%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU545^1 : 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 : n026.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 09:35:11 AM UTC 2026
% Result : Theorem 2.13s 1.49s
% Output : Refutation 2.13s
% Verified :
% SZS Type : Refutation
% Derivation depth : 20
% Number of leaves : 4
% Syntax : Number of formulae : 35 ( 13 unt; 0 typ; 0 def)
% Number of atoms : 1136 ( 332 equ; 0 cnn)
% Maximal formula atoms : 66 ( 32 avg)
% Number of connectives : 661 ( 30 ~; 26 |; 201 &; 120 @)
% ( 0 <=>; 273 =>; 0 <=; 0 <~>)
% Maximal formula depth : 71 ( 16 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 12 ( 12 >; 0 *; 0 +; 0 <<)
% Number of symbols : 120 ( 115 usr; 66 con; 0-3 aty)
% ( 7 !!; 4 ??; 0 @@+; 0 @@-)
% Number of variables : 81 ( 29 ^; 38 !; 14 ?; 81 :)
% 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_90,type,
sK0: $i > $o ).
thf(func_def_93,type,
sK3: ( $i > $o ) > $i ).
thf(func_def_94,type,
sK4: ( $i > $o ) > $i > $i ).
thf(func_def_95,type,
sK5: ( $i > $o ) > $i > $i ).
thf(func_def_96,type,
sK6: $i > $i > $i ).
thf(func_def_97,type,
sK7: $i > ( $i > $o ) > $i ).
thf(func_def_98,type,
sK8: $i > ( $i > $o ) > $i ).
thf(func_def_99,type,
sK9: $i > $i > $i ).
thf(func_def_100,type,
sK10: $i > $i ).
thf(func_def_101,type,
sK11: ( $i > $o ) > $i > $i ).
thf(func_def_102,type,
sK12: $i > $i ).
thf(func_def_104,type,
sK13: ( $i > $i > $o ) > $i > $i ).
thf(func_def_105,type,
sK14: ( $i > $i > $o ) > $i > $i ).
thf(func_def_106,type,
sK15: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_107,type,
sK16: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_108,type,
sK17: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_109,type,
sK18: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_110,type,
sK19: $i > $i ).
thf(func_def_111,type,
sK20: $i > ( $i > $o ) > $i ).
thf(func_def_112,type,
sK21: $i > $i ).
thf(func_def_113,type,
sK22: ( $i > $o ) > $i > $i ).
thf(func_def_114,type,
sK23: $i > $i > $i ).
thf(func_def_115,type,
sK24: $i > $i > $i ).
thf(func_def_116,type,
sK25: $i > $i > $i ).
thf(func_def_117,type,
sK26: $i > ( $i > $o ) > $i ).
thf(func_def_118,type,
sK27: ( $i > $o ) > $i ).
thf(func_def_119,type,
sK28: ( $i > $o ) > $i > $i ).
thf(func_def_120,type,
sK29: $i > $i ).
thf(func_def_121,type,
sK30: $i > $i ).
thf(func_def_122,type,
sK31: $o > $i > $i > $i ).
thf(func_def_123,type,
sK32: ( $i > $o ) > $i ).
thf(func_def_124,type,
sK33: ( $i > $o ) > $i > $i ).
thf(func_def_125,type,
sK34: ( $i > $o ) > $i > $i ).
thf(func_def_126,type,
sK35: $i > $i ).
thf(func_def_127,type,
sK36: $i > $i > $i ).
thf(func_def_128,type,
sK37: ( $i > $o ) > $i ).
thf(func_def_129,type,
sK38: ( $i > $o ) > $i > $i ).
thf(func_def_130,type,
sK39: ( $i > $o ) > $i > $i ).
thf(func_def_131,type,
sK40: $i > $i ).
thf(func_def_132,type,
sK41: $i > $i > $i ).
thf(func_def_133,type,
sK42: $i > $i > $i ).
thf(func_def_134,type,
sK43: $i > $i > $i ).
thf(func_def_135,type,
sK44: $i > $i > $i > $i ).
thf(f1208,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1206,f1145]) ).
thf(f1145,plain,
( sK1
!= ( sK21 @ sK1 ) ),
inference(trivial_inequality_removal,[status(thm)],[f1140]) ).
thf(f1140,plain,
( ( sK1
!= ( sK21 @ sK1 ) )
| ( $false = $true ) ),
inference(superposition,[status(thm)],[f663,f260]) ).
thf(f260,plain,
( ( sK0 @ sK1 )
= $true ),
inference(cnf_transformation,[status(thm)],[f195]) ).
thf(f195,plain,
( ( setunionI = $true )
& ( omegaIndAx = $true )
& ( quantDeMorgan4 = $true )
& ( setadjoinE = $true )
& ( notinsingleton = $true )
& ( nonemptyI1 = $true )
& ( setunionE = $true )
& ( vacuousDall = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( upairsetIL = $true )
& ( emptyInPowerset = $true )
& ( setadjoinIL = $true )
& ( eqinunit = $true )
& ( notinemptyset = $true )
& ( wellorderingAx = $true )
& ( powersetI = $true )
& ( noeltsimpempty = $true )
& ( emptyE1 = $true )
& ( omega0Ax = $true )
& ( setadjoinAx = $true )
& ( emptysetimpfalse = $true )
& ( setadjoinOr = $true )
& ( ( exu
@ ^ [Y0: $i] : ( sK0 @ Y0 ) )
!= $true )
& ( ( sK0 @ sK1 )
= $true )
& ! [X2: $i,X3: $i] :
( ( $true
!= ( sK0 @ X3 ) )
| ( ( sK0 @ X2 )
!= $true )
| ( X2 = X3 ) )
& ( nonemptyImpWitness = $true )
& ( dsetconstrI = $true )
& ( prop2set2propI = $true )
& ( setadjoinIR = $true )
& ( dsetconstrEL = $true )
& ( prop2setI = $true )
& ( quantDeMorgan2 = $true )
& ( emptyinunitempty = $true )
& ( foundationAx = $true )
& ( setoftrueEq = $true )
& ( powersetAx = $true )
& ( setext = $true )
& ( nonemptyI = $true )
& ( exuE1 = $true )
& ( upairsetE = $true )
& ( emptyI = $true )
& ( exuE2 = $true )
& ( singletonsswitch = $true )
& ( setunionAx = $true )
& ( upairsetIR = $true )
& ( setextAx = $true )
& ( emptysetAx = $true )
& ( exuI1 = $true )
& ( subPowSU = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan1 = $true )
& ( powersetE = $true )
& ( notdexE = $true )
& ( descrp = $true )
& ( emptyinPowerset = $true )
& ( replAx = $true )
& ( dsetconstrER = $true )
& ( omegaSAx = $true )
& ( uniqinunit = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( notdallE = $true )
& ( exuE3e = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f192,f194,f193]) ).
thf(f193,plain,
( ? [X0: $i > $o] :
( ( $true
!= ( exu
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
& ? [X1: $i] :
( ( X0 @ X1 )
= $true )
& ! [X2: $i,X3: $i] :
( ( ( X0 @ X3 )
!= $true )
| ( ( X0 @ X2 )
!= $true )
| ( X2 = X3 ) ) )
=> ( ( ( exu
@ ^ [Y0: $i] : ( sK0 @ Y0 ) )
!= $true )
& ? [X1: $i] :
( ( sK0 @ X1 )
= $true )
& ! [X3: $i,X2: $i] :
( ( $true
!= ( sK0 @ X3 ) )
| ( ( sK0 @ X2 )
!= $true )
| ( X2 = X3 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f194,plain,
( ? [X1: $i] :
( ( sK0 @ X1 )
= $true )
=> ( ( sK0 @ sK1 )
= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f192,plain,
( ( setunionI = $true )
& ( omegaIndAx = $true )
& ( quantDeMorgan4 = $true )
& ( setadjoinE = $true )
& ( notinsingleton = $true )
& ( nonemptyI1 = $true )
& ( setunionE = $true )
& ( vacuousDall = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( upairsetIL = $true )
& ( emptyInPowerset = $true )
& ( setadjoinIL = $true )
& ( eqinunit = $true )
& ( notinemptyset = $true )
& ( wellorderingAx = $true )
& ( powersetI = $true )
& ( noeltsimpempty = $true )
& ( emptyE1 = $true )
& ( omega0Ax = $true )
& ( setadjoinAx = $true )
& ( emptysetimpfalse = $true )
& ( setadjoinOr = $true )
& ? [X0: $i > $o] :
( ( $true
!= ( exu
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
& ? [X1: $i] :
( ( X0 @ X1 )
= $true )
& ! [X2: $i,X3: $i] :
( ( ( X0 @ X3 )
!= $true )
| ( ( X0 @ X2 )
!= $true )
| ( X2 = X3 ) ) )
& ( nonemptyImpWitness = $true )
& ( dsetconstrI = $true )
& ( prop2set2propI = $true )
& ( setadjoinIR = $true )
& ( dsetconstrEL = $true )
& ( prop2setI = $true )
& ( quantDeMorgan2 = $true )
& ( emptyinunitempty = $true )
& ( foundationAx = $true )
& ( setoftrueEq = $true )
& ( powersetAx = $true )
& ( setext = $true )
& ( nonemptyI = $true )
& ( exuE1 = $true )
& ( upairsetE = $true )
& ( emptyI = $true )
& ( exuE2 = $true )
& ( singletonsswitch = $true )
& ( setunionAx = $true )
& ( upairsetIR = $true )
& ( setextAx = $true )
& ( emptysetAx = $true )
& ( exuI1 = $true )
& ( subPowSU = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan1 = $true )
& ( powersetE = $true )
& ( notdexE = $true )
& ( descrp = $true )
& ( emptyinPowerset = $true )
& ( replAx = $true )
& ( dsetconstrER = $true )
& ( omegaSAx = $true )
& ( uniqinunit = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( notdallE = $true )
& ( exuE3e = $true ) ),
inference(flattening,[status(thm)],[f191]) ).
thf(f191,plain,
( ? [X0: $i > $o] :
( ( $true
!= ( exu
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) )
& ! [X2: $i,X3: $i] :
( ( X2 = X3 )
| ( ( X0 @ X3 )
!= $true )
| ( ( X0 @ X2 )
!= $true ) )
& ? [X1: $i] :
( ( X0 @ X1 )
= $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)],[f86]) ).
thf(f86,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 )
=> ! [X0: $i > $o] :
( ? [X1: $i] :
( ( X0 @ X1 )
= $true )
=> ( ! [X2: $i,X3: $i] :
( ( ( X0 @ X2 )
= $true )
=> ( ( ( X0 @ X3 )
= $true )
=> ( X2 = X3 ) ) )
=> ( $true
= ( exu
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f85]) ).
thf(f85,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
=> ! [X0: $i > $o] :
( ? [X1: $i] : ( X0 @ X1 )
=> ( ! [X2: $i,X3: $i] :
( ( X0 @ X2 )
=> ( ( X0 @ X3 )
=> ( X2 = X3 ) ) )
=> ( exu
@ ^ [X4: $i] : ( X0 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f65]) ).
thf(f65,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
=> ! [X0: $i > $o] :
( ? [X1: $i] : ( X0 @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( X0 @ X1 )
=> ( ( X0 @ X2 )
=> ( X1 = X2 ) ) )
=> ( exu
@ ^ [X1: $i] : ( X0 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f64]) ).
thf(f64,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
=> ! [X0: $i > $o] :
( ? [X1: $i] : ( X0 @ X1 )
=> ( ! [X1: $i,X2: $i] :
( ( X0 @ X1 )
=> ( ( X0 @ X2 )
=> ( X1 = X2 ) ) )
=> ( exu
@ ^ [X1: $i] : ( X0 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.V2DC10xSo5/DTF2THF_32358.p',exuI3) ).
thf(f663,plain,
! [X1: $i] :
( ( $false
= ( sK0 @ X1 ) )
| ( ( sK21 @ X1 )
!= X1 ) ),
inference(equality_proxy_clausification,[status(thm)],[f661]) ).
thf(f661,plain,
! [X1: $i] :
( ( ( X1
= ( sK21 @ X1 ) )
= $false )
| ( $false
= ( sK0 @ X1 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f660]) ).
thf(f660,plain,
! [X1: $i] :
( ( $false
= ( ( sK0 @ ( sK21 @ X1 ) )
=> ( X1
= ( sK21 @ X1 ) ) ) )
| ( $false
= ( sK0 @ X1 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f659]) ).
thf(f659,plain,
! [X1: $i] :
( ( ( ^ [Y0: $i] :
( ( sK0 @ Y0 )
=> ( X1 = Y0 ) )
@ ( sK21 @ X1 ) )
= $false )
| ( $false
= ( sK0 @ X1 ) ) ),
inference(sigma_clausification,[status(thm)],[f658]) ).
thf(f658,plain,
! [X1: $i] :
( ( $false
= ( !! @ $i
@ ^ [Y0: $i] :
( ( sK0 @ Y0 )
=> ( X1 = Y0 ) ) ) )
| ( $false
= ( sK0 @ X1 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f657]) ).
thf(f657,plain,
! [X1: $i] :
( ( ( sK0 @ X1 )
& ( !! @ $i
@ ^ [Y0: $i] :
( ( sK0 @ Y0 )
=> ( X1 = Y0 ) ) ) )
= $false ),
inference(beta_eta_normalization,[status(thm)],[f656]) ).
