o ).
% 0.18/0.48
% 0.18/0.48 thf(cartprodpairmemER,definition,
% 0.18/0.48 ( cartprodpairmemER
% 0.18/0.48 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.18/0.48 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.18/0.48 => ( in @ Xy @ B ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 thf(cartprodmempaircEq,conjecture,
% 0.18/0.48 ( setextAx
% 0.18/0.48 => ( emptysetAx
% 0.18/0.48 => ( setadjoinAx
% 0.18/0.48 => ( powersetAx
% 0.18/0.48 => ( setunionAx
% 0.18/0.48 => ( omega0Ax
% 0.18/0.48 => ( omegaSAx
% 0.18/0.48 => ( omegaIndAx
% 0.18/0.48 => ( replAx
% 0.18/0.48 => ( foundationAx
% 0.18/0.48 => ( wellorderingAx
% 0.18/0.48 => ( descrp
% 0.18/0.48 => ( dsetconstrI
% 0.18/0.48 => ( dsetconstrEL
% 0.18/0.48 => ( dsetconstrER
% 0.18/0.48 => ( exuE1
% 0.18/0.48 => ( prop2setE
% 0.18/0.48 => ( emptysetE
% 0.18/0.48 => ( emptysetimpfalse
% 0.18/0.48 => ( notinemptyset
% 0.18/0.48 => ( exuE3e
% 0.18/0.48 => ( setext
% 0.18/0.48 => ( emptyI
% 0.18/0.48 => ( noeltsimpempty
% 0.18/0.48 => ( setbeta
% 0.18/0.48 => ( nonemptyE1
% 0.18/0.48 => ( nonemptyI
% 0.18/0.48 => ( nonemptyI1
% 0.18/0.48 => ( setadjoinIL
% 0.18/0.48 => ( emptyinunitempty
% 0.18/0.48 => ( setadjoinIR
% 0.18/0.48 => ( setadjoinE
% 0.18/0.48 => ( setadjoinOr
% 0.18/0.48 => ( setoftrueEq
% 0.18/0.48 => ( powersetI
% 0.18/0.48 => ( emptyinPowerset
% 0.18/0.48 => ( emptyInPowerset
% 0.18/0.48 => ( powersetE
% 0.18/0.48 => ( setunionI
% 0.18/0.48 => ( setunionE
% 0.18/0.48 => ( subPowSU
% 0.18/0.48 => ( exuE2
% 0.18/0.48 => ( nonemptyImpWitness
% 0.18/0.48 => ( uniqinunit
% 0.18/0.48 => ( notinsingleton
% 0.18/0.48 => ( eqinunit
% 0.18/0.48 => ( singletonsswitch
% 0.18/0.48 => ( upairsetE
% 0.18/0.48 => ( upairsetIL
% 0.18/0.48 => ( upairsetIR
% 0.18/0.48 => ( emptyE1
% 0.18/0.48 => ( vacuousDall
% 0.18/0.48 => ( quantDeMorgan1
% 0.18/0.48 => ( quantDeMorgan2
% 0.18/0.48 => ( quantDeMorgan3
% 0.18/0.48 => ( quantDeMorgan4
% 0.18/0.48 => ( prop2setI
% 0.18/0.48 => ( prop2set2propI
% 0.18/0.48 => ( notdexE
% 0.18/0.48 => ( notdallE
% 0.18/0.48 => ( exuI1
% 0.18/0.48 => ( exuI3
% 0.18/0.48 => ( exuI2
% 0.18/0.48 => ( inCongP
% 0.18/0.48 => ( in__Cong
% 0.18/0.48 => ( exuE3u
% 0.18/0.48 => ( exu__Cong
% 0.18/0.48 => ( emptyset__Cong
% 0.18/0.48 => ( setadjoin__Cong
% 0.18/0.48 => ( powerset__Cong
% 0.18/0.48 => ( setunion__Cong
% 0.18/0.48 => ( omega__Cong
% 0.18/0.48 => ( exuEu
% 0.18/0.48 => ( descr__Cong
% 0.18/0.48 => ( dsetconstr__Cong
% 0.18/0.48 => ( subsetI1
% 0.18/0.48 => ( eqimpsubset2
% 0.18/0.48 => ( eqimpsubset1
% 0.18/0.48 => ( subsetI2
% 0.18/0.48 => ( emptysetsubset
% 0.18/0.48 => ( subsetE
% 0.18/0.48 => ( subsetE2
% 0.18/0.48 => ( notsubsetI
% 0.18/0.48 => ( notequalI1
% 0.18/0.48 => ( notequalI2
% 0.18/0.48 => ( subsetRefl
% 0.18/0.48 => ( subsetTrans
% 0.18/0.48 => ( setadjoinSub
% 0.18/0.48 => ( setadjoinSub2
% 0.18/0.48 => ( subset2powerset
% 0.18/0.48 => ( setextsub
% 0.18/0.48 => ( subsetemptysetimpeq
% 0.18/0.48 => ( powersetI1
% 0.18/0.48 => ( powersetE1
% 0.18/0.48 => ( inPowerset
% 0.18/0.48 => ( powersetsubset
% 0.18/0.48 => ( sepInPowerset
% 0.18/0.48 => ( sepSubset
% 0.18/0.48 => ( binunionIL
% 0.18/0.48 => ( upairset2IR
% 0.18/0.48 => ( binunionIR
% 0.18/0.48 => ( binunionEcases
% 0.18/0.48 => ( binunionE
% 0.18/0.48 => ( binunionLsub
% 0.18/0.48 => ( binunionRsub
% 0.18/0.48 => ( binintersectI
% 0.18/0.48 => ( binintersectSubset5
% 0.18/0.48 => ( binintersectEL
% 0.18/0.48 => ( binintersectLsub
% 0.18/0.48 => ( binintersectSubset2
% 0.18/0.48 => ( binintersectSubset3
% 0.18/0.48 => ( binintersectER
% 0.18/0.48 => ( disjointsetsI1
% 0.18/0.48 => ( binintersectRsub
% 0.18/0.48 => ( binintersectSubset4
% 0.18/0.48 => ( binintersectSubset1
% 0.18/0.48 => ( bs114d
% 0.18/0.48 => ( setminusI
% 0.18/0.48 => ( setminusEL
% 0.18/0.48 => ( setminusER
% 0.18/0.48 => ( setminusSubset2
% 0.18/0.48 => ( setminusERneg
% 0.18/0.48 => ( setminusELneg
% 0.18/0.48 => ( setminusILneg
% 0.18/0.48 => ( setminusIRneg
% 0.18/0.48 => ( setminusLsub
% 0.18/0.48 => ( setminusSubset1
% 0.18/0.48 => ( symdiffE
% 0.18/0.48 => ( symdiffI1
% 0.18/0.48 => ( symdiffI2
% 0.18/0.48 => ( symdiffIneg1
% 0.18/0.48 => ( symdiffIneg2
% 0.18/0.48 => ( secondinupair
% 0.18/0.48 => ( setukpairIL
% 0.18/0.48 => ( setukpairIR
% 0.18/0.48 => ( kpairiskpair
% 0.18/0.48 => ( kpairp
% 0.18/0.48 => ( singletonsubset
% 0.18/0.48 => ( singletoninpowerset
% 0.18/0.48 => ( singletoninpowunion
% 0.18/0.48 => ( upairset2E
% 0.18/0.48 => ( upairsubunion
% 0.18/0.48 => ( upairinpowunion
% 0.18/0.48 => ( ubforcartprodlem1
% 0.18/0.48 => ( ubforcartprodlem2
% 0.18/0.48 => ( ubforcartprodlem3
% 0.18/0.48 => ( cartprodpairin
% 0.18/0.48 => ( cartprodmempair1
% 0.18/0.48 => ( cartprodmempair
% 0.18/0.48 => ( setunionE2
% 0.18/0.48 => ( setunionsingleton1
% 0.18/0.48 => ( setunionsingleton2
% 0.18/0.48 => ( setunionsingleton
% 0.18/0.48 => ( singletonprop
% 0.18/0.48 => ( ex1E1
% 0.18/0.48 => ( ex1I
% 0.18/0.48 => ( ex1I2
% 0.18/0.48 => ( singletonsuniq
% 0.18/0.48 => ( setukpairinjL1
% 0.18/0.48 => ( kfstsingleton
% 0.18/0.48 => ( theprop
% 0.18/0.48 => ( kfstpairEq
% 0.18/0.48 => ( cartprodfstin
% 0.18/0.48 => ( setukpairinjL2
% 0.18/0.48 => ( setukpairinjL
% 0.18/0.48 => ( setukpairinjR11
% 0.18/0.48 => ( setukpairinjR12
% 0.18/0.48 => ( setukpairinjR1
% 0.18/0.48 => ( upairequniteq
% 0.18/0.48 => ( setukpairinjR2
% 0.18/0.48 => ( setukpairinjR
% 0.18/0.48 => ( ksndsingleton
% 0.18/0.48 => ( ksndpairEq
% 0.18/0.48 => ( kpairsurjEq
% 0.18/0.48 => ( cartprodsndin
% 0.18/0.48 => ( cartprodpairmemEL
% 0.18/0.48 => ( cartprodpairmemER
% 0.18/0.48 => ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.48 ( ( in @ Xx @ A )
% 0.18/0.48 => ! [Xy: $i] :
% 0.18/0.48 ( ( in @ Xy @ B )
% 0.18/0.48 => ( ( kpair @ Xx @ Xy )
% 0.18/0.48 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.18/0.48
% 0.18/0.48 %------------------------------------------------------------------------------
% 0.18/0.48 ------------------------
% 0.32/1.70 ---- Embedded in THF ---
% 0.32/1.70 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.32/1.70 %%% Generated on Tue Feb 24 12:29:46 EST 2026
% 0.32/1.70 %%% using '$$dhol' embedding, version 1.3.0.
% 0.32/1.70 %%% Logic specification used:
% 0.32/1.70 %%% thf(spec, logic, $$dhol).
% 0.32/1.70
% 0.32/1.70 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.CrCPL8bmZ3/DTF2DTF9448.p
% See solution above
% 0.32/1.70 ------------------------
% 0.32/1.74 ---- Cleaned THF ---
% 0.32/1.74 thf(in_type,type,
% 0.32/1.74 in: $i > $i > $o ).
% 0.32/1.74
% 0.32/1.74 thf(exu_type,type,
% 0.32/1.74 exu: ( $i > $o ) > $o ).
% 0.32/1.74
% 0.32/1.74 thf(exu,definition,
% 0.32/1.74 ( exu
% 0.32/1.74 = ( ^ [Xphi: $i > $o] :
% 0.32/1.74 ? [Xx: $i] :
% 0.32/1.74 ( ( Xphi @ Xx )
% 0.32/1.74 & ! [Xy: $i] :
% 0.32/1.74 ( ( Xphi @ Xy )
% 0.32/1.74 => ( Xx = Xy ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setextAx_type,type,
% 0.32/1.74 setextAx: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setextAx,definition,
% 0.32/1.74 ( setextAx
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 <=> ( in @ Xx @ B ) )
% 0.32/1.74 => ( A = B ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(emptyset_type,type,
% 0.32/1.74 emptyset: $i ).
% 0.32/1.74
% 0.32/1.74 thf(emptysetAx_type,type,
% 0.32/1.74 emptysetAx: $o ).
% 0.32/1.74
% 0.32/1.74 thf(emptysetAx,definition,
% 0.32/1.74 ( emptysetAx
% 0.32/1.74 = ( ! [Xx: $i] :
% 0.32/1.74 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setadjoin_type,type,
% 0.32/1.74 setadjoin: $i > $i > $i ).
% 0.32/1.74
% 0.32/1.74 thf(setadjoinAx_type,type,
% 0.32/1.74 setadjoinAx: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setadjoinAx,definition,
% 0.32/1.74 ( setadjoinAx
% 0.32/1.74 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.32/1.74 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.32/1.74 <=> ( ( Xy = Xx )
% 0.32/1.74 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(powerset_type,type,
% 0.32/1.74 powerset: $i > $i ).
% 0.32/1.74
% 0.32/1.74 thf(powersetAx_type,type,
% 0.32/1.74 powersetAx: $o ).
% 0.32/1.74
% 0.32/1.74 thf(powersetAx,definition,
% 0.32/1.74 ( powersetAx
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ( in @ B @ ( powerset @ A ) )
% 0.32/1.74 <=> ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ B )
% 0.32/1.74 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setunion_type,type,
% 0.32/1.74 setunion: $i > $i ).
% 0.32/1.74
% 0.32/1.74 thf(setunionAx_type,type,
% 0.32/1.74 setunionAx: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setunionAx,definition,
% 0.32/1.74 ( setunionAx
% 0.32/1.74 = ( ! [A: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ ( setunion @ A ) )
% 0.32/1.74 <=> ? [B: $i] :
% 0.32/1.74 ( ( in @ Xx @ B )
% 0.32/1.74 & ( in @ B @ A ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(omega_type,type,
% 0.32/1.74 omega: $i ).
% 0.32/1.74
% 0.32/1.74 thf(omega0Ax_type,type,
% 0.32/1.74 omega0Ax: $o ).
% 0.32/1.74
% 0.32/1.74 thf(omega0Ax,definition,
% 0.32/1.74 ( omega0Ax
% 0.32/1.74 = ( in @ emptyset @ omega ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(omegaSAx_type,type,
% 0.32/1.74 omegaSAx: $o ).
% 0.32/1.74
% 0.32/1.74 thf(omegaSAx,definition,
% 0.32/1.74 ( omegaSAx
% 0.32/1.74 = ( ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ omega )
% 0.32/1.74 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(omegaIndAx_type,type,
% 0.32/1.74 omegaIndAx: $o ).
% 0.32/1.74
% 0.32/1.74 thf(omegaIndAx,definition,
% 0.32/1.74 ( omegaIndAx
% 0.32/1.74 = ( ! [A: $i] :
% 0.32/1.74 ( ( ( in @ emptyset @ A )
% 0.32/1.74 & ! [Xx: $i] :
% 0.32/1.74 ( ( ( in @ Xx @ omega )
% 0.32/1.74 & ( in @ Xx @ A ) )
% 0.32/1.74 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.32/1.74 => ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ omega )
% 0.32/1.74 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(replAx_type,type,
% 0.32/1.74 replAx: $o ).
