%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU508^1 : 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 : n031.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 09:34:59 AM UTC 2026
% Result : Theorem 0.31s 1.37s
% Output : Refutation 0.31s
% Verified :
% SZS Type : Refutation
% Derivation depth : 24
% Number of leaves : 8
% Syntax : Number of formulae : 66 ( 30 unt; 0 typ; 4 def)
% Number of atoms : 717 ( 168 equ; 0 cnn)
% Maximal formula atoms : 26 ( 10 avg)
% Number of connectives : 699 ( 30 ~; 34 |; 77 &; 377 @)
% ( 7 <=>; 138 =>; 0 <=; 0 <~>)
% Maximal formula depth : 35 ( 6 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 50 ( 50 >; 0 *; 0 +; 0 <<)
% Number of symbols : 74 ( 70 usr; 33 con; 0-3 aty)
% ( 36 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 128 ( 62 ^; 57 !; 9 ?; 128 :)
% 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_51,type,
sK0: $i > $o ).
thf(func_def_55,type,
sK4: $i > $i > $i ).
thf(func_def_56,type,
sK5: $i > ( $i > $i > $o ) > $i ).
thf(func_def_57,type,
sK6: $i > ( $i > $i > $o ) > $i ).
thf(func_def_58,type,
sK7: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_59,type,
sK8: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_60,type,
sK9: $i > ( $i > $i > $o ) > $i ).
thf(func_def_61,type,
sK10: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_62,type,
sK11: $i > ( $i > $i > $o ) > $i ).
thf(func_def_63,type,
sK12: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_64,type,
sK13: $i > $i ).
thf(func_def_65,type,
sK14: $i > $i > $i > $i ).
thf(func_def_66,type,
sK15: $i > $i > $i ).
thf(func_def_67,type,
sK16: $i > $i > $i ).
thf(func_def_68,type,
sK17: $i > $i > $i ).
thf(func_def_69,type,
sK18: $i > $i > $i ).
thf(func_def_70,type,
sK19: $i > $i > $i ).
thf(func_def_71,type,
sK20: $i > $i > $i ).
thf(func_def_72,type,
sK21: $i > $i ).
thf(func_def_73,type,
sK22: $i > $i ).
thf(func_def_74,type,
sK23: $i > $i > $i ).
thf(func_def_75,type,
sK24: ( $i > $o ) > $i > $i ).
thf(func_def_76,type,
sK25: $i > $i > $i ).
thf(func_def_77,type,
sK26: $i > $i ).
thf(func_def_78,type,
sK27: $i > $i > $i ).
thf(func_def_79,type,
sK28: $i > $i > $i ).
thf(func_def_80,type,
sK29: $i > $i > $i ).
thf(func_def_81,type,
sK30: ( $i > $o ) > $i ).
thf(func_def_82,type,
sK31: ( $i > $o ) > $i > $i ).
thf(func_def_83,type,
sK32: $i > $i ).
thf(func_def_84,type,
ph33:
!>[X0: $tType] : X0 ).
thf(func_def_85,type,
sK34: $i > $i > $i ).
thf(f903,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f539,f548,f577,f583,f899]) ).
thf(f899,plain,
( spl3_3
| ~ spl3_4 ),
inference(avatar_contradiction_clause,[status(thm)],[f898]) ).
thf(f898,plain,
( $false
| spl3_3
| ~ spl3_4 ),
inference(trivial_inequality_removal,[status(thm)],[f886]) ).
thf(f886,plain,
( ( $false = $true )
| spl3_3
| ~ spl3_4 ),
inference(superposition,[status(thm)],[f103,f881]) ).
thf(f881,plain,
( ( $false
= ( in @ sK1 @ sK2 ) )
| spl3_3
| ~ spl3_4 ),
inference(subsumption_resolution,[status(thm)],[f879,f542]) ).
thf(f542,plain,
( ( $false
!= ( sK0 @ sK1 ) )
| spl3_3 ),
inference(avatar_component_clause,[status(thm)],[f541]) ).
thf(f541,definition,
( spl3_3
<=> ( $false
= ( sK0 @ sK1 ) ) ),
introduced(definition,[new_symbols(naming,[spl3_3])],[avatar_definition]) ).
thf(f879,plain,
( ( $false
= ( sK0 @ sK1 ) )
| ( $false
= ( in @ sK1 @ sK2 ) )
| ~ spl3_4 ),
inference(trivial_inequality_removal,[status(thm)],[f869]) ).
thf(f869,plain,
( ( $false = $true )
| ( $false
= ( sK0 @ sK1 ) )
| ( $false
= ( in @ sK1 @ sK2 ) )
| ~ spl3_4 ),
inference(superposition,[status(thm)],[f547,f502]) ).
thf(f502,plain,
! [X2: $i,X3: $i > $o,X1: $i] :
( ( ( in @ X2 @ ( dsetconstr @ X1 @ X3 ) )
= $true )
| ( $false
= ( in @ X2 @ X1 ) )
| ( ( X3 @ X2 )
= $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f501]) ).
thf(f501,plain,
! [X2: $i,X3: $i > $o,X1: $i] :
( ( $true
= ( ( X3 @ X2 )
=> ( in @ X2 @ ( dsetconstr @ X1 @ X3 ) ) ) )
| ( $false
= ( in @ X2 @ X1 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f500]) ).
thf(f500,plain,
! [X2: $i,X3: $i > $o,X1: $i] :
( $true
= ( ( in @ X2 @ X1 )
=> ( ( X3 @ X2 )
=> ( in @ X2 @ ( dsetconstr @ X1 @ X3 ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f499]) ).
thf(f499,plain,
! [X2: $i,X3: $i > $o,X1: $i] :
( ( ^ [Y0: $i > $o] :
( ( in @ X2 @ X1 )
=> ( ( Y0 @ X2 )
=> ( in @ X2 @ ( dsetconstr @ X1 @ Y0 ) ) ) )
@ X3 )
= $true ),
inference(pi_clausification,[status(thm)],[f498]) ).
thf(f498,plain,
! [X2: $i,X1: $i] :
( ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( ( in @ X2 @ X1 )
=> ( ( Y0 @ X2 )
=> ( in @ X2 @ ( dsetconstr @ X1 @ Y0 ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f497]) ).
thf(f497,plain,
! [X2: $i,X1: $i] :
( ( ^ [Y0: $i] :
( !! @ ( $i > $o )
@ ^ [Y1: $i > $o] :
( ( in @ Y0 @ X1 )
=> ( ( Y1 @ Y0 )
=> ( in @ Y0 @ ( dsetconstr @ X1 @ Y1 ) ) ) ) )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f496]) ).
thf(f496,plain,
! [X1: $i] :
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ ( $i > $o )
@ ^ [Y1: $i > $o] :
( ( in @ Y0 @ X1 )
=> ( ( Y1 @ Y0 )
=> ( in @ Y0 @ ( dsetconstr @ X1 @ Y1 ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f495]) ).
thf(f495,plain,
! [X1: $i] :
( $true
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ ( $i > $o )
@ ^ [Y2: $i > $o] :
( ( in @ Y1 @ Y0 )
=> ( ( Y2 @ Y1 )
=> ( in @ Y1 @ ( dsetconstr @ Y0 @ Y2 ) ) ) ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f494]) ).
thf(f494,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ ( $i > $o )
@ ^ [Y2: $i > $o] :
( ( in @ Y1 @ Y0 )
=> ( ( Y2 @ Y1 )
=> ( in @ Y1 @ ( dsetconstr @ Y0 @ Y2 ) ) ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f156]) ).
thf(f156,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ ( $i > $o )
@ ^ [Y2: $i > $o] :
( ( in @ Y1 @ Y0 )
=> ( ( Y2 @ Y1 )
=> ( in @ Y1
@ ( dsetconstr @ Y0
@ ^ [Y3: $i] : ( Y2 @ Y3 ) ) ) ) ) ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f129,f116]) ).
thf(f116,plain,
dsetconstrI = $true,
inference(cnf_transformation,[status(thm)],[f84]) ).
thf(f84,plain,
( ( emptysetAx = $true )
& ( emptysetimpfalse = $true )
& ( setunionAx = $true )
& ( omegaIndAx = $true )
& ( dsetconstrI = $true )
& ( omega0Ax = $true )
& ( setext = $true )
& ( setextAx = $true )
& ( notinemptyset = $true )
& ( descrp = $true )
& ( foundationAx = $true )
& ( noeltsimpempty = $true )
& ( powersetAx = $true )
& ( prop2setE = $true )
& ( omegaSAx = $true )
& ( dsetconstrER = $true )
& ( exuE3e = $true )
& ( $true
= ( in @ sK1 @ sK2 ) )
& ( ( in @ sK1
@ ( dsetconstr @ sK2
@ ^ [Y0: $i] : ( sK0 @ Y0 ) ) )
!= ( sK0 @ sK1 ) )
& ( setadjoinAx = $true )
& ( emptyI = $true )
& ( exuE1 = $true )
& ( wellorderingAx = $true )
& ( emptysetE = $true )
& ( dsetconstrEL = $true )
& ( replAx = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f82,f83]) ).
thf(f83,plain,
( ? [X0: $i > $o,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
= $true )
& ( ( X0 @ X1 )
!= ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) ) ) )
=> ( ( $true
= ( in @ sK1 @ sK2 ) )
& ( ( in @ sK1
@ ( dsetconstr @ sK2
@ ^ [Y0: $i] : ( sK0 @ Y0 ) ) )
!= ( sK0 @ sK1 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f82,plain,
( ( emptysetAx = $true )
& ( emptysetimpfalse = $true )
& ( setunionAx = $true )
& ( omegaIndAx = $true )
& ( dsetconstrI = $true )
& ( omega0Ax = $true )
& ( setext = $true )
& ( setextAx = $true )
& ( notinemptyset = $true )
& ( descrp = $true )
& ( foundationAx = $true )
& ( noeltsimpempty = $true )
& ( powersetAx = $true )
& ( prop2setE = $true )
& ( omegaSAx = $true )
& ( dsetconstrER = $true )
& ( exuE3e = $true )
& ? [X0: $i > $o,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
= $true )
& ( ( X0 @ X1 )
!= ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) ) ) )
& ( setadjoinAx = $true )
& ( emptyI = $true )
& ( exuE1 = $true )
& ( wellorderingAx = $true )
& ( emptysetE = $true )
& ( dsetconstrEL = $true )
& ( replAx = $true ) ),
inference(flattening,[status(thm)],[f81]) ).
thf(f81,plain,
( ? [X0: $i > $o,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
= $true )
& ( ( X0 @ X1 )
!= ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) ) ) )
& ( 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)],[f41]) ).
thf(f41,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 )
=> ! [X0: $i > $o,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
= $true )
=> ( ( X0 @ X1 )
= ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f40]) ).
thf(f40,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
=> ! [X0: $i > $o,X1: $i,X2: $i] :
( ( in @ X1 @ X2 )
=> ( ( X0 @ X1 )
<=> ( in @ X1
@ ( dsetconstr @ X2
@ ^ [X3: $i] : ( X0 @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f28]) ).
thf(f28,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
=> ! [X0: $i > $o,X1: $i,X3: $i] :
( ( in @ X1 @ X3 )
=> ( ( X0 @ X1 )
<=> ( in @ X1
@ ( dsetconstr @ X3
@ ^ [X2: $i] : ( X0 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f27]) ).
