%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU544^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n001.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 2.10s 1.52s
% Output : Refutation 2.10s
% Verified :
% SZS Type : Refutation
% Derivation depth : 20
% Number of leaves : 4
% Syntax : Number of formulae : 33 ( 13 unt; 0 typ; 0 def)
% Number of atoms : 1102 ( 320 equ; 0 cnn)
% Maximal formula atoms : 64 ( 33 avg)
% Number of connectives : 628 ( 23 ~; 19 |; 204 &; 110 @)
% ( 0 <=>; 261 =>; 0 <=; 0 <~>)
% Maximal formula depth : 69 ( 16 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 12 ( 12 >; 0 *; 0 +; 0 <<)
% Number of symbols : 120 ( 115 usr; 65 con; 0-3 aty)
% ( 7 !!; 4 ??; 0 @@+; 0 @@-)
% Number of variables : 71 ( 29 ^; 28 !; 14 ?; 71 :)
% 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_89,type,
sK0: $i > $o ).
thf(func_def_92,type,
sK3: $i > $i ).
thf(func_def_93,type,
sK4: $i > $i > $i ).
thf(func_def_94,type,
sK5: $i > $i > $i ).
thf(func_def_96,type,
sK6: ( $i > $i > $o ) > $i > $i ).
thf(func_def_97,type,
sK7: ( $i > $i > $o ) > $i > $i ).
thf(func_def_98,type,
sK8: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_99,type,
sK9: ( $i > $i > $o ) > $i > $i ).
thf(func_def_100,type,
sK10: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_101,type,
sK11: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_102,type,
sK12: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_103,type,
sK13: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_104,type,
sK14: $i > $i ).
thf(func_def_105,type,
sK15: $o > $i > $i > $i ).
thf(func_def_106,type,
sK16: $i > $i > $i ).
thf(func_def_107,type,
sK17: ( $i > $o ) > $i > $i ).
thf(func_def_108,type,
sK18: $i > $i > $i ).
thf(func_def_109,type,
sK19: $i > $i > $i ).
thf(func_def_110,type,
sK20: $i > $i > $i ).
thf(func_def_111,type,
sK21: $i > $i > $i ).
thf(func_def_112,type,
sK22: $i > $i ).
thf(func_def_113,type,
sK23: ( $i > $o ) > $i > $i ).
thf(func_def_114,type,
sK24: ( $i > $o ) > $i > $i ).
thf(func_def_115,type,
sK25: $i > $i ).
thf(func_def_116,type,
sK26: $i > $i ).
thf(func_def_117,type,
sK27: $i > $i ).
thf(func_def_118,type,
sK28: $i > $i ).
thf(func_def_119,type,
sK29: $i > ( $i > $o ) > $i ).
thf(func_def_120,type,
sK30: ( $i > $o ) > $i ).
thf(func_def_121,type,
sK31: ( $i > $o ) > $i > $i ).
thf(func_def_122,type,
sK32: ( $i > $o ) > $i ).
thf(func_def_123,type,
sK33: ( $i > $o ) > $i > $i ).
thf(func_def_124,type,
sK34: ( $i > $o ) > $i > $i ).
thf(func_def_125,type,
sK35: ( $i > $o ) > $i > $i ).
thf(func_def_126,type,
sK36: $i > ( $i > $o ) > $i ).
thf(func_def_127,type,
sK37: ( $i > $o ) > $i > $i ).
thf(func_def_128,type,
sK38: ( $i > $o ) > $i ).
thf(func_def_129,type,
sK39: ( $i > $o ) > $i > $i ).
thf(func_def_130,type,
sK40: $i > $i ).
thf(func_def_131,type,
sK41: $i > $i > $i ).
thf(func_def_132,type,
sK42: $i > $i > $i ).
thf(func_def_133,type,
sK43: $i > $i > $i ).
thf(func_def_134,type,
sK44: $i > $i > $i ).
thf(func_def_135,type,
sK45: $i > $i > $i > $i ).
thf(f1134,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1131,f1110]) ).
thf(f1110,plain,
( ( sK28 @ sK1 )
!= sK1 ),
inference(trivial_inequality_removal,[status(thm)],[f1106]) ).
thf(f1106,plain,
( ( ( sK28 @ sK1 )
!= sK1 )
| ( $false = $true ) ),
inference(superposition,[status(thm)],[f794,f250]) ).
thf(f250,plain,
( ( sK0 @ sK1 )
= $true ),
inference(cnf_transformation,[status(thm)],[f192]) ).
thf(f192,plain,
( ( prop2setI = $true )
& ( dsetconstrI = $true )
& ( emptyE1 = $true )
& ( setunionAx = $true )
& ( setoftrueEq = $true )
& ( nonemptyE1 = $true )
& ( dsetconstrER = $true )
& ( uniqinunit = $true )
& ( setbeta = $true )
& ( notdallE = $true )
& ( emptysetE = $true )
& ( omegaSAx = $true )
& ( wellorderingAx = $true )
& ( singletonsswitch = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( quantDeMorgan4 = $true )
& ( omegaIndAx = $true )
& ! [X2: $i] :
( ( sK1 = X2 )
| ( ( sK0 @ X2 )
!= $true ) )
& ( ( sK0 @ sK1 )
= $true )
& ( ( exu
@ ^ [Y0: $i] : ( sK0 @ Y0 ) )
!= $true )
& ( vacuousDall = $true )
& ( setadjoinIL = $true )
& ( upairsetIL = $true )
& ( exuE1 = $true )
& ( setadjoinIR = $true )
& ( emptyinPowerset = $true )
& ( exuE3e = $true )
& ( eqinunit = $true )
& ( setextAx = $true )
& ( replAx = $true )
& ( powersetAx = $true )
& ( setadjoinE = $true )
& ( omega0Ax = $true )
& ( powersetI = $true )
& ( notinsingleton = $true )
& ( nonemptyImpWitness = $true )
& ( emptysetimpfalse = $true )
& ( quantDeMorgan2 = $true )
& ( notinemptyset = $true )
& ( setadjoinOr = $true )
& ( exuE2 = $true )
& ( setunionE = $true )
& ( descrp = $true )
& ( nonemptyI = $true )
& ( nonemptyI1 = $true )
& ( quantDeMorgan3 = $true )
& ( emptyinunitempty = $true )
& ( setadjoinAx = $true )
& ( prop2setE = $true )
& ( quantDeMorgan1 = $true )
& ( upairsetE = $true )
& ( foundationAx = $true )
& ( prop2set2propI = $true )
& ( noeltsimpempty = $true )
& ( setext = $true )
& ( upairsetIR = $true )
& ( notdexE = $true )
& ( emptyI = $true )
& ( setunionI = $true )
& ( dsetconstrEL = $true )
& ( emptysetAx = $true )
& ( subPowSU = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f189,f191,f190]) ).
thf(f190,plain,
( ? [X0: $i > $o] :
( ? [X1: $i] :
( ! [X2: $i] :
( ( X1 = X2 )
| ( ( X0 @ X2 )
!= $true ) )
& ( ( X0 @ X1 )
= $true ) )
& ( ( exu
@ ^ [Y0: $i] : ( X0 @ Y0 ) )
!= $true ) )
=> ( ? [X1: $i] :
( ! [X2: $i] :
( ( X1 = X2 )
| ( ( sK0 @ X2 )
!= $true ) )
& ( $true
= ( sK0 @ X1 ) ) )
& ( ( exu
@ ^ [Y0: $i] : ( sK0 @ Y0 ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f191,plain,
( ? [X1: $i] :
( ! [X2: $i] :
( ( X1 = X2 )
| ( ( sK0 @ X2 )
!= $true ) )
& ( $true
= ( sK0 @ X1 ) ) )
=> ( ! [X2: $i] :
( ( sK1 = X2 )
| ( ( sK0 @ X2 )
!= $true ) )
& ( ( sK0 @ sK1 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f189,plain,
( ( prop2setI = $true )
& ( dsetconstrI = $true )
& ( emptyE1 = $true )
& ( setunionAx = $true )
& ( setoftrueEq = $true )
& ( nonemptyE1 = $true )
& ( dsetconstrER = $true )
& ( uniqinunit = $true )
& ( setbeta = $true )
& ( notdallE = $true )
& ( emptysetE = $true )
& ( omegaSAx = $true )
& ( wellorderingAx = $true )
& ( singletonsswitch = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( quantDeMorgan4 = $true )
& ( omegaIndAx = $true )
& ? [X0: $i > $o] :
( ? [X1: $i] :
( ! [X2: $i] :
( ( X1 = X2 )
| ( ( X0 @ X2 )
!= $true ) )
& ( ( X0 @ X1 )
= $true ) )
& ( ( exu
@ ^ [Y0: $i] : ( X0 @ Y0 ) )
!= $true ) )
& ( vacuousDall = $true )
& ( setadjoinIL = $true )
& ( upairsetIL = $true )
& ( exuE1 = $true )
& ( setadjoinIR = $true )
& ( emptyinPowerset = $true )
& ( exuE3e = $true )
& ( eqinunit = $true )
& ( setextAx = $true )
& ( replAx = $true )
& ( powersetAx = $true )
& ( setadjoinE = $true )
& ( omega0Ax = $true )
& ( powersetI = $true )
& ( notinsingleton = $true )
& ( nonemptyImpWitness = $true )
& ( emptysetimpfalse = $true )
& ( quantDeMorgan2 = $true )
& ( notinemptyset = $true )
& ( setadjoinOr = $true )
& ( exuE2 = $true )
& ( setunionE = $true )
& ( descrp = $true )
& ( nonemptyI = $true )
& ( nonemptyI1 = $true )
& ( quantDeMorgan3 = $true )
& ( emptyinunitempty = $true )
& ( setadjoinAx = $true )
& ( prop2setE = $true )
& ( quantDeMorgan1 = $true )
& ( upairsetE = $true )
& ( foundationAx = $true )
& ( prop2set2propI = $true )
& ( noeltsimpempty = $true )
& ( setext = $true )
& ( upairsetIR = $true )
& ( notdexE = $true )
& ( emptyI = $true )
& ( setunionI = $true )
& ( dsetconstrEL = $true )
& ( emptysetAx = $true )
& ( subPowSU = $true ) ),
inference(flattening,[status(thm)],[f188]) ).
thf(f188,plain,
( ? [X0: $i > $o] :
( ? [X1: $i] :
( ! [X2: $i] :
( ( X1 = X2 )
| ( ( X0 @ X2 )
!= $true ) )
& ( ( X0 @ X1 )
= $true ) )
& ( ( exu
@ ^ [Y0: $i] : ( X0 @ Y0 ) )
!= $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)],[f169]) ).
thf(f169,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 )
=> ! [X0: $i > $o] :
( ? [X1: $i] :
( ( ( X0 @ X1 )
= $true )
& ! [X2: $i] :
( ( ( X0 @ X2 )
= $true )
=> ( X1 = X2 ) ) )
=> ( ( exu
@ ^ [Y0: $i] : ( X0 @ Y0 ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f168]) ).
thf(f168,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
=> ! [X0: $i > $o] :
( ? [X1: $i] :
( ! [X2: $i] :
( ( X0 @ X2 )
=> ( X1 = X2 ) )
& ( X0 @ X1 ) )
=> ( exu
@ ^ [X3: $i] : ( X0 @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f64]) ).
thf(f64,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
=> ! [X0: $i > $o] :
( ? [X1: $i] :
( ! [X2: $i] :
( ( X0 @ X2 )
=> ( X1 = X2 ) )
& ( X0 @ X1 ) )
=> ( exu
@ ^ [X1: $i] : ( X0 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f63]) ).
thf(f63,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
=> ! [X0: $i > $o] :
( ? [X1: $i] :
( ! [X2: $i] :
( ( X0 @ X2 )
=> ( X1 = X2 ) )
& ( X0 @ X1 ) )
=> ( exu
@ ^ [X1: $i] : ( X0 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.ujh7G4tgox/DTF2THF_11924.p',exuI1) ).
thf(f794,plain,
! [X1: $i] :
( ( $false
= ( sK0 @ X1 ) )
| ( ( sK28 @ X1 )
!= X1 ) ),
inference(equality_proxy_clausification,[status(thm)],[f792]) ).
thf(f792,plain,
! [X1: $i] :
( ( ( X1
= ( sK28 @ X1 ) )
= $false )
| ( $false
= ( sK0 @ X1 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f791]) ).
thf(f791,plain,
! [X1: $i] :
( ( $false
= ( sK0 @ X1 ) )
| ( $false
= ( ( sK0 @ ( sK28 @ X1 ) )
=> ( X1
= ( sK28 @ X1 ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f790]) ).
thf(f790,plain,
! [X1: $i] :
( ( $false
= ( sK0 @ X1 ) )
| ( $false
= ( ^ [Y0: $i] :
( ( sK0 @ Y0 )
=> ( X1 = Y0 ) )
@ ( sK28 @ X1 ) ) ) ),
inference(sigma_clausification,[status(thm)],[f789]) ).
thf(f789,plain,
! [X1: $i] :
( ( $false
= ( !! @ $i
@ ^ [Y0: $i] :
( ( sK0 @ Y0 )
=> ( X1 = Y0 ) ) ) )
| ( $false
= ( sK0 @ X1 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f788]) ).
thf(f788,plain,
! [X1: $i] :
( $false
= ( ( sK0 @ X1 )
& ( !! @ $i
@ ^ [Y0: $i] :
( ( sK0 @ Y0 )
=> ( X1 = Y0 ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f787]) ).
