%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU510^1 : 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 : n006.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 09:35:00 AM UTC 2026
% Result : Theorem 1.89s 1.49s
% Output : Refutation 1.89s
% Verified :
% SZS Type : Refutation
% Derivation depth : 22
% Number of leaves : 7
% Syntax : Number of formulae : 58 ( 36 unt; 0 typ; 2 def)
% Number of atoms : 700 ( 181 equ; 0 cnn)
% Maximal formula atoms : 29 ( 12 avg)
% Number of connectives : 558 ( 32 ~; 15 |; 88 &; 262 @)
% ( 2 <=>; 138 =>; 0 <=; 0 <~>)
% Maximal formula depth : 37 ( 6 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 30 ( 30 >; 0 *; 0 +; 0 <<)
% Number of symbols : 74 ( 70 usr; 34 con; 0-3 aty)
% ( 21 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 97 ( 46 ^; 42 !; 9 ?; 97 :)
% 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_55,type,
sK1: $i > $o ).
thf(func_def_58,type,
sK4: $i > $i > $i ).
thf(func_def_60,type,
sK5: ( $i > $i > $o ) > $i > $i ).
thf(func_def_61,type,
sK6: ( $i > $i > $o ) > $i > $i ).
thf(func_def_62,type,
sK7: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_63,type,
sK8: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_64,type,
sK9: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_65,type,
sK10: $i > $i > $i ).
thf(func_def_66,type,
sK11: ( $i > $o ) > $i > $i ).
thf(func_def_67,type,
sK12: $i > $i ).
thf(func_def_68,type,
sK13: $i > $i ).
thf(func_def_69,type,
sK14: $i > $i ).
thf(func_def_70,type,
sK15: $i > $i ).
thf(func_def_71,type,
sK16: $i > $i ).
thf(func_def_72,type,
sK17: $i > $i > $i > $i ).
thf(func_def_73,type,
sK18: $i > $i > $i ).
thf(func_def_74,type,
sK19: $i > $i > $i ).
thf(func_def_75,type,
sK20: $i > $i > $i ).
thf(func_def_76,type,
sK21: $i > $i > $i ).
thf(func_def_77,type,
sK22: $i > $i > $i ).
thf(func_def_78,type,
sK23: $i > $i > $i ).
thf(func_def_79,type,
sK24: ( $i > $o ) > $i ).
thf(func_def_80,type,
sK25: ( $i > $o ) > $i > $i ).
thf(func_def_81,type,
sK26: $i > ( $i > $o ) > $i ).
thf(func_def_82,type,
sK27: $i > $i > $i ).
thf(func_def_83,type,
sK28: ( $i > $o ) > $i ).
thf(func_def_84,type,
sK29: ( $i > $o ) > $i > $i ).
thf(func_def_85,type,
sK30: $i > $i ).
thf(func_def_86,type,
ph31:
!>[X0: $tType] : X0 ).
thf(func_def_87,type,
sK32: $i > $i > $i ).
thf(f1138,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f1054,f1099,f1137]) ).
thf(f1137,plain,
~ spl3_1,
inference(avatar_contradiction_clause,[status(thm)],[f1136]) ).
thf(f1136,plain,
( $false
| ~ spl3_1 ),
inference(trivial_inequality_removal,[status(thm)],[f1102]) ).
thf(f1102,plain,
( ( $false = $true )
| ~ spl3_1 ),
inference(superposition,[status(thm)],[f1005,f133]) ).
thf(f133,plain,
( ( sK1 @ sK0 )
= $true ),
inference(cnf_transformation,[status(thm)],[f93]) ).
thf(f93,plain,
( ( noeltsimpempty = $true )
& ( notinemptyset = $true )
& ( emptysetE = $true )
& ( setext = $true )
& ( foundationAx = $true )
& ( ( nonempty
@ ( dsetconstr @ sK2
@ ^ [Y0: $i] : ( sK1 @ Y0 ) ) )
!= $true )
& ( ( sK1 @ sK0 )
= $true )
& ( ( in @ sK0 @ sK2 )
= $true )
& ( emptysetimpfalse = $true )
& ( setadjoinAx = $true )
& ( setbeta = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( prop2setE = $true )
& ( setunionAx = $true )
& ( omega0Ax = $true )
& ( emptysetAx = $true )
& ( dsetconstrI = $true )
& ( setextAx = $true )
& ( powersetAx = $true )
& ( descrp = $true )
& ( replAx = $true )
& ( omegaSAx = $true )
& ( exuE3e = $true )
& ( emptyI = $true )
& ( nonemptyE1 = $true )
& ( omegaIndAx = $true )
& ( wellorderingAx = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f91,f92]) ).
thf(f92,plain,
( ? [X0: $i,X1: $i > $o,X2: $i] :
( ( ( nonempty
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X1 @ Y0 ) ) )
!= $true )
& ( ( X1 @ X0 )
= $true )
& ( ( in @ X0 @ X2 )
= $true ) )
=> ( ( ( nonempty
@ ( dsetconstr @ sK2
@ ^ [Y0: $i] : ( sK1 @ Y0 ) ) )
!= $true )
& ( ( sK1 @ sK0 )
= $true )
& ( ( in @ sK0 @ sK2 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f91,plain,
( ( noeltsimpempty = $true )
& ( notinemptyset = $true )
& ( emptysetE = $true )
& ( setext = $true )
& ( foundationAx = $true )
& ? [X0: $i,X1: $i > $o,X2: $i] :
( ( ( nonempty
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X1 @ Y0 ) ) )
!= $true )
& ( ( X1 @ X0 )
= $true )
& ( ( in @ X0 @ X2 )
= $true ) )
& ( emptysetimpfalse = $true )
& ( setadjoinAx = $true )
& ( setbeta = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( prop2setE = $true )
& ( setunionAx = $true )
& ( omega0Ax = $true )
& ( emptysetAx = $true )
& ( dsetconstrI = $true )
& ( setextAx = $true )
& ( powersetAx = $true )
& ( descrp = $true )
& ( replAx = $true )
& ( omegaSAx = $true )
& ( exuE3e = $true )
& ( emptyI = $true )
& ( nonemptyE1 = $true )
& ( omegaIndAx = $true )
& ( wellorderingAx = $true ) ),
inference(flattening,[status(thm)],[f90]) ).