thf(f656,plain,
! [X1: $i] :
( $false
= ( ^ [Y0: $i] :
( ( sK0 @ Y0 )
& ( !! @ $i
@ ^ [Y1: $i] :
( ( sK0 @ Y1 )
=> ( Y0 = Y1 ) ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f655]) ).
thf(f655,plain,
( $true
!= ( ?? @ $i
@ ^ [Y0: $i] :
( ( sK0 @ Y0 )
& ( !! @ $i
@ ^ [Y1: $i] :
( ( sK0 @ Y1 )
=> ( Y0 = Y1 ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f336]) ).
thf(f336,plain,
( ( ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( !! @ $i
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 = Y2 ) ) ) ) )
@ ^ [Y0: $i] : ( sK0 @ Y0 ) )
!= $true ),
inference(definition_unfolding,[status(thm)],[f261,f312]) ).
thf(f312,plain,
( exu
= ( ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( !! @ $i
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 = Y2 ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f124]) ).
thf(f124,plain,
( exu
= ( ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( !! @ $i
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 = Y2 ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f123]) ).
thf(f123,plain,
( exu
= ( ^ [X0: $i > $o] :
? [X1: $i] :
( ! [X2: $i] :
( ( X0 @ X2 )
=> ( X1 = X2 ) )
& ( X0 @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
( exu
= ( ^ [X0: $i > $o] :
? [X1: $i] :
( ! [X2: $i] :
( ( X0 @ X2 )
=> ( X1 = X2 ) )
& ( X0 @ X1 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.V2DC10xSo5/DTF2THF_32358.p',exu) ).
thf(f261,plain,
( ( exu
@ ^ [Y0: $i] : ( sK0 @ Y0 ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f195]) ).
thf(f1206,plain,
( sK1
= ( sK21 @ sK1 ) ),
inference(trivial_inequality_removal,[status(thm)],[f1196]) ).
thf(f1196,plain,
( ( sK1
= ( sK21 @ sK1 ) )
| ( $false = $true ) ),
inference(superposition,[status(thm)],[f1194,f260]) ).
thf(f1194,plain,
! [X0: $i] :
( ( $false
= ( sK0 @ X0 ) )
| ( ( sK21 @ X0 )
= sK1 ) ),
inference(trivial_inequality_removal,[status(thm)],[f1185]) ).
thf(f1185,plain,
! [X0: $i] :
( ( $false
= ( sK0 @ X0 ) )
| ( ( sK21 @ X0 )
= sK1 )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f1155,f662]) ).
thf(f662,plain,
! [X1: $i] :
( ( ( sK0 @ ( sK21 @ X1 ) )
= $true )
| ( $false
= ( sK0 @ X1 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f660]) ).
thf(f1155,plain,
! [X0: $i] :
( ( ( sK0 @ X0 )
!= $true )
| ( sK1 = X0 ) ),
inference(trivial_inequality_removal,[status(thm)],[f1152]) ).
thf(f1152,plain,
! [X0: $i] :
( ( ( sK0 @ X0 )
!= $true )
| ( $true != $true )
| ( sK1 = X0 ) ),
inference(superposition,[status(thm)],[f259,f260]) ).
thf(f259,plain,
! [X2: $i,X3: $i] :
( ( $true
!= ( sK0 @ X3 ) )
| ( ( sK0 @ X2 )
!= $true )
| ( X2 = X3 ) ),
inference(cnf_transformation,[status(thm)],[f195]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU545^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n026.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 12:26:55 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.20/0.46 ---- Original DTF file ---
% 0.20/0.46 thf(spec,logic,$$dhol).
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 % File : SEU545^1 : TPTP v9.2.1. Released v3.7.0.
% 0.20/0.46 % Domain : Set Theory
% 0.20/0.46 % Problem : Preliminary Notions - Equivalence Laws
% 0.20/0.46 % Version : Especial.
% 0.20/0.46 % English : (! phi:i>o.(? x:i.phi x) -> (! x:i.! y:i.phi x -> phi y -> x = y)
% 0.20/0.46 % -> exu (^ x:i.phi x))
% 0.20/0.46
% 0.20/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.20/0.46 % Source : [Bro08]
% 0.20/0.46 % Names : ZFC047g [Bro08]
% 0.20/0.46
% 0.20/0.46 % Status : Theorem
% 0.20/0.46 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.31 v8.1.0, 0.09 v7.5.0, 0.00 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.67 v3.7.0
% 0.20/0.46 % Syntax : Number of formulae : 137 ( 63 unt; 73 typ; 63 def)
% 0.20/0.46 % Number of atoms : 344 ( 84 equ; 0 cnn)
% 0.20/0.46 % Maximal formula atoms : 63 ( 5 avg)
% 0.20/0.46 % Number of connectives : 579 ( 21 ~; 5 |; 27 &; 365 @)
% 0.20/0.46 % ( 8 <=>; 153 =>; 0 <=; 0 <~>)
% 0.20/0.46 % Maximal formula depth : 70 ( 2 avg)
% 0.20/0.46 % Number of types : 2 ( 0 usr)
% 0.20/0.46 % Number of type conns : 38 ( 38 >; 0 *; 0 +; 0 <<)
% 0.20/0.46 % Number of symbols : 76 ( 73 usr; 65 con; 0-2 aty)
% 0.20/0.46 % Number of variables : 192 ( 17 ^; 148 !; 27 ?; 192 :)
% 0.20/0.46 % SPC : TH0_THM_EQU_NAR
% 0.20/0.46
% 0.20/0.46 % Comments : http://mathgate.info/detsetitem.php?id=410
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 thf(in_type,type,
% 0.20/0.46 in: $i > $i > $o ).
% 0.20/0.46
% 0.20/0.46 thf(exu_type,type,
% 0.20/0.46 exu: ( $i > $o ) > $o ).
% 0.20/0.46
% 0.20/0.46 thf(exu,definition,
% 0.20/0.46 ( exu
% 0.20/0.46 = ( ^ [Xphi: $i > $o] :
% 0.20/0.46 ? [Xx: $i] :
% 0.20/0.46 ( ( Xphi @ Xx )
% 0.20/0.46 & ! [Xy: $i] :
% 0.20/0.46 ( ( Xphi @ Xy )
% 0.20/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setextAx_type,type,
% 0.20/0.46 setextAx: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setextAx,definition,
% 0.20/0.46 ( setextAx
% 0.20/0.46 = ( ! [A: $i,B: $i] :
% 0.20/0.46 ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 <=> ( in @ Xx @ B ) )
% 0.20/0.46 => ( A = B ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(emptyset_type,type,
% 0.20/0.46 emptyset: $i ).
% 0.20/0.46
% 0.20/0.46 thf(emptysetAx_type,type,
% 0.20/0.46 emptysetAx: $o ).
% 0.20/0.46
% 0.20/0.46 thf(emptysetAx,definition,
% 0.20/0.46 ( emptysetAx
% 0.20/0.46 = ( ! [Xx: $i] :
% 0.20/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setadjoin_type,type,
% 0.20/0.46 setadjoin: $i > $i > $i ).
% 0.20/0.46
% 0.20/0.46 thf(setadjoinAx_type,type,
% 0.20/0.46 setadjoinAx: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setadjoinAx,definition,
% 0.20/0.46 ( setadjoinAx
% 0.20/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.20/0.46 <=> ( ( Xy = Xx )
% 0.20/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(powerset_type,type,
% 0.20/0.46 powerset: $i > $i ).
% 0.20/0.46
% 0.20/0.46 thf(powersetAx_type,type,
% 0.20/0.46 powersetAx: $o ).
% 0.20/0.46
% 0.20/0.46 thf(powersetAx,definition,
% 0.20/0.46 ( powersetAx
% 0.20/0.46 = ( ! [A: $i,B: $i] :
% 0.20/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.20/0.46 <=> ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ B )
% 0.20/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setunion_type,type,
% 0.20/0.46 setunion: $i > $i ).
% 0.20/0.46
% 0.20/0.46 thf(setunionAx_type,type,
% 0.20/0.46 setunionAx: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setunionAx,definition,
% 0.20/0.46 ( setunionAx
% 0.20/0.46 = ( ! [A: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.20/0.46 <=> ? [B: $i] :
% 0.20/0.46 ( ( in @ Xx @ B )
% 0.20/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(omega_type,type,
% 0.20/0.46 omega: $i ).
% 0.20/0.46
% 0.20/0.46 thf(omega0Ax_type,type,
% 0.20/0.46 omega0Ax: $o ).
% 0.20/0.46
% 0.20/0.46 thf(omega0Ax,definition,
% 0.20/0.46 ( omega0Ax
% 0.20/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(omegaSAx_type,type,
% 0.20/0.46 omegaSAx: $o ).
% 0.20/0.46
% 0.20/0.46 thf(omegaSAx,definition,
% 0.20/0.46 ( omegaSAx
% 0.20/0.46 = ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ omega )
% 0.20/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(omegaIndAx_type,type,
% 0.20/0.46 omegaIndAx: $o ).
% 0.20/0.46
% 0.20/0.46 thf(omegaIndAx,definition,
% 0.20/0.46 ( omegaIndAx
% 0.20/0.46 = ( ! [A: $i] :
% 0.20/0.46 ( ( ( in @ emptyset @ A )
% 0.20/0.46 & ! [Xx: $i] :
% 0.20/0.46 ( ( ( in @ Xx @ omega )
% 0.20/0.46 & ( in @ Xx @ A ) )
% 0.20/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ omega )
% 0.20/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(replAx_type,type,
% 0.20/0.46 replAx: $o ).
% 0.20/0.46
% 0.20/0.46 thf(replAx,definition,
% 0.20/0.46 ( replAx
% 0.20/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.20/0.46 ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( exu
% 0.20/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.20/0.46 => ? [B: $i] :
% 0.20/0.46 ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ B )
% 0.20/0.46 <=> ? [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ A )
% 0.20/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(foundationAx_type,type,
% 0.20/0.46 foundationAx: $o ).
% 0.20/0.46
% 0.20/0.46 thf(foundationAx,definition,
% 0.20/0.46 ( foundationAx
% 0.20/0.46 = ( ! [A: $i] :
% 0.20/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.20/0.46 => ? [B: $i] :
% 0.20/0.46 ( ( in @ B @ A )
% 0.20/0.46 & ~ ? [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ B )
% 0.20/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(wellorderingAx_type,type,
% 0.20/0.46 wellorderingAx: $o ).
% 0.20/0.46
% 0.20/0.46 thf(wellorderingAx,definition,
% 0.20/0.46 ( wellorderingAx
% 0.20/0.46 = ( ! [A: $i] :
% 0.20/0.46 ? [B: $i] :
% 0.20/0.46 ( ! [C: $i] :
% 0.20/0.46 ( ( in @ C @ B )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ C )
% 0.20/0.46 => ( in @ Xx @ A ) ) )
% 0.20/0.46 & ! [Xx: $i,Xy: $i] :
% 0.20/0.46 ( ( ( in @ Xx @ A )
% 0.20/0.46 & ( in @ Xy @ A ) )
% 0.20/0.46 => ( ! [C: $i] :
% 0.20/0.46 ( ( in @ C @ B )
% 0.20/0.46 => ( ( in @ Xx @ C )
% 0.20/0.46 <=> ( in @ Xy @ C ) ) )
% 0.20/0.46 => ( Xx = Xy ) ) )
% 0.20/0.46 & ! [C: $i,D: $i] :
% 0.20/0.46 ( ( ( in @ C @ B )
% 0.20/0.46 & ( in @ D @ B ) )
% 0.20/0.46 => ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ C )
% 0.20/0.46 => ( in @ Xx @ D ) )
% 0.20/0.46 | ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ D )
% 0.20/0.46 => ( in @ Xx @ C ) ) ) )
% 0.20/0.46 & ! [C: $i] :
% 0.20/0.46 ( ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ C )
% 0.20/0.46 => ( in @ Xx @ A ) )
% 0.20/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.20/0.46 => ? [D: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ D @ B )
% 0.20/0.46 & ( in @ Xx @ C )
% 0.20/0.46 & ~ ? [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ D )
% 0.20/0.46 & ( in @ Xy @ C ) )
% 0.20/0.46 & ! [E: $i] :
% 0.20/0.46 ( ( in @ E @ B )
% 0.20/0.46 => ( ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ E )
% 0.20/0.46 => ( in @ Xy @ D ) )
% 0.20/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(descr_type,type,
% 0.20/0.46 descr: ( $i > $o ) > $i ).
% 0.20/0.46
% 0.20/0.46 thf(descrp_type,type,
% 0.20/0.46 descrp: $o ).
% 0.20/0.46
% 0.20/0.46 thf(descrp,definition,
% 0.20/0.46 ( descrp
% 0.20/0.46 = ( ! [Xphi: $i > $o] :
% 0.20/0.46 ( ( exu
% 0.20/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.46 => ( Xphi
% 0.20/0.46 @ ( descr
% 0.20/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(dsetconstr_type,type,
% 0.20/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.20/0.46
% 0.20/0.46 thf(dsetconstrI_type,type,
% 0.20/0.46 dsetconstrI: $o ).