% 0.32/1.74
% 0.32/1.74 thf(replAx,definition,
% 0.32/1.74 ( replAx
% 0.32/1.74 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.32/1.74 ( ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ( exu
% 0.32/1.74 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.32/1.74 => ? [B: $i] :
% 0.32/1.74 ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ B )
% 0.32/1.74 <=> ? [Xy: $i] :
% 0.32/1.74 ( ( in @ Xy @ A )
% 0.32/1.74 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(foundationAx_type,type,
% 0.32/1.74 foundationAx: $o ).
% 0.32/1.74
% 0.32/1.74 thf(foundationAx,definition,
% 0.32/1.74 ( foundationAx
% 0.32/1.74 = ( ! [A: $i] :
% 0.32/1.74 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.32/1.74 => ? [B: $i] :
% 0.32/1.74 ( ( in @ B @ A )
% 0.32/1.74 & ~ ? [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ B )
% 0.32/1.74 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(wellorderingAx_type,type,
% 0.32/1.74 wellorderingAx: $o ).
% 0.32/1.74
% 0.32/1.74 thf(wellorderingAx,definition,
% 0.32/1.74 ( wellorderingAx
% 0.32/1.74 = ( ! [A: $i] :
% 0.32/1.74 ? [B: $i] :
% 0.32/1.74 ( ! [C: $i] :
% 0.32/1.74 ( ( in @ C @ B )
% 0.32/1.74 => ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ C )
% 0.32/1.74 => ( in @ Xx @ A ) ) )
% 0.32/1.74 & ! [Xx: $i,Xy: $i] :
% 0.32/1.74 ( ( ( in @ Xx @ A )
% 0.32/1.74 & ( in @ Xy @ A ) )
% 0.32/1.74 => ( ! [C: $i] :
% 0.32/1.74 ( ( in @ C @ B )
% 0.32/1.74 => ( ( in @ Xx @ C )
% 0.32/1.74 <=> ( in @ Xy @ C ) ) )
% 0.32/1.74 => ( Xx = Xy ) ) )
% 0.32/1.74 & ! [C: $i,D: $i] :
% 0.32/1.74 ( ( ( in @ C @ B )
% 0.32/1.74 & ( in @ D @ B ) )
% 0.32/1.74 => ( ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ C )
% 0.32/1.74 => ( in @ Xx @ D ) )
% 0.32/1.74 | ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ D )
% 0.32/1.74 => ( in @ Xx @ C ) ) ) )
% 0.32/1.74 & ! [C: $i] :
% 0.32/1.74 ( ( ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ C )
% 0.32/1.74 => ( in @ Xx @ A ) )
% 0.32/1.74 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.32/1.74 => ? [D: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ D @ B )
% 0.32/1.74 & ( in @ Xx @ C )
% 0.32/1.74 & ~ ? [Xy: $i] :
% 0.32/1.74 ( ( in @ Xy @ D )
% 0.32/1.74 & ( in @ Xy @ C ) )
% 0.32/1.74 & ! [E: $i] :
% 0.32/1.74 ( ( in @ E @ B )
% 0.32/1.74 => ( ! [Xy: $i] :
% 0.32/1.74 ( ( in @ Xy @ E )
% 0.32/1.74 => ( in @ Xy @ D ) )
% 0.32/1.74 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(descr_type,type,
% 0.32/1.74 descr: ( $i > $o ) > $i ).
% 0.32/1.74
% 0.32/1.74 thf(descrp_type,type,
% 0.32/1.74 descrp: $o ).
% 0.32/1.74
% 0.32/1.74 thf(descrp,definition,
% 0.32/1.74 ( descrp
% 0.32/1.74 = ( ! [Xphi: $i > $o] :
% 0.32/1.74 ( ( exu
% 0.32/1.74 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/1.74 => ( Xphi
% 0.32/1.74 @ ( descr
% 0.32/1.74 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(dsetconstr_type,type,
% 0.32/1.74 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.32/1.74
% 0.32/1.74 thf(dsetconstrI_type,type,
% 0.32/1.74 dsetconstrI: $o ).
% 0.32/1.74
% 0.32/1.74 thf(dsetconstrI,definition,
% 0.32/1.74 ( dsetconstrI
% 0.32/1.74 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ( ( Xphi @ Xx )
% 0.32/1.74 => ( in @ Xx
% 0.32/1.74 @ ( dsetconstr @ A
% 0.32/1.74 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(dsetconstrEL_type,type,
% 0.32/1.74 dsetconstrEL: $o ).
% 0.32/1.74
% 0.32/1.74 thf(dsetconstrEL,definition,
% 0.32/1.74 ( dsetconstrEL
% 0.32/1.74 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx
% 0.32/1.74 @ ( dsetconstr @ A
% 0.32/1.74 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.32/1.74 => ( in @ Xx @ A ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(dsetconstrER_type,type,
% 0.32/1.74 dsetconstrER: $o ).
% 0.32/1.74
% 0.32/1.74 thf(dsetconstrER,definition,
% 0.32/1.74 ( dsetconstrER
% 0.32/1.74 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx
% 0.32/1.74 @ ( dsetconstr @ A
% 0.32/1.74 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.32/1.74 => ( Xphi @ Xx ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(exuE1_type,type,
% 0.32/1.74 exuE1: $o ).
% 0.32/1.74
% 0.32/1.74 thf(exuE1,definition,
% 0.32/1.74 ( exuE1
% 0.32/1.74 = ( ! [Xphi: $i > $o] :
% 0.32/1.74 ( ( exu
% 0.32/1.74 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/1.74 => ? [Xx: $i] :
% 0.32/1.74 ( ( Xphi @ Xx )
% 0.32/1.74 & ! [Xy: $i] :
% 0.32/1.74 ( ( Xphi @ Xy )
% 0.32/1.74 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(prop2set_type,type,
% 0.32/1.74 prop2set: $o > $i ).
% 0.32/1.74
% 0.32/1.74 thf(prop2setE_type,type,
% 0.32/1.74 prop2setE: $o ).
% 0.32/1.74
% 0.32/1.74 thf(prop2setE,definition,
% 0.32/1.74 ( prop2setE
% 0.32/1.74 = ( ! [Xphi: $o,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.32/1.74 => Xphi ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(emptysetE_type,type,
% 0.32/1.74 emptysetE: $o ).
% 0.32/1.74
% 0.32/1.74 thf(emptysetE,definition,
% 0.32/1.74 ( emptysetE
% 0.32/1.74 = ( ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ emptyset )
% 0.32/1.74 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(emptysetimpfalse_type,type,
% 0.32/1.74 emptysetimpfalse: $o ).
% 0.32/1.74
% 0.32/1.74 thf(emptysetimpfalse,definition,
% 0.32/1.74 ( emptysetimpfalse
% 0.32/1.74 = ( ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ emptyset )
% 0.32/1.74 => $false ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(notinemptyset_type,type,
% 0.32/1.74 notinemptyset: $o ).
% 0.32/1.74
% 0.32/1.74 thf(notinemptyset,definition,
% 0.32/1.74 ( notinemptyset
% 0.32/1.74 = ( ! [Xx: $i] :
% 0.32/1.74 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(exuE3e_type,type,
% 0.32/1.74 exuE3e: $o ).
% 0.32/1.74
% 0.32/1.74 thf(exuE3e,definition,
% 0.32/1.74 ( exuE3e
% 0.32/1.74 = ( ! [Xphi: $i > $o] :
% 0.32/1.74 ( ( exu
% 0.32/1.74 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/1.74 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setext_type,type,
% 0.32/1.74 setext: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setext,definition,
% 0.32/1.74 ( setext
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ( in @ Xx @ B ) )
% 0.32/1.74 => ( ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ B )
% 0.32/1.74 => ( in @ Xx @ A ) )
% 0.32/1.74 => ( A = B ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(emptyI_type,type,
% 0.32/1.74 emptyI: $o ).
% 0.32/1.74
% 0.32/1.74 thf(emptyI,definition,
% 0.32/1.74 ( emptyI
% 0.32/1.74 = ( ! [A: $i] :
% 0.32/1.74 ( ! [Xx: $i] :
% 0.32/1.74 ~ ( in @ Xx @ A )
% 0.32/1.74 => ( A = emptyset ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(noeltsimpempty_type,type,
% 0.32/1.74 noeltsimpempty: $o ).
% 0.32/1.74
% 0.32/1.74 thf(noeltsimpempty,definition,
% 0.32/1.74 ( noeltsimpempty
% 0.32/1.74 = ( ! [A: $i] :
% 0.32/1.74 ( ! [Xx: $i] :
% 0.32/1.74 ~ ( in @ Xx @ A )
% 0.32/1.74 => ( A = emptyset ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setbeta_type,type,
% 0.32/1.74 setbeta: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setbeta,definition,
% 0.32/1.74 ( setbeta
% 0.32/1.74 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ( ( in @ Xx
% 0.32/1.74 @ ( dsetconstr @ A
% 0.32/1.74 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.32/1.74 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(nonempty_type,type,
% 0.32/1.74 nonempty: $i > $o ).
% 0.32/1.74
% 0.32/1.74 thf(nonempty,definition,
% 0.32/1.74 ( nonempty
% 0.32/1.74 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(nonemptyE1_type,type,
% 0.32/1.74 nonemptyE1: $o ).
% 0.32/1.74
% 0.32/1.74 thf(nonemptyE1,definition,
% 0.32/1.74 ( nonemptyE1
% 0.32/1.74 = ( ! [A: $i] :
% 0.32/1.74 ( ( nonempty @ A )
% 0.32/1.74 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(nonemptyI_type,type,
% 0.32/1.74 nonemptyI: $o ).
% 0.32/1.74
% 0.32/1.74 thf(nonemptyI,definition,
% 0.32/1.74 ( nonemptyI
% 0.32/1.74 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ( ( Xphi @ Xx )
% 0.32/1.74 => ( nonempty
% 0.32/1.74 @ ( dsetconstr @ A
% 0.32/1.74 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(nonemptyI1_type,type,
% 0.32/1.74 nonemptyI1: $o ).
% 0.32/1.74
% 0.32/1.74 thf(nonemptyI1,definition,
% 0.32/1.74 ( nonemptyI1
% 0.32/1.74 = ( ! [A: $i] :
% 0.32/1.74 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.32/1.74 => ( nonempty @ A ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setadjoinIL_type,type,
% 0.32/1.74 setadjoinIL: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setadjoinIL,definition,
% 0.32/1.74 ( setadjoinIL
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(emptyinunitempty_type,type,
% 0.32/1.74 emptyinunitempty: $o ).
% 0.32/1.74
% 0.32/1.74 thf(emptyinunitempty,definition,
% 0.32/1.74 ( emptyinunitempty
% 0.32/1.74 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setadjoinIR_type,type,
% 0.32/1.74 setadjoinIR: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setadjoinIR,definition,
% 0.32/1.74 ( setadjoinIR
% 0.32/1.74 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.32/1.74 ( ( in @ Xy @ A )
% 0.32/1.74 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setadjoinE_type,type,
% 0.32/1.74 setadjoinE: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setadjoinE,definition,
% 0.32/1.74 ( setadjoinE
% 0.32/1.74 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.32/1.74 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.32/1.74 => ! [Xphi: $o] :
% 0.32/1.74 ( ( ( Xy = Xx )
% 0.32/1.74 => Xphi )
% 0.32/1.74 => ( ( ( in @ Xy @ A )
% 0.32/1.74 => Xphi )
% 0.32/1.74 => Xphi ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setadjoinOr_type,type,
% 0.32/1.74 setadjoinOr: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setadjoinOr,definition,
% 0.32/1.74 ( setadjoinOr
% 0.32/1.74 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.32/1.74 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.32/1.74 => ( ( Xy = Xx )
% 0.32/1.74 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setoftrueEq_type,type,
% 0.32/1.74 setoftrueEq: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setoftrueEq,definition,
% 0.32/1.74 ( setoftrueEq
% 0.32/1.74 = ( ! [A: $i] :
% 0.32/1.74 ( ( dsetconstr @ A
% 0.32/1.74 @ ^ [Xx: $i] : $true )
% 0.32/1.74 = A ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(powersetI_type,type,
% 0.32/1.74 powersetI: $o ).
% 0.32/1.74
% 0.32/1.74 thf(powersetI,definition,
% 0.32/1.74 ( powersetI
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ B )
% 0.32/1.74 => ( in @ Xx @ A ) )
% 0.32/1.74 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(emptyinPowerset_type,type,
% 0.32/1.74 emptyinPowerset: $o ).
% 0.32/1.74
% 0.32/1.74 thf(emptyinPowerset,definition,
% 0.32/1.74 ( emptyinPowerset
% 0.32/1.74 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(emptyInPowerset_type,type,
% 0.32/1.74 emptyInPowerset: $o ).
% 0.32/1.74
% 0.32/1.74 thf(emptyInPowerset,definition,
% 0.32/1.74 ( emptyInPowerset
% 0.32/1.74 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(powersetE_type,type,
% 0.32/1.74 powersetE: $o ).
% 0.32/1.74
% 0.32/1.74 thf(powersetE,definition,
% 0.32/1.74 ( powersetE
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ B @ ( powerset @ A ) )
% 0.32/1.74 => ( ( in @ Xx @ B )
% 0.32/1.74 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setunionI_type,type,
% 0.32/1.74 setunionI: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setunionI,definition,
% 0.32/1.74 ( setunionI
% 0.32/1.74 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.32/1.74 ( ( in @ Xx @ B )
% 0.32/1.74 => ( ( in @ B @ A )
% 0.32/1.74 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setunionE_type,type,
% 0.32/1.74 setunionE: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setunionE,definition,
% 0.32/1.74 ( setunionE
% 0.32/1.74 = ( ! [A: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ ( setunion @ A ) )
% 0.32/1.74 => ! [Xphi: $o] :
% 0.32/1.74 ( ! [B: $i] :
% 0.32/1.74 ( ( in @ Xx @ B )
% 0.32/1.74 => ( ( in @ B @ A )
% 0.32/1.74 => Xphi ) )
% 0.32/1.74 => Xphi ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(subPowSU_type,type,
% 0.32/1.74 subPowSU: $o ).