thf(f787,plain,
! [X1: $i] :
( $false
= ( ^ [Y0: $i] :
( ( sK0 @ Y0 )
& ( !! @ $i
@ ^ [Y1: $i] :
( ( sK0 @ Y1 )
=> ( Y0 = Y1 ) ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f786]) ).
thf(f786,plain,
( ( ?? @ $i
@ ^ [Y0: $i] :
( ( sK0 @ Y0 )
& ( !! @ $i
@ ^ [Y1: $i] :
( ( sK0 @ Y1 )
=> ( Y0 = Y1 ) ) ) ) )
!= $true ),
inference(beta_eta_normalization,[status(thm)],[f326]) ).
thf(f326,plain,
( $true
!= ( ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( !! @ $i
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 = Y2 ) ) ) ) )
@ ^ [Y0: $i] : ( sK0 @ Y0 ) ) ),
inference(definition_unfolding,[status(thm)],[f249,f285]) ).
thf(f285,plain,
( exu
= ( ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( !! @ $i
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 = Y2 ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f83]) ).
thf(f83,plain,
( exu
= ( ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( !! @ $i
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 = Y2 ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f82]) ).
thf(f82,plain,
( ( ^ [X0: $i > $o] :
? [X1: $i] :
( ! [X2: $i] :
( ( X0 @ X2 )
=> ( X1 = X2 ) )
& ( X0 @ X1 ) ) )
= exu ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
( ( ^ [X0: $i > $o] :
? [X1: $i] :
( ! [X2: $i] :
( ( X0 @ X2 )
=> ( X1 = X2 ) )
& ( X0 @ X1 ) ) )
= exu ),
file('/export/starexec/sandbox2/tmp/tmp.ujh7G4tgox/DTF2THF_11924.p',exu) ).
thf(f249,plain,
( ( exu
@ ^ [Y0: $i] : ( sK0 @ Y0 ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f192]) ).
thf(f1131,plain,
( ( sK28 @ sK1 )
= sK1 ),
inference(trivial_inequality_removal,[status(thm)],[f1127]) ).
thf(f1127,plain,
( ( ( sK28 @ sK1 )
= sK1 )
| ( $false = $true ) ),
inference(superposition,[status(thm)],[f250,f1123]) ).
thf(f1123,plain,
! [X0: $i] :
( ( ( sK0 @ X0 )
= $false )
| ( ( sK28 @ X0 )
= sK1 ) ),
inference(trivial_inequality_removal,[status(thm)],[f1121]) ).
thf(f1121,plain,
! [X0: $i] :
( ( ( sK28 @ X0 )
= sK1 )
| ( ( sK0 @ X0 )
= $false )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f251,f793]) ).
thf(f793,plain,
! [X1: $i] :
( ( ( sK0 @ ( sK28 @ X1 ) )
= $true )
| ( $false
= ( sK0 @ X1 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f791]) ).
thf(f251,plain,
! [X2: $i] :
( ( ( sK0 @ X2 )
!= $true )
| ( sK1 = X2 ) ),
inference(cnf_transformation,[status(thm)],[f192]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : SEU544^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.10 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.11/0.30 % Computer : n001.cluster.edu
% 0.11/0.30 % Model : x86_64 x86_64
% 0.11/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.30 % Memory : 8042.1875MB
% 0.11/0.30 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.30 % CPULimit : 300
% 0.11/0.30 % WCLimit : 300
% 0.11/0.30 % DateTime : Tue Feb 24 12:27:05 EST 2026
% 0.11/0.30 % CPUTime :
% 0.11/0.30 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.39 ---- Original DTF file ---
% 0.17/0.39 thf(spec,logic,$$dhol).
% 0.17/0.39 %------------------------------------------------------------------------------
% 0.17/0.39 % File : SEU544^1 : TPTP v9.2.1. Released v3.7.0.
% 0.17/0.39 % Domain : Set Theory
% 0.17/0.39 % Problem : Preliminary Notions - Equivalence Laws
% 0.17/0.39 % Version : Especial.
% 0.17/0.39 % English : (! phi:i>o.(? x:i.phi x & (! y:i.phi y -> x = y)) ->
% 0.17/0.39 % exu (^ x:i.phi x))
% 0.17/0.39
% 0.17/0.39 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.17/0.39 % Source : [Bro08]
% 0.17/0.39 % Names : ZFC046g [Bro08]
% 0.17/0.39
% 0.17/0.39 % Status : Theorem
% 0.17/0.39 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.40 v8.2.0, 0.38 v8.1.0, 0.09 v7.5.0, 0.00 v7.4.0, 0.22 v7.2.0, 0.12 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.33 v4.0.1, 0.67 v4.0.0, 0.33 v3.7.0
% 0.17/0.39 % Syntax : Number of formulae : 135 ( 62 unt; 72 typ; 62 def)
% 0.17/0.39 % Number of atoms : 339 ( 82 equ; 0 cnn)
% 0.17/0.39 % Maximal formula atoms : 62 ( 5 avg)
% 0.17/0.39 % Number of connectives : 569 ( 21 ~; 5 |; 27 &; 360 @)
% 0.17/0.39 % ( 8 <=>; 148 =>; 0 <=; 0 <~>)
% 0.17/0.39 % Maximal formula depth : 68 ( 2 avg)
% 0.17/0.39 % Number of types : 2 ( 0 usr)
% 0.17/0.39 % Number of type conns : 37 ( 37 >; 0 *; 0 +; 0 <<)
% 0.17/0.39 % Number of symbols : 75 ( 72 usr; 64 con; 0-2 aty)
% 0.17/0.39 % Number of variables : 187 ( 16 ^; 145 !; 26 ?; 187 :)
% 0.17/0.39 % SPC : TH0_THM_EQU_NAR
% 0.17/0.39
% 0.17/0.39 % Comments : http://mathgate.info/detsetitem.php?id=408
% 0.17/0.39 %------------------------------------------------------------------------------
% 0.17/0.39 thf(in_type,type,
% 0.17/0.39 in: $i > $i > $o ).
% 0.17/0.39
% 0.17/0.39 thf(exu_type,type,
% 0.17/0.39 exu: ( $i > $o ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(exu,definition,
% 0.17/0.39 ( exu
% 0.17/0.39 = ( ^ [Xphi: $i > $o] :
% 0.17/0.39 ? [Xx: $i] :
% 0.17/0.39 ( ( Xphi @ Xx )
% 0.17/0.39 & ! [Xy: $i] :
% 0.17/0.39 ( ( Xphi @ Xy )
% 0.17/0.39 => ( Xx = Xy ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setextAx_type,type,
% 0.17/0.39 setextAx: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setextAx,definition,
% 0.17/0.39 ( setextAx
% 0.17/0.39 = ( ! [A: $i,B: $i] :
% 0.17/0.39 ( ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 <=> ( in @ Xx @ B ) )
% 0.17/0.39 => ( A = B ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(emptyset_type,type,
% 0.17/0.39 emptyset: $i ).
% 0.17/0.39
% 0.17/0.39 thf(emptysetAx_type,type,
% 0.17/0.39 emptysetAx: $o ).
% 0.17/0.39
% 0.17/0.39 thf(emptysetAx,definition,
% 0.17/0.39 ( emptysetAx
% 0.17/0.39 = ( ! [Xx: $i] :
% 0.17/0.39 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoin_type,type,
% 0.17/0.39 setadjoin: $i > $i > $i ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoinAx_type,type,
% 0.17/0.39 setadjoinAx: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoinAx,definition,
% 0.17/0.39 ( setadjoinAx
% 0.17/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.39 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.17/0.39 <=> ( ( Xy = Xx )
% 0.17/0.39 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(powerset_type,type,
% 0.17/0.39 powerset: $i > $i ).
% 0.17/0.39
% 0.17/0.39 thf(powersetAx_type,type,
% 0.17/0.39 powersetAx: $o ).
% 0.17/0.39
% 0.17/0.39 thf(powersetAx,definition,
% 0.17/0.39 ( powersetAx
% 0.17/0.39 = ( ! [A: $i,B: $i] :
% 0.17/0.39 ( ( in @ B @ ( powerset @ A ) )
% 0.17/0.39 <=> ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ B )
% 0.17/0.39 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setunion_type,type,
% 0.17/0.39 setunion: $i > $i ).
% 0.17/0.39
% 0.17/0.39 thf(setunionAx_type,type,
% 0.17/0.39 setunionAx: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setunionAx,definition,
% 0.17/0.39 ( setunionAx
% 0.17/0.39 = ( ! [A: $i,Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ ( setunion @ A ) )
% 0.17/0.39 <=> ? [B: $i] :
% 0.17/0.39 ( ( in @ Xx @ B )
% 0.17/0.39 & ( in @ B @ A ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(omega_type,type,
% 0.17/0.39 omega: $i ).
% 0.17/0.39
% 0.17/0.39 thf(omega0Ax_type,type,
% 0.17/0.39 omega0Ax: $o ).
% 0.17/0.39
% 0.17/0.39 thf(omega0Ax,definition,
% 0.17/0.39 ( omega0Ax
% 0.17/0.39 = ( in @ emptyset @ omega ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(omegaSAx_type,type,
% 0.17/0.39 omegaSAx: $o ).
% 0.17/0.39
% 0.17/0.39 thf(omegaSAx,definition,
% 0.17/0.39 ( omegaSAx
% 0.17/0.39 = ( ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ omega )
% 0.17/0.39 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(omegaIndAx_type,type,
% 0.17/0.39 omegaIndAx: $o ).
% 0.17/0.39
% 0.17/0.39 thf(omegaIndAx,definition,
% 0.17/0.39 ( omegaIndAx
% 0.17/0.39 = ( ! [A: $i] :
% 0.17/0.39 ( ( ( in @ emptyset @ A )
% 0.17/0.39 & ! [Xx: $i] :
% 0.17/0.39 ( ( ( in @ Xx @ omega )
% 0.17/0.39 & ( in @ Xx @ A ) )
% 0.17/0.39 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.17/0.39 => ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ omega )
% 0.17/0.39 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(replAx_type,type,
% 0.17/0.39 replAx: $o ).
% 0.17/0.39
% 0.17/0.39 thf(replAx,definition,
% 0.17/0.39 ( replAx
% 0.17/0.39 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.17/0.39 ( ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ( exu
% 0.17/0.39 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.17/0.39 => ? [B: $i] :
% 0.17/0.39 ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ B )
% 0.17/0.39 <=> ? [Xy: $i] :
% 0.17/0.39 ( ( in @ Xy @ A )
% 0.17/0.39 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(foundationAx_type,type,
% 0.17/0.39 foundationAx: $o ).
% 0.17/0.39
% 0.17/0.39 thf(foundationAx,definition,
% 0.17/0.39 ( foundationAx
% 0.17/0.39 = ( ! [A: $i] :
% 0.17/0.39 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.17/0.39 => ? [B: $i] :
% 0.17/0.39 ( ( in @ B @ A )
% 0.17/0.39 & ~ ? [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ B )
% 0.17/0.39 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(wellorderingAx_type,type,
% 0.17/0.39 wellorderingAx: $o ).
% 0.17/0.39
% 0.17/0.39 thf(wellorderingAx,definition,
% 0.17/0.39 ( wellorderingAx
% 0.17/0.39 = ( ! [A: $i] :
% 0.17/0.39 ? [B: $i] :
% 0.17/0.39 ( ! [C: $i] :
% 0.17/0.39 ( ( in @ C @ B )
% 0.17/0.39 => ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ C )
% 0.17/0.39 => ( in @ Xx @ A ) ) )
% 0.17/0.39 & ! [Xx: $i,Xy: $i] :
% 0.17/0.39 ( ( ( in @ Xx @ A )
% 0.17/0.39 & ( in @ Xy @ A ) )
% 0.17/0.39 => ( ! [C: $i] :
% 0.17/0.39 ( ( in @ C @ B )
% 0.17/0.39 => ( ( in @ Xx @ C )
% 0.17/0.39 <=> ( in @ Xy @ C ) ) )
% 0.17/0.39 => ( Xx = Xy ) ) )
% 0.17/0.39 & ! [C: $i,D: $i] :
% 0.17/0.39 ( ( ( in @ C @ B )
% 0.17/0.39 & ( in @ D @ B ) )
% 0.17/0.39 => ( ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ C )
% 0.17/0.39 => ( in @ Xx @ D ) )
% 0.17/0.39 | ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ D )
% 0.17/0.39 => ( in @ Xx @ C ) ) ) )
% 0.17/0.39 & ! [C: $i] :
% 0.17/0.39 ( ( ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ C )
% 0.17/0.39 => ( in @ Xx @ A ) )
% 0.17/0.39 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.17/0.39 => ? [D: $i,Xx: $i] :
% 0.17/0.39 ( ( in @ D @ B )
% 0.17/0.39 & ( in @ Xx @ C )
% 0.17/0.39 & ~ ? [Xy: $i] :
% 0.17/0.39 ( ( in @ Xy @ D )
% 0.17/0.39 & ( in @ Xy @ C ) )
% 0.17/0.39 & ! [E: $i] :
% 0.17/0.39 ( ( in @ E @ B )
% 0.17/0.39 => ( ! [Xy: $i] :
% 0.17/0.39 ( ( in @ Xy @ E )
% 0.17/0.39 => ( in @ Xy @ D ) )
% 0.17/0.39 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(descr_type,type,
% 0.17/0.39 descr: ( $i > $o ) > $i ).
% 0.17/0.39
% 0.17/0.39 thf(descrp_type,type,
% 0.17/0.39 descrp: $o ).