% 0.20/0.46
% 0.20/0.46 thf(dsetconstrI,definition,
% 0.20/0.46 ( dsetconstrI
% 0.20/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( Xphi @ Xx )
% 0.20/0.46 => ( in @ Xx
% 0.20/0.46 @ ( dsetconstr @ A
% 0.20/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(dsetconstrEL_type,type,
% 0.20/0.46 dsetconstrEL: $o ).
% 0.20/0.46
% 0.20/0.46 thf(dsetconstrEL,definition,
% 0.20/0.46 ( dsetconstrEL
% 0.20/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx
% 0.20/0.46 @ ( dsetconstr @ A
% 0.20/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.20/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(dsetconstrER_type,type,
% 0.20/0.46 dsetconstrER: $o ).
% 0.20/0.46
% 0.20/0.46 thf(dsetconstrER,definition,
% 0.20/0.46 ( dsetconstrER
% 0.20/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx
% 0.20/0.46 @ ( dsetconstr @ A
% 0.20/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.20/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(exuE1_type,type,
% 0.20/0.46 exuE1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(exuE1,definition,
% 0.20/0.46 ( exuE1
% 0.20/0.46 = ( ! [Xphi: $i > $o] :
% 0.20/0.46 ( ( exu
% 0.20/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.46 => ? [Xx: $i] :
% 0.20/0.46 ( ( Xphi @ Xx )
% 0.20/0.46 & ! [Xy: $i] :
% 0.20/0.46 ( ( Xphi @ Xy )
% 0.20/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(prop2set_type,type,
% 0.20/0.46 prop2set: $o > $i ).
% 0.20/0.46
% 0.20/0.46 thf(prop2setE_type,type,
% 0.20/0.46 prop2setE: $o ).
% 0.20/0.46
% 0.20/0.46 thf(prop2setE,definition,
% 0.20/0.46 ( prop2setE
% 0.20/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.20/0.46 => Xphi ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(emptysetE_type,type,
% 0.20/0.46 emptysetE: $o ).
% 0.20/0.46
% 0.20/0.46 thf(emptysetE,definition,
% 0.20/0.46 ( emptysetE
% 0.20/0.46 = ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ emptyset )
% 0.20/0.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(emptysetimpfalse_type,type,
% 0.20/0.46 emptysetimpfalse: $o ).
% 0.20/0.46
% 0.20/0.46 thf(emptysetimpfalse,definition,
% 0.20/0.46 ( emptysetimpfalse
% 0.20/0.46 = ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ emptyset )
% 0.20/0.46 => $false ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(notinemptyset_type,type,
% 0.20/0.46 notinemptyset: $o ).
% 0.20/0.46
% 0.20/0.46 thf(notinemptyset,definition,
% 0.20/0.46 ( notinemptyset
% 0.20/0.46 = ( ! [Xx: $i] :
% 0.20/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(exuE3e_type,type,
% 0.20/0.46 exuE3e: $o ).
% 0.20/0.46
% 0.20/0.46 thf(exuE3e,definition,
% 0.20/0.46 ( exuE3e
% 0.20/0.46 = ( ! [Xphi: $i > $o] :
% 0.20/0.46 ( ( exu
% 0.20/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setext_type,type,
% 0.20/0.46 setext: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setext,definition,
% 0.20/0.46 ( setext
% 0.20/0.46 = ( ! [A: $i,B: $i] :
% 0.20/0.46 ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( in @ Xx @ B ) )
% 0.20/0.46 => ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ B )
% 0.20/0.46 => ( in @ Xx @ A ) )
% 0.20/0.46 => ( A = B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(emptyI_type,type,
% 0.20/0.46 emptyI: $o ).
% 0.20/0.46
% 0.20/0.46 thf(emptyI,definition,
% 0.20/0.46 ( emptyI
% 0.20/0.46 = ( ! [A: $i] :
% 0.20/0.46 ( ! [Xx: $i] :
% 0.20/0.46 ~ ( in @ Xx @ A )
% 0.20/0.46 => ( A = emptyset ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(noeltsimpempty_type,type,
% 0.20/0.46 noeltsimpempty: $o ).
% 0.20/0.46
% 0.20/0.46 thf(noeltsimpempty,definition,
% 0.20/0.46 ( noeltsimpempty
% 0.20/0.46 = ( ! [A: $i] :
% 0.20/0.46 ( ! [Xx: $i] :
% 0.20/0.46 ~ ( in @ Xx @ A )
% 0.20/0.46 => ( A = emptyset ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setbeta_type,type,
% 0.20/0.46 setbeta: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setbeta,definition,
% 0.20/0.46 ( setbeta
% 0.20/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx
% 0.20/0.46 @ ( dsetconstr @ A
% 0.20/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.20/0.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(nonempty_type,type,
% 0.20/0.46 nonempty: $i > $o ).
% 0.20/0.46
% 0.20/0.46 thf(nonempty,definition,
% 0.20/0.46 ( nonempty
% 0.20/0.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(nonemptyE1_type,type,
% 0.20/0.46 nonemptyE1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(nonemptyE1,definition,
% 0.20/0.46 ( nonemptyE1
% 0.20/0.46 = ( ! [A: $i] :
% 0.20/0.46 ( ( nonempty @ A )
% 0.20/0.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(nonemptyI_type,type,
% 0.20/0.46 nonemptyI: $o ).
% 0.20/0.46
% 0.20/0.46 thf(nonemptyI,definition,
% 0.20/0.46 ( nonemptyI
% 0.20/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( Xphi @ Xx )
% 0.20/0.46 => ( nonempty
% 0.20/0.46 @ ( dsetconstr @ A
% 0.20/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(nonemptyI1_type,type,
% 0.20/0.46 nonemptyI1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(nonemptyI1,definition,
% 0.20/0.46 ( nonemptyI1
% 0.20/0.46 = ( ! [A: $i] :
% 0.20/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.20/0.46 => ( nonempty @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setadjoinIL_type,type,
% 0.20/0.46 setadjoinIL: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setadjoinIL,definition,
% 0.20/0.46 ( setadjoinIL
% 0.20/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(emptyinunitempty_type,type,
% 0.20/0.46 emptyinunitempty: $o ).
% 0.20/0.46
% 0.20/0.46 thf(emptyinunitempty,definition,
% 0.20/0.46 ( emptyinunitempty
% 0.20/0.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setadjoinIR_type,type,
% 0.20/0.47 setadjoinIR: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setadjoinIR,definition,
% 0.20/0.47 ( setadjoinIR
% 0.20/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.20/0.47 ( ( in @ Xy @ A )
% 0.20/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setadjoinE_type,type,
% 0.20/0.47 setadjoinE: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setadjoinE,definition,
% 0.20/0.47 ( setadjoinE
% 0.20/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.20/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.20/0.47 => ! [Xphi: $o] :
% 0.20/0.47 ( ( ( Xy = Xx )
% 0.20/0.47 => Xphi )
% 0.20/0.47 => ( ( ( in @ Xy @ A )
% 0.20/0.47 => Xphi )
% 0.20/0.47 => Xphi ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setadjoinOr_type,type,
% 0.20/0.47 setadjoinOr: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setadjoinOr,definition,
% 0.20/0.47 ( setadjoinOr
% 0.20/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.20/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.20/0.47 => ( ( Xy = Xx )
% 0.20/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setoftrueEq_type,type,
% 0.20/0.47 setoftrueEq: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setoftrueEq,definition,
% 0.20/0.47 ( setoftrueEq
% 0.20/0.47 = ( ! [A: $i] :
% 0.20/0.47 ( ( dsetconstr @ A
% 0.20/0.47 @ ^ [Xx: $i] : $true )
% 0.20/0.47 = A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(powersetI_type,type,
% 0.20/0.47 powersetI: $o ).
% 0.20/0.47
% 0.20/0.47 thf(powersetI,definition,
% 0.20/0.47 ( powersetI
% 0.20/0.47 = ( ! [A: $i,B: $i] :
% 0.20/0.47 ( ! [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ B )
% 0.20/0.47 => ( in @ Xx @ A ) )
% 0.20/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(emptyinPowerset_type,type,
% 0.20/0.47 emptyinPowerset: $o ).
% 0.20/0.47
% 0.20/0.47 thf(emptyinPowerset,definition,
% 0.20/0.47 ( emptyinPowerset
% 0.20/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(emptyInPowerset_type,type,
% 0.20/0.47 emptyInPowerset: $o ).
% 0.20/0.47
% 0.20/0.47 thf(emptyInPowerset,definition,
% 0.20/0.47 ( emptyInPowerset
% 0.20/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(powersetE_type,type,
% 0.20/0.47 powersetE: $o ).
% 0.20/0.47
% 0.20/0.47 thf(powersetE,definition,
% 0.20/0.47 ( powersetE
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.20/0.47 => ( ( in @ Xx @ B )
% 0.20/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setunionI_type,type,
% 0.20/0.47 setunionI: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setunionI,definition,
% 0.20/0.47 ( setunionI
% 0.20/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.20/0.47 ( ( in @ Xx @ B )
% 0.20/0.47 => ( ( in @ B @ A )
% 0.20/0.47 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setunionE_type,type,
% 0.20/0.47 setunionE: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setunionE,definition,
% 0.20/0.47 ( setunionE
% 0.20/0.47 = ( ! [A: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.20/0.47 => ! [Xphi: $o] :
% 0.20/0.47 ( ! [B: $i] :
% 0.20/0.47 ( ( in @ Xx @ B )
% 0.20/0.47 => ( ( in @ B @ A )
% 0.20/0.47 => Xphi ) )
% 0.20/0.47 => Xphi ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(subPowSU_type,type,
% 0.20/0.47 subPowSU: $o ).
% 0.20/0.47
% 0.20/0.47 thf(subPowSU,definition,
% 0.20/0.47 ( subPowSU
% 0.20/0.47 = ( ! [A: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(exuE2_type,type,
% 0.20/0.47 exuE2: $o ).
% 0.20/0.47
% 0.20/0.47 thf(exuE2,definition,
% 0.20/0.47 ( exuE2
% 0.20/0.47 = ( ! [Xphi: $i > $o] :
% 0.20/0.47 ( ( exu
% 0.20/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.47 => ? [Xx: $i] :
% 0.20/0.47 ! [Xy: $i] :
% 0.20/0.47 ( ( Xphi @ Xy )
% 0.20/0.47 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(nonemptyImpWitness_type,type,
% 0.20/0.47 nonemptyImpWitness: $o ).
% 0.20/0.47
% 0.20/0.47 thf(nonemptyImpWitness,definition,
% 0.20/0.47 ( nonemptyImpWitness
% 0.20/0.47 = ( ! [A: $i] :
% 0.20/0.47 ( ( nonempty @ A )
% 0.20/0.47 => ? [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 & $true ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(uniqinunit_type,type,
% 0.20/0.47 uniqinunit: $o ).
% 0.20/0.47
% 0.20/0.47 thf(uniqinunit,definition,
% 0.20/0.47 ( uniqinunit
% 0.20/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.20/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.20/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(notinsingleton_type,type,
% 0.20/0.47 notinsingleton: $o ).
% 0.20/0.47
% 0.20/0.47 thf(notinsingleton,definition,
% 0.20/0.47 ( notinsingleton
% 0.20/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.20/0.47 ( ( Xx != Xy )
% 0.20/0.47 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(eqinunit_type,type,
% 0.20/0.47 eqinunit: $o ).
% 0.20/0.47
% 0.20/0.47 thf(eqinunit,definition,
% 0.20/0.47 ( eqinunit
% 0.20/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.20/0.47 ( ( Xx = Xy )
% 0.20/0.47 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(singletonsswitch_type,type,
% 0.20/0.47 singletonsswitch: $o ).
% 0.20/0.47
% 0.20/0.47 thf(singletonsswitch,definition,
% 0.20/0.47 ( singletonsswitch
% 0.20/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.20/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.20/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(upairsetE_type,type,
% 0.20/0.47 upairsetE: $o ).
% 0.20/0.47
% 0.20/0.47 thf(upairsetE,definition,
% 0.20/0.47 ( upairsetE
% 0.20/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.20/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.20/0.47 => ( ( Xz = Xx )
% 0.20/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(upairsetIL_type,type,
% 0.20/0.47 upairsetIL: $o ).
% 0.20/0.47
% 0.20/0.47 thf(upairsetIL,definition,
% 0.20/0.47 ( upairsetIL
% 0.20/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(upairsetIR_type,type,
% 0.20/0.47 upairsetIR: $o ).
% 0.20/0.47
% 0.20/0.47 thf(upairsetIR,definition,
% 0.20/0.47 ( upairsetIR
% 0.20/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(emptyE1_type,type,
% 0.20/0.47 emptyE1: $o ).
% 0.20/0.47
% 0.20/0.47 thf(emptyE1,definition,
% 0.20/0.47 ( emptyE1
% 0.20/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.47 ( ? [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 & ( Xphi @ Xx ) )
% 0.20/0.47 => ( ( ( dsetconstr @ A
% 0.20/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.47 = emptyset )
% 0.20/0.47 => $false ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(vacuousDall_type,type,
% 0.20/0.47 vacuousDall: $o ).
% 0.20/0.47
% 0.20/0.47 thf(vacuousDall,definition,
% 0.20/0.47 ( vacuousDall
% 0.20/0.47 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ emptyset )
% 0.20/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(quantDeMorgan1_type,type,
% 0.20/0.47 quantDeMorgan1: $o ).