% 0.32/1.74
% 0.32/1.74 thf(subPowSU,definition,
% 0.32/1.74 ( subPowSU
% 0.32/1.74 = ( ! [A: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(exuE2_type,type,
% 0.32/1.74 exuE2: $o ).
% 0.32/1.74
% 0.32/1.74 thf(exuE2,definition,
% 0.32/1.74 ( exuE2
% 0.32/1.74 = ( ! [Xphi: $i > $o] :
% 0.32/1.74 ( ( exu
% 0.32/1.74 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/1.74 => ? [Xx: $i] :
% 0.32/1.74 ! [Xy: $i] :
% 0.32/1.74 ( ( Xphi @ Xy )
% 0.32/1.74 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(nonemptyImpWitness_type,type,
% 0.32/1.74 nonemptyImpWitness: $o ).
% 0.32/1.74
% 0.32/1.74 thf(nonemptyImpWitness,definition,
% 0.32/1.74 ( nonemptyImpWitness
% 0.32/1.74 = ( ! [A: $i] :
% 0.32/1.74 ( ( nonempty @ A )
% 0.32/1.74 => ? [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 & $true ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(uniqinunit_type,type,
% 0.32/1.74 uniqinunit: $o ).
% 0.32/1.74
% 0.32/1.74 thf(uniqinunit,definition,
% 0.32/1.74 ( uniqinunit
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i] :
% 0.32/1.74 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.32/1.74 => ( Xx = Xy ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(notinsingleton_type,type,
% 0.32/1.74 notinsingleton: $o ).
% 0.32/1.74
% 0.32/1.74 thf(notinsingleton,definition,
% 0.32/1.74 ( notinsingleton
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i] :
% 0.32/1.74 ( ( Xx != Xy )
% 0.32/1.74 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(eqinunit_type,type,
% 0.32/1.74 eqinunit: $o ).
% 0.32/1.74
% 0.32/1.74 thf(eqinunit,definition,
% 0.32/1.74 ( eqinunit
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i] :
% 0.32/1.74 ( ( Xx = Xy )
% 0.32/1.74 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(singletonsswitch_type,type,
% 0.32/1.74 singletonsswitch: $o ).
% 0.32/1.74
% 0.32/1.74 thf(singletonsswitch,definition,
% 0.32/1.74 ( singletonsswitch
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i] :
% 0.32/1.74 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.32/1.74 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(upairsetE_type,type,
% 0.32/1.74 upairsetE: $o ).
% 0.32/1.74
% 0.32/1.74 thf(upairsetE,definition,
% 0.32/1.74 ( upairsetE
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.32/1.74 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.32/1.74 => ( ( Xz = Xx )
% 0.32/1.74 | ( Xz = Xy ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(upairsetIL_type,type,
% 0.32/1.74 upairsetIL: $o ).
% 0.32/1.74
% 0.32/1.74 thf(upairsetIL,definition,
% 0.32/1.74 ( upairsetIL
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(upairsetIR_type,type,
% 0.32/1.74 upairsetIR: $o ).
% 0.32/1.74
% 0.32/1.74 thf(upairsetIR,definition,
% 0.32/1.74 ( upairsetIR
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(emptyE1_type,type,
% 0.32/1.74 emptyE1: $o ).
% 0.32/1.74
% 0.32/1.74 thf(emptyE1,definition,
% 0.32/1.74 ( emptyE1
% 0.32/1.74 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/1.74 ( ? [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 & ( Xphi @ Xx ) )
% 0.32/1.74 => ( ( ( dsetconstr @ A
% 0.32/1.74 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/1.74 = emptyset )
% 0.32/1.74 => $false ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(vacuousDall_type,type,
% 0.32/1.74 vacuousDall: $o ).
% 0.32/1.74
% 0.32/1.74 thf(vacuousDall,definition,
% 0.32/1.74 ( vacuousDall
% 0.32/1.74 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ emptyset )
% 0.32/1.74 => ( Xphi @ Xx ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(quantDeMorgan1_type,type,
% 0.32/1.74 quantDeMorgan1: $o ).
% 0.32/1.74
% 0.32/1.74 thf(quantDeMorgan1,definition,
% 0.32/1.74 ( quantDeMorgan1
% 0.32/1.74 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/1.74 ( ~ ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ( Xphi @ Xx ) )
% 0.32/1.74 => ? [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(quantDeMorgan2_type,type,
% 0.32/1.74 quantDeMorgan2: $o ).
% 0.32/1.74
% 0.32/1.74 thf(quantDeMorgan2,definition,
% 0.32/1.74 ( quantDeMorgan2
% 0.32/1.74 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/1.74 ( ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ~ ( Xphi @ Xx ) )
% 0.32/1.74 => ~ ? [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(quantDeMorgan3_type,type,
% 0.32/1.74 quantDeMorgan3: $o ).
% 0.32/1.74
% 0.32/1.74 thf(quantDeMorgan3,definition,
% 0.32/1.74 ( quantDeMorgan3
% 0.32/1.74 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/1.74 ( ~ ? [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 & ( Xphi @ Xx ) )
% 0.32/1.74 => ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(quantDeMorgan4_type,type,
% 0.32/1.74 quantDeMorgan4: $o ).
% 0.32/1.74
% 0.32/1.74 thf(quantDeMorgan4,definition,
% 0.32/1.74 ( quantDeMorgan4
% 0.32/1.74 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/1.74 ( ? [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 & ~ ( Xphi @ Xx ) )
% 0.32/1.74 => ~ ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(prop2setI_type,type,
% 0.32/1.74 prop2setI: $o ).
% 0.32/1.74
% 0.32/1.74 thf(prop2setI,definition,
% 0.32/1.74 ( prop2setI
% 0.32/1.74 = ( ! [Xphi: $o] :
% 0.32/1.74 ( Xphi
% 0.32/1.74 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(set2prop_type,type,
% 0.32/1.74 set2prop: $i > $o ).
% 0.32/1.74
% 0.32/1.74 thf(prop2set2propI_type,type,
% 0.32/1.74 prop2set2propI: $o ).
% 0.32/1.74
% 0.32/1.74 thf(prop2set2propI,definition,
% 0.32/1.74 ( prop2set2propI
% 0.32/1.74 = ( ! [Xphi: $o] :
% 0.32/1.74 ( Xphi
% 0.32/1.74 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(notdexE_type,type,
% 0.32/1.74 notdexE: $o ).
% 0.32/1.74
% 0.32/1.74 thf(notdexE,definition,
% 0.32/1.74 ( notdexE
% 0.32/1.74 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/1.74 ( ~ ? [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 & ( Xphi @ Xx ) )
% 0.32/1.74 => ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(notdallE_type,type,
% 0.32/1.74 notdallE: $o ).
% 0.32/1.74
% 0.32/1.74 thf(notdallE,definition,
% 0.32/1.74 ( notdallE
% 0.32/1.74 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/1.74 ( ~ ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ( Xphi @ Xx ) )
% 0.32/1.74 => ? [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(exuI1_type,type,
% 0.32/1.74 exuI1: $o ).
% 0.32/1.74
% 0.32/1.74 thf(exuI1,definition,
% 0.32/1.74 ( exuI1
% 0.32/1.74 = ( ! [Xphi: $i > $o] :
% 0.32/1.74 ( ? [Xx: $i] :
% 0.32/1.74 ( ( Xphi @ Xx )
% 0.32/1.74 & ! [Xy: $i] :
% 0.32/1.74 ( ( Xphi @ Xy )
% 0.32/1.74 => ( Xx = Xy ) ) )
% 0.32/1.74 => ( exu
% 0.32/1.74 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(exuI3_type,type,
% 0.32/1.74 exuI3: $o ).
% 0.32/1.74
% 0.32/1.74 thf(exuI3,definition,
% 0.32/1.74 ( exuI3
% 0.32/1.74 = ( ! [Xphi: $i > $o] :
% 0.32/1.74 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.32/1.74 => ( ! [Xx: $i,Xy: $i] :
% 0.32/1.74 ( ( Xphi @ Xx )
% 0.32/1.74 => ( ( Xphi @ Xy )
% 0.32/1.74 => ( Xx = Xy ) ) )
% 0.32/1.74 => ( exu
% 0.32/1.74 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(exuI2_type,type,
% 0.32/1.74 exuI2: $o ).
% 0.32/1.74
% 0.32/1.74 thf(exuI2,definition,
% 0.32/1.74 ( exuI2
% 0.32/1.74 = ( ! [Xphi: $i > $o] :
% 0.32/1.74 ( ? [Xx: $i] :
% 0.32/1.74 ! [Xy: $i] :
% 0.32/1.74 ( ( Xphi @ Xy )
% 0.32/1.74 <=> ( Xy = Xx ) )
% 0.32/1.74 => ( exu
% 0.32/1.74 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(inCongP_type,type,
% 0.32/1.74 inCongP: $o ).
% 0.32/1.74
% 0.32/1.74 thf(inCongP,definition,
% 0.32/1.74 ( inCongP
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ( A = B )
% 0.32/1.74 => ! [Xx: $i,Xy: $i] :
% 0.32/1.74 ( ( Xx = Xy )
% 0.32/1.74 => ( ( in @ Xx @ A )
% 0.32/1.74 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(in__Cong_type,type,
% 0.32/1.74 in__Cong: $o ).
% 0.32/1.74
% 0.32/1.74 thf(in__Cong,definition,
% 0.32/1.74 ( in__Cong
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ( A = B )
% 0.32/1.74 => ! [Xx: $i,Xy: $i] :
% 0.32/1.74 ( ( Xx = Xy )
% 0.32/1.74 => ( ( in @ Xx @ A )
% 0.32/1.74 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(exuE3u_type,type,
% 0.32/1.74 exuE3u: $o ).
% 0.32/1.74
% 0.32/1.74 thf(exuE3u,definition,
% 0.32/1.74 ( exuE3u
% 0.32/1.74 = ( ! [Xphi: $i > $o] :
% 0.32/1.74 ( ( exu
% 0.32/1.74 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/1.74 => ! [Xx: $i,Xy: $i] :
% 0.32/1.74 ( ( Xphi @ Xx )
% 0.32/1.74 => ( ( Xphi @ Xy )
% 0.32/1.74 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(exu__Cong_type,type,
% 0.32/1.74 exu__Cong: $o ).
% 0.32/1.74
% 0.32/1.74 thf(exu__Cong,definition,
% 0.32/1.74 ( exu__Cong
% 0.32/1.74 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.32/1.74 ( ! [Xx: $i,Xy: $i] :
% 0.32/1.74 ( ( Xx = Xy )
% 0.32/1.74 => ( ( Xphi @ Xx )
% 0.32/1.74 <=> ( Xpsi @ Xy ) ) )
% 0.32/1.74 => ( ( exu
% 0.32/1.74 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/1.74 <=> ( exu
% 0.32/1.74 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(emptyset__Cong_type,type,
% 0.32/1.74 emptyset__Cong: $o ).
% 0.32/1.74
% 0.32/1.74 thf(emptyset__Cong,definition,
% 0.32/1.74 ( emptyset__Cong
% 0.32/1.74 = ( emptyset = emptyset ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setadjoin__Cong_type,type,
% 0.32/1.74 setadjoin__Cong: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setadjoin__Cong,definition,
% 0.32/1.74 ( setadjoin__Cong
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i] :
% 0.32/1.74 ( ( Xx = Xy )
% 0.32/1.74 => ! [Xz: $i,Xu: $i] :
% 0.32/1.74 ( ( Xz = Xu )
% 0.32/1.74 => ( ( setadjoin @ Xx @ Xz )
% 0.32/1.74 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(powerset__Cong_type,type,
% 0.32/1.74 powerset__Cong: $o ).
% 0.32/1.74
% 0.32/1.74 thf(powerset__Cong,definition,
% 0.32/1.74 ( powerset__Cong
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ( A = B )
% 0.32/1.74 => ( ( powerset @ A )
% 0.32/1.74 = ( powerset @ B ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setunion__Cong_type,type,
% 0.32/1.74 setunion__Cong: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setunion__Cong,definition,
% 0.32/1.74 ( setunion__Cong
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ( A = B )
% 0.32/1.74 => ( ( setunion @ A )
% 0.32/1.74 = ( setunion @ B ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(omega__Cong_type,type,
% 0.32/1.74 omega__Cong: $o ).
% 0.32/1.74
% 0.32/1.74 thf(omega__Cong,definition,
% 0.32/1.74 ( omega__Cong
% 0.32/1.74 = ( omega = omega ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(exuEu_type,type,
% 0.32/1.74 exuEu: $o ).
% 0.32/1.74
% 0.32/1.74 thf(exuEu,definition,
% 0.32/1.74 ( exuEu
% 0.32/1.74 = ( ! [Xphi: $i > $o] :
% 0.32/1.74 ( ( exu
% 0.32/1.74 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/1.74 => ! [Xx: $i,Xy: $i] :
% 0.32/1.74 ( ( Xphi @ Xx )
% 0.32/1.74 => ( ( Xphi @ Xy )
% 0.32/1.74 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(descr__Cong_type,type,
% 0.32/1.74 descr__Cong: $o ).
% 0.32/1.74
% 0.32/1.74 thf(descr__Cong,definition,
% 0.32/1.74 ( descr__Cong
% 0.32/1.74 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.32/1.74 ( ! [Xx: $i,Xy: $i] :
% 0.32/1.74 ( ( Xx = Xy )
% 0.32/1.74 => ( ( Xphi @ Xx )
% 0.32/1.74 <=> ( Xpsi @ Xy ) ) )
% 0.32/1.74 => ( ( exu
% 0.32/1.74 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/1.74 => ( ( exu
% 0.32/1.74 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.32/1.74 => ( ( descr
% 0.32/1.74 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/1.74 = ( descr
% 0.32/1.74 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(dsetconstr__Cong_type,type,
% 0.32/1.74 dsetconstr__Cong: $o ).
