%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU535^1 : 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 : n004.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 09:35:08 AM UTC 2026
% Result : Theorem 2.09s 1.40s
% Output : Refutation 2.09s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 4
% Syntax : Number of formulae : 26 ( 14 unt; 0 typ; 1 def)
% Number of atoms : 885 ( 230 equ; 0 cnn)
% Maximal formula atoms : 53 ( 34 avg)
% Number of connectives : 455 ( 15 ~; 3 |; 158 &; 60 @)
% ( 1 <=>; 214 =>; 0 <=; 0 <~>)
% Maximal formula depth : 58 ( 16 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 7 ( 7 >; 0 *; 0 +; 0 <<)
% Number of symbols : 97 ( 93 usr; 57 con; 0-3 aty)
% ( 4 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 26 ( 5 ^; 15 !; 6 ?; 26 :)
% 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_79,type,
sK0: $i > $o ).
thf(func_def_82,type,
sK3: $i > $i ).
thf(func_def_83,type,
sK4: $i > $i > $i ).
thf(func_def_84,type,
sK5: $i > $i ).
thf(func_def_85,type,
sK6: $o > $i > $i > $i ).
thf(func_def_86,type,
sK7: ( $i > $o ) > $i ).
thf(func_def_87,type,
sK8: ( $i > $o ) > $i > $i ).
thf(func_def_88,type,
sK9: $i > $i > $i ).
thf(func_def_89,type,
sK10: $i > $i ).
thf(func_def_90,type,
sK11: $i > $i > $i > $i ).
thf(func_def_91,type,
sK12: $i > $i > $i ).
thf(func_def_92,type,
sK13: $i > $i > $i ).
thf(func_def_93,type,
sK14: $i > $i > $i ).
thf(func_def_94,type,
sK15: $i > $i > $i ).
thf(func_def_95,type,
sK16: $i > ( $i > $i > $o ) > $i ).
thf(func_def_96,type,
sK17: $i > ( $i > $i > $o ) > $i ).
thf(func_def_97,type,
sK18: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_98,type,
sK19: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_99,type,
sK20: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_100,type,
sK21: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_101,type,
sK22: $i > $i ).
thf(func_def_102,type,
sK23: $i > $i ).
thf(func_def_103,type,
sK24: $i > $i > $i ).
thf(func_def_104,type,
sK25: $i > $i ).
thf(func_def_105,type,
sK26: $i > $i > $i ).
thf(func_def_106,type,
sK27: $i > $i > $i ).
thf(func_def_107,type,
sK28: $i > $i ).
thf(func_def_108,type,
sK29: ( $i > $o ) > $i ).
thf(func_def_109,type,
sK30: ( $i > $o ) > $i > $i ).
thf(func_def_110,type,
sK31: ( $i > $o ) > $i > $i ).
thf(f947,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f907,f946]) ).
thf(f946,plain,
~ spl2_1,
inference(avatar_contradiction_clause,[status(thm)],[f945]) ).
thf(f945,plain,
( $false
| ~ spl2_1 ),
inference(trivial_inequality_removal,[status(thm)],[f941]) ).
thf(f941,plain,
( ( $false = $true )
| ~ spl2_1 ),
inference(superposition,[status(thm)],[f228,f900]) ).
thf(f900,plain,
( ! [X1: $i] :
( ( in @ X1 @ emptyset )
= $false )
| ~ spl2_1 ),
inference(avatar_component_clause,[status(thm)],[f899]) ).
thf(f899,definition,
( spl2_1
<=> ! [X1: $i] :
( ( in @ X1 @ emptyset )
= $false ) ),
introduced(definition,[new_symbols(naming,[spl2_1])],[avatar_definition]) ).
thf(f228,plain,
( ( in @ sK1 @ emptyset )
= $true ),
inference(cnf_transformation,[status(thm)],[f166]) ).
thf(f166,plain,
( ( upairsetE = $true )
& ( powersetI = $true )
& ( omega0Ax = $true )
& ( notinsingleton = $true )
& ( eqinunit = $true )
& ( noeltsimpempty = $true )
& ( notinemptyset = $true )
& ( nonemptyI1 = $true )
& ( upairsetIL = $true )
& ( upairsetIR = $true )
& ( emptyinunitempty = $true )
& ( replAx = $true )
& ( nonemptyI = $true )
& ( setoftrueEq = $true )
& ( exuE3e = $true )
& ( emptyInPowerset = $true )
& ( subPowSU = $true )
& ( dsetconstrI = $true )
& ( dsetconstrEL = $true )
& ( setextAx = $true )
& ( setadjoinIL = $true )
& ( dsetconstrER = $true )
& ( exuE2 = $true )
& ( emptysetAx = $true )
& ( setunionI = $true )
& ( emptyI = $true )
& ( omegaIndAx = $true )
& ( powersetAx = $true )
& ( emptysetE = $true )
& ( wellorderingAx = $true )
& ( setadjoinIR = $true )
& ( omegaSAx = $true )
& ( uniqinunit = $true )
& ( setadjoinAx = $true )
& ( ( sK0 @ sK1 )
!= $true )
& ( ( in @ sK1 @ emptyset )
= $true )
& ( descrp = $true )
& ( setunionAx = $true )
& ( nonemptyImpWitness = $true )
& ( emptysetimpfalse = $true )
& ( setadjoinE = $true )
& ( setext = $true )
& ( nonemptyE1 = $true )
& ( emptyinPowerset = $true )
& ( setbeta = $true )
& ( prop2setE = $true )
& ( powersetE = $true )
& ( emptyE1 = $true )
& ( singletonsswitch = $true )
& ( foundationAx = $true )
& ( setunionE = $true )
& ( exuE1 = $true )
& ( setadjoinOr = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f164,f165]) ).
thf(f165,plain,
( ? [X0: $i > $o,X1: $i] :
( ( ( X0 @ X1 )
!= $true )
& ( ( in @ X1 @ emptyset )
= $true ) )
=> ( ( ( sK0 @ sK1 )
!= $true )
& ( ( in @ sK1 @ emptyset )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f164,plain,
( ( upairsetE = $true )
& ( powersetI = $true )
& ( omega0Ax = $true )
& ( notinsingleton = $true )
& ( eqinunit = $true )
& ( noeltsimpempty = $true )
& ( notinemptyset = $true )
& ( nonemptyI1 = $true )
& ( upairsetIL = $true )
& ( upairsetIR = $true )
& ( emptyinunitempty = $true )
& ( replAx = $true )
& ( nonemptyI = $true )
& ( setoftrueEq = $true )
& ( exuE3e = $true )
& ( emptyInPowerset = $true )
& ( subPowSU = $true )
& ( dsetconstrI = $true )
& ( dsetconstrEL = $true )
& ( setextAx = $true )
& ( setadjoinIL = $true )
& ( dsetconstrER = $true )
& ( exuE2 = $true )
& ( emptysetAx = $true )
& ( setunionI = $true )
& ( emptyI = $true )
& ( omegaIndAx = $true )
& ( powersetAx = $true )
& ( emptysetE = $true )
& ( wellorderingAx = $true )
& ( setadjoinIR = $true )
& ( omegaSAx = $true )
& ( uniqinunit = $true )
& ( setadjoinAx = $true )
& ? [X0: $i > $o,X1: $i] :
( ( ( X0 @ X1 )
!= $true )
& ( ( in @ X1 @ emptyset )
= $true ) )
& ( descrp = $true )
& ( setunionAx = $true )
& ( nonemptyImpWitness = $true )
& ( emptysetimpfalse = $true )
& ( setadjoinE = $true )
& ( setext = $true )
& ( nonemptyE1 = $true )
& ( emptyinPowerset = $true )
& ( setbeta = $true )
& ( prop2setE = $true )
& ( powersetE = $true )
& ( emptyE1 = $true )
& ( singletonsswitch = $true )
& ( foundationAx = $true )
& ( setunionE = $true )
& ( exuE1 = $true )
& ( setadjoinOr = $true ) ),
inference(flattening,[status(thm)],[f163]) ).
thf(f163,plain,
( ? [X0: $i > $o,X1: $i] :
( ( ( X0 @ X1 )
!= $true )
& ( ( in @ X1 @ emptyset )
= $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)],[f116]) ).
thf(f116,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 )
=> ! [X0: $i > $o,X1: $i] :
( ( ( in @ X1 @ emptyset )
= $true )
=> ( ( X0 @ X1 )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f115]) ).
thf(f115,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
=> ! [X0: $i > $o,X1: $i] :
( ( in @ X1 @ emptyset )
=> ( X0 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f56]) ).
thf(f56,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
=> ! [X0: $i > $o,X1: $i] :
( ( in @ X1 @ emptyset )
=> ( X0 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f55]) ).
thf(f55,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
=> ! [X0: $i > $o,X1: $i] :
( ( in @ X1 @ emptyset )
=> ( X0 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.ivwA5omCSE/DTF2THF_27016.p',vacuousDall) ).
thf(f907,plain,
spl2_1,
inference(avatar_split_clause,[status(thm)],[f476,f899]) ).
thf(f476,plain,
! [X1: $i] :
( ( in @ X1 @ emptyset )
= $false ),
inference(not_proxy_clausification,[status(thm)],[f475]) ).
thf(f475,plain,
! [X1: $i] :
( ( ~ ( in @ X1 @ emptyset ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f474]) ).
thf(f474,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
~ ( in @ Y0 @ emptyset )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f473]) ).
thf(f473,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
~ ( in @ Y0 @ emptyset ) )
= $true ),
inference(boolean_simplification,[status(thm)],[f311]) ).
thf(f311,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ emptyset )
=> $false ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f224,f184]) ).
thf(f184,plain,
( emptysetimpfalse
= ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ emptyset )
=> $false ) ) ),
inference(cnf_transformation,[status(thm)],[f98]) ).
thf(f98,plain,
( emptysetimpfalse
= ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ emptyset )
=> $false ) ) ),
inference(fool_elimination,[status(thm)],[f97]) ).
thf(f97,plain,
! [X0: $i] :
( ( ( in @ X0 @ emptyset )
=> $false )
= emptysetimpfalse ),
inference(rectify,[status(thm)],[f21]) ).
thf(f21,axiom,
! [X1: $i] :
( ( ( in @ X1 @ emptyset )
=> $false )
= emptysetimpfalse ),
file('/export/starexec/sandbox2/tmp/tmp.ivwA5omCSE/DTF2THF_27016.p',emptysetimpfalse) ).
thf(f224,plain,
emptysetimpfalse = $true,
inference(cnf_transformation,[status(thm)],[f166]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU535^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n004.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 12:25:56 EST 2026
% 0.16/0.33 % CPUTime :
% 0.16/0.33 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.20/0.44 ---- Original DTF file ---
% 0.20/0.44 thf(spec,logic,$$dhol).