% 0.20/0.47
% 0.20/0.47 thf(quantDeMorgan1,definition,
% 0.20/0.47 ( quantDeMorgan1
% 0.20/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.47 ( ~ ! [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ( Xphi @ Xx ) )
% 0.20/0.47 => ? [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(quantDeMorgan2_type,type,
% 0.20/0.47 quantDeMorgan2: $o ).
% 0.20/0.47
% 0.20/0.47 thf(quantDeMorgan2,definition,
% 0.20/0.47 ( quantDeMorgan2
% 0.20/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.47 ( ! [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ~ ( Xphi @ Xx ) )
% 0.20/0.47 => ~ ? [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(quantDeMorgan3_type,type,
% 0.20/0.47 quantDeMorgan3: $o ).
% 0.20/0.47
% 0.20/0.47 thf(quantDeMorgan3,definition,
% 0.20/0.47 ( quantDeMorgan3
% 0.20/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.47 ( ~ ? [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 & ( Xphi @ Xx ) )
% 0.20/0.47 => ! [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(quantDeMorgan4_type,type,
% 0.20/0.47 quantDeMorgan4: $o ).
% 0.20/0.47
% 0.20/0.47 thf(quantDeMorgan4,definition,
% 0.20/0.47 ( quantDeMorgan4
% 0.20/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.47 ( ? [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 & ~ ( Xphi @ Xx ) )
% 0.20/0.47 => ~ ! [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(prop2setI_type,type,
% 0.20/0.47 prop2setI: $o ).
% 0.20/0.47
% 0.20/0.47 thf(prop2setI,definition,
% 0.20/0.47 ( prop2setI
% 0.20/0.47 = ( ! [Xphi: $o] :
% 0.20/0.47 ( Xphi
% 0.20/0.47 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(set2prop_type,type,
% 0.20/0.47 set2prop: $i > $o ).
% 0.20/0.47
% 0.20/0.47 thf(prop2set2propI_type,type,
% 0.20/0.47 prop2set2propI: $o ).
% 0.20/0.47
% 0.20/0.47 thf(prop2set2propI,definition,
% 0.20/0.47 ( prop2set2propI
% 0.20/0.47 = ( ! [Xphi: $o] :
% 0.20/0.47 ( Xphi
% 0.20/0.47 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(notdexE_type,type,
% 0.20/0.47 notdexE: $o ).
% 0.20/0.47
% 0.20/0.47 thf(notdexE,definition,
% 0.20/0.47 ( notdexE
% 0.20/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.47 ( ~ ? [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 & ( Xphi @ Xx ) )
% 0.20/0.47 => ! [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(notdallE_type,type,
% 0.20/0.47 notdallE: $o ).
% 0.20/0.47
% 0.20/0.47 thf(notdallE,definition,
% 0.20/0.47 ( notdallE
% 0.20/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.47 ( ~ ! [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ( Xphi @ Xx ) )
% 0.20/0.47 => ? [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(exuI1_type,type,
% 0.20/0.47 exuI1: $o ).
% 0.20/0.47
% 0.20/0.47 thf(exuI1,definition,
% 0.20/0.47 ( exuI1
% 0.20/0.47 = ( ! [Xphi: $i > $o] :
% 0.20/0.47 ( ? [Xx: $i] :
% 0.20/0.47 ( ( Xphi @ Xx )
% 0.20/0.47 & ! [Xy: $i] :
% 0.20/0.47 ( ( Xphi @ Xy )
% 0.20/0.47 => ( Xx = Xy ) ) )
% 0.20/0.47 => ( exu
% 0.20/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(exuI3,conjecture,
% 0.20/0.47 ( setextAx
% 0.20/0.47 => ( emptysetAx
% 0.20/0.47 => ( setadjoinAx
% 0.20/0.47 => ( powersetAx
% 0.20/0.47 => ( setunionAx
% 0.20/0.47 => ( omega0Ax
% 0.20/0.47 => ( omegaSAx
% 0.20/0.47 => ( omegaIndAx
% 0.20/0.47 => ( replAx
% 0.20/0.47 => ( foundationAx
% 0.20/0.47 => ( wellorderingAx
% 0.20/0.47 => ( descrp
% 0.20/0.47 => ( dsetconstrI
% 0.20/0.47 => ( dsetconstrEL
% 0.20/0.47 => ( dsetconstrER
% 0.20/0.47 => ( exuE1
% 0.20/0.47 => ( prop2setE
% 0.20/0.47 => ( emptysetE
% 0.20/0.47 => ( emptysetimpfalse
% 0.20/0.47 => ( notinemptyset
% 0.20/0.47 => ( exuE3e
% 0.20/0.47 => ( setext
% 0.20/0.47 => ( emptyI
% 0.20/0.47 => ( noeltsimpempty
% 0.20/0.47 => ( setbeta
% 0.20/0.47 => ( nonemptyE1
% 0.20/0.47 => ( nonemptyI
% 0.20/0.47 => ( nonemptyI1
% 0.20/0.47 => ( setadjoinIL
% 0.20/0.47 => ( emptyinunitempty
% 0.20/0.47 => ( setadjoinIR
% 0.20/0.47 => ( setadjoinE
% 0.20/0.47 => ( setadjoinOr
% 0.20/0.47 => ( setoftrueEq
% 0.20/0.47 => ( powersetI
% 0.20/0.47 => ( emptyinPowerset
% 0.20/0.47 => ( emptyInPowerset
% 0.20/0.47 => ( powersetE
% 0.20/0.47 => ( setunionI
% 0.20/0.47 => ( setunionE
% 0.20/0.47 => ( subPowSU
% 0.20/0.47 => ( exuE2
% 0.20/0.47 => ( nonemptyImpWitness
% 0.20/0.47 => ( uniqinunit
% 0.20/0.47 => ( notinsingleton
% 0.20/0.47 => ( eqinunit
% 0.20/0.47 => ( singletonsswitch
% 0.20/0.47 => ( upairsetE
% 0.20/0.47 => ( upairsetIL
% 0.20/0.47 => ( upairsetIR
% 0.20/0.47 => ( emptyE1
% 0.20/0.47 => ( vacuousDall
% 0.20/0.47 => ( quantDeMorgan1
% 0.20/0.47 => ( quantDeMorgan2
% 0.20/0.47 => ( quantDeMorgan3
% 0.20/0.47 => ( quantDeMorgan4
% 0.20/0.47 => ( prop2setI
% 0.20/0.47 => ( prop2set2propI
% 0.20/0.47 => ( notdexE
% 0.20/0.47 => ( notdallE
% 0.20/0.47 => ( exuI1
% 0.20/0.47 => ! [Xphi: $i > $o] :
% 0.20/0.47 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.20/0.47 => ( ! [Xx: $i,Xy: $i] :
% 0.20/0.47 ( ( Xphi @ Xx )
% 0.20/0.47 => ( ( Xphi @ Xy )
% 0.20/0.47 => ( Xx = Xy ) ) )
% 0.20/0.47 => ( exu
% 0.20/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 %------------------------------------------------------------------------------
% 0.20/0.47 ------------------------
% 2.13/1.42 ---- Embedded in THF ---
% 2.13/1.42 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.13/1.42 %%% Generated on Tue Feb 24 12:26:56 EST 2026
% 2.13/1.42 %%% using '$$dhol' embedding, version 1.3.0.
% 2.13/1.42 %%% Logic specification used:
% 2.13/1.42 %%% thf(spec, logic, $$dhol).
% 2.13/1.42
% 2.13/1.42 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.V2DC10xSo5/DTF2DTF32358.p
% See solution above
% 2.13/1.42 ------------------------
% 2.13/1.44 ---- Cleaned THF ---
% 2.13/1.44 thf(in_type,type,
% 2.13/1.44 in: $i > $i > $o ).
% 2.13/1.44
% 2.13/1.44 thf(exu_type,type,
% 2.13/1.44 exu: ( $i > $o ) > $o ).
% 2.13/1.44
% 2.13/1.44 thf(exu,definition,
% 2.13/1.44 ( exu
% 2.13/1.44 = ( ^ [Xphi: $i > $o] :
% 2.13/1.44 ? [Xx: $i] :
% 2.13/1.44 ( ( Xphi @ Xx )
% 2.13/1.44 & ! [Xy: $i] :
% 2.13/1.44 ( ( Xphi @ Xy )
% 2.13/1.44 => ( Xx = Xy ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(setextAx_type,type,
% 2.13/1.44 setextAx: $o ).
% 2.13/1.44
% 2.13/1.44 thf(setextAx,definition,
% 2.13/1.44 ( setextAx
% 2.13/1.44 = ( ! [A: $i,B: $i] :
% 2.13/1.44 ( ! [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ A )
% 2.13/1.44 <=> ( in @ Xx @ B ) )
% 2.13/1.44 => ( A = B ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(emptyset_type,type,
% 2.13/1.44 emptyset: $i ).
% 2.13/1.44
% 2.13/1.44 thf(emptysetAx_type,type,
% 2.13/1.44 emptysetAx: $o ).
% 2.13/1.44
% 2.13/1.44 thf(emptysetAx,definition,
% 2.13/1.44 ( emptysetAx
% 2.13/1.44 = ( ! [Xx: $i] :
% 2.13/1.44 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(setadjoin_type,type,
% 2.13/1.44 setadjoin: $i > $i > $i ).
% 2.13/1.44
% 2.13/1.44 thf(setadjoinAx_type,type,
% 2.13/1.44 setadjoinAx: $o ).
% 2.13/1.44
% 2.13/1.44 thf(setadjoinAx,definition,
% 2.13/1.44 ( setadjoinAx
% 2.13/1.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.13/1.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.13/1.44 <=> ( ( Xy = Xx )
% 2.13/1.44 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(powerset_type,type,
% 2.13/1.44 powerset: $i > $i ).
% 2.13/1.44
% 2.13/1.44 thf(powersetAx_type,type,
% 2.13/1.44 powersetAx: $o ).
% 2.13/1.44
% 2.13/1.44 thf(powersetAx,definition,
% 2.13/1.44 ( powersetAx
% 2.13/1.44 = ( ! [A: $i,B: $i] :
% 2.13/1.44 ( ( in @ B @ ( powerset @ A ) )
% 2.13/1.44 <=> ! [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ B )
% 2.13/1.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(setunion_type,type,
% 2.13/1.44 setunion: $i > $i ).
% 2.13/1.44
% 2.13/1.44 thf(setunionAx_type,type,
% 2.13/1.44 setunionAx: $o ).
% 2.13/1.44
% 2.13/1.44 thf(setunionAx,definition,
% 2.13/1.44 ( setunionAx
% 2.13/1.44 = ( ! [A: $i,Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ ( setunion @ A ) )
% 2.13/1.44 <=> ? [B: $i] :
% 2.13/1.44 ( ( in @ Xx @ B )
% 2.13/1.44 & ( in @ B @ A ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(omega_type,type,
% 2.13/1.44 omega: $i ).
% 2.13/1.44
% 2.13/1.44 thf(omega0Ax_type,type,
% 2.13/1.44 omega0Ax: $o ).
% 2.13/1.44
% 2.13/1.44 thf(omega0Ax,definition,
% 2.13/1.44 ( omega0Ax
% 2.13/1.44 = ( in @ emptyset @ omega ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(omegaSAx_type,type,
% 2.13/1.44 omegaSAx: $o ).
% 2.13/1.44
% 2.13/1.44 thf(omegaSAx,definition,
% 2.13/1.44 ( omegaSAx
% 2.13/1.44 = ( ! [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ omega )
% 2.13/1.44 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(omegaIndAx_type,type,
% 2.13/1.44 omegaIndAx: $o ).
% 2.13/1.44
% 2.13/1.44 thf(omegaIndAx,definition,
% 2.13/1.44 ( omegaIndAx
% 2.13/1.44 = ( ! [A: $i] :
% 2.13/1.44 ( ( ( in @ emptyset @ A )
% 2.13/1.44 & ! [Xx: $i] :
% 2.13/1.44 ( ( ( in @ Xx @ omega )
% 2.13/1.44 & ( in @ Xx @ A ) )
% 2.13/1.44 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.13/1.44 => ! [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ omega )
% 2.13/1.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(replAx_type,type,
% 2.13/1.44 replAx: $o ).
% 2.13/1.44
% 2.13/1.44 thf(replAx,definition,
% 2.13/1.44 ( replAx
% 2.13/1.44 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.13/1.44 ( ! [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ A )
% 2.13/1.44 => ( exu
% 2.13/1.44 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.13/1.44 => ? [B: $i] :
% 2.13/1.44 ! [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ B )
% 2.13/1.44 <=> ? [Xy: $i] :
% 2.13/1.44 ( ( in @ Xy @ A )
% 2.13/1.44 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(foundationAx_type,type,
% 2.13/1.44 foundationAx: $o ).
% 2.13/1.44
% 2.13/1.44 thf(foundationAx,definition,
% 2.13/1.44 ( foundationAx
% 2.13/1.44 = ( ! [A: $i] :
% 2.13/1.44 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.13/1.44 => ? [B: $i] :
% 2.13/1.44 ( ( in @ B @ A )
% 2.13/1.44 & ~ ? [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ B )
% 2.13/1.44 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(wellorderingAx_type,type,
% 2.13/1.44 wellorderingAx: $o ).