% 0.32/1.74
% 0.32/1.74 thf(dsetconstr__Cong,definition,
% 0.32/1.74 ( dsetconstr__Cong
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ( A = B )
% 0.32/1.74 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.32/1.74 ( ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ! [Xy: $i] :
% 0.32/1.74 ( ( in @ Xy @ B )
% 0.32/1.74 => ( ( Xx = Xy )
% 0.32/1.74 => ( ( Xphi @ Xx )
% 0.32/1.74 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.32/1.74 => ( ( dsetconstr @ A
% 0.32/1.74 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/1.74 = ( dsetconstr @ B
% 0.32/1.74 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(subset_type,type,
% 0.32/1.74 subset: $i > $i > $o ).
% 0.32/1.74
% 0.32/1.74 thf(disjoint_type,type,
% 0.32/1.74 disjoint: $i > $i > $o ).
% 0.32/1.74
% 0.32/1.74 thf(setsmeet_type,type,
% 0.32/1.74 setsmeet: $i > $i > $o ).
% 0.32/1.74
% 0.32/1.74 thf(subsetI1_type,type,
% 0.32/1.74 subsetI1: $o ).
% 0.32/1.74
% 0.32/1.74 thf(subsetI1,definition,
% 0.32/1.74 ( subsetI1
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ( in @ Xx @ B ) )
% 0.32/1.74 => ( subset @ A @ B ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(eqimpsubset2_type,type,
% 0.32/1.74 eqimpsubset2: $o ).
% 0.32/1.74
% 0.32/1.74 thf(eqimpsubset2,definition,
% 0.32/1.74 ( eqimpsubset2
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ( A = B )
% 0.32/1.74 => ( subset @ B @ A ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(eqimpsubset1_type,type,
% 0.32/1.74 eqimpsubset1: $o ).
% 0.32/1.74
% 0.32/1.74 thf(eqimpsubset1,definition,
% 0.32/1.74 ( eqimpsubset1
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ( A = B )
% 0.32/1.74 => ( subset @ A @ B ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(subsetI2_type,type,
% 0.32/1.74 subsetI2: $o ).
% 0.32/1.74
% 0.32/1.74 thf(subsetI2,definition,
% 0.32/1.74 ( subsetI2
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ( in @ Xx @ B ) )
% 0.32/1.74 => ( subset @ A @ B ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(emptysetsubset_type,type,
% 0.32/1.74 emptysetsubset: $o ).
% 0.32/1.74
% 0.32/1.74 thf(emptysetsubset,definition,
% 0.32/1.74 ( emptysetsubset
% 0.32/1.74 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(subsetE_type,type,
% 0.32/1.74 subsetE: $o ).
% 0.32/1.74
% 0.32/1.74 thf(subsetE,definition,
% 0.32/1.74 ( subsetE
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ( subset @ A @ B )
% 0.32/1.74 => ( ( in @ Xx @ A )
% 0.32/1.74 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(subsetE2_type,type,
% 0.32/1.74 subsetE2: $o ).
% 0.32/1.74
% 0.32/1.74 thf(subsetE2,definition,
% 0.32/1.74 ( subsetE2
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ( subset @ A @ B )
% 0.32/1.74 => ( ~ ( in @ Xx @ B )
% 0.32/1.74 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(notsubsetI_type,type,
% 0.32/1.74 notsubsetI: $o ).
% 0.32/1.74
% 0.32/1.74 thf(notsubsetI,definition,
% 0.32/1.74 ( notsubsetI
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ( ~ ( in @ Xx @ B )
% 0.32/1.74 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(notequalI1_type,type,
% 0.32/1.74 notequalI1: $o ).
% 0.32/1.74
% 0.32/1.74 thf(notequalI1,definition,
% 0.32/1.74 ( notequalI1
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ~ ( subset @ A @ B )
% 0.32/1.74 => ( A != B ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(notequalI2_type,type,
% 0.32/1.74 notequalI2: $o ).
% 0.32/1.74
% 0.32/1.74 thf(notequalI2,definition,
% 0.32/1.74 ( notequalI2
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ( ~ ( in @ Xx @ B )
% 0.32/1.74 => ( A != B ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(subsetRefl_type,type,
% 0.32/1.74 subsetRefl: $o ).
% 0.32/1.74
% 0.32/1.74 thf(subsetRefl,definition,
% 0.32/1.74 ( subsetRefl
% 0.32/1.74 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(subsetTrans_type,type,
% 0.32/1.74 subsetTrans: $o ).
% 0.32/1.74
% 0.32/1.74 thf(subsetTrans,definition,
% 0.32/1.74 ( subsetTrans
% 0.32/1.74 = ( ! [A: $i,B: $i,C: $i] :
% 0.32/1.74 ( ( subset @ A @ B )
% 0.32/1.74 => ( ( subset @ B @ C )
% 0.32/1.74 => ( subset @ A @ C ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setadjoinSub_type,type,
% 0.32/1.74 setadjoinSub: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setadjoinSub,definition,
% 0.32/1.74 ( setadjoinSub
% 0.32/1.74 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setadjoinSub2_type,type,
% 0.32/1.74 setadjoinSub2: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setadjoinSub2,definition,
% 0.32/1.74 ( setadjoinSub2
% 0.32/1.74 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.32/1.74 ( ( subset @ A @ B )
% 0.32/1.74 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(subset2powerset_type,type,
% 0.32/1.74 subset2powerset: $o ).
% 0.32/1.74
% 0.32/1.74 thf(subset2powerset,definition,
% 0.32/1.74 ( subset2powerset
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ( subset @ A @ B )
% 0.32/1.74 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setextsub_type,type,
% 0.32/1.74 setextsub: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setextsub,definition,
% 0.32/1.74 ( setextsub
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ( subset @ A @ B )
% 0.32/1.74 => ( ( subset @ B @ A )
% 0.32/1.74 => ( A = B ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(subsetemptysetimpeq_type,type,
% 0.32/1.74 subsetemptysetimpeq: $o ).
% 0.32/1.74
% 0.32/1.74 thf(subsetemptysetimpeq,definition,
% 0.32/1.74 ( subsetemptysetimpeq
% 0.32/1.74 = ( ! [A: $i] :
% 0.32/1.74 ( ( subset @ A @ emptyset )
% 0.32/1.74 => ( A = emptyset ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(powersetI1_type,type,
% 0.32/1.74 powersetI1: $o ).
% 0.32/1.74
% 0.32/1.74 thf(powersetI1,definition,
% 0.32/1.74 ( powersetI1
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ( subset @ B @ A )
% 0.32/1.74 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(powersetE1_type,type,
% 0.32/1.74 powersetE1: $o ).
% 0.32/1.74
% 0.32/1.74 thf(powersetE1,definition,
% 0.32/1.74 ( powersetE1
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ( in @ B @ ( powerset @ A ) )
% 0.32/1.74 => ( subset @ B @ A ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(inPowerset_type,type,
% 0.32/1.74 inPowerset: $o ).
% 0.32/1.74
% 0.32/1.74 thf(inPowerset,definition,
% 0.32/1.74 ( inPowerset
% 0.32/1.74 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(powersetsubset_type,type,
% 0.32/1.74 powersetsubset: $o ).
% 0.32/1.74
% 0.32/1.74 thf(powersetsubset,definition,
% 0.32/1.74 ( powersetsubset
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ( subset @ A @ B )
% 0.32/1.74 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(sepInPowerset_type,type,
% 0.32/1.74 sepInPowerset: $o ).
% 0.32/1.74
% 0.32/1.74 thf(sepInPowerset,definition,
% 0.32/1.74 ( sepInPowerset
% 0.32/1.74 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/1.74 ( in
% 0.32/1.74 @ ( dsetconstr @ A
% 0.32/1.74 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/1.74 @ ( powerset @ A ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(sepSubset_type,type,
% 0.32/1.74 sepSubset: $o ).
% 0.32/1.74
% 0.32/1.74 thf(sepSubset,definition,
% 0.32/1.74 ( sepSubset
% 0.32/1.74 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/1.74 ( subset
% 0.32/1.74 @ ( dsetconstr @ A
% 0.32/1.74 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/1.74 @ A ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(binunion_type,type,
% 0.32/1.74 binunion: $i > $i > $i ).
% 0.32/1.74
% 0.32/1.74 thf(binunionIL_type,type,
% 0.32/1.74 binunionIL: $o ).
% 0.32/1.74
% 0.32/1.74 thf(binunionIL,definition,
% 0.32/1.74 ( binunionIL
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(upairset2IR_type,type,
% 0.32/1.74 upairset2IR: $o ).
% 0.32/1.74
% 0.32/1.74 thf(upairset2IR,definition,
% 0.32/1.74 ( upairset2IR
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(binunionIR_type,type,
% 0.32/1.74 binunionIR: $o ).
% 0.32/1.74
% 0.32/1.74 thf(binunionIR,definition,
% 0.32/1.74 ( binunionIR
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ B )
% 0.32/1.74 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(binunionEcases_type,type,
% 0.32/1.74 binunionEcases: $o ).
% 0.32/1.74
% 0.32/1.74 thf(binunionEcases,definition,
% 0.32/1.74 ( binunionEcases
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.32/1.74 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.32/1.74 => ( ( ( in @ Xx @ A )
% 0.32/1.74 => Xphi )
% 0.32/1.74 => ( ( ( in @ Xx @ B )
% 0.32/1.74 => Xphi )
% 0.32/1.74 => Xphi ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(binunionE_type,type,
% 0.32/1.74 binunionE: $o ).
% 0.32/1.74
% 0.32/1.74 thf(binunionE,definition,
% 0.32/1.74 ( binunionE
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.32/1.74 => ( ( in @ Xx @ A )
% 0.32/1.74 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(binunionLsub_type,type,
% 0.32/1.74 binunionLsub: $o ).
% 0.32/1.74
% 0.32/1.74 thf(binunionLsub,definition,
% 0.32/1.74 ( binunionLsub
% 0.32/1.74 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(binunionRsub_type,type,
% 0.32/1.74 binunionRsub: $o ).
% 0.32/1.74
% 0.32/1.74 thf(binunionRsub,definition,
% 0.32/1.74 ( binunionRsub
% 0.32/1.74 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(binintersect_type,type,
% 0.32/1.74 binintersect: $i > $i > $i ).
% 0.32/1.74
% 0.32/1.74 thf(binintersectI_type,type,
% 0.32/1.74 binintersectI: $o ).
% 0.32/1.74
% 0.32/1.74 thf(binintersectI,definition,
% 0.32/1.74 ( binintersectI
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ( ( in @ Xx @ B )
% 0.32/1.74 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(binintersectSubset5_type,type,
% 0.32/1.74 binintersectSubset5: $o ).
% 0.32/1.74
% 0.32/1.74 thf(binintersectSubset5,definition,
% 0.32/1.74 ( binintersectSubset5
% 0.32/1.74 = ( ! [A: $i,B: $i,C: $i] :
% 0.32/1.74 ( ( subset @ C @ A )
% 0.32/1.74 => ( ( subset @ C @ B )
% 0.32/1.74 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(binintersectEL_type,type,
% 0.32/1.74 binintersectEL: $o ).
% 0.32/1.74
% 0.32/1.74 thf(binintersectEL,definition,
% 0.32/1.74 ( binintersectEL
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.32/1.74 => ( in @ Xx @ A ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(binintersectLsub_type,type,
% 0.32/1.74 binintersectLsub: $o ).
% 0.32/1.74
% 0.32/1.74 thf(binintersectLsub,definition,
% 0.32/1.74 ( binintersectLsub
% 0.32/1.74 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(binintersectSubset2_type,type,
% 0.32/1.74 binintersectSubset2: $o ).
% 0.32/1.74
% 0.32/1.74 thf(binintersectSubset2,definition,
% 0.32/1.74 ( binintersectSubset2
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ( subset @ A @ B )
% 0.32/1.74 => ( ( binintersect @ A @ B )
% 0.32/1.74 = A ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(binintersectSubset3_type,type,
% 0.32/1.74 binintersectSubset3: $o ).
% 0.32/1.74
% 0.32/1.74 thf(binintersectSubset3,definition,
% 0.32/1.74 ( binintersectSubset3
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ( ( binintersect @ A @ B )
% 0.32/1.74 = B )
% 0.32/1.74 => ( subset @ B @ A ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(binintersectER_type,type,
% 0.32/1.74 binintersectER: $o ).
% 0.32/1.74
% 0.32/1.74 thf(binintersectER,definition,
% 0.32/1.74 ( binintersectER
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.32/1.74 => ( in @ Xx @ B ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(disjointsetsI1_type,type,
% 0.32/1.74 disjointsetsI1: $o ).
% 0.32/1.74
% 0.32/1.74 thf(disjointsetsI1,definition,
% 0.32/1.74 ( disjointsetsI1
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ~ ? [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 & ( in @ Xx @ B ) )
% 0.32/1.74 => ( ( binintersect @ A @ B )
% 0.32/1.74 = emptyset ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(binintersectRsub_type,type,
% 0.32/1.74 binintersectRsub: $o ).
% 0.32/1.74
% 0.32/1.74 thf(binintersectRsub,definition,
% 0.32/1.74 ( binintersectRsub
% 0.32/1.74 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(binintersectSubset4_type,type,
% 0.32/1.74 binintersectSubset4: $o ).
% 0.32/1.74
% 0.32/1.74 thf(binintersectSubset4,definition,
% 0.32/1.74 ( binintersectSubset4
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ( subset @ B @ A )
% 0.32/1.74 => ( ( binintersect @ A @ B )
% 0.32/1.74 = B ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(binintersectSubset1_type,type,
% 0.32/1.74 binintersectSubset1: $o ).
% 0.32/1.74
% 0.32/1.74 thf(binintersectSubset1,definition,
% 0.32/1.74 ( binintersectSubset1
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ( ( binintersect @ A @ B )
% 0.32/1.74 = A )
% 0.32/1.74 => ( subset @ A @ B ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(bs114d_type,type,
% 0.32/1.74 bs114d: $o ).
% 0.32/1.74
% 0.32/1.74 thf(bs114d,definition,
% 0.32/1.74 ( bs114d
% 0.32/1.74 = ( ! [A: $i,B: $i,C: $i] :
% 0.32/1.74 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.32/1.74 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(regular_type,type,
% 0.32/1.74 regular: $i > $o ).
% 0.32/1.74
% 0.32/1.74 thf(setminus_type,type,
% 0.32/1.74 setminus: $i > $i > $i ).