thf(f90,plain,
( ? [X0: $i,X1: $i > $o,X2: $i] :
( ( ( nonempty
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X1 @ Y0 ) ) )
!= $true )
& ( ( X1 @ X0 )
= $true )
& ( ( in @ X0 @ X2 )
= $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)],[f71]) ).
thf(f71,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 )
=> ! [X0: $i,X1: $i > $o,X2: $i] :
( ( ( in @ X0 @ X2 )
= $true )
=> ( ( ( X1 @ X0 )
= $true )
=> ( ( nonempty
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X1 @ Y0 ) ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f70]) ).
thf(f70,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
=> ! [X0: $i,X1: $i > $o,X2: $i] :
( ( in @ X0 @ X2 )
=> ( ( X1 @ X0 )
=> ( nonempty
@ ( dsetconstr @ X2
@ ^ [X3: $i] : ( X1 @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f31]) ).
thf(f31,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
=> ! [X1: $i,X0: $i > $o,X3: $i] :
( ( in @ X1 @ X3 )
=> ( ( X0 @ X1 )
=> ( nonempty
@ ( dsetconstr @ X3
@ ^ [X2: $i] : ( X0 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f30]) ).
thf(f30,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
=> ! [X1: $i,X0: $i > $o,X3: $i] :
( ( in @ X1 @ X3 )
=> ( ( X0 @ X1 )
=> ( nonempty
@ ( dsetconstr @ X3
@ ^ [X2: $i] : ( X0 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.vEp0jiah3y/DTF2THF_17746.p',nonemptyI) ).
thf(f1005,plain,
( ( ( sK1 @ sK0 )
= $false )
| ~ spl3_1 ),
inference(avatar_component_clause,[status(thm)],[f1003]) ).
thf(f1003,definition,
( spl3_1
<=> ( ( sK1 @ sK0 )
= $false ) ),
introduced(definition,[new_symbols(naming,[spl3_1])],[avatar_definition]) ).
thf(f1099,plain,
~ spl3_2,
inference(avatar_contradiction_clause,[status(thm)],[f1098]) ).
thf(f1098,plain,
( $false
| ~ spl3_2 ),
inference(trivial_inequality_removal,[status(thm)],[f1090]) ).
thf(f1090,plain,
( ( $false = $true )
| ~ spl3_2 ),
inference(superposition,[status(thm)],[f1009,f558]) ).
thf(f558,plain,
! [X1: $i] :
( ( in @ X1 @ emptyset )
= $false ),
inference(not_proxy_clausification,[status(thm)],[f557]) ).
thf(f557,plain,
! [X1: $i] :
( ( ~ ( in @ X1 @ emptyset ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f556]) ).
thf(f556,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
~ ( in @ Y0 @ emptyset )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f158]) ).
thf(f158,plain,
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
~ ( in @ Y0 @ emptyset ) ) ),
inference(definition_unfolding,[status(thm)],[f138,f100]) ).
thf(f100,plain,
( notinemptyset
= ( !! @ $i
@ ^ [Y0: $i] :
~ ( in @ Y0 @ emptyset ) ) ),
inference(cnf_transformation,[status(thm)],[f56]) ).
thf(f56,plain,
( notinemptyset
= ( !! @ $i
@ ^ [Y0: $i] :
~ ( in @ Y0 @ emptyset ) ) ),
inference(fool_elimination,[status(thm)],[f55]) ).
thf(f55,plain,
( notinemptyset
= ( ! [X0: $i] :
~ ( in @ X0 @ emptyset ) ) ),
inference(rectify,[status(thm)],[f22]) ).
thf(f22,axiom,
( notinemptyset
= ( ! [X1: $i] :
~ ( in @ X1 @ emptyset ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.vEp0jiah3y/DTF2THF_17746.p',notinemptyset) ).
thf(f138,plain,
notinemptyset = $true,
inference(cnf_transformation,[status(thm)],[f93]) ).
thf(f1009,plain,
( ( ( in @ sK0 @ emptyset )
= $true )
| ~ spl3_2 ),
inference(avatar_component_clause,[status(thm)],[f1007]) ).
thf(f1007,definition,
( spl3_2
<=> ( ( in @ sK0 @ emptyset )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl3_2])],[avatar_definition]) ).
thf(f1054,plain,
( spl3_2
| spl3_1 ),
inference(avatar_split_clause,[status(thm)],[f983,f1003,f1007]) ).
thf(f983,plain,
( ( ( in @ sK0 @ emptyset )
= $true )
| ( ( sK1 @ sK0 )
= $false ) ),
inference(trivial_inequality_removal,[status(thm)],[f952]) ).
thf(f952,plain,
( ( $false = $true )
| ( ( sK1 @ sK0 )
= $false )
| ( ( in @ sK0 @ emptyset )
= $true ) ),
inference(superposition,[status(thm)],[f697,f132]) ).
thf(f132,plain,
( ( in @ sK0 @ sK2 )
= $true ),
inference(cnf_transformation,[status(thm)],[f93]) ).
thf(f697,plain,
! [X0: $i] :
( ( ( in @ X0 @ sK2 )
= $false )
| ( ( in @ X0 @ emptyset )
= $true )
| ( ( sK1 @ X0 )
= $false ) ),
inference(superposition,[status(thm)],[f538,f507]) ).
thf(f507,plain,
( emptyset
= ( dsetconstr @ sK2 @ sK1 ) ),
inference(equality_proxy_clausification,[status(thm)],[f506]) ).
thf(f506,plain,
( ( emptyset
= ( dsetconstr @ sK2 @ sK1 ) )
= $true ),
inference(not_proxy_clausification,[status(thm)],[f505]) ).