% 0.17/0.39
% 0.17/0.39 thf(descrp,definition,
% 0.17/0.39 ( descrp
% 0.17/0.39 = ( ! [Xphi: $i > $o] :
% 0.17/0.39 ( ( exu
% 0.17/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.39 => ( Xphi
% 0.17/0.39 @ ( descr
% 0.17/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(dsetconstr_type,type,
% 0.17/0.39 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.17/0.39
% 0.17/0.39 thf(dsetconstrI_type,type,
% 0.17/0.39 dsetconstrI: $o ).
% 0.17/0.39
% 0.17/0.39 thf(dsetconstrI,definition,
% 0.17/0.39 ( dsetconstrI
% 0.17/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ( ( Xphi @ Xx )
% 0.17/0.39 => ( in @ Xx
% 0.17/0.39 @ ( dsetconstr @ A
% 0.17/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(dsetconstrEL_type,type,
% 0.17/0.39 dsetconstrEL: $o ).
% 0.17/0.39
% 0.17/0.39 thf(dsetconstrEL,definition,
% 0.17/0.39 ( dsetconstrEL
% 0.17/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.39 ( ( in @ Xx
% 0.17/0.39 @ ( dsetconstr @ A
% 0.17/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.17/0.39 => ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(dsetconstrER_type,type,
% 0.17/0.39 dsetconstrER: $o ).
% 0.17/0.39
% 0.17/0.39 thf(dsetconstrER,definition,
% 0.17/0.39 ( dsetconstrER
% 0.17/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.39 ( ( in @ Xx
% 0.17/0.39 @ ( dsetconstr @ A
% 0.17/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.17/0.39 => ( Xphi @ Xx ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(exuE1_type,type,
% 0.17/0.39 exuE1: $o ).
% 0.17/0.39
% 0.17/0.39 thf(exuE1,definition,
% 0.17/0.39 ( exuE1
% 0.17/0.39 = ( ! [Xphi: $i > $o] :
% 0.17/0.39 ( ( exu
% 0.17/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.39 => ? [Xx: $i] :
% 0.17/0.39 ( ( Xphi @ Xx )
% 0.17/0.39 & ! [Xy: $i] :
% 0.17/0.39 ( ( Xphi @ Xy )
% 0.17/0.39 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(prop2set_type,type,
% 0.17/0.39 prop2set: $o > $i ).
% 0.17/0.39
% 0.17/0.39 thf(prop2setE_type,type,
% 0.17/0.39 prop2setE: $o ).
% 0.17/0.39
% 0.17/0.39 thf(prop2setE,definition,
% 0.17/0.39 ( prop2setE
% 0.17/0.39 = ( ! [Xphi: $o,Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.17/0.39 => Xphi ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(emptysetE_type,type,
% 0.17/0.39 emptysetE: $o ).
% 0.17/0.39
% 0.17/0.39 thf(emptysetE,definition,
% 0.17/0.39 ( emptysetE
% 0.17/0.39 = ( ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ emptyset )
% 0.17/0.39 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(emptysetimpfalse_type,type,
% 0.17/0.39 emptysetimpfalse: $o ).
% 0.17/0.39
% 0.17/0.39 thf(emptysetimpfalse,definition,
% 0.17/0.39 ( emptysetimpfalse
% 0.17/0.39 = ( ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ emptyset )
% 0.17/0.39 => $false ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(notinemptyset_type,type,
% 0.17/0.39 notinemptyset: $o ).
% 0.17/0.39
% 0.17/0.39 thf(notinemptyset,definition,
% 0.17/0.39 ( notinemptyset
% 0.17/0.39 = ( ! [Xx: $i] :
% 0.17/0.39 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(exuE3e_type,type,
% 0.17/0.39 exuE3e: $o ).
% 0.17/0.39
% 0.17/0.39 thf(exuE3e,definition,
% 0.17/0.39 ( exuE3e
% 0.17/0.39 = ( ! [Xphi: $i > $o] :
% 0.17/0.39 ( ( exu
% 0.17/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.39 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setext_type,type,
% 0.17/0.39 setext: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setext,definition,
% 0.17/0.39 ( setext
% 0.17/0.39 = ( ! [A: $i,B: $i] :
% 0.17/0.39 ( ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ( in @ Xx @ B ) )
% 0.17/0.39 => ( ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ B )
% 0.17/0.39 => ( in @ Xx @ A ) )
% 0.17/0.39 => ( A = B ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(emptyI_type,type,
% 0.17/0.39 emptyI: $o ).
% 0.17/0.39
% 0.17/0.39 thf(emptyI,definition,
% 0.17/0.39 ( emptyI
% 0.17/0.39 = ( ! [A: $i] :
% 0.17/0.39 ( ! [Xx: $i] :
% 0.17/0.39 ~ ( in @ Xx @ A )
% 0.17/0.39 => ( A = emptyset ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(noeltsimpempty_type,type,
% 0.17/0.39 noeltsimpempty: $o ).
% 0.17/0.39
% 0.17/0.39 thf(noeltsimpempty,definition,
% 0.17/0.39 ( noeltsimpempty
% 0.17/0.39 = ( ! [A: $i] :
% 0.17/0.39 ( ! [Xx: $i] :
% 0.17/0.39 ~ ( in @ Xx @ A )
% 0.17/0.39 => ( A = emptyset ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setbeta_type,type,
% 0.17/0.39 setbeta: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setbeta,definition,
% 0.17/0.39 ( setbeta
% 0.17/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ( ( in @ Xx
% 0.17/0.39 @ ( dsetconstr @ A
% 0.17/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.17/0.39 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(nonempty_type,type,
% 0.17/0.39 nonempty: $i > $o ).
% 0.17/0.39
% 0.17/0.39 thf(nonempty,definition,
% 0.17/0.39 ( nonempty
% 0.17/0.39 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(nonemptyE1_type,type,
% 0.17/0.39 nonemptyE1: $o ).
% 0.17/0.39
% 0.17/0.39 thf(nonemptyE1,definition,
% 0.17/0.39 ( nonemptyE1
% 0.17/0.39 = ( ! [A: $i] :
% 0.17/0.39 ( ( nonempty @ A )
% 0.17/0.39 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(nonemptyI_type,type,
% 0.17/0.39 nonemptyI: $o ).
% 0.17/0.39
% 0.17/0.39 thf(nonemptyI,definition,
% 0.17/0.39 ( nonemptyI
% 0.17/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ( ( Xphi @ Xx )
% 0.17/0.39 => ( nonempty
% 0.17/0.39 @ ( dsetconstr @ A
% 0.17/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(nonemptyI1_type,type,
% 0.17/0.39 nonemptyI1: $o ).
% 0.17/0.39
% 0.17/0.39 thf(nonemptyI1,definition,
% 0.17/0.39 ( nonemptyI1
% 0.17/0.39 = ( ! [A: $i] :
% 0.17/0.39 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.17/0.39 => ( nonempty @ A ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoinIL_type,type,
% 0.17/0.39 setadjoinIL: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoinIL,definition,
% 0.17/0.39 ( setadjoinIL
% 0.17/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(emptyinunitempty_type,type,
% 0.17/0.39 emptyinunitempty: $o ).
% 0.17/0.39
% 0.17/0.39 thf(emptyinunitempty,definition,
% 0.17/0.39 ( emptyinunitempty
% 0.17/0.39 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoinIR_type,type,
% 0.17/0.39 setadjoinIR: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoinIR,definition,
% 0.17/0.39 ( setadjoinIR
% 0.17/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.39 ( ( in @ Xy @ A )
% 0.17/0.39 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoinE_type,type,
% 0.17/0.39 setadjoinE: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoinE,definition,
% 0.17/0.39 ( setadjoinE
% 0.17/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.39 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.17/0.39 => ! [Xphi: $o] :
% 0.17/0.39 ( ( ( Xy = Xx )
% 0.17/0.39 => Xphi )
% 0.17/0.39 => ( ( ( in @ Xy @ A )
% 0.17/0.39 => Xphi )
% 0.17/0.39 => Xphi ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoinOr_type,type,
% 0.17/0.39 setadjoinOr: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoinOr,definition,
% 0.17/0.39 ( setadjoinOr
% 0.17/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.39 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.17/0.39 => ( ( Xy = Xx )
% 0.17/0.39 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setoftrueEq_type,type,
% 0.17/0.39 setoftrueEq: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setoftrueEq,definition,
% 0.17/0.39 ( setoftrueEq
% 0.17/0.39 = ( ! [A: $i] :
% 0.17/0.39 ( ( dsetconstr @ A
% 0.17/0.39 @ ^ [Xx: $i] : $true )
% 0.17/0.39 = A ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(powersetI_type,type,
% 0.17/0.39 powersetI: $o ).
% 0.17/0.39
% 0.17/0.39 thf(powersetI,definition,
% 0.17/0.39 ( powersetI
% 0.17/0.39 = ( ! [A: $i,B: $i] :
% 0.17/0.39 ( ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ B )
% 0.17/0.39 => ( in @ Xx @ A ) )
% 0.17/0.39 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(emptyinPowerset_type,type,
% 0.17/0.39 emptyinPowerset: $o ).
% 0.17/0.39
% 0.17/0.39 thf(emptyinPowerset,definition,
% 0.17/0.39 ( emptyinPowerset
% 0.17/0.39 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(emptyInPowerset_type,type,
% 0.17/0.39 emptyInPowerset: $o ).
% 0.17/0.39
% 0.17/0.39 thf(emptyInPowerset,definition,
% 0.17/0.39 ( emptyInPowerset
% 0.17/0.39 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(powersetE_type,type,
% 0.17/0.39 powersetE: $o ).
% 0.17/0.39
% 0.17/0.39 thf(powersetE,definition,
% 0.17/0.39 ( powersetE
% 0.17/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.39 ( ( in @ B @ ( powerset @ A ) )
% 0.17/0.39 => ( ( in @ Xx @ B )
% 0.17/0.39 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setunionI_type,type,
% 0.17/0.39 setunionI: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setunionI,definition,
% 0.17/0.39 ( setunionI
% 0.17/0.39 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.17/0.39 ( ( in @ Xx @ B )
% 0.17/0.39 => ( ( in @ B @ A )
% 0.17/0.39 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setunionE_type,type,
% 0.17/0.39 setunionE: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setunionE,definition,
% 0.17/0.39 ( setunionE
% 0.17/0.39 = ( ! [A: $i,Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ ( setunion @ A ) )
% 0.17/0.39 => ! [Xphi: $o] :
% 0.17/0.39 ( ! [B: $i] :
% 0.17/0.39 ( ( in @ Xx @ B )
% 0.17/0.39 => ( ( in @ B @ A )
% 0.17/0.39 => Xphi ) )
% 0.17/0.39 => Xphi ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(subPowSU_type,type,
% 0.17/0.39 subPowSU: $o ).
% 0.17/0.39
% 0.17/0.39 thf(subPowSU,definition,
% 0.17/0.39 ( subPowSU
% 0.17/0.39 = ( ! [A: $i,Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(exuE2_type,type,
% 0.17/0.39 exuE2: $o ).
% 0.17/0.39
% 0.17/0.39 thf(exuE2,definition,
% 0.17/0.39 ( exuE2
% 0.17/0.39 = ( ! [Xphi: $i > $o] :
% 0.17/0.39 ( ( exu
% 0.17/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.39 => ? [Xx: $i] :
% 0.17/0.39 ! [Xy: $i] :
% 0.17/0.39 ( ( Xphi @ Xy )
% 0.17/0.39 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(nonemptyImpWitness_type,type,
% 0.17/0.39 nonemptyImpWitness: $o ).
% 0.17/0.39
% 0.17/0.39 thf(nonemptyImpWitness,definition,
% 0.17/0.39 ( nonemptyImpWitness
% 0.17/0.39 = ( ! [A: $i] :
% 0.17/0.39 ( ( nonempty @ A )
% 0.17/0.39 => ? [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 & $true ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(uniqinunit_type,type,
% 0.17/0.39 uniqinunit: $o ).
% 0.17/0.39
% 0.17/0.39 thf(uniqinunit,definition,
% 0.17/0.39 ( uniqinunit
% 0.17/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.39 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.39 => ( Xx = Xy ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(notinsingleton_type,type,
% 0.17/0.39 notinsingleton: $o ).
% 0.17/0.39
% 0.17/0.39 thf(notinsingleton,definition,
% 0.17/0.39 ( notinsingleton
% 0.17/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.39 ( ( Xx != Xy )
% 0.17/0.39 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(eqinunit_type,type,
% 0.17/0.39 eqinunit: $o ).
% 0.17/0.39
% 0.17/0.39 thf(eqinunit,definition,
% 0.17/0.39 ( eqinunit
% 0.17/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.39 ( ( Xx = Xy )
% 0.17/0.39 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(singletonsswitch_type,type,
% 0.17/0.39 singletonsswitch: $o ).
% 0.17/0.39
% 0.17/0.39 thf(singletonsswitch,definition,
% 0.17/0.39 ( singletonsswitch
% 0.17/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.39 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.39 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(upairsetE_type,type,
% 0.17/0.39 upairsetE: $o ).
% 0.17/0.39
% 0.17/0.39 thf(upairsetE,definition,
% 0.17/0.39 ( upairsetE
% 0.17/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.17/0.39 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.17/0.39 => ( ( Xz = Xx )
% 0.17/0.39 | ( Xz = Xy ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(upairsetIL_type,type,
% 0.17/0.39 upairsetIL: $o ).
% 0.17/0.39
% 0.17/0.39 thf(upairsetIL,definition,
% 0.17/0.39 ( upairsetIL
% 0.17/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(upairsetIR_type,type,
% 0.17/0.39 upairsetIR: $o ).