% 0.20/0.44 %------------------------------------------------------------------------------
% 0.20/0.44 % File : SEU535^1 : TPTP v9.2.1. Released v3.7.0.
% 0.20/0.44 % Domain : Set Theory
% 0.20/0.44 % Problem : Preliminary Notions - Bounded Quantifier Laws
% 0.20/0.44 % Version : Especial.
% 0.20/0.44 % English : (! phi:i>o.! x:i.in x emptyset -> phi x)
% 0.20/0.44
% 0.20/0.44 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.20/0.44 % Source : [Bro08]
% 0.20/0.44 % Names : ZFC037g [Bro08]
% 0.20/0.44
% 0.20/0.44 % Status : Theorem
% 0.20/0.44 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.31 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, 0.33 v5.4.0, 0.40 v5.2.0, 0.20 v5.1.0, 0.40 v5.0.0, 0.20 v4.1.0, 0.33 v4.0.0, 0.00 v3.7.0
% 0.20/0.44 % Syntax : Number of formulae : 117 ( 54 unt; 62 typ; 54 def)
% 0.20/0.44 % Number of atoms : 297 ( 73 equ; 0 cnn)
% 0.20/0.44 % Maximal formula atoms : 52 ( 5 avg)
% 0.20/0.44 % Number of connectives : 483 ( 9 ~; 5 |; 20 &; 318 @)
% 0.20/0.44 % ( 8 <=>; 123 =>; 0 <=; 0 <~>)
% 0.20/0.44 % Maximal formula depth : 57 ( 2 avg)
% 0.20/0.44 % Number of types : 2 ( 0 usr)
% 0.20/0.44 % Number of type conns : 29 ( 29 >; 0 *; 0 +; 0 <<)
% 0.20/0.44 % Number of symbols : 65 ( 62 usr; 55 con; 0-2 aty)
% 0.20/0.44 % Number of variables : 159 ( 17 ^; 123 !; 19 ?; 159 :)
% 0.20/0.44 % SPC : TH0_THM_EQU_NAR
% 0.20/0.44
% 0.20/0.44 % Comments : http://mathgate.info/detsetitem.php?id=508
% 0.20/0.44 %------------------------------------------------------------------------------
% 0.20/0.44 thf(in_type,type,
% 0.20/0.44 in: $i > $i > $o ).
% 0.20/0.44
% 0.20/0.44 thf(exu_type,type,
% 0.20/0.44 exu: ( $i > $o ) > $o ).
% 0.20/0.44
% 0.20/0.44 thf(exu,definition,
% 0.20/0.44 ( exu
% 0.20/0.44 = ( ^ [Xphi: $i > $o] :
% 0.20/0.44 ? [Xx: $i] :
% 0.20/0.44 ( ( Xphi @ Xx )
% 0.20/0.44 & ! [Xy: $i] :
% 0.20/0.44 ( ( Xphi @ Xy )
% 0.20/0.44 => ( Xx = Xy ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(setextAx_type,type,
% 0.20/0.44 setextAx: $o ).
% 0.20/0.44
% 0.20/0.44 thf(setextAx,definition,
% 0.20/0.44 ( setextAx
% 0.20/0.44 = ( ! [A: $i,B: $i] :
% 0.20/0.44 ( ! [Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ A )
% 0.20/0.44 <=> ( in @ Xx @ B ) )
% 0.20/0.44 => ( A = B ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(emptyset_type,type,
% 0.20/0.44 emptyset: $i ).
% 0.20/0.44
% 0.20/0.44 thf(emptysetAx_type,type,
% 0.20/0.44 emptysetAx: $o ).
% 0.20/0.44
% 0.20/0.44 thf(emptysetAx,definition,
% 0.20/0.44 ( emptysetAx
% 0.20/0.44 = ( ! [Xx: $i] :
% 0.20/0.44 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(setadjoin_type,type,
% 0.20/0.44 setadjoin: $i > $i > $i ).
% 0.20/0.44
% 0.20/0.44 thf(setadjoinAx_type,type,
% 0.20/0.44 setadjoinAx: $o ).
% 0.20/0.44
% 0.20/0.44 thf(setadjoinAx,definition,
% 0.20/0.44 ( setadjoinAx
% 0.20/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.20/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.20/0.44 <=> ( ( Xy = Xx )
% 0.20/0.44 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(powerset_type,type,
% 0.20/0.44 powerset: $i > $i ).
% 0.20/0.44
% 0.20/0.44 thf(powersetAx_type,type,
% 0.20/0.44 powersetAx: $o ).
% 0.20/0.44
% 0.20/0.44 thf(powersetAx,definition,
% 0.20/0.44 ( powersetAx
% 0.20/0.44 = ( ! [A: $i,B: $i] :
% 0.20/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.20/0.44 <=> ! [Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ B )
% 0.20/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(setunion_type,type,
% 0.20/0.44 setunion: $i > $i ).
% 0.20/0.44
% 0.20/0.44 thf(setunionAx_type,type,
% 0.20/0.44 setunionAx: $o ).
% 0.20/0.44
% 0.20/0.44 thf(setunionAx,definition,
% 0.20/0.44 ( setunionAx
% 0.20/0.44 = ( ! [A: $i,Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.20/0.44 <=> ? [B: $i] :
% 0.20/0.44 ( ( in @ Xx @ B )
% 0.20/0.44 & ( in @ B @ A ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(omega_type,type,
% 0.20/0.44 omega: $i ).
% 0.20/0.44
% 0.20/0.44 thf(omega0Ax_type,type,
% 0.20/0.44 omega0Ax: $o ).
% 0.20/0.44
% 0.20/0.44 thf(omega0Ax,definition,
% 0.20/0.44 ( omega0Ax
% 0.20/0.44 = ( in @ emptyset @ omega ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(omegaSAx_type,type,
% 0.20/0.44 omegaSAx: $o ).
% 0.20/0.44
% 0.20/0.44 thf(omegaSAx,definition,
% 0.20/0.44 ( omegaSAx
% 0.20/0.44 = ( ! [Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ omega )
% 0.20/0.44 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(omegaIndAx_type,type,
% 0.20/0.44 omegaIndAx: $o ).
% 0.20/0.44
% 0.20/0.44 thf(omegaIndAx,definition,
% 0.20/0.44 ( omegaIndAx
% 0.20/0.44 = ( ! [A: $i] :
% 0.20/0.44 ( ( ( in @ emptyset @ A )
% 0.20/0.44 & ! [Xx: $i] :
% 0.20/0.44 ( ( ( in @ Xx @ omega )
% 0.20/0.44 & ( in @ Xx @ A ) )
% 0.20/0.44 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.20/0.44 => ! [Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ omega )
% 0.20/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(replAx_type,type,
% 0.20/0.44 replAx: $o ).
% 0.20/0.44
% 0.20/0.44 thf(replAx,definition,
% 0.20/0.44 ( replAx
% 0.20/0.44 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.20/0.44 ( ! [Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ A )
% 0.20/0.44 => ( exu
% 0.20/0.44 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.20/0.44 => ? [B: $i] :
% 0.20/0.44 ! [Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ B )
% 0.20/0.44 <=> ? [Xy: $i] :
% 0.20/0.44 ( ( in @ Xy @ A )
% 0.20/0.44 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(foundationAx_type,type,
% 0.20/0.44 foundationAx: $o ).
% 0.20/0.44
% 0.20/0.44 thf(foundationAx,definition,
% 0.20/0.44 ( foundationAx
% 0.20/0.44 = ( ! [A: $i] :
% 0.20/0.44 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.20/0.44 => ? [B: $i] :
% 0.20/0.44 ( ( in @ B @ A )
% 0.20/0.44 & ~ ? [Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ B )
% 0.20/0.44 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(wellorderingAx_type,type,
% 0.20/0.44 wellorderingAx: $o ).
% 0.20/0.44
% 0.20/0.44 thf(wellorderingAx,definition,
% 0.20/0.44 ( wellorderingAx
% 0.20/0.44 = ( ! [A: $i] :
% 0.20/0.44 ? [B: $i] :
% 0.20/0.44 ( ! [C: $i] :
% 0.20/0.44 ( ( in @ C @ B )
% 0.20/0.44 => ! [Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ C )
% 0.20/0.44 => ( in @ Xx @ A ) ) )
% 0.20/0.44 & ! [Xx: $i,Xy: $i] :
% 0.20/0.44 ( ( ( in @ Xx @ A )
% 0.20/0.44 & ( in @ Xy @ A ) )
% 0.20/0.44 => ( ! [C: $i] :
% 0.20/0.44 ( ( in @ C @ B )
% 0.20/0.44 => ( ( in @ Xx @ C )
% 0.20/0.44 <=> ( in @ Xy @ C ) ) )
% 0.20/0.44 => ( Xx = Xy ) ) )
% 0.20/0.44 & ! [C: $i,D: $i] :
% 0.20/0.44 ( ( ( in @ C @ B )
% 0.20/0.44 & ( in @ D @ B ) )
% 0.20/0.44 => ( ! [Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ C )
% 0.20/0.44 => ( in @ Xx @ D ) )
% 0.20/0.44 | ! [Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ D )
% 0.20/0.44 => ( in @ Xx @ C ) ) ) )
% 0.20/0.44 & ! [C: $i] :
% 0.20/0.44 ( ( ! [Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ C )
% 0.20/0.44 => ( in @ Xx @ A ) )
% 0.20/0.44 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.20/0.44 => ? [D: $i,Xx: $i] :
% 0.20/0.44 ( ( in @ D @ B )
% 0.20/0.44 & ( in @ Xx @ C )
% 0.20/0.44 & ~ ? [Xy: $i] :
% 0.20/0.44 ( ( in @ Xy @ D )
% 0.20/0.44 & ( in @ Xy @ C ) )
% 0.20/0.44 & ! [E: $i] :
% 0.20/0.44 ( ( in @ E @ B )
% 0.20/0.44 => ( ! [Xy: $i] :
% 0.20/0.44 ( ( in @ Xy @ E )
% 0.20/0.44 => ( in @ Xy @ D ) )
% 0.20/0.44 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(descr_type,type,
% 0.20/0.44 descr: ( $i > $o ) > $i ).
% 0.20/0.44
% 0.20/0.44 thf(descrp_type,type,
% 0.20/0.44 descrp: $o ).
% 0.20/0.44
% 0.20/0.44 thf(descrp,definition,
% 0.20/0.44 ( descrp
% 0.20/0.44 = ( ! [Xphi: $i > $o] :
% 0.20/0.44 ( ( exu
% 0.20/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.44 => ( Xphi
% 0.20/0.44 @ ( descr
% 0.20/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(dsetconstr_type,type,
% 0.20/0.44 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.20/0.44
% 0.20/0.44 thf(dsetconstrI_type,type,
% 0.20/0.44 dsetconstrI: $o ).