thf(f505,plain,
( ( emptyset
!= ( dsetconstr @ sK2 @ sK1 ) )
!= $true ),
inference(beta_eta_normalization,[status(thm)],[f159]) ).
thf(f159,plain,
( ( ^ [Y0: $i] : ( emptyset != Y0 )
@ ( dsetconstr @ sK2
@ ^ [Y0: $i] : ( sK1 @ Y0 ) ) )
!= $true ),
inference(definition_unfolding,[status(thm)],[f134,f99]) ).
thf(f99,plain,
( nonempty
= ( ^ [Y0: $i] : ( emptyset != Y0 ) ) ),
inference(cnf_transformation,[status(thm)],[f40]) ).
thf(f40,plain,
( nonempty
= ( ^ [Y0: $i] : ( emptyset != Y0 ) ) ),
inference(fool_elimination,[status(thm)],[f39]) ).
thf(f39,plain,
( ( ^ [X0: $i] : ( emptyset != X0 ) )
= nonempty ),
inference(rectify,[status(thm)],[f28]) ).
thf(f28,axiom,
( ( ^ [X1: $i] : ( emptyset != X1 ) )
= nonempty ),
file('/export/starexec/sandbox2/tmp/tmp.vEp0jiah3y/DTF2THF_17746.p',nonempty) ).
thf(f134,plain,
( ( nonempty
@ ( dsetconstr @ sK2
@ ^ [Y0: $i] : ( sK1 @ Y0 ) ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f93]) ).
thf(f538,plain,
! [X2: $i,X3: $i > $o,X1: $i] :
( ( ( in @ X2 @ ( dsetconstr @ X1 @ X3 ) )
= $true )
| ( ( X3 @ X2 )
= $false )
| ( ( in @ X2 @ X1 )
= $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f537]) ).
thf(f537,plain,
! [X2: $i,X3: $i > $o,X1: $i] :
( ( ( in @ X2 @ X1 )
= $false )
| ( ( ( X3 @ X2 )
=> ( in @ X2 @ ( dsetconstr @ X1 @ X3 ) ) )
= $true ) ),
inference(binary_proxy_clausification,[status(thm)],[f536]) ).
thf(f536,plain,
! [X2: $i,X3: $i > $o,X1: $i] :
( ( ( in @ X2 @ X1 )
=> ( ( X3 @ X2 )
=> ( in @ X2 @ ( dsetconstr @ X1 @ X3 ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f535]) ).
thf(f535,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)],[f534]) ).
thf(f534,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)],[f533]) ).
thf(f533,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)],[f532]) ).
thf(f532,plain,
! [X1: $i] :
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ ( $i > $o )
@ ^ [Y1: $i > $o] :
( ( in @ Y0 @ X1 )
=> ( ( Y1 @ Y0 )
=> ( in @ Y0 @ ( dsetconstr @ X1 @ Y1 ) ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f531]) ).
thf(f531,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ ( $i > $o )
@ ^ [Y2: $i > $o] :
( ( in @ Y1 @ Y0 )
=> ( ( Y2 @ Y1 )
=> ( in @ Y1 @ ( dsetconstr @ Y0 @ Y2 ) ) ) ) ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f530]) ).
thf(f530,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)],[f175]) ).
thf(f175,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)],[f143,f121]) ).
thf(f121,plain,
dsetconstrI = $true,
inference(cnf_transformation,[status(thm)],[f93]) ).
thf(f143,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)],[f38]) ).
thf(f38,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)],[f37]) ).
thf(f37,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/sandbox2/tmp/tmp.vEp0jiah3y/DTF2THF_17746.p',dsetconstrI) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : SEU510^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.09 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.11/0.29 % Computer : n006.cluster.edu
% 0.11/0.29 % Model : x86_64 x86_64
% 0.11/0.29 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.29 % Memory : 8042.1875MB
% 0.11/0.29 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.29 % CPULimit : 300
% 0.11/0.29 % WCLimit : 300
% 0.11/0.29 % DateTime : Tue Feb 24 12:25:00 EST 2026
% 0.11/0.29 % CPUTime :
% 0.11/0.29 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.38 ---- Original DTF file ---
% 0.16/0.38 thf(spec,logic,$$dhol).
% 0.16/0.38 %------------------------------------------------------------------------------
% 0.16/0.38 % File : SEU510^1 : TPTP v9.2.1. Released v3.7.0.
% 0.16/0.38 % Domain : Set Theory
% 0.16/0.38 % Problem : Preliminary Notions - Basic Laws of Logic
% 0.16/0.38 % Version : Especial.
% 0.16/0.38 % English : (! A:i.! phi:i>o.! x:i.in x A -> phi x -> nonempty
% 0.16/0.38 % (dsetconstr A (^ y:i.phi y)))
% 0.16/0.38
% 0.16/0.38 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.16/0.38 % Source : [Bro08]
% 0.16/0.38 % Names : ZFC012g [Bro08]
% 0.16/0.38
% 0.16/0.38 % Status : Theorem
% 0.16/0.38 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.40 v8.2.0, 0.38 v8.1.0, 0.36 v7.5.0, 0.29 v7.4.0, 0.33 v7.2.0, 0.38 v7.1.0, 0.50 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.40 v6.2.0, 0.43 v6.1.0, 0.57 v5.5.0, 0.67 v5.4.0, 0.80 v4.1.0, 1.00 v3.7.0
% 0.16/0.38 % Syntax : Number of formulae : 67 ( 29 unt; 37 typ; 29 def)
% 0.16/0.38 % Number of atoms : 170 ( 38 equ; 0 cnn)
% 0.16/0.38 % Maximal formula atoms : 28 ( 5 avg)
% 0.16/0.38 % Number of connectives : 298 ( 7 ~; 3 |; 18 &; 194 @)
% 0.16/0.38 % ( 7 <=>; 69 =>; 0 <=; 0 <~>)
% 0.16/0.38 % Maximal formula depth : 36 ( 2 avg)
% 0.16/0.38 % Number of types : 2 ( 0 usr)
% 0.16/0.38 % Number of type conns : 26 ( 26 >; 0 *; 0 +; 0 <<)
% 0.16/0.38 % Number of symbols : 39 ( 37 usr; 29 con; 0-2 aty)
% 0.16/0.38 % Number of variables : 99 ( 14 ^; 70 !; 15 ?; 99 :)
% 0.16/0.38 % SPC : TH0_THM_EQU_NAR
% 0.16/0.38
% 0.16/0.38 % Comments : http://mathgate.info/detsetitem.php?id=430
% 0.16/0.38 %------------------------------------------------------------------------------
% 0.16/0.38 thf(in_type,type,
% 0.16/0.38 in: $i > $i > $o ).