thf(f27,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
=> ! [X0: $i > $o,X1: $i,X3: $i] :
( ( in @ X1 @ X3 )
=> ( ( X0 @ X1 )
<=> ( in @ X1
@ ( dsetconstr @ X3
@ ^ [X2: $i] : ( X0 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.hL7X93FyCR/DTF2THF_31925.p',setbeta) ).
thf(f129,plain,
( dsetconstrI
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ ( $i > $o )
@ ^ [Y2: $i > $o] :
( ( in @ Y1 @ Y0 )
=> ( ( Y2 @ Y1 )
=> ( in @ Y1
@ ( dsetconstr @ Y0
@ ^ [Y3: $i] : ( Y2 @ Y3 ) ) ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f51]) ).
thf(f51,plain,
( dsetconstrI
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ ( $i > $o )
@ ^ [Y2: $i > $o] :
( ( in @ Y1 @ Y0 )
=> ( ( Y2 @ Y1 )
=> ( in @ Y1
@ ( dsetconstr @ Y0
@ ^ [Y3: $i] : ( Y2 @ Y3 ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f50]) ).
thf(f50,plain,
! [X0: $i > $o,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
=> ( ( X0 @ X1 )
=> ( in @ X1
@ ( dsetconstr @ X2
@ ^ [X3: $i] : ( X0 @ X3 ) ) ) ) )
= dsetconstrI ),
inference(rectify,[status(thm)],[f14]) ).
thf(f14,axiom,
! [X0: $i > $o,X1: $i,X3: $i] :
( ( ( in @ X1 @ X3 )
=> ( ( X0 @ X1 )
=> ( in @ X1
@ ( dsetconstr @ X3
@ ^ [X2: $i] : ( X0 @ X2 ) ) ) ) )
= dsetconstrI ),
file('/export/starexec/sandbox/tmp/tmp.hL7X93FyCR/DTF2THF_31925.p',dsetconstrI) ).
thf(f547,plain,
( ( $false
= ( in @ sK1 @ ( dsetconstr @ sK2 @ sK0 ) ) )
| ~ spl3_4 ),
inference(avatar_component_clause,[status(thm)],[f545]) ).
thf(f545,definition,
( spl3_4
<=> ( $false
= ( in @ sK1 @ ( dsetconstr @ sK2 @ sK0 ) ) ) ),
introduced(definition,[new_symbols(naming,[spl3_4])],[avatar_definition]) ).
thf(f103,plain,
( $true
= ( in @ sK1 @ sK2 ) ),
inference(cnf_transformation,[status(thm)],[f84]) ).
thf(f583,plain,
( ~ spl3_1
| ~ spl3_3 ),
inference(avatar_contradiction_clause,[status(thm)],[f582]) ).
thf(f582,plain,
( $false
| ~ spl3_1
| ~ spl3_3 ),
inference(trivial_inequality_removal,[status(thm)],[f578]) ).
thf(f578,plain,
( ( $false = $true )
| ~ spl3_1
| ~ spl3_3 ),
inference(superposition,[status(thm)],[f534,f543]) ).
thf(f543,plain,
( ( $false
= ( sK0 @ sK1 ) )
| ~ spl3_3 ),
inference(avatar_component_clause,[status(thm)],[f541]) ).
thf(f534,plain,
( ( ( sK0 @ sK1 )
= $true )
| ~ spl3_1 ),
inference(avatar_component_clause,[status(thm)],[f532]) ).
thf(f532,definition,
( spl3_1
<=> ( ( sK0 @ sK1 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl3_1])],[avatar_definition]) ).
thf(f577,plain,
( spl3_1
| ~ spl3_2 ),
inference(avatar_split_clause,[status(thm)],[f569,f536,f532]) ).
thf(f536,definition,
( spl3_2
<=> ( $true
= ( in @ sK1 @ ( dsetconstr @ sK2 @ sK0 ) ) ) ),
introduced(definition,[new_symbols(naming,[spl3_2])],[avatar_definition]) ).
thf(f569,plain,
( ( ( sK0 @ sK1 )
= $true )
| ~ spl3_2 ),
inference(trivial_inequality_removal,[status(thm)],[f558]) ).
thf(f558,plain,
( ( ( sK0 @ sK1 )
= $true )
| ( $false = $true )
| ~ spl3_2 ),
inference(superposition,[status(thm)],[f418,f538]) ).
thf(f538,plain,
( ( $true
= ( in @ sK1 @ ( dsetconstr @ sK2 @ sK0 ) ) )
| ~ spl3_2 ),
inference(avatar_component_clause,[status(thm)],[f536]) ).
thf(f418,plain,
! [X2: $i > $o,X3: $i,X1: $i] :
( ( $false
= ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) )
| ( $true
= ( X2 @ X3 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f417]) ).
thf(f417,plain,
! [X2: $i > $o,X3: $i,X1: $i] :
( ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f416]) ).
thf(f416,plain,
! [X2: $i > $o,X3: $i,X1: $i] :
( $true
= ( ^ [Y0: $i] :
( ( in @ Y0 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ Y0 ) )
@ X3 ) ),
inference(pi_clausification,[status(thm)],[f415]) ).
thf(f415,plain,
! [X2: $i > $o,X1: $i] :
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ Y0 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f414]) ).
thf(f414,plain,
! [X2: $i > $o,X1: $i] :
( ( ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ ( dsetconstr @ X1 @ Y0 ) )
=> ( Y0 @ Y1 ) ) )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f413]) ).
thf(f413,plain,
! [X1: $i] :
( $true
= ( !! @ ( $i > $o )
@ ^ [Y0: $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ ( dsetconstr @ X1 @ Y0 ) )
=> ( Y0 @ Y1 ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f412]) ).
thf(f412,plain,
! [X1: $i] :
( $true
= ( ^ [Y0: $i] :
( !! @ ( $i > $o )
@ ^ [Y1: $i > $o] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( dsetconstr @ Y0 @ Y1 ) )
=> ( Y1 @ Y2 ) ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f411]) ).
thf(f411,plain,
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ ( $i > $o )
@ ^ [Y1: $i > $o] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( dsetconstr @ Y0 @ Y1 ) )
=> ( Y1 @ Y2 ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f157]) ).
thf(f157,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ ( $i > $o )
@ ^ [Y1: $i > $o] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2
@ ( dsetconstr @ Y0
@ ^ [Y3: $i] : ( Y1 @ Y3 ) ) )
=> ( Y1 @ Y2 ) ) ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f130,f105]) ).
thf(f105,plain,
dsetconstrER = $true,
inference(cnf_transformation,[status(thm)],[f84]) ).
thf(f130,plain,
( dsetconstrER
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ ( $i > $o )
@ ^ [Y1: $i > $o] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2
@ ( dsetconstr @ Y0
@ ^ [Y3: $i] : ( Y1 @ Y3 ) ) )
=> ( Y1 @ Y2 ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f72]) ).
thf(f72,plain,
( dsetconstrER
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ ( $i > $o )
@ ^ [Y1: $i > $o] :
( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2
@ ( dsetconstr @ Y0
@ ^ [Y3: $i] : ( Y1 @ Y3 ) ) )
=> ( Y1 @ Y2 ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f71]) ).
thf(f71,plain,
! [X0: $i,X1: $i > $o,X2: $i] :
( ( ( in @ X0
@ ( dsetconstr @ X2
@ ^ [X3: $i] : ( X1 @ X3 ) ) )
=> ( X1 @ X0 ) )
= dsetconstrER ),
inference(rectify,[status(thm)],[f16]) ).
thf(f16,axiom,
! [X1: $i,X0: $i > $o,X3: $i] :
( ( ( in @ X1
@ ( dsetconstr @ X3
@ ^ [X2: $i] : ( X0 @ X2 ) ) )
=> ( X0 @ X1 ) )
= dsetconstrER ),
file('/export/starexec/sandbox/tmp/tmp.hL7X93FyCR/DTF2THF_31925.p',dsetconstrER) ).
thf(f548,plain,
( spl3_3
| spl3_4 ),
inference(avatar_split_clause,[status(thm)],[f410,f545,f541]) ).
thf(f410,plain,
( ( $false
= ( in @ sK1 @ ( dsetconstr @ sK2 @ sK0 ) ) )
| ( $false
= ( sK0 @ sK1 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f408]) ).
thf(f408,plain,
( ( sK0 @ sK1 )
!= ( in @ sK1 @ ( dsetconstr @ sK2 @ sK0 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f102]) ).
thf(f102,plain,
( ( in @ sK1
@ ( dsetconstr @ sK2
@ ^ [Y0: $i] : ( sK0 @ Y0 ) ) )
!= ( sK0 @ sK1 ) ),
inference(cnf_transformation,[status(thm)],[f84]) ).
thf(f539,plain,
( spl3_1
| spl3_2 ),
inference(avatar_split_clause,[status(thm)],[f409,f536,f532]) ).
thf(f409,plain,
( ( $true
= ( in @ sK1 @ ( dsetconstr @ sK2 @ sK0 ) ) )
| ( ( sK0 @ sK1 )
= $true ) ),
inference(binary_proxy_clausification,[status(thm)],[f408]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU508^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n031.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 12:25:33 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
% 0.31/0.45 ---- Original DTF file ---
% 0.31/0.45 thf(spec,logic,$$dhol).
% 0.31/0.45 %------------------------------------------------------------------------------
% 0.31/0.45 % File : SEU508^1 : TPTP v9.2.1. Released v3.7.0.
% 0.31/0.45 % Domain : Set Theory
% 0.31/0.45 % Problem : Preliminary Notions - Basic Laws of Logic
% 0.31/0.45 % Version : Especial.
% 0.31/0.45 % English : (! A:i.! phi:i>o.! x:i.in x A -> (in x (dsetconstr A
% 0.31/0.45 % (^ y:i.phi y)) <-> phi x))
% 0.31/0.45
% 0.31/0.45 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.31/0.45 % Source : [Bro08]
% 0.31/0.45 % Names : ZFC010g [Bro08]
% 0.31/0.45
% 0.31/0.45 % Status : Theorem
% 0.31/0.45 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.38 v8.1.0, 0.27 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 0.40 v4.1.0, 0.67 v4.0.0, 1.00 v3.7.0
% 0.31/0.45 % Syntax : Number of formulae : 61 ( 26 unt; 34 typ; 26 def)
% 0.31/0.45 % Number of atoms : 157 ( 34 equ; 0 cnn)
% 0.31/0.45 % Maximal formula atoms : 26 ( 5 avg)
% 0.31/0.45 % Number of connectives : 282 ( 6 ~; 3 |; 18 &; 184 @)
% 0.31/0.45 % ( 7 <=>; 64 =>; 0 <=; 0 <~>)
% 0.31/0.45 % Maximal formula depth : 34 ( 2 avg)
% 0.31/0.45 % Number of types : 2 ( 0 usr)
% 0.31/0.45 % Number of type conns : 24 ( 24 >; 0 *; 0 +; 0 <<)
% 0.31/0.45 % Number of symbols : 36 ( 34 usr; 27 con; 0-2 aty)
% 0.31/0.45 % Number of variables : 92 ( 12 ^; 66 !; 14 ?; 92 :)
% 0.31/0.45 % SPC : TH0_THM_EQU_NAR
% 0.31/0.45
% 0.31/0.45 % Comments : http://mathgate.info/detsetitem.php?id=459
% 0.31/0.45 %------------------------------------------------------------------------------
% 0.31/0.45 thf(in_type,type,
% 0.31/0.45 in: $i > $i > $o ).
% 0.31/0.45
% 0.31/0.45 thf(exu_type,type,
% 0.31/0.45 exu: ( $i > $o ) > $o ).