% 0.20/0.44
% 0.20/0.44 thf(dsetconstrI,definition,
% 0.20/0.44 ( dsetconstrI
% 0.20/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ A )
% 0.20/0.44 => ( ( Xphi @ Xx )
% 0.20/0.44 => ( in @ Xx
% 0.20/0.44 @ ( dsetconstr @ A
% 0.20/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(dsetconstrEL_type,type,
% 0.20/0.44 dsetconstrEL: $o ).
% 0.20/0.44
% 0.20/0.44 thf(dsetconstrEL,definition,
% 0.20/0.44 ( dsetconstrEL
% 0.20/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.20/0.44 ( ( in @ Xx
% 0.20/0.44 @ ( dsetconstr @ A
% 0.20/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.20/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(dsetconstrER_type,type,
% 0.20/0.44 dsetconstrER: $o ).
% 0.20/0.44
% 0.20/0.44 thf(dsetconstrER,definition,
% 0.20/0.44 ( dsetconstrER
% 0.20/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.20/0.44 ( ( in @ Xx
% 0.20/0.44 @ ( dsetconstr @ A
% 0.20/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.20/0.44 => ( Xphi @ Xx ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(exuE1_type,type,
% 0.20/0.44 exuE1: $o ).
% 0.20/0.44
% 0.20/0.44 thf(exuE1,definition,
% 0.20/0.44 ( exuE1
% 0.20/0.44 = ( ! [Xphi: $i > $o] :
% 0.20/0.44 ( ( exu
% 0.20/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.44 => ? [Xx: $i] :
% 0.20/0.44 ( ( Xphi @ Xx )
% 0.20/0.44 & ! [Xy: $i] :
% 0.20/0.44 ( ( Xphi @ Xy )
% 0.20/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(prop2set_type,type,
% 0.20/0.44 prop2set: $o > $i ).
% 0.20/0.44
% 0.20/0.44 thf(prop2set,definition,
% 0.20/0.44 ( prop2set
% 0.20/0.44 = ( ^ [Xphi: $o] :
% 0.20/0.44 ( dsetconstr @ ( powerset @ emptyset )
% 0.20/0.44 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(prop2setE_type,type,
% 0.20/0.44 prop2setE: $o ).
% 0.20/0.44
% 0.20/0.44 thf(prop2setE,definition,
% 0.20/0.44 ( prop2setE
% 0.20/0.44 = ( ! [Xphi: $o,Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.20/0.44 => Xphi ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(emptysetE_type,type,
% 0.20/0.44 emptysetE: $o ).
% 0.20/0.44
% 0.20/0.44 thf(emptysetE,definition,
% 0.20/0.44 ( emptysetE
% 0.20/0.44 = ( ! [Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ emptyset )
% 0.20/0.44 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(emptysetimpfalse_type,type,
% 0.20/0.44 emptysetimpfalse: $o ).
% 0.20/0.44
% 0.20/0.44 thf(emptysetimpfalse,definition,
% 0.20/0.44 ( emptysetimpfalse
% 0.20/0.44 = ( ! [Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ emptyset )
% 0.20/0.44 => $false ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(notinemptyset_type,type,
% 0.20/0.44 notinemptyset: $o ).
% 0.20/0.44
% 0.20/0.44 thf(notinemptyset,definition,
% 0.20/0.44 ( notinemptyset
% 0.20/0.44 = ( ! [Xx: $i] :
% 0.20/0.44 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(exuE3e_type,type,
% 0.20/0.44 exuE3e: $o ).
% 0.20/0.44
% 0.20/0.44 thf(exuE3e,definition,
% 0.20/0.44 ( exuE3e
% 0.20/0.44 = ( ! [Xphi: $i > $o] :
% 0.20/0.44 ( ( exu
% 0.20/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.44 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(setext_type,type,
% 0.20/0.44 setext: $o ).
% 0.20/0.44
% 0.20/0.44 thf(setext,definition,
% 0.20/0.44 ( setext
% 0.20/0.44 = ( ! [A: $i,B: $i] :
% 0.20/0.44 ( ! [Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ A )
% 0.20/0.44 => ( in @ Xx @ B ) )
% 0.20/0.44 => ( ! [Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ B )
% 0.20/0.44 => ( in @ Xx @ A ) )
% 0.20/0.44 => ( A = B ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(emptyI_type,type,
% 0.20/0.44 emptyI: $o ).
% 0.20/0.44
% 0.20/0.44 thf(emptyI,definition,
% 0.20/0.44 ( emptyI
% 0.20/0.44 = ( ! [A: $i] :
% 0.20/0.44 ( ! [Xx: $i] :
% 0.20/0.44 ~ ( in @ Xx @ A )
% 0.20/0.44 => ( A = emptyset ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(noeltsimpempty_type,type,
% 0.20/0.44 noeltsimpempty: $o ).
% 0.20/0.44
% 0.20/0.44 thf(noeltsimpempty,definition,
% 0.20/0.44 ( noeltsimpempty
% 0.20/0.44 = ( ! [A: $i] :
% 0.20/0.44 ( ! [Xx: $i] :
% 0.20/0.44 ~ ( in @ Xx @ A )
% 0.20/0.44 => ( A = emptyset ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(setbeta_type,type,
% 0.20/0.44 setbeta: $o ).
% 0.20/0.44
% 0.20/0.44 thf(setbeta,definition,
% 0.20/0.44 ( setbeta
% 0.20/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ A )
% 0.20/0.44 => ( ( in @ Xx
% 0.20/0.44 @ ( dsetconstr @ A
% 0.20/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.20/0.44 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(nonempty_type,type,
% 0.20/0.44 nonempty: $i > $o ).
% 0.20/0.44
% 0.20/0.44 thf(nonempty,definition,
% 0.20/0.44 ( nonempty
% 0.20/0.44 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(nonemptyE1_type,type,
% 0.20/0.44 nonemptyE1: $o ).
% 0.20/0.44
% 0.20/0.44 thf(nonemptyE1,definition,
% 0.20/0.44 ( nonemptyE1
% 0.20/0.44 = ( ! [A: $i] :
% 0.20/0.44 ( ( nonempty @ A )
% 0.20/0.44 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(nonemptyI_type,type,
% 0.20/0.44 nonemptyI: $o ).
% 0.20/0.44
% 0.20/0.44 thf(nonemptyI,definition,
% 0.20/0.44 ( nonemptyI
% 0.20/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ A )
% 0.20/0.44 => ( ( Xphi @ Xx )
% 0.20/0.44 => ( nonempty
% 0.20/0.44 @ ( dsetconstr @ A
% 0.20/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(nonemptyI1_type,type,
% 0.20/0.44 nonemptyI1: $o ).
% 0.20/0.44
% 0.20/0.44 thf(nonemptyI1,definition,
% 0.20/0.44 ( nonemptyI1
% 0.20/0.44 = ( ! [A: $i] :
% 0.20/0.44 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.20/0.44 => ( nonempty @ A ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(setadjoinIL_type,type,
% 0.20/0.44 setadjoinIL: $o ).
% 0.20/0.44
% 0.20/0.44 thf(setadjoinIL,definition,
% 0.20/0.44 ( setadjoinIL
% 0.20/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(emptyinunitempty_type,type,
% 0.20/0.44 emptyinunitempty: $o ).
% 0.20/0.44
% 0.20/0.44 thf(emptyinunitempty,definition,
% 0.20/0.44 ( emptyinunitempty
% 0.20/0.44 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(setadjoinIR_type,type,
% 0.20/0.44 setadjoinIR: $o ).
% 0.20/0.44
% 0.20/0.44 thf(setadjoinIR,definition,
% 0.20/0.44 ( setadjoinIR
% 0.20/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.20/0.44 ( ( in @ Xy @ A )
% 0.20/0.44 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(setadjoinE_type,type,
% 0.20/0.44 setadjoinE: $o ).
% 0.20/0.44
% 0.20/0.44 thf(setadjoinE,definition,
% 0.20/0.44 ( setadjoinE
% 0.20/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.20/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.20/0.44 => ! [Xphi: $o] :
% 0.20/0.44 ( ( ( Xy = Xx )
% 0.20/0.44 => Xphi )
% 0.20/0.44 => ( ( ( in @ Xy @ A )
% 0.20/0.44 => Xphi )
% 0.20/0.44 => Xphi ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(setadjoinOr_type,type,
% 0.20/0.44 setadjoinOr: $o ).
% 0.20/0.44
% 0.20/0.44 thf(setadjoinOr,definition,
% 0.20/0.44 ( setadjoinOr
% 0.20/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.20/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.20/0.44 => ( ( Xy = Xx )
% 0.20/0.44 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(setoftrueEq_type,type,
% 0.20/0.44 setoftrueEq: $o ).
% 0.20/0.44
% 0.20/0.44 thf(setoftrueEq,definition,
% 0.20/0.44 ( setoftrueEq
% 0.20/0.44 = ( ! [A: $i] :
% 0.20/0.44 ( ( dsetconstr @ A
% 0.20/0.44 @ ^ [Xx: $i] : $true )
% 0.20/0.44 = A ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(powersetI_type,type,
% 0.20/0.44 powersetI: $o ).
% 0.20/0.44
% 0.20/0.44 thf(powersetI,definition,
% 0.20/0.44 ( powersetI
% 0.20/0.44 = ( ! [A: $i,B: $i] :
% 0.20/0.44 ( ! [Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ B )
% 0.20/0.44 => ( in @ Xx @ A ) )
% 0.20/0.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(emptyinPowerset_type,type,
% 0.20/0.44 emptyinPowerset: $o ).
% 0.20/0.44
% 0.20/0.44 thf(emptyinPowerset,definition,
% 0.20/0.44 ( emptyinPowerset
% 0.20/0.44 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(emptyInPowerset_type,type,
% 0.20/0.44 emptyInPowerset: $o ).
% 0.20/0.44
% 0.20/0.44 thf(emptyInPowerset,definition,
% 0.20/0.44 ( emptyInPowerset
% 0.20/0.44 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(powersetE_type,type,
% 0.20/0.44 powersetE: $o ).
% 0.20/0.44
% 0.20/0.44 thf(powersetE,definition,
% 0.20/0.44 ( powersetE
% 0.20/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.20/0.44 => ( ( in @ Xx @ B )
% 0.20/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(setunionI_type,type,
% 0.20/0.44 setunionI: $o ).
% 0.20/0.44
% 0.20/0.44 thf(setunionI,definition,
% 0.20/0.44 ( setunionI
% 0.20/0.44 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.20/0.44 ( ( in @ Xx @ B )
% 0.20/0.44 => ( ( in @ B @ A )
% 0.20/0.44 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(setunionE_type,type,
% 0.20/0.44 setunionE: $o ).