% 0.16/0.38
% 0.16/0.38 thf(exu_type,type,
% 0.16/0.38 exu: ( $i > $o ) > $o ).
% 0.16/0.38
% 0.16/0.38 thf(exu,definition,
% 0.16/0.38 ( exu
% 0.16/0.38 = ( ^ [Xphi: $i > $o] :
% 0.16/0.38 ? [Xx: $i] :
% 0.16/0.38 ( ( Xphi @ Xx )
% 0.16/0.38 & ! [Xy: $i] :
% 0.16/0.38 ( ( Xphi @ Xy )
% 0.16/0.38 => ( Xx = Xy ) ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(setextAx_type,type,
% 0.16/0.38 setextAx: $o ).
% 0.16/0.38
% 0.16/0.38 thf(setextAx,definition,
% 0.16/0.38 ( setextAx
% 0.16/0.38 = ( ! [A: $i,B: $i] :
% 0.16/0.38 ( ! [Xx: $i] :
% 0.16/0.38 ( ( in @ Xx @ A )
% 0.16/0.38 <=> ( in @ Xx @ B ) )
% 0.16/0.38 => ( A = B ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(emptyset_type,type,
% 0.16/0.38 emptyset: $i ).
% 0.16/0.38
% 0.16/0.38 thf(emptysetAx_type,type,
% 0.16/0.38 emptysetAx: $o ).
% 0.16/0.38
% 0.16/0.38 thf(emptysetAx,definition,
% 0.16/0.38 ( emptysetAx
% 0.16/0.38 = ( ! [Xx: $i] :
% 0.16/0.38 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(setadjoin_type,type,
% 0.16/0.38 setadjoin: $i > $i > $i ).
% 0.16/0.38
% 0.16/0.38 thf(setadjoinAx_type,type,
% 0.16/0.38 setadjoinAx: $o ).
% 0.16/0.38
% 0.16/0.38 thf(setadjoinAx,definition,
% 0.16/0.38 ( setadjoinAx
% 0.16/0.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.38 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.38 <=> ( ( Xy = Xx )
% 0.16/0.38 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(powerset_type,type,
% 0.16/0.38 powerset: $i > $i ).
% 0.16/0.38
% 0.16/0.38 thf(powersetAx_type,type,
% 0.16/0.38 powersetAx: $o ).
% 0.16/0.38
% 0.16/0.38 thf(powersetAx,definition,
% 0.16/0.38 ( powersetAx
% 0.16/0.38 = ( ! [A: $i,B: $i] :
% 0.16/0.38 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.38 <=> ! [Xx: $i] :
% 0.16/0.38 ( ( in @ Xx @ B )
% 0.16/0.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(setunion_type,type,
% 0.16/0.38 setunion: $i > $i ).
% 0.16/0.38
% 0.16/0.38 thf(setunionAx_type,type,
% 0.16/0.38 setunionAx: $o ).
% 0.16/0.38
% 0.16/0.38 thf(setunionAx,definition,
% 0.16/0.38 ( setunionAx
% 0.16/0.38 = ( ! [A: $i,Xx: $i] :
% 0.16/0.38 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.38 <=> ? [B: $i] :
% 0.16/0.38 ( ( in @ Xx @ B )
% 0.16/0.38 & ( in @ B @ A ) ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(omega_type,type,
% 0.16/0.38 omega: $i ).
% 0.16/0.38
% 0.16/0.38 thf(omega0Ax_type,type,
% 0.16/0.38 omega0Ax: $o ).
% 0.16/0.38
% 0.16/0.38 thf(omega0Ax,definition,
% 0.16/0.38 ( omega0Ax
% 0.16/0.38 = ( in @ emptyset @ omega ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(omegaSAx_type,type,
% 0.16/0.38 omegaSAx: $o ).
% 0.16/0.38
% 0.16/0.38 thf(omegaSAx,definition,
% 0.16/0.38 ( omegaSAx
% 0.16/0.38 = ( ! [Xx: $i] :
% 0.16/0.38 ( ( in @ Xx @ omega )
% 0.16/0.38 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(omegaIndAx_type,type,
% 0.16/0.38 omegaIndAx: $o ).
% 0.16/0.38
% 0.16/0.38 thf(omegaIndAx,definition,
% 0.16/0.38 ( omegaIndAx
% 0.16/0.38 = ( ! [A: $i] :
% 0.16/0.38 ( ( ( in @ emptyset @ A )
% 0.16/0.38 & ! [Xx: $i] :
% 0.16/0.38 ( ( ( in @ Xx @ omega )
% 0.16/0.38 & ( in @ Xx @ A ) )
% 0.16/0.38 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.16/0.38 => ! [Xx: $i] :
% 0.16/0.38 ( ( in @ Xx @ omega )
% 0.16/0.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(replAx_type,type,
% 0.16/0.38 replAx: $o ).
% 0.16/0.38
% 0.16/0.38 thf(replAx,definition,
% 0.16/0.38 ( replAx
% 0.16/0.38 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.16/0.38 ( ! [Xx: $i] :
% 0.16/0.38 ( ( in @ Xx @ A )
% 0.16/0.38 => ( exu
% 0.16/0.38 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.16/0.38 => ? [B: $i] :
% 0.16/0.38 ! [Xx: $i] :
% 0.16/0.38 ( ( in @ Xx @ B )
% 0.16/0.38 <=> ? [Xy: $i] :
% 0.16/0.38 ( ( in @ Xy @ A )
% 0.16/0.38 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(foundationAx_type,type,
% 0.16/0.38 foundationAx: $o ).