% 0.31/0.45
% 0.31/0.45 thf(exu,definition,
% 0.31/0.45 ( exu
% 0.31/0.45 = ( ^ [Xphi: $i > $o] :
% 0.31/0.45 ? [Xx: $i] :
% 0.31/0.45 ( ( Xphi @ Xx )
% 0.31/0.45 & ! [Xy: $i] :
% 0.31/0.45 ( ( Xphi @ Xy )
% 0.31/0.45 => ( Xx = Xy ) ) ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(setextAx_type,type,
% 0.31/0.45 setextAx: $o ).
% 0.31/0.45
% 0.31/0.45 thf(setextAx,definition,
% 0.31/0.45 ( setextAx
% 0.31/0.45 = ( ! [A: $i,B: $i] :
% 0.31/0.45 ( ! [Xx: $i] :
% 0.31/0.45 ( ( in @ Xx @ A )
% 0.31/0.45 <=> ( in @ Xx @ B ) )
% 0.31/0.45 => ( A = B ) ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(emptyset_type,type,
% 0.31/0.45 emptyset: $i ).
% 0.31/0.45
% 0.31/0.45 thf(emptysetAx_type,type,
% 0.31/0.45 emptysetAx: $o ).
% 0.31/0.45
% 0.31/0.45 thf(emptysetAx,definition,
% 0.31/0.45 ( emptysetAx
% 0.31/0.45 = ( ! [Xx: $i] :
% 0.31/0.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(setadjoin_type,type,
% 0.31/0.45 setadjoin: $i > $i > $i ).
% 0.31/0.45
% 0.31/0.45 thf(setadjoinAx_type,type,
% 0.31/0.45 setadjoinAx: $o ).
% 0.31/0.45
% 0.31/0.45 thf(setadjoinAx,definition,
% 0.31/0.45 ( setadjoinAx
% 0.31/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/0.45 <=> ( ( Xy = Xx )
% 0.31/0.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(powerset_type,type,
% 0.31/0.45 powerset: $i > $i ).
% 0.31/0.45
% 0.31/0.45 thf(powersetAx_type,type,
% 0.31/0.45 powersetAx: $o ).
% 0.31/0.45
% 0.31/0.45 thf(powersetAx,definition,
% 0.31/0.45 ( powersetAx
% 0.31/0.45 = ( ! [A: $i,B: $i] :
% 0.31/0.45 ( ( in @ B @ ( powerset @ A ) )
% 0.31/0.45 <=> ! [Xx: $i] :
% 0.31/0.45 ( ( in @ Xx @ B )
% 0.31/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(setunion_type,type,
% 0.31/0.45 setunion: $i > $i ).
% 0.31/0.45
% 0.31/0.45 thf(setunionAx_type,type,
% 0.31/0.45 setunionAx: $o ).
% 0.31/0.45
% 0.31/0.45 thf(setunionAx,definition,
% 0.31/0.45 ( setunionAx
% 0.31/0.45 = ( ! [A: $i,Xx: $i] :
% 0.31/0.45 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.45 <=> ? [B: $i] :
% 0.31/0.45 ( ( in @ Xx @ B )
% 0.31/0.45 & ( in @ B @ A ) ) ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(omega_type,type,
% 0.31/0.45 omega: $i ).
% 0.31/0.45
% 0.31/0.45 thf(omega0Ax_type,type,
% 0.31/0.45 omega0Ax: $o ).
% 0.31/0.45
% 0.31/0.45 thf(omega0Ax,definition,
% 0.31/0.45 ( omega0Ax
% 0.31/0.45 = ( in @ emptyset @ omega ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(omegaSAx_type,type,
% 0.31/0.45 omegaSAx: $o ).
% 0.31/0.45
% 0.31/0.45 thf(omegaSAx,definition,
% 0.31/0.45 ( omegaSAx
% 0.31/0.45 = ( ! [Xx: $i] :
% 0.31/0.45 ( ( in @ Xx @ omega )
% 0.31/0.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(omegaIndAx_type,type,
% 0.31/0.45 omegaIndAx: $o ).
% 0.31/0.45
% 0.31/0.45 thf(omegaIndAx,definition,
% 0.31/0.45 ( omegaIndAx
% 0.31/0.45 = ( ! [A: $i] :
% 0.31/0.45 ( ( ( in @ emptyset @ A )
% 0.31/0.45 & ! [Xx: $i] :
% 0.31/0.45 ( ( ( in @ Xx @ omega )
% 0.31/0.45 & ( in @ Xx @ A ) )
% 0.31/0.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.31/0.45 => ! [Xx: $i] :
% 0.31/0.45 ( ( in @ Xx @ omega )
% 0.31/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(replAx_type,type,
% 0.31/0.45 replAx: $o ).
% 0.31/0.45
% 0.31/0.45 thf(replAx,definition,
% 0.31/0.45 ( replAx
% 0.31/0.45 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.31/0.45 ( ! [Xx: $i] :
% 0.31/0.45 ( ( in @ Xx @ A )
% 0.31/0.45 => ( exu
% 0.31/0.45 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.31/0.45 => ? [B: $i] :
% 0.31/0.45 ! [Xx: $i] :
% 0.31/0.45 ( ( in @ Xx @ B )
% 0.31/0.45 <=> ? [Xy: $i] :
% 0.31/0.45 ( ( in @ Xy @ A )
% 0.31/0.45 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(foundationAx_type,type,
% 0.31/0.45 foundationAx: $o ).
% 0.31/0.45
% 0.31/0.45 thf(foundationAx,definition,
% 0.31/0.45 ( foundationAx
% 0.31/0.45 = ( ! [A: $i] :
% 0.31/0.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.31/0.45 => ? [B: $i] :
% 0.31/0.45 ( ( in @ B @ A )
% 0.31/0.45 & ~ ? [Xx: $i] :
% 0.31/0.45 ( ( in @ Xx @ B )
% 0.31/0.45 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(wellorderingAx_type,type,
% 0.31/0.45 wellorderingAx: $o ).
% 0.31/0.45
% 0.31/0.45 thf(wellorderingAx,definition,
% 0.31/0.45 ( wellorderingAx
% 0.31/0.45 = ( ! [A: $i] :
% 0.31/0.45 ? [B: $i] :
% 0.31/0.45 ( ! [C: $i] :
% 0.31/0.45 ( ( in @ C @ B )
% 0.31/0.45 => ! [Xx: $i] :
% 0.31/0.45 ( ( in @ Xx @ C )
% 0.31/0.45 => ( in @ Xx @ A ) ) )
% 0.31/0.45 & ! [Xx: $i,Xy: $i] :
% 0.31/0.45 ( ( ( in @ Xx @ A )
% 0.31/0.45 & ( in @ Xy @ A ) )
% 0.31/0.45 => ( ! [C: $i] :
% 0.31/0.45 ( ( in @ C @ B )
% 0.31/0.45 => ( ( in @ Xx @ C )
% 0.31/0.45 <=> ( in @ Xy @ C ) ) )
% 0.31/0.45 => ( Xx = Xy ) ) )
% 0.31/0.45 & ! [C: $i,D: $i] :
% 0.31/0.45 ( ( ( in @ C @ B )
% 0.31/0.45 & ( in @ D @ B ) )
% 0.31/0.45 => ( ! [Xx: $i] :
% 0.31/0.45 ( ( in @ Xx @ C )
% 0.31/0.45 => ( in @ Xx @ D ) )
% 0.31/0.45 | ! [Xx: $i] :
% 0.31/0.45 ( ( in @ Xx @ D )
% 0.31/0.45 => ( in @ Xx @ C ) ) ) )
% 0.31/0.45 & ! [C: $i] :
% 0.31/0.45 ( ( ! [Xx: $i] :
% 0.31/0.45 ( ( in @ Xx @ C )
% 0.31/0.45 => ( in @ Xx @ A ) )
% 0.31/0.45 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.31/0.45 => ? [D: $i,Xx: $i] :
% 0.31/0.45 ( ( in @ D @ B )
% 0.31/0.45 & ( in @ Xx @ C )
% 0.31/0.45 & ~ ? [Xy: $i] :
% 0.31/0.45 ( ( in @ Xy @ D )
% 0.31/0.45 & ( in @ Xy @ C ) )
% 0.31/0.45 & ! [E: $i] :
% 0.31/0.45 ( ( in @ E @ B )
% 0.31/0.45 => ( ! [Xy: $i] :
% 0.31/0.45 ( ( in @ Xy @ E )
% 0.31/0.45 => ( in @ Xy @ D ) )
% 0.31/0.45 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(descr_type,type,
% 0.31/0.45 descr: ( $i > $o ) > $i ).
% 0.31/0.45
% 0.31/0.45 thf(descrp_type,type,
% 0.31/0.45 descrp: $o ).
% 0.31/0.45
% 0.31/0.45 thf(descrp,definition,
% 0.31/0.45 ( descrp
% 0.31/0.45 = ( ! [Xphi: $i > $o] :
% 0.31/0.45 ( ( exu
% 0.31/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.45 => ( Xphi
% 0.31/0.45 @ ( descr
% 0.31/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(dsetconstr_type,type,
% 0.31/0.45 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.31/0.45
% 0.31/0.45 thf(dsetconstrI_type,type,
% 0.31/0.45 dsetconstrI: $o ).
% 0.31/0.45
% 0.31/0.45 thf(dsetconstrI,definition,
% 0.31/0.45 ( dsetconstrI
% 0.31/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.45 ( ( in @ Xx @ A )
% 0.31/0.45 => ( ( Xphi @ Xx )
% 0.31/0.45 => ( in @ Xx
% 0.31/0.45 @ ( dsetconstr @ A
% 0.31/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(dsetconstrEL_type,type,
% 0.31/0.45 dsetconstrEL: $o ).
% 0.31/0.45
% 0.31/0.45 thf(dsetconstrEL,definition,
% 0.31/0.45 ( dsetconstrEL
% 0.31/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.45 ( ( in @ Xx
% 0.31/0.45 @ ( dsetconstr @ A
% 0.31/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(dsetconstrER_type,type,
% 0.31/0.45 dsetconstrER: $o ).
% 0.31/0.45
% 0.31/0.45 thf(dsetconstrER,definition,
% 0.31/0.45 ( dsetconstrER
% 0.31/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.45 ( ( in @ Xx
% 0.31/0.45 @ ( dsetconstr @ A
% 0.31/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/0.45 => ( Xphi @ Xx ) ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(exuE1_type,type,
% 0.31/0.45 exuE1: $o ).
% 0.31/0.45
% 0.31/0.45 thf(exuE1,definition,
% 0.31/0.45 ( exuE1
% 0.31/0.45 = ( ! [Xphi: $i > $o] :
% 0.31/0.45 ( ( exu
% 0.31/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.45 => ? [Xx: $i] :
% 0.31/0.45 ( ( Xphi @ Xx )
% 0.31/0.45 & ! [Xy: $i] :
% 0.31/0.45 ( ( Xphi @ Xy )
% 0.31/0.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(prop2set_type,type,
% 0.31/0.45 prop2set: $o > $i ).
% 0.31/0.45
% 0.31/0.45 thf(prop2set,definition,
% 0.31/0.45 ( prop2set
% 0.31/0.45 = ( ^ [Xphi: $o] :
% 0.31/0.45 ( dsetconstr @ ( powerset @ emptyset )
% 0.31/0.45 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(prop2setE_type,type,
% 0.31/0.45 prop2setE: $o ).