% 0.17/0.39
% 0.17/0.39 thf(upairsetIR,definition,
% 0.17/0.39 ( upairsetIR
% 0.17/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(emptyE1_type,type,
% 0.17/0.39 emptyE1: $o ).
% 0.17/0.39
% 0.17/0.39 thf(emptyE1,definition,
% 0.17/0.39 ( emptyE1
% 0.17/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.39 ( ? [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 & ( Xphi @ Xx ) )
% 0.17/0.39 => ( ( ( dsetconstr @ A
% 0.17/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.39 = emptyset )
% 0.17/0.39 => $false ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(vacuousDall_type,type,
% 0.17/0.39 vacuousDall: $o ).
% 0.17/0.39
% 0.17/0.39 thf(vacuousDall,definition,
% 0.17/0.39 ( vacuousDall
% 0.17/0.39 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ emptyset )
% 0.17/0.39 => ( Xphi @ Xx ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(quantDeMorgan1_type,type,
% 0.17/0.39 quantDeMorgan1: $o ).
% 0.17/0.39
% 0.17/0.39 thf(quantDeMorgan1,definition,
% 0.17/0.39 ( quantDeMorgan1
% 0.17/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.39 ( ~ ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ( Xphi @ Xx ) )
% 0.17/0.39 => ? [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(quantDeMorgan2_type,type,
% 0.17/0.39 quantDeMorgan2: $o ).
% 0.17/0.39
% 0.17/0.39 thf(quantDeMorgan2,definition,
% 0.17/0.39 ( quantDeMorgan2
% 0.17/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.39 ( ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ~ ( Xphi @ Xx ) )
% 0.17/0.39 => ~ ? [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(quantDeMorgan3_type,type,
% 0.17/0.39 quantDeMorgan3: $o ).
% 0.17/0.39
% 0.17/0.39 thf(quantDeMorgan3,definition,
% 0.17/0.39 ( quantDeMorgan3
% 0.17/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.39 ( ~ ? [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 & ( Xphi @ Xx ) )
% 0.17/0.39 => ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(quantDeMorgan4_type,type,
% 0.17/0.39 quantDeMorgan4: $o ).
% 0.17/0.39
% 0.17/0.39 thf(quantDeMorgan4,definition,
% 0.17/0.39 ( quantDeMorgan4
% 0.17/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.39 ( ? [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 & ~ ( Xphi @ Xx ) )
% 0.17/0.39 => ~ ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(prop2setI_type,type,
% 0.17/0.39 prop2setI: $o ).
% 0.17/0.39
% 0.17/0.39 thf(prop2setI,definition,
% 0.17/0.39 ( prop2setI
% 0.17/0.39 = ( ! [Xphi: $o] :
% 0.17/0.39 ( Xphi
% 0.17/0.39 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(set2prop_type,type,
% 0.17/0.39 set2prop: $i > $o ).
% 0.17/0.39
% 0.17/0.39 thf(prop2set2propI_type,type,
% 0.17/0.39 prop2set2propI: $o ).
% 0.17/0.39
% 0.17/0.39 thf(prop2set2propI,definition,
% 0.17/0.39 ( prop2set2propI
% 0.17/0.39 = ( ! [Xphi: $o] :
% 0.17/0.39 ( Xphi
% 0.17/0.39 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(notdexE_type,type,
% 0.17/0.39 notdexE: $o ).
% 0.17/0.39
% 0.17/0.39 thf(notdexE,definition,
% 0.17/0.39 ( notdexE
% 0.17/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.39 ( ~ ? [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 & ( Xphi @ Xx ) )
% 0.17/0.39 => ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(notdallE_type,type,
% 0.17/0.39 notdallE: $o ).
% 0.17/0.39
% 0.17/0.39 thf(notdallE,definition,
% 0.17/0.39 ( notdallE
% 0.17/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.39 ( ~ ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ( Xphi @ Xx ) )
% 0.17/0.39 => ? [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(exuI1,conjecture,
% 0.17/0.39 ( setextAx
% 0.17/0.39 => ( emptysetAx
% 0.17/0.39 => ( setadjoinAx
% 0.17/0.39 => ( powersetAx
% 0.17/0.39 => ( setunionAx
% 0.17/0.39 => ( omega0Ax
% 0.17/0.39 => ( omegaSAx
% 0.17/0.39 => ( omegaIndAx
% 0.17/0.39 => ( replAx
% 0.17/0.39 => ( foundationAx
% 0.17/0.39 => ( wellorderingAx
% 0.17/0.39 => ( descrp
% 0.17/0.39 => ( dsetconstrI
% 0.17/0.39 => ( dsetconstrEL
% 0.17/0.39 => ( dsetconstrER
% 0.17/0.39 => ( exuE1
% 0.17/0.39 => ( prop2setE
% 0.17/0.39 => ( emptysetE
% 0.17/0.39 => ( emptysetimpfalse
% 0.17/0.39 => ( notinemptyset
% 0.17/0.39 => ( exuE3e
% 0.17/0.39 => ( setext
% 0.17/0.39 => ( emptyI
% 0.17/0.39 => ( noeltsimpempty
% 0.17/0.39 => ( setbeta
% 0.17/0.39 => ( nonemptyE1
% 0.17/0.39 => ( nonemptyI
% 0.17/0.39 => ( nonemptyI1
% 0.17/0.39 => ( setadjoinIL
% 0.17/0.39 => ( emptyinunitempty
% 0.17/0.39 => ( setadjoinIR
% 0.17/0.39 => ( setadjoinE
% 0.17/0.39 => ( setadjoinOr
% 0.17/0.39 => ( setoftrueEq
% 0.17/0.39 => ( powersetI
% 0.17/0.39 => ( emptyinPowerset
% 0.17/0.39 => ( emptyInPowerset
% 0.17/0.39 => ( powersetE
% 0.17/0.39 => ( setunionI
% 0.17/0.39 => ( setunionE
% 0.17/0.39 => ( subPowSU
% 0.17/0.39 => ( exuE2
% 0.17/0.39 => ( nonemptyImpWitness
% 0.17/0.39 => ( uniqinunit
% 0.17/0.39 => ( notinsingleton
% 0.17/0.39 => ( eqinunit
% 0.17/0.39 => ( singletonsswitch
% 0.17/0.39 => ( upairsetE
% 0.17/0.39 => ( upairsetIL
% 0.17/0.39 => ( upairsetIR
% 0.17/0.39 => ( emptyE1
% 0.17/0.39 => ( vacuousDall
% 0.17/0.39 => ( quantDeMorgan1
% 0.17/0.39 => ( quantDeMorgan2
% 0.17/0.39 => ( quantDeMorgan3
% 0.17/0.39 => ( quantDeMorgan4
% 0.17/0.39 => ( prop2setI
% 0.17/0.39 => ( prop2set2propI
% 0.17/0.39 => ( notdexE
% 0.17/0.39 => ( notdallE
% 0.17/0.39 => ! [Xphi: $i > $o] :
% 0.17/0.39 ( ? [Xx: $i] :
% 0.17/0.39 ( ( Xphi @ Xx )
% 0.17/0.39 & ! [Xy: $i] :
% 0.17/0.39 ( ( Xphi @ Xy )
% 0.17/0.39 => ( Xx = Xy ) ) )
% 0.17/0.39 => ( exu
% 0.17/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 %------------------------------------------------------------------------------
% 0.17/0.39 ------------------------
% 1.99/1.44 ---- Embedded in THF ---
% 1.99/1.44 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.99/1.44 %%% Generated on Tue Feb 24 12:27:06 EST 2026
% 1.99/1.44 %%% using '$$dhol' embedding, version 1.3.0.
% 1.99/1.44 %%% Logic specification used:
% 1.99/1.44 %%% thf(spec, logic, $$dhol).
% 1.99/1.44
% 1.99/1.44 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.ujh7G4tgox/DTF2DTF11924.p
% See solution above
% 1.99/1.44 ------------------------
% 1.99/1.45 ---- Cleaned THF ---
% 1.99/1.45 thf(in_type,type,
% 1.99/1.45 in: $i > $i > $o ).
% 1.99/1.45
% 1.99/1.45 thf(exu_type,type,
% 1.99/1.45 exu: ( $i > $o ) > $o ).
% 1.99/1.45
% 1.99/1.45 thf(exu,definition,
% 1.99/1.45 ( exu
% 1.99/1.45 = ( ^ [Xphi: $i > $o] :
% 1.99/1.45 ? [Xx: $i] :
% 1.99/1.45 ( ( Xphi @ Xx )
% 1.99/1.45 & ! [Xy: $i] :
% 1.99/1.45 ( ( Xphi @ Xy )
% 1.99/1.45 => ( Xx = Xy ) ) ) ) ) ).
% 1.99/1.45
% 1.99/1.45 thf(setextAx_type,type,
% 1.99/1.45 setextAx: $o ).
% 1.99/1.45
% 1.99/1.45 thf(setextAx,definition,
% 1.99/1.45 ( setextAx
% 1.99/1.45 = ( ! [A: $i,B: $i] :
% 1.99/1.45 ( ! [Xx: $i] :
% 1.99/1.45 ( ( in @ Xx @ A )
% 1.99/1.45 <=> ( in @ Xx @ B ) )
% 1.99/1.45 => ( A = B ) ) ) ) ).
% 1.99/1.45
% 1.99/1.45 thf(emptyset_type,type,
% 1.99/1.45 emptyset: $i ).
% 1.99/1.45
% 1.99/1.45 thf(emptysetAx_type,type,
% 1.99/1.45 emptysetAx: $o ).
% 1.99/1.45
% 1.99/1.45 thf(emptysetAx,definition,
% 1.99/1.45 ( emptysetAx
% 1.99/1.45 = ( ! [Xx: $i] :
% 1.99/1.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 1.99/1.45
% 1.99/1.45 thf(setadjoin_type,type,
% 1.99/1.45 setadjoin: $i > $i > $i ).
% 1.99/1.45
% 1.99/1.45 thf(setadjoinAx_type,type,
% 1.99/1.45 setadjoinAx: $o ).
% 1.99/1.45
% 1.99/1.45 thf(setadjoinAx,definition,
% 1.99/1.45 ( setadjoinAx
% 1.99/1.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.99/1.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.99/1.45 <=> ( ( Xy = Xx )
% 1.99/1.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 1.99/1.45
% 1.99/1.45 thf(powerset_type,type,
% 1.99/1.45 powerset: $i > $i ).
% 1.99/1.45
% 1.99/1.45 thf(powersetAx_type,type,
% 1.99/1.45 powersetAx: $o ).
% 1.99/1.45
% 1.99/1.45 thf(powersetAx,definition,
% 1.99/1.45 ( powersetAx
% 1.99/1.45 = ( ! [A: $i,B: $i] :
% 1.99/1.45 ( ( in @ B @ ( powerset @ A ) )
% 1.99/1.45 <=> ! [Xx: $i] :
% 1.99/1.45 ( ( in @ Xx @ B )
% 1.99/1.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.99/1.45
% 1.99/1.45 thf(setunion_type,type,
% 1.99/1.45 setunion: $i > $i ).
% 1.99/1.45
% 1.99/1.45 thf(setunionAx_type,type,
% 1.99/1.45 setunionAx: $o ).
% 1.99/1.45
% 1.99/1.45 thf(setunionAx,definition,
% 1.99/1.45 ( setunionAx
% 1.99/1.45 = ( ! [A: $i,Xx: $i] :
% 1.99/1.45 ( ( in @ Xx @ ( setunion @ A ) )
% 1.99/1.45 <=> ? [B: $i] :
% 1.99/1.45 ( ( in @ Xx @ B )
% 1.99/1.45 & ( in @ B @ A ) ) ) ) ) ).
% 1.99/1.45
% 1.99/1.45 thf(omega_type,type,
% 1.99/1.45 omega: $i ).
% 1.99/1.45
% 1.99/1.45 thf(omega0Ax_type,type,
% 1.99/1.45 omega0Ax: $o ).
% 1.99/1.45
% 1.99/1.45 thf(omega0Ax,definition,
% 1.99/1.45 ( omega0Ax
% 1.99/1.45 = ( in @ emptyset @ omega ) ) ).
% 1.99/1.45
% 1.99/1.45 thf(omegaSAx_type,type,
% 1.99/1.45 omegaSAx: $o ).
% 1.99/1.45
% 1.99/1.45 thf(omegaSAx,definition,
% 1.99/1.45 ( omegaSAx
% 1.99/1.45 = ( ! [Xx: $i] :
% 1.99/1.45 ( ( in @ Xx @ omega )
% 1.99/1.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 1.99/1.45
% 1.99/1.45 thf(omegaIndAx_type,type,
% 1.99/1.45 omegaIndAx: $o ).
% 1.99/1.46
% 1.99/1.46 thf(omegaIndAx,definition,
% 1.99/1.46 ( omegaIndAx
% 1.99/1.46 = ( ! [A: $i] :
% 1.99/1.46 ( ( ( in @ emptyset @ A )
% 1.99/1.46 & ! [Xx: $i] :
% 1.99/1.46 ( ( ( in @ Xx @ omega )
% 1.99/1.46 & ( in @ Xx @ A ) )
% 1.99/1.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 1.99/1.46 => ! [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ omega )
% 1.99/1.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(replAx_type,type,
% 1.99/1.46 replAx: $o ).
% 1.99/1.46
% 1.99/1.46 thf(replAx,definition,
% 1.99/1.46 ( replAx
% 1.99/1.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 1.99/1.46 ( ! [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ A )
% 1.99/1.46 => ( exu
% 1.99/1.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 1.99/1.46 => ? [B: $i] :
% 1.99/1.46 ! [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ B )
% 1.99/1.46 <=> ? [Xy: $i] :
% 1.99/1.46 ( ( in @ Xy @ A )
% 1.99/1.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(foundationAx_type,type,
% 1.99/1.46 foundationAx: $o ).
