%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU536^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n022.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 09:35:09 AM UTC 2026
% Result : Theorem 2.09s 1.48s
% Output : Refutation 2.09s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 3
% Syntax : Number of formulae : 17 ( 4 unt; 0 typ; 0 def)
% Number of atoms : 1232 ( 355 equ; 0 cnn)
% Maximal formula atoms : 56 ( 72 avg)
% Number of connectives : 618 ( 31 ~; 8 |; 227 &; 80 @)
% ( 0 <=>; 272 =>; 0 <=; 0 <~>)
% Maximal formula depth : 62 ( 32 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 9 ( 9 >; 0 *; 0 +; 0 <<)
% Number of symbols : 99 ( 96 usr; 56 con; 0-3 aty)
% Number of variables : 41 ( 0 ^; 22 !; 19 ?; 41 :)
% 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_81,type,
sK1: $i > $o ).
thf(func_def_84,type,
sK4: $i > ( $i > $i > $o ) > $i ).
thf(func_def_85,type,
sK5: $i > ( $i > $i > $o ) > $i ).
thf(func_def_86,type,
sK6: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_87,type,
sK7: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_88,type,
sK8: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_89,type,
sK9: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_90,type,
sK10: ( $i > $o ) > $i > $i ).
thf(func_def_91,type,
sK11: $i > $i ).
thf(func_def_92,type,
sK12: $o > $i > $i > $i ).
thf(func_def_93,type,
sK13: ( $i > $o ) > $i ).
thf(func_def_94,type,
sK14: ( $i > $o ) > $i > $i ).
thf(func_def_95,type,
sK15: $i > $i > $i ).
thf(func_def_96,type,
sK16: $i > $i > $i ).
thf(func_def_97,type,
sK17: $i > $i > $i ).
thf(func_def_98,type,
sK18: $i > $i > $i ).
thf(func_def_99,type,
sK19: $i > $i ).
thf(func_def_100,type,
sK20: $i > $i ).
thf(func_def_101,type,
sK21: ( $i > $o ) > $i > $i ).
thf(func_def_102,type,
sK22: ( $i > $o ) > $i ).
thf(func_def_103,type,
sK23: ( $i > $o ) > $i ).
thf(func_def_104,type,
sK24: ( $i > $o ) > $i ).
thf(func_def_105,type,
sK25: ( $i > $o ) > $i > $i ).
thf(func_def_106,type,
sK26: $i > $i ).
thf(func_def_107,type,
sK27: $i > $i > $i ).
thf(func_def_108,type,
sK28: $i > $i > $i ).
thf(func_def_109,type,
sK29: $i > $i ).
thf(func_def_110,type,
sK30: $i > $i ).
thf(func_def_111,type,
sK31: $i > $i ).
thf(func_def_112,type,
sK32: $i > $i > $i ).
thf(func_def_113,type,
sK33: $i > $i > $i ).
thf(func_def_114,type,
sK34: $i > $i > $i ).
thf(func_def_115,type,
sK35: $i > $i > $i > $i ).
thf(f973,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f972,f243]) ).
thf(f243,plain,
( $true
!= ( sK1 @ sK2 ) ),
inference(cnf_transformation,[status(thm)],[f172]) ).
thf(f172,plain,
( ( nonemptyImpWitness = $true )
& ( setunionE = $true )
& ( nonemptyE1 = $true )
& ( exuE2 = $true )
& ( exuE3e = $true )
& ( replAx = $true )
& ( exuE1 = $true )
& ( setext = $true )
& ( emptyinPowerset = $true )
& ( powersetE = $true )
& ( emptysetE = $true )
& ( dsetconstrI = $true )
& ( powersetI = $true )
& ( setadjoinOr = $true )
& ( emptyInPowerset = $true )
& ( omega0Ax = $true )
& ( setoftrueEq = $true )
& ( descrp = $true )
& ( subPowSU = $true )
& ( emptysetAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( setadjoinIR = $true )
& ( dsetconstrEL = $true )
& ( notinsingleton = $true )
& ( emptyE1 = $true )
& ( powersetAx = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( setadjoinIL = $true )
& ( dsetconstrER = $true )
& ( setadjoinE = $true )
& ( upairsetIR = $true )
& ! [X2: $i] :
( ( $true
= ( sK1 @ X2 ) )
| ( ( in @ X2 @ sK0 )
!= $true ) )
& ( $true
= ( in @ sK2 @ sK0 ) )
& ( $true
!= ( sK1 @ sK2 ) )
& ( setextAx = $true )
& ( prop2setE = $true )
& ( notinemptyset = $true )
& ( eqinunit = $true )
& ( emptyinunitempty = $true )
& ( wellorderingAx = $true )
& ( noeltsimpempty = $true )
& ( emptysetimpfalse = $true )
& ( setunionAx = $true )
& ( setadjoinAx = $true )
& ( setbeta = $true )
& ( emptyI = $true )
& ( uniqinunit = $true )
& ( upairsetIL = $true )
& ( vacuousDall = $true )
& ( nonemptyI1 = $true )
& ( foundationAx = $true )
& ( nonemptyI = $true )
& ( setunionI = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f169,f171,f170]) ).
thf(f170,plain,
( ? [X0: $i,X1: $i > $o] :
( ! [X2: $i] :
( ( $true
= ( X1 @ X2 ) )
| ( $true
!= ( in @ X2 @ X0 ) ) )
& ? [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
& ( $true
!= ( X1 @ X3 ) ) ) )
=> ( ! [X2: $i] :
( ( $true
= ( sK1 @ X2 ) )
| ( ( in @ X2 @ sK0 )
!= $true ) )
& ? [X3: $i] :
( ( ( in @ X3 @ sK0 )
= $true )
& ( $true
!= ( sK1 @ X3 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f171,plain,
( ? [X3: $i] :
( ( ( in @ X3 @ sK0 )
= $true )
& ( $true
!= ( sK1 @ X3 ) ) )
=> ( ( $true
= ( in @ sK2 @ sK0 ) )
& ( $true
!= ( sK1 @ sK2 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f169,plain,
( ( nonemptyImpWitness = $true )
& ( setunionE = $true )
& ( nonemptyE1 = $true )
& ( exuE2 = $true )
& ( exuE3e = $true )
& ( replAx = $true )
& ( exuE1 = $true )
& ( setext = $true )
& ( emptyinPowerset = $true )
& ( powersetE = $true )
& ( emptysetE = $true )
& ( dsetconstrI = $true )
& ( powersetI = $true )
& ( setadjoinOr = $true )
& ( emptyInPowerset = $true )
& ( omega0Ax = $true )
& ( setoftrueEq = $true )
& ( descrp = $true )
& ( subPowSU = $true )
& ( emptysetAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( setadjoinIR = $true )
& ( dsetconstrEL = $true )
& ( notinsingleton = $true )
& ( emptyE1 = $true )
& ( powersetAx = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( setadjoinIL = $true )
& ( dsetconstrER = $true )
& ( setadjoinE = $true )
& ( upairsetIR = $true )
& ? [X0: $i,X1: $i > $o] :
( ! [X2: $i] :
( ( $true
= ( X1 @ X2 ) )
| ( $true
!= ( in @ X2 @ X0 ) ) )
& ? [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
& ( $true
!= ( X1 @ X3 ) ) ) )
& ( setextAx = $true )
& ( prop2setE = $true )
& ( notinemptyset = $true )
& ( eqinunit = $true )
& ( emptyinunitempty = $true )
& ( wellorderingAx = $true )
& ( noeltsimpempty = $true )
& ( emptysetimpfalse = $true )
& ( setunionAx = $true )
& ( setadjoinAx = $true )
& ( setbeta = $true )
& ( emptyI = $true )
& ( uniqinunit = $true )
& ( upairsetIL = $true )
& ( vacuousDall = $true )
& ( nonemptyI1 = $true )
& ( foundationAx = $true )
& ( nonemptyI = $true )
& ( setunionI = $true ) ),
inference(rectify,[status(thm)],[f168]) ).
thf(f168,plain,
( ( nonemptyImpWitness = $true )
& ( setunionE = $true )
& ( nonemptyE1 = $true )
& ( exuE2 = $true )
& ( exuE3e = $true )
& ( replAx = $true )
& ( exuE1 = $true )
& ( setext = $true )
& ( emptyinPowerset = $true )
& ( powersetE = $true )
& ( emptysetE = $true )
& ( dsetconstrI = $true )
& ( powersetI = $true )
& ( setadjoinOr = $true )
& ( emptyInPowerset = $true )
& ( omega0Ax = $true )
& ( setoftrueEq = $true )
& ( descrp = $true )
& ( subPowSU = $true )
& ( emptysetAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( setadjoinIR = $true )
& ( dsetconstrEL = $true )
& ( notinsingleton = $true )
& ( emptyE1 = $true )
& ( powersetAx = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( setadjoinIL = $true )
& ( dsetconstrER = $true )
& ( setadjoinE = $true )
& ( upairsetIR = $true )
& ? [X0: $i,X1: $i > $o] :
( ! [X3: $i] :
( ( $true
= ( X1 @ X3 ) )
| ( ( in @ X3 @ X0 )
!= $true ) )
& ? [X2: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
& ( $true
!= ( X1 @ X2 ) ) ) )
& ( setextAx = $true )
& ( prop2setE = $true )
& ( notinemptyset = $true )
& ( eqinunit = $true )
& ( emptyinunitempty = $true )
& ( wellorderingAx = $true )
& ( noeltsimpempty = $true )
& ( emptysetimpfalse = $true )
& ( setunionAx = $true )
& ( setadjoinAx = $true )
& ( setbeta = $true )
& ( emptyI = $true )
& ( uniqinunit = $true )
& ( upairsetIL = $true )
& ( vacuousDall = $true )
& ( nonemptyI1 = $true )
& ( foundationAx = $true )
& ( nonemptyI = $true )
& ( setunionI = $true ) ),
inference(flattening,[status(thm)],[f167]) ).
thf(f167,plain,
( ? [X0: $i,X1: $i > $o] :
( ! [X3: $i] :
( ( $true
= ( X1 @ X3 ) )
| ( ( in @ X3 @ X0 )
!= $true ) )
& ? [X2: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
& ( $true
!= ( X1 @ X2 ) ) ) )
& ( 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)],[f166]) ).
thf(f166,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 )
=> ! [X0: $i,X1: $i > $o] :
( ~ ! [X2: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
=> ( $true
= ( X1 @ X2 ) ) )
=> ? [X3: $i] :
( ( $true
!= ( X1 @ X3 ) )
& ( ( in @ X3 @ X0 )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(flattening,[status(thm)],[f125]) ).
thf(f125,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 )
=> ! [X0: $i,X1: $i > $o] :
( ~ ! [X2: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
=> ( $true
= ( X1 @ X2 ) ) )
=> ? [X3: $i] :
( ( $true
!= ( X1 @ X3 ) )
& ( ( in @ X3 @ X0 )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f124]) ).
thf(f124,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
=> ! [X0: $i,X1: $i > $o] :
( ~ ! [X2: $i] :
( ( in @ X2 @ X0 )
=> ( X1 @ X2 ) )
=> ? [X3: $i] :
( ~ ( X1 @ X3 )
& ( in @ X3 @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f57]) ).
thf(f57,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
=> ! [X3: $i,X0: $i > $o] :
( ~ ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( X0 @ X1 ) )
=> ? [X1: $i] :
( ~ ( X0 @ X1 )
& ( in @ X1 @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f56]) ).
thf(f56,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
=> ! [X3: $i,X0: $i > $o] :
( ~ ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( X0 @ X1 ) )
=> ? [X1: $i] :
( ~ ( X0 @ X1 )
& ( in @ X1 @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.T76pLv6kux/DTF2THF_32465.p',quantDeMorgan1) ).
thf(f972,plain,
( $true
= ( sK1 @ sK2 ) ),
inference(trivial_inequality_removal,[status(thm)],[f970]) ).
thf(f970,plain,
( ( $true
= ( sK1 @ sK2 ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f245,f244]) ).
thf(f244,plain,
( $true
= ( in @ sK2 @ sK0 ) ),
inference(cnf_transformation,[status(thm)],[f172]) ).
thf(f245,plain,
! [X2: $i] :
( ( ( in @ X2 @ sK0 )
!= $true )
| ( $true
= ( sK1 @ X2 ) ) ),
inference(cnf_transformation,[status(thm)],[f172]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU536^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n022.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 12:26:06 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.31/0.47 ---- Original DTF file ---
% 0.31/0.47 thf(spec,logic,$$dhol).