% 0.31/0.45
% 0.31/0.45 thf(prop2setE,definition,
% 0.31/0.45 ( prop2setE
% 0.31/0.45 = ( ! [Xphi: $o,Xx: $i] :
% 0.31/0.45 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.31/0.45 => Xphi ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(emptysetE_type,type,
% 0.31/0.45 emptysetE: $o ).
% 0.31/0.45
% 0.31/0.45 thf(emptysetE,definition,
% 0.31/0.45 ( emptysetE
% 0.31/0.45 = ( ! [Xx: $i] :
% 0.31/0.45 ( ( in @ Xx @ emptyset )
% 0.31/0.45 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(emptysetimpfalse_type,type,
% 0.31/0.45 emptysetimpfalse: $o ).
% 0.31/0.45
% 0.31/0.45 thf(emptysetimpfalse,definition,
% 0.31/0.45 ( emptysetimpfalse
% 0.31/0.45 = ( ! [Xx: $i] :
% 0.31/0.45 ( ( in @ Xx @ emptyset )
% 0.31/0.45 => $false ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(notinemptyset_type,type,
% 0.31/0.45 notinemptyset: $o ).
% 0.31/0.45
% 0.31/0.45 thf(notinemptyset,definition,
% 0.31/0.45 ( notinemptyset
% 0.31/0.45 = ( ! [Xx: $i] :
% 0.31/0.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(exuE3e_type,type,
% 0.31/0.45 exuE3e: $o ).
% 0.31/0.45
% 0.31/0.45 thf(exuE3e,definition,
% 0.31/0.45 ( exuE3e
% 0.31/0.45 = ( ! [Xphi: $i > $o] :
% 0.31/0.45 ( ( exu
% 0.31/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.45 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(setext_type,type,
% 0.31/0.45 setext: $o ).
% 0.31/0.45
% 0.31/0.45 thf(setext,definition,
% 0.31/0.45 ( setext
% 0.31/0.45 = ( ! [A: $i,B: $i] :
% 0.31/0.45 ( ! [Xx: $i] :
% 0.31/0.45 ( ( in @ Xx @ A )
% 0.31/0.45 => ( in @ Xx @ B ) )
% 0.31/0.45 => ( ! [Xx: $i] :
% 0.31/0.45 ( ( in @ Xx @ B )
% 0.31/0.45 => ( in @ Xx @ A ) )
% 0.31/0.45 => ( A = B ) ) ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(emptyI_type,type,
% 0.31/0.45 emptyI: $o ).
% 0.31/0.45
% 0.31/0.45 thf(emptyI,definition,
% 0.31/0.45 ( emptyI
% 0.31/0.45 = ( ! [A: $i] :
% 0.31/0.45 ( ! [Xx: $i] :
% 0.31/0.45 ~ ( in @ Xx @ A )
% 0.31/0.45 => ( A = emptyset ) ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(noeltsimpempty_type,type,
% 0.31/0.45 noeltsimpempty: $o ).
% 0.31/0.45
% 0.31/0.45 thf(noeltsimpempty,definition,
% 0.31/0.45 ( noeltsimpempty
% 0.31/0.45 = ( ! [A: $i] :
% 0.31/0.45 ( ! [Xx: $i] :
% 0.31/0.45 ~ ( in @ Xx @ A )
% 0.31/0.45 => ( A = emptyset ) ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(setbeta,conjecture,
% 0.31/0.45 ( setextAx
% 0.31/0.45 => ( emptysetAx
% 0.31/0.45 => ( setadjoinAx
% 0.31/0.45 => ( powersetAx
% 0.31/0.45 => ( setunionAx
% 0.31/0.45 => ( omega0Ax
% 0.31/0.45 => ( omegaSAx
% 0.31/0.45 => ( omegaIndAx
% 0.31/0.45 => ( replAx
% 0.31/0.45 => ( foundationAx
% 0.31/0.45 => ( wellorderingAx
% 0.31/0.45 => ( descrp
% 0.31/0.45 => ( dsetconstrI
% 0.31/0.45 => ( dsetconstrEL
% 0.31/0.45 => ( dsetconstrER
% 0.31/0.45 => ( exuE1
% 0.31/0.45 => ( prop2setE
% 0.31/0.45 => ( emptysetE
% 0.31/0.45 => ( emptysetimpfalse
% 0.31/0.45 => ( notinemptyset
% 0.31/0.45 => ( exuE3e
% 0.31/0.45 => ( setext
% 0.31/0.45 => ( emptyI
% 0.31/0.45 => ( noeltsimpempty
% 0.31/0.45 => ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.45 ( ( in @ Xx @ A )
% 0.31/0.45 => ( ( in @ Xx
% 0.31/0.45 @ ( dsetconstr @ A
% 0.31/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/0.45 <=> ( Xphi @ Xx ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.45
% 0.31/0.45 %------------------------------------------------------------------------------
% 0.31/0.45 ------------------------
% 0.31/1.30 ---- Embedded in THF ---
% 0.31/1.30 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.31/1.30 %%% Generated on Tue Feb 24 12:25:34 EST 2026
% 0.31/1.30 %%% using '$$dhol' embedding, version 1.3.0.
% 0.31/1.30 %%% Logic specification used:
% 0.31/1.30 %%% thf(spec, logic, $$dhol).
% 0.31/1.30
% 0.31/1.30 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.hL7X93FyCR/DTF2DTF31925.p
% See solution above
% 0.31/1.30 ------------------------
% 0.31/1.31 ---- Cleaned THF ---
% 0.31/1.31 thf(in_type,type,
% 0.31/1.31 in: $i > $i > $o ).
% 0.31/1.31
% 0.31/1.31 thf(exu_type,type,
% 0.31/1.31 exu: ( $i > $o ) > $o ).
% 0.31/1.31
% 0.31/1.31 thf(exu,definition,
% 0.31/1.31 ( exu
% 0.31/1.31 = ( ^ [Xphi: $i > $o] :
% 0.31/1.31 ? [Xx: $i] :
% 0.31/1.31 ( ( Xphi @ Xx )
% 0.31/1.31 & ! [Xy: $i] :
% 0.31/1.31 ( ( Xphi @ Xy )
% 0.31/1.31 => ( Xx = Xy ) ) ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(setextAx_type,type,
% 0.31/1.31 setextAx: $o ).
% 0.31/1.31
% 0.31/1.31 thf(setextAx,definition,
% 0.31/1.31 ( setextAx
% 0.31/1.31 = ( ! [A: $i,B: $i] :
% 0.31/1.31 ( ! [Xx: $i] :
% 0.31/1.31 ( ( in @ Xx @ A )
% 0.31/1.31 <=> ( in @ Xx @ B ) )
% 0.31/1.31 => ( A = B ) ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(emptyset_type,type,
% 0.31/1.31 emptyset: $i ).
% 0.31/1.31
% 0.31/1.31 thf(emptysetAx_type,type,
% 0.31/1.31 emptysetAx: $o ).
% 0.31/1.31
% 0.31/1.31 thf(emptysetAx,definition,
% 0.31/1.31 ( emptysetAx
% 0.31/1.31 = ( ! [Xx: $i] :
% 0.31/1.31 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(setadjoin_type,type,
% 0.31/1.31 setadjoin: $i > $i > $i ).
% 0.31/1.31
% 0.31/1.31 thf(setadjoinAx_type,type,
% 0.31/1.31 setadjoinAx: $o ).
% 0.31/1.31
% 0.31/1.31 thf(setadjoinAx,definition,
% 0.31/1.31 ( setadjoinAx
% 0.31/1.31 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/1.31 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/1.31 <=> ( ( Xy = Xx )
% 0.31/1.31 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(powerset_type,type,
% 0.31/1.31 powerset: $i > $i ).
% 0.31/1.31
% 0.31/1.31 thf(powersetAx_type,type,
% 0.31/1.31 powersetAx: $o ).
% 0.31/1.31
% 0.31/1.31 thf(powersetAx,definition,
% 0.31/1.31 ( powersetAx
% 0.31/1.31 = ( ! [A: $i,B: $i] :
% 0.31/1.31 ( ( in @ B @ ( powerset @ A ) )
% 0.31/1.31 <=> ! [Xx: $i] :
% 0.31/1.31 ( ( in @ Xx @ B )
% 0.31/1.31 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(setunion_type,type,
% 0.31/1.31 setunion: $i > $i ).
% 0.31/1.31
% 0.31/1.31 thf(setunionAx_type,type,
% 0.31/1.31 setunionAx: $o ).
% 0.31/1.31
% 0.31/1.31 thf(setunionAx,definition,
% 0.31/1.31 ( setunionAx
% 0.31/1.31 = ( ! [A: $i,Xx: $i] :
% 0.31/1.31 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/1.31 <=> ? [B: $i] :
% 0.31/1.31 ( ( in @ Xx @ B )
% 0.31/1.31 & ( in @ B @ A ) ) ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(omega_type,type,
% 0.31/1.31 omega: $i ).
% 0.31/1.31
% 0.31/1.31 thf(omega0Ax_type,type,
% 0.31/1.31 omega0Ax: $o ).
% 0.31/1.31
% 0.31/1.31 thf(omega0Ax,definition,
% 0.31/1.31 ( omega0Ax
% 0.31/1.31 = ( in @ emptyset @ omega ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(omegaSAx_type,type,
% 0.31/1.31 omegaSAx: $o ).
% 0.31/1.31
% 0.31/1.31 thf(omegaSAx,definition,
% 0.31/1.31 ( omegaSAx
% 0.31/1.31 = ( ! [Xx: $i] :
% 0.31/1.31 ( ( in @ Xx @ omega )
% 0.31/1.31 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(omegaIndAx_type,type,
% 0.31/1.31 omegaIndAx: $o ).
% 0.31/1.31
% 0.31/1.31 thf(omegaIndAx,definition,
% 0.31/1.31 ( omegaIndAx
% 0.31/1.31 = ( ! [A: $i] :
% 0.31/1.31 ( ( ( in @ emptyset @ A )
% 0.31/1.31 & ! [Xx: $i] :
% 0.31/1.31 ( ( ( in @ Xx @ omega )
% 0.31/1.31 & ( in @ Xx @ A ) )
% 0.31/1.31 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.31/1.31 => ! [Xx: $i] :
% 0.31/1.31 ( ( in @ Xx @ omega )
% 0.31/1.31 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(replAx_type,type,
% 0.31/1.31 replAx: $o ).
% 0.31/1.31
% 0.31/1.31 thf(replAx,definition,
% 0.31/1.31 ( replAx
% 0.31/1.31 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.31/1.31 ( ! [Xx: $i] :
% 0.31/1.31 ( ( in @ Xx @ A )
% 0.31/1.31 => ( exu
% 0.31/1.31 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.31/1.31 => ? [B: $i] :
% 0.31/1.31 ! [Xx: $i] :
% 0.31/1.31 ( ( in @ Xx @ B )
% 0.31/1.31 <=> ? [Xy: $i] :
% 0.31/1.31 ( ( in @ Xy @ A )
% 0.31/1.31 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(foundationAx_type,type,
% 0.31/1.31 foundationAx: $o ).
% 0.31/1.31
% 0.31/1.31 thf(foundationAx,definition,
% 0.31/1.31 ( foundationAx
% 0.31/1.31 = ( ! [A: $i] :
% 0.31/1.31 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.31/1.31 => ? [B: $i] :
% 0.31/1.31 ( ( in @ B @ A )
% 0.31/1.31 & ~ ? [Xx: $i] :
% 0.31/1.31 ( ( in @ Xx @ B )
% 0.31/1.31 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(wellorderingAx_type,type,
% 0.31/1.31 wellorderingAx: $o ).