% 0.20/0.44
% 0.20/0.44 thf(setunionE,definition,
% 0.20/0.44 ( setunionE
% 0.20/0.44 = ( ! [A: $i,Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.20/0.44 => ! [Xphi: $o] :
% 0.20/0.44 ( ! [B: $i] :
% 0.20/0.44 ( ( in @ Xx @ B )
% 0.20/0.44 => ( ( in @ B @ A )
% 0.20/0.44 => Xphi ) )
% 0.20/0.44 => Xphi ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(subPowSU_type,type,
% 0.20/0.44 subPowSU: $o ).
% 0.20/0.44
% 0.20/0.44 thf(subPowSU,definition,
% 0.20/0.44 ( subPowSU
% 0.20/0.44 = ( ! [A: $i,Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ A )
% 0.20/0.44 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(exuE2_type,type,
% 0.20/0.44 exuE2: $o ).
% 0.20/0.44
% 0.20/0.44 thf(exuE2,definition,
% 0.20/0.44 ( exuE2
% 0.20/0.44 = ( ! [Xphi: $i > $o] :
% 0.20/0.44 ( ( exu
% 0.20/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.44 => ? [Xx: $i] :
% 0.20/0.44 ! [Xy: $i] :
% 0.20/0.44 ( ( Xphi @ Xy )
% 0.20/0.44 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(nonemptyImpWitness_type,type,
% 0.20/0.44 nonemptyImpWitness: $o ).
% 0.20/0.44
% 0.20/0.44 thf(nonemptyImpWitness,definition,
% 0.20/0.44 ( nonemptyImpWitness
% 0.20/0.44 = ( ! [A: $i] :
% 0.20/0.44 ( ( nonempty @ A )
% 0.20/0.44 => ? [Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ A )
% 0.20/0.44 & $true ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(uniqinunit_type,type,
% 0.20/0.44 uniqinunit: $o ).
% 0.20/0.44
% 0.20/0.44 thf(uniqinunit,definition,
% 0.20/0.44 ( uniqinunit
% 0.20/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.20/0.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.20/0.44 => ( Xx = Xy ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(notinsingleton_type,type,
% 0.20/0.44 notinsingleton: $o ).
% 0.20/0.44
% 0.20/0.44 thf(notinsingleton,definition,
% 0.20/0.44 ( notinsingleton
% 0.20/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.20/0.44 ( ( Xx != Xy )
% 0.20/0.44 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(eqinunit_type,type,
% 0.20/0.44 eqinunit: $o ).
% 0.20/0.44
% 0.20/0.44 thf(eqinunit,definition,
% 0.20/0.44 ( eqinunit
% 0.20/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.20/0.44 ( ( Xx = Xy )
% 0.20/0.44 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(singletonsswitch_type,type,
% 0.20/0.44 singletonsswitch: $o ).
% 0.20/0.44
% 0.20/0.44 thf(singletonsswitch,definition,
% 0.20/0.44 ( singletonsswitch
% 0.20/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.20/0.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.20/0.44 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(upairsetE_type,type,
% 0.20/0.44 upairsetE: $o ).
% 0.20/0.44
% 0.20/0.44 thf(upairsetE,definition,
% 0.20/0.44 ( upairsetE
% 0.20/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.20/0.44 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.20/0.44 => ( ( Xz = Xx )
% 0.20/0.44 | ( Xz = Xy ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(upairsetIL_type,type,
% 0.20/0.44 upairsetIL: $o ).
% 0.20/0.44
% 0.20/0.44 thf(upairsetIL,definition,
% 0.20/0.44 ( upairsetIL
% 0.20/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(upairsetIR_type,type,
% 0.20/0.44 upairsetIR: $o ).
% 0.20/0.44
% 0.20/0.44 thf(upairsetIR,definition,
% 0.20/0.44 ( upairsetIR
% 0.20/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(emptyE1_type,type,
% 0.20/0.44 emptyE1: $o ).
% 0.20/0.44
% 0.20/0.44 thf(emptyE1,definition,
% 0.20/0.44 ( emptyE1
% 0.20/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.44 ( ? [Xx: $i] :
% 0.20/0.44 ( ( in @ Xx @ A )
% 0.20/0.44 & ( Xphi @ Xx ) )
% 0.20/0.44 => ( ( ( dsetconstr @ A
% 0.20/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.44 = emptyset )
% 0.20/0.44 => $false ) ) ) ) ).
% 0.20/0.44
% 0.20/0.44 thf(vacuousDall,conjecture,
% 0.20/0.44 ( setextAx
% 0.20/0.44 => ( emptysetAx
% 0.20/0.44 => ( setadjoinAx
% 0.20/0.44 => ( powersetAx
% 0.20/0.44 => ( setunionAx
% 0.20/0.44 => ( omega0Ax
% 0.20/0.44 => ( omegaSAx
% 0.20/0.44 => ( omegaIndAx
% 0.20/0.44 => ( replAx
% 0.20/0.44 => ( foundationAx
% 0.20/0.44 => ( wellorderingAx
% 0.20/0.44 => ( descrp
% 0.20/0.44 => ( dsetconstrI
% 0.20/0.44 => ( dsetconstrEL
% 0.20/0.44 => ( dsetconstrER
% 0.20/0.44 => ( exuE1
% 0.20/0.44 => ( prop2setE
% 0.20/0.44 => ( emptysetE
% 0.20/0.44 => ( emptysetimpfalse
% 0.20/0.44 => ( notinemptyset
% 0.20/0.44 => ( exuE3e
% 0.20/0.44 => ( setext
% 0.20/0.44 => ( emptyI
% 0.20/0.44 => ( noeltsimpempty
% 0.20/0.44 => ( setbeta
% 0.20/0.44 => ( nonemptyE1
% 0.20/0.44 => ( nonemptyI
% 0.20/0.44 => ( nonemptyI1
% 0.20/0.44 => ( setadjoinIL
% 0.20/0.44 => ( emptyinunitempty
% 0.20/0.44 => ( setadjoinIR
% 0.20/0.44 => ( setadjoinE
% 0.20/0.44 => ( setadjoinOr
% 0.20/0.44 => ( setoftrueEq
% 0.20/0.44 => ( powersetI
% 0.20/0.44 => ( emptyinPowerset
% 0.20/0.44 => ( emptyInPowerset
% 0.20/0.45 => ( powersetE
% 0.20/0.45 => ( setunionI
% 0.20/0.45 => ( setunionE
% 0.20/0.45 => ( subPowSU
% 0.20/0.45 => ( exuE2
% 0.20/0.45 => ( nonemptyImpWitness
% 0.20/0.45 => ( uniqinunit
% 0.20/0.45 => ( notinsingleton
% 0.20/0.45 => ( eqinunit
% 0.20/0.45 => ( singletonsswitch
% 0.20/0.45 => ( upairsetE
% 0.20/0.45 => ( upairsetIL
% 0.20/0.45 => ( upairsetIR
% 0.20/0.45 => ( emptyE1
% 0.20/0.45 => ! [Xphi: $i > $o,Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ emptyset )
% 0.20/0.45 => ( Xphi @ Xx ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 %------------------------------------------------------------------------------
% 0.20/0.45 ------------------------
% 1.95/1.35 ---- Embedded in THF ---
% 1.95/1.35 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.95/1.35 %%% Generated on Tue Feb 24 12:25:57 EST 2026
% 1.95/1.35 %%% using '$$dhol' embedding, version 1.3.0.
% 1.95/1.35 %%% Logic specification used:
% 1.95/1.35 %%% thf(spec, logic, $$dhol).
% 1.95/1.35
% 1.95/1.35 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.ivwA5omCSE/DTF2DTF27016.p
% See solution above
% 1.95/1.35 ------------------------
% 1.95/1.36 ---- Cleaned THF ---
% 1.95/1.36 thf(in_type,type,
% 1.95/1.36 in: $i > $i > $o ).
% 1.95/1.36
% 1.95/1.36 thf(exu_type,type,
% 1.95/1.36 exu: ( $i > $o ) > $o ).
% 1.95/1.36
% 1.95/1.36 thf(exu,definition,
% 1.95/1.36 ( exu
% 1.95/1.36 = ( ^ [Xphi: $i > $o] :
% 1.95/1.36 ? [Xx: $i] :
% 1.95/1.36 ( ( Xphi @ Xx )
% 1.95/1.36 & ! [Xy: $i] :
% 1.95/1.36 ( ( Xphi @ Xy )
% 1.95/1.36 => ( Xx = Xy ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(setextAx_type,type,
% 1.95/1.36 setextAx: $o ).
% 1.95/1.36
% 1.95/1.36 thf(setextAx,definition,
% 1.95/1.36 ( setextAx
% 1.95/1.36 = ( ! [A: $i,B: $i] :
% 1.95/1.36 ( ! [Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ A )
% 1.95/1.36 <=> ( in @ Xx @ B ) )
% 1.95/1.36 => ( A = B ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(emptyset_type,type,
% 1.95/1.36 emptyset: $i ).
% 1.95/1.36
% 1.95/1.36 thf(emptysetAx_type,type,
% 1.95/1.36 emptysetAx: $o ).
% 1.95/1.36
% 1.95/1.36 thf(emptysetAx,definition,
% 1.95/1.36 ( emptysetAx
% 1.95/1.36 = ( ! [Xx: $i] :
% 1.95/1.36 ~ ( in @ Xx @ emptyset ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(setadjoin_type,type,
% 1.95/1.36 setadjoin: $i > $i > $i ).
% 1.95/1.36
% 1.95/1.36 thf(setadjoinAx_type,type,
% 1.95/1.36 setadjoinAx: $o ).
% 1.95/1.36
% 1.95/1.36 thf(setadjoinAx,definition,
% 1.95/1.36 ( setadjoinAx
% 1.95/1.36 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.95/1.36 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.95/1.36 <=> ( ( Xy = Xx )
% 1.95/1.36 | ( in @ Xy @ A ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(powerset_type,type,
% 1.95/1.36 powerset: $i > $i ).
% 1.95/1.36
% 1.95/1.36 thf(powersetAx_type,type,
% 1.95/1.36 powersetAx: $o ).
% 1.95/1.36
% 1.95/1.36 thf(powersetAx,definition,
% 1.95/1.36 ( powersetAx
% 1.95/1.36 = ( ! [A: $i,B: $i] :
% 1.95/1.36 ( ( in @ B @ ( powerset @ A ) )
% 1.95/1.36 <=> ! [Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ B )
% 1.95/1.36 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(setunion_type,type,
% 1.95/1.36 setunion: $i > $i ).
% 1.95/1.36
% 1.95/1.36 thf(setunionAx_type,type,
% 1.95/1.36 setunionAx: $o ).