% 0.31/0.47 %------------------------------------------------------------------------------
% 0.31/0.47 % File : SEU536^1 : TPTP v9.2.1. Released v3.7.0.
% 0.31/0.47 % Domain : Set Theory
% 0.31/0.47 % Problem : Preliminary Notions - Bounded Quantifier Laws
% 0.31/0.47 % Version : Especial.
% 0.31/0.47 % English : (! A:i.! phi:i>o.~(! x:i.in x A -> phi x) ->
% 0.31/0.47 % (? x:i.in x A & ~(phi x)))
% 0.31/0.47
% 0.31/0.47 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.31/0.47 % Source : [Bro08]
% 0.31/0.47 % Names : ZFC038g [Bro08]
% 0.31/0.47
% 0.31/0.47 % Status : Theorem
% 0.31/0.47 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.23 v8.1.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.17 v5.4.0, 0.20 v5.3.0, 0.40 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.67 v4.0.0, 0.33 v3.7.0
% 0.31/0.47 % Syntax : Number of formulae : 119 ( 55 unt; 63 typ; 55 def)
% 0.31/0.47 % Number of atoms : 302 ( 74 equ; 0 cnn)
% 0.31/0.47 % Maximal formula atoms : 54 ( 5 avg)
% 0.31/0.47 % Number of connectives : 495 ( 11 ~; 5 |; 21 &; 324 @)
% 0.31/0.47 % ( 8 <=>; 126 =>; 0 <=; 0 <~>)
% 0.31/0.47 % Maximal formula depth : 61 ( 2 avg)
% 0.31/0.47 % Number of types : 2 ( 0 usr)
% 0.31/0.47 % Number of type conns : 30 ( 30 >; 0 *; 0 +; 0 <<)
% 0.31/0.47 % Number of symbols : 66 ( 63 usr; 56 con; 0-2 aty)
% 0.31/0.47 % Number of variables : 163 ( 17 ^; 126 !; 20 ?; 163 :)
% 0.31/0.47 % SPC : TH0_THM_EQU_NAR
% 0.31/0.47
% 0.31/0.47 % Comments : http://mathgate.info/detsetitem.php?id=449
% 0.31/0.47 %------------------------------------------------------------------------------
% 0.31/0.47 thf(in_type,type,
% 0.31/0.47 in: $i > $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(exu_type,type,
% 0.31/0.47 exu: ( $i > $o ) > $o ).
% 0.31/0.47
% 0.31/0.47 thf(exu,definition,
% 0.31/0.47 ( exu
% 0.31/0.47 = ( ^ [Xphi: $i > $o] :
% 0.31/0.47 ? [Xx: $i] :
% 0.31/0.47 ( ( Xphi @ Xx )
% 0.31/0.47 & ! [Xy: $i] :
% 0.31/0.47 ( ( Xphi @ Xy )
% 0.31/0.47 => ( Xx = Xy ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setextAx_type,type,
% 0.31/0.47 setextAx: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setextAx,definition,
% 0.31/0.47 ( setextAx
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 <=> ( in @ Xx @ B ) )
% 0.31/0.47 => ( A = B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptyset_type,type,
% 0.31/0.47 emptyset: $i ).
% 0.31/0.47
% 0.31/0.47 thf(emptysetAx_type,type,
% 0.31/0.47 emptysetAx: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptysetAx,definition,
% 0.31/0.47 ( emptysetAx
% 0.31/0.47 = ( ! [Xx: $i] :
% 0.31/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoin_type,type,
% 0.31/0.47 setadjoin: $i > $i > $i ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinAx_type,type,
% 0.31/0.47 setadjoinAx: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinAx,definition,
% 0.31/0.47 ( setadjoinAx
% 0.31/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/0.47 <=> ( ( Xy = Xx )
% 0.31/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(powerset_type,type,
% 0.31/0.47 powerset: $i > $i ).
% 0.31/0.47
% 0.31/0.47 thf(powersetAx_type,type,
% 0.31/0.47 powersetAx: $o ).
% 0.31/0.47
% 0.31/0.47 thf(powersetAx,definition,
% 0.31/0.47 ( powersetAx
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.31/0.47 <=> ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ B )
% 0.31/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setunion_type,type,
% 0.31/0.47 setunion: $i > $i ).
% 0.31/0.47
% 0.31/0.47 thf(setunionAx_type,type,
% 0.31/0.47 setunionAx: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setunionAx,definition,
% 0.31/0.47 ( setunionAx
% 0.31/0.47 = ( ! [A: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.47 <=> ? [B: $i] :
% 0.31/0.47 ( ( in @ Xx @ B )
% 0.31/0.47 & ( in @ B @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(omega_type,type,
% 0.31/0.47 omega: $i ).
% 0.31/0.47
% 0.31/0.47 thf(omega0Ax_type,type,
% 0.31/0.47 omega0Ax: $o ).
% 0.31/0.47
% 0.31/0.47 thf(omega0Ax,definition,
% 0.31/0.47 ( omega0Ax
% 0.31/0.47 = ( in @ emptyset @ omega ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(omegaSAx_type,type,
% 0.31/0.47 omegaSAx: $o ).
% 0.31/0.47
% 0.31/0.47 thf(omegaSAx,definition,
% 0.31/0.47 ( omegaSAx
% 0.31/0.47 = ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ omega )
% 0.31/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(omegaIndAx_type,type,
% 0.31/0.47 omegaIndAx: $o ).
% 0.31/0.47
% 0.31/0.47 thf(omegaIndAx,definition,
% 0.31/0.47 ( omegaIndAx
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ( ( ( in @ emptyset @ A )
% 0.31/0.47 & ! [Xx: $i] :
% 0.31/0.47 ( ( ( in @ Xx @ omega )
% 0.31/0.47 & ( in @ Xx @ A ) )
% 0.31/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ omega )
% 0.31/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(replAx_type,type,
% 0.31/0.47 replAx: $o ).
% 0.31/0.47
% 0.31/0.47 thf(replAx,definition,
% 0.31/0.47 ( replAx
% 0.31/0.47 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( exu
% 0.31/0.47 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.31/0.47 => ? [B: $i] :
% 0.31/0.47 ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ B )
% 0.31/0.47 <=> ? [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ A )
% 0.31/0.47 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(foundationAx_type,type,
% 0.31/0.47 foundationAx: $o ).
% 0.31/0.47
% 0.31/0.47 thf(foundationAx,definition,
% 0.31/0.47 ( foundationAx
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.31/0.47 => ? [B: $i] :
% 0.31/0.47 ( ( in @ B @ A )
% 0.31/0.47 & ~ ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ B )
% 0.31/0.47 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(wellorderingAx_type,type,
% 0.31/0.47 wellorderingAx: $o ).
% 0.31/0.47
% 0.31/0.47 thf(wellorderingAx,definition,
% 0.31/0.47 ( wellorderingAx
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ? [B: $i] :
% 0.31/0.47 ( ! [C: $i] :
% 0.31/0.47 ( ( in @ C @ B )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ C )
% 0.31/0.47 => ( in @ Xx @ A ) ) )
% 0.31/0.47 & ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( ( in @ Xx @ A )
% 0.31/0.47 & ( in @ Xy @ A ) )
% 0.31/0.47 => ( ! [C: $i] :
% 0.31/0.47 ( ( in @ C @ B )
% 0.31/0.47 => ( ( in @ Xx @ C )
% 0.31/0.47 <=> ( in @ Xy @ C ) ) )
% 0.31/0.47 => ( Xx = Xy ) ) )
% 0.31/0.47 & ! [C: $i,D: $i] :
% 0.31/0.47 ( ( ( in @ C @ B )
% 0.31/0.47 & ( in @ D @ B ) )
% 0.31/0.47 => ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ C )
% 0.31/0.47 => ( in @ Xx @ D ) )
% 0.31/0.47 | ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ D )
% 0.31/0.47 => ( in @ Xx @ C ) ) ) )
% 0.31/0.47 & ! [C: $i] :
% 0.31/0.47 ( ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ C )
% 0.31/0.47 => ( in @ Xx @ A ) )
% 0.31/0.47 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.31/0.47 => ? [D: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ D @ B )
% 0.31/0.47 & ( in @ Xx @ C )
% 0.31/0.47 & ~ ? [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ D )
% 0.31/0.47 & ( in @ Xy @ C ) )
% 0.31/0.47 & ! [E: $i] :
% 0.31/0.47 ( ( in @ E @ B )
% 0.31/0.47 => ( ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ E )
% 0.31/0.47 => ( in @ Xy @ D ) )
% 0.31/0.47 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(descr_type,type,
% 0.31/0.47 descr: ( $i > $o ) > $i ).
% 0.31/0.47
% 0.31/0.47 thf(descrp_type,type,
% 0.31/0.47 descrp: $o ).
% 0.31/0.47
% 0.31/0.47 thf(descrp,definition,
% 0.31/0.47 ( descrp
% 0.31/0.47 = ( ! [Xphi: $i > $o] :
% 0.31/0.47 ( ( exu
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 => ( Xphi
% 0.31/0.47 @ ( descr
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(dsetconstr_type,type,
% 0.31/0.47 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.31/0.47
% 0.31/0.47 thf(dsetconstrI_type,type,
% 0.31/0.47 dsetconstrI: $o ).
% 0.31/0.47
% 0.31/0.47 thf(dsetconstrI,definition,
% 0.31/0.47 ( dsetconstrI
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( Xphi @ Xx )
% 0.31/0.47 => ( in @ Xx
% 0.31/0.47 @ ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(dsetconstrEL_type,type,
% 0.31/0.47 dsetconstrEL: $o ).
% 0.31/0.47
% 0.31/0.47 thf(dsetconstrEL,definition,
% 0.31/0.47 ( dsetconstrEL
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx
% 0.31/0.47 @ ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(dsetconstrER_type,type,
% 0.31/0.47 dsetconstrER: $o ).