% 0.31/1.31
% 0.31/1.31 thf(wellorderingAx,definition,
% 0.31/1.31 ( wellorderingAx
% 0.31/1.31 = ( ! [A: $i] :
% 0.31/1.31 ? [B: $i] :
% 0.31/1.31 ( ! [C: $i] :
% 0.31/1.31 ( ( in @ C @ B )
% 0.31/1.31 => ! [Xx: $i] :
% 0.31/1.31 ( ( in @ Xx @ C )
% 0.31/1.31 => ( in @ Xx @ A ) ) )
% 0.31/1.31 & ! [Xx: $i,Xy: $i] :
% 0.31/1.31 ( ( ( in @ Xx @ A )
% 0.31/1.31 & ( in @ Xy @ A ) )
% 0.31/1.31 => ( ! [C: $i] :
% 0.31/1.31 ( ( in @ C @ B )
% 0.31/1.31 => ( ( in @ Xx @ C )
% 0.31/1.31 <=> ( in @ Xy @ C ) ) )
% 0.31/1.31 => ( Xx = Xy ) ) )
% 0.31/1.31 & ! [C: $i,D: $i] :
% 0.31/1.31 ( ( ( in @ C @ B )
% 0.31/1.31 & ( in @ D @ B ) )
% 0.31/1.31 => ( ! [Xx: $i] :
% 0.31/1.31 ( ( in @ Xx @ C )
% 0.31/1.31 => ( in @ Xx @ D ) )
% 0.31/1.31 | ! [Xx: $i] :
% 0.31/1.31 ( ( in @ Xx @ D )
% 0.31/1.31 => ( in @ Xx @ C ) ) ) )
% 0.31/1.31 & ! [C: $i] :
% 0.31/1.31 ( ( ! [Xx: $i] :
% 0.31/1.31 ( ( in @ Xx @ C )
% 0.31/1.31 => ( in @ Xx @ A ) )
% 0.31/1.31 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.31/1.31 => ? [D: $i,Xx: $i] :
% 0.31/1.31 ( ( in @ D @ B )
% 0.31/1.31 & ( in @ Xx @ C )
% 0.31/1.31 & ~ ? [Xy: $i] :
% 0.31/1.31 ( ( in @ Xy @ D )
% 0.31/1.31 & ( in @ Xy @ C ) )
% 0.31/1.31 & ! [E: $i] :
% 0.31/1.31 ( ( in @ E @ B )
% 0.31/1.31 => ( ! [Xy: $i] :
% 0.31/1.31 ( ( in @ Xy @ E )
% 0.31/1.31 => ( in @ Xy @ D ) )
% 0.31/1.31 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(descr_type,type,
% 0.31/1.31 descr: ( $i > $o ) > $i ).
% 0.31/1.31
% 0.31/1.31 thf(descrp_type,type,
% 0.31/1.31 descrp: $o ).
% 0.31/1.31
% 0.31/1.31 thf(descrp,definition,
% 0.31/1.31 ( descrp
% 0.31/1.31 = ( ! [Xphi: $i > $o] :
% 0.31/1.31 ( ( exu
% 0.31/1.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.31 => ( Xphi
% 0.31/1.31 @ ( descr
% 0.31/1.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(dsetconstr_type,type,
% 0.31/1.31 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.31/1.31
% 0.31/1.31 thf(dsetconstrI_type,type,
% 0.31/1.31 dsetconstrI: $o ).
% 0.31/1.31
% 0.31/1.31 thf(dsetconstrI,definition,
% 0.31/1.31 ( dsetconstrI
% 0.31/1.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/1.31 ( ( in @ Xx @ A )
% 0.31/1.31 => ( ( Xphi @ Xx )
% 0.31/1.31 => ( in @ Xx
% 0.31/1.31 @ ( dsetconstr @ A
% 0.31/1.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(dsetconstrEL_type,type,
% 0.31/1.31 dsetconstrEL: $o ).
% 0.31/1.31
% 0.31/1.31 thf(dsetconstrEL,definition,
% 0.31/1.31 ( dsetconstrEL
% 0.31/1.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/1.31 ( ( in @ Xx
% 0.31/1.31 @ ( dsetconstr @ A
% 0.31/1.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/1.31 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(dsetconstrER_type,type,
% 0.31/1.31 dsetconstrER: $o ).
% 0.31/1.31
% 0.31/1.31 thf(dsetconstrER,definition,
% 0.31/1.31 ( dsetconstrER
% 0.31/1.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/1.31 ( ( in @ Xx
% 0.31/1.31 @ ( dsetconstr @ A
% 0.31/1.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/1.31 => ( Xphi @ Xx ) ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(exuE1_type,type,
% 0.31/1.31 exuE1: $o ).
% 0.31/1.31
% 0.31/1.31 thf(exuE1,definition,
% 0.31/1.31 ( exuE1
% 0.31/1.31 = ( ! [Xphi: $i > $o] :
% 0.31/1.31 ( ( exu
% 0.31/1.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.31 => ? [Xx: $i] :
% 0.31/1.31 ( ( Xphi @ Xx )
% 0.31/1.31 & ! [Xy: $i] :
% 0.31/1.31 ( ( Xphi @ Xy )
% 0.31/1.31 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(prop2set_type,type,
% 0.31/1.31 prop2set: $o > $i ).
% 0.31/1.31
% 0.31/1.31 thf(prop2set,definition,
% 0.31/1.31 ( prop2set
% 0.31/1.31 = ( ^ [Xphi: $o] :
% 0.31/1.31 ( dsetconstr @ ( powerset @ emptyset )
% 0.31/1.31 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(prop2setE_type,type,
% 0.31/1.31 prop2setE: $o ).
% 0.31/1.31
% 0.31/1.31 thf(prop2setE,definition,
% 0.31/1.31 ( prop2setE
% 0.31/1.31 = ( ! [Xphi: $o,Xx: $i] :
% 0.31/1.31 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.31/1.31 => Xphi ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(emptysetE_type,type,
% 0.31/1.31 emptysetE: $o ).
% 0.31/1.31
% 0.31/1.31 thf(emptysetE,definition,
% 0.31/1.31 ( emptysetE
% 0.31/1.31 = ( ! [Xx: $i] :
% 0.31/1.31 ( ( in @ Xx @ emptyset )
% 0.31/1.31 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(emptysetimpfalse_type,type,
% 0.31/1.31 emptysetimpfalse: $o ).
% 0.31/1.31
% 0.31/1.31 thf(emptysetimpfalse,definition,
% 0.31/1.31 ( emptysetimpfalse
% 0.31/1.31 = ( ! [Xx: $i] :
% 0.31/1.31 ( ( in @ Xx @ emptyset )
% 0.31/1.31 => $false ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(notinemptyset_type,type,
% 0.31/1.31 notinemptyset: $o ).
% 0.31/1.31
% 0.31/1.31 thf(notinemptyset,definition,
% 0.31/1.31 ( notinemptyset
% 0.31/1.31 = ( ! [Xx: $i] :
% 0.31/1.31 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(exuE3e_type,type,
% 0.31/1.31 exuE3e: $o ).
% 0.31/1.31
% 0.31/1.31 thf(exuE3e,definition,
% 0.31/1.31 ( exuE3e
% 0.31/1.31 = ( ! [Xphi: $i > $o] :
% 0.31/1.31 ( ( exu
% 0.31/1.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.31 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(setext_type,type,
% 0.31/1.31 setext: $o ).
% 0.31/1.31
% 0.31/1.31 thf(setext,definition,
% 0.31/1.31 ( setext
% 0.31/1.31 = ( ! [A: $i,B: $i] :
% 0.31/1.31 ( ! [Xx: $i] :
% 0.31/1.31 ( ( in @ Xx @ A )
% 0.31/1.31 => ( in @ Xx @ B ) )
% 0.31/1.31 => ( ! [Xx: $i] :
% 0.31/1.31 ( ( in @ Xx @ B )
% 0.31/1.31 => ( in @ Xx @ A ) )
% 0.31/1.31 => ( A = B ) ) ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(emptyI_type,type,
% 0.31/1.31 emptyI: $o ).
% 0.31/1.31
% 0.31/1.31 thf(emptyI,definition,
% 0.31/1.31 ( emptyI
% 0.31/1.31 = ( ! [A: $i] :
% 0.31/1.31 ( ! [Xx: $i] :
% 0.31/1.31 ~ ( in @ Xx @ A )
% 0.31/1.31 => ( A = emptyset ) ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(noeltsimpempty_type,type,
% 0.31/1.31 noeltsimpempty: $o ).
% 0.31/1.31
% 0.31/1.31 thf(noeltsimpempty,definition,
% 0.31/1.31 ( noeltsimpempty
% 0.31/1.31 = ( ! [A: $i] :
% 0.31/1.31 ( ! [Xx: $i] :
% 0.31/1.31 ~ ( in @ Xx @ A )
% 0.31/1.31 => ( A = emptyset ) ) ) ) ).
% 0.31/1.31
% 0.31/1.31 thf(setbeta,conjecture,
% 0.31/1.31 ( setextAx
% 0.31/1.31 => ( emptysetAx
% 0.31/1.31 => ( setadjoinAx
% 0.31/1.31 => ( powersetAx
% 0.31/1.31 => ( setunionAx
% 0.31/1.31 => ( omega0Ax
% 0.31/1.31 => ( omegaSAx
% 0.31/1.31 => ( omegaIndAx
% 0.31/1.31 => ( replAx
% 0.31/1.31 => ( foundationAx
% 0.31/1.31 => ( wellorderingAx
% 0.31/1.31 => ( descrp
% 0.31/1.31 => ( dsetconstrI
% 0.31/1.31 => ( dsetconstrEL
% 0.31/1.31 => ( dsetconstrER
% 0.31/1.31 => ( exuE1
% 0.31/1.31 => ( prop2setE
% 0.31/1.31 => ( emptysetE
% 0.31/1.31 => ( emptysetimpfalse
% 0.31/1.31 => ( notinemptyset
% 0.31/1.31 => ( exuE3e
% 0.31/1.31 => ( setext
% 0.31/1.31 => ( emptyI
% 0.31/1.31 => ( noeltsimpempty
% 0.31/1.31 => ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/1.31 ( ( in @ Xx @ A )
% 0.31/1.31 => ( ( in @ Xx
% 0.31/1.31 @ ( dsetconstr @ A
% 0.31/1.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/1.31 <=> ( Xphi @ Xx ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.31/1.31 ------------------------
% 0.31/1.32 % (32046)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_31925 for (2999ds/18Mi)
% 0.31/1.32 % (32045)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_31925 for (2999ds/275Mi)
% 0.31/1.32 % (32040)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_31925 for (2999ds/183Mi)
% 0.31/1.33 % (32046)Instruction limit reached!
% 0.31/1.33 % (32046)------------------------------
% 0.31/1.33 % (32046)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.33 % (32046)Termination reason: Unknown
% 0.31/1.33 % (32046)Termination phase: Property scanning
% 0.31/1.33
% 0.31/1.33 % (32046)Memory used [KB]: 1407
% 0.31/1.33 % (32046)Time elapsed: 0.007 s
% 0.31/1.33 % (32046)Instructions burned: 20 (million)
% 0.31/1.33 % (32046)------------------------------
% 0.31/1.33 % (32046)------------------------------
% 0.31/1.33 % (32041)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_31925 for (2999ds/4Mi)
% 0.31/1.33 % (32042)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_31925 for (2999ds/27Mi)
% 0.31/1.33 % (32043)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_31925 for (2999ds/2Mi)
% 0.31/1.33 % (32044)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_31925 for (2999ds/2Mi)
% 0.31/1.33 % (32043)Instruction limit reached!