% 0.32/1.74
% 0.32/1.74 thf(setminusI_type,type,
% 0.32/1.74 setminusI: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setminusI,definition,
% 0.32/1.74 ( setminusI
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ( ~ ( in @ Xx @ B )
% 0.32/1.74 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setminusEL_type,type,
% 0.32/1.74 setminusEL: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setminusEL,definition,
% 0.32/1.74 ( setminusEL
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.32/1.74 => ( in @ Xx @ A ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setminusER_type,type,
% 0.32/1.74 setminusER: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setminusER,definition,
% 0.32/1.74 ( setminusER
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.32/1.74 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setminusSubset2_type,type,
% 0.32/1.74 setminusSubset2: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setminusSubset2,definition,
% 0.32/1.74 ( setminusSubset2
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ( subset @ A @ B )
% 0.32/1.74 => ( ( setminus @ A @ B )
% 0.32/1.74 = emptyset ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setminusERneg_type,type,
% 0.32/1.74 setminusERneg: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setminusERneg,definition,
% 0.32/1.74 ( setminusERneg
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.32/1.74 => ( ( in @ Xx @ A )
% 0.32/1.74 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setminusELneg_type,type,
% 0.32/1.74 setminusELneg: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setminusELneg,definition,
% 0.32/1.74 ( setminusELneg
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.32/1.74 => ( ~ ( in @ Xx @ B )
% 0.32/1.74 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setminusILneg_type,type,
% 0.32/1.74 setminusILneg: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setminusILneg,definition,
% 0.32/1.74 ( setminusILneg
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ~ ( in @ Xx @ A )
% 0.32/1.74 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setminusIRneg_type,type,
% 0.32/1.74 setminusIRneg: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setminusIRneg,definition,
% 0.32/1.74 ( setminusIRneg
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ B )
% 0.32/1.74 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setminusLsub_type,type,
% 0.32/1.74 setminusLsub: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setminusLsub,definition,
% 0.32/1.74 ( setminusLsub
% 0.32/1.74 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setminusSubset1_type,type,
% 0.32/1.74 setminusSubset1: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setminusSubset1,definition,
% 0.32/1.74 ( setminusSubset1
% 0.32/1.74 = ( ! [A: $i,B: $i] :
% 0.32/1.74 ( ( ( setminus @ A @ B )
% 0.32/1.74 = emptyset )
% 0.32/1.74 => ( subset @ A @ B ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(symdiff_type,type,
% 0.32/1.74 symdiff: $i > $i > $i ).
% 0.32/1.74
% 0.32/1.74 thf(symdiffE_type,type,
% 0.32/1.74 symdiffE: $o ).
% 0.32/1.74
% 0.32/1.74 thf(symdiffE,definition,
% 0.32/1.74 ( symdiffE
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.32/1.74 => ! [Xphi: $o] :
% 0.32/1.74 ( ( ( in @ Xx @ A )
% 0.32/1.74 => ( ~ ( in @ Xx @ B )
% 0.32/1.74 => Xphi ) )
% 0.32/1.74 => ( ( ~ ( in @ Xx @ A )
% 0.32/1.74 => ( ( in @ Xx @ B )
% 0.32/1.74 => Xphi ) )
% 0.32/1.74 => Xphi ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(symdiffI1_type,type,
% 0.32/1.74 symdiffI1: $o ).
% 0.32/1.74
% 0.32/1.74 thf(symdiffI1,definition,
% 0.32/1.74 ( symdiffI1
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ( ~ ( in @ Xx @ B )
% 0.32/1.74 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(symdiffI2_type,type,
% 0.32/1.74 symdiffI2: $o ).
% 0.32/1.74
% 0.32/1.74 thf(symdiffI2,definition,
% 0.32/1.74 ( symdiffI2
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ~ ( in @ Xx @ A )
% 0.32/1.74 => ( ( in @ Xx @ B )
% 0.32/1.74 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(symdiffIneg1_type,type,
% 0.32/1.74 symdiffIneg1: $o ).
% 0.32/1.74
% 0.32/1.74 thf(symdiffIneg1,definition,
% 0.32/1.74 ( symdiffIneg1
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ( ( in @ Xx @ B )
% 0.32/1.74 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(symdiffIneg2_type,type,
% 0.32/1.74 symdiffIneg2: $o ).
% 0.32/1.74
% 0.32/1.74 thf(symdiffIneg2,definition,
% 0.32/1.74 ( symdiffIneg2
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ~ ( in @ Xx @ A )
% 0.32/1.74 => ( ~ ( in @ Xx @ B )
% 0.32/1.74 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(iskpair_type,type,
% 0.32/1.74 iskpair: $i > $o ).
% 0.32/1.74
% 0.32/1.74 thf(secondinupair_type,type,
% 0.32/1.74 secondinupair: $o ).
% 0.32/1.74
% 0.32/1.74 thf(secondinupair,definition,
% 0.32/1.74 ( secondinupair
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setukpairIL_type,type,
% 0.32/1.74 setukpairIL: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setukpairIL,definition,
% 0.32/1.74 ( setukpairIL
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setukpairIR_type,type,
% 0.32/1.74 setukpairIR: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setukpairIR,definition,
% 0.32/1.74 ( setukpairIR
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(kpairiskpair_type,type,
% 0.32/1.74 kpairiskpair: $o ).
% 0.32/1.74
% 0.32/1.74 thf(kpairiskpair,definition,
% 0.32/1.74 ( kpairiskpair
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(kpair_type,type,
% 0.32/1.74 kpair: $i > $i > $i ).
% 0.32/1.74
% 0.32/1.74 thf(kpairp_type,type,
% 0.32/1.74 kpairp: $o ).
% 0.32/1.74
% 0.32/1.74 thf(kpairp,definition,
% 0.32/1.74 ( kpairp
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(cartprod_type,type,
% 0.32/1.74 cartprod: $i > $i > $i ).
% 0.32/1.74
% 0.32/1.74 thf(singletonsubset_type,type,
% 0.32/1.74 singletonsubset: $o ).
% 0.32/1.74
% 0.32/1.74 thf(singletonsubset,definition,
% 0.32/1.74 ( singletonsubset
% 0.32/1.74 = ( ! [A: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(singletoninpowerset_type,type,
% 0.32/1.74 singletoninpowerset: $o ).
% 0.32/1.74
% 0.32/1.74 thf(singletoninpowerset,definition,
% 0.32/1.74 ( singletoninpowerset
% 0.32/1.74 = ( ! [A: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(singletoninpowunion_type,type,
% 0.32/1.74 singletoninpowunion: $o ).
% 0.32/1.74
% 0.32/1.74 thf(singletoninpowunion,definition,
% 0.32/1.74 ( singletoninpowunion
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(upairset2E_type,type,
% 0.32/1.74 upairset2E: $o ).
% 0.32/1.74
% 0.32/1.74 thf(upairset2E,definition,
% 0.32/1.74 ( upairset2E
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.32/1.74 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.32/1.74 => ( ( Xz = Xx )
% 0.32/1.74 | ( Xz = Xy ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(upairsubunion_type,type,
% 0.32/1.74 upairsubunion: $o ).
% 0.32/1.74
% 0.32/1.74 thf(upairsubunion,definition,
% 0.32/1.74 ( upairsubunion
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ! [Xy: $i] :
% 0.32/1.74 ( ( in @ Xy @ B )
% 0.32/1.74 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(upairinpowunion_type,type,
% 0.32/1.74 upairinpowunion: $o ).
% 0.32/1.74
% 0.32/1.74 thf(upairinpowunion,definition,
% 0.32/1.74 ( upairinpowunion
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ! [Xy: $i] :
% 0.32/1.74 ( ( in @ Xy @ B )
% 0.32/1.74 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(ubforcartprodlem1_type,type,
% 0.32/1.74 ubforcartprodlem1: $o ).
% 0.32/1.74
% 0.32/1.74 thf(ubforcartprodlem1,definition,
% 0.32/1.74 ( ubforcartprodlem1
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ! [Xy: $i] :
% 0.32/1.74 ( ( in @ Xy @ B )
% 0.32/1.74 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(ubforcartprodlem2_type,type,
% 0.32/1.74 ubforcartprodlem2: $o ).
% 0.32/1.74
% 0.32/1.74 thf(ubforcartprodlem2,definition,
% 0.32/1.74 ( ubforcartprodlem2
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ! [Xy: $i] :
% 0.32/1.74 ( ( in @ Xy @ B )
% 0.32/1.74 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(ubforcartprodlem3_type,type,
% 0.32/1.74 ubforcartprodlem3: $o ).
% 0.32/1.74
% 0.32/1.74 thf(ubforcartprodlem3,definition,
% 0.32/1.74 ( ubforcartprodlem3
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ! [Xy: $i] :
% 0.32/1.74 ( ( in @ Xy @ B )
% 0.32/1.74 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(cartprodpairin_type,type,
% 0.32/1.74 cartprodpairin: $o ).
% 0.32/1.74
% 0.32/1.74 thf(cartprodpairin,definition,
% 0.32/1.74 ( cartprodpairin
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ! [Xy: $i] :
% 0.32/1.74 ( ( in @ Xy @ B )
% 0.32/1.74 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(cartprodmempair1_type,type,
% 0.32/1.74 cartprodmempair1: $o ).
% 0.32/1.74
% 0.32/1.74 thf(cartprodmempair1,definition,
% 0.32/1.74 ( cartprodmempair1
% 0.32/1.74 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.32/1.74 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.32/1.74 => ? [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 & ? [Xy: $i] :
% 0.32/1.74 ( ( in @ Xy @ B )
% 0.32/1.74 & ( Xu
% 0.32/1.74 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(cartprodmempair_type,type,
% 0.32/1.74 cartprodmempair: $o ).
% 0.32/1.74
% 0.32/1.74 thf(cartprodmempair,definition,
% 0.32/1.74 ( cartprodmempair
% 0.32/1.74 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.32/1.74 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.32/1.74 => ( iskpair @ Xu ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setunionE2_type,type,
% 0.32/1.74 setunionE2: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setunionE2,definition,
% 0.32/1.74 ( setunionE2
% 0.32/1.74 = ( ! [A: $i,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ ( setunion @ A ) )
% 0.32/1.74 => ? [X: $i] :
% 0.32/1.74 ( ( in @ X @ A )
% 0.32/1.74 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setunionsingleton1_type,type,
% 0.32/1.74 setunionsingleton1: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setunionsingleton1,definition,
% 0.32/1.74 ( setunionsingleton1
% 0.32/1.74 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setunionsingleton2_type,type,
% 0.32/1.74 setunionsingleton2: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setunionsingleton2,definition,
% 0.32/1.74 ( setunionsingleton2
% 0.32/1.74 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setunionsingleton_type,type,
% 0.32/1.74 setunionsingleton: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setunionsingleton,definition,
% 0.32/1.74 ( setunionsingleton
% 0.32/1.74 = ( ! [Xx: $i] :
% 0.32/1.74 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.32/1.74 = Xx ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(singleton_type,type,
% 0.32/1.74 singleton: $i > $o ).
% 0.32/1.74
% 0.32/1.74 thf(singleton,definition,
% 0.32/1.74 ( singleton
% 0.32/1.74 = ( ^ [A: $i] :
% 0.32/1.74 ? [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 & ( A
% 0.32/1.74 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(singletonprop_type,type,
% 0.32/1.74 singletonprop: $o ).
% 0.32/1.74
% 0.32/1.74 thf(singletonprop,definition,
% 0.32/1.74 ( singletonprop
% 0.32/1.74 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/1.74 ( ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ! [Xy: $i] :
% 0.32/1.74 ( ( in @ Xy @ A )
% 0.32/1.74 => ( ( Xphi @ Xx )
% 0.32/1.74 => ( ( Xphi @ Xy )
% 0.32/1.74 => ( Xx = Xy ) ) ) ) )
% 0.32/1.74 => ( ? [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 & ( Xphi @ Xx ) )
% 0.32/1.74 => ( singleton
% 0.32/1.74 @ ( dsetconstr @ A
% 0.32/1.74 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(ex1_type,type,
% 0.32/1.74 ex1: $i > ( $i > $o ) > $o ).
% 0.32/1.74
% 0.32/1.74 thf(ex1,definition,
% 0.32/1.74 ( ex1
% 0.32/1.74 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.32/1.74 ( singleton
% 0.32/1.74 @ ( dsetconstr @ A
% 0.32/1.74 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(ex1E1_type,type,
% 0.32/1.74 ex1E1: $o ).
% 0.32/1.74
% 0.32/1.74 thf(ex1E1,definition,
% 0.32/1.74 ( ex1E1
% 0.32/1.74 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/1.74 ( ( ex1 @ A
% 0.32/1.74 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/1.74 => ? [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(ex1I_type,type,
% 0.32/1.74 ex1I: $o ).
% 0.32/1.74
% 0.32/1.74 thf(ex1I,definition,
% 0.32/1.74 ( ex1I
% 0.32/1.74 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ( ( Xphi @ Xx )
% 0.32/1.74 => ( ! [Xy: $i] :
% 0.32/1.74 ( ( in @ Xy @ A )
% 0.32/1.74 => ( ( Xphi @ Xy )
% 0.32/1.74 => ( Xy = Xx ) ) )
% 0.32/1.74 => ( ex1 @ A
% 0.32/1.74 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(ex1I2_type,type,
% 0.32/1.74 ex1I2: $o ).
% 0.32/1.74
% 0.32/1.74 thf(ex1I2,definition,
% 0.32/1.74 ( ex1I2
% 0.32/1.74 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/1.74 ( ! [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 => ! [Xy: $i] :
% 0.32/1.74 ( ( in @ Xy @ A )
% 0.32/1.74 => ( ( Xphi @ Xx )
% 0.32/1.74 => ( ( Xphi @ Xy )
% 0.32/1.74 => ( Xx = Xy ) ) ) ) )
% 0.32/1.74 => ( ? [Xx: $i] :
% 0.32/1.74 ( ( in @ Xx @ A )
% 0.32/1.74 & ( Xphi @ Xx ) )
% 0.32/1.74 => ( ex1 @ A
% 0.32/1.74 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(singletonsuniq_type,type,
% 0.32/1.74 singletonsuniq: $o ).