% 0.31/0.47
% 0.31/0.47 thf(dsetconstrER,definition,
% 0.31/0.47 ( dsetconstrER
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx
% 0.31/0.47 @ ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(exuE1_type,type,
% 0.31/0.47 exuE1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(exuE1,definition,
% 0.31/0.47 ( exuE1
% 0.31/0.47 = ( ! [Xphi: $i > $o] :
% 0.31/0.47 ( ( exu
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 => ? [Xx: $i] :
% 0.31/0.47 ( ( Xphi @ Xx )
% 0.31/0.47 & ! [Xy: $i] :
% 0.31/0.47 ( ( Xphi @ Xy )
% 0.31/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(prop2set_type,type,
% 0.31/0.47 prop2set: $o > $i ).
% 0.31/0.47
% 0.31/0.47 thf(prop2set,definition,
% 0.31/0.47 ( prop2set
% 0.31/0.47 = ( ^ [Xphi: $o] :
% 0.31/0.47 ( dsetconstr @ ( powerset @ emptyset )
% 0.31/0.47 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(prop2setE_type,type,
% 0.31/0.47 prop2setE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(prop2setE,definition,
% 0.31/0.47 ( prop2setE
% 0.31/0.47 = ( ! [Xphi: $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.31/0.47 => Xphi ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptysetE_type,type,
% 0.31/0.47 emptysetE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptysetE,definition,
% 0.31/0.47 ( emptysetE
% 0.31/0.47 = ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ emptyset )
% 0.31/0.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptysetimpfalse_type,type,
% 0.31/0.47 emptysetimpfalse: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptysetimpfalse,definition,
% 0.31/0.47 ( emptysetimpfalse
% 0.31/0.47 = ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ emptyset )
% 0.31/0.47 => $false ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(notinemptyset_type,type,
% 0.31/0.47 notinemptyset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(notinemptyset,definition,
% 0.31/0.47 ( notinemptyset
% 0.31/0.47 = ( ! [Xx: $i] :
% 0.31/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(exuE3e_type,type,
% 0.31/0.47 exuE3e: $o ).
% 0.31/0.47
% 0.31/0.47 thf(exuE3e,definition,
% 0.31/0.47 ( exuE3e
% 0.31/0.47 = ( ! [Xphi: $i > $o] :
% 0.31/0.47 ( ( exu
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setext_type,type,
% 0.31/0.47 setext: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setext,definition,
% 0.31/0.47 ( setext
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ Xx @ B ) )
% 0.31/0.47 => ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ B )
% 0.31/0.47 => ( in @ Xx @ A ) )
% 0.31/0.47 => ( A = B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptyI_type,type,
% 0.31/0.47 emptyI: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptyI,definition,
% 0.31/0.47 ( emptyI
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ~ ( in @ Xx @ A )
% 0.31/0.47 => ( A = emptyset ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(noeltsimpempty_type,type,
% 0.31/0.47 noeltsimpempty: $o ).
% 0.31/0.47
% 0.31/0.47 thf(noeltsimpempty,definition,
% 0.31/0.47 ( noeltsimpempty
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ~ ( in @ Xx @ A )
% 0.31/0.47 => ( A = emptyset ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setbeta_type,type,
% 0.31/0.47 setbeta: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setbeta,definition,
% 0.31/0.47 ( setbeta
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx
% 0.31/0.47 @ ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/0.47 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(nonempty_type,type,
% 0.31/0.47 nonempty: $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(nonempty,definition,
% 0.31/0.47 ( nonempty
% 0.31/0.47 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(nonemptyE1_type,type,
% 0.31/0.47 nonemptyE1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(nonemptyE1,definition,
% 0.31/0.47 ( nonemptyE1
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ( ( nonempty @ A )
% 0.31/0.47 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(nonemptyI_type,type,
% 0.31/0.47 nonemptyI: $o ).
% 0.31/0.47
% 0.31/0.47 thf(nonemptyI,definition,
% 0.31/0.47 ( nonemptyI
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( Xphi @ Xx )
% 0.31/0.47 => ( nonempty
% 0.31/0.47 @ ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(nonemptyI1_type,type,
% 0.31/0.47 nonemptyI1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(nonemptyI1,definition,
% 0.31/0.47 ( nonemptyI1
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.31/0.47 => ( nonempty @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinIL_type,type,
% 0.31/0.47 setadjoinIL: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinIL,definition,
% 0.31/0.47 ( setadjoinIL
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptyinunitempty_type,type,
% 0.31/0.47 emptyinunitempty: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptyinunitempty,definition,
% 0.31/0.47 ( emptyinunitempty
% 0.31/0.47 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinIR_type,type,
% 0.31/0.47 setadjoinIR: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinIR,definition,
% 0.31/0.47 ( setadjoinIR
% 0.31/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ A )
% 0.31/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinE_type,type,
% 0.31/0.47 setadjoinE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinE,definition,
% 0.31/0.47 ( setadjoinE
% 0.31/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/0.47 => ! [Xphi: $o] :
% 0.31/0.47 ( ( ( Xy = Xx )
% 0.31/0.47 => Xphi )
% 0.31/0.47 => ( ( ( in @ Xy @ A )
% 0.31/0.47 => Xphi )
% 0.31/0.47 => Xphi ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinOr_type,type,
% 0.31/0.47 setadjoinOr: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinOr,definition,
% 0.31/0.47 ( setadjoinOr
% 0.31/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/0.47 => ( ( Xy = Xx )
% 0.31/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setoftrueEq_type,type,
% 0.31/0.47 setoftrueEq: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setoftrueEq,definition,
% 0.31/0.47 ( setoftrueEq
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ( ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xx: $i] : $true )
% 0.31/0.47 = A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(powersetI_type,type,
% 0.31/0.47 powersetI: $o ).
% 0.31/0.47
% 0.31/0.47 thf(powersetI,definition,
% 0.31/0.47 ( powersetI
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ B )
% 0.31/0.47 => ( in @ Xx @ A ) )
% 0.31/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptyinPowerset_type,type,
% 0.31/0.47 emptyinPowerset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptyinPowerset,definition,
% 0.31/0.47 ( emptyinPowerset
% 0.31/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptyInPowerset_type,type,
% 0.31/0.47 emptyInPowerset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptyInPowerset,definition,
% 0.31/0.47 ( emptyInPowerset
% 0.31/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(powersetE_type,type,
% 0.31/0.47 powersetE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(powersetE,definition,
% 0.31/0.47 ( powersetE
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.31/0.47 => ( ( in @ Xx @ B )
% 0.31/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setunionI_type,type,
% 0.31/0.47 setunionI: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setunionI,definition,
% 0.31/0.47 ( setunionI
% 0.31/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.31/0.47 ( ( in @ Xx @ B )
% 0.31/0.47 => ( ( in @ B @ A )
% 0.31/0.47 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setunionE_type,type,
% 0.31/0.47 setunionE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setunionE,definition,
% 0.31/0.47 ( setunionE
% 0.31/0.47 = ( ! [A: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.47 => ! [Xphi: $o] :
% 0.31/0.47 ( ! [B: $i] :
% 0.31/0.47 ( ( in @ Xx @ B )
% 0.31/0.47 => ( ( in @ B @ A )
% 0.31/0.47 => Xphi ) )
% 0.31/0.47 => Xphi ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subPowSU_type,type,
% 0.31/0.47 subPowSU: $o ).
% 0.31/0.47
% 0.31/0.47 thf(subPowSU,definition,
% 0.31/0.47 ( subPowSU
% 0.31/0.47 = ( ! [A: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(exuE2_type,type,
% 0.31/0.47 exuE2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(exuE2,definition,
% 0.31/0.47 ( exuE2
% 0.31/0.47 = ( ! [Xphi: $i > $o] :
% 0.31/0.47 ( ( exu
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 => ? [Xx: $i] :
% 0.31/0.47 ! [Xy: $i] :
% 0.31/0.47 ( ( Xphi @ Xy )
% 0.31/0.47 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(nonemptyImpWitness_type,type,
% 0.31/0.47 nonemptyImpWitness: $o ).
% 0.31/0.47
% 0.31/0.47 thf(nonemptyImpWitness,definition,
% 0.31/0.47 ( nonemptyImpWitness
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ( ( nonempty @ A )
% 0.31/0.47 => ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & $true ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(uniqinunit_type,type,
% 0.31/0.47 uniqinunit: $o ).
% 0.31/0.47
% 0.31/0.47 thf(uniqinunit,definition,
% 0.31/0.47 ( uniqinunit
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(notinsingleton_type,type,
% 0.31/0.47 notinsingleton: $o ).
% 0.31/0.47
% 0.31/0.47 thf(notinsingleton,definition,
% 0.31/0.47 ( notinsingleton
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( Xx != Xy )
% 0.31/0.47 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(eqinunit_type,type,
% 0.31/0.47 eqinunit: $o ).
% 0.31/0.47
% 0.31/0.47 thf(eqinunit,definition,
% 0.31/0.47 ( eqinunit
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( Xx = Xy )
% 0.31/0.47 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(singletonsswitch_type,type,
% 0.31/0.47 singletonsswitch: $o ).
% 0.31/0.47
% 0.31/0.47 thf(singletonsswitch,definition,
% 0.31/0.47 ( singletonsswitch
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(upairsetE_type,type,
% 0.31/0.47 upairsetE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(upairsetE,definition,
% 0.31/0.47 ( upairsetE
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.31/0.47 => ( ( Xz = Xx )
% 0.31/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(upairsetIL_type,type,
% 0.31/0.47 upairsetIL: $o ).
% 0.31/0.47
% 0.31/0.47 thf(upairsetIL,definition,
% 0.31/0.47 ( upairsetIL
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(upairsetIR_type,type,
% 0.31/0.47 upairsetIR: $o ).
% 0.31/0.47
% 0.31/0.47 thf(upairsetIR,definition,
% 0.31/0.47 ( upairsetIR
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptyE1_type,type,
% 0.31/0.47 emptyE1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptyE1,definition,
% 0.31/0.47 ( emptyE1
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.47 ( ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & ( Xphi @ Xx ) )
% 0.31/0.47 => ( ( ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 = emptyset )
% 0.31/0.47 => $false ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(vacuousDall_type,type,
% 0.31/0.47 vacuousDall: $o ).