% 0.31/1.33 % (32043)------------------------------
% 0.31/1.33 % (32043)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.33 % (32044)Instruction limit reached!
% 0.31/1.33 % (32044)------------------------------
% 0.31/1.33 % (32044)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.33 % (32043)Termination reason: Unknown
% 0.31/1.33 % (32043)Termination phase: shuffling
% 0.31/1.33
% 0.31/1.33 % (32043)Memory used [KB]: 1023
% 0.31/1.33 % (32043)Time elapsed: 0.003 s
% 0.31/1.33 % (32043)Instructions burned: 2 (million)
% 0.31/1.33 % (32043)------------------------------
% 0.31/1.33 % (32043)------------------------------
% 0.31/1.33 % (32044)Termination reason: Unknown
% 0.31/1.33 % (32044)Termination phase: shuffling
% 0.31/1.33
% 0.31/1.33 % (32044)Memory used [KB]: 1023
% 0.31/1.33 % (32044)Time elapsed: 0.003 s
% 0.31/1.33 % (32044)Instructions burned: 2 (million)
% 0.31/1.33 % (32044)------------------------------
% 0.31/1.33 % (32044)------------------------------
% 0.31/1.33 % (32041)Instruction limit reached!
% 0.31/1.33 % (32041)------------------------------
% 0.31/1.33 % (32041)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.33 % (32041)Termination reason: Unknown
% 0.31/1.33 % (32041)Termination phase: shuffling
% 0.31/1.33
% 0.31/1.33 % (32041)Memory used [KB]: 1023
% 0.31/1.33 % (32041)Time elapsed: 0.004 s
% 0.31/1.33 % (32041)Instructions burned: 4 (million)
% 0.31/1.33 % (32041)------------------------------
% 0.31/1.33 % (32041)------------------------------
% 0.31/1.34 % (32047)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_31925 for (2999ds/3Mi)
% 0.31/1.34 % (32047)Instruction limit reached!
% 0.31/1.34 % (32047)------------------------------
% 0.31/1.34 % (32047)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.34 % (32047)Termination reason: Unknown
% 0.31/1.34 % (32047)Termination phase: shuffling
% 0.31/1.34
% 0.31/1.34 % (32047)Memory used [KB]: 1023
% 0.31/1.34 % (32047)Time elapsed: 0.002 s
% 0.31/1.34 % (32047)Instructions burned: 4 (million)
% 0.31/1.34 % (32047)------------------------------
% 0.31/1.34 % (32047)------------------------------
% 0.31/1.35 % (32042)Instruction limit reached!
% 0.31/1.35 % (32042)------------------------------
% 0.31/1.35 % (32042)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.35 % (32042)Termination reason: Unknown
% 0.31/1.35 % (32042)Termination phase: Saturation
% 0.31/1.35
% 0.31/1.35 % (32042)Memory used [KB]: 5756
% 0.31/1.35 % (32042)Time elapsed: 0.016 s
% 0.31/1.35 % (32042)Instructions burned: 27 (million)
% 0.31/1.35 % (32042)------------------------------
% 0.31/1.35 % (32042)------------------------------
% 0.31/1.35 % (32051)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_31925 for (2999ds/1041Mi)
% 0.31/1.35 % (32048)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_31925 for (2999ds/37Mi)
% 0.31/1.35 % (32049)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_31925 for (2999ds/15Mi)
% 0.31/1.35 % (32050)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_31925 for (2999ds/3Mi)
% 0.31/1.35 % (32050)Instruction limit reached!
% 0.31/1.35 % (32050)------------------------------
% 0.31/1.35 % (32050)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.35 % (32050)Termination reason: Unknown
% 0.31/1.35 % (32050)Termination phase: shuffling
% 0.31/1.35
% 0.31/1.35 % (32050)Memory used [KB]: 1023
% 0.31/1.35 % (32050)Time elapsed: 0.004 s
% 0.31/1.35 % (32050)Instructions burned: 4 (million)
% 0.31/1.35 % (32050)------------------------------
% 0.31/1.35 % (32050)------------------------------
% 0.31/1.36 % (32049)Instruction limit reached!
% 0.31/1.36 % (32049)------------------------------
% 0.31/1.36 % (32049)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.36 % (32049)Termination reason: Unknown
% 0.31/1.36 % (32049)Termination phase: Property scanning
% 0.31/1.36
% 0.31/1.36 % (32049)Memory used [KB]: 1407
% 0.31/1.36 % (32049)Time elapsed: 0.010 s
% 0.31/1.36 % (32049)Instructions burned: 15 (million)
% 0.31/1.36 % (32049)------------------------------
% 0.31/1.36 % (32049)------------------------------
% 0.31/1.36 % (32052)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_31925 for (2999ds/7Mi)
% 0.31/1.37 % (32052)Instruction limit reached!
% 0.31/1.37 % (32052)------------------------------
% 0.31/1.37 % (32052)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.37 % (32052)Termination reason: Unknown
% 0.31/1.37 % (32052)Termination phase: Property scanning
% 0.31/1.37
% 0.31/1.37 % (32052)Memory used [KB]: 1151
% 0.31/1.37 % (32052)Time elapsed: 0.005 s
% 0.31/1.37 % (32052)Instructions burned: 8 (million)
% 0.31/1.37 % (32052)------------------------------
% 0.31/1.37 % (32052)------------------------------
% 0.31/1.37 % (32048)Instruction limit reached!
% 0.31/1.37 % (32048)------------------------------
% 0.31/1.37 % (32048)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.37 % (32048)Termination reason: Unknown
% 0.31/1.37 % (32048)Termination phase: Saturation
% 0.31/1.37
% 0.31/1.37 % (32048)Memory used [KB]: 5756
% 0.31/1.37 % (32048)Time elapsed: 0.020 s
% 0.31/1.37 % (32048)Instructions burned: 37 (million)
% 0.31/1.37 % (32048)------------------------------
% 0.31/1.37 % (32048)------------------------------
% 0.31/1.37 % (32040)First to succeed.
% 0.31/1.37 % (32053)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_31925 for (2999ds/16Mi)
% 0.31/1.37 % (32040)Refutation found. Thanks to Tanya!
% 0.31/1.37 % SZS status Theorem for DTF2THF_31925
% 0.31/1.37 % SZS output start Proof for DTF2THF_31925
% 0.31/1.37 thf(func_def_0, type, in: $i > $i > $o).
% 0.31/1.37 thf(func_def_1, type, exu: ($i > $o) > $o).
% 0.31/1.37 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 0.31/1.37 thf(func_def_8, type, powerset: $i > $i).
% 0.31/1.37 thf(func_def_10, type, setunion: $i > $i).
% 0.31/1.37 thf(func_def_19, type, descr: ($i > $o) > $i).
% 0.31/1.37 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 0.31/1.37 thf(func_def_26, type, prop2set: $o > $i).
% 0.31/1.37 thf(func_def_51, type, sK0: $i > $o).
% 0.31/1.37 thf(func_def_55, type, sK4: $i > $i > $i).
% 0.31/1.37 thf(func_def_56, type, sK5: $i > ($i > $i > $o) > $i).
% 0.31/1.37 thf(func_def_57, type, sK6: $i > ($i > $i > $o) > $i).
% 0.31/1.37 thf(func_def_58, type, sK7: $i > $i > ($i > $i > $o) > $i).
% 0.31/1.37 thf(func_def_59, type, sK8: $i > $i > ($i > $i > $o) > $i).
% 0.31/1.37 thf(func_def_60, type, sK9: $i > ($i > $i > $o) > $i).
% 0.31/1.37 thf(func_def_61, type, sK10: $i > $i > ($i > $i > $o) > $i).
% 0.31/1.37 thf(func_def_62, type, sK11: $i > ($i > $i > $o) > $i).
% 0.31/1.37 thf(func_def_63, type, sK12: $i > $i > ($i > $i > $o) > $i).
% 0.31/1.37 thf(func_def_64, type, sK13: $i > $i).
% 0.31/1.37 thf(func_def_65, type, sK14: $i > $i > $i > $i).
% 0.31/1.37 thf(func_def_66, type, sK15: $i > $i > $i).
% 0.31/1.37 thf(func_def_67, type, sK16: $i > $i > $i).
% 0.31/1.37 thf(func_def_68, type, sK17: $i > $i > $i).
% 0.31/1.37 thf(func_def_69, type, sK18: $i > $i > $i).
% 0.31/1.37 thf(func_def_70, type, sK19: $i > $i > $i).
% 0.31/1.37 thf(func_def_71, type, sK20: $i > $i > $i).
% 0.31/1.37 thf(func_def_72, type, sK21: $i > $i).
% 0.31/1.37 thf(func_def_73, type, sK22: $i > $i).
% 0.31/1.37 thf(func_def_74, type, sK23: $i > $i > $i).
% 0.31/1.37 thf(func_def_75, type, sK24: ($i > $o) > $i > $i).
% 0.31/1.37 thf(func_def_76, type, sK25: $i > $i > $i).
% 0.31/1.37 thf(func_def_77, type, sK26: $i > $i).
% 0.31/1.37 thf(func_def_78, type, sK27: $i > $i > $i).
% 0.31/1.37 thf(func_def_79, type, sK28: $i > $i > $i).
% 0.31/1.37 thf(func_def_80, type, sK29: $i > $i > $i).
% 0.31/1.37 thf(func_def_81, type, sK30: ($i > $o) > $i).
% 0.31/1.37 thf(func_def_82, type, sK31: ($i > $o) > $i > $i).
% 0.31/1.37 thf(func_def_83, type, sK32: $i > $i).
% 0.31/1.37 thf(func_def_84, type, ph33: !>[X0: $tType]:(X0)).
% 0.31/1.37 thf(func_def_85, type, sK34: $i > $i > $i).
% 0.31/1.37 thf(f903,plain,(
% 0.31/1.37 $false),
% 0.31/1.37 inference(avatar_sat_refutation,[status(thm)],[f539,f548,f577,f583,f899])).
% 0.31/1.37 thf(f899,plain,(
% 0.31/1.37 spl3_3 | ~spl3_4),
% 0.31/1.37 inference(avatar_contradiction_clause,[status(thm)],[f898])).
% 0.31/1.37 thf(f898,plain,(
% 0.31/1.37 $false | (spl3_3 | ~spl3_4)),
% 0.31/1.37 inference(trivial_inequality_removal,[status(thm)],[f886])).
% 0.31/1.37 thf(f886,plain,(
% 0.31/1.37 ($false = $true) | (spl3_3 | ~spl3_4)),
% 0.31/1.37 inference(superposition,[status(thm)],[f103,f881])).
% 0.31/1.37 thf(f881,plain,(
% 0.31/1.37 ($false = (in @ sK1 @ sK2)) | (spl3_3 | ~spl3_4)),
% 0.31/1.37 inference(subsumption_resolution,[status(thm)],[f879,f542])).
% 0.31/1.37 thf(f542,plain,(
% 0.31/1.37 ($false != (sK0 @ sK1)) | spl3_3),
% 0.31/1.37 inference(avatar_component_clause,[status(thm)],[f541])).