% 1.95/1.36
% 1.95/1.36 thf(setunionAx,definition,
% 1.95/1.36 ( setunionAx
% 1.95/1.36 = ( ! [A: $i,Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ ( setunion @ A ) )
% 1.95/1.36 <=> ? [B: $i] :
% 1.95/1.36 ( ( in @ Xx @ B )
% 1.95/1.36 & ( in @ B @ A ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(omega_type,type,
% 1.95/1.36 omega: $i ).
% 1.95/1.36
% 1.95/1.36 thf(omega0Ax_type,type,
% 1.95/1.36 omega0Ax: $o ).
% 1.95/1.36
% 1.95/1.36 thf(omega0Ax,definition,
% 1.95/1.36 ( omega0Ax
% 1.95/1.36 = ( in @ emptyset @ omega ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(omegaSAx_type,type,
% 1.95/1.36 omegaSAx: $o ).
% 1.95/1.36
% 1.95/1.36 thf(omegaSAx,definition,
% 1.95/1.36 ( omegaSAx
% 1.95/1.36 = ( ! [Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ omega )
% 1.95/1.36 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(omegaIndAx_type,type,
% 1.95/1.36 omegaIndAx: $o ).
% 1.95/1.36
% 1.95/1.36 thf(omegaIndAx,definition,
% 1.95/1.36 ( omegaIndAx
% 1.95/1.36 = ( ! [A: $i] :
% 1.95/1.36 ( ( ( in @ emptyset @ A )
% 1.95/1.36 & ! [Xx: $i] :
% 1.95/1.36 ( ( ( in @ Xx @ omega )
% 1.95/1.36 & ( in @ Xx @ A ) )
% 1.95/1.36 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 1.95/1.36 => ! [Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ omega )
% 1.95/1.36 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(replAx_type,type,
% 1.95/1.36 replAx: $o ).
% 1.95/1.36
% 1.95/1.36 thf(replAx,definition,
% 1.95/1.36 ( replAx
% 1.95/1.36 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 1.95/1.36 ( ! [Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ A )
% 1.95/1.36 => ( exu
% 1.95/1.36 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 1.95/1.36 => ? [B: $i] :
% 1.95/1.36 ! [Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ B )
% 1.95/1.36 <=> ? [Xy: $i] :
% 1.95/1.36 ( ( in @ Xy @ A )
% 1.95/1.36 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(foundationAx_type,type,
% 1.95/1.36 foundationAx: $o ).
% 1.95/1.36
% 1.95/1.36 thf(foundationAx,definition,
% 1.95/1.36 ( foundationAx
% 1.95/1.36 = ( ! [A: $i] :
% 1.95/1.36 ( ? [Xx: $i] : ( in @ Xx @ A )
% 1.95/1.36 => ? [B: $i] :
% 1.95/1.36 ( ( in @ B @ A )
% 1.95/1.36 & ~ ? [Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ B )
% 1.95/1.36 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(wellorderingAx_type,type,
% 1.95/1.36 wellorderingAx: $o ).
% 1.95/1.36
% 1.95/1.36 thf(wellorderingAx,definition,
% 1.95/1.36 ( wellorderingAx
% 1.95/1.36 = ( ! [A: $i] :
% 1.95/1.36 ? [B: $i] :
% 1.95/1.36 ( ! [C: $i] :
% 1.95/1.36 ( ( in @ C @ B )
% 1.95/1.36 => ! [Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ C )
% 1.95/1.36 => ( in @ Xx @ A ) ) )
% 1.95/1.36 & ! [Xx: $i,Xy: $i] :
% 1.95/1.36 ( ( ( in @ Xx @ A )
% 1.95/1.36 & ( in @ Xy @ A ) )
% 1.95/1.36 => ( ! [C: $i] :
% 1.95/1.36 ( ( in @ C @ B )
% 1.95/1.36 => ( ( in @ Xx @ C )
% 1.95/1.36 <=> ( in @ Xy @ C ) ) )
% 1.95/1.36 => ( Xx = Xy ) ) )
% 1.95/1.36 & ! [C: $i,D: $i] :
% 1.95/1.36 ( ( ( in @ C @ B )
% 1.95/1.36 & ( in @ D @ B ) )
% 1.95/1.36 => ( ! [Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ C )
% 1.95/1.36 => ( in @ Xx @ D ) )
% 1.95/1.36 | ! [Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ D )
% 1.95/1.36 => ( in @ Xx @ C ) ) ) )
% 1.95/1.36 & ! [C: $i] :
% 1.95/1.36 ( ( ! [Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ C )
% 1.95/1.36 => ( in @ Xx @ A ) )
% 1.95/1.36 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 1.95/1.36 => ? [D: $i,Xx: $i] :
% 1.95/1.36 ( ( in @ D @ B )
% 1.95/1.36 & ( in @ Xx @ C )
% 1.95/1.36 & ~ ? [Xy: $i] :
% 1.95/1.36 ( ( in @ Xy @ D )
% 1.95/1.36 & ( in @ Xy @ C ) )
% 1.95/1.36 & ! [E: $i] :
% 1.95/1.36 ( ( in @ E @ B )
% 1.95/1.36 => ( ! [Xy: $i] :
% 1.95/1.36 ( ( in @ Xy @ E )
% 1.95/1.36 => ( in @ Xy @ D ) )
% 1.95/1.36 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(descr_type,type,
% 1.95/1.36 descr: ( $i > $o ) > $i ).
% 1.95/1.36
% 1.95/1.36 thf(descrp_type,type,
% 1.95/1.36 descrp: $o ).
% 1.95/1.36
% 1.95/1.36 thf(descrp,definition,
% 1.95/1.36 ( descrp
% 1.95/1.36 = ( ! [Xphi: $i > $o] :
% 1.95/1.36 ( ( exu
% 1.95/1.36 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.95/1.36 => ( Xphi
% 1.95/1.36 @ ( descr
% 1.95/1.36 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(dsetconstr_type,type,
% 1.95/1.36 dsetconstr: $i > ( $i > $o ) > $i ).
% 1.95/1.36
% 1.95/1.36 thf(dsetconstrI_type,type,
% 1.95/1.36 dsetconstrI: $o ).
% 1.95/1.36
% 1.95/1.36 thf(dsetconstrI,definition,
% 1.95/1.36 ( dsetconstrI
% 1.95/1.36 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ A )
% 1.95/1.36 => ( ( Xphi @ Xx )
% 1.95/1.36 => ( in @ Xx
% 1.95/1.36 @ ( dsetconstr @ A
% 1.95/1.36 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(dsetconstrEL_type,type,
% 1.95/1.36 dsetconstrEL: $o ).
% 1.95/1.36
% 1.95/1.36 thf(dsetconstrEL,definition,
% 1.95/1.36 ( dsetconstrEL
% 1.95/1.36 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.95/1.36 ( ( in @ Xx
% 1.95/1.36 @ ( dsetconstr @ A
% 1.95/1.36 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.95/1.36 => ( in @ Xx @ A ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(dsetconstrER_type,type,
% 1.95/1.36 dsetconstrER: $o ).
% 1.95/1.36
% 1.95/1.36 thf(dsetconstrER,definition,
% 1.95/1.36 ( dsetconstrER
% 1.95/1.36 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.95/1.36 ( ( in @ Xx
% 1.95/1.36 @ ( dsetconstr @ A
% 1.95/1.36 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.95/1.36 => ( Xphi @ Xx ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(exuE1_type,type,
% 1.95/1.36 exuE1: $o ).
% 1.95/1.36
% 1.95/1.36 thf(exuE1,definition,
% 1.95/1.36 ( exuE1
% 1.95/1.36 = ( ! [Xphi: $i > $o] :
% 1.95/1.36 ( ( exu
% 1.95/1.36 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.95/1.36 => ? [Xx: $i] :
% 1.95/1.36 ( ( Xphi @ Xx )
% 1.95/1.36 & ! [Xy: $i] :
% 1.95/1.36 ( ( Xphi @ Xy )
% 1.95/1.36 => ( Xx = Xy ) ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(prop2set_type,type,
% 1.95/1.36 prop2set: $o > $i ).
% 1.95/1.36
% 1.95/1.36 thf(prop2set,definition,
% 1.95/1.36 ( prop2set
% 1.95/1.36 = ( ^ [Xphi: $o] :
% 1.95/1.36 ( dsetconstr @ ( powerset @ emptyset )
% 1.95/1.36 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(prop2setE_type,type,
% 1.95/1.36 prop2setE: $o ).
% 1.95/1.36
% 1.95/1.36 thf(prop2setE,definition,
% 1.95/1.36 ( prop2setE
% 1.95/1.36 = ( ! [Xphi: $o,Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 1.95/1.36 => Xphi ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(emptysetE_type,type,
% 1.95/1.36 emptysetE: $o ).
% 1.95/1.36
% 1.95/1.36 thf(emptysetE,definition,
% 1.95/1.36 ( emptysetE
% 1.95/1.36 = ( ! [Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ emptyset )
% 1.95/1.36 => ! [Xphi: $o] : Xphi ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(emptysetimpfalse_type,type,
% 1.95/1.36 emptysetimpfalse: $o ).
% 1.95/1.36
% 1.95/1.36 thf(emptysetimpfalse,definition,
% 1.95/1.36 ( emptysetimpfalse
% 1.95/1.36 = ( ! [Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ emptyset )
% 1.95/1.36 => $false ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(notinemptyset_type,type,
% 1.95/1.36 notinemptyset: $o ).
% 1.95/1.36
% 1.95/1.36 thf(notinemptyset,definition,
% 1.95/1.36 ( notinemptyset
% 1.95/1.36 = ( ! [Xx: $i] :
% 1.95/1.36 ~ ( in @ Xx @ emptyset ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(exuE3e_type,type,
% 1.95/1.36 exuE3e: $o ).
% 1.95/1.36
% 1.95/1.36 thf(exuE3e,definition,
% 1.95/1.36 ( exuE3e
% 1.95/1.36 = ( ! [Xphi: $i > $o] :
% 1.95/1.36 ( ( exu
% 1.95/1.36 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.95/1.36 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(setext_type,type,
% 1.95/1.36 setext: $o ).
% 1.95/1.36
% 1.95/1.36 thf(setext,definition,
% 1.95/1.36 ( setext
% 1.95/1.36 = ( ! [A: $i,B: $i] :
% 1.95/1.36 ( ! [Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ A )
% 1.95/1.36 => ( in @ Xx @ B ) )
% 1.95/1.36 => ( ! [Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ B )
% 1.95/1.36 => ( in @ Xx @ A ) )
% 1.95/1.36 => ( A = B ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(emptyI_type,type,
% 1.95/1.36 emptyI: $o ).
% 1.95/1.36
% 1.95/1.36 thf(emptyI,definition,
% 1.95/1.36 ( emptyI
% 1.95/1.36 = ( ! [A: $i] :
% 1.95/1.36 ( ! [Xx: $i] :
% 1.95/1.36 ~ ( in @ Xx @ A )
% 1.95/1.36 => ( A = emptyset ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(noeltsimpempty_type,type,
% 1.95/1.36 noeltsimpempty: $o ).