% 0.31/0.47
% 0.31/0.47 thf(vacuousDall,definition,
% 0.31/0.47 ( vacuousDall
% 0.31/0.47 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ emptyset )
% 0.31/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(quantDeMorgan1,conjecture,
% 0.31/0.47 ( setextAx
% 0.31/0.47 => ( emptysetAx
% 0.31/0.47 => ( setadjoinAx
% 0.31/0.47 => ( powersetAx
% 0.31/0.47 => ( setunionAx
% 0.31/0.47 => ( omega0Ax
% 0.31/0.47 => ( omegaSAx
% 0.31/0.47 => ( omegaIndAx
% 0.31/0.47 => ( replAx
% 0.31/0.47 => ( foundationAx
% 0.31/0.47 => ( wellorderingAx
% 0.31/0.47 => ( descrp
% 0.31/0.47 => ( dsetconstrI
% 0.31/0.47 => ( dsetconstrEL
% 0.31/0.47 => ( dsetconstrER
% 0.31/0.47 => ( exuE1
% 0.31/0.47 => ( prop2setE
% 0.31/0.47 => ( emptysetE
% 0.31/0.47 => ( emptysetimpfalse
% 0.31/0.47 => ( notinemptyset
% 0.31/0.47 => ( exuE3e
% 0.31/0.47 => ( setext
% 0.31/0.47 => ( emptyI
% 0.31/0.47 => ( noeltsimpempty
% 0.31/0.47 => ( setbeta
% 0.31/0.47 => ( nonemptyE1
% 0.31/0.47 => ( nonemptyI
% 0.31/0.47 => ( nonemptyI1
% 0.31/0.47 => ( setadjoinIL
% 0.31/0.47 => ( emptyinunitempty
% 0.31/0.47 => ( setadjoinIR
% 0.31/0.47 => ( setadjoinE
% 0.31/0.47 => ( setadjoinOr
% 0.31/0.47 => ( setoftrueEq
% 0.31/0.47 => ( powersetI
% 0.31/0.47 => ( emptyinPowerset
% 0.31/0.47 => ( emptyInPowerset
% 0.31/0.47 => ( powersetE
% 0.31/0.47 => ( setunionI
% 0.31/0.47 => ( setunionE
% 0.31/0.47 => ( subPowSU
% 0.31/0.47 => ( exuE2
% 0.31/0.47 => ( nonemptyImpWitness
% 0.31/0.47 => ( uniqinunit
% 0.31/0.47 => ( notinsingleton
% 0.31/0.47 => ( eqinunit
% 0.31/0.47 => ( singletonsswitch
% 0.31/0.47 => ( upairsetE
% 0.31/0.47 => ( upairsetIL
% 0.31/0.47 => ( upairsetIR
% 0.31/0.47 => ( emptyE1
% 0.31/0.47 => ( vacuousDall
% 0.31/0.47 => ! [A: $i,Xphi: $i > $o] :
% 0.31/0.47 ( ~ ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( Xphi @ Xx ) )
% 0.31/0.47 => ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 %------------------------------------------------------------------------------
% 0.31/0.47 ------------------------
% 1.87/1.39 ---- Embedded in THF ---
% 1.87/1.39 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.87/1.39 %%% Generated on Tue Feb 24 12:26:07 EST 2026
% 1.87/1.39 %%% using '$$dhol' embedding, version 1.3.0.
% 1.87/1.39 %%% Logic specification used:
% 1.87/1.39 %%% thf(spec, logic, $$dhol).
% 1.87/1.39
% 1.87/1.39 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.T76pLv6kux/DTF2DTF32465.p
% See solution above
% 1.87/1.39 ------------------------
% 2.09/1.41 ---- Cleaned THF ---
% 2.09/1.41 thf(in_type,type,
% 2.09/1.41 in: $i > $i > $o ).
% 2.09/1.41
% 2.09/1.41 thf(exu_type,type,
% 2.09/1.41 exu: ( $i > $o ) > $o ).
% 2.09/1.41
% 2.09/1.41 thf(exu,definition,
% 2.09/1.41 ( exu
% 2.09/1.41 = ( ^ [Xphi: $i > $o] :
% 2.09/1.41 ? [Xx: $i] :
% 2.09/1.41 ( ( Xphi @ Xx )
% 2.09/1.41 & ! [Xy: $i] :
% 2.09/1.41 ( ( Xphi @ Xy )
% 2.09/1.41 => ( Xx = Xy ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(setextAx_type,type,
% 2.09/1.41 setextAx: $o ).
% 2.09/1.41
% 2.09/1.41 thf(setextAx,definition,
% 2.09/1.41 ( setextAx
% 2.09/1.41 = ( ! [A: $i,B: $i] :
% 2.09/1.41 ( ! [Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ A )
% 2.09/1.41 <=> ( in @ Xx @ B ) )
% 2.09/1.41 => ( A = B ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(emptyset_type,type,
% 2.09/1.41 emptyset: $i ).
% 2.09/1.41
% 2.09/1.41 thf(emptysetAx_type,type,
% 2.09/1.41 emptysetAx: $o ).
% 2.09/1.41
% 2.09/1.41 thf(emptysetAx,definition,
% 2.09/1.41 ( emptysetAx
% 2.09/1.41 = ( ! [Xx: $i] :
% 2.09/1.41 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(setadjoin_type,type,
% 2.09/1.41 setadjoin: $i > $i > $i ).
% 2.09/1.41
% 2.09/1.41 thf(setadjoinAx_type,type,
% 2.09/1.41 setadjoinAx: $o ).
% 2.09/1.41
% 2.09/1.41 thf(setadjoinAx,definition,
% 2.09/1.41 ( setadjoinAx
% 2.09/1.41 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.09/1.41 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.09/1.41 <=> ( ( Xy = Xx )
% 2.09/1.41 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(powerset_type,type,
% 2.09/1.41 powerset: $i > $i ).
% 2.09/1.41
% 2.09/1.41 thf(powersetAx_type,type,
% 2.09/1.41 powersetAx: $o ).
% 2.09/1.41
% 2.09/1.41 thf(powersetAx,definition,
% 2.09/1.41 ( powersetAx
% 2.09/1.41 = ( ! [A: $i,B: $i] :
% 2.09/1.41 ( ( in @ B @ ( powerset @ A ) )
% 2.09/1.41 <=> ! [Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ B )
% 2.09/1.41 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(setunion_type,type,
% 2.09/1.41 setunion: $i > $i ).
% 2.09/1.41
% 2.09/1.41 thf(setunionAx_type,type,
% 2.09/1.41 setunionAx: $o ).
% 2.09/1.41
% 2.09/1.41 thf(setunionAx,definition,
% 2.09/1.41 ( setunionAx
% 2.09/1.41 = ( ! [A: $i,Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ ( setunion @ A ) )
% 2.09/1.41 <=> ? [B: $i] :
% 2.09/1.41 ( ( in @ Xx @ B )
% 2.09/1.41 & ( in @ B @ A ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(omega_type,type,
% 2.09/1.41 omega: $i ).
% 2.09/1.41
% 2.09/1.41 thf(omega0Ax_type,type,
% 2.09/1.41 omega0Ax: $o ).
% 2.09/1.41
% 2.09/1.41 thf(omega0Ax,definition,
% 2.09/1.41 ( omega0Ax
% 2.09/1.41 = ( in @ emptyset @ omega ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(omegaSAx_type,type,
% 2.09/1.41 omegaSAx: $o ).
% 2.09/1.41
% 2.09/1.41 thf(omegaSAx,definition,
% 2.09/1.41 ( omegaSAx
% 2.09/1.41 = ( ! [Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ omega )
% 2.09/1.41 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(omegaIndAx_type,type,
% 2.09/1.41 omegaIndAx: $o ).
% 2.09/1.41
% 2.09/1.41 thf(omegaIndAx,definition,
% 2.09/1.41 ( omegaIndAx
% 2.09/1.41 = ( ! [A: $i] :
% 2.09/1.41 ( ( ( in @ emptyset @ A )
% 2.09/1.41 & ! [Xx: $i] :
% 2.09/1.41 ( ( ( in @ Xx @ omega )
% 2.09/1.41 & ( in @ Xx @ A ) )
% 2.09/1.41 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.09/1.41 => ! [Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ omega )
% 2.09/1.41 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(replAx_type,type,
% 2.09/1.41 replAx: $o ).
% 2.09/1.41
% 2.09/1.41 thf(replAx,definition,
% 2.09/1.41 ( replAx
% 2.09/1.41 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.09/1.41 ( ! [Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ A )
% 2.09/1.41 => ( exu
% 2.09/1.41 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.09/1.41 => ? [B: $i] :
% 2.09/1.41 ! [Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ B )
% 2.09/1.41 <=> ? [Xy: $i] :
% 2.09/1.41 ( ( in @ Xy @ A )
% 2.09/1.41 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(foundationAx_type,type,
% 2.09/1.41 foundationAx: $o ).
% 2.09/1.41
% 2.09/1.41 thf(foundationAx,definition,
% 2.09/1.41 ( foundationAx
% 2.09/1.41 = ( ! [A: $i] :
% 2.09/1.41 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.09/1.41 => ? [B: $i] :
% 2.09/1.41 ( ( in @ B @ A )
% 2.09/1.41 & ~ ? [Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ B )
% 2.09/1.41 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(wellorderingAx_type,type,
% 2.09/1.41 wellorderingAx: $o ).
% 2.09/1.41
% 2.09/1.41 thf(wellorderingAx,definition,
% 2.09/1.41 ( wellorderingAx
% 2.09/1.41 = ( ! [A: $i] :
% 2.09/1.41 ? [B: $i] :
% 2.09/1.41 ( ! [C: $i] :
% 2.09/1.41 ( ( in @ C @ B )
% 2.09/1.41 => ! [Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ C )
% 2.09/1.41 => ( in @ Xx @ A ) ) )
% 2.09/1.41 & ! [Xx: $i,Xy: $i] :
% 2.09/1.41 ( ( ( in @ Xx @ A )
% 2.09/1.41 & ( in @ Xy @ A ) )
% 2.09/1.41 => ( ! [C: $i] :
% 2.09/1.41 ( ( in @ C @ B )
% 2.09/1.41 => ( ( in @ Xx @ C )
% 2.09/1.41 <=> ( in @ Xy @ C ) ) )
% 2.09/1.41 => ( Xx = Xy ) ) )
% 2.09/1.41 & ! [C: $i,D: $i] :
% 2.09/1.41 ( ( ( in @ C @ B )
% 2.09/1.41 & ( in @ D @ B ) )
% 2.09/1.41 => ( ! [Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ C )
% 2.09/1.41 => ( in @ Xx @ D ) )
% 2.09/1.41 | ! [Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ D )
% 2.09/1.41 => ( in @ Xx @ C ) ) ) )
% 2.09/1.41 & ! [C: $i] :
% 2.09/1.41 ( ( ! [Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ C )
% 2.09/1.41 => ( in @ Xx @ A ) )
% 2.09/1.41 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.09/1.41 => ? [D: $i,Xx: $i] :
% 2.09/1.41 ( ( in @ D @ B )
% 2.09/1.41 & ( in @ Xx @ C )
% 2.09/1.41 & ~ ? [Xy: $i] :
% 2.09/1.41 ( ( in @ Xy @ D )
% 2.09/1.41 & ( in @ Xy @ C ) )
% 2.09/1.41 & ! [E: $i] :
% 2.09/1.41 ( ( in @ E @ B )
% 2.09/1.41 => ( ! [Xy: $i] :
% 2.09/1.41 ( ( in @ Xy @ E )
% 2.09/1.41 => ( in @ Xy @ D ) )
% 2.09/1.41 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(descr_type,type,
% 2.09/1.41 descr: ( $i > $o ) > $i ).
% 2.09/1.41
% 2.09/1.41 thf(descrp_type,type,
% 2.09/1.41 descrp: $o ).
% 2.09/1.41
% 2.09/1.41 thf(descrp,definition,
% 2.09/1.41 ( descrp
% 2.09/1.41 = ( ! [Xphi: $i > $o] :
% 2.09/1.41 ( ( exu
% 2.09/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.09/1.41 => ( Xphi
% 2.09/1.41 @ ( descr
% 2.09/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(dsetconstr_type,type,
% 2.09/1.41 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.09/1.41
% 2.09/1.41 thf(dsetconstrI_type,type,
% 2.09/1.41 dsetconstrI: $o ).