% 0.31/1.37 thf(f541,definition,(
% 0.31/1.37 spl3_3 <=> ($false = (sK0 @ sK1))),
% 0.31/1.37 introduced(definition,[new_symbols(naming,[spl3_3])],[avatar_definition])).
% 0.31/1.37 thf(f879,plain,(
% 0.31/1.37 ($false = (sK0 @ sK1)) | ($false = (in @ sK1 @ sK2)) | ~spl3_4),
% 0.31/1.37 inference(trivial_inequality_removal,[status(thm)],[f869])).
% 0.31/1.37 thf(f869,plain,(
% 0.31/1.37 ($false = $true) | ($false = (sK0 @ sK1)) | ($false = (in @ sK1 @ sK2)) | ~spl3_4),
% 0.31/1.37 inference(superposition,[status(thm)],[f547,f502])).
% 0.31/1.37 thf(f502,plain,(
% 0.31/1.37 ( ! [X2 : $i,X3 : $i > $o,X1 : $i] : (((in @ X2 @ (dsetconstr @ X1 @ X3)) = $true) | ($false = (in @ X2 @ X1)) | ((X3 @ X2) = $false)) )),
% 0.31/1.37 inference(binary_proxy_clausification,[status(thm)],[f501])).
% 0.31/1.37 thf(f501,plain,(
% 0.31/1.37 ( ! [X2 : $i,X3 : $i > $o,X1 : $i] : (($true = ((X3 @ X2) => (in @ X2 @ (dsetconstr @ X1 @ X3)))) | ($false = (in @ X2 @ X1))) )),
% 0.31/1.37 inference(binary_proxy_clausification,[status(thm)],[f500])).
% 0.31/1.37 thf(f500,plain,(
% 0.31/1.37 ( ! [X2 : $i,X3 : $i > $o,X1 : $i] : (($true = ((in @ X2 @ X1) => ((X3 @ X2) => (in @ X2 @ (dsetconstr @ X1 @ X3)))))) )),
% 0.31/1.37 inference(beta_eta_normalization,[status(thm)],[f499])).
% 0.31/1.37 thf(f499,plain,(
% 0.31/1.37 ( ! [X2 : $i,X3 : $i > $o,X1 : $i] : ((((^[Y0 : $i > $o]: ((in @ X2 @ X1) => ((Y0 @ X2) => (in @ X2 @ (dsetconstr @ X1 @ Y0))))) @ X3) = $true)) )),
% 0.31/1.37 inference(pi_clausification,[status(thm)],[f498])).
% 0.31/1.37 thf(f498,plain,(
% 0.31/1.37 ( ! [X2 : $i,X1 : $i] : (((!! @ ($i > $o) @ (^[Y0 : $i > $o]: ((in @ X2 @ X1) => ((Y0 @ X2) => (in @ X2 @ (dsetconstr @ X1 @ Y0)))))) = $true)) )),
% 0.31/1.37 inference(beta_eta_normalization,[status(thm)],[f497])).
% 0.31/1.37 thf(f497,plain,(
% 0.31/1.37 ( ! [X2 : $i,X1 : $i] : ((((^[Y0 : $i]: (!! @ ($i > $o) @ (^[Y1 : $i > $o]: ((in @ Y0 @ X1) => ((Y1 @ Y0) => (in @ Y0 @ (dsetconstr @ X1 @ Y1))))))) @ X2) = $true)) )),
% 0.31/1.37 inference(pi_clausification,[status(thm)],[f496])).
% 0.31/1.37 thf(f496,plain,(
% 0.31/1.37 ( ! [X1 : $i] : (($true = (!! @ $i @ (^[Y0 : $i]: (!! @ ($i > $o) @ (^[Y1 : $i > $o]: ((in @ Y0 @ X1) => ((Y1 @ Y0) => (in @ Y0 @ (dsetconstr @ X1 @ Y1)))))))))) )),
% 0.31/1.37 inference(beta_eta_normalization,[status(thm)],[f495])).
% 0.31/1.37 thf(f495,plain,(
% 0.31/1.37 ( ! [X1 : $i] : (($true = ((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ ($i > $o) @ (^[Y2 : $i > $o]: ((in @ Y1 @ Y0) => ((Y2 @ Y1) => (in @ Y1 @ (dsetconstr @ Y0 @ Y2))))))))) @ X1))) )),
% 0.31/1.37 inference(pi_clausification,[status(thm)],[f494])).
% 0.31/1.37 thf(f494,plain,(
% 0.31/1.37 ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ ($i > $o) @ (^[Y2 : $i > $o]: ((in @ Y1 @ Y0) => ((Y2 @ Y1) => (in @ Y1 @ (dsetconstr @ Y0 @ Y2)))))))))) = $true)),
% 0.31/1.37 inference(beta_eta_normalization,[status(thm)],[f156])).
% 0.31/1.37 thf(f156,plain,(
% 0.31/1.37 ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ ($i > $o) @ (^[Y2 : $i > $o]: ((in @ Y1 @ Y0) => ((Y2 @ Y1) => (in @ Y1 @ (dsetconstr @ Y0 @ (^[Y3 : $i]: (Y2 @ Y3)))))))))))) = $true)),
% 0.31/1.37 inference(definition_unfolding,[status(thm)],[f129,f116])).
% 0.31/1.37 thf(f116,plain,(
% 0.31/1.37 (dsetconstrI = $true)),
% 0.31/1.37 inference(cnf_transformation,[status(thm)],[f84])).
% 0.31/1.37 thf(f84,plain,(
% 0.31/1.37 (emptysetAx = $true) & (emptysetimpfalse = $true) & (setunionAx = $true) & (omegaIndAx = $true) & (dsetconstrI = $true) & (omega0Ax = $true) & (setext = $true) & (setextAx = $true) & (notinemptyset = $true) & (descrp = $true) & (foundationAx = $true) & (noeltsimpempty = $true) & (powersetAx = $true) & (prop2setE = $true) & (omegaSAx = $true) & (dsetconstrER = $true) & (exuE3e = $true) & (($true = (in @ sK1 @ sK2)) & ((in @ sK1 @ (dsetconstr @ sK2 @ (^[Y0 : $i]: (sK0 @ Y0)))) != (sK0 @ sK1))) & (setadjoinAx = $true) & (emptyI = $true) & (exuE1 = $true) & (wellorderingAx = $true) & (emptysetE = $true) & (dsetconstrEL = $true) & (replAx = $true)),
% 0.31/1.37 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f82,f83])).
% 0.31/1.37 thf(f83,plain,(
% 0.31/1.37 ? [X0 : $i > $o,X1 : $i,X2 : $i] : (((in @ X1 @ X2) = $true) & ((X0 @ X1) != (in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))))) => (($true = (in @ sK1 @ sK2)) & ((in @ sK1 @ (dsetconstr @ sK2 @ (^[Y0 : $i]: (sK0 @ Y0)))) != (sK0 @ sK1)))),
% 0.31/1.37 introduced(definition,[],[choice_axiom])).
% 0.31/1.37 thf(f82,plain,(
% 0.31/1.37 (emptysetAx = $true) & (emptysetimpfalse = $true) & (setunionAx = $true) & (omegaIndAx = $true) & (dsetconstrI = $true) & (omega0Ax = $true) & (setext = $true) & (setextAx = $true) & (notinemptyset = $true) & (descrp = $true) & (foundationAx = $true) & (noeltsimpempty = $true) & (powersetAx = $true) & (prop2setE = $true) & (omegaSAx = $true) & (dsetconstrER = $true) & (exuE3e = $true) & ? [X0 : $i > $o,X1 : $i,X2 : $i] : (((in @ X1 @ X2) = $true) & ((X0 @ X1) != (in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))))) & (setadjoinAx = $true) & (emptyI = $true) & (exuE1 = $true) & (wellorderingAx = $true) & (emptysetE = $true) & (dsetconstrEL = $true) & (replAx = $true)),
% 0.31/1.37 inference(flattening,[status(thm)],[f81])).
% 0.31/1.37 thf(f81,plain,(
% 0.31/1.37 (((((((((((((((((((((((? [X0 : $i > $o,X1 : $i,X2 : $i] : (((in @ X1 @ X2) = $true) & ((X0 @ X1) != (in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0)))))) & (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)),
% 0.31/1.37 inference(ennf_transformation,[status(thm)],[f41])).
% 0.31/1.37 thf(f41,plain,(
% 0.31/1.37 ~((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) => ! [X0 : $i > $o,X1 : $i,X2 : $i] : (((in @ X1 @ X2) = $true) => ((X0 @ X1) = (in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0))))))))))))))))))))))))))))))),
% 0.31/1.37 inference(fool_elimination,[status(thm)],[f40])).
% 0.31/1.37 thf(f40,plain,(
% 0.31/1.37 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => ! [X0 : $i > $o,X1 : $i,X2 : $i] : ((in @ X1 @ X2) => ((X0 @ X1) <=> (in @ X1 @ (dsetconstr @ X2 @ (^[X3 : $i] : (X0 @ X3))))))))))))))))))))))))))))))),
% 0.31/1.37 inference(rectify,[status(thm)],[f28])).
% 0.31/1.37 thf(f28,negated_conjecture,(
% 0.31/1.37 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => ! [X0 : $i > $o,X1 : $i,X3 : $i] : ((in @ X1 @ X3) => ((X0 @ X1) <=> (in @ X1 @ (dsetconstr @ X3 @ (^[X2 : $i] : (X0 @ X2))))))))))))))))))))))))))))))),
% 0.31/1.37 inference(negated_conjecture,[status(cth)],[f27])).
% 0.31/1.37 thf(f27,conjecture,(
% 0.31/1.37 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => ! [X0 : $i > $o,X1 : $i,X3 : $i] : ((in @ X1 @ X3) => ((X0 @ X1) <=> (in @ X1 @ (dsetconstr @ X3 @ (^[X2 : $i] : (X0 @ X2)))))))))))))))))))))))))))))),
% 0.31/1.37 file('/export/starexec/sandbox/tmp/tmp.hL7X93FyCR/DTF2THF_31925.p',setbeta)).
% 0.31/1.37 thf(f129,plain,(
% 0.31/1.37 (dsetconstrI = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ ($i > $o) @ (^[Y2 : $i > $o]: ((in @ Y1 @ Y0) => ((Y2 @ Y1) => (in @ Y1 @ (dsetconstr @ Y0 @ (^[Y3 : $i]: (Y2 @ Y3)))))))))))))),
% 0.31/1.37 inference(cnf_transformation,[status(thm)],[f51])).
% 0.31/1.37 thf(f51,plain,(
% 0.31/1.37 (dsetconstrI = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ ($i > $o) @ (^[Y2 : $i > $o]: ((in @ Y1 @ Y0) => ((Y2 @ Y1) => (in @ Y1 @ (dsetconstr @ Y0 @ (^[Y3 : $i]: (Y2 @ Y3)))))))))))))),
% 0.31/1.37 inference(fool_elimination,[status(thm)],[f50])).
% 0.31/1.37 thf(f50,plain,(
% 0.31/1.37 (! [X0 : $i > $o,X1 : $i,X2 : $i] : ((in @ X1 @ X2) => ((X0 @ X1) => (in @ X1 @ (dsetconstr @ X2 @ (^[X3 : $i] : (X0 @ X3)))))) = dsetconstrI)),
% 0.31/1.37 inference(rectify,[status(thm)],[f14])).