% 2.13/1.44
% 2.13/1.44 thf(wellorderingAx,definition,
% 2.13/1.44 ( wellorderingAx
% 2.13/1.44 = ( ! [A: $i] :
% 2.13/1.44 ? [B: $i] :
% 2.13/1.44 ( ! [C: $i] :
% 2.13/1.44 ( ( in @ C @ B )
% 2.13/1.44 => ! [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ C )
% 2.13/1.44 => ( in @ Xx @ A ) ) )
% 2.13/1.44 & ! [Xx: $i,Xy: $i] :
% 2.13/1.44 ( ( ( in @ Xx @ A )
% 2.13/1.44 & ( in @ Xy @ A ) )
% 2.13/1.44 => ( ! [C: $i] :
% 2.13/1.44 ( ( in @ C @ B )
% 2.13/1.44 => ( ( in @ Xx @ C )
% 2.13/1.44 <=> ( in @ Xy @ C ) ) )
% 2.13/1.44 => ( Xx = Xy ) ) )
% 2.13/1.44 & ! [C: $i,D: $i] :
% 2.13/1.44 ( ( ( in @ C @ B )
% 2.13/1.44 & ( in @ D @ B ) )
% 2.13/1.44 => ( ! [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ C )
% 2.13/1.44 => ( in @ Xx @ D ) )
% 2.13/1.44 | ! [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ D )
% 2.13/1.44 => ( in @ Xx @ C ) ) ) )
% 2.13/1.44 & ! [C: $i] :
% 2.13/1.44 ( ( ! [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ C )
% 2.13/1.44 => ( in @ Xx @ A ) )
% 2.13/1.44 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.13/1.44 => ? [D: $i,Xx: $i] :
% 2.13/1.44 ( ( in @ D @ B )
% 2.13/1.44 & ( in @ Xx @ C )
% 2.13/1.44 & ~ ? [Xy: $i] :
% 2.13/1.44 ( ( in @ Xy @ D )
% 2.13/1.44 & ( in @ Xy @ C ) )
% 2.13/1.44 & ! [E: $i] :
% 2.13/1.44 ( ( in @ E @ B )
% 2.13/1.44 => ( ! [Xy: $i] :
% 2.13/1.44 ( ( in @ Xy @ E )
% 2.13/1.44 => ( in @ Xy @ D ) )
% 2.13/1.44 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(descr_type,type,
% 2.13/1.44 descr: ( $i > $o ) > $i ).
% 2.13/1.44
% 2.13/1.44 thf(descrp_type,type,
% 2.13/1.44 descrp: $o ).
% 2.13/1.44
% 2.13/1.44 thf(descrp,definition,
% 2.13/1.44 ( descrp
% 2.13/1.44 = ( ! [Xphi: $i > $o] :
% 2.13/1.44 ( ( exu
% 2.13/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.13/1.44 => ( Xphi
% 2.13/1.44 @ ( descr
% 2.13/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(dsetconstr_type,type,
% 2.13/1.44 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.13/1.44
% 2.13/1.44 thf(dsetconstrI_type,type,
% 2.13/1.44 dsetconstrI: $o ).
% 2.13/1.44
% 2.13/1.44 thf(dsetconstrI,definition,
% 2.13/1.44 ( dsetconstrI
% 2.13/1.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ A )
% 2.13/1.44 => ( ( Xphi @ Xx )
% 2.13/1.44 => ( in @ Xx
% 2.13/1.44 @ ( dsetconstr @ A
% 2.13/1.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(dsetconstrEL_type,type,
% 2.13/1.44 dsetconstrEL: $o ).
% 2.13/1.44
% 2.13/1.44 thf(dsetconstrEL,definition,
% 2.13/1.44 ( dsetconstrEL
% 2.13/1.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.13/1.44 ( ( in @ Xx
% 2.13/1.44 @ ( dsetconstr @ A
% 2.13/1.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.13/1.44 => ( in @ Xx @ A ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(dsetconstrER_type,type,
% 2.13/1.44 dsetconstrER: $o ).
% 2.13/1.44
% 2.13/1.44 thf(dsetconstrER,definition,
% 2.13/1.44 ( dsetconstrER
% 2.13/1.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.13/1.44 ( ( in @ Xx
% 2.13/1.44 @ ( dsetconstr @ A
% 2.13/1.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.13/1.44 => ( Xphi @ Xx ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(exuE1_type,type,
% 2.13/1.44 exuE1: $o ).
% 2.13/1.44
% 2.13/1.44 thf(exuE1,definition,
% 2.13/1.44 ( exuE1
% 2.13/1.44 = ( ! [Xphi: $i > $o] :
% 2.13/1.44 ( ( exu
% 2.13/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.13/1.44 => ? [Xx: $i] :
% 2.13/1.44 ( ( Xphi @ Xx )
% 2.13/1.44 & ! [Xy: $i] :
% 2.13/1.44 ( ( Xphi @ Xy )
% 2.13/1.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(prop2set_type,type,
% 2.13/1.44 prop2set: $o > $i ).
% 2.13/1.44
% 2.13/1.44 thf(prop2setE_type,type,
% 2.13/1.44 prop2setE: $o ).
% 2.13/1.44
% 2.13/1.44 thf(prop2setE,definition,
% 2.13/1.44 ( prop2setE
% 2.13/1.44 = ( ! [Xphi: $o,Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.13/1.44 => Xphi ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(emptysetE_type,type,
% 2.13/1.44 emptysetE: $o ).
% 2.13/1.44
% 2.13/1.44 thf(emptysetE,definition,
% 2.13/1.44 ( emptysetE
% 2.13/1.44 = ( ! [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ emptyset )
% 2.13/1.44 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(emptysetimpfalse_type,type,
% 2.13/1.44 emptysetimpfalse: $o ).
% 2.13/1.44
% 2.13/1.44 thf(emptysetimpfalse,definition,
% 2.13/1.44 ( emptysetimpfalse
% 2.13/1.44 = ( ! [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ emptyset )
% 2.13/1.44 => $false ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(notinemptyset_type,type,
% 2.13/1.44 notinemptyset: $o ).
% 2.13/1.44
% 2.13/1.44 thf(notinemptyset,definition,
% 2.13/1.44 ( notinemptyset
% 2.13/1.44 = ( ! [Xx: $i] :
% 2.13/1.44 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(exuE3e_type,type,
% 2.13/1.44 exuE3e: $o ).
% 2.13/1.44
% 2.13/1.44 thf(exuE3e,definition,
% 2.13/1.44 ( exuE3e
% 2.13/1.44 = ( ! [Xphi: $i > $o] :
% 2.13/1.44 ( ( exu
% 2.13/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.13/1.44 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(setext_type,type,
% 2.13/1.44 setext: $o ).
% 2.13/1.44
% 2.13/1.44 thf(setext,definition,
% 2.13/1.44 ( setext
% 2.13/1.44 = ( ! [A: $i,B: $i] :
% 2.13/1.44 ( ! [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ A )
% 2.13/1.44 => ( in @ Xx @ B ) )
% 2.13/1.44 => ( ! [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ B )
% 2.13/1.44 => ( in @ Xx @ A ) )
% 2.13/1.44 => ( A = B ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(emptyI_type,type,
% 2.13/1.44 emptyI: $o ).
% 2.13/1.44
% 2.13/1.44 thf(emptyI,definition,
% 2.13/1.44 ( emptyI
% 2.13/1.44 = ( ! [A: $i] :
% 2.13/1.44 ( ! [Xx: $i] :
% 2.13/1.44 ~ ( in @ Xx @ A )
% 2.13/1.44 => ( A = emptyset ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(noeltsimpempty_type,type,
% 2.13/1.44 noeltsimpempty: $o ).
% 2.13/1.44
% 2.13/1.44 thf(noeltsimpempty,definition,
% 2.13/1.44 ( noeltsimpempty
% 2.13/1.44 = ( ! [A: $i] :
% 2.13/1.44 ( ! [Xx: $i] :
% 2.13/1.44 ~ ( in @ Xx @ A )
% 2.13/1.44 => ( A = emptyset ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(setbeta_type,type,
% 2.13/1.44 setbeta: $o ).
% 2.13/1.44
% 2.13/1.44 thf(setbeta,definition,
% 2.13/1.44 ( setbeta
% 2.13/1.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ A )
% 2.13/1.44 => ( ( in @ Xx
% 2.13/1.44 @ ( dsetconstr @ A
% 2.13/1.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.13/1.44 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(nonempty_type,type,
% 2.13/1.44 nonempty: $i > $o ).
% 2.13/1.44
% 2.13/1.44 thf(nonempty,definition,
% 2.13/1.44 ( nonempty
% 2.13/1.44 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(nonemptyE1_type,type,
% 2.13/1.44 nonemptyE1: $o ).
% 2.13/1.44
% 2.13/1.44 thf(nonemptyE1,definition,
% 2.13/1.44 ( nonemptyE1
% 2.13/1.44 = ( ! [A: $i] :
% 2.13/1.44 ( ( nonempty @ A )
% 2.13/1.44 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(nonemptyI_type,type,
% 2.13/1.44 nonemptyI: $o ).
% 2.13/1.44
% 2.13/1.44 thf(nonemptyI,definition,
% 2.13/1.44 ( nonemptyI
% 2.13/1.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ A )
% 2.13/1.44 => ( ( Xphi @ Xx )
% 2.13/1.44 => ( nonempty
% 2.13/1.44 @ ( dsetconstr @ A
% 2.13/1.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(nonemptyI1_type,type,
% 2.13/1.44 nonemptyI1: $o ).
% 2.13/1.44
% 2.13/1.44 thf(nonemptyI1,definition,
% 2.13/1.44 ( nonemptyI1
% 2.13/1.44 = ( ! [A: $i] :
% 2.13/1.44 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.13/1.44 => ( nonempty @ A ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(setadjoinIL_type,type,
% 2.13/1.44 setadjoinIL: $o ).
% 2.13/1.44
% 2.13/1.44 thf(setadjoinIL,definition,
% 2.13/1.44 ( setadjoinIL
% 2.13/1.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(emptyinunitempty_type,type,
% 2.13/1.44 emptyinunitempty: $o ).
% 2.13/1.44
% 2.13/1.44 thf(emptyinunitempty,definition,
% 2.13/1.44 ( emptyinunitempty
% 2.13/1.44 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(setadjoinIR_type,type,
% 2.13/1.44 setadjoinIR: $o ).
% 2.13/1.44
% 2.13/1.44 thf(setadjoinIR,definition,
% 2.13/1.44 ( setadjoinIR
% 2.13/1.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.13/1.44 ( ( in @ Xy @ A )
% 2.13/1.44 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(setadjoinE_type,type,
% 2.13/1.44 setadjoinE: $o ).
% 2.13/1.44
% 2.13/1.44 thf(setadjoinE,definition,
% 2.13/1.44 ( setadjoinE
% 2.13/1.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.13/1.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.13/1.44 => ! [Xphi: $o] :
% 2.13/1.44 ( ( ( Xy = Xx )
% 2.13/1.44 => Xphi )
% 2.13/1.44 => ( ( ( in @ Xy @ A )
% 2.13/1.44 => Xphi )
% 2.13/1.44 => Xphi ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(setadjoinOr_type,type,
% 2.13/1.44 setadjoinOr: $o ).
% 2.13/1.44
% 2.13/1.44 thf(setadjoinOr,definition,
% 2.13/1.44 ( setadjoinOr
% 2.13/1.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.13/1.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.13/1.44 => ( ( Xy = Xx )
% 2.13/1.44 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(setoftrueEq_type,type,
% 2.13/1.44 setoftrueEq: $o ).
% 2.13/1.44
% 2.13/1.44 thf(setoftrueEq,definition,
% 2.13/1.44 ( setoftrueEq
% 2.13/1.44 = ( ! [A: $i] :
% 2.13/1.44 ( ( dsetconstr @ A
% 2.13/1.44 @ ^ [Xx: $i] : $true )
% 2.13/1.44 = A ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(powersetI_type,type,
% 2.13/1.44 powersetI: $o ).
% 2.13/1.44
% 2.13/1.44 thf(powersetI,definition,
% 2.13/1.44 ( powersetI
% 2.13/1.44 = ( ! [A: $i,B: $i] :
% 2.13/1.44 ( ! [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ B )
% 2.13/1.44 => ( in @ Xx @ A ) )
% 2.13/1.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(emptyinPowerset_type,type,
% 2.13/1.44 emptyinPowerset: $o ).
% 2.13/1.44
% 2.13/1.44 thf(emptyinPowerset,definition,
% 2.13/1.44 ( emptyinPowerset
% 2.13/1.44 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(emptyInPowerset_type,type,
% 2.13/1.44 emptyInPowerset: $o ).
% 2.13/1.44
% 2.13/1.44 thf(emptyInPowerset,definition,
% 2.13/1.44 ( emptyInPowerset
% 2.13/1.44 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(powersetE_type,type,
% 2.13/1.44 powersetE: $o ).
% 2.13/1.44
% 2.13/1.44 thf(powersetE,definition,
% 2.13/1.44 ( powersetE
% 2.13/1.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.44 ( ( in @ B @ ( powerset @ A ) )
% 2.13/1.44 => ( ( in @ Xx @ B )
% 2.13/1.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(setunionI_type,type,
% 2.13/1.44 setunionI: $o ).