% 2.09/1.41
% 2.09/1.41 thf(dsetconstrI,definition,
% 2.09/1.41 ( dsetconstrI
% 2.09/1.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ A )
% 2.09/1.41 => ( ( Xphi @ Xx )
% 2.09/1.41 => ( in @ Xx
% 2.09/1.41 @ ( dsetconstr @ A
% 2.09/1.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(dsetconstrEL_type,type,
% 2.09/1.41 dsetconstrEL: $o ).
% 2.09/1.41
% 2.09/1.41 thf(dsetconstrEL,definition,
% 2.09/1.41 ( dsetconstrEL
% 2.09/1.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.09/1.41 ( ( in @ Xx
% 2.09/1.41 @ ( dsetconstr @ A
% 2.09/1.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.09/1.41 => ( in @ Xx @ A ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(dsetconstrER_type,type,
% 2.09/1.41 dsetconstrER: $o ).
% 2.09/1.41
% 2.09/1.41 thf(dsetconstrER,definition,
% 2.09/1.41 ( dsetconstrER
% 2.09/1.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.09/1.41 ( ( in @ Xx
% 2.09/1.41 @ ( dsetconstr @ A
% 2.09/1.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.09/1.41 => ( Xphi @ Xx ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(exuE1_type,type,
% 2.09/1.41 exuE1: $o ).
% 2.09/1.41
% 2.09/1.41 thf(exuE1,definition,
% 2.09/1.41 ( exuE1
% 2.09/1.41 = ( ! [Xphi: $i > $o] :
% 2.09/1.41 ( ( exu
% 2.09/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.09/1.41 => ? [Xx: $i] :
% 2.09/1.41 ( ( Xphi @ Xx )
% 2.09/1.41 & ! [Xy: $i] :
% 2.09/1.41 ( ( Xphi @ Xy )
% 2.09/1.41 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(prop2set_type,type,
% 2.09/1.41 prop2set: $o > $i ).
% 2.09/1.41
% 2.09/1.41 thf(prop2set,definition,
% 2.09/1.41 ( prop2set
% 2.09/1.41 = ( ^ [Xphi: $o] :
% 2.09/1.41 ( dsetconstr @ ( powerset @ emptyset )
% 2.09/1.41 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(prop2setE_type,type,
% 2.09/1.41 prop2setE: $o ).
% 2.09/1.41
% 2.09/1.41 thf(prop2setE,definition,
% 2.09/1.41 ( prop2setE
% 2.09/1.41 = ( ! [Xphi: $o,Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.09/1.41 => Xphi ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(emptysetE_type,type,
% 2.09/1.41 emptysetE: $o ).
% 2.09/1.41
% 2.09/1.41 thf(emptysetE,definition,
% 2.09/1.41 ( emptysetE
% 2.09/1.41 = ( ! [Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ emptyset )
% 2.09/1.41 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(emptysetimpfalse_type,type,
% 2.09/1.41 emptysetimpfalse: $o ).
% 2.09/1.41
% 2.09/1.41 thf(emptysetimpfalse,definition,
% 2.09/1.41 ( emptysetimpfalse
% 2.09/1.41 = ( ! [Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ emptyset )
% 2.09/1.41 => $false ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(notinemptyset_type,type,
% 2.09/1.41 notinemptyset: $o ).
% 2.09/1.41
% 2.09/1.41 thf(notinemptyset,definition,
% 2.09/1.41 ( notinemptyset
% 2.09/1.41 = ( ! [Xx: $i] :
% 2.09/1.41 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(exuE3e_type,type,
% 2.09/1.41 exuE3e: $o ).
% 2.09/1.41
% 2.09/1.41 thf(exuE3e,definition,
% 2.09/1.41 ( exuE3e
% 2.09/1.41 = ( ! [Xphi: $i > $o] :
% 2.09/1.41 ( ( exu
% 2.09/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.09/1.41 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(setext_type,type,
% 2.09/1.41 setext: $o ).
% 2.09/1.41
% 2.09/1.41 thf(setext,definition,
% 2.09/1.41 ( setext
% 2.09/1.41 = ( ! [A: $i,B: $i] :
% 2.09/1.41 ( ! [Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ A )
% 2.09/1.41 => ( in @ Xx @ B ) )
% 2.09/1.41 => ( ! [Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ B )
% 2.09/1.41 => ( in @ Xx @ A ) )
% 2.09/1.41 => ( A = B ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(emptyI_type,type,
% 2.09/1.41 emptyI: $o ).
% 2.09/1.41
% 2.09/1.41 thf(emptyI,definition,
% 2.09/1.41 ( emptyI
% 2.09/1.41 = ( ! [A: $i] :
% 2.09/1.41 ( ! [Xx: $i] :
% 2.09/1.41 ~ ( in @ Xx @ A )
% 2.09/1.41 => ( A = emptyset ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(noeltsimpempty_type,type,
% 2.09/1.41 noeltsimpempty: $o ).
% 2.09/1.41
% 2.09/1.41 thf(noeltsimpempty,definition,
% 2.09/1.41 ( noeltsimpempty
% 2.09/1.41 = ( ! [A: $i] :
% 2.09/1.41 ( ! [Xx: $i] :
% 2.09/1.41 ~ ( in @ Xx @ A )
% 2.09/1.41 => ( A = emptyset ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(setbeta_type,type,
% 2.09/1.41 setbeta: $o ).
% 2.09/1.41
% 2.09/1.41 thf(setbeta,definition,
% 2.09/1.41 ( setbeta
% 2.09/1.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ A )
% 2.09/1.41 => ( ( in @ Xx
% 2.09/1.41 @ ( dsetconstr @ A
% 2.09/1.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.09/1.41 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(nonempty_type,type,
% 2.09/1.41 nonempty: $i > $o ).
% 2.09/1.41
% 2.09/1.41 thf(nonempty,definition,
% 2.09/1.41 ( nonempty
% 2.09/1.41 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(nonemptyE1_type,type,
% 2.09/1.41 nonemptyE1: $o ).
% 2.09/1.41
% 2.09/1.41 thf(nonemptyE1,definition,
% 2.09/1.41 ( nonemptyE1
% 2.09/1.41 = ( ! [A: $i] :
% 2.09/1.41 ( ( nonempty @ A )
% 2.09/1.41 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(nonemptyI_type,type,
% 2.09/1.41 nonemptyI: $o ).
% 2.09/1.41
% 2.09/1.41 thf(nonemptyI,definition,
% 2.09/1.41 ( nonemptyI
% 2.09/1.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ A )
% 2.09/1.41 => ( ( Xphi @ Xx )
% 2.09/1.41 => ( nonempty
% 2.09/1.41 @ ( dsetconstr @ A
% 2.09/1.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(nonemptyI1_type,type,
% 2.09/1.41 nonemptyI1: $o ).
% 2.09/1.41
% 2.09/1.41 thf(nonemptyI1,definition,
% 2.09/1.41 ( nonemptyI1
% 2.09/1.41 = ( ! [A: $i] :
% 2.09/1.41 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.09/1.41 => ( nonempty @ A ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(setadjoinIL_type,type,
% 2.09/1.41 setadjoinIL: $o ).
% 2.09/1.41
% 2.09/1.41 thf(setadjoinIL,definition,
% 2.09/1.41 ( setadjoinIL
% 2.09/1.41 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(emptyinunitempty_type,type,
% 2.09/1.41 emptyinunitempty: $o ).
% 2.09/1.41
% 2.09/1.41 thf(emptyinunitempty,definition,
% 2.09/1.41 ( emptyinunitempty
% 2.09/1.41 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(setadjoinIR_type,type,
% 2.09/1.41 setadjoinIR: $o ).
% 2.09/1.41
% 2.09/1.41 thf(setadjoinIR,definition,
% 2.09/1.41 ( setadjoinIR
% 2.09/1.41 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.09/1.41 ( ( in @ Xy @ A )
% 2.09/1.41 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(setadjoinE_type,type,
% 2.09/1.41 setadjoinE: $o ).
% 2.09/1.41
% 2.09/1.41 thf(setadjoinE,definition,
% 2.09/1.41 ( setadjoinE
% 2.09/1.41 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.09/1.41 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.09/1.41 => ! [Xphi: $o] :
% 2.09/1.41 ( ( ( Xy = Xx )
% 2.09/1.41 => Xphi )
% 2.09/1.41 => ( ( ( in @ Xy @ A )
% 2.09/1.41 => Xphi )
% 2.09/1.41 => Xphi ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(setadjoinOr_type,type,
% 2.09/1.41 setadjoinOr: $o ).
% 2.09/1.41
% 2.09/1.41 thf(setadjoinOr,definition,
% 2.09/1.41 ( setadjoinOr
% 2.09/1.41 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.09/1.41 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.09/1.41 => ( ( Xy = Xx )
% 2.09/1.41 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(setoftrueEq_type,type,
% 2.09/1.41 setoftrueEq: $o ).
% 2.09/1.41
% 2.09/1.41 thf(setoftrueEq,definition,
% 2.09/1.41 ( setoftrueEq
% 2.09/1.41 = ( ! [A: $i] :
% 2.09/1.41 ( ( dsetconstr @ A
% 2.09/1.41 @ ^ [Xx: $i] : $true )
% 2.09/1.41 = A ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(powersetI_type,type,
% 2.09/1.41 powersetI: $o ).
% 2.09/1.41
% 2.09/1.41 thf(powersetI,definition,
% 2.09/1.41 ( powersetI
% 2.09/1.41 = ( ! [A: $i,B: $i] :
% 2.09/1.41 ( ! [Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ B )
% 2.09/1.41 => ( in @ Xx @ A ) )
% 2.09/1.41 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(emptyinPowerset_type,type,
% 2.09/1.41 emptyinPowerset: $o ).
% 2.09/1.41
% 2.09/1.41 thf(emptyinPowerset,definition,
% 2.09/1.41 ( emptyinPowerset
% 2.09/1.41 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(emptyInPowerset_type,type,
% 2.09/1.41 emptyInPowerset: $o ).
% 2.09/1.41
% 2.09/1.41 thf(emptyInPowerset,definition,
% 2.09/1.41 ( emptyInPowerset
% 2.09/1.41 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(powersetE_type,type,
% 2.09/1.41 powersetE: $o ).
% 2.09/1.41
% 2.09/1.41 thf(powersetE,definition,
% 2.09/1.41 ( powersetE
% 2.09/1.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.09/1.41 ( ( in @ B @ ( powerset @ A ) )
% 2.09/1.41 => ( ( in @ Xx @ B )
% 2.09/1.41 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(setunionI_type,type,
% 2.09/1.41 setunionI: $o ).
% 2.09/1.41
% 2.09/1.41 thf(setunionI,definition,
% 2.09/1.41 ( setunionI
% 2.09/1.41 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.09/1.41 ( ( in @ Xx @ B )
% 2.09/1.41 => ( ( in @ B @ A )
% 2.09/1.41 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(setunionE_type,type,
% 2.09/1.41 setunionE: $o ).
% 2.09/1.41
% 2.09/1.41 thf(setunionE,definition,
% 2.09/1.41 ( setunionE
% 2.09/1.41 = ( ! [A: $i,Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ ( setunion @ A ) )
% 2.09/1.41 => ! [Xphi: $o] :
% 2.09/1.41 ( ! [B: $i] :
% 2.09/1.41 ( ( in @ Xx @ B )
% 2.09/1.41 => ( ( in @ B @ A )
% 2.09/1.41 => Xphi ) )
% 2.09/1.41 => Xphi ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(subPowSU_type,type,
% 2.09/1.41 subPowSU: $o ).