% 1.95/1.36
% 1.95/1.36 thf(noeltsimpempty,definition,
% 1.95/1.36 ( noeltsimpempty
% 1.95/1.36 = ( ! [A: $i] :
% 1.95/1.36 ( ! [Xx: $i] :
% 1.95/1.36 ~ ( in @ Xx @ A )
% 1.95/1.36 => ( A = emptyset ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(setbeta_type,type,
% 1.95/1.36 setbeta: $o ).
% 1.95/1.36
% 1.95/1.36 thf(setbeta,definition,
% 1.95/1.36 ( setbeta
% 1.95/1.36 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ A )
% 1.95/1.36 => ( ( in @ Xx
% 1.95/1.36 @ ( dsetconstr @ A
% 1.95/1.36 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.95/1.36 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(nonempty_type,type,
% 1.95/1.36 nonempty: $i > $o ).
% 1.95/1.36
% 1.95/1.36 thf(nonempty,definition,
% 1.95/1.36 ( nonempty
% 1.95/1.36 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(nonemptyE1_type,type,
% 1.95/1.36 nonemptyE1: $o ).
% 1.95/1.36
% 1.95/1.36 thf(nonemptyE1,definition,
% 1.95/1.36 ( nonemptyE1
% 1.95/1.36 = ( ! [A: $i] :
% 1.95/1.36 ( ( nonempty @ A )
% 1.95/1.36 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(nonemptyI_type,type,
% 1.95/1.36 nonemptyI: $o ).
% 1.95/1.36
% 1.95/1.36 thf(nonemptyI,definition,
% 1.95/1.36 ( nonemptyI
% 1.95/1.36 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ A )
% 1.95/1.36 => ( ( Xphi @ Xx )
% 1.95/1.36 => ( nonempty
% 1.95/1.36 @ ( dsetconstr @ A
% 1.95/1.36 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(nonemptyI1_type,type,
% 1.95/1.36 nonemptyI1: $o ).
% 1.95/1.36
% 1.95/1.36 thf(nonemptyI1,definition,
% 1.95/1.36 ( nonemptyI1
% 1.95/1.36 = ( ! [A: $i] :
% 1.95/1.36 ( ? [Xx: $i] : ( in @ Xx @ A )
% 1.95/1.36 => ( nonempty @ A ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(setadjoinIL_type,type,
% 1.95/1.36 setadjoinIL: $o ).
% 1.95/1.36
% 1.95/1.36 thf(setadjoinIL,definition,
% 1.95/1.36 ( setadjoinIL
% 1.95/1.36 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(emptyinunitempty_type,type,
% 1.95/1.36 emptyinunitempty: $o ).
% 1.95/1.36
% 1.95/1.36 thf(emptyinunitempty,definition,
% 1.95/1.36 ( emptyinunitempty
% 1.95/1.36 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(setadjoinIR_type,type,
% 1.95/1.36 setadjoinIR: $o ).
% 1.95/1.36
% 1.95/1.36 thf(setadjoinIR,definition,
% 1.95/1.36 ( setadjoinIR
% 1.95/1.36 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.95/1.36 ( ( in @ Xy @ A )
% 1.95/1.36 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(setadjoinE_type,type,
% 1.95/1.36 setadjoinE: $o ).
% 1.95/1.36
% 1.95/1.36 thf(setadjoinE,definition,
% 1.95/1.36 ( setadjoinE
% 1.95/1.36 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.95/1.36 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.95/1.36 => ! [Xphi: $o] :
% 1.95/1.36 ( ( ( Xy = Xx )
% 1.95/1.36 => Xphi )
% 1.95/1.36 => ( ( ( in @ Xy @ A )
% 1.95/1.36 => Xphi )
% 1.95/1.36 => Xphi ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(setadjoinOr_type,type,
% 1.95/1.36 setadjoinOr: $o ).
% 1.95/1.36
% 1.95/1.36 thf(setadjoinOr,definition,
% 1.95/1.36 ( setadjoinOr
% 1.95/1.36 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.95/1.36 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.95/1.36 => ( ( Xy = Xx )
% 1.95/1.36 | ( in @ Xy @ A ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(setoftrueEq_type,type,
% 1.95/1.36 setoftrueEq: $o ).
% 1.95/1.36
% 1.95/1.36 thf(setoftrueEq,definition,
% 1.95/1.36 ( setoftrueEq
% 1.95/1.36 = ( ! [A: $i] :
% 1.95/1.36 ( ( dsetconstr @ A
% 1.95/1.36 @ ^ [Xx: $i] : $true )
% 1.95/1.36 = A ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(powersetI_type,type,
% 1.95/1.36 powersetI: $o ).
% 1.95/1.36
% 1.95/1.36 thf(powersetI,definition,
% 1.95/1.36 ( powersetI
% 1.95/1.36 = ( ! [A: $i,B: $i] :
% 1.95/1.36 ( ! [Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ B )
% 1.95/1.36 => ( in @ Xx @ A ) )
% 1.95/1.36 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(emptyinPowerset_type,type,
% 1.95/1.36 emptyinPowerset: $o ).
% 1.95/1.36
% 1.95/1.36 thf(emptyinPowerset,definition,
% 1.95/1.36 ( emptyinPowerset
% 1.95/1.36 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(emptyInPowerset_type,type,
% 1.95/1.36 emptyInPowerset: $o ).
% 1.95/1.36
% 1.95/1.36 thf(emptyInPowerset,definition,
% 1.95/1.36 ( emptyInPowerset
% 1.95/1.36 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(powersetE_type,type,
% 1.95/1.36 powersetE: $o ).
% 1.95/1.36
% 1.95/1.36 thf(powersetE,definition,
% 1.95/1.36 ( powersetE
% 1.95/1.36 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.95/1.36 ( ( in @ B @ ( powerset @ A ) )
% 1.95/1.36 => ( ( in @ Xx @ B )
% 1.95/1.36 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(setunionI_type,type,
% 1.95/1.36 setunionI: $o ).
% 1.95/1.36
% 1.95/1.36 thf(setunionI,definition,
% 1.95/1.36 ( setunionI
% 1.95/1.36 = ( ! [A: $i,Xx: $i,B: $i] :
% 1.95/1.36 ( ( in @ Xx @ B )
% 1.95/1.36 => ( ( in @ B @ A )
% 1.95/1.36 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(setunionE_type,type,
% 1.95/1.36 setunionE: $o ).
% 1.95/1.36
% 1.95/1.36 thf(setunionE,definition,
% 1.95/1.36 ( setunionE
% 1.95/1.36 = ( ! [A: $i,Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ ( setunion @ A ) )
% 1.95/1.36 => ! [Xphi: $o] :
% 1.95/1.36 ( ! [B: $i] :
% 1.95/1.36 ( ( in @ Xx @ B )
% 1.95/1.36 => ( ( in @ B @ A )
% 1.95/1.36 => Xphi ) )
% 1.95/1.36 => Xphi ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(subPowSU_type,type,
% 1.95/1.36 subPowSU: $o ).
% 1.95/1.36
% 1.95/1.36 thf(subPowSU,definition,
% 1.95/1.36 ( subPowSU
% 1.95/1.36 = ( ! [A: $i,Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ A )
% 1.95/1.36 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(exuE2_type,type,
% 1.95/1.36 exuE2: $o ).
% 1.95/1.36
% 1.95/1.36 thf(exuE2,definition,
% 1.95/1.36 ( exuE2
% 1.95/1.36 = ( ! [Xphi: $i > $o] :
% 1.95/1.36 ( ( exu
% 1.95/1.36 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.95/1.36 => ? [Xx: $i] :
% 1.95/1.36 ! [Xy: $i] :
% 1.95/1.36 ( ( Xphi @ Xy )
% 1.95/1.36 <=> ( Xy = Xx ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(nonemptyImpWitness_type,type,
% 1.95/1.36 nonemptyImpWitness: $o ).
% 1.95/1.36
% 1.95/1.36 thf(nonemptyImpWitness,definition,
% 1.95/1.36 ( nonemptyImpWitness
% 1.95/1.36 = ( ! [A: $i] :
% 1.95/1.36 ( ( nonempty @ A )
% 1.95/1.36 => ? [Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ A )
% 1.95/1.36 & $true ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(uniqinunit_type,type,
% 1.95/1.36 uniqinunit: $o ).
% 1.95/1.36
% 1.95/1.36 thf(uniqinunit,definition,
% 1.95/1.36 ( uniqinunit
% 1.95/1.36 = ( ! [Xx: $i,Xy: $i] :
% 1.95/1.36 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 1.95/1.36 => ( Xx = Xy ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(notinsingleton_type,type,
% 1.95/1.36 notinsingleton: $o ).
% 1.95/1.36
% 1.95/1.36 thf(notinsingleton,definition,
% 1.95/1.36 ( notinsingleton
% 1.95/1.36 = ( ! [Xx: $i,Xy: $i] :
% 1.95/1.36 ( ( Xx != Xy )
% 1.95/1.36 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(eqinunit_type,type,
% 1.95/1.36 eqinunit: $o ).
% 1.95/1.36
% 1.95/1.36 thf(eqinunit,definition,
% 1.95/1.36 ( eqinunit
% 1.95/1.36 = ( ! [Xx: $i,Xy: $i] :
% 1.95/1.36 ( ( Xx = Xy )
% 1.95/1.36 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(singletonsswitch_type,type,
% 1.95/1.36 singletonsswitch: $o ).
% 1.95/1.36
% 1.95/1.36 thf(singletonsswitch,definition,
% 1.95/1.36 ( singletonsswitch
% 1.95/1.36 = ( ! [Xx: $i,Xy: $i] :
% 1.95/1.36 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 1.95/1.36 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(upairsetE_type,type,
% 1.95/1.36 upairsetE: $o ).
% 1.95/1.36
% 1.95/1.36 thf(upairsetE,definition,
% 1.95/1.36 ( upairsetE
% 1.95/1.36 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 1.95/1.36 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 1.95/1.36 => ( ( Xz = Xx )
% 1.95/1.36 | ( Xz = Xy ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(upairsetIL_type,type,
% 1.95/1.36 upairsetIL: $o ).
% 1.95/1.36
% 1.95/1.36 thf(upairsetIL,definition,
% 1.95/1.36 ( upairsetIL
% 1.95/1.36 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(upairsetIR_type,type,
% 1.95/1.36 upairsetIR: $o ).
% 1.95/1.36
% 1.95/1.36 thf(upairsetIR,definition,
% 1.95/1.36 ( upairsetIR
% 1.95/1.36 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(emptyE1_type,type,
% 1.95/1.36 emptyE1: $o ).