% 0.16/0.38
% 0.16/0.38 thf(foundationAx,definition,
% 0.16/0.38 ( foundationAx
% 0.16/0.38 = ( ! [A: $i] :
% 0.16/0.38 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.38 => ? [B: $i] :
% 0.16/0.38 ( ( in @ B @ A )
% 0.16/0.38 & ~ ? [Xx: $i] :
% 0.16/0.38 ( ( in @ Xx @ B )
% 0.16/0.38 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(wellorderingAx_type,type,
% 0.16/0.38 wellorderingAx: $o ).
% 0.16/0.38
% 0.16/0.38 thf(wellorderingAx,definition,
% 0.16/0.38 ( wellorderingAx
% 0.16/0.38 = ( ! [A: $i] :
% 0.16/0.38 ? [B: $i] :
% 0.16/0.38 ( ! [C: $i] :
% 0.16/0.38 ( ( in @ C @ B )
% 0.16/0.38 => ! [Xx: $i] :
% 0.16/0.38 ( ( in @ Xx @ C )
% 0.16/0.38 => ( in @ Xx @ A ) ) )
% 0.16/0.38 & ! [Xx: $i,Xy: $i] :
% 0.16/0.38 ( ( ( in @ Xx @ A )
% 0.16/0.38 & ( in @ Xy @ A ) )
% 0.16/0.38 => ( ! [C: $i] :
% 0.16/0.38 ( ( in @ C @ B )
% 0.16/0.38 => ( ( in @ Xx @ C )
% 0.16/0.38 <=> ( in @ Xy @ C ) ) )
% 0.16/0.38 => ( Xx = Xy ) ) )
% 0.16/0.38 & ! [C: $i,D: $i] :
% 0.16/0.38 ( ( ( in @ C @ B )
% 0.16/0.38 & ( in @ D @ B ) )
% 0.16/0.38 => ( ! [Xx: $i] :
% 0.16/0.38 ( ( in @ Xx @ C )
% 0.16/0.38 => ( in @ Xx @ D ) )
% 0.16/0.38 | ! [Xx: $i] :
% 0.16/0.38 ( ( in @ Xx @ D )
% 0.16/0.38 => ( in @ Xx @ C ) ) ) )
% 0.16/0.38 & ! [C: $i] :
% 0.16/0.38 ( ( ! [Xx: $i] :
% 0.16/0.38 ( ( in @ Xx @ C )
% 0.16/0.38 => ( in @ Xx @ A ) )
% 0.16/0.38 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.16/0.38 => ? [D: $i,Xx: $i] :
% 0.16/0.38 ( ( in @ D @ B )
% 0.16/0.38 & ( in @ Xx @ C )
% 0.16/0.38 & ~ ? [Xy: $i] :
% 0.16/0.38 ( ( in @ Xy @ D )
% 0.16/0.38 & ( in @ Xy @ C ) )
% 0.16/0.38 & ! [E: $i] :
% 0.16/0.38 ( ( in @ E @ B )
% 0.16/0.38 => ( ! [Xy: $i] :
% 0.16/0.38 ( ( in @ Xy @ E )
% 0.16/0.38 => ( in @ Xy @ D ) )
% 0.16/0.38 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(descr_type,type,
% 0.16/0.38 descr: ( $i > $o ) > $i ).
% 0.16/0.38
% 0.16/0.38 thf(descrp_type,type,
% 0.16/0.38 descrp: $o ).
% 0.16/0.38
% 0.16/0.38 thf(descrp,definition,
% 0.16/0.38 ( descrp
% 0.16/0.38 = ( ! [Xphi: $i > $o] :
% 0.16/0.38 ( ( exu
% 0.16/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.38 => ( Xphi
% 0.16/0.38 @ ( descr
% 0.16/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(dsetconstr_type,type,
% 0.16/0.38 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.16/0.38
% 0.16/0.38 thf(dsetconstrI_type,type,
% 0.16/0.38 dsetconstrI: $o ).
% 0.16/0.38
% 0.16/0.38 thf(dsetconstrI,definition,
% 0.16/0.38 ( dsetconstrI
% 0.16/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.38 ( ( in @ Xx @ A )
% 0.16/0.38 => ( ( Xphi @ Xx )
% 0.16/0.38 => ( in @ Xx
% 0.16/0.38 @ ( dsetconstr @ A
% 0.16/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(dsetconstrEL_type,type,
% 0.16/0.38 dsetconstrEL: $o ).
% 0.16/0.38
% 0.16/0.38 thf(dsetconstrEL,definition,
% 0.16/0.38 ( dsetconstrEL
% 0.16/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.38 ( ( in @ Xx
% 0.16/0.38 @ ( dsetconstr @ A
% 0.16/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.38 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(dsetconstrER_type,type,
% 0.16/0.38 dsetconstrER: $o ).
% 0.16/0.38
% 0.16/0.38 thf(dsetconstrER,definition,
% 0.16/0.38 ( dsetconstrER
% 0.16/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.38 ( ( in @ Xx
% 0.16/0.38 @ ( dsetconstr @ A
% 0.16/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.38 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(exuE1_type,type,
% 0.16/0.38 exuE1: $o ).
% 0.16/0.38
% 0.16/0.38 thf(exuE1,definition,
% 0.16/0.38 ( exuE1
% 0.16/0.38 = ( ! [Xphi: $i > $o] :
% 0.16/0.38 ( ( exu
% 0.16/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.38 => ? [Xx: $i] :
% 0.16/0.38 ( ( Xphi @ Xx )
% 0.16/0.38 & ! [Xy: $i] :
% 0.16/0.38 ( ( Xphi @ Xy )
% 0.16/0.38 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(prop2set_type,type,
% 0.16/0.38 prop2set: $o > $i ).