% 1.99/1.46
% 1.99/1.46 thf(foundationAx,definition,
% 1.99/1.46 ( foundationAx
% 1.99/1.46 = ( ! [A: $i] :
% 1.99/1.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 1.99/1.46 => ? [B: $i] :
% 1.99/1.46 ( ( in @ B @ A )
% 1.99/1.46 & ~ ? [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ B )
% 1.99/1.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(wellorderingAx_type,type,
% 1.99/1.46 wellorderingAx: $o ).
% 1.99/1.46
% 1.99/1.46 thf(wellorderingAx,definition,
% 1.99/1.46 ( wellorderingAx
% 1.99/1.46 = ( ! [A: $i] :
% 1.99/1.46 ? [B: $i] :
% 1.99/1.46 ( ! [C: $i] :
% 1.99/1.46 ( ( in @ C @ B )
% 1.99/1.46 => ! [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ C )
% 1.99/1.46 => ( in @ Xx @ A ) ) )
% 1.99/1.46 & ! [Xx: $i,Xy: $i] :
% 1.99/1.46 ( ( ( in @ Xx @ A )
% 1.99/1.46 & ( in @ Xy @ A ) )
% 1.99/1.46 => ( ! [C: $i] :
% 1.99/1.46 ( ( in @ C @ B )
% 1.99/1.46 => ( ( in @ Xx @ C )
% 1.99/1.46 <=> ( in @ Xy @ C ) ) )
% 1.99/1.46 => ( Xx = Xy ) ) )
% 1.99/1.46 & ! [C: $i,D: $i] :
% 1.99/1.46 ( ( ( in @ C @ B )
% 1.99/1.46 & ( in @ D @ B ) )
% 1.99/1.46 => ( ! [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ C )
% 1.99/1.46 => ( in @ Xx @ D ) )
% 1.99/1.46 | ! [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ D )
% 1.99/1.46 => ( in @ Xx @ C ) ) ) )
% 1.99/1.46 & ! [C: $i] :
% 1.99/1.46 ( ( ! [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ C )
% 1.99/1.46 => ( in @ Xx @ A ) )
% 1.99/1.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 1.99/1.46 => ? [D: $i,Xx: $i] :
% 1.99/1.46 ( ( in @ D @ B )
% 1.99/1.46 & ( in @ Xx @ C )
% 1.99/1.46 & ~ ? [Xy: $i] :
% 1.99/1.46 ( ( in @ Xy @ D )
% 1.99/1.46 & ( in @ Xy @ C ) )
% 1.99/1.46 & ! [E: $i] :
% 1.99/1.46 ( ( in @ E @ B )
% 1.99/1.46 => ( ! [Xy: $i] :
% 1.99/1.46 ( ( in @ Xy @ E )
% 1.99/1.46 => ( in @ Xy @ D ) )
% 1.99/1.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(descr_type,type,
% 1.99/1.46 descr: ( $i > $o ) > $i ).
% 1.99/1.46
% 1.99/1.46 thf(descrp_type,type,
% 1.99/1.46 descrp: $o ).
% 1.99/1.46
% 1.99/1.46 thf(descrp,definition,
% 1.99/1.46 ( descrp
% 1.99/1.46 = ( ! [Xphi: $i > $o] :
% 1.99/1.46 ( ( exu
% 1.99/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.99/1.46 => ( Xphi
% 1.99/1.46 @ ( descr
% 1.99/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(dsetconstr_type,type,
% 1.99/1.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 1.99/1.46
% 1.99/1.46 thf(dsetconstrI_type,type,
% 1.99/1.46 dsetconstrI: $o ).
% 1.99/1.46
% 1.99/1.46 thf(dsetconstrI,definition,
% 1.99/1.46 ( dsetconstrI
% 1.99/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ A )
% 1.99/1.46 => ( ( Xphi @ Xx )
% 1.99/1.46 => ( in @ Xx
% 1.99/1.46 @ ( dsetconstr @ A
% 1.99/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(dsetconstrEL_type,type,
% 1.99/1.46 dsetconstrEL: $o ).
% 1.99/1.46
% 1.99/1.46 thf(dsetconstrEL,definition,
% 1.99/1.46 ( dsetconstrEL
% 1.99/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.99/1.46 ( ( in @ Xx
% 1.99/1.46 @ ( dsetconstr @ A
% 1.99/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.99/1.46 => ( in @ Xx @ A ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(dsetconstrER_type,type,
% 1.99/1.46 dsetconstrER: $o ).
% 1.99/1.46
% 1.99/1.46 thf(dsetconstrER,definition,
% 1.99/1.46 ( dsetconstrER
% 1.99/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.99/1.46 ( ( in @ Xx
% 1.99/1.46 @ ( dsetconstr @ A
% 1.99/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.99/1.46 => ( Xphi @ Xx ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(exuE1_type,type,
% 1.99/1.46 exuE1: $o ).
% 1.99/1.46
% 1.99/1.46 thf(exuE1,definition,
% 1.99/1.46 ( exuE1
% 1.99/1.46 = ( ! [Xphi: $i > $o] :
% 1.99/1.46 ( ( exu
% 1.99/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.99/1.46 => ? [Xx: $i] :
% 1.99/1.46 ( ( Xphi @ Xx )
% 1.99/1.46 & ! [Xy: $i] :
% 1.99/1.46 ( ( Xphi @ Xy )
% 1.99/1.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(prop2set_type,type,
% 1.99/1.46 prop2set: $o > $i ).
% 1.99/1.46
% 1.99/1.46 thf(prop2setE_type,type,
% 1.99/1.46 prop2setE: $o ).
% 1.99/1.46
% 1.99/1.46 thf(prop2setE,definition,
% 1.99/1.46 ( prop2setE
% 1.99/1.46 = ( ! [Xphi: $o,Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 1.99/1.46 => Xphi ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(emptysetE_type,type,
% 1.99/1.46 emptysetE: $o ).
% 1.99/1.46
% 1.99/1.46 thf(emptysetE,definition,
% 1.99/1.46 ( emptysetE
% 1.99/1.46 = ( ! [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ emptyset )
% 1.99/1.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(emptysetimpfalse_type,type,
% 1.99/1.46 emptysetimpfalse: $o ).
% 1.99/1.46
% 1.99/1.46 thf(emptysetimpfalse,definition,
% 1.99/1.46 ( emptysetimpfalse
% 1.99/1.46 = ( ! [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ emptyset )
% 1.99/1.46 => $false ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(notinemptyset_type,type,
% 1.99/1.46 notinemptyset: $o ).
% 1.99/1.46
% 1.99/1.46 thf(notinemptyset,definition,
% 1.99/1.46 ( notinemptyset
% 1.99/1.46 = ( ! [Xx: $i] :
% 1.99/1.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(exuE3e_type,type,
% 1.99/1.46 exuE3e: $o ).
% 1.99/1.46
% 1.99/1.46 thf(exuE3e,definition,
% 1.99/1.46 ( exuE3e
% 1.99/1.46 = ( ! [Xphi: $i > $o] :
% 1.99/1.46 ( ( exu
% 1.99/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.99/1.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(setext_type,type,
% 1.99/1.46 setext: $o ).
% 1.99/1.46
% 1.99/1.46 thf(setext,definition,
% 1.99/1.46 ( setext
% 1.99/1.46 = ( ! [A: $i,B: $i] :
% 1.99/1.46 ( ! [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ A )
% 1.99/1.46 => ( in @ Xx @ B ) )
% 1.99/1.46 => ( ! [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ B )
% 1.99/1.46 => ( in @ Xx @ A ) )
% 1.99/1.46 => ( A = B ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(emptyI_type,type,
% 1.99/1.46 emptyI: $o ).
% 1.99/1.46
% 1.99/1.46 thf(emptyI,definition,
% 1.99/1.46 ( emptyI
% 1.99/1.46 = ( ! [A: $i] :
% 1.99/1.46 ( ! [Xx: $i] :
% 1.99/1.46 ~ ( in @ Xx @ A )
% 1.99/1.46 => ( A = emptyset ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(noeltsimpempty_type,type,
% 1.99/1.46 noeltsimpempty: $o ).
% 1.99/1.46
% 1.99/1.46 thf(noeltsimpempty,definition,
% 1.99/1.46 ( noeltsimpempty
% 1.99/1.46 = ( ! [A: $i] :
% 1.99/1.46 ( ! [Xx: $i] :
% 1.99/1.46 ~ ( in @ Xx @ A )
% 1.99/1.46 => ( A = emptyset ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(setbeta_type,type,
% 1.99/1.46 setbeta: $o ).
% 1.99/1.46
% 1.99/1.46 thf(setbeta,definition,
% 1.99/1.46 ( setbeta
% 1.99/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ A )
% 1.99/1.46 => ( ( in @ Xx
% 1.99/1.46 @ ( dsetconstr @ A
% 1.99/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.99/1.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(nonempty_type,type,
% 1.99/1.46 nonempty: $i > $o ).
% 1.99/1.46
% 1.99/1.46 thf(nonempty,definition,
% 1.99/1.46 ( nonempty
% 1.99/1.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(nonemptyE1_type,type,
% 1.99/1.46 nonemptyE1: $o ).
% 1.99/1.46
% 1.99/1.46 thf(nonemptyE1,definition,
% 1.99/1.46 ( nonemptyE1
% 1.99/1.46 = ( ! [A: $i] :
% 1.99/1.46 ( ( nonempty @ A )
% 1.99/1.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(nonemptyI_type,type,
% 1.99/1.46 nonemptyI: $o ).
% 1.99/1.46
% 1.99/1.46 thf(nonemptyI,definition,
% 1.99/1.46 ( nonemptyI
% 1.99/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ A )
% 1.99/1.46 => ( ( Xphi @ Xx )
% 1.99/1.46 => ( nonempty
% 1.99/1.46 @ ( dsetconstr @ A
% 1.99/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(nonemptyI1_type,type,
% 1.99/1.46 nonemptyI1: $o ).
% 1.99/1.46
% 1.99/1.46 thf(nonemptyI1,definition,
% 1.99/1.46 ( nonemptyI1
% 1.99/1.46 = ( ! [A: $i] :
% 1.99/1.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 1.99/1.46 => ( nonempty @ A ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(setadjoinIL_type,type,
% 1.99/1.46 setadjoinIL: $o ).
% 1.99/1.46
% 1.99/1.46 thf(setadjoinIL,definition,
% 1.99/1.46 ( setadjoinIL
% 1.99/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(emptyinunitempty_type,type,
% 1.99/1.46 emptyinunitempty: $o ).
% 1.99/1.46
% 1.99/1.46 thf(emptyinunitempty,definition,
% 1.99/1.46 ( emptyinunitempty
% 1.99/1.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(setadjoinIR_type,type,
% 1.99/1.46 setadjoinIR: $o ).
% 1.99/1.46
% 1.99/1.46 thf(setadjoinIR,definition,
% 1.99/1.46 ( setadjoinIR
% 1.99/1.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.99/1.46 ( ( in @ Xy @ A )
% 1.99/1.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(setadjoinE_type,type,
% 1.99/1.46 setadjoinE: $o ).
% 1.99/1.46
% 1.99/1.46 thf(setadjoinE,definition,
% 1.99/1.46 ( setadjoinE
% 1.99/1.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.99/1.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.99/1.46 => ! [Xphi: $o] :
% 1.99/1.46 ( ( ( Xy = Xx )
% 1.99/1.46 => Xphi )
% 1.99/1.46 => ( ( ( in @ Xy @ A )
% 1.99/1.46 => Xphi )
% 1.99/1.46 => Xphi ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(setadjoinOr_type,type,
% 1.99/1.46 setadjoinOr: $o ).
% 1.99/1.46
% 1.99/1.46 thf(setadjoinOr,definition,
% 1.99/1.46 ( setadjoinOr
% 1.99/1.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.99/1.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.99/1.46 => ( ( Xy = Xx )
% 1.99/1.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(setoftrueEq_type,type,
% 1.99/1.46 setoftrueEq: $o ).
% 1.99/1.46
% 1.99/1.46 thf(setoftrueEq,definition,
% 1.99/1.46 ( setoftrueEq
% 1.99/1.46 = ( ! [A: $i] :
% 1.99/1.46 ( ( dsetconstr @ A
% 1.99/1.46 @ ^ [Xx: $i] : $true )
% 1.99/1.46 = A ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(powersetI_type,type,
% 1.99/1.46 powersetI: $o ).
% 1.99/1.46
% 1.99/1.46 thf(powersetI,definition,
% 1.99/1.46 ( powersetI
% 1.99/1.46 = ( ! [A: $i,B: $i] :
% 1.99/1.46 ( ! [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ B )
% 1.99/1.46 => ( in @ Xx @ A ) )
% 1.99/1.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(emptyinPowerset_type,type,
% 1.99/1.46 emptyinPowerset: $o ).
% 1.99/1.46
% 1.99/1.46 thf(emptyinPowerset,definition,
% 1.99/1.46 ( emptyinPowerset
% 1.99/1.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(emptyInPowerset_type,type,
% 1.99/1.46 emptyInPowerset: $o ).
% 1.99/1.46
% 1.99/1.46 thf(emptyInPowerset,definition,
% 1.99/1.46 ( emptyInPowerset
% 1.99/1.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(powersetE_type,type,
% 1.99/1.46 powersetE: $o ).