% 0.32/1.74
% 0.32/1.74 thf(singletonsuniq,definition,
% 0.32/1.74 ( singletonsuniq
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i] :
% 0.32/1.74 ( ( ( setadjoin @ Xx @ emptyset )
% 0.32/1.74 = ( setadjoin @ Xy @ emptyset ) )
% 0.32/1.74 => ( Xx = Xy ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(atmost1p_type,type,
% 0.32/1.74 atmost1p: $i > $o ).
% 0.32/1.74
% 0.32/1.74 thf(atleast2p_type,type,
% 0.32/1.74 atleast2p: $i > $o ).
% 0.32/1.74
% 0.32/1.74 thf(atmost2p_type,type,
% 0.32/1.74 atmost2p: $i > $o ).
% 0.32/1.74
% 0.32/1.74 thf(upairsetp_type,type,
% 0.32/1.74 upairsetp: $i > $o ).
% 0.32/1.74
% 0.32/1.74 thf(setukpairinjL1_type,type,
% 0.32/1.74 setukpairinjL1: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setukpairinjL1,definition,
% 0.32/1.74 ( setukpairinjL1
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.32/1.74 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.32/1.74 => ( Xx = Xz ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(kfstsingleton_type,type,
% 0.32/1.74 kfstsingleton: $o ).
% 0.32/1.74
% 0.32/1.74 thf(kfstsingleton,definition,
% 0.32/1.74 ( kfstsingleton
% 0.32/1.74 = ( ! [Xu: $i] :
% 0.32/1.74 ( ( iskpair @ Xu )
% 0.32/1.74 => ( singleton
% 0.32/1.74 @ ( dsetconstr @ ( setunion @ Xu )
% 0.32/1.74 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(theprop_type,type,
% 0.32/1.74 theprop: $o ).
% 0.32/1.74
% 0.32/1.74 thf(theprop,definition,
% 0.32/1.74 ( theprop
% 0.32/1.74 = ( ! [X: $i] :
% 0.32/1.74 ( ( singleton @ X )
% 0.32/1.74 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(kfst_type,type,
% 0.32/1.74 kfst: $i > $i ).
% 0.32/1.74
% 0.32/1.74 thf(kfstpairEq_type,type,
% 0.32/1.74 kfstpairEq: $o ).
% 0.32/1.74
% 0.32/1.74 thf(kfstpairEq,definition,
% 0.32/1.74 ( kfstpairEq
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i] :
% 0.32/1.74 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.32/1.74 = Xx ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(cartprodfstin_type,type,
% 0.32/1.74 cartprodfstin: $o ).
% 0.32/1.74
% 0.32/1.74 thf(cartprodfstin,definition,
% 0.32/1.74 ( cartprodfstin
% 0.32/1.74 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.32/1.74 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.32/1.74 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setukpairinjL2_type,type,
% 0.32/1.74 setukpairinjL2: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setukpairinjL2,definition,
% 0.32/1.74 ( setukpairinjL2
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/1.74 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.32/1.74 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.32/1.74 => ( Xx = Xz ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setukpairinjL_type,type,
% 0.32/1.74 setukpairinjL: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setukpairinjL,definition,
% 0.32/1.74 ( setukpairinjL
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/1.74 ( ( ( kpair @ Xx @ Xy )
% 0.32/1.74 = ( kpair @ Xz @ Xu ) )
% 0.32/1.74 => ( Xx = Xz ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setukpairinjR11_type,type,
% 0.32/1.74 setukpairinjR11: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setukpairinjR11,definition,
% 0.32/1.74 ( setukpairinjR11
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i] :
% 0.32/1.74 ( ( Xx = Xy )
% 0.32/1.74 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.32/1.74 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setukpairinjR12_type,type,
% 0.32/1.74 setukpairinjR12: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setukpairinjR12,definition,
% 0.32/1.74 ( setukpairinjR12
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i] :
% 0.32/1.74 ( ( Xx = Xy )
% 0.32/1.74 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.32/1.74 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setukpairinjR1_type,type,
% 0.32/1.74 setukpairinjR1: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setukpairinjR1,definition,
% 0.32/1.74 ( setukpairinjR1
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/1.74 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.32/1.74 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.32/1.74 => ( ( Xz = Xu )
% 0.32/1.74 => ( Xy = Xu ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(upairequniteq_type,type,
% 0.32/1.74 upairequniteq: $o ).
% 0.32/1.74
% 0.32/1.74 thf(upairequniteq,definition,
% 0.32/1.74 ( upairequniteq
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.32/1.74 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.32/1.74 = ( setadjoin @ Xz @ emptyset ) )
% 0.32/1.74 => ( Xx = Xy ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setukpairinjR2_type,type,
% 0.32/1.74 setukpairinjR2: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setukpairinjR2,definition,
% 0.32/1.74 ( setukpairinjR2
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/1.74 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.32/1.74 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.32/1.74 => ( Xy = Xu ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(setukpairinjR_type,type,
% 0.32/1.74 setukpairinjR: $o ).
% 0.32/1.74
% 0.32/1.74 thf(setukpairinjR,definition,
% 0.32/1.74 ( setukpairinjR
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/1.74 ( ( ( kpair @ Xx @ Xy )
% 0.32/1.74 = ( kpair @ Xz @ Xu ) )
% 0.32/1.74 => ( Xy = Xu ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(ksndsingleton_type,type,
% 0.32/1.74 ksndsingleton: $o ).
% 0.32/1.74
% 0.32/1.74 thf(ksndsingleton,definition,
% 0.32/1.74 ( ksndsingleton
% 0.32/1.74 = ( ! [Xu: $i] :
% 0.32/1.74 ( ( iskpair @ Xu )
% 0.32/1.74 => ( singleton
% 0.32/1.74 @ ( dsetconstr @ ( setunion @ Xu )
% 0.32/1.74 @ ^ [Xx: $i] :
% 0.32/1.74 ( Xu
% 0.32/1.74 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(ksnd_type,type,
% 0.32/1.74 ksnd: $i > $i ).
% 0.32/1.74
% 0.32/1.74 thf(ksndpairEq_type,type,
% 0.32/1.74 ksndpairEq: $o ).
% 0.32/1.74
% 0.32/1.74 thf(ksndpairEq,definition,
% 0.32/1.74 ( ksndpairEq
% 0.32/1.74 = ( ! [Xx: $i,Xy: $i] :
% 0.32/1.74 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.32/1.74 = Xy ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(kpairsurjEq_type,type,
% 0.32/1.74 kpairsurjEq: $o ).
% 0.32/1.74
% 0.32/1.74 thf(kpairsurjEq,definition,
% 0.32/1.74 ( kpairsurjEq
% 0.32/1.74 = ( ! [Xu: $i] :
% 0.32/1.74 ( ( iskpair @ Xu )
% 0.32/1.74 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.32/1.74 = Xu ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(cartprodsndin_type,type,
% 0.32/1.74 cartprodsndin: $o ).
% 0.32/1.74
% 0.32/1.74 thf(cartprodsndin,definition,
% 0.32/1.74 ( cartprodsndin
% 0.32/1.74 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.32/1.74 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.32/1.74 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(cartprodpairmemEL_type,type,
% 0.32/1.74 cartprodpairmemEL: $o ).
% 0.32/1.74
% 0.32/1.74 thf(cartprodpairmemEL,definition,
% 0.32/1.74 ( cartprodpairmemEL
% 0.32/1.74 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.32/1.74 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.32/1.74 => ( in @ Xx @ A ) ) ) ) ).
% 0.32/1.74
% 0.32/1.74 thf(cartprodpairmemER_type,type,
% 0.32/1.75 cartprodpairmemER: $o ).
% 0.32/1.75
% 0.32/1.75 thf(cartprodpairmemER,definition,
% 0.32/1.75 ( cartprodpairmemER
% 0.32/1.75 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.32/1.75 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.32/1.75 => ( in @ Xy @ B ) ) ) ) ).
% 0.32/1.75
% 0.32/1.75 thf(cartprodmempaircEq,conjecture,
% 0.32/1.75 ( setextAx
% 0.32/1.75 => ( emptysetAx
% 0.32/1.75 => ( setadjoinAx
% 0.32/1.75 => ( powersetAx
% 0.32/1.75 => ( setunionAx
% 0.32/1.75 => ( omega0Ax
% 0.32/1.75 => ( omegaSAx
% 0.32/1.75 => ( omegaIndAx
% 0.32/1.75 => ( replAx
% 0.32/1.75 => ( foundationAx
% 0.32/1.75 => ( wellorderingAx
% 0.32/1.75 => ( descrp
% 0.32/1.75 => ( dsetconstrI
% 0.32/1.75 => ( dsetconstrEL
% 0.32/1.75 => ( dsetconstrER
% 0.32/1.75 => ( exuE1
% 0.32/1.75 => ( prop2setE
% 0.32/1.75 => ( emptysetE
% 0.32/1.75 => ( emptysetimpfalse
% 0.32/1.75 => ( notinemptyset
% 0.32/1.75 => ( exuE3e
% 0.32/1.75 => ( setext
% 0.32/1.75 => ( emptyI
% 0.32/1.75 => ( noeltsimpempty
% 0.32/1.75 => ( setbeta
% 0.32/1.75 => ( nonemptyE1
% 0.32/1.75 => ( nonemptyI
% 0.32/1.75 => ( nonemptyI1
% 0.32/1.75 => ( setadjoinIL
% 0.32/1.75 => ( emptyinunitempty
% 0.32/1.75 => ( setadjoinIR
% 0.32/1.75 => ( setadjoinE
% 0.32/1.75 => ( setadjoinOr
% 0.32/1.75 => ( setoftrueEq
% 0.32/1.75 => ( powersetI
% 0.32/1.75 => ( emptyinPowerset
% 0.32/1.75 => ( emptyInPowerset
% 0.32/1.75 => ( powersetE
% 0.32/1.75 => ( setunionI
% 0.32/1.75 => ( setunionE
% 0.32/1.75 => ( subPowSU
% 0.32/1.75 => ( exuE2
% 0.32/1.75 => ( nonemptyImpWitness
% 0.32/1.75 => ( uniqinunit
% 0.32/1.75 => ( notinsingleton
% 0.32/1.75 => ( eqinunit
% 0.32/1.75 => ( singletonsswitch
% 0.32/1.75 => ( upairsetE
% 0.32/1.75 => ( upairsetIL
% 0.32/1.75 => ( upairsetIR
% 0.32/1.75 => ( emptyE1
% 0.32/1.75 => ( vacuousDall
% 0.32/1.75 => ( quantDeMorgan1
% 0.32/1.75 => ( quantDeMorgan2
% 0.32/1.75 => ( quantDeMorgan3
% 0.32/1.75 => ( quantDeMorgan4
% 0.32/1.75 => ( prop2setI
% 0.32/1.75 => ( prop2set2propI
% 0.32/1.75 => ( notdexE
% 0.32/1.75 => ( notdallE
% 0.32/1.75 => ( exuI1
% 0.32/1.75 => ( exuI3
% 0.32/1.75 => ( exuI2
% 0.32/1.75 => ( inCongP
% 0.32/1.75 => ( in__Cong
% 0.32/1.75 => ( exuE3u
% 0.32/1.75 => ( exu__Cong
% 0.32/1.75 => ( emptyset__Cong
% 0.32/1.75 => ( setadjoin__Cong
% 0.32/1.75 => ( powerset__Cong
% 0.32/1.75 => ( setunion__Cong
% 0.32/1.75 => ( omega__Cong
% 0.32/1.75 => ( exuEu
% 0.32/1.75 => ( descr__Cong
% 0.32/1.75 => ( dsetconstr__Cong
% 0.32/1.75 => ( subsetI1
% 0.32/1.75 => ( eqimpsubset2
% 0.32/1.75 => ( eqimpsubset1
% 0.32/1.75 => ( subsetI2
% 0.32/1.75 => ( emptysetsubset
% 0.32/1.75 => ( subsetE
% 0.32/1.75 => ( subsetE2
% 0.32/1.75 => ( notsubsetI
% 0.32/1.75 => ( notequalI1
% 0.32/1.75 => ( notequalI2
% 0.32/1.75 => ( subsetRefl
% 0.32/1.75 => ( subsetTrans
% 0.32/1.75 => ( setadjoinSub
% 0.32/1.75 => ( setadjoinSub2
% 0.32/1.75 => ( subset2powerset
% 0.32/1.75 => ( setextsub
% 0.32/1.75 => ( subsetemptysetimpeq
% 0.32/1.75 => ( powersetI1
% 0.32/1.75 => ( powersetE1
% 0.32/1.75 => ( inPowerset
% 0.32/1.75 => ( powersetsubset
% 0.32/1.75 => ( sepInPowerset
% 0.32/1.75 => ( sepSubset
% 0.32/1.75 => ( binunionIL
% 0.32/1.75 => ( upairset2IR
% 0.32/1.75 => ( binunionIR
% 0.32/1.75 => ( binunionEcases
% 0.32/1.75 => ( binunionE
% 0.32/1.75 => ( binunionLsub
% 0.32/1.75 => ( binunionRsub
% 0.32/1.75 => ( binintersectI
% 0.32/1.75 => ( binintersectSubset5
% 0.32/1.75 => ( binintersectEL
% 0.32/1.75 => ( binintersectLsub
% 0.32/1.75 => ( binintersectSubset2
% 0.32/1.75 => ( binintersectSubset3
% 0.32/1.75 => ( binintersectER
% 0.32/1.75 => ( disjointsetsI1
% 0.32/1.75 => ( binintersectRsub
% 0.32/1.75 => ( binintersectSubset4
% 0.32/1.75 => ( binintersectSubset1
% 0.32/1.75 => ( bs114d
% 0.32/1.75 => ( setminusI
% 0.32/1.75 => ( setminusEL
% 0.32/1.75 => ( setminusER
% 0.32/1.75 => ( setminusSubset2
% 0.32/1.75 => ( setminusERneg
% 0.32/1.75 => ( setminusELneg
% 0.32/1.75 => ( setminusILneg
% 0.32/1.75 => ( setminusIRneg
% 0.32/1.75 => ( setminusLsub
% 0.32/1.75 => ( setminusSubset1
% 0.32/1.75 => ( symdiffE
% 0.32/1.75 => ( symdiffI1
% 0.32/1.75 => ( symdiffI2
% 0.32/1.75 => ( symdiffIneg1
% 0.32/1.75 => ( symdiffIneg2
% 0.32/1.75 => ( secondinupair
% 0.32/1.75 => ( setukpairIL
% 0.32/1.75 => ( setukpairIR
% 0.32/1.75 => ( kpairiskpair
% 0.32/1.75 => ( kpairp
% 0.32/1.75 => ( singletonsubset
% 0.32/1.75 => ( singletoninpowerset
% 0.32/1.75 => ( singletoninpowunion
% 0.32/1.75 => ( upairset2E
% 0.32/1.75 => ( upairsubunion
% 0.32/1.75 => ( upairinpowunion
% 0.32/1.75 => ( ubforcartprodlem1
% 0.32/1.75 => ( ubforcartprodlem2
% 0.32/1.75 => ( ubforcartprodlem3
% 0.32/1.75 => ( cartprodpairin
% 0.32/1.75 => ( cartprodmempair1
% 0.32/1.75 => ( cartprodmempair
% 0.32/1.75 => ( setunionE2
% 0.32/1.75 => ( setunionsingleton1
% 0.32/1.75 => ( setunionsingleton2
% 0.32/1.75 => ( setunionsingleton
% 0.32/1.75 => ( singletonprop
% 0.32/1.75 => ( ex1E1
% 0.32/1.75 => ( ex1I
% 0.32/1.75 => ( ex1I2
% 0.32/1.75 => ( singletonsuniq
% 0.32/1.75 => ( setukpairinjL1
% 0.32/1.75 => ( kfstsingleton
% 0.32/1.75 => ( theprop
% 0.32/1.75 => ( kfstpairEq
% 0.32/1.75 => ( cartprodfstin
% 0.32/1.75 => ( setukpairinjL2
% 0.32/1.75 => ( setukpairinjL
% 0.32/1.75 => ( setukpairinjR11
% 0.32/1.75 => ( setukpairinjR12
% 0.32/1.75 => ( setukpairinjR1
% 0.32/1.75 => ( upairequniteq
% 0.32/1.75 => ( setukpairinjR2
% 0.32/1.75 => ( setukpairinjR
% 0.32/1.75 => ( ksndsingleton
% 0.32/1.75 => ( ksndpairEq
% 0.32/1.75 => ( kpairsurjEq
% 0.32/1.75 => ( cartprodsndin
% 0.32/1.75 => ( cartprodpairmemEL
% 0.32/1.75 => ( cartprodpairmemER
% 0.32/1.75 => ! [A: $i,B: $i,Xx: $i] :
% 0.32/1.75 ( ( in @ Xx @ A )
% 0.32/1.75 => ! [Xy: $i] :
% 0.32/1.75 ( ( in @ Xy @ B )
% 0.32/1.75 => ( ( kpair @ Xx @ Xy )
% 0.32/1.75 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/1.75 ------------------------
% 0.32/1.77 % (9552)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_9448 for (2999ds/275Mi)
% 0.32/1.77 % (9548)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_9448 for (2999ds/4Mi)
% 0.32/1.77 % (9551)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_9448 for (2999ds/2Mi)
% 0.32/1.77 % (9549)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_9448 for (2999ds/27Mi)
% 0.32/1.77 % (9547)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_9448 for (2999ds/183Mi)
% 0.32/1.77 % (9550)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_9448 for (2999ds/2Mi)
% 0.32/1.77 % (9553)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_9448 for (2999ds/18Mi)
% 0.32/1.78 % (9550)Instruction limit reached!