% 0.16/0.38
% 0.16/0.38 thf(prop2set,definition,
% 0.16/0.38 ( prop2set
% 0.16/0.38 = ( ^ [Xphi: $o] :
% 0.16/0.38 ( dsetconstr @ ( powerset @ emptyset )
% 0.16/0.38 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(prop2setE_type,type,
% 0.16/0.38 prop2setE: $o ).
% 0.16/0.38
% 0.16/0.38 thf(prop2setE,definition,
% 0.16/0.38 ( prop2setE
% 0.16/0.38 = ( ! [Xphi: $o,Xx: $i] :
% 0.16/0.38 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.16/0.38 => Xphi ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(emptysetE_type,type,
% 0.16/0.38 emptysetE: $o ).
% 0.16/0.38
% 0.16/0.38 thf(emptysetE,definition,
% 0.16/0.38 ( emptysetE
% 0.16/0.38 = ( ! [Xx: $i] :
% 0.16/0.38 ( ( in @ Xx @ emptyset )
% 0.16/0.38 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(emptysetimpfalse_type,type,
% 0.16/0.38 emptysetimpfalse: $o ).
% 0.16/0.38
% 0.16/0.38 thf(emptysetimpfalse,definition,
% 0.16/0.38 ( emptysetimpfalse
% 0.16/0.38 = ( ! [Xx: $i] :
% 0.16/0.38 ( ( in @ Xx @ emptyset )
% 0.16/0.38 => $false ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(notinemptyset_type,type,
% 0.16/0.38 notinemptyset: $o ).
% 0.16/0.38
% 0.16/0.38 thf(notinemptyset,definition,
% 0.16/0.38 ( notinemptyset
% 0.16/0.38 = ( ! [Xx: $i] :
% 0.16/0.38 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(exuE3e_type,type,
% 0.16/0.38 exuE3e: $o ).
% 0.16/0.38
% 0.16/0.38 thf(exuE3e,definition,
% 0.16/0.38 ( exuE3e
% 0.16/0.38 = ( ! [Xphi: $i > $o] :
% 0.16/0.38 ( ( exu
% 0.16/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.38 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(setext_type,type,
% 0.16/0.38 setext: $o ).
% 0.16/0.38
% 0.16/0.38 thf(setext,definition,
% 0.16/0.38 ( setext
% 0.16/0.38 = ( ! [A: $i,B: $i] :
% 0.16/0.38 ( ! [Xx: $i] :
% 0.16/0.38 ( ( in @ Xx @ A )
% 0.16/0.38 => ( in @ Xx @ B ) )
% 0.16/0.38 => ( ! [Xx: $i] :
% 0.16/0.38 ( ( in @ Xx @ B )
% 0.16/0.38 => ( in @ Xx @ A ) )
% 0.16/0.38 => ( A = B ) ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(emptyI_type,type,
% 0.16/0.38 emptyI: $o ).
% 0.16/0.38
% 0.16/0.38 thf(emptyI,definition,
% 0.16/0.38 ( emptyI
% 0.16/0.38 = ( ! [A: $i] :
% 0.16/0.38 ( ! [Xx: $i] :
% 0.16/0.38 ~ ( in @ Xx @ A )
% 0.16/0.38 => ( A = emptyset ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(noeltsimpempty_type,type,
% 0.16/0.38 noeltsimpempty: $o ).
% 0.16/0.38
% 0.16/0.38 thf(noeltsimpempty,definition,
% 0.16/0.38 ( noeltsimpempty
% 0.16/0.38 = ( ! [A: $i] :
% 0.16/0.38 ( ! [Xx: $i] :
% 0.16/0.38 ~ ( in @ Xx @ A )
% 0.16/0.38 => ( A = emptyset ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(setbeta_type,type,
% 0.16/0.38 setbeta: $o ).
% 0.16/0.38
% 0.16/0.38 thf(setbeta,definition,
% 0.16/0.38 ( setbeta
% 0.16/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.38 ( ( in @ Xx @ A )
% 0.16/0.38 => ( ( in @ Xx
% 0.16/0.38 @ ( dsetconstr @ A
% 0.16/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.38 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(nonempty_type,type,
% 0.16/0.38 nonempty: $i > $o ).
% 0.16/0.38
% 0.16/0.38 thf(nonempty,definition,
% 0.16/0.38 ( nonempty
% 0.16/0.38 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(nonemptyE1_type,type,
% 0.16/0.38 nonemptyE1: $o ).
% 0.16/0.38
% 0.16/0.38 thf(nonemptyE1,definition,
% 0.16/0.38 ( nonemptyE1
% 0.16/0.38 = ( ! [A: $i] :
% 0.16/0.38 ( ( nonempty @ A )
% 0.16/0.38 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 thf(nonemptyI,conjecture,
% 0.16/0.38 ( setextAx
% 0.16/0.38 => ( emptysetAx
% 0.16/0.38 => ( setadjoinAx
% 0.16/0.38 => ( powersetAx
% 0.16/0.38 => ( setunionAx
% 0.16/0.38 => ( omega0Ax
% 0.16/0.38 => ( omegaSAx
% 0.16/0.38 => ( omegaIndAx
% 0.16/0.38 => ( replAx
% 0.16/0.38 => ( foundationAx
% 0.16/0.38 => ( wellorderingAx
% 0.16/0.38 => ( descrp
% 0.16/0.38 => ( dsetconstrI
% 0.16/0.38 => ( dsetconstrEL
% 0.16/0.38 => ( dsetconstrER
% 0.16/0.38 => ( exuE1
% 0.16/0.38 => ( prop2setE
% 0.16/0.38 => ( emptysetE
% 0.16/0.38 => ( emptysetimpfalse
% 0.16/0.38 => ( notinemptyset
% 0.16/0.38 => ( exuE3e
% 0.16/0.38 => ( setext
% 0.16/0.38 => ( emptyI
% 0.16/0.38 => ( noeltsimpempty
% 0.16/0.38 => ( setbeta
% 0.16/0.38 => ( nonemptyE1
% 0.16/0.38 => ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.38 ( ( in @ Xx @ A )
% 0.16/0.38 => ( ( Xphi @ Xx )
% 0.16/0.38 => ( nonempty
% 0.16/0.38 @ ( dsetconstr @ A
% 0.16/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.38
% 0.16/0.38 %------------------------------------------------------------------------------
% 0.16/0.38 ------------------------
% 1.71/1.37 ---- Embedded in THF ---
% 1.71/1.37 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.71/1.37 %%% Generated on Tue Feb 24 12:25:01 EST 2026
% 1.71/1.37 %%% using '$$dhol' embedding, version 1.3.0.