% 1.99/1.46
% 1.99/1.46 thf(powersetE,definition,
% 1.99/1.46 ( powersetE
% 1.99/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.99/1.46 ( ( in @ B @ ( powerset @ A ) )
% 1.99/1.46 => ( ( in @ Xx @ B )
% 1.99/1.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(setunionI_type,type,
% 1.99/1.46 setunionI: $o ).
% 1.99/1.46
% 1.99/1.46 thf(setunionI,definition,
% 1.99/1.46 ( setunionI
% 1.99/1.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 1.99/1.46 ( ( in @ Xx @ B )
% 1.99/1.46 => ( ( in @ B @ A )
% 1.99/1.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(setunionE_type,type,
% 1.99/1.46 setunionE: $o ).
% 1.99/1.46
% 1.99/1.46 thf(setunionE,definition,
% 1.99/1.46 ( setunionE
% 1.99/1.46 = ( ! [A: $i,Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ ( setunion @ A ) )
% 1.99/1.46 => ! [Xphi: $o] :
% 1.99/1.46 ( ! [B: $i] :
% 1.99/1.46 ( ( in @ Xx @ B )
% 1.99/1.46 => ( ( in @ B @ A )
% 1.99/1.46 => Xphi ) )
% 1.99/1.46 => Xphi ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(subPowSU_type,type,
% 1.99/1.46 subPowSU: $o ).
% 1.99/1.46
% 1.99/1.46 thf(subPowSU,definition,
% 1.99/1.46 ( subPowSU
% 1.99/1.46 = ( ! [A: $i,Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ A )
% 1.99/1.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(exuE2_type,type,
% 1.99/1.46 exuE2: $o ).
% 1.99/1.46
% 1.99/1.46 thf(exuE2,definition,
% 1.99/1.46 ( exuE2
% 1.99/1.46 = ( ! [Xphi: $i > $o] :
% 1.99/1.46 ( ( exu
% 1.99/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.99/1.46 => ? [Xx: $i] :
% 1.99/1.46 ! [Xy: $i] :
% 1.99/1.46 ( ( Xphi @ Xy )
% 1.99/1.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(nonemptyImpWitness_type,type,
% 1.99/1.46 nonemptyImpWitness: $o ).
% 1.99/1.46
% 1.99/1.46 thf(nonemptyImpWitness,definition,
% 1.99/1.46 ( nonemptyImpWitness
% 1.99/1.46 = ( ! [A: $i] :
% 1.99/1.46 ( ( nonempty @ A )
% 1.99/1.46 => ? [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ A )
% 1.99/1.46 & $true ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(uniqinunit_type,type,
% 1.99/1.46 uniqinunit: $o ).
% 1.99/1.46
% 1.99/1.46 thf(uniqinunit,definition,
% 1.99/1.46 ( uniqinunit
% 1.99/1.46 = ( ! [Xx: $i,Xy: $i] :
% 1.99/1.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 1.99/1.46 => ( Xx = Xy ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(notinsingleton_type,type,
% 1.99/1.46 notinsingleton: $o ).
% 1.99/1.46
% 1.99/1.46 thf(notinsingleton,definition,
% 1.99/1.46 ( notinsingleton
% 1.99/1.46 = ( ! [Xx: $i,Xy: $i] :
% 1.99/1.46 ( ( Xx != Xy )
% 1.99/1.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(eqinunit_type,type,
% 1.99/1.46 eqinunit: $o ).
% 1.99/1.46
% 1.99/1.46 thf(eqinunit,definition,
% 1.99/1.46 ( eqinunit
% 1.99/1.46 = ( ! [Xx: $i,Xy: $i] :
% 1.99/1.46 ( ( Xx = Xy )
% 1.99/1.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(singletonsswitch_type,type,
% 1.99/1.46 singletonsswitch: $o ).
% 1.99/1.46
% 1.99/1.46 thf(singletonsswitch,definition,
% 1.99/1.46 ( singletonsswitch
% 1.99/1.46 = ( ! [Xx: $i,Xy: $i] :
% 1.99/1.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 1.99/1.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(upairsetE_type,type,
% 1.99/1.46 upairsetE: $o ).
% 1.99/1.46
% 1.99/1.46 thf(upairsetE,definition,
% 1.99/1.46 ( upairsetE
% 1.99/1.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 1.99/1.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 1.99/1.46 => ( ( Xz = Xx )
% 1.99/1.46 | ( Xz = Xy ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(upairsetIL_type,type,
% 1.99/1.46 upairsetIL: $o ).
% 1.99/1.46
% 1.99/1.46 thf(upairsetIL,definition,
% 1.99/1.46 ( upairsetIL
% 1.99/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(upairsetIR_type,type,
% 1.99/1.46 upairsetIR: $o ).
% 1.99/1.46
% 1.99/1.46 thf(upairsetIR,definition,
% 1.99/1.46 ( upairsetIR
% 1.99/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(emptyE1_type,type,
% 1.99/1.46 emptyE1: $o ).
% 1.99/1.46
% 1.99/1.46 thf(emptyE1,definition,
% 1.99/1.46 ( emptyE1
% 1.99/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 1.99/1.46 ( ? [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ A )
% 1.99/1.46 & ( Xphi @ Xx ) )
% 1.99/1.46 => ( ( ( dsetconstr @ A
% 1.99/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.99/1.46 = emptyset )
% 1.99/1.46 => $false ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(vacuousDall_type,type,
% 1.99/1.46 vacuousDall: $o ).
% 1.99/1.46
% 1.99/1.46 thf(vacuousDall,definition,
% 1.99/1.46 ( vacuousDall
% 1.99/1.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ emptyset )
% 1.99/1.46 => ( Xphi @ Xx ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(quantDeMorgan1_type,type,
% 1.99/1.46 quantDeMorgan1: $o ).
% 1.99/1.46
% 1.99/1.46 thf(quantDeMorgan1,definition,
% 1.99/1.46 ( quantDeMorgan1
% 1.99/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 1.99/1.46 ( ~ ! [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ A )
% 1.99/1.46 => ( Xphi @ Xx ) )
% 1.99/1.46 => ? [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ A )
% 1.99/1.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(quantDeMorgan2_type,type,
% 1.99/1.46 quantDeMorgan2: $o ).
% 1.99/1.46
% 1.99/1.46 thf(quantDeMorgan2,definition,
% 1.99/1.46 ( quantDeMorgan2
% 1.99/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 1.99/1.46 ( ! [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ A )
% 1.99/1.46 => ~ ( Xphi @ Xx ) )
% 1.99/1.46 => ~ ? [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ A )
% 1.99/1.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(quantDeMorgan3_type,type,
% 1.99/1.46 quantDeMorgan3: $o ).
% 1.99/1.46
% 1.99/1.46 thf(quantDeMorgan3,definition,
% 1.99/1.46 ( quantDeMorgan3
% 1.99/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 1.99/1.46 ( ~ ? [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ A )
% 1.99/1.46 & ( Xphi @ Xx ) )
% 1.99/1.46 => ! [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ A )
% 1.99/1.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(quantDeMorgan4_type,type,
% 1.99/1.46 quantDeMorgan4: $o ).
% 1.99/1.46
% 1.99/1.46 thf(quantDeMorgan4,definition,
% 1.99/1.46 ( quantDeMorgan4
% 1.99/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 1.99/1.46 ( ? [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ A )
% 1.99/1.46 & ~ ( Xphi @ Xx ) )
% 1.99/1.46 => ~ ! [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ A )
% 1.99/1.46 => ( Xphi @ Xx ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(prop2setI_type,type,
% 1.99/1.46 prop2setI: $o ).
% 1.99/1.46
% 1.99/1.46 thf(prop2setI,definition,
% 1.99/1.46 ( prop2setI
% 1.99/1.46 = ( ! [Xphi: $o] :
% 1.99/1.46 ( Xphi
% 1.99/1.46 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(set2prop_type,type,
% 1.99/1.46 set2prop: $i > $o ).
% 1.99/1.46
% 1.99/1.46 thf(prop2set2propI_type,type,
% 1.99/1.46 prop2set2propI: $o ).
% 1.99/1.46
% 1.99/1.46 thf(prop2set2propI,definition,
% 1.99/1.46 ( prop2set2propI
% 1.99/1.46 = ( ! [Xphi: $o] :
% 1.99/1.46 ( Xphi
% 1.99/1.46 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(notdexE_type,type,
% 1.99/1.46 notdexE: $o ).
% 1.99/1.46
% 1.99/1.46 thf(notdexE,definition,
% 1.99/1.46 ( notdexE
% 1.99/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 1.99/1.46 ( ~ ? [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ A )
% 1.99/1.46 & ( Xphi @ Xx ) )
% 1.99/1.46 => ! [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ A )
% 1.99/1.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(notdallE_type,type,
% 1.99/1.46 notdallE: $o ).
% 1.99/1.46
% 1.99/1.46 thf(notdallE,definition,
% 1.99/1.46 ( notdallE
% 1.99/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 1.99/1.46 ( ~ ! [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ A )
% 1.99/1.46 => ( Xphi @ Xx ) )
% 1.99/1.46 => ? [Xx: $i] :
% 1.99/1.46 ( ( in @ Xx @ A )
% 1.99/1.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 1.99/1.46
% 1.99/1.46 thf(exuI1,conjecture,
% 1.99/1.46 ( setextAx
% 1.99/1.46 => ( emptysetAx
% 1.99/1.46 => ( setadjoinAx
% 1.99/1.46 => ( powersetAx
% 1.99/1.46 => ( setunionAx
% 1.99/1.46 => ( omega0Ax
% 1.99/1.46 => ( omegaSAx
% 1.99/1.46 => ( omegaIndAx
% 1.99/1.46 => ( replAx
% 1.99/1.46 => ( foundationAx
% 1.99/1.46 => ( wellorderingAx
% 1.99/1.46 => ( descrp
% 1.99/1.46 => ( dsetconstrI
% 1.99/1.46 => ( dsetconstrEL
% 1.99/1.46 => ( dsetconstrER
% 1.99/1.46 => ( exuE1
% 1.99/1.46 => ( prop2setE
% 1.99/1.46 => ( emptysetE
% 1.99/1.46 => ( emptysetimpfalse
% 1.99/1.46 => ( notinemptyset
% 1.99/1.46 => ( exuE3e
% 1.99/1.46 => ( setext
% 1.99/1.46 => ( emptyI
% 1.99/1.46 => ( noeltsimpempty
% 1.99/1.46 => ( setbeta
% 1.99/1.46 => ( nonemptyE1
% 1.99/1.46 => ( nonemptyI
% 1.99/1.46 => ( nonemptyI1
% 1.99/1.46 => ( setadjoinIL
% 1.99/1.46 => ( emptyinunitempty
% 1.99/1.46 => ( setadjoinIR
% 1.99/1.46 => ( setadjoinE
% 1.99/1.46 => ( setadjoinOr
% 1.99/1.46 => ( setoftrueEq
% 1.99/1.46 => ( powersetI
% 1.99/1.46 => ( emptyinPowerset
% 1.99/1.46 => ( emptyInPowerset
% 1.99/1.46 => ( powersetE
% 1.99/1.46 => ( setunionI
% 1.99/1.46 => ( setunionE
% 1.99/1.46 => ( subPowSU
% 1.99/1.46 => ( exuE2
% 1.99/1.46 => ( nonemptyImpWitness
% 1.99/1.46 => ( uniqinunit
% 1.99/1.46 => ( notinsingleton
% 1.99/1.46 => ( eqinunit
% 1.99/1.46 => ( singletonsswitch
% 1.99/1.46 => ( upairsetE
% 1.99/1.46 => ( upairsetIL
% 1.99/1.46 => ( upairsetIR
% 1.99/1.46 => ( emptyE1
% 1.99/1.46 => ( vacuousDall
% 1.99/1.46 => ( quantDeMorgan1
% 1.99/1.46 => ( quantDeMorgan2
% 1.99/1.46 => ( quantDeMorgan3
% 1.99/1.46 => ( quantDeMorgan4
% 1.99/1.46 => ( prop2setI
% 1.99/1.46 => ( prop2set2propI
% 1.99/1.46 => ( notdexE
% 1.99/1.46 => ( notdallE
% 1.99/1.46 => ! [Xphi: $i > $o] :
% 1.99/1.46 ( ? [Xx: $i] :
% 1.99/1.46 ( ( Xphi @ Xx )
% 1.99/1.46 & ! [Xy: $i] :
% 1.99/1.46 ( ( Xphi @ Xy )
% 1.99/1.46 => ( Xx = Xy ) ) )
% 1.99/1.46 => ( exu
% 1.99/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 1.99/1.46 ------------------------
% 1.99/1.48 % (12022)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_11924 for (2999ds/183Mi)
% 1.99/1.48 % (12028)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_11924 for (2999ds/18Mi)
% 1.99/1.48 % (12024)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_11924 for (2999ds/27Mi)
% 1.99/1.48 % (12023)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_11924 for (2999ds/4Mi)
% 1.99/1.48 % (12027)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_11924 for (2999ds/275Mi)
% 1.99/1.48 % (12025)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_11924 for (2999ds/2Mi)
% 1.99/1.48 % (12026)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_11924 for (2999ds/2Mi)
% 1.99/1.48 % (12025)Instruction limit reached!
% 1.99/1.48 % (12025)------------------------------
% 1.99/1.48 % (12025)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.99/1.48 % (12025)Termination reason: Unknown
% 1.99/1.48 % (12025)Termination phase: shuffling
% 1.99/1.48
% 1.99/1.48 % (12026)Instruction limit reached!