% 2.09/1.41
% 2.09/1.41 thf(subPowSU,definition,
% 2.09/1.41 ( subPowSU
% 2.09/1.41 = ( ! [A: $i,Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ A )
% 2.09/1.41 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(exuE2_type,type,
% 2.09/1.41 exuE2: $o ).
% 2.09/1.41
% 2.09/1.41 thf(exuE2,definition,
% 2.09/1.41 ( exuE2
% 2.09/1.41 = ( ! [Xphi: $i > $o] :
% 2.09/1.41 ( ( exu
% 2.09/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.09/1.41 => ? [Xx: $i] :
% 2.09/1.41 ! [Xy: $i] :
% 2.09/1.41 ( ( Xphi @ Xy )
% 2.09/1.41 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(nonemptyImpWitness_type,type,
% 2.09/1.41 nonemptyImpWitness: $o ).
% 2.09/1.41
% 2.09/1.41 thf(nonemptyImpWitness,definition,
% 2.09/1.41 ( nonemptyImpWitness
% 2.09/1.41 = ( ! [A: $i] :
% 2.09/1.41 ( ( nonempty @ A )
% 2.09/1.41 => ? [Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ A )
% 2.09/1.41 & $true ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(uniqinunit_type,type,
% 2.09/1.41 uniqinunit: $o ).
% 2.09/1.41
% 2.09/1.41 thf(uniqinunit,definition,
% 2.09/1.41 ( uniqinunit
% 2.09/1.41 = ( ! [Xx: $i,Xy: $i] :
% 2.09/1.41 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.09/1.41 => ( Xx = Xy ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(notinsingleton_type,type,
% 2.09/1.41 notinsingleton: $o ).
% 2.09/1.41
% 2.09/1.41 thf(notinsingleton,definition,
% 2.09/1.41 ( notinsingleton
% 2.09/1.41 = ( ! [Xx: $i,Xy: $i] :
% 2.09/1.41 ( ( Xx != Xy )
% 2.09/1.41 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(eqinunit_type,type,
% 2.09/1.41 eqinunit: $o ).
% 2.09/1.41
% 2.09/1.41 thf(eqinunit,definition,
% 2.09/1.41 ( eqinunit
% 2.09/1.41 = ( ! [Xx: $i,Xy: $i] :
% 2.09/1.41 ( ( Xx = Xy )
% 2.09/1.41 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(singletonsswitch_type,type,
% 2.09/1.41 singletonsswitch: $o ).
% 2.09/1.41
% 2.09/1.41 thf(singletonsswitch,definition,
% 2.09/1.41 ( singletonsswitch
% 2.09/1.41 = ( ! [Xx: $i,Xy: $i] :
% 2.09/1.41 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.09/1.41 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(upairsetE_type,type,
% 2.09/1.41 upairsetE: $o ).
% 2.09/1.41
% 2.09/1.41 thf(upairsetE,definition,
% 2.09/1.41 ( upairsetE
% 2.09/1.41 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.09/1.41 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.09/1.41 => ( ( Xz = Xx )
% 2.09/1.41 | ( Xz = Xy ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(upairsetIL_type,type,
% 2.09/1.41 upairsetIL: $o ).
% 2.09/1.41
% 2.09/1.41 thf(upairsetIL,definition,
% 2.09/1.41 ( upairsetIL
% 2.09/1.41 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(upairsetIR_type,type,
% 2.09/1.41 upairsetIR: $o ).
% 2.09/1.41
% 2.09/1.41 thf(upairsetIR,definition,
% 2.09/1.41 ( upairsetIR
% 2.09/1.41 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(emptyE1_type,type,
% 2.09/1.41 emptyE1: $o ).
% 2.09/1.41
% 2.09/1.41 thf(emptyE1,definition,
% 2.09/1.41 ( emptyE1
% 2.09/1.41 = ( ! [A: $i,Xphi: $i > $o] :
% 2.09/1.41 ( ? [Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ A )
% 2.09/1.41 & ( Xphi @ Xx ) )
% 2.09/1.41 => ( ( ( dsetconstr @ A
% 2.09/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.09/1.41 = emptyset )
% 2.09/1.41 => $false ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(vacuousDall_type,type,
% 2.09/1.41 vacuousDall: $o ).
% 2.09/1.41
% 2.09/1.41 thf(vacuousDall,definition,
% 2.09/1.41 ( vacuousDall
% 2.09/1.41 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ emptyset )
% 2.09/1.41 => ( Xphi @ Xx ) ) ) ) ).
% 2.09/1.41
% 2.09/1.41 thf(quantDeMorgan1,conjecture,
% 2.09/1.41 ( setextAx
% 2.09/1.41 => ( emptysetAx
% 2.09/1.41 => ( setadjoinAx
% 2.09/1.41 => ( powersetAx
% 2.09/1.41 => ( setunionAx
% 2.09/1.41 => ( omega0Ax
% 2.09/1.41 => ( omegaSAx
% 2.09/1.41 => ( omegaIndAx
% 2.09/1.41 => ( replAx
% 2.09/1.41 => ( foundationAx
% 2.09/1.41 => ( wellorderingAx
% 2.09/1.41 => ( descrp
% 2.09/1.41 => ( dsetconstrI
% 2.09/1.41 => ( dsetconstrEL
% 2.09/1.41 => ( dsetconstrER
% 2.09/1.41 => ( exuE1
% 2.09/1.41 => ( prop2setE
% 2.09/1.41 => ( emptysetE
% 2.09/1.41 => ( emptysetimpfalse
% 2.09/1.41 => ( notinemptyset
% 2.09/1.41 => ( exuE3e
% 2.09/1.41 => ( setext
% 2.09/1.41 => ( emptyI
% 2.09/1.41 => ( noeltsimpempty
% 2.09/1.41 => ( setbeta
% 2.09/1.41 => ( nonemptyE1
% 2.09/1.41 => ( nonemptyI
% 2.09/1.41 => ( nonemptyI1
% 2.09/1.41 => ( setadjoinIL
% 2.09/1.41 => ( emptyinunitempty
% 2.09/1.41 => ( setadjoinIR
% 2.09/1.41 => ( setadjoinE
% 2.09/1.41 => ( setadjoinOr
% 2.09/1.41 => ( setoftrueEq
% 2.09/1.41 => ( powersetI
% 2.09/1.41 => ( emptyinPowerset
% 2.09/1.41 => ( emptyInPowerset
% 2.09/1.41 => ( powersetE
% 2.09/1.41 => ( setunionI
% 2.09/1.41 => ( setunionE
% 2.09/1.41 => ( subPowSU
% 2.09/1.41 => ( exuE2
% 2.09/1.41 => ( nonemptyImpWitness
% 2.09/1.41 => ( uniqinunit
% 2.09/1.41 => ( notinsingleton
% 2.09/1.41 => ( eqinunit
% 2.09/1.41 => ( singletonsswitch
% 2.09/1.41 => ( upairsetE
% 2.09/1.41 => ( upairsetIL
% 2.09/1.41 => ( upairsetIR
% 2.09/1.41 => ( emptyE1
% 2.09/1.41 => ( vacuousDall
% 2.09/1.41 => ! [A: $i,Xphi: $i > $o] :
% 2.09/1.41 ( ~ ! [Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ A )
% 2.09/1.41 => ( Xphi @ Xx ) )
% 2.09/1.41 => ? [Xx: $i] :
% 2.09/1.41 ( ( in @ Xx @ A )
% 2.09/1.41 & ~ ( Xphi @ Xx ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.09/1.41 ------------------------
% 2.09/1.42 % (32601)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_32465 for (2999ds/18Mi)
% 2.09/1.42 % (32598)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_32465 for (2999ds/2Mi)
% 2.09/1.43 % (32598)Instruction limit reached!
% 2.09/1.43 % (32598)------------------------------
% 2.09/1.43 % (32598)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.43 % (32598)Termination reason: Unknown
% 2.09/1.43 % (32598)Termination phase: shuffling
% 2.09/1.43
% 2.09/1.43 % (32598)Memory used [KB]: 1023
% 2.09/1.43 % (32598)Time elapsed: 0.002 s
% 2.09/1.43 % (32598)Instructions burned: 3 (million)
% 2.09/1.43 % (32598)------------------------------
% 2.09/1.43 % (32598)------------------------------
% 2.09/1.43 % (32596)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_32465 for (2999ds/4Mi)
% 2.09/1.43 % (32596)Instruction limit reached!
% 2.09/1.43 % (32596)------------------------------
% 2.09/1.43 % (32596)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.43 % (32596)Termination reason: Unknown
% 2.09/1.43 % (32596)Termination phase: shuffling
% 2.09/1.43
% 2.09/1.43 % (32596)Memory used [KB]: 1151
% 2.09/1.43 % (32596)Time elapsed: 0.003 s
% 2.09/1.43 % (32596)Instructions burned: 5 (million)
% 2.09/1.43 % (32596)------------------------------
% 2.09/1.43 % (32596)------------------------------
% 2.09/1.43 % (32601)Instruction limit reached!
% 2.09/1.43 % (32601)------------------------------
% 2.09/1.43 % (32601)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.43 % (32601)Termination reason: Unknown
% 2.09/1.43 % (32601)Termination phase: Preprocessing 3
% 2.09/1.43
% 2.09/1.43 % (32601)Memory used [KB]: 1535
% 2.09/1.43 % (32601)Time elapsed: 0.007 s
% 2.09/1.43 % (32601)Instructions burned: 19 (million)
% 2.09/1.43 % (32601)------------------------------
% 2.09/1.43 % (32601)------------------------------
% 2.09/1.43 % (32597)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_32465 for (2999ds/27Mi)
% 2.09/1.43 % (32600)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_32465 for (2999ds/275Mi)
% 2.09/1.43 % (32599)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_32465 for (2999ds/2Mi)
% 2.09/1.43 % (32595)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_32465 for (2999ds/183Mi)
% 2.09/1.43 % (32599)Instruction limit reached!
% 2.09/1.43 % (32599)------------------------------
% 2.09/1.43 % (32599)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.43 % (32599)Termination reason: Unknown
% 2.09/1.43 % (32599)Termination phase: shuffling
% 2.09/1.43
% 2.09/1.43 % (32599)Memory used [KB]: 1023
% 2.09/1.43 % (32599)Time elapsed: 0.003 s
% 2.09/1.43 % (32599)Instructions burned: 2 (million)
% 2.09/1.43 % (32599)------------------------------
% 2.09/1.43 % (32599)------------------------------
% 2.09/1.44 % (32602)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_32465 for (2999ds/3Mi)
% 2.09/1.44 % (32602)Instruction limit reached!
% 2.09/1.44 % (32602)------------------------------
% 2.09/1.44 % (32602)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.44 % (32602)Termination reason: Unknown
% 2.09/1.44 % (32602)Termination phase: shuffling
% 2.09/1.44
% 2.09/1.44 % (32602)Memory used [KB]: 1151
% 2.09/1.44 % (32602)Time elapsed: 0.003 s
% 2.09/1.44 % (32602)Instructions burned: 4 (million)
% 2.09/1.44 % (32602)------------------------------
% 2.09/1.44 % (32602)------------------------------
% 2.09/1.44 % (32603)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_32465 for (2999ds/37Mi)
% 2.09/1.44 % (32604)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_32465 for (2999ds/15Mi)
% 2.09/1.44 % (32604)Instruction limit reached!