% 1.71/1.37 %%% Logic specification used:
% 1.71/1.37 %%% thf(spec, logic, $$dhol).
% 1.71/1.37
% 1.71/1.37 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.vEp0jiah3y/DTF2DTF17746.p
% See solution above
% 1.71/1.37 ------------------------
% 1.71/1.38 ---- Cleaned THF ---
% 1.71/1.38 thf(in_type,type,
% 1.71/1.38 in: $i > $i > $o ).
% 1.71/1.38
% 1.71/1.38 thf(exu_type,type,
% 1.71/1.38 exu: ( $i > $o ) > $o ).
% 1.71/1.38
% 1.71/1.38 thf(exu,definition,
% 1.71/1.38 ( exu
% 1.71/1.38 = ( ^ [Xphi: $i > $o] :
% 1.71/1.38 ? [Xx: $i] :
% 1.71/1.38 ( ( Xphi @ Xx )
% 1.71/1.38 & ! [Xy: $i] :
% 1.71/1.38 ( ( Xphi @ Xy )
% 1.71/1.38 => ( Xx = Xy ) ) ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(setextAx_type,type,
% 1.71/1.38 setextAx: $o ).
% 1.71/1.38
% 1.71/1.38 thf(setextAx,definition,
% 1.71/1.38 ( setextAx
% 1.71/1.38 = ( ! [A: $i,B: $i] :
% 1.71/1.38 ( ! [Xx: $i] :
% 1.71/1.38 ( ( in @ Xx @ A )
% 1.71/1.38 <=> ( in @ Xx @ B ) )
% 1.71/1.38 => ( A = B ) ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(emptyset_type,type,
% 1.71/1.38 emptyset: $i ).
% 1.71/1.38
% 1.71/1.38 thf(emptysetAx_type,type,
% 1.71/1.38 emptysetAx: $o ).
% 1.71/1.38
% 1.71/1.38 thf(emptysetAx,definition,
% 1.71/1.38 ( emptysetAx
% 1.71/1.38 = ( ! [Xx: $i] :
% 1.71/1.38 ~ ( in @ Xx @ emptyset ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(setadjoin_type,type,
% 1.71/1.38 setadjoin: $i > $i > $i ).
% 1.71/1.38
% 1.71/1.38 thf(setadjoinAx_type,type,
% 1.71/1.38 setadjoinAx: $o ).
% 1.71/1.38
% 1.71/1.38 thf(setadjoinAx,definition,
% 1.71/1.38 ( setadjoinAx
% 1.71/1.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.71/1.38 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.71/1.38 <=> ( ( Xy = Xx )
% 1.71/1.38 | ( in @ Xy @ A ) ) ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(powerset_type,type,
% 1.71/1.38 powerset: $i > $i ).
% 1.71/1.38
% 1.71/1.38 thf(powersetAx_type,type,
% 1.71/1.38 powersetAx: $o ).
% 1.71/1.38
% 1.71/1.38 thf(powersetAx,definition,
% 1.71/1.38 ( powersetAx
% 1.71/1.38 = ( ! [A: $i,B: $i] :
% 1.71/1.38 ( ( in @ B @ ( powerset @ A ) )
% 1.71/1.38 <=> ! [Xx: $i] :
% 1.71/1.38 ( ( in @ Xx @ B )
% 1.71/1.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(setunion_type,type,
% 1.71/1.38 setunion: $i > $i ).
% 1.71/1.38
% 1.71/1.38 thf(setunionAx_type,type,
% 1.71/1.38 setunionAx: $o ).
% 1.71/1.38
% 1.71/1.38 thf(setunionAx,definition,
% 1.71/1.38 ( setunionAx
% 1.71/1.38 = ( ! [A: $i,Xx: $i] :
% 1.71/1.38 ( ( in @ Xx @ ( setunion @ A ) )
% 1.71/1.38 <=> ? [B: $i] :
% 1.71/1.38 ( ( in @ Xx @ B )
% 1.71/1.38 & ( in @ B @ A ) ) ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(omega_type,type,
% 1.71/1.38 omega: $i ).
% 1.71/1.38
% 1.71/1.38 thf(omega0Ax_type,type,
% 1.71/1.38 omega0Ax: $o ).
% 1.71/1.38
% 1.71/1.38 thf(omega0Ax,definition,
% 1.71/1.38 ( omega0Ax
% 1.71/1.38 = ( in @ emptyset @ omega ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(omegaSAx_type,type,
% 1.71/1.38 omegaSAx: $o ).
% 1.71/1.38
% 1.71/1.38 thf(omegaSAx,definition,
% 1.71/1.38 ( omegaSAx
% 1.71/1.38 = ( ! [Xx: $i] :
% 1.71/1.38 ( ( in @ Xx @ omega )
% 1.71/1.38 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(omegaIndAx_type,type,
% 1.71/1.38 omegaIndAx: $o ).
% 1.71/1.38
% 1.71/1.38 thf(omegaIndAx,definition,
% 1.71/1.38 ( omegaIndAx
% 1.71/1.38 = ( ! [A: $i] :
% 1.71/1.38 ( ( ( in @ emptyset @ A )
% 1.71/1.38 & ! [Xx: $i] :
% 1.71/1.38 ( ( ( in @ Xx @ omega )
% 1.71/1.38 & ( in @ Xx @ A ) )
% 1.71/1.38 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 1.71/1.38 => ! [Xx: $i] :
% 1.71/1.38 ( ( in @ Xx @ omega )
% 1.71/1.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(replAx_type,type,
% 1.71/1.38 replAx: $o ).