% 1.99/1.48 % (12026)------------------------------
% 1.99/1.48 % (12026)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.99/1.48 % (12026)Termination reason: Unknown
% 1.99/1.48 % (12026)Termination phase: shuffling
% 1.99/1.48
% 1.99/1.48 % (12026)Memory used [KB]: 1151
% 1.99/1.48 % (12026)Time elapsed: 0.003 s
% 1.99/1.48 % (12026)Instructions burned: 3 (million)
% 1.99/1.48 % (12026)------------------------------
% 1.99/1.48 % (12026)------------------------------
% 1.99/1.48 % (12025)Memory used [KB]: 1151
% 1.99/1.48 % (12025)Time elapsed: 0.003 s
% 1.99/1.48 % (12025)Instructions burned: 3 (million)
% 1.99/1.48 % (12025)------------------------------
% 1.99/1.48 % (12025)------------------------------
% 1.99/1.48 % (12023)Instruction limit reached!
% 1.99/1.48 % (12023)------------------------------
% 1.99/1.48 % (12023)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.99/1.48 % (12023)Termination reason: Unknown
% 1.99/1.48 % (12023)Termination phase: shuffling
% 1.99/1.48
% 1.99/1.48 % (12023)Memory used [KB]: 1151
% 1.99/1.48 % (12023)Time elapsed: 0.004 s
% 1.99/1.48 % (12023)Instructions burned: 6 (million)
% 1.99/1.48 % (12023)------------------------------
% 1.99/1.48 % (12023)------------------------------
% 1.99/1.49 % (12028)Instruction limit reached!
% 1.99/1.49 % (12028)------------------------------
% 1.99/1.49 % (12028)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.99/1.49 % (12028)Termination reason: Unknown
% 1.99/1.49 % (12028)Termination phase: Preprocessing 3
% 1.99/1.49
% 1.99/1.49 % (12028)Memory used [KB]: 1663
% 1.99/1.49 % (12028)Time elapsed: 0.011 s
% 1.99/1.49 % (12028)Instructions burned: 19 (million)
% 1.99/1.49 % (12028)------------------------------
% 1.99/1.49 % (12028)------------------------------
% 1.99/1.49 % (12024)Instruction limit reached!
% 1.99/1.49 % (12024)------------------------------
% 1.99/1.49 % (12024)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.99/1.49 % (12024)Termination reason: Unknown
% 1.99/1.49 % (12024)Termination phase: Property scanning
% 1.99/1.49
% 1.99/1.49 % (12024)Memory used [KB]: 1407
% 1.99/1.49 % (12024)Time elapsed: 0.013 s
% 1.99/1.49 % (12024)Instructions burned: 28 (million)
% 1.99/1.49 % (12024)------------------------------
% 1.99/1.49 % (12024)------------------------------
% 1.99/1.50 % (12029)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_11924 for (2999ds/3Mi)
% 1.99/1.50 % (12030)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_11924 for (2999ds/37Mi)
% 1.99/1.50 % (12031)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_11924 for (2999ds/15Mi)
% 2.10/1.50 % (12029)Instruction limit reached!
% 2.10/1.50 % (12029)------------------------------
% 2.10/1.50 % (12029)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.50 % (12029)Termination reason: Unknown
% 2.10/1.50 % (12029)Termination phase: shuffling
% 2.10/1.50
% 2.10/1.50 % (12029)Memory used [KB]: 1151
% 2.10/1.50 % (12029)Time elapsed: 0.004 s
% 2.10/1.50 % (12029)Instructions burned: 4 (million)
% 2.10/1.50 % (12029)------------------------------
% 2.10/1.50 % (12029)------------------------------
% 2.10/1.50 % (12032)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_11924 for (2999ds/3Mi)
% 2.10/1.50 % (12032)Instruction limit reached!
% 2.10/1.50 % (12032)------------------------------
% 2.10/1.50 % (12032)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.50 % (12032)Termination reason: Unknown
% 2.10/1.50 % (12032)Termination phase: shuffling
% 2.10/1.50
% 2.10/1.50 % (12032)Memory used [KB]: 1151
% 2.10/1.50 % (12032)Time elapsed: 0.004 s
% 2.10/1.50 % (12032)Instructions burned: 4 (million)
% 2.10/1.50 % (12032)------------------------------
% 2.10/1.50 % (12032)------------------------------
% 2.10/1.50 % (12031)Instruction limit reached!
% 2.10/1.50 % (12031)------------------------------
% 2.10/1.50 % (12031)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.50 % (12031)Termination reason: Unknown
% 2.10/1.50 % (12031)Termination phase: Preprocessing 3
% 2.10/1.50
% 2.10/1.50 % (12031)Memory used [KB]: 1535
% 2.10/1.50 % (12031)Time elapsed: 0.009 s
% 2.10/1.50 % (12031)Instructions burned: 16 (million)
% 2.10/1.50 % (12031)------------------------------
% 2.10/1.50 % (12031)------------------------------
% 2.10/1.50 % (12033)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_11924 for (2999ds/1041Mi)
% 2.10/1.51 % (12034)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_11924 for (2999ds/7Mi)
% 2.10/1.51 % (12030)Instruction limit reached!
% 2.10/1.51 % (12030)------------------------------
% 2.10/1.51 % (12030)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.51 % (12030)Termination reason: Unknown
% 2.10/1.51 % (12030)Termination phase: Saturation
% 2.10/1.51
% 2.10/1.51 % (12030)Memory used [KB]: 5884
% 2.10/1.51 % (12030)Time elapsed: 0.018 s
% 2.10/1.51 % (12030)Instructions burned: 38 (million)
% 2.10/1.51 % (12030)------------------------------
% 2.10/1.51 % (12030)------------------------------
% 2.10/1.52 % (12034)Instruction limit reached!
% 2.10/1.52 % (12034)------------------------------
% 2.10/1.52 % (12034)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.52 % (12034)Termination reason: Unknown
% 2.10/1.52 % (12034)Termination phase: shuffling
% 2.10/1.52
% 2.10/1.52 % (12034)Memory used [KB]: 1151
% 2.10/1.52 % (12034)Time elapsed: 0.005 s
% 2.10/1.52 % (12034)Instructions burned: 8 (million)
% 2.10/1.52 % (12034)------------------------------
% 2.10/1.52 % (12034)------------------------------
% 2.10/1.52 % (12035)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_11924 for (2999ds/16Mi)
% 2.10/1.52 % (12022)First to succeed.
% 2.10/1.52 % (12036)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_11924 for (2999ds/3Mi)
% 2.10/1.52 % (12036)Instruction limit reached!
% 2.10/1.52 % (12036)------------------------------
% 2.10/1.52 % (12036)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.52 % (12036)Termination reason: Unknown
% 2.10/1.52 % (12036)Termination phase: shuffling
% 2.10/1.52
% 2.10/1.52 % (12036)Memory used [KB]: 1151
% 2.10/1.52 % (12036)Time elapsed: 0.003 s
% 2.10/1.52 % (12036)Instructions burned: 4 (million)
% 2.10/1.52 % (12036)------------------------------
% 2.10/1.52 % (12036)------------------------------
% 2.10/1.52 % (12022)Refutation found. Thanks to Tanya!
% 2.10/1.52 % SZS status Theorem for DTF2THF_11924
% 2.10/1.52 % SZS output start Proof for DTF2THF_11924
% 2.10/1.52 thf(func_def_0, type, in: $i > $i > $o).
% 2.10/1.52 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.10/1.52 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.10/1.52 thf(func_def_8, type, powerset: $i > $i).
% 2.10/1.52 thf(func_def_10, type, setunion: $i > $i).
% 2.10/1.52 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.10/1.52 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.10/1.52 thf(func_def_26, type, prop2set: $o > $i).
% 2.10/1.52 thf(func_def_36, type, nonempty: $i > $o).
% 2.10/1.52 thf(func_def_69, type, set2prop: $i > $o).
% 2.10/1.52 thf(func_def_89, type, sK0: $i > $o).
% 2.10/1.52 thf(func_def_92, type, sK3: $i > $i).
% 2.10/1.52 thf(func_def_93, type, sK4: $i > $i > $i).
% 2.10/1.52 thf(func_def_94, type, sK5: $i > $i > $i).
% 2.10/1.52 thf(func_def_96, type, sK6: ($i > $i > $o) > $i > $i).
% 2.10/1.52 thf(func_def_97, type, sK7: ($i > $i > $o) > $i > $i).
% 2.10/1.52 thf(func_def_98, type, sK8: ($i > $i > $o) > $i > $i > $i).
% 2.10/1.52 thf(func_def_99, type, sK9: ($i > $i > $o) > $i > $i).
% 2.10/1.52 thf(func_def_100, type, sK10: $i > ($i > $i > $o) > $i > $i).
% 2.10/1.52 thf(func_def_101, type, sK11: $i > ($i > $i > $o) > $i > $i).
% 2.10/1.52 thf(func_def_102, type, sK12: $i > ($i > $i > $o) > $i > $i).
% 2.10/1.52 thf(func_def_103, type, sK13: $i > ($i > $i > $o) > $i > $i).
% 2.10/1.52 thf(func_def_104, type, sK14: $i > $i).
% 2.10/1.52 thf(func_def_105, type, sK15: $o > $i > $i > $i).
% 2.10/1.52 thf(func_def_106, type, sK16: $i > $i > $i).
% 2.10/1.52 thf(func_def_107, type, sK17: ($i > $o) > $i > $i).
% 2.10/1.52 thf(func_def_108, type, sK18: $i > $i > $i).
% 2.10/1.52 thf(func_def_109, type, sK19: $i > $i > $i).
% 2.10/1.52 thf(func_def_110, type, sK20: $i > $i > $i).
% 2.10/1.52 thf(func_def_111, type, sK21: $i > $i > $i).
% 2.10/1.52 thf(func_def_112, type, sK22: $i > $i).
% 2.10/1.52 thf(func_def_113, type, sK23: ($i > $o) > $i > $i).
% 2.10/1.52 thf(func_def_114, type, sK24: ($i > $o) > $i > $i).
% 2.10/1.52 thf(func_def_115, type, sK25: $i > $i).
% 2.10/1.52 thf(func_def_116, type, sK26: $i > $i).
% 2.10/1.52 thf(func_def_117, type, sK27: $i > $i).
% 2.10/1.52 thf(func_def_118, type, sK28: $i > $i).
% 2.10/1.52 thf(func_def_119, type, sK29: $i > ($i > $o) > $i).
% 2.10/1.52 thf(func_def_120, type, sK30: ($i > $o) > $i).
% 2.10/1.52 thf(func_def_121, type, sK31: ($i > $o) > $i > $i).
% 2.10/1.52 thf(func_def_122, type, sK32: ($i > $o) > $i).
% 2.10/1.52 thf(func_def_123, type, sK33: ($i > $o) > $i > $i).
% 2.10/1.52 thf(func_def_124, type, sK34: ($i > $o) > $i > $i).
% 2.10/1.52 thf(func_def_125, type, sK35: ($i > $o) > $i > $i).
% 2.10/1.52 thf(func_def_126, type, sK36: $i > ($i > $o) > $i).
% 2.10/1.52 thf(func_def_127, type, sK37: ($i > $o) > $i > $i).
% 2.10/1.52 thf(func_def_128, type, sK38: ($i > $o) > $i).
% 2.10/1.52 thf(func_def_129, type, sK39: ($i > $o) > $i > $i).
% 2.10/1.52 thf(func_def_130, type, sK40: $i > $i).
% 2.10/1.52 thf(func_def_131, type, sK41: $i > $i > $i).
% 2.10/1.52 thf(func_def_132, type, sK42: $i > $i > $i).
% 2.10/1.52 thf(func_def_133, type, sK43: $i > $i > $i).
% 2.10/1.52 thf(func_def_134, type, sK44: $i > $i > $i).
% 2.10/1.52 thf(func_def_135, type, sK45: $i > $i > $i > $i).
% 2.10/1.52 thf(f1134,plain,(
% 2.10/1.52 $false),
% 2.10/1.52 inference(subsumption_resolution,[status(thm)],[f1131,f1110])).
% 2.10/1.52 thf(f1110,plain,(
% 2.10/1.52 ((sK28 @ sK1) != sK1)),
% 2.10/1.52 inference(trivial_inequality_removal,[status(thm)],[f1106])).
% 2.10/1.52 thf(f1106,plain,(
% 2.10/1.52 ((sK28 @ sK1) != sK1) | ($false = $true)),
% 2.10/1.52 inference(superposition,[status(thm)],[f794,f250])).
% 2.10/1.52 thf(f250,plain,(
% 2.10/1.52 ((sK0 @ sK1) = $true)),
% 2.10/1.52 inference(cnf_transformation,[status(thm)],[f192])).