% 1.95/1.36
% 1.95/1.36 thf(emptyE1,definition,
% 1.95/1.36 ( emptyE1
% 1.95/1.36 = ( ! [A: $i,Xphi: $i > $o] :
% 1.95/1.36 ( ? [Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ A )
% 1.95/1.36 & ( Xphi @ Xx ) )
% 1.95/1.36 => ( ( ( dsetconstr @ A
% 1.95/1.36 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.95/1.36 = emptyset )
% 1.95/1.36 => $false ) ) ) ) ).
% 1.95/1.36
% 1.95/1.36 thf(vacuousDall,conjecture,
% 1.95/1.36 ( setextAx
% 1.95/1.36 => ( emptysetAx
% 1.95/1.36 => ( setadjoinAx
% 1.95/1.36 => ( powersetAx
% 1.95/1.36 => ( setunionAx
% 1.95/1.36 => ( omega0Ax
% 1.95/1.36 => ( omegaSAx
% 1.95/1.36 => ( omegaIndAx
% 1.95/1.36 => ( replAx
% 1.95/1.36 => ( foundationAx
% 1.95/1.36 => ( wellorderingAx
% 1.95/1.36 => ( descrp
% 1.95/1.36 => ( dsetconstrI
% 1.95/1.36 => ( dsetconstrEL
% 1.95/1.36 => ( dsetconstrER
% 1.95/1.36 => ( exuE1
% 1.95/1.36 => ( prop2setE
% 1.95/1.36 => ( emptysetE
% 1.95/1.36 => ( emptysetimpfalse
% 1.95/1.36 => ( notinemptyset
% 1.95/1.36 => ( exuE3e
% 1.95/1.36 => ( setext
% 1.95/1.36 => ( emptyI
% 1.95/1.36 => ( noeltsimpempty
% 1.95/1.36 => ( setbeta
% 1.95/1.36 => ( nonemptyE1
% 1.95/1.36 => ( nonemptyI
% 1.95/1.36 => ( nonemptyI1
% 1.95/1.36 => ( setadjoinIL
% 1.95/1.36 => ( emptyinunitempty
% 1.95/1.36 => ( setadjoinIR
% 1.95/1.36 => ( setadjoinE
% 1.95/1.36 => ( setadjoinOr
% 1.95/1.36 => ( setoftrueEq
% 1.95/1.36 => ( powersetI
% 1.95/1.36 => ( emptyinPowerset
% 1.95/1.36 => ( emptyInPowerset
% 1.95/1.36 => ( powersetE
% 1.95/1.36 => ( setunionI
% 1.95/1.36 => ( setunionE
% 1.95/1.36 => ( subPowSU
% 1.95/1.36 => ( exuE2
% 1.95/1.36 => ( nonemptyImpWitness
% 1.95/1.36 => ( uniqinunit
% 1.95/1.36 => ( notinsingleton
% 1.95/1.36 => ( eqinunit
% 1.95/1.36 => ( singletonsswitch
% 1.95/1.36 => ( upairsetE
% 1.95/1.36 => ( upairsetIL
% 1.95/1.36 => ( upairsetIR
% 1.95/1.36 => ( emptyE1
% 1.95/1.36 => ! [Xphi: $i > $o,Xx: $i] :
% 1.95/1.36 ( ( in @ Xx @ emptyset )
% 1.95/1.36 => ( Xphi @ Xx ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 1.95/1.36 ------------------------
% 1.95/1.38 % (27235)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_27016 for (2999ds/183Mi)
% 1.95/1.38 % (27241)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_27016 for (2999ds/18Mi)
% 1.95/1.39 % (27241)Instruction limit reached!
% 1.95/1.39 % (27241)------------------------------
% 1.95/1.39 % (27241)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.95/1.39 % (27241)Termination reason: Unknown
% 1.95/1.39 % (27241)Termination phase: Preprocessing 3
% 1.95/1.39
% 1.95/1.39 % (27241)Memory used [KB]: 1535
% 1.95/1.39 % (27241)Time elapsed: 0.007 s
% 1.95/1.39 % (27241)Instructions burned: 19 (million)
% 1.95/1.39 % (27241)------------------------------
% 1.95/1.39 % (27241)------------------------------
% 1.95/1.39 % (27236)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_27016 for (2999ds/4Mi)
% 1.95/1.39 % (27238)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_27016 for (2999ds/2Mi)
% 1.95/1.39 % (27237)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_27016 for (2999ds/27Mi)
% 1.95/1.39 % (27239)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_27016 for (2999ds/2Mi)
% 1.95/1.39 % (27240)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_27016 for (2999ds/275Mi)
% 1.95/1.39 % (27238)Instruction limit reached!
% 1.95/1.39 % (27238)------------------------------
% 1.95/1.39 % (27238)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.95/1.39 % (27238)Termination reason: Unknown
% 1.95/1.39 % (27238)Termination phase: shuffling
% 1.95/1.39
% 1.95/1.39 % (27238)Memory used [KB]: 1023
% 1.95/1.39 % (27238)Time elapsed: 0.003 s
% 1.95/1.39 % (27238)Instructions burned: 2 (million)
% 1.95/1.39 % (27238)------------------------------
% 1.95/1.39 % (27238)------------------------------
% 1.95/1.39 % (27239)Instruction limit reached!
% 1.95/1.39 % (27239)------------------------------
% 1.95/1.39 % (27239)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.95/1.39 % (27239)Termination reason: Unknown
% 1.95/1.39 % (27239)Termination phase: shuffling
% 1.95/1.39
% 1.95/1.39 % (27239)Memory used [KB]: 1023
% 1.95/1.39 % (27239)Time elapsed: 0.003 s
% 1.95/1.39 % (27239)Instructions burned: 2 (million)
% 1.95/1.39 % (27239)------------------------------
% 1.95/1.39 % (27239)------------------------------
% 1.95/1.39 % (27236)Instruction limit reached!
% 1.95/1.39 % (27236)------------------------------
% 1.95/1.39 % (27236)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.95/1.39 % (27236)Termination reason: Unknown
% 1.95/1.39 % (27236)Termination phase: shuffling
% 1.95/1.39
% 1.95/1.39 % (27236)Memory used [KB]: 1151
% 1.95/1.39 % (27236)Time elapsed: 0.004 s
% 1.95/1.39 % (27236)Instructions burned: 4 (million)
% 1.95/1.39 % (27236)------------------------------
% 1.95/1.39 % (27236)------------------------------
% 1.95/1.40 % (27242)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_27016 for (2999ds/3Mi)
% 2.09/1.40 % (27242)Instruction limit reached!
% 2.09/1.40 % (27242)------------------------------
% 2.09/1.40 % (27242)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.40 % (27242)Termination reason: Unknown
% 2.09/1.40 % (27242)Termination phase: shuffling
% 2.09/1.40
% 2.09/1.40 % (27242)Memory used [KB]: 1151
% 2.09/1.40 % (27242)Time elapsed: 0.003 s
% 2.09/1.40 % (27242)Instructions burned: 4 (million)
% 2.09/1.40 % (27242)------------------------------
% 2.09/1.40 % (27242)------------------------------
% 2.09/1.40 % (27237)Instruction limit reached!
% 2.09/1.40 % (27237)------------------------------
% 2.09/1.40 % (27237)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.40 % (27237)Termination reason: Unknown
% 2.09/1.40 % (27237)Termination phase: Saturation
% 2.09/1.40
% 2.09/1.40 % (27237)Memory used [KB]: 5756
% 2.09/1.40 % (27237)Time elapsed: 0.016 s
% 2.09/1.40 % (27237)Instructions burned: 27 (million)
% 2.09/1.40 % (27237)------------------------------
% 2.09/1.40 % (27237)------------------------------
% 2.09/1.40 % (27235)First to succeed.
% 2.09/1.40 % (27235)Refutation found. Thanks to Tanya!
% 2.09/1.40 % SZS status Theorem for DTF2THF_27016
% 2.09/1.40 % SZS output start Proof for DTF2THF_27016
% 2.09/1.40 thf(func_def_0, type, in: $i > $i > $o).
% 2.09/1.40 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.09/1.40 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.09/1.40 thf(func_def_8, type, powerset: $i > $i).
% 2.09/1.40 thf(func_def_10, type, setunion: $i > $i).
% 2.09/1.40 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.09/1.40 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.09/1.40 thf(func_def_26, type, prop2set: $o > $i).
% 2.09/1.40 thf(func_def_36, type, nonempty: $i > $o).
% 2.09/1.40 thf(func_def_79, type, sK0: $i > $o).
% 2.09/1.40 thf(func_def_82, type, sK3: $i > $i).
% 2.09/1.40 thf(func_def_83, type, sK4: $i > $i > $i).
% 2.09/1.40 thf(func_def_84, type, sK5: $i > $i).
% 2.09/1.40 thf(func_def_85, type, sK6: $o > $i > $i > $i).
% 2.09/1.40 thf(func_def_86, type, sK7: ($i > $o) > $i).
% 2.09/1.40 thf(func_def_87, type, sK8: ($i > $o) > $i > $i).
% 2.09/1.40 thf(func_def_88, type, sK9: $i > $i > $i).
% 2.09/1.40 thf(func_def_89, type, sK10: $i > $i).
% 2.09/1.40 thf(func_def_90, type, sK11: $i > $i > $i > $i).
% 2.09/1.40 thf(func_def_91, type, sK12: $i > $i > $i).
% 2.09/1.40 thf(func_def_92, type, sK13: $i > $i > $i).
% 2.09/1.40 thf(func_def_93, type, sK14: $i > $i > $i).
% 2.09/1.40 thf(func_def_94, type, sK15: $i > $i > $i).
% 2.09/1.40 thf(func_def_95, type, sK16: $i > ($i > $i > $o) > $i).
% 2.09/1.40 thf(func_def_96, type, sK17: $i > ($i > $i > $o) > $i).
% 2.09/1.40 thf(func_def_97, type, sK18: $i > $i > ($i > $i > $o) > $i).
% 2.09/1.40 thf(func_def_98, type, sK19: $i > $i > ($i > $i > $o) > $i).
% 2.09/1.40 thf(func_def_99, type, sK20: $i > $i > ($i > $i > $o) > $i).
% 2.09/1.40 thf(func_def_100, type, sK21: $i > $i > ($i > $i > $o) > $i).
% 2.09/1.40 thf(func_def_101, type, sK22: $i > $i).
% 2.09/1.40 thf(func_def_102, type, sK23: $i > $i).
% 2.09/1.40 thf(func_def_103, type, sK24: $i > $i > $i).
% 2.09/1.40 thf(func_def_104, type, sK25: $i > $i).
% 2.09/1.40 thf(func_def_105, type, sK26: $i > $i > $i).