% 0.31/1.37 thf(f14,axiom,(
% 0.31/1.37 (! [X0 : $i > $o,X1 : $i,X3 : $i] : ((in @ X1 @ X3) => ((X0 @ X1) => (in @ X1 @ (dsetconstr @ X3 @ (^[X2 : $i] : (X0 @ X2)))))) = dsetconstrI)),
% 0.31/1.37 file('/export/starexec/sandbox/tmp/tmp.hL7X93FyCR/DTF2THF_31925.p',dsetconstrI)).
% 0.31/1.37 thf(f547,plain,(
% 0.31/1.37 ($false = (in @ sK1 @ (dsetconstr @ sK2 @ sK0))) | ~spl3_4),
% 0.31/1.37 inference(avatar_component_clause,[status(thm)],[f545])).
% 0.31/1.37 thf(f545,definition,(
% 0.31/1.37 spl3_4 <=> ($false = (in @ sK1 @ (dsetconstr @ sK2 @ sK0)))),
% 0.31/1.37 introduced(definition,[new_symbols(naming,[spl3_4])],[avatar_definition])).
% 0.31/1.37 thf(f103,plain,(
% 0.31/1.37 ($true = (in @ sK1 @ sK2))),
% 0.31/1.37 inference(cnf_transformation,[status(thm)],[f84])).
% 0.31/1.37 thf(f583,plain,(
% 0.31/1.37 ~spl3_1 | ~spl3_3),
% 0.31/1.37 inference(avatar_contradiction_clause,[status(thm)],[f582])).
% 0.31/1.37 thf(f582,plain,(
% 0.31/1.37 $false | (~spl3_1 | ~spl3_3)),
% 0.31/1.37 inference(trivial_inequality_removal,[status(thm)],[f578])).
% 0.31/1.37 thf(f578,plain,(
% 0.31/1.37 ($false = $true) | (~spl3_1 | ~spl3_3)),
% 0.31/1.37 inference(superposition,[status(thm)],[f534,f543])).
% 0.31/1.37 thf(f543,plain,(
% 0.31/1.37 ($false = (sK0 @ sK1)) | ~spl3_3),
% 0.31/1.37 inference(avatar_component_clause,[status(thm)],[f541])).
% 0.31/1.37 thf(f534,plain,(
% 0.31/1.37 ((sK0 @ sK1) = $true) | ~spl3_1),
% 0.31/1.37 inference(avatar_component_clause,[status(thm)],[f532])).
% 0.31/1.37 thf(f532,definition,(
% 0.31/1.37 spl3_1 <=> ((sK0 @ sK1) = $true)),
% 0.31/1.37 introduced(definition,[new_symbols(naming,[spl3_1])],[avatar_definition])).
% 0.31/1.37 thf(f577,plain,(
% 0.31/1.37 spl3_1 | ~spl3_2),
% 0.31/1.37 inference(avatar_split_clause,[status(thm)],[f569,f536,f532])).
% 0.31/1.37 thf(f536,definition,(
% 0.31/1.37 spl3_2 <=> ($true = (in @ sK1 @ (dsetconstr @ sK2 @ sK0)))),
% 0.31/1.37 introduced(definition,[new_symbols(naming,[spl3_2])],[avatar_definition])).
% 0.31/1.37 thf(f569,plain,(
% 0.31/1.37 ((sK0 @ sK1) = $true) | ~spl3_2),
% 0.31/1.37 inference(trivial_inequality_removal,[status(thm)],[f558])).
% 0.31/1.37 thf(f558,plain,(
% 0.31/1.37 ((sK0 @ sK1) = $true) | ($false = $true) | ~spl3_2),
% 0.31/1.37 inference(superposition,[status(thm)],[f418,f538])).
% 0.31/1.37 thf(f538,plain,(
% 0.31/1.37 ($true = (in @ sK1 @ (dsetconstr @ sK2 @ sK0))) | ~spl3_2),
% 0.31/1.37 inference(avatar_component_clause,[status(thm)],[f536])).
% 0.31/1.37 thf(f418,plain,(
% 0.31/1.37 ( ! [X2 : $i > $o,X3 : $i,X1 : $i] : (($false = (in @ X3 @ (dsetconstr @ X1 @ X2))) | ($true = (X2 @ X3))) )),
% 0.31/1.37 inference(binary_proxy_clausification,[status(thm)],[f417])).
% 0.31/1.37 thf(f417,plain,(
% 0.31/1.37 ( ! [X2 : $i > $o,X3 : $i,X1 : $i] : ((((in @ X3 @ (dsetconstr @ X1 @ X2)) => (X2 @ X3)) = $true)) )),
% 0.31/1.37 inference(beta_eta_normalization,[status(thm)],[f416])).
% 0.31/1.37 thf(f416,plain,(
% 0.31/1.37 ( ! [X2 : $i > $o,X3 : $i,X1 : $i] : (($true = ((^[Y0 : $i]: ((in @ Y0 @ (dsetconstr @ X1 @ X2)) => (X2 @ Y0))) @ X3))) )),
% 0.31/1.37 inference(pi_clausification,[status(thm)],[f415])).
% 0.31/1.37 thf(f415,plain,(
% 0.31/1.37 ( ! [X2 : $i > $o,X1 : $i] : (($true = (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ (dsetconstr @ X1 @ X2)) => (X2 @ Y0)))))) )),
% 0.31/1.37 inference(beta_eta_normalization,[status(thm)],[f414])).
% 0.31/1.37 thf(f414,plain,(
% 0.31/1.37 ( ! [X2 : $i > $o,X1 : $i] : ((((^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ (dsetconstr @ X1 @ Y0)) => (Y0 @ Y1))))) @ X2) = $true)) )),
% 0.31/1.37 inference(pi_clausification,[status(thm)],[f413])).
% 0.31/1.37 thf(f413,plain,(
% 0.31/1.37 ( ! [X1 : $i] : (($true = (!! @ ($i > $o) @ (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ (dsetconstr @ X1 @ Y0)) => (Y0 @ Y1)))))))) )),
% 0.31/1.37 inference(beta_eta_normalization,[status(thm)],[f412])).
% 0.31/1.37 thf(f412,plain,(
% 0.31/1.37 ( ! [X1 : $i] : (($true = ((^[Y0 : $i]: (!! @ ($i > $o) @ (^[Y1 : $i > $o]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ (dsetconstr @ Y0 @ Y1)) => (Y1 @ Y2))))))) @ X1))) )),
% 0.31/1.37 inference(pi_clausification,[status(thm)],[f411])).
% 0.31/1.37 thf(f411,plain,(
% 0.31/1.37 ($true = (!! @ $i @ (^[Y0 : $i]: (!! @ ($i > $o) @ (^[Y1 : $i > $o]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ (dsetconstr @ Y0 @ Y1)) => (Y1 @ Y2)))))))))),
% 0.31/1.37 inference(beta_eta_normalization,[status(thm)],[f157])).
% 0.31/1.37 thf(f157,plain,(
% 0.31/1.37 ((!! @ $i @ (^[Y0 : $i]: (!! @ ($i > $o) @ (^[Y1 : $i > $o]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ (dsetconstr @ Y0 @ (^[Y3 : $i]: (Y1 @ Y3)))) => (Y1 @ Y2)))))))) = $true)),
% 0.31/1.37 inference(definition_unfolding,[status(thm)],[f130,f105])).
% 0.31/1.37 thf(f105,plain,(
% 0.31/1.37 (dsetconstrER = $true)),
% 0.31/1.37 inference(cnf_transformation,[status(thm)],[f84])).
% 0.31/1.37 thf(f130,plain,(
% 0.31/1.37 (dsetconstrER = (!! @ $i @ (^[Y0 : $i]: (!! @ ($i > $o) @ (^[Y1 : $i > $o]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ (dsetconstr @ Y0 @ (^[Y3 : $i]: (Y1 @ Y3)))) => (Y1 @ Y2)))))))))),
% 0.31/1.37 inference(cnf_transformation,[status(thm)],[f72])).
% 0.31/1.37 thf(f72,plain,(
% 0.31/1.37 (dsetconstrER = (!! @ $i @ (^[Y0 : $i]: (!! @ ($i > $o) @ (^[Y1 : $i > $o]: (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ (dsetconstr @ Y0 @ (^[Y3 : $i]: (Y1 @ Y3)))) => (Y1 @ Y2)))))))))),
% 0.31/1.37 inference(fool_elimination,[status(thm)],[f71])).
% 0.31/1.37 thf(f71,plain,(
% 0.31/1.37 (! [X0 : $i,X1 : $i > $o,X2 : $i] : ((in @ X0 @ (dsetconstr @ X2 @ (^[X3 : $i] : (X1 @ X3)))) => (X1 @ X0)) = dsetconstrER)),
% 0.31/1.37 inference(rectify,[status(thm)],[f16])).
% 0.31/1.37 thf(f16,axiom,(
% 0.31/1.37 (! [X1 : $i,X0 : $i > $o,X3 : $i] : ((in @ X1 @ (dsetconstr @ X3 @ (^[X2 : $i] : (X0 @ X2)))) => (X0 @ X1)) = dsetconstrER)),
% 0.31/1.37 file('/export/starexec/sandbox/tmp/tmp.hL7X93FyCR/DTF2THF_31925.p',dsetconstrER)).
% 0.31/1.37 thf(f548,plain,(
% 0.31/1.37 spl3_3 | spl3_4),
% 0.31/1.37 inference(avatar_split_clause,[status(thm)],[f410,f545,f541])).
% 0.31/1.37 thf(f410,plain,(
% 0.31/1.37 ($false = (in @ sK1 @ (dsetconstr @ sK2 @ sK0))) | ($false = (sK0 @ sK1))),
% 0.31/1.37 inference(binary_proxy_clausification,[status(thm)],[f408])).
% 0.31/1.37 thf(f408,plain,(
% 0.31/1.37 ((sK0 @ sK1) != (in @ sK1 @ (dsetconstr @ sK2 @ sK0)))),
% 0.31/1.37 inference(beta_eta_normalization,[status(thm)],[f102])).
% 0.31/1.37 thf(f102,plain,(
% 0.31/1.37 ((in @ sK1 @ (dsetconstr @ sK2 @ (^[Y0 : $i]: (sK0 @ Y0)))) != (sK0 @ sK1))),
% 0.31/1.37 inference(cnf_transformation,[status(thm)],[f84])).
% 0.31/1.37 thf(f539,plain,(
% 0.31/1.37 spl3_1 | spl3_2),
% 0.31/1.37 inference(avatar_split_clause,[status(thm)],[f409,f536,f532])).
% 0.31/1.37 thf(f409,plain,(
% 0.31/1.37 ($true = (in @ sK1 @ (dsetconstr @ sK2 @ sK0))) | ((sK0 @ sK1) = $true)),
% 0.31/1.37 inference(binary_proxy_clausification,[status(thm)],[f408])).
% 0.31/1.37 % SZS output end Proof for DTF2THF_31925
% 0.31/1.37 % (32040)------------------------------
% 0.31/1.37 % (32040)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.31/1.37 % (32040)Termination reason: Refutation
% 0.31/1.37
% 0.31/1.37 % (32040)Memory used [KB]: 6524
% 0.31/1.37 % (32040)Time elapsed: 0.046 s
% 0.31/1.37 % (32040)Instructions burned: 136 (million)
% 0.31/1.37 % (32040)------------------------------
% 0.31/1.37 % (32040)------------------------------
% 0.31/1.37 % (32039)Success in time 0.051 s
%------------------------------------------------------------------------------