% 2.10/1.52 thf(f192,plain,(
% 2.10/1.52 (prop2setI = $true) & (dsetconstrI = $true) & (emptyE1 = $true) & (setunionAx = $true) & (setoftrueEq = $true) & (nonemptyE1 = $true) & (dsetconstrER = $true) & (uniqinunit = $true) & (setbeta = $true) & (notdallE = $true) & (emptysetE = $true) & (omegaSAx = $true) & (wellorderingAx = $true) & (singletonsswitch = $true) & (powersetE = $true) & (emptyInPowerset = $true) & (quantDeMorgan4 = $true) & (omegaIndAx = $true) & ((! [X2 : $i] : ((sK1 = X2) | ((sK0 @ X2) != $true)) & ((sK0 @ sK1) = $true)) & ((exu @ (^[Y0 : $i]: (sK0 @ Y0))) != $true)) & (vacuousDall = $true) & (setadjoinIL = $true) & (upairsetIL = $true) & (exuE1 = $true) & (setadjoinIR = $true) & (emptyinPowerset = $true) & (exuE3e = $true) & (eqinunit = $true) & (setextAx = $true) & (replAx = $true) & (powersetAx = $true) & (setadjoinE = $true) & (omega0Ax = $true) & (powersetI = $true) & (notinsingleton = $true) & (nonemptyImpWitness = $true) & (emptysetimpfalse = $true) & (quantDeMorgan2 = $true) & (notinemptyset = $true) & (setadjoinOr = $true) & (exuE2 = $true) & (setunionE = $true) & (descrp = $true) & (nonemptyI = $true) & (nonemptyI1 = $true) & (quantDeMorgan3 = $true) & (emptyinunitempty = $true) & (setadjoinAx = $true) & (prop2setE = $true) & (quantDeMorgan1 = $true) & (upairsetE = $true) & (foundationAx = $true) & (prop2set2propI = $true) & (noeltsimpempty = $true) & (setext = $true) & (upairsetIR = $true) & (notdexE = $true) & (emptyI = $true) & (setunionI = $true) & (dsetconstrEL = $true) & (emptysetAx = $true) & (subPowSU = $true)),
% 2.10/1.52 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f189,f191,f190])).
% 2.10/1.52 thf(f190,plain,(
% 2.10/1.52 ? [X0 : $i > $o] : (? [X1 : $i] : (! [X2 : $i] : ((X1 = X2) | ((X0 @ X2) != $true)) & ((X0 @ X1) = $true)) & ((exu @ (^[Y0 : $i]: (X0 @ Y0))) != $true)) => (? [X1 : $i] : (! [X2 : $i] : ((X1 = X2) | ((sK0 @ X2) != $true)) & ($true = (sK0 @ X1))) & ((exu @ (^[Y0 : $i]: (sK0 @ Y0))) != $true))),
% 2.10/1.52 introduced(definition,[],[choice_axiom])).
% 2.10/1.52 thf(f191,plain,(
% 2.10/1.52 ? [X1 : $i] : (! [X2 : $i] : ((X1 = X2) | ((sK0 @ X2) != $true)) & ($true = (sK0 @ X1))) => (! [X2 : $i] : ((sK1 = X2) | ((sK0 @ X2) != $true)) & ((sK0 @ sK1) = $true))),
% 2.10/1.52 introduced(definition,[],[choice_axiom])).
% 2.10/1.52 thf(f189,plain,(
% 2.10/1.52 (prop2setI = $true) & (dsetconstrI = $true) & (emptyE1 = $true) & (setunionAx = $true) & (setoftrueEq = $true) & (nonemptyE1 = $true) & (dsetconstrER = $true) & (uniqinunit = $true) & (setbeta = $true) & (notdallE = $true) & (emptysetE = $true) & (omegaSAx = $true) & (wellorderingAx = $true) & (singletonsswitch = $true) & (powersetE = $true) & (emptyInPowerset = $true) & (quantDeMorgan4 = $true) & (omegaIndAx = $true) & ? [X0 : $i > $o] : (? [X1 : $i] : (! [X2 : $i] : ((X1 = X2) | ((X0 @ X2) != $true)) & ((X0 @ X1) = $true)) & ((exu @ (^[Y0 : $i]: (X0 @ Y0))) != $true)) & (vacuousDall = $true) & (setadjoinIL = $true) & (upairsetIL = $true) & (exuE1 = $true) & (setadjoinIR = $true) & (emptyinPowerset = $true) & (exuE3e = $true) & (eqinunit = $true) & (setextAx = $true) & (replAx = $true) & (powersetAx = $true) & (setadjoinE = $true) & (omega0Ax = $true) & (powersetI = $true) & (notinsingleton = $true) & (nonemptyImpWitness = $true) & (emptysetimpfalse = $true) & (quantDeMorgan2 = $true) & (notinemptyset = $true) & (setadjoinOr = $true) & (exuE2 = $true) & (setunionE = $true) & (descrp = $true) & (nonemptyI = $true) & (nonemptyI1 = $true) & (quantDeMorgan3 = $true) & (emptyinunitempty = $true) & (setadjoinAx = $true) & (prop2setE = $true) & (quantDeMorgan1 = $true) & (upairsetE = $true) & (foundationAx = $true) & (prop2set2propI = $true) & (noeltsimpempty = $true) & (setext = $true) & (upairsetIR = $true) & (notdexE = $true) & (emptyI = $true) & (setunionI = $true) & (dsetconstrEL = $true) & (emptysetAx = $true) & (subPowSU = $true)),
% 2.10/1.52 inference(flattening,[status(thm)],[f188])).
% 2.10/1.52 thf(f188,plain,(
% 2.10/1.52 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i > $o] : (? [X1 : $i] : (! [X2 : $i] : ((X1 = X2) | ((X0 @ X2) != $true)) & ((X0 @ X1) = $true)) & ((exu @ (^[Y0 : $i]: (X0 @ Y0))) != $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)),
% 2.10/1.52 inference(ennf_transformation,[status(thm)],[f169])).
% 2.10/1.52 thf(f169,plain,(
% 2.10/1.52 ~((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) => ! [X0 : $i > $o] : (? [X1 : $i] : (((X0 @ X1) = $true) & ! [X2 : $i] : (((X0 @ X2) = $true) => (X1 = X2))) => ((exu @ (^[Y0 : $i]: (X0 @ Y0))) = $true))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.10/1.52 inference(fool_elimination,[status(thm)],[f168])).
% 2.10/1.52 thf(f168,plain,(
% 2.10/1.52 ~(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 => ! [X0 : $i > $o] : (? [X1 : $i] : (! [X2 : $i] : ((X0 @ X2) => (X1 = X2)) & (X0 @ X1)) => (exu @ (^[X3 : $i] : (X0 @ X3))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.10/1.52 inference(rectify,[status(thm)],[f64])).
% 2.10/1.52 thf(f64,negated_conjecture,(
% 2.10/1.52 ~(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 => ! [X0 : $i > $o] : (? [X1 : $i] : (! [X2 : $i] : ((X0 @ X2) => (X1 = X2)) & (X0 @ X1)) => (exu @ (^[X1 : $i] : (X0 @ X1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.10/1.52 inference(negated_conjecture,[status(cth)],[f63])).
% 2.10/1.52 thf(f63,conjecture,(
% 2.10/1.52 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 => ! [X0 : $i > $o] : (? [X1 : $i] : (! [X2 : $i] : ((X0 @ X2) => (X1 = X2)) & (X0 @ X1)) => (exu @ (^[X1 : $i] : (X0 @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.10/1.52 file('/export/starexec/sandbox2/tmp/tmp.ujh7G4tgox/DTF2THF_11924.p',exuI1)).
% 2.10/1.52 thf(f794,plain,(
% 2.10/1.52 ( ! [X1 : $i] : (($false = (sK0 @ X1)) | ((sK28 @ X1) != X1)) )),
% 2.10/1.52 inference(equality_proxy_clausification,[status(thm)],[f792])).
% 2.10/1.52 thf(f792,plain,(
% 2.10/1.52 ( ! [X1 : $i] : (((X1 = (sK28 @ X1)) = $false) | ($false = (sK0 @ X1))) )),
% 2.10/1.52 inference(binary_proxy_clausification,[status(thm)],[f791])).
% 2.10/1.52 thf(f791,plain,(
% 2.10/1.52 ( ! [X1 : $i] : (($false = (sK0 @ X1)) | ($false = ((sK0 @ (sK28 @ X1)) => (X1 = (sK28 @ X1))))) )),
% 2.10/1.52 inference(beta_eta_normalization,[status(thm)],[f790])).
% 2.10/1.52 thf(f790,plain,(
% 2.10/1.52 ( ! [X1 : $i] : (($false = (sK0 @ X1)) | ($false = ((^[Y0 : $i]: ((sK0 @ Y0) => (X1 = Y0))) @ (sK28 @ X1)))) )),
% 2.10/1.52 inference(sigma_clausification,[status(thm)],[f789])).
% 2.10/1.52 thf(f789,plain,(
% 2.10/1.52 ( ! [X1 : $i] : (($false = (!! @ $i @ (^[Y0 : $i]: ((sK0 @ Y0) => (X1 = Y0))))) | ($false = (sK0 @ X1))) )),
% 2.10/1.52 inference(binary_proxy_clausification,[status(thm)],[f788])).
% 2.10/1.52 thf(f788,plain,(
% 2.10/1.52 ( ! [X1 : $i] : (($false = ((sK0 @ X1) & (!! @ $i @ (^[Y0 : $i]: ((sK0 @ Y0) => (X1 = Y0))))))) )),
% 2.10/1.52 inference(beta_eta_normalization,[status(thm)],[f787])).
% 2.10/1.52 thf(f787,plain,(
% 2.10/1.52 ( ! [X1 : $i] : (($false = ((^[Y0 : $i]: ((sK0 @ Y0) & (!! @ $i @ (^[Y1 : $i]: ((sK0 @ Y1) => (Y0 = Y1)))))) @ X1))) )),
% 2.10/1.52 inference(pi_clausification,[status(thm)],[f786])).
% 2.10/1.52 thf(f786,plain,(
% 2.10/1.52 ((?? @ $i @ (^[Y0 : $i]: ((sK0 @ Y0) & (!! @ $i @ (^[Y1 : $i]: ((sK0 @ Y1) => (Y0 = Y1))))))) != $true)),
% 2.10/1.52 inference(beta_eta_normalization,[status(thm)],[f326])).
% 2.10/1.52 thf(f326,plain,(
% 2.10/1.52 ($true != ((^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) & (!! @ $i @ (^[Y2 : $i]: ((Y0 @ Y2) => (Y1 = Y2)))))))) @ (^[Y0 : $i]: (sK0 @ Y0))))),
% 2.10/1.52 inference(definition_unfolding,[status(thm)],[f249,f285])).
% 2.10/1.52 thf(f285,plain,(
% 2.10/1.52 (exu = (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) & (!! @ $i @ (^[Y2 : $i]: ((Y0 @ Y2) => (Y1 = Y2)))))))))),
% 2.10/1.52 inference(cnf_transformation,[status(thm)],[f83])).
% 2.10/1.52 thf(f83,plain,(
% 2.10/1.52 (exu = (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) & (!! @ $i @ (^[Y2 : $i]: ((Y0 @ Y2) => (Y1 = Y2)))))))))),
% 2.10/1.52 inference(fool_elimination,[status(thm)],[f82])).
% 2.10/1.52 thf(f82,plain,(
% 2.10/1.52 ((^[X0 : $i > $o] : (? [X1 : $i] : (! [X2 : $i] : ((X0 @ X2) => (X1 = X2)) & (X0 @ X1)))) = exu)),
% 2.10/1.52 inference(rectify,[status(thm)],[f1])).
% 2.10/1.52 thf(f1,axiom,(
% 2.10/1.52 ((^[X0 : $i > $o] : (? [X1 : $i] : (! [X2 : $i] : ((X0 @ X2) => (X1 = X2)) & (X0 @ X1)))) = exu)),
% 2.10/1.52 file('/export/starexec/sandbox2/tmp/tmp.ujh7G4tgox/DTF2THF_11924.p',exu)).
% 2.10/1.52 thf(f249,plain,(
% 2.10/1.52 ((exu @ (^[Y0 : $i]: (sK0 @ Y0))) != $true)),
% 2.10/1.52 inference(cnf_transformation,[status(thm)],[f192])).
% 2.10/1.52 thf(f1131,plain,(
% 2.10/1.52 ((sK28 @ sK1) = sK1)),
% 2.10/1.52 inference(trivial_inequality_removal,[status(thm)],[f1127])).
% 2.10/1.52 thf(f1127,plain,(
% 2.10/1.52 ((sK28 @ sK1) = sK1) | ($false = $true)),
% 2.10/1.52 inference(superposition,[status(thm)],[f250,f1123])).
% 2.10/1.52 thf(f1123,plain,(
% 2.10/1.52 ( ! [X0 : $i] : (((sK0 @ X0) = $false) | ((sK28 @ X0) = sK1)) )),
% 2.10/1.52 inference(trivial_inequality_removal,[status(thm)],[f1121])).
% 2.10/1.52 thf(f1121,plain,(
% 2.10/1.52 ( ! [X0 : $i] : (((sK28 @ X0) = sK1) | ((sK0 @ X0) = $false) | ($true != $true)) )),
% 2.10/1.52 inference(superposition,[status(thm)],[f251,f793])).
% 2.10/1.52 thf(f793,plain,(
% 2.10/1.52 ( ! [X1 : $i] : (((sK0 @ (sK28 @ X1)) = $true) | ($false = (sK0 @ X1))) )),
% 2.10/1.52 inference(binary_proxy_clausification,[status(thm)],[f791])).
% 2.10/1.52 thf(f251,plain,(
% 2.10/1.52 ( ! [X2 : $i] : (((sK0 @ X2) != $true) | (sK1 = X2)) )),
% 2.10/1.52 inference(cnf_transformation,[status(thm)],[f192])).
% 2.10/1.52 % SZS output end Proof for DTF2THF_11924
% 2.10/1.52 % (12022)------------------------------
% 2.10/1.52 % (12022)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.10/1.52 % (12022)Termination reason: Refutation
% 2.10/1.52
% 2.10/1.52 % (12022)Memory used [KB]: 6396
% 2.10/1.52 % (12022)Time elapsed: 0.045 s
% 2.10/1.52 % (12022)Instructions burned: 93 (million)
% 2.10/1.52 % (12022)------------------------------
% 2.10/1.52 % (12022)------------------------------
% 2.10/1.52 % (12021)Success in time 0.058 s
%------------------------------------------------------------------------------