% 0.32/1.78 % (9550)------------------------------
% 0.32/1.78 % (9550)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.78 % (9550)Termination reason: Unknown
% 0.32/1.78 % (9550)Termination phase: shuffling
% 0.32/1.78
% 0.32/1.78 % (9550)Memory used [KB]: 1279
% 0.32/1.78 % (9550)Time elapsed: 0.003 s
% 0.32/1.78 % (9550)Instructions burned: 2 (million)
% 0.32/1.78 % (9550)------------------------------
% 0.32/1.78 % (9550)------------------------------
% 0.32/1.78 % (9551)Instruction limit reached!
% 0.32/1.78 % (9551)------------------------------
% 0.32/1.78 % (9551)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.78 % (9551)Termination reason: Unknown
% 0.32/1.78 % (9551)Termination phase: shuffling
% 0.32/1.78
% 0.32/1.78 % (9551)Memory used [KB]: 1279
% 0.32/1.78 % (9551)Time elapsed: 0.003 s
% 0.32/1.78 % (9551)Instructions burned: 2 (million)
% 0.32/1.78 % (9551)------------------------------
% 0.32/1.78 % (9551)------------------------------
% 0.32/1.78 % (9548)Instruction limit reached!
% 0.32/1.78 % (9548)------------------------------
% 0.32/1.78 % (9548)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.78 % (9548)Termination reason: Unknown
% 0.32/1.78 % (9548)Termination phase: shuffling
% 0.32/1.78
% 0.32/1.78 % (9548)Memory used [KB]: 1407
% 0.32/1.78 % (9548)Time elapsed: 0.006 s
% 0.32/1.78 % (9548)Instructions burned: 5 (million)
% 0.32/1.78 % (9548)------------------------------
% 0.32/1.78 % (9548)------------------------------
% 0.32/1.78 % (9553)Instruction limit reached!
% 0.32/1.78 % (9553)------------------------------
% 0.32/1.78 % (9553)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.78 % (9553)Termination reason: Unknown
% 0.32/1.78 % (9553)Termination phase: Property scanning
% 0.32/1.78
% 0.32/1.78 % (9553)Memory used [KB]: 1663
% 0.32/1.78 % (9553)Time elapsed: 0.011 s
% 0.32/1.78 % (9553)Instructions burned: 19 (million)
% 0.32/1.78 % (9553)------------------------------
% 0.32/1.78 % (9553)------------------------------
% 0.32/1.79 % (9549)Instruction limit reached!
% 0.32/1.79 % (9549)------------------------------
% 0.32/1.79 % (9549)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.79 % (9549)Termination reason: Unknown
% 0.32/1.79 % (9549)Termination phase: Property scanning
% 0.32/1.79
% 0.32/1.79 % (9549)Memory used [KB]: 1791
% 0.32/1.79 % (9549)Time elapsed: 0.015 s
% 0.32/1.79 % (9549)Instructions burned: 28 (million)
% 0.32/1.79 % (9549)------------------------------
% 0.32/1.79 % (9549)------------------------------
% 0.32/1.79 % (9554)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_9448 for (2999ds/3Mi)
% 0.32/1.79 % (9555)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_9448 for (2999ds/37Mi)
% 0.32/1.79 % (9554)Instruction limit reached!
% 0.32/1.79 % (9554)------------------------------
% 0.32/1.79 % (9554)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.79 % (9554)Termination reason: Unknown
% 0.32/1.79 % (9554)Termination phase: shuffling
% 0.32/1.79
% 0.32/1.79 % (9554)Memory used [KB]: 1279
% 0.32/1.79 % (9554)Time elapsed: 0.003 s
% 0.32/1.79 % (9554)Instructions burned: 3 (million)
% 0.32/1.79 % (9554)------------------------------
% 0.32/1.79 % (9554)------------------------------
% 0.32/1.79 % (9556)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_9448 for (2999ds/15Mi)
% 3.02/1.80 % (9557)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_9448 for (2999ds/3Mi)
% 3.02/1.80 % (9556)Instruction limit reached!
% 3.02/1.80 % (9556)------------------------------
% 3.02/1.80 % (9556)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.80 % (9556)Termination reason: Unknown
% 3.02/1.80 % (9556)Termination phase: shuffling
% 3.02/1.80
% 3.02/1.80 % (9556)Memory used [KB]: 1663
% 3.02/1.80 % (9556)Time elapsed: 0.009 s
% 3.02/1.80 % (9556)Instructions burned: 15 (million)
% 3.02/1.80 % (9556)------------------------------
% 3.02/1.80 % (9556)------------------------------
% 3.02/1.80 % (9557)Instruction limit reached!
% 3.02/1.80 % (9557)------------------------------
% 3.02/1.80 % (9557)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.80 % (9557)Termination reason: Unknown
% 3.02/1.80 % (9557)Termination phase: shuffling
% 3.02/1.80
% 3.02/1.80 % (9557)Memory used [KB]: 1279
% 3.02/1.80 % (9557)Time elapsed: 0.004 s
% 3.02/1.80 % (9557)Instructions burned: 4 (million)
% 3.02/1.80 % (9557)------------------------------
% 3.02/1.80 % (9557)------------------------------
% 3.02/1.80 % (9558)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_9448 for (2999ds/1041Mi)
% 3.02/1.81 % (9555)Instruction limit reached!
% 3.02/1.81 % (9555)------------------------------
% 3.02/1.81 % (9555)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.81 % (9555)Termination reason: Unknown
% 3.02/1.81 % (9555)Termination phase: Property scanning
% 3.02/1.81
% 3.02/1.81 % (9555)Memory used [KB]: 1918
% 3.02/1.81 % (9555)Time elapsed: 0.018 s
% 3.02/1.81 % (9555)Instructions burned: 37 (million)
% 3.02/1.81 % (9555)------------------------------
% 3.02/1.81 % (9555)------------------------------
% 3.02/1.81 % (9559)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_9448 for (2999ds/7Mi)
% 3.02/1.81 % (9559)Instruction limit reached!
% 3.02/1.81 % (9559)------------------------------
% 3.02/1.81 % (9559)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.81 % (9559)Termination reason: Unknown
% 3.02/1.81 % (9559)Termination phase: shuffling
% 3.02/1.81
% 3.02/1.81 % (9559)Memory used [KB]: 1407
% 3.02/1.81 % (9559)Time elapsed: 0.005 s
% 3.02/1.81 % (9559)Instructions burned: 8 (million)
% 3.02/1.81 % (9559)------------------------------
% 3.02/1.81 % (9559)------------------------------
% 3.02/1.82 % (9547)First to succeed.
% 3.02/1.82 % (9560)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_9448 for (2999ds/16Mi)
% 3.02/1.82 % (9561)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_9448 for (2999ds/3Mi)
% 3.02/1.82 % (9547)Refutation found. Thanks to Tanya!
% 3.02/1.82 % SZS status Theorem for DTF2THF_9448
% 3.02/1.82 % SZS output start Proof for DTF2THF_9448
% 3.02/1.82 thf(func_def_0, type, in: $i > $i > $o).
% 3.02/1.82 thf(func_def_1, type, exu: ($i > $o) > $o).
% 3.02/1.82 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 3.02/1.82 thf(func_def_8, type, powerset: $i > $i).
% 3.02/1.82 thf(func_def_10, type, setunion: $i > $i).
% 3.02/1.82 thf(func_def_19, type, descr: ($i > $o) > $i).
% 3.02/1.82 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 3.02/1.82 thf(func_def_26, type, prop2set: $o > $i).
% 3.02/1.82 thf(func_def_36, type, nonempty: $i > $o).
% 3.02/1.82 thf(func_def_69, type, set2prop: $i > $o).
% 3.02/1.82 thf(func_def_88, type, subset: $i > $i > $o).
% 3.02/1.82 thf(func_def_89, type, disjoint: $i > $i > $o).
% 3.02/1.82 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 3.02/1.82 thf(func_def_114, type, binunion: $i > $i > $i).
% 3.02/1.82 thf(func_def_122, type, binintersect: $i > $i > $i).
% 3.02/1.82 thf(func_def_135, type, regular: $i > $o).
% 3.02/1.82 thf(func_def_136, type, setminus: $i > $i > $i).
% 3.02/1.82 thf(func_def_147, type, symdiff: $i > $i > $i).
% 3.02/1.82 thf(func_def_153, type, iskpair: $i > $o).
% 3.02/1.82 thf(func_def_158, type, kpair: $i > $i > $i).
% 3.02/1.82 thf(func_def_160, type, cartprod: $i > $i > $i).
% 3.02/1.82 thf(func_def_177, type, singleton: $i > $o).
% 3.02/1.82 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 3.02/1.82 thf(func_def_184, type, atmost1p: $i > $o).
% 3.02/1.82 thf(func_def_185, type, atleast2p: $i > $o).
% 3.02/1.82 thf(func_def_186, type, atmost2p: $i > $o).
% 3.02/1.82 thf(func_def_187, type, upairsetp: $i > $o).
% 3.02/1.82 thf(func_def_191, type, kfst: $i > $i).
% 3.02/1.82 thf(func_def_203, type, ksnd: $i > $i).
% 3.02/1.82 thf(func_def_231, type, sK5: $i > $i).
% 3.02/1.82 thf(func_def_232, type, sK6: $i > $i > $i).
% 3.02/1.82 thf(func_def_233, type, sK7: $i > $i > $i).
% 3.02/1.82 thf(func_def_234, type, sK8: $i > $i > $i).
% 3.02/1.82 thf(func_def_235, type, sK9: $o > $i > $i > $i).
% 3.02/1.82 thf(func_def_236, type, sK10: $i > ($i > $o) > $i).
% 3.02/1.82 thf(f1264,plain,(
% 3.02/1.82 $false),
% 3.02/1.82 inference(trivial_inequality_removal,[status(thm)],[f723])).
% 3.02/1.82 thf(f723,plain,(
% 3.02/1.82 ((kpair @ sK2 @ sK3) != (kpair @ sK2 @ sK3))),
% 3.02/1.82 inference(cnf_transformation,[status(thm)],[f547])).