% 2.09/1.44 % (32604)------------------------------
% 2.09/1.44 % (32604)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.44 % (32604)Termination reason: Unknown
% 2.09/1.44 % (32604)Termination phase: Preprocessing 3
% 2.09/1.44
% 2.09/1.44 % (32604)Memory used [KB]: 1535
% 2.09/1.44 % (32604)Time elapsed: 0.006 s
% 2.09/1.44 % (32604)Instructions burned: 17 (million)
% 2.09/1.44 % (32604)------------------------------
% 2.09/1.44 % (32604)------------------------------
% 2.09/1.45 % (32597)Instruction limit reached!
% 2.09/1.45 % (32597)------------------------------
% 2.09/1.45 % (32597)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.45 % (32597)Termination reason: Unknown
% 2.09/1.45 % (32597)Termination phase: Saturation
% 2.09/1.45
% 2.09/1.45 % (32597)Memory used [KB]: 5756
% 2.09/1.45 % (32597)Time elapsed: 0.016 s
% 2.09/1.45 % (32597)Instructions burned: 28 (million)
% 2.09/1.45 % (32597)------------------------------
% 2.09/1.45 % (32597)------------------------------
% 2.09/1.45 % (32606)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_32465 for (2999ds/1041Mi)
% 2.09/1.45 % (32603)Instruction limit reached!
% 2.09/1.45 % (32603)------------------------------
% 2.09/1.45 % (32603)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.45 % (32603)Termination reason: Unknown
% 2.09/1.45 % (32603)Termination phase: Saturation
% 2.09/1.45
% 2.09/1.45 % (32603)Memory used [KB]: 5756
% 2.09/1.45 % (32603)Time elapsed: 0.012 s
% 2.09/1.45 % (32603)Instructions burned: 40 (million)
% 2.09/1.45 % (32603)------------------------------
% 2.09/1.45 % (32603)------------------------------
% 2.09/1.45 % (32605)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_32465 for (2999ds/3Mi)
% 2.09/1.45 % (32605)Instruction limit reached!
% 2.09/1.45 % (32605)------------------------------
% 2.09/1.45 % (32605)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.45 % (32605)Termination reason: Unknown
% 2.09/1.45 % (32605)Termination phase: shuffling
% 2.09/1.45
% 2.09/1.45 % (32605)Memory used [KB]: 1023
% 2.09/1.45 % (32605)Time elapsed: 0.004 s
% 2.09/1.45 % (32605)Instructions burned: 3 (million)
% 2.09/1.45 % (32605)------------------------------
% 2.09/1.45 % (32605)------------------------------
% 2.09/1.45 % (32607)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_32465 for (2999ds/7Mi)
% 2.09/1.46 % (32607)Instruction limit reached!
% 2.09/1.46 % (32607)------------------------------
% 2.09/1.46 % (32607)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.46 % (32607)Termination reason: Unknown
% 2.09/1.46 % (32607)Termination phase: shuffling
% 2.09/1.46
% 2.09/1.46 % (32607)Memory used [KB]: 1151
% 2.09/1.46 % (32607)Time elapsed: 0.003 s
% 2.09/1.46 % (32607)Instructions burned: 7 (million)
% 2.09/1.46 % (32607)------------------------------
% 2.09/1.46 % (32607)------------------------------
% 2.09/1.46 % (32609)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_32465 for (2999ds/3Mi)
% 2.09/1.46 % (32609)Instruction limit reached!
% 2.09/1.46 % (32609)------------------------------
% 2.09/1.46 % (32609)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.46 % (32609)Termination reason: Unknown
% 2.09/1.46 % (32609)Termination phase: shuffling
% 2.09/1.46
% 2.09/1.46 % (32609)Memory used [KB]: 1151
% 2.09/1.46 % (32609)Time elapsed: 0.002 s
% 2.09/1.46 % (32609)Instructions burned: 4 (million)
% 2.09/1.46 % (32609)------------------------------
% 2.09/1.46 % (32609)------------------------------
% 2.09/1.46 % (32608)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_32465 for (2999ds/16Mi)
% 2.09/1.47 % (32611)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_32465 for (2999ds/7Mi)
% 2.09/1.47 % (32608)Instruction limit reached!
% 2.09/1.47 % (32608)------------------------------
% 2.09/1.47 % (32608)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.47 % (32608)Termination reason: Unknown
% 2.09/1.47 % (32608)Termination phase: Function definition elimination
% 2.09/1.47
% 2.09/1.47 % (32608)Memory used [KB]: 1279
% 2.09/1.47 % (32608)Time elapsed: 0.007 s
% 2.09/1.47 % (32608)Instructions burned: 19 (million)
% 2.09/1.47 % (32608)------------------------------
% 2.09/1.47 % (32608)------------------------------
% 2.09/1.47 % (32611)Instruction limit reached!
% 2.09/1.47 % (32611)------------------------------
% 2.09/1.47 % (32611)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.47 % (32611)Termination reason: Unknown
% 2.09/1.47 % (32611)Termination phase: shuffling
% 2.09/1.47
% 2.09/1.47 % (32611)Memory used [KB]: 1151
% 2.09/1.47 % (32611)Time elapsed: 0.003 s
% 2.09/1.47 % (32611)Instructions burned: 8 (million)
% 2.09/1.47 % (32611)------------------------------
% 2.09/1.47 % (32611)------------------------------
% 2.09/1.47 % (32610)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_32465 for (2999ds/3Mi)
% 2.09/1.47 % (32612)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_32465 for (2999ds/3Mi)
% 2.09/1.47 % (32612)Instruction limit reached!
% 2.09/1.47 % (32612)------------------------------
% 2.09/1.47 % (32612)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.47 % (32612)Termination reason: Unknown
% 2.09/1.47 % (32612)Termination phase: shuffling
% 2.09/1.47
% 2.09/1.47 % (32612)Memory used [KB]: 1151
% 2.09/1.47 % (32612)Time elapsed: 0.003 s
% 2.09/1.47 % (32612)Instructions burned: 4 (million)
% 2.09/1.47 % (32612)------------------------------
% 2.09/1.47 % (32612)------------------------------
% 2.09/1.47 % (32610)Instruction limit reached!
% 2.09/1.47 % (32610)------------------------------
% 2.09/1.47 % (32610)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.47 % (32610)Termination reason: Unknown
% 2.09/1.47 % (32610)Termination phase: shuffling
% 2.09/1.47
% 2.09/1.47 % (32610)Memory used [KB]: 1151
% 2.09/1.47 % (32610)Time elapsed: 0.004 s
% 2.09/1.47 % (32610)Instructions burned: 4 (million)
% 2.09/1.47 % (32610)------------------------------
% 2.09/1.47 % (32610)------------------------------
% 2.09/1.48 % (32595)First to succeed.
% 2.09/1.48 % (32613)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_32465 for (2999ds/4Mi)
% 2.09/1.48 % (32614)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_32465 for (2999ds/18Mi)
% 2.09/1.48 % (32613)Instruction limit reached!
% 2.09/1.48 % (32613)------------------------------
% 2.09/1.48 % (32613)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.48 % (32613)Termination reason: Unknown
% 2.09/1.48 % (32613)Termination phase: shuffling
% 2.09/1.48
% 2.09/1.48 % (32613)Memory used [KB]: 1151
% 2.09/1.48 % (32613)Time elapsed: 0.003 s
% 2.09/1.48 % (32613)Instructions burned: 4 (million)
% 2.09/1.48 % (32613)------------------------------
% 2.09/1.48 % (32613)------------------------------
% 2.09/1.48 % (32595)Refutation found. Thanks to Tanya!
% 2.09/1.48 % SZS status Theorem for DTF2THF_32465
% 2.09/1.48 % SZS output start Proof for DTF2THF_32465
% 2.09/1.48 thf(func_def_0, type, in: $i > $i > $o).
% 2.09/1.48 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.09/1.48 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.09/1.48 thf(func_def_8, type, powerset: $i > $i).
% 2.09/1.48 thf(func_def_10, type, setunion: $i > $i).
% 2.09/1.48 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.09/1.48 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.09/1.48 thf(func_def_26, type, prop2set: $o > $i).
% 2.09/1.48 thf(func_def_36, type, nonempty: $i > $o).
% 2.09/1.48 thf(func_def_81, type, sK1: $i > $o).
% 2.09/1.48 thf(func_def_84, type, sK4: $i > ($i > $i > $o) > $i).
% 2.09/1.48 thf(func_def_85, type, sK5: $i > ($i > $i > $o) > $i).
% 2.09/1.48 thf(func_def_86, type, sK6: $i > $i > ($i > $i > $o) > $i).
% 2.09/1.48 thf(func_def_87, type, sK7: $i > $i > ($i > $i > $o) > $i).
% 2.09/1.48 thf(func_def_88, type, sK8: $i > $i > ($i > $i > $o) > $i).
% 2.09/1.48 thf(func_def_89, type, sK9: $i > $i > ($i > $i > $o) > $i).
% 2.09/1.48 thf(func_def_90, type, sK10: ($i > $o) > $i > $i).
% 2.09/1.48 thf(func_def_91, type, sK11: $i > $i).
% 2.09/1.48 thf(func_def_92, type, sK12: $o > $i > $i > $i).
% 2.09/1.48 thf(func_def_93, type, sK13: ($i > $o) > $i).
% 2.09/1.48 thf(func_def_94, type, sK14: ($i > $o) > $i > $i).
% 2.09/1.48 thf(func_def_95, type, sK15: $i > $i > $i).
% 2.09/1.48 thf(func_def_96, type, sK16: $i > $i > $i).
% 2.09/1.48 thf(func_def_97, type, sK17: $i > $i > $i).
% 2.09/1.48 thf(func_def_98, type, sK18: $i > $i > $i).
% 2.09/1.48 thf(func_def_99, type, sK19: $i > $i).
% 2.09/1.48 thf(func_def_100, type, sK20: $i > $i).
% 2.09/1.48 thf(func_def_101, type, sK21: ($i > $o) > $i > $i).
% 2.09/1.48 thf(func_def_102, type, sK22: ($i > $o) > $i).
% 2.09/1.48 thf(func_def_103, type, sK23: ($i > $o) > $i).
% 2.09/1.48 thf(func_def_104, type, sK24: ($i > $o) > $i).
% 2.09/1.48 thf(func_def_105, type, sK25: ($i > $o) > $i > $i).
% 2.09/1.48 thf(func_def_106, type, sK26: $i > $i).
% 2.09/1.48 thf(func_def_107, type, sK27: $i > $i > $i).
% 2.09/1.48 thf(func_def_108, type, sK28: $i > $i > $i).
% 2.09/1.48 thf(func_def_109, type, sK29: $i > $i).
% 2.09/1.48 thf(func_def_110, type, sK30: $i > $i).
% 2.09/1.48 thf(func_def_111, type, sK31: $i > $i).
% 2.09/1.48 thf(func_def_112, type, sK32: $i > $i > $i).
% 2.09/1.48 thf(func_def_113, type, sK33: $i > $i > $i).
% 2.09/1.48 thf(func_def_114, type, sK34: $i > $i > $i).
% 2.09/1.48 thf(func_def_115, type, sK35: $i > $i > $i > $i).