% 2.13/1.44
% 2.13/1.44 thf(setunionI,definition,
% 2.13/1.44 ( setunionI
% 2.13/1.44 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.13/1.44 ( ( in @ Xx @ B )
% 2.13/1.44 => ( ( in @ B @ A )
% 2.13/1.44 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(setunionE_type,type,
% 2.13/1.44 setunionE: $o ).
% 2.13/1.44
% 2.13/1.44 thf(setunionE,definition,
% 2.13/1.44 ( setunionE
% 2.13/1.44 = ( ! [A: $i,Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ ( setunion @ A ) )
% 2.13/1.44 => ! [Xphi: $o] :
% 2.13/1.44 ( ! [B: $i] :
% 2.13/1.44 ( ( in @ Xx @ B )
% 2.13/1.44 => ( ( in @ B @ A )
% 2.13/1.44 => Xphi ) )
% 2.13/1.44 => Xphi ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(subPowSU_type,type,
% 2.13/1.44 subPowSU: $o ).
% 2.13/1.44
% 2.13/1.44 thf(subPowSU,definition,
% 2.13/1.44 ( subPowSU
% 2.13/1.44 = ( ! [A: $i,Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ A )
% 2.13/1.44 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(exuE2_type,type,
% 2.13/1.44 exuE2: $o ).
% 2.13/1.44
% 2.13/1.44 thf(exuE2,definition,
% 2.13/1.44 ( exuE2
% 2.13/1.44 = ( ! [Xphi: $i > $o] :
% 2.13/1.44 ( ( exu
% 2.13/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.13/1.44 => ? [Xx: $i] :
% 2.13/1.44 ! [Xy: $i] :
% 2.13/1.44 ( ( Xphi @ Xy )
% 2.13/1.44 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(nonemptyImpWitness_type,type,
% 2.13/1.44 nonemptyImpWitness: $o ).
% 2.13/1.44
% 2.13/1.44 thf(nonemptyImpWitness,definition,
% 2.13/1.44 ( nonemptyImpWitness
% 2.13/1.44 = ( ! [A: $i] :
% 2.13/1.44 ( ( nonempty @ A )
% 2.13/1.44 => ? [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ A )
% 2.13/1.44 & $true ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(uniqinunit_type,type,
% 2.13/1.44 uniqinunit: $o ).
% 2.13/1.44
% 2.13/1.44 thf(uniqinunit,definition,
% 2.13/1.44 ( uniqinunit
% 2.13/1.44 = ( ! [Xx: $i,Xy: $i] :
% 2.13/1.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.13/1.44 => ( Xx = Xy ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(notinsingleton_type,type,
% 2.13/1.44 notinsingleton: $o ).
% 2.13/1.44
% 2.13/1.44 thf(notinsingleton,definition,
% 2.13/1.44 ( notinsingleton
% 2.13/1.44 = ( ! [Xx: $i,Xy: $i] :
% 2.13/1.44 ( ( Xx != Xy )
% 2.13/1.44 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(eqinunit_type,type,
% 2.13/1.44 eqinunit: $o ).
% 2.13/1.44
% 2.13/1.44 thf(eqinunit,definition,
% 2.13/1.44 ( eqinunit
% 2.13/1.44 = ( ! [Xx: $i,Xy: $i] :
% 2.13/1.44 ( ( Xx = Xy )
% 2.13/1.44 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(singletonsswitch_type,type,
% 2.13/1.44 singletonsswitch: $o ).
% 2.13/1.44
% 2.13/1.44 thf(singletonsswitch,definition,
% 2.13/1.44 ( singletonsswitch
% 2.13/1.44 = ( ! [Xx: $i,Xy: $i] :
% 2.13/1.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.13/1.44 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(upairsetE_type,type,
% 2.13/1.44 upairsetE: $o ).
% 2.13/1.44
% 2.13/1.44 thf(upairsetE,definition,
% 2.13/1.44 ( upairsetE
% 2.13/1.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.13/1.44 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.13/1.44 => ( ( Xz = Xx )
% 2.13/1.44 | ( Xz = Xy ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(upairsetIL_type,type,
% 2.13/1.44 upairsetIL: $o ).
% 2.13/1.44
% 2.13/1.44 thf(upairsetIL,definition,
% 2.13/1.44 ( upairsetIL
% 2.13/1.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(upairsetIR_type,type,
% 2.13/1.44 upairsetIR: $o ).
% 2.13/1.44
% 2.13/1.44 thf(upairsetIR,definition,
% 2.13/1.44 ( upairsetIR
% 2.13/1.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(emptyE1_type,type,
% 2.13/1.44 emptyE1: $o ).
% 2.13/1.44
% 2.13/1.44 thf(emptyE1,definition,
% 2.13/1.44 ( emptyE1
% 2.13/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 2.13/1.44 ( ? [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ A )
% 2.13/1.44 & ( Xphi @ Xx ) )
% 2.13/1.44 => ( ( ( dsetconstr @ A
% 2.13/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.13/1.44 = emptyset )
% 2.13/1.44 => $false ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(vacuousDall_type,type,
% 2.13/1.44 vacuousDall: $o ).
% 2.13/1.44
% 2.13/1.44 thf(vacuousDall,definition,
% 2.13/1.44 ( vacuousDall
% 2.13/1.44 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ emptyset )
% 2.13/1.44 => ( Xphi @ Xx ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(quantDeMorgan1_type,type,
% 2.13/1.44 quantDeMorgan1: $o ).
% 2.13/1.44
% 2.13/1.44 thf(quantDeMorgan1,definition,
% 2.13/1.44 ( quantDeMorgan1
% 2.13/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 2.13/1.44 ( ~ ! [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ A )
% 2.13/1.44 => ( Xphi @ Xx ) )
% 2.13/1.44 => ? [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ A )
% 2.13/1.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(quantDeMorgan2_type,type,
% 2.13/1.44 quantDeMorgan2: $o ).
% 2.13/1.44
% 2.13/1.44 thf(quantDeMorgan2,definition,
% 2.13/1.44 ( quantDeMorgan2
% 2.13/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 2.13/1.44 ( ! [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ A )
% 2.13/1.44 => ~ ( Xphi @ Xx ) )
% 2.13/1.44 => ~ ? [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ A )
% 2.13/1.44 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(quantDeMorgan3_type,type,
% 2.13/1.44 quantDeMorgan3: $o ).
% 2.13/1.44
% 2.13/1.44 thf(quantDeMorgan3,definition,
% 2.13/1.44 ( quantDeMorgan3
% 2.13/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 2.13/1.44 ( ~ ? [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ A )
% 2.13/1.44 & ( Xphi @ Xx ) )
% 2.13/1.44 => ! [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ A )
% 2.13/1.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(quantDeMorgan4_type,type,
% 2.13/1.44 quantDeMorgan4: $o ).
% 2.13/1.44
% 2.13/1.44 thf(quantDeMorgan4,definition,
% 2.13/1.44 ( quantDeMorgan4
% 2.13/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 2.13/1.44 ( ? [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ A )
% 2.13/1.44 & ~ ( Xphi @ Xx ) )
% 2.13/1.44 => ~ ! [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ A )
% 2.13/1.44 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(prop2setI_type,type,
% 2.13/1.44 prop2setI: $o ).
% 2.13/1.44
% 2.13/1.44 thf(prop2setI,definition,
% 2.13/1.44 ( prop2setI
% 2.13/1.44 = ( ! [Xphi: $o] :
% 2.13/1.44 ( Xphi
% 2.13/1.44 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(set2prop_type,type,
% 2.13/1.44 set2prop: $i > $o ).
% 2.13/1.44
% 2.13/1.44 thf(prop2set2propI_type,type,
% 2.13/1.44 prop2set2propI: $o ).
% 2.13/1.44
% 2.13/1.44 thf(prop2set2propI,definition,
% 2.13/1.44 ( prop2set2propI
% 2.13/1.44 = ( ! [Xphi: $o] :
% 2.13/1.44 ( Xphi
% 2.13/1.44 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(notdexE_type,type,
% 2.13/1.44 notdexE: $o ).
% 2.13/1.44
% 2.13/1.44 thf(notdexE,definition,
% 2.13/1.44 ( notdexE
% 2.13/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 2.13/1.44 ( ~ ? [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ A )
% 2.13/1.44 & ( Xphi @ Xx ) )
% 2.13/1.44 => ! [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ A )
% 2.13/1.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(notdallE_type,type,
% 2.13/1.44 notdallE: $o ).
% 2.13/1.44
% 2.13/1.44 thf(notdallE,definition,
% 2.13/1.44 ( notdallE
% 2.13/1.44 = ( ! [A: $i,Xphi: $i > $o] :
% 2.13/1.44 ( ~ ! [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ A )
% 2.13/1.44 => ( Xphi @ Xx ) )
% 2.13/1.44 => ? [Xx: $i] :
% 2.13/1.44 ( ( in @ Xx @ A )
% 2.13/1.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(exuI1_type,type,
% 2.13/1.44 exuI1: $o ).
% 2.13/1.44
% 2.13/1.44 thf(exuI1,definition,
% 2.13/1.44 ( exuI1
% 2.13/1.44 = ( ! [Xphi: $i > $o] :
% 2.13/1.44 ( ? [Xx: $i] :
% 2.13/1.44 ( ( Xphi @ Xx )
% 2.13/1.44 & ! [Xy: $i] :
% 2.13/1.44 ( ( Xphi @ Xy )
% 2.13/1.44 => ( Xx = Xy ) ) )
% 2.13/1.44 => ( exu
% 2.13/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.13/1.44
% 2.13/1.44 thf(exuI3,conjecture,
% 2.13/1.44 ( setextAx
% 2.13/1.44 => ( emptysetAx
% 2.13/1.44 => ( setadjoinAx
% 2.13/1.44 => ( powersetAx
% 2.13/1.44 => ( setunionAx
% 2.13/1.44 => ( omega0Ax
% 2.13/1.44 => ( omegaSAx
% 2.13/1.44 => ( omegaIndAx
% 2.13/1.44 => ( replAx
% 2.13/1.44 => ( foundationAx
% 2.13/1.44 => ( wellorderingAx
% 2.13/1.44 => ( descrp
% 2.13/1.44 => ( dsetconstrI
% 2.13/1.44 => ( dsetconstrEL
% 2.13/1.44 => ( dsetconstrER
% 2.13/1.44 => ( exuE1
% 2.13/1.44 => ( prop2setE
% 2.13/1.44 => ( emptysetE
% 2.13/1.44 => ( emptysetimpfalse
% 2.13/1.44 => ( notinemptyset
% 2.13/1.44 => ( exuE3e
% 2.13/1.44 => ( setext
% 2.13/1.44 => ( emptyI
% 2.13/1.44 => ( noeltsimpempty
% 2.13/1.44 => ( setbeta
% 2.13/1.44 => ( nonemptyE1
% 2.13/1.44 => ( nonemptyI
% 2.13/1.44 => ( nonemptyI1
% 2.13/1.44 => ( setadjoinIL
% 2.13/1.44 => ( emptyinunitempty
% 2.13/1.44 => ( setadjoinIR
% 2.13/1.44 => ( setadjoinE
% 2.13/1.44 => ( setadjoinOr
% 2.13/1.44 => ( setoftrueEq
% 2.13/1.44 => ( powersetI
% 2.13/1.44 => ( emptyinPowerset
% 2.13/1.44 => ( emptyInPowerset
% 2.13/1.44 => ( powersetE
% 2.13/1.44 => ( setunionI
% 2.13/1.44 => ( setunionE
% 2.13/1.44 => ( subPowSU
% 2.13/1.44 => ( exuE2
% 2.13/1.44 => ( nonemptyImpWitness
% 2.13/1.44 => ( uniqinunit
% 2.13/1.44 => ( notinsingleton
% 2.13/1.44 => ( eqinunit
% 2.13/1.44 => ( singletonsswitch
% 2.13/1.44 => ( upairsetE
% 2.13/1.44 => ( upairsetIL
% 2.13/1.44 => ( upairsetIR
% 2.13/1.44 => ( emptyE1
% 2.13/1.44 => ( vacuousDall
% 2.13/1.44 => ( quantDeMorgan1
% 2.13/1.44 => ( quantDeMorgan2
% 2.13/1.44 => ( quantDeMorgan3
% 2.13/1.44 => ( quantDeMorgan4
% 2.13/1.44 => ( prop2setI
% 2.13/1.44 => ( prop2set2propI
% 2.13/1.44 => ( notdexE
% 2.13/1.44 => ( notdallE
% 2.13/1.44 => ( exuI1
% 2.13/1.44 => ! [Xphi: $i > $o] :
% 2.13/1.44 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.13/1.44 => ( ! [Xx: $i,Xy: $i] :
% 2.13/1.44 ( ( Xphi @ Xx )
% 2.13/1.44 => ( ( Xphi @ Xy )
% 2.13/1.44 => ( Xx = Xy ) ) )
% 2.13/1.44 => ( exu
% 2.13/1.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.13/1.44 ------------------------
% 2.13/1.46 % (32549)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_32358 for (2999ds/183Mi)
% 2.13/1.46 % (32550)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_32358 for (2999ds/4Mi)
% 2.13/1.46 % (32550)Instruction limit reached!