% 2.09/1.40 thf(func_def_106, type, sK27: $i > $i > $i).
% 2.09/1.40 thf(func_def_107, type, sK28: $i > $i).
% 2.09/1.40 thf(func_def_108, type, sK29: ($i > $o) > $i).
% 2.09/1.40 thf(func_def_109, type, sK30: ($i > $o) > $i > $i).
% 2.09/1.40 thf(func_def_110, type, sK31: ($i > $o) > $i > $i).
% 2.09/1.40 thf(f947,plain,(
% 2.09/1.40 $false),
% 2.09/1.40 inference(avatar_sat_refutation,[status(thm)],[f907,f946])).
% 2.09/1.40 thf(f946,plain,(
% 2.09/1.40 ~spl2_1),
% 2.09/1.40 inference(avatar_contradiction_clause,[status(thm)],[f945])).
% 2.09/1.40 thf(f945,plain,(
% 2.09/1.40 $false | ~spl2_1),
% 2.09/1.40 inference(trivial_inequality_removal,[status(thm)],[f941])).
% 2.09/1.40 thf(f941,plain,(
% 2.09/1.40 ($false = $true) | ~spl2_1),
% 2.09/1.40 inference(superposition,[status(thm)],[f228,f900])).
% 2.09/1.40 thf(f900,plain,(
% 2.09/1.40 ( ! [X1 : $i] : (((in @ X1 @ emptyset) = $false)) ) | ~spl2_1),
% 2.09/1.40 inference(avatar_component_clause,[status(thm)],[f899])).
% 2.09/1.40 thf(f899,definition,(
% 2.09/1.40 spl2_1 <=> ! [X1 : $i] : ((in @ X1 @ emptyset) = $false)),
% 2.09/1.40 introduced(definition,[new_symbols(naming,[spl2_1])],[avatar_definition])).
% 2.09/1.40 thf(f228,plain,(
% 2.09/1.40 ((in @ sK1 @ emptyset) = $true)),
% 2.09/1.40 inference(cnf_transformation,[status(thm)],[f166])).
% 2.09/1.40 thf(f166,plain,(
% 2.09/1.40 (upairsetE = $true) & (powersetI = $true) & (omega0Ax = $true) & (notinsingleton = $true) & (eqinunit = $true) & (noeltsimpempty = $true) & (notinemptyset = $true) & (nonemptyI1 = $true) & (upairsetIL = $true) & (upairsetIR = $true) & (emptyinunitempty = $true) & (replAx = $true) & (nonemptyI = $true) & (setoftrueEq = $true) & (exuE3e = $true) & (emptyInPowerset = $true) & (subPowSU = $true) & (dsetconstrI = $true) & (dsetconstrEL = $true) & (setextAx = $true) & (setadjoinIL = $true) & (dsetconstrER = $true) & (exuE2 = $true) & (emptysetAx = $true) & (setunionI = $true) & (emptyI = $true) & (omegaIndAx = $true) & (powersetAx = $true) & (emptysetE = $true) & (wellorderingAx = $true) & (setadjoinIR = $true) & (omegaSAx = $true) & (uniqinunit = $true) & (setadjoinAx = $true) & (((sK0 @ sK1) != $true) & ((in @ sK1 @ emptyset) = $true)) & (descrp = $true) & (setunionAx = $true) & (nonemptyImpWitness = $true) & (emptysetimpfalse = $true) & (setadjoinE = $true) & (setext = $true) & (nonemptyE1 = $true) & (emptyinPowerset = $true) & (setbeta = $true) & (prop2setE = $true) & (powersetE = $true) & (emptyE1 = $true) & (singletonsswitch = $true) & (foundationAx = $true) & (setunionE = $true) & (exuE1 = $true) & (setadjoinOr = $true)),
% 2.09/1.40 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f164,f165])).
% 2.09/1.40 thf(f165,plain,(
% 2.09/1.40 ? [X0 : $i > $o,X1 : $i] : (((X0 @ X1) != $true) & ((in @ X1 @ emptyset) = $true)) => (((sK0 @ sK1) != $true) & ((in @ sK1 @ emptyset) = $true))),
% 2.09/1.40 introduced(definition,[],[choice_axiom])).
% 2.09/1.40 thf(f164,plain,(
% 2.09/1.40 (upairsetE = $true) & (powersetI = $true) & (omega0Ax = $true) & (notinsingleton = $true) & (eqinunit = $true) & (noeltsimpempty = $true) & (notinemptyset = $true) & (nonemptyI1 = $true) & (upairsetIL = $true) & (upairsetIR = $true) & (emptyinunitempty = $true) & (replAx = $true) & (nonemptyI = $true) & (setoftrueEq = $true) & (exuE3e = $true) & (emptyInPowerset = $true) & (subPowSU = $true) & (dsetconstrI = $true) & (dsetconstrEL = $true) & (setextAx = $true) & (setadjoinIL = $true) & (dsetconstrER = $true) & (exuE2 = $true) & (emptysetAx = $true) & (setunionI = $true) & (emptyI = $true) & (omegaIndAx = $true) & (powersetAx = $true) & (emptysetE = $true) & (wellorderingAx = $true) & (setadjoinIR = $true) & (omegaSAx = $true) & (uniqinunit = $true) & (setadjoinAx = $true) & ? [X0 : $i > $o,X1 : $i] : (((X0 @ X1) != $true) & ((in @ X1 @ emptyset) = $true)) & (descrp = $true) & (setunionAx = $true) & (nonemptyImpWitness = $true) & (emptysetimpfalse = $true) & (setadjoinE = $true) & (setext = $true) & (nonemptyE1 = $true) & (emptyinPowerset = $true) & (setbeta = $true) & (prop2setE = $true) & (powersetE = $true) & (emptyE1 = $true) & (singletonsswitch = $true) & (foundationAx = $true) & (setunionE = $true) & (exuE1 = $true) & (setadjoinOr = $true)),
% 2.09/1.40 inference(flattening,[status(thm)],[f163])).
% 2.09/1.40 thf(f163,plain,(
% 2.09/1.40 ((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i > $o,X1 : $i] : (((X0 @ X1) != $true) & ((in @ X1 @ emptyset) = $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)),
% 2.09/1.40 inference(ennf_transformation,[status(thm)],[f116])).
% 2.09/1.40 thf(f116,plain,(
% 2.09/1.40 ~((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) => ! [X0 : $i > $o,X1 : $i] : (((in @ X1 @ emptyset) = $true) => ((X0 @ X1) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.09/1.40 inference(fool_elimination,[status(thm)],[f115])).
% 2.09/1.40 thf(f115,plain,(
% 2.09/1.40 ~(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 => ! [X0 : $i > $o,X1 : $i] : ((in @ X1 @ emptyset) => (X0 @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.09/1.40 inference(rectify,[status(thm)],[f56])).
% 2.09/1.40 thf(f56,negated_conjecture,(
% 2.09/1.40 ~(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 => ! [X0 : $i > $o,X1 : $i] : ((in @ X1 @ emptyset) => (X0 @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.09/1.40 inference(negated_conjecture,[status(cth)],[f55])).
% 2.09/1.40 thf(f55,conjecture,(
% 2.09/1.40 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 => ! [X0 : $i > $o,X1 : $i] : ((in @ X1 @ emptyset) => (X0 @ X1))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.09/1.40 file('/export/starexec/sandbox2/tmp/tmp.ivwA5omCSE/DTF2THF_27016.p',vacuousDall)).
% 2.09/1.40 thf(f907,plain,(
% 2.09/1.40 spl2_1),
% 2.09/1.40 inference(avatar_split_clause,[status(thm)],[f476,f899])).
% 2.09/1.40 thf(f476,plain,(
% 2.09/1.40 ( ! [X1 : $i] : (((in @ X1 @ emptyset) = $false)) )),
% 2.09/1.40 inference(not_proxy_clausification,[status(thm)],[f475])).
% 2.09/1.40 thf(f475,plain,(
% 2.09/1.40 ( ! [X1 : $i] : (((~ (in @ X1 @ emptyset)) = $true)) )),
% 2.09/1.40 inference(beta_eta_normalization,[status(thm)],[f474])).
% 2.09/1.40 thf(f474,plain,(
% 2.09/1.40 ( ! [X1 : $i] : ((((^[Y0 : $i]: (~ (in @ Y0 @ emptyset))) @ X1) = $true)) )),
% 2.09/1.40 inference(pi_clausification,[status(thm)],[f473])).
% 2.09/1.40 thf(f473,plain,(
% 2.09/1.40 ((!! @ $i @ (^[Y0 : $i]: (~ (in @ Y0 @ emptyset)))) = $true)),
% 2.09/1.40 inference(boolean_simplification,[status(thm)],[f311])).
% 2.09/1.40 thf(f311,plain,(
% 2.09/1.40 ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ emptyset) => $false))) = $true)),
% 2.09/1.40 inference(definition_unfolding,[status(thm)],[f224,f184])).
% 2.09/1.40 thf(f184,plain,(
% 2.09/1.40 (emptysetimpfalse = (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ emptyset) => $false))))),
% 2.09/1.40 inference(cnf_transformation,[status(thm)],[f98])).
% 2.09/1.40 thf(f98,plain,(
% 2.09/1.40 (emptysetimpfalse = (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ emptyset) => $false))))),
% 2.09/1.40 inference(fool_elimination,[status(thm)],[f97])).
% 2.09/1.40 thf(f97,plain,(
% 2.09/1.40 (! [X0 : $i] : ((in @ X0 @ emptyset) => $false) = emptysetimpfalse)),
% 2.09/1.40 inference(rectify,[status(thm)],[f21])).
% 2.09/1.40 thf(f21,axiom,(
% 2.09/1.40 (! [X1 : $i] : ((in @ X1 @ emptyset) => $false) = emptysetimpfalse)),
% 2.09/1.40 file('/export/starexec/sandbox2/tmp/tmp.ivwA5omCSE/DTF2THF_27016.p',emptysetimpfalse)).
% 2.09/1.40 thf(f224,plain,(
% 2.09/1.40 (emptysetimpfalse = $true)),
% 2.09/1.40 inference(cnf_transformation,[status(thm)],[f166])).
% 2.09/1.40 % SZS output end Proof for DTF2THF_27016
% 2.09/1.40 % (27235)------------------------------
% 2.09/1.40 % (27235)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.09/1.40 % (27235)Termination reason: Refutation
% 2.09/1.40
% 2.09/1.40 % (27235)Memory used [KB]: 6268
% 2.09/1.40 % (27235)Time elapsed: 0.024 s
% 2.09/1.40 % (27235)Instructions burned: 77 (million)
% 2.09/1.40 % (27235)------------------------------
% 2.09/1.40 % (27235)------------------------------
% 2.09/1.40 % (27234)Success in time 0.034 s
%------------------------------------------------------------------------------