% 3.02/1.82 thf(f547,plain,(
% 3.02/1.82 (emptyset__Cong = $true) & (setadjoinSub2 = $true) & (setunionE2 = $true) & (quantDeMorgan1 = $true) & (ubforcartprodlem1 = $true) & (nonemptyI1 = $true) & (dsetconstrER = $true) & (upairequniteq = $true) & (emptyI = $true) & (setext = $true) & (foundationAx = $true) & (subsetE = $true) & (in__Cong = $true) & (singletonsuniq = $true) & (setadjoinOr = $true) & (setunionsingleton = $true) & (upairsetE = $true) & (eqimpsubset2 = $true) & (cartprodmempair1 = $true) & (upairsubunion = $true) & (setukpairinjR = $true) & (notdallE = $true) & (setadjoinAx = $true) & (binintersectSubset2 = $true) & (setminusEL = $true) & (upairinpowunion = $true) & (inPowerset = $true) & (setukpairinjL = $true) & (exu__Cong = $true) & (eqimpsubset1 = $true) & (descr__Cong = $true) & (symdiffI2 = $true) & (omegaSAx = $true) & (setunionAx = $true) & (omegaIndAx = $true) & (binintersectI = $true) & (exuE3e = $true) & (emptyinPowerset = $true) & (setukpairinjL1 = $true) & (emptysetimpfalse = $true) & (omega0Ax = $true) & (emptysetsubset = $true) & (dsetconstrEL = $true) & (cartprodfstin = $true) & (powersetI = $true) & (wellorderingAx = $true) & (exuEu = $true) & (binintersectSubset1 = $true) & (ksndsingleton = $true) & (emptyE1 = $true) & ((((kpair @ sK2 @ sK3) != (kpair @ sK2 @ sK3)) & ((in @ sK3 @ sK1) = $true)) & ((in @ sK2 @ sK0) = $true)) & (setadjoinE = $true) & (emptyinunitempty = $true) & (dsetconstr__Cong = $true) & (setadjoinIL = $true) & (setukpairinjR11 = $true) & (setunionE = $true) & (exuI3 = $true) & (notdexE = $true) & (eqinunit = $true) & (exuI2 = $true) & (powersetAx = $true) & (dsetconstrI = $true) & (setunionsingleton2 = $true) & (setukpairIL = $true) & (setextsub = $true) & (setminusI = $true) & (binunionLsub = $true) & (setukpairinjR12 = $true) & (subset2powerset = $true) & (powersetE = $true) & (quantDeMorgan4 = $true) & (quantDeMorgan2 = $true) & (binintersectLsub = $true) & (powersetE1 = $true) & (binintersectSubset4 = $true) & (omega__Cong = $true) & (bs114d = $true) & (prop2set2propI = $true) & (subsetI2 = $true) & (setunionI = $true) & (quantDeMorgan3 = $true) & (upairset2E = $true) & (notequalI2 = $true) & (symdiffE = $true) & (binunionEcases = $true) & (upairsetIL = $true) & (ex1I = $true) & (binintersectRsub = $true) & (kfstpairEq = $true) & (exuI1 = $true) & (kpairp = $true) & (singletoninpowerset = $true) & (descrp = $true) & (setminusELneg = $true) & (setadjoinSub = $true) & (notinemptyset = $true) & (ex1I2 = $true) & (cartprodpairmemER = $true) & (symdiffIneg2 = $true) & (subsetRefl = $true) & (exuE1 = $true) & (binintersectER = $true) & (powersetsubset = $true) & (disjointsetsI1 = $true) & (binintersectSubset5 = $true) & (nonemptyI = $true) & (setukpairinjL2 = $true) & (prop2setI = $true) & (nonemptyImpWitness = $true) & (setminusSubset2 = $true) & (setminusERneg = $true) & (singletonsswitch = $true) & (binunionRsub = $true) & (singletonsubset = $true) & (exuE2 = $true) & (setunionsingleton1 = $true) & (setadjoinIR = $true) & (subsetTrans = $true) & (cartprodpairin = $true) & (setminusSubset1 = $true) & (cartprodpairmemEL = $true) & (binunionE = $true) & (subsetE2 = $true) & (ksndpairEq = $true) & (binunionIR = $true) & (notsubsetI = $true) & (subsetI1 = $true) & (binintersectEL = $true) & (secondinupair = $true) & (setminusIRneg = $true) & (setextAx = $true) & (upairset2IR = $true) & (emptysetE = $true) & (replAx = $true) & (setadjoin__Cong = $true) & (ex1E1 = $true) & (symdiffIneg1 = $true) & (notequalI1 = $true) & (setminusER = $true) & (sepInPowerset = $true) & (setunion__Cong = $true) & (setbeta = $true) & (setukpairIR = $true) & (nonemptyE1 = $true) & (subPowSU = $true) & (kfstsingleton = $true) & (kpairiskpair = $true) & (noeltsimpempty = $true) & (ubforcartprodlem2 = $true) & (uniqinunit = $true) & (prop2setE = $true) & (setukpairinjR1 = $true) & (upairsetIR = $true) & (setminusILneg = $true) & (setoftrueEq = $true) & (notinsingleton = $true) & (sepSubset = $true) & (exuE3u = $true) & (emptyInPowerset = $true) & (powersetI1 = $true) & (inCongP = $true) & (ubforcartprodlem3 = $true) & (kpairsurjEq = $true) & (subsetemptysetimpeq = $true) & (binintersectSubset3 = $true) & (singletoninpowunion = $true) & (powerset__Cong = $true) & (emptysetAx = $true) & (vacuousDall = $true) & (cartprodmempair = $true) & (cartprodsndin = $true) & (symdiffI1 = $true) & (theprop = $true) & (setukpairinjR2 = $true) & (setminusLsub = $true) & (binunionIL = $true) & (singletonprop = $true)),
% 3.02/1.82 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3])],[f544,f546,f545])).
% 3.02/1.82 thf(f545,plain,(
% 3.02/1.82 ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((kpair @ X2 @ X3) != (kpair @ X2 @ X3)) & ((in @ X3 @ X1) = $true)) & ($true = (in @ X2 @ X0))) => (? [X3 : $i] : (((kpair @ sK2 @ X3) != (kpair @ sK2 @ X3)) & ((in @ X3 @ sK1) = $true)) & ((in @ sK2 @ sK0) = $true))),
% 3.02/1.82 introduced(definition,[],[choice_axiom])).
% 3.02/1.82 thf(f546,plain,(
% 3.02/1.82 ? [X3 : $i] : (((kpair @ sK2 @ X3) != (kpair @ sK2 @ X3)) & ((in @ X3 @ sK1) = $true)) => (((kpair @ sK2 @ sK3) != (kpair @ sK2 @ sK3)) & ((in @ sK3 @ sK1) = $true))),
% 3.02/1.82 introduced(definition,[],[choice_axiom])).
% 3.02/1.82 thf(f544,plain,(
% 3.02/1.82 (emptyset__Cong = $true) & (setadjoinSub2 = $true) & (setunionE2 = $true) & (quantDeMorgan1 = $true) & (ubforcartprodlem1 = $true) & (nonemptyI1 = $true) & (dsetconstrER = $true) & (upairequniteq = $true) & (emptyI = $true) & (setext = $true) & (foundationAx = $true) & (subsetE = $true) & (in__Cong = $true) & (singletonsuniq = $true) & (setadjoinOr = $true) & (setunionsingleton = $true) & (upairsetE = $true) & (eqimpsubset2 = $true) & (cartprodmempair1 = $true) & (upairsubunion = $true) & (setukpairinjR = $true) & (notdallE = $true) & (setadjoinAx = $true) & (binintersectSubset2 = $true) & (setminusEL = $true) & (upairinpowunion = $true) & (inPowerset = $true) & (setukpairinjL = $true) & (exu__Cong = $true) & (eqimpsubset1 = $true) & (descr__Cong = $true) & (symdiffI2 = $true) & (omegaSAx = $true) & (setunionAx = $true) & (omegaIndAx = $true) & (binintersectI = $true) & (exuE3e = $true) & (emptyinPowerset = $true) & (setukpairinjL1 = $true) & (emptysetimpfalse = $true) & (omega0Ax = $true) & (emptysetsubset = $true) & (dsetconstrEL = $true) & (cartprodfstin = $true) & (powersetI = $true) & (wellorderingAx = $true) & (exuEu = $true) & (binintersectSubset1 = $true) & (ksndsingleton = $true) & (emptyE1 = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((kpair @ X2 @ X3) != (kpair @ X2 @ X3)) & ((in @ X3 @ X1) = $true)) & ($true = (in @ X2 @ X0))) & (setadjoinE = $true) & (emptyinunitempty = $true) & (dsetconstr__Cong = $true) & (setadjoinIL = $true) & (setukpairinjR11 = $true) & (setunionE = $true) & (exuI3 = $true) & (notdexE = $true) & (eqinunit = $true) & (exuI2 = $true) & (powersetAx = $true) & (dsetconstrI = $true) & (setunionsingleton2 = $true) & (setukpairIL = $true) & (setextsub = $true) & (setminusI = $true) & (binunionLsub = $true) & (setukpairinjR12 = $true) & (subset2powerset = $true) & (powersetE = $true) & (quantDeMorgan4 = $true) & (quantDeMorgan2 = $true) & (binintersectLsub = $true) & (powersetE1 = $true) & (binintersectSubset4 = $true) & (omega__Cong = $true) & (bs114d = $true) & (prop2set2propI = $true) & (subsetI2 = $true) & (setunionI = $true) & (quantDeMorgan3 = $true) & (upairset2E = $true) & (notequalI2 = $true) & (symdiffE = $true) & (binunionEcases = $true) & (upairsetIL = $true) & (ex1I = $true) & (binintersectRsub = $true) & (kfstpairEq = $true) & (exuI1 = $true) & (kpairp = $true) & (singletoninpowerset = $true) & (descrp = $true) & (setminusELneg = $true) & (setadjoinSub = $true) & (notinemptyset = $true) & (ex1I2 = $true) & (cartprodpairmemER = $true) & (symdiffIneg2 = $true) & (subsetRefl = $true) & (exuE1 = $true) & (binintersectER = $true) & (powersetsubset = $true) & (disjointsetsI1 = $true) & (binintersectSubset5 = $true) & (nonemptyI = $true) & (setukpairinjL2 = $true) & (prop2setI = $true) & (nonemptyImpWitness = $true) & (setminusSubset2 = $true) & (setminusERneg = $true) & (singletonsswitch = $true) & (binunionRsub = $true) & (singletonsubset = $true) & (exuE2 = $true) & (setunionsingleton1 = $true) & (setadjoinIR = $true) & (subsetTrans = $true) & (cartprodpairin = $true) & (setminusSubset1 = $true) & (cartprodpairmemEL = $true) & (binunionE = $true) & (subsetE2 = $true) & (ksndpairEq = $true) & (binunionIR = $true) & (notsubsetI = $true) & (subsetI1 = $true) & (binintersectEL = $true) & (secondinupair = $true) & (setminusIRneg = $true) & (setextAx = $true) & (upairset2IR = $true) & (emptysetE = $true) & (replAx = $true) & (setadjoin__Cong = $true) & (ex1E1 = $true) & (symdiffIneg1 = $true) & (notequalI1 = $true) & (setminusER = $true) & (sepInPowerset = $true) & (setunion__Cong = $true) & (setbeta = $true) & (setukpairIR = $true) & (nonemptyE1 = $true) & (subPowSU = $true) & (kfstsingleton = $true) & (kpairiskpair = $true) & (noeltsimpempty = $true) & (ubforcartprodlem2 = $true) & (uniqinunit = $true) & (prop2setE = $true) & (setukpairinjR1 = $true) & (upairsetIR = $true) & (setminusILneg = $true) & (setoftrueEq = $true) & (notinsingleton = $true) & (sepSubset = $true) & (exuE3u = $true) & (emptyInPowerset = $true) & (powersetI1 = $true) & (inCongP = $true) & (ubforcartprodlem3 = $true) & (kpairsurjEq = $true) & (subsetemptysetimpeq = $true) & (binintersectSubset3 = $true) & (singletoninpowunion = $true) & (powerset__Cong = $true) & (emptysetAx = $true) & (vacuousDall = $true) & (cartprodmempair = $true) & (cartprodsndin = $true) & (symdiffI1 = $true) & (theprop = $true) & (setukpairinjR2 = $true) & (setminusLsub = $true) & (binunionIL = $true) & (singletonprop = $true)),
% 3.02/1.82 inference(flattening,[status(thm)],[f543])).
% 3.02/1.82 thf(f543,plain,(
% 3.02/1.82 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (((kpair @ X2 @ X3) != (kpair @ X2 @ X3)) & ((in @ X3 @ X1) = $true)) & ($true = (in @ X2 @ X0))) & (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)),
% 3.02/1.82 inference(ennf_transformation,[status(thm)],[f290])).
% 3.02/1.82 thf(f290,plain,(
% 3.02/1.82 ~((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) => ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ X0)) => ! [X3 : $i] : (((in @ X3 @ X1) = $true) => ((kpair @ X2 @ X3) = (kpair @ X2 @ X3)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 3.02/1.82 inference(fool_elimination,[status(thm)],[f289])).
% 3.02/1.82 thf(f289,plain,(
% 3.02/1.82 ~(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 => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ X0) => ! [X3 : $i] : ((in @ X3 @ X1) => ((kpair @ X2 @ X3) = (kpair @ X2 @ X3)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 3.02/1.82 inference(rectify,[status(thm)],[f183])).
% 3.02/1.82 thf(f183,negated_conjecture,(
% 3.02/1.82 ~(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 => ! [X3 : $i,X4 : $i,X1 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X4) => ((kpair @ X1 @ X2) = (kpair @ X1 @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 3.02/1.82 inference(negated_conjecture,[status(cth)],[f182])).
% 3.02/1.82 thf(f182,conjecture,(
% 3.02/1.82 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 => ! [X3 : $i,X4 : $i,X1 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X4) => ((kpair @ X1 @ X2) = (kpair @ X1 @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 3.02/1.82 file('/export/starexec/sandbox/tmp/tmp.CrCPL8bmZ3/DTF2THF_9448.p',cartprodmempaircEq)).
% 3.02/1.82 % SZS output end Proof for DTF2THF_9448
% 3.02/1.82 % (9547)------------------------------
% 3.02/1.82 % (9547)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.02/1.82 % (9547)Termination reason: Refutation
% 3.02/1.82
% 3.02/1.82 % (9547)Memory used [KB]: 6780
% 3.02/1.82 % (9547)Time elapsed: 0.046 s
% 3.02/1.82 % (9547)Instructions burned: 92 (million)
% 3.02/1.82 % (9547)------------------------------
% 3.02/1.82 % (9547)------------------------------
% 3.02/1.82 % (9546)Success in time 0.062 s
%------------------------------------------------------------------------------