% 2.13/1.46 % (32550)------------------------------
% 2.13/1.46 % (32550)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.13/1.46 % (32550)Termination reason: Unknown
% 2.13/1.46 % (32550)Termination phase: shuffling
% 2.13/1.46
% 2.13/1.46 % (32550)Memory used [KB]: 1151
% 2.13/1.46 % (32550)Time elapsed: 0.003 s
% 2.13/1.46 % (32554)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_32358 for (2999ds/275Mi)
% 2.13/1.46 % (32550)Instructions burned: 7 (million)
% 2.13/1.46 % (32550)------------------------------
% 2.13/1.46 % (32550)------------------------------
% 2.13/1.46 % (32552)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_32358 for (2999ds/2Mi)
% 2.13/1.46 % (32552)Instruction limit reached!
% 2.13/1.46 % (32552)------------------------------
% 2.13/1.46 % (32552)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.13/1.46 % (32552)Termination reason: Unknown
% 2.13/1.46 % (32552)Termination phase: shuffling
% 2.13/1.46
% 2.13/1.46 % (32552)Memory used [KB]: 1023
% 2.13/1.46 % (32552)Time elapsed: 0.002 s
% 2.13/1.46 % (32552)Instructions burned: 2 (million)
% 2.13/1.46 % (32552)------------------------------
% 2.13/1.46 % (32552)------------------------------
% 2.13/1.47 % (32551)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_32358 for (2999ds/27Mi)
% 2.13/1.47 % (32553)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_32358 for (2999ds/2Mi)
% 2.13/1.47 % (32553)Instruction limit reached!
% 2.13/1.47 % (32553)------------------------------
% 2.13/1.47 % (32553)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.13/1.47 % (32553)Termination reason: Unknown
% 2.13/1.47 % (32553)Termination phase: shuffling
% 2.13/1.47
% 2.13/1.47 % (32553)Memory used [KB]: 1151
% 2.13/1.47 % (32553)Time elapsed: 0.004 s
% 2.13/1.47 % (32553)Instructions burned: 3 (million)
% 2.13/1.47 % (32553)------------------------------
% 2.13/1.47 % (32553)------------------------------
% 2.13/1.47 % (32555)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_32358 for (2999ds/18Mi)
% 2.13/1.47 % (32556)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_32358 for (2999ds/3Mi)
% 2.13/1.47 % (32556)Instruction limit reached!
% 2.13/1.47 % (32556)------------------------------
% 2.13/1.47 % (32556)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.13/1.47 % (32556)Termination reason: Unknown
% 2.13/1.47 % (32556)Termination phase: shuffling
% 2.13/1.47
% 2.13/1.47 % (32556)Memory used [KB]: 1151
% 2.13/1.47 % (32556)Time elapsed: 0.002 s
% 2.13/1.47 % (32556)Instructions burned: 4 (million)
% 2.13/1.47 % (32556)------------------------------
% 2.13/1.47 % (32556)------------------------------
% 2.13/1.48 % (32555)Instruction limit reached!
% 2.13/1.48 % (32555)------------------------------
% 2.13/1.48 % (32555)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.13/1.48 % (32555)Termination reason: Unknown
% 2.13/1.48 % (32555)Termination phase: Preprocessing 3
% 2.13/1.48
% 2.13/1.48 % (32555)Memory used [KB]: 1535
% 2.13/1.48 % (32555)Time elapsed: 0.009 s
% 2.13/1.48 % (32555)Instructions burned: 18 (million)
% 2.13/1.48 % (32555)------------------------------
% 2.13/1.48 % (32555)------------------------------
% 2.13/1.48 % (32557)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_32358 for (2999ds/37Mi)
% 2.13/1.48 % (32551)Instruction limit reached!
% 2.13/1.48 % (32551)------------------------------
% 2.13/1.48 % (32551)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.13/1.48 % (32551)Termination reason: Unknown
% 2.13/1.48 % (32551)Termination phase: Property scanning
% 2.13/1.48
% 2.13/1.48 % (32551)Memory used [KB]: 1407
% 2.13/1.48 % (32551)Time elapsed: 0.016 s
% 2.13/1.48 % (32551)Instructions burned: 28 (million)
% 2.13/1.48 % (32551)------------------------------
% 2.13/1.48 % (32551)------------------------------
% 2.13/1.49 % (32559)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_32358 for (2999ds/3Mi)
% 2.13/1.49 % (32559)Instruction limit reached!
% 2.13/1.49 % (32559)------------------------------
% 2.13/1.49 % (32559)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.13/1.49 % (32559)Termination reason: Unknown
% 2.13/1.49 % (32559)Termination phase: shuffling
% 2.13/1.49
% 2.13/1.49 % (32559)Memory used [KB]: 1151
% 2.13/1.49 % (32559)Time elapsed: 0.002 s
% 2.13/1.49 % (32559)Instructions burned: 4 (million)
% 2.13/1.49 % (32559)------------------------------
% 2.13/1.49 % (32559)------------------------------
% 2.13/1.49 % (32558)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_32358 for (2999ds/15Mi)
% 2.13/1.49 % (32549)First to succeed.
% 2.13/1.49 % (32557)Instruction limit reached!
% 2.13/1.49 % (32557)------------------------------
% 2.13/1.49 % (32557)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.13/1.49 % (32557)Termination reason: Unknown
% 2.13/1.49 % (32557)Termination phase: Saturation
% 2.13/1.49
% 2.13/1.49 % (32557)Memory used [KB]: 5884
% 2.13/1.49 % (32557)Time elapsed: 0.013 s
% 2.13/1.49 % (32557)Instructions burned: 39 (million)
% 2.13/1.49 % (32557)------------------------------
% 2.13/1.49 % (32557)------------------------------
% 2.13/1.49 % (32560)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_32358 for (2999ds/1041Mi)
% 2.13/1.49 % (32549)Refutation found. Thanks to Tanya!
% 2.13/1.49 % SZS status Theorem for DTF2THF_32358
% 2.13/1.49 % SZS output start Proof for DTF2THF_32358
% 2.13/1.49 thf(func_def_0, type, in: $i > $i > $o).
% 2.13/1.49 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.13/1.49 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.13/1.49 thf(func_def_8, type, powerset: $i > $i).
% 2.13/1.49 thf(func_def_10, type, setunion: $i > $i).
% 2.13/1.49 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.13/1.49 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.13/1.49 thf(func_def_26, type, prop2set: $o > $i).
% 2.13/1.49 thf(func_def_36, type, nonempty: $i > $o).
% 2.13/1.49 thf(func_def_69, type, set2prop: $i > $o).
% 2.13/1.49 thf(func_def_90, type, sK0: $i > $o).
% 2.13/1.49 thf(func_def_93, type, sK3: ($i > $o) > $i).
% 2.13/1.49 thf(func_def_94, type, sK4: ($i > $o) > $i > $i).
% 2.13/1.49 thf(func_def_95, type, sK5: ($i > $o) > $i > $i).
% 2.13/1.49 thf(func_def_96, type, sK6: $i > $i > $i).
% 2.13/1.49 thf(func_def_97, type, sK7: $i > ($i > $o) > $i).
% 2.13/1.49 thf(func_def_98, type, sK8: $i > ($i > $o) > $i).
% 2.13/1.49 thf(func_def_99, type, sK9: $i > $i > $i).
% 2.13/1.49 thf(func_def_100, type, sK10: $i > $i).
% 2.13/1.49 thf(func_def_101, type, sK11: ($i > $o) > $i > $i).
% 2.13/1.49 thf(func_def_102, type, sK12: $i > $i).
% 2.13/1.49 thf(func_def_104, type, sK13: ($i > $i > $o) > $i > $i).
% 2.13/1.49 thf(func_def_105, type, sK14: ($i > $i > $o) > $i > $i).
% 2.13/1.49 thf(func_def_106, type, sK15: $i > ($i > $i > $o) > $i > $i).
% 2.13/1.49 thf(func_def_107, type, sK16: $i > ($i > $i > $o) > $i > $i).
% 2.13/1.49 thf(func_def_108, type, sK17: $i > ($i > $i > $o) > $i > $i).
% 2.13/1.49 thf(func_def_109, type, sK18: $i > ($i > $i > $o) > $i > $i).
% 2.13/1.49 thf(func_def_110, type, sK19: $i > $i).
% 2.13/1.49 thf(func_def_111, type, sK20: $i > ($i > $o) > $i).
% 2.13/1.49 thf(func_def_112, type, sK21: $i > $i).
% 2.13/1.49 thf(func_def_113, type, sK22: ($i > $o) > $i > $i).
% 2.13/1.49 thf(func_def_114, type, sK23: $i > $i > $i).
% 2.13/1.49 thf(func_def_115, type, sK24: $i > $i > $i).
% 2.13/1.49 thf(func_def_116, type, sK25: $i > $i > $i).
% 2.13/1.49 thf(func_def_117, type, sK26: $i > ($i > $o) > $i).
% 2.13/1.49 thf(func_def_118, type, sK27: ($i > $o) > $i).
% 2.13/1.49 thf(func_def_119, type, sK28: ($i > $o) > $i > $i).
% 2.13/1.49 thf(func_def_120, type, sK29: $i > $i).
% 2.13/1.49 thf(func_def_121, type, sK30: $i > $i).
% 2.13/1.49 thf(func_def_122, type, sK31: $o > $i > $i > $i).
% 2.13/1.49 thf(func_def_123, type, sK32: ($i > $o) > $i).
% 2.13/1.49 thf(func_def_124, type, sK33: ($i > $o) > $i > $i).
% 2.13/1.49 thf(func_def_125, type, sK34: ($i > $o) > $i > $i).
% 2.13/1.49 thf(func_def_126, type, sK35: $i > $i).
% 2.13/1.49 thf(func_def_127, type, sK36: $i > $i > $i).
% 2.13/1.49 thf(func_def_128, type, sK37: ($i > $o) > $i).
% 2.13/1.49 thf(func_def_129, type, sK38: ($i > $o) > $i > $i).
% 2.13/1.49 thf(func_def_130, type, sK39: ($i > $o) > $i > $i).
% 2.13/1.49 thf(func_def_131, type, sK40: $i > $i).
% 2.13/1.49 thf(func_def_132, type, sK41: $i > $i > $i).
% 2.13/1.49 thf(func_def_133, type, sK42: $i > $i > $i).
% 2.13/1.49 thf(func_def_134, type, sK43: $i > $i > $i).
% 2.13/1.49 thf(func_def_135, type, sK44: $i > $i > $i > $i).
% 2.13/1.49 thf(f1208,plain,(
% 2.13/1.49 $false),
% 2.13/1.49 inference(subsumption_resolution,[status(thm)],[f1206,f1145])).
% 2.13/1.49 thf(f1145,plain,(
% 2.13/1.49 (sK1 != (sK21 @ sK1))),
% 2.13/1.49 inference(trivial_inequality_removal,[status(thm)],[f1140])).
% 2.13/1.49 thf(f1140,plain,(
% 2.13/1.49 (sK1 != (sK21 @ sK1)) | ($false = $true)),
% 2.13/1.49 inference(superposition,[status(thm)],[f663,f260])).
% 2.13/1.49 thf(f260,plain,(
% 2.13/1.49 ((sK0 @ sK1) = $true)),
% 2.13/1.49 inference(cnf_transformation,[status(thm)],[f195])).
% 2.13/1.49 thf(f195,plain,(
% 2.13/1.49 (setunionI = $true) & (omegaIndAx = $true) & (quantDeMorgan4 = $true) & (setadjoinE = $true) & (notinsingleton = $true) & (nonemptyI1 = $true) & (setunionE = $true) & (vacuousDall = $true) & (emptysetE = $true) & (prop2setE = $true) & (upairsetIL = $true) & (emptyInPowerset = $true) & (setadjoinIL = $true) & (eqinunit = $true) & (notinemptyset = $true) & (wellorderingAx = $true) & (powersetI = $true) & (noeltsimpempty = $true) & (emptyE1 = $true) & (omega0Ax = $true) & (setadjoinAx = $true) & (emptysetimpfalse = $true) & (setadjoinOr = $true) & (((exu @ (^[Y0 : $i]: (sK0 @ Y0))) != $true) & ((sK0 @ sK1) = $true) & ! [X2 : $i,X3 : $i] : (($true != (sK0 @ X3)) | ((sK0 @ X2) != $true) | (X2 = X3))) & (nonemptyImpWitness = $true) & (dsetconstrI = $true) & (prop2set2propI = $true) & (setadjoinIR = $true) & (dsetconstrEL = $true) & (prop2setI = $true) & (quantDeMorgan2 = $true) & (emptyinunitempty = $true) & (foundationAx = $true) & (setoftrueEq = $true) & (powersetAx = $true) & (setext = $true) & (nonemptyI = $true) & (exuE1 = $true) & (upairsetE = $true) & (emptyI = $true) & (exuE2 = $true) & (singletonsswitch = $true) & (setunionAx = $true) & (upairsetIR = $true) & (setextAx = $true) & (emptysetAx = $true) & (exuI1 = $true) & (subPowSU = $true) & (quantDeMorgan3 = $true) & (quantDeMorgan1 = $true) & (powersetE = $true) & (notdexE = $true) & (descrp = $true) & (emptyinPowerset = $true) & (replAx = $true) & (dsetconstrER = $true) & (omegaSAx = $true) & (uniqinunit = $true) & (nonemptyE1 = $true) & (setbeta = $true) & (notdallE = $true) & (exuE3e = $true)),
% 2.13/1.49 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f192,f194,f193])).