% 2.09/1.48 thf(f973,plain,(
% 2.09/1.48 $false),
% 2.09/1.48 inference(subsumption_resolution,[status(thm)],[f972,f243])).
% 2.09/1.48 thf(f243,plain,(
% 2.09/1.48 ($true != (sK1 @ sK2))),
% 2.09/1.48 inference(cnf_transformation,[status(thm)],[f172])).
% 2.09/1.48 thf(f172,plain,(
% 2.09/1.48 (nonemptyImpWitness = $true) & (setunionE = $true) & (nonemptyE1 = $true) & (exuE2 = $true) & (exuE3e = $true) & (replAx = $true) & (exuE1 = $true) & (setext = $true) & (emptyinPowerset = $true) & (powersetE = $true) & (emptysetE = $true) & (dsetconstrI = $true) & (powersetI = $true) & (setadjoinOr = $true) & (emptyInPowerset = $true) & (omega0Ax = $true) & (setoftrueEq = $true) & (descrp = $true) & (subPowSU = $true) & (emptysetAx = $true) & (omegaIndAx = $true) & (omegaSAx = $true) & (setadjoinIR = $true) & (dsetconstrEL = $true) & (notinsingleton = $true) & (emptyE1 = $true) & (powersetAx = $true) & (upairsetE = $true) & (singletonsswitch = $true) & (setadjoinIL = $true) & (dsetconstrER = $true) & (setadjoinE = $true) & (upairsetIR = $true) & (! [X2 : $i] : (($true = (sK1 @ X2)) | ((in @ X2 @ sK0) != $true)) & (($true = (in @ sK2 @ sK0)) & ($true != (sK1 @ sK2)))) & (setextAx = $true) & (prop2setE = $true) & (notinemptyset = $true) & (eqinunit = $true) & (emptyinunitempty = $true) & (wellorderingAx = $true) & (noeltsimpempty = $true) & (emptysetimpfalse = $true) & (setunionAx = $true) & (setadjoinAx = $true) & (setbeta = $true) & (emptyI = $true) & (uniqinunit = $true) & (upairsetIL = $true) & (vacuousDall = $true) & (nonemptyI1 = $true) & (foundationAx = $true) & (nonemptyI = $true) & (setunionI = $true)),
% 2.09/1.48 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f169,f171,f170])).
% 2.09/1.48 thf(f170,plain,(
% 2.09/1.48 ? [X0 : $i,X1 : $i > $o] : (! [X2 : $i] : (($true = (X1 @ X2)) | ($true != (in @ X2 @ X0))) & ? [X3 : $i] : (((in @ X3 @ X0) = $true) & ($true != (X1 @ X3)))) => (! [X2 : $i] : (($true = (sK1 @ X2)) | ((in @ X2 @ sK0) != $true)) & ? [X3 : $i] : (((in @ X3 @ sK0) = $true) & ($true != (sK1 @ X3))))),
% 2.09/1.48 introduced(definition,[],[choice_axiom])).
% 2.09/1.48 thf(f171,plain,(
% 2.09/1.48 ? [X3 : $i] : (((in @ X3 @ sK0) = $true) & ($true != (sK1 @ X3))) => (($true = (in @ sK2 @ sK0)) & ($true != (sK1 @ sK2)))),
% 2.09/1.48 introduced(definition,[],[choice_axiom])).
% 2.09/1.48 thf(f169,plain,(
% 2.09/1.48 (nonemptyImpWitness = $true) & (setunionE = $true) & (nonemptyE1 = $true) & (exuE2 = $true) & (exuE3e = $true) & (replAx = $true) & (exuE1 = $true) & (setext = $true) & (emptyinPowerset = $true) & (powersetE = $true) & (emptysetE = $true) & (dsetconstrI = $true) & (powersetI = $true) & (setadjoinOr = $true) & (emptyInPowerset = $true) & (omega0Ax = $true) & (setoftrueEq = $true) & (descrp = $true) & (subPowSU = $true) & (emptysetAx = $true) & (omegaIndAx = $true) & (omegaSAx = $true) & (setadjoinIR = $true) & (dsetconstrEL = $true) & (notinsingleton = $true) & (emptyE1 = $true) & (powersetAx = $true) & (upairsetE = $true) & (singletonsswitch = $true) & (setadjoinIL = $true) & (dsetconstrER = $true) & (setadjoinE = $true) & (upairsetIR = $true) & ? [X0 : $i,X1 : $i > $o] : (! [X2 : $i] : (($true = (X1 @ X2)) | ($true != (in @ X2 @ X0))) & ? [X3 : $i] : (((in @ X3 @ X0) = $true) & ($true != (X1 @ X3)))) & (setextAx = $true) & (prop2setE = $true) & (notinemptyset = $true) & (eqinunit = $true) & (emptyinunitempty = $true) & (wellorderingAx = $true) & (noeltsimpempty = $true) & (emptysetimpfalse = $true) & (setunionAx = $true) & (setadjoinAx = $true) & (setbeta = $true) & (emptyI = $true) & (uniqinunit = $true) & (upairsetIL = $true) & (vacuousDall = $true) & (nonemptyI1 = $true) & (foundationAx = $true) & (nonemptyI = $true) & (setunionI = $true)),
% 2.09/1.48 inference(rectify,[status(thm)],[f168])).
% 2.09/1.48 thf(f168,plain,(
% 2.09/1.48 (nonemptyImpWitness = $true) & (setunionE = $true) & (nonemptyE1 = $true) & (exuE2 = $true) & (exuE3e = $true) & (replAx = $true) & (exuE1 = $true) & (setext = $true) & (emptyinPowerset = $true) & (powersetE = $true) & (emptysetE = $true) & (dsetconstrI = $true) & (powersetI = $true) & (setadjoinOr = $true) & (emptyInPowerset = $true) & (omega0Ax = $true) & (setoftrueEq = $true) & (descrp = $true) & (subPowSU = $true) & (emptysetAx = $true) & (omegaIndAx = $true) & (omegaSAx = $true) & (setadjoinIR = $true) & (dsetconstrEL = $true) & (notinsingleton = $true) & (emptyE1 = $true) & (powersetAx = $true) & (upairsetE = $true) & (singletonsswitch = $true) & (setadjoinIL = $true) & (dsetconstrER = $true) & (setadjoinE = $true) & (upairsetIR = $true) & ? [X0 : $i,X1 : $i > $o] : (! [X3 : $i] : (($true = (X1 @ X3)) | ((in @ X3 @ X0) != $true)) & ? [X2 : $i] : (($true = (in @ X2 @ X0)) & ($true != (X1 @ X2)))) & (setextAx = $true) & (prop2setE = $true) & (notinemptyset = $true) & (eqinunit = $true) & (emptyinunitempty = $true) & (wellorderingAx = $true) & (noeltsimpempty = $true) & (emptysetimpfalse = $true) & (setunionAx = $true) & (setadjoinAx = $true) & (setbeta = $true) & (emptyI = $true) & (uniqinunit = $true) & (upairsetIL = $true) & (vacuousDall = $true) & (nonemptyI1 = $true) & (foundationAx = $true) & (nonemptyI = $true) & (setunionI = $true)),
% 2.09/1.48 inference(flattening,[status(thm)],[f167])).
% 2.09/1.48 thf(f167,plain,(
% 2.09/1.48 (((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i > $o] : (! [X3 : $i] : (($true = (X1 @ X3)) | ((in @ X3 @ X0) != $true)) & ? [X2 : $i] : (($true = (in @ X2 @ X0)) & ($true != (X1 @ X2)))) & (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)),
% 2.09/1.48 inference(ennf_transformation,[status(thm)],[f166])).
% 2.09/1.48 thf(f166,plain,(
% 2.09/1.48 ~((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) => ! [X0 : $i,X1 : $i > $o] : (~! [X2 : $i] : (($true = (in @ X2 @ X0)) => ($true = (X1 @ X2))) => ? [X3 : $i] : (($true != (X1 @ X3)) & ((in @ X3 @ X0) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.09/1.48 inference(flattening,[status(thm)],[f125])).
% 2.09/1.48 thf(f125,plain,(
% 2.09/1.48 ~((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) => ! [X0 : $i,X1 : $i > $o] : (~! [X2 : $i] : (($true = (in @ X2 @ X0)) => ($true = (X1 @ X2))) => ? [X3 : $i] : (~($true = (X1 @ X3)) & ((in @ X3 @ X0) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.09/1.48 inference(fool_elimination,[status(thm)],[f124])).
% 2.09/1.48 thf(f124,plain,(
% 2.09/1.48 ~(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 => ! [X0 : $i,X1 : $i > $o] : (~! [X2 : $i] : ((in @ X2 @ X0) => (X1 @ X2)) => ? [X3 : $i] : (~(X1 @ X3) & (in @ X3 @ X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.09/1.48 inference(rectify,[status(thm)],[f57])).
% 2.09/1.48 thf(f57,negated_conjecture,(
% 2.09/1.48 ~(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 => ! [X3 : $i,X0 : $i > $o] : (~! [X1 : $i] : ((in @ X1 @ X3) => (X0 @ X1)) => ? [X1 : $i] : (~(X0 @ X1) & (in @ X1 @ X3)))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.09/1.48 inference(negated_conjecture,[status(cth)],[f56])).
% 2.09/1.48 thf(f56,conjecture,(
% 2.09/1.48 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 => ! [X3 : $i,X0 : $i > $o] : (~! [X1 : $i] : ((in @ X1 @ X3) => (X0 @ X1)) => ? [X1 : $i] : (~(X0 @ X1) & (in @ X1 @ X3))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.09/1.48 file('/export/starexec/sandbox/tmp/tmp.T76pLv6kux/DTF2THF_32465.p',quantDeMorgan1)).
% 2.09/1.48 thf(f972,plain,(
% 2.09/1.48 ($true = (sK1 @ sK2))),
% 2.09/1.48 inference(trivial_inequality_removal,[status(thm)],[f970])).
% 2.09/1.48 thf(f970,plain,(
% 2.09/1.48 ($true = (sK1 @ sK2)) | ($true != $true)),
% 2.09/1.48 inference(superposition,[status(thm)],[f245,f244])).
% 2.09/1.48 thf(f244,plain,(
% 2.09/1.48 ($true = (in @ sK2 @ sK0))),
% 2.09/1.48 inference(cnf_transformation,[status(thm)],[f172])).
% 2.09/1.48 thf(f245,plain,(
% 2.09/1.48 ( ! [X2 : $i] : (((in @ X2 @ sK0) != $true) | ($true = (sK1 @ X2))) )),
% 2.09/1.48 inference(cnf_transformation,[status(thm)],[f172])).
% 2.09/1.48 % SZS output end Proof for DTF2THF_32465
% 2.09/1.48 % (32595)------------------------------
% 2.09/1.48 % (32595)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.48 % (32595)Termination reason: Refutation
% 2.09/1.48
% 2.09/1.48 % (32595)Memory used [KB]: 6268
% 2.09/1.48 % (32595)Time elapsed: 0.046 s
% 2.09/1.48 % (32595)Instructions burned: 79 (million)
% 2.09/1.48 % (32595)------------------------------
% 2.09/1.48 % (32595)------------------------------
% 2.09/1.48 % (32594)Success in time 0.048 s
%------------------------------------------------------------------------------