% 2.13/1.49 thf(f193,plain,(
% 2.13/1.49 ? [X0 : $i > $o] : (($true != (exu @ (^[Y0 : $i]: (X0 @ Y0)))) & ? [X1 : $i] : ((X0 @ X1) = $true) & ! [X2 : $i,X3 : $i] : (((X0 @ X3) != $true) | ((X0 @ X2) != $true) | (X2 = X3))) => (((exu @ (^[Y0 : $i]: (sK0 @ Y0))) != $true) & ? [X1 : $i] : ((sK0 @ X1) = $true) & ! [X3 : $i,X2 : $i] : (($true != (sK0 @ X3)) | ((sK0 @ X2) != $true) | (X2 = X3)))),
% 2.13/1.49 introduced(definition,[],[choice_axiom])).
% 2.13/1.49 thf(f194,plain,(
% 2.13/1.49 ? [X1 : $i] : ((sK0 @ X1) = $true) => ((sK0 @ sK1) = $true)),
% 2.13/1.49 introduced(definition,[],[choice_axiom])).
% 2.13/1.49 thf(f192,plain,(
% 2.13/1.49 (setunionI = $true) & (omegaIndAx = $true) & (quantDeMorgan4 = $true) & (setadjoinE = $true) & (notinsingleton = $true) & (nonemptyI1 = $true) & (setunionE = $true) & (vacuousDall = $true) & (emptysetE = $true) & (prop2setE = $true) & (upairsetIL = $true) & (emptyInPowerset = $true) & (setadjoinIL = $true) & (eqinunit = $true) & (notinemptyset = $true) & (wellorderingAx = $true) & (powersetI = $true) & (noeltsimpempty = $true) & (emptyE1 = $true) & (omega0Ax = $true) & (setadjoinAx = $true) & (emptysetimpfalse = $true) & (setadjoinOr = $true) & ? [X0 : $i > $o] : (($true != (exu @ (^[Y0 : $i]: (X0 @ Y0)))) & ? [X1 : $i] : ((X0 @ X1) = $true) & ! [X2 : $i,X3 : $i] : (((X0 @ X3) != $true) | ((X0 @ X2) != $true) | (X2 = X3))) & (nonemptyImpWitness = $true) & (dsetconstrI = $true) & (prop2set2propI = $true) & (setadjoinIR = $true) & (dsetconstrEL = $true) & (prop2setI = $true) & (quantDeMorgan2 = $true) & (emptyinunitempty = $true) & (foundationAx = $true) & (setoftrueEq = $true) & (powersetAx = $true) & (setext = $true) & (nonemptyI = $true) & (exuE1 = $true) & (upairsetE = $true) & (emptyI = $true) & (exuE2 = $true) & (singletonsswitch = $true) & (setunionAx = $true) & (upairsetIR = $true) & (setextAx = $true) & (emptysetAx = $true) & (exuI1 = $true) & (subPowSU = $true) & (quantDeMorgan3 = $true) & (quantDeMorgan1 = $true) & (powersetE = $true) & (notdexE = $true) & (descrp = $true) & (emptyinPowerset = $true) & (replAx = $true) & (dsetconstrER = $true) & (omegaSAx = $true) & (uniqinunit = $true) & (nonemptyE1 = $true) & (setbeta = $true) & (notdallE = $true) & (exuE3e = $true)),
% 2.13/1.49 inference(flattening,[status(thm)],[f191])).
% 2.13/1.49 thf(f191,plain,(
% 2.13/1.49 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i > $o] : ((($true != (exu @ (^[Y0 : $i]: (X0 @ Y0)))) & ! [X2 : $i,X3 : $i] : (((X2 = X3) | ((X0 @ X3) != $true)) | ((X0 @ X2) != $true))) & ? [X1 : $i] : ((X0 @ X1) = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.13/1.49 inference(ennf_transformation,[status(thm)],[f86])).
% 2.13/1.49 thf(f86,plain,(
% 2.13/1.49 ~((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) => ! [X0 : $i > $o] : (? [X1 : $i] : ((X0 @ X1) = $true) => (! [X2 : $i,X3 : $i] : (((X0 @ X2) = $true) => (((X0 @ X3) = $true) => (X2 = X3))) => ($true = (exu @ (^[Y0 : $i]: (X0 @ Y0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.13/1.49 inference(fool_elimination,[status(thm)],[f85])).
% 2.13/1.49 thf(f85,plain,(
% 2.13/1.49 ~(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 => ! [X0 : $i > $o] : (? [X1 : $i] : (X0 @ X1) => (! [X2 : $i,X3 : $i] : ((X0 @ X2) => ((X0 @ X3) => (X2 = X3))) => (exu @ (^[X4 : $i] : (X0 @ X4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.13/1.49 inference(rectify,[status(thm)],[f65])).
% 2.13/1.49 thf(f65,negated_conjecture,(
% 2.13/1.49 ~(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 => ! [X0 : $i > $o] : (? [X1 : $i] : (X0 @ X1) => (! [X1 : $i,X2 : $i] : ((X0 @ X1) => ((X0 @ X2) => (X1 = X2))) => (exu @ (^[X1 : $i] : (X0 @ X1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.13/1.49 inference(negated_conjecture,[status(cth)],[f64])).
% 2.13/1.49 thf(f64,conjecture,(
% 2.13/1.49 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 => ! [X0 : $i > $o] : (? [X1 : $i] : (X0 @ X1) => (! [X1 : $i,X2 : $i] : ((X0 @ X1) => ((X0 @ X2) => (X1 = X2))) => (exu @ (^[X1 : $i] : (X0 @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.13/1.49 file('/export/starexec/sandbox/tmp/tmp.V2DC10xSo5/DTF2THF_32358.p',exuI3)).
% 2.13/1.49 thf(f663,plain,(
% 2.13/1.49 ( ! [X1 : $i] : (($false = (sK0 @ X1)) | ((sK21 @ X1) != X1)) )),
% 2.13/1.49 inference(equality_proxy_clausification,[status(thm)],[f661])).
% 2.13/1.49 thf(f661,plain,(
% 2.13/1.49 ( ! [X1 : $i] : (((X1 = (sK21 @ X1)) = $false) | ($false = (sK0 @ X1))) )),
% 2.13/1.49 inference(binary_proxy_clausification,[status(thm)],[f660])).
% 2.13/1.49 thf(f660,plain,(
% 2.13/1.49 ( ! [X1 : $i] : (($false = ((sK0 @ (sK21 @ X1)) => (X1 = (sK21 @ X1)))) | ($false = (sK0 @ X1))) )),
% 2.13/1.49 inference(beta_eta_normalization,[status(thm)],[f659])).
% 2.13/1.49 thf(f659,plain,(
% 2.13/1.49 ( ! [X1 : $i] : ((((^[Y0 : $i]: ((sK0 @ Y0) => (X1 = Y0))) @ (sK21 @ X1)) = $false) | ($false = (sK0 @ X1))) )),
% 2.13/1.49 inference(sigma_clausification,[status(thm)],[f658])).
% 2.13/1.49 thf(f658,plain,(
% 2.13/1.49 ( ! [X1 : $i] : (($false = (!! @ $i @ (^[Y0 : $i]: ((sK0 @ Y0) => (X1 = Y0))))) | ($false = (sK0 @ X1))) )),
% 2.13/1.49 inference(binary_proxy_clausification,[status(thm)],[f657])).
% 2.13/1.49 thf(f657,plain,(
% 2.13/1.49 ( ! [X1 : $i] : ((((sK0 @ X1) & (!! @ $i @ (^[Y0 : $i]: ((sK0 @ Y0) => (X1 = Y0))))) = $false)) )),
% 2.13/1.49 inference(beta_eta_normalization,[status(thm)],[f656])).
% 2.13/1.49 thf(f656,plain,(
% 2.13/1.49 ( ! [X1 : $i] : (($false = ((^[Y0 : $i]: ((sK0 @ Y0) & (!! @ $i @ (^[Y1 : $i]: ((sK0 @ Y1) => (Y0 = Y1)))))) @ X1))) )),
% 2.13/1.49 inference(pi_clausification,[status(thm)],[f655])).
% 2.13/1.49 thf(f655,plain,(
% 2.13/1.49 ($true != (?? @ $i @ (^[Y0 : $i]: ((sK0 @ Y0) & (!! @ $i @ (^[Y1 : $i]: ((sK0 @ Y1) => (Y0 = Y1))))))))),
% 2.13/1.49 inference(beta_eta_normalization,[status(thm)],[f336])).
% 2.13/1.49 thf(f336,plain,(
% 2.13/1.49 (((^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) & (!! @ $i @ (^[Y2 : $i]: ((Y0 @ Y2) => (Y1 = Y2)))))))) @ (^[Y0 : $i]: (sK0 @ Y0))) != $true)),
% 2.13/1.49 inference(definition_unfolding,[status(thm)],[f261,f312])).
% 2.13/1.49 thf(f312,plain,(
% 2.13/1.49 (exu = (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) & (!! @ $i @ (^[Y2 : $i]: ((Y0 @ Y2) => (Y1 = Y2)))))))))),
% 2.13/1.49 inference(cnf_transformation,[status(thm)],[f124])).
% 2.13/1.49 thf(f124,plain,(
% 2.13/1.49 (exu = (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) & (!! @ $i @ (^[Y2 : $i]: ((Y0 @ Y2) => (Y1 = Y2)))))))))),
% 2.13/1.49 inference(fool_elimination,[status(thm)],[f123])).
% 2.13/1.49 thf(f123,plain,(
% 2.13/1.49 (exu = (^[X0 : $i > $o] : (? [X1 : $i] : (! [X2 : $i] : ((X0 @ X2) => (X1 = X2)) & (X0 @ X1)))))),
% 2.13/1.49 inference(rectify,[status(thm)],[f1])).
% 2.13/1.49 thf(f1,axiom,(
% 2.13/1.49 (exu = (^[X0 : $i > $o] : (? [X1 : $i] : (! [X2 : $i] : ((X0 @ X2) => (X1 = X2)) & (X0 @ X1)))))),
% 2.13/1.49 file('/export/starexec/sandbox/tmp/tmp.V2DC10xSo5/DTF2THF_32358.p',exu)).
% 2.13/1.49 thf(f261,plain,(
% 2.13/1.49 ((exu @ (^[Y0 : $i]: (sK0 @ Y0))) != $true)),
% 2.13/1.49 inference(cnf_transformation,[status(thm)],[f195])).
% 2.13/1.49 thf(f1206,plain,(
% 2.13/1.49 (sK1 = (sK21 @ sK1))),
% 2.13/1.49 inference(trivial_inequality_removal,[status(thm)],[f1196])).
% 2.13/1.49 thf(f1196,plain,(
% 2.13/1.49 (sK1 = (sK21 @ sK1)) | ($false = $true)),
% 2.13/1.49 inference(superposition,[status(thm)],[f1194,f260])).
% 2.13/1.49 thf(f1194,plain,(
% 2.13/1.49 ( ! [X0 : $i] : (($false = (sK0 @ X0)) | ((sK21 @ X0) = sK1)) )),
% 2.13/1.49 inference(trivial_inequality_removal,[status(thm)],[f1185])).
% 2.13/1.49 thf(f1185,plain,(
% 2.13/1.49 ( ! [X0 : $i] : (($false = (sK0 @ X0)) | ((sK21 @ X0) = sK1) | ($true != $true)) )),
% 2.13/1.49 inference(superposition,[status(thm)],[f1155,f662])).
% 2.13/1.49 thf(f662,plain,(
% 2.13/1.49 ( ! [X1 : $i] : (((sK0 @ (sK21 @ X1)) = $true) | ($false = (sK0 @ X1))) )),
% 2.13/1.49 inference(binary_proxy_clausification,[status(thm)],[f660])).
% 2.13/1.49 thf(f1155,plain,(
% 2.13/1.49 ( ! [X0 : $i] : (((sK0 @ X0) != $true) | (sK1 = X0)) )),
% 2.13/1.49 inference(trivial_inequality_removal,[status(thm)],[f1152])).
% 2.13/1.49 thf(f1152,plain,(
% 2.13/1.49 ( ! [X0 : $i] : (((sK0 @ X0) != $true) | ($true != $true) | (sK1 = X0)) )),
% 2.13/1.49 inference(superposition,[status(thm)],[f259,f260])).
% 2.13/1.49 thf(f259,plain,(
% 2.13/1.49 ( ! [X2 : $i,X3 : $i] : (($true != (sK0 @ X3)) | ((sK0 @ X2) != $true) | (X2 = X3)) )),
% 2.13/1.49 inference(cnf_transformation,[status(thm)],[f195])).
% 2.13/1.49 % SZS output end Proof for DTF2THF_32358
% 2.13/1.49 % (32549)------------------------------
% 2.13/1.49 % (32549)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.13/1.49 % (32549)Termination reason: Refutation
% 2.13/1.49
% 2.13/1.49 % (32549)Memory used [KB]: 6396
% 2.13/1.49 % (32549)Time elapsed: 0.032 s
% 2.13/1.49 % (32549)Instructions burned: 102 (million)
% 2.13/1.49 % (32549)------------------------------
% 2.13/1.49 % (32549)------------------------------
% 2.13/1.49 % (32548)Success in time 0.048 s
%------------------------------------------------------------------------------