% 1.71/1.38
% 1.71/1.38 thf(replAx,definition,
% 1.71/1.38 ( replAx
% 1.71/1.38 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 1.71/1.38 ( ! [Xx: $i] :
% 1.71/1.38 ( ( in @ Xx @ A )
% 1.71/1.38 => ( exu
% 1.71/1.38 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 1.71/1.38 => ? [B: $i] :
% 1.71/1.38 ! [Xx: $i] :
% 1.71/1.38 ( ( in @ Xx @ B )
% 1.71/1.38 <=> ? [Xy: $i] :
% 1.71/1.38 ( ( in @ Xy @ A )
% 1.71/1.38 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(foundationAx_type,type,
% 1.71/1.38 foundationAx: $o ).
% 1.71/1.38
% 1.71/1.38 thf(foundationAx,definition,
% 1.71/1.38 ( foundationAx
% 1.71/1.38 = ( ! [A: $i] :
% 1.71/1.38 ( ? [Xx: $i] : ( in @ Xx @ A )
% 1.71/1.38 => ? [B: $i] :
% 1.71/1.38 ( ( in @ B @ A )
% 1.71/1.38 & ~ ? [Xx: $i] :
% 1.71/1.38 ( ( in @ Xx @ B )
% 1.71/1.38 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(wellorderingAx_type,type,
% 1.71/1.38 wellorderingAx: $o ).
% 1.71/1.38
% 1.71/1.38 thf(wellorderingAx,definition,
% 1.71/1.38 ( wellorderingAx
% 1.71/1.38 = ( ! [A: $i] :
% 1.71/1.38 ? [B: $i] :
% 1.71/1.38 ( ! [C: $i] :
% 1.71/1.38 ( ( in @ C @ B )
% 1.71/1.38 => ! [Xx: $i] :
% 1.71/1.38 ( ( in @ Xx @ C )
% 1.71/1.38 => ( in @ Xx @ A ) ) )
% 1.71/1.38 & ! [Xx: $i,Xy: $i] :
% 1.71/1.38 ( ( ( in @ Xx @ A )
% 1.71/1.38 & ( in @ Xy @ A ) )
% 1.71/1.38 => ( ! [C: $i] :
% 1.71/1.38 ( ( in @ C @ B )
% 1.71/1.38 => ( ( in @ Xx @ C )
% 1.71/1.38 <=> ( in @ Xy @ C ) ) )
% 1.71/1.38 => ( Xx = Xy ) ) )
% 1.71/1.38 & ! [C: $i,D: $i] :
% 1.71/1.38 ( ( ( in @ C @ B )
% 1.71/1.38 & ( in @ D @ B ) )
% 1.71/1.38 => ( ! [Xx: $i] :
% 1.71/1.38 ( ( in @ Xx @ C )
% 1.71/1.38 => ( in @ Xx @ D ) )
% 1.71/1.38 | ! [Xx: $i] :
% 1.71/1.38 ( ( in @ Xx @ D )
% 1.71/1.38 => ( in @ Xx @ C ) ) ) )
% 1.71/1.38 & ! [C: $i] :
% 1.71/1.38 ( ( ! [Xx: $i] :
% 1.71/1.38 ( ( in @ Xx @ C )
% 1.71/1.38 => ( in @ Xx @ A ) )
% 1.71/1.38 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 1.71/1.38 => ? [D: $i,Xx: $i] :
% 1.71/1.38 ( ( in @ D @ B )
% 1.71/1.38 & ( in @ Xx @ C )
% 1.71/1.38 & ~ ? [Xy: $i] :
% 1.71/1.38 ( ( in @ Xy @ D )
% 1.71/1.38 & ( in @ Xy @ C ) )
% 1.71/1.38 & ! [E: $i] :
% 1.71/1.38 ( ( in @ E @ B )
% 1.71/1.38 => ( ! [Xy: $i] :
% 1.71/1.38 ( ( in @ Xy @ E )
% 1.71/1.38 => ( in @ Xy @ D ) )
% 1.71/1.38 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(descr_type,type,
% 1.71/1.38 descr: ( $i > $o ) > $i ).
% 1.71/1.38
% 1.71/1.38 thf(descrp_type,type,
% 1.71/1.38 descrp: $o ).
% 1.71/1.38
% 1.71/1.38 thf(descrp,definition,
% 1.71/1.38 ( descrp
% 1.71/1.38 = ( ! [Xphi: $i > $o] :
% 1.71/1.38 ( ( exu
% 1.71/1.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.71/1.38 => ( Xphi
% 1.71/1.38 @ ( descr
% 1.71/1.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(dsetconstr_type,type,
% 1.71/1.38 dsetconstr: $i > ( $i > $o ) > $i ).
% 1.71/1.38
% 1.71/1.38 thf(dsetconstrI_type,type,
% 1.71/1.38 dsetconstrI: $o ).
% 1.71/1.38
% 1.71/1.38 thf(dsetconstrI,definition,
% 1.71/1.38 ( dsetconstrI
% 1.71/1.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.71/1.38 ( ( in @ Xx @ A )
% 1.71/1.38 => ( ( Xphi @ Xx )
% 1.71/1.38 => ( in @ Xx
% 1.71/1.38 @ ( dsetconstr @ A
% 1.71/1.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(dsetconstrEL_type,type,
% 1.71/1.38 dsetconstrEL: $o ).
% 1.71/1.38
% 1.71/1.38 thf(dsetconstrEL,definition,
% 1.71/1.38 ( dsetconstrEL
% 1.71/1.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.71/1.38 ( ( in @ Xx
% 1.71/1.38 @ ( dsetconstr @ A
% 1.71/1.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.71/1.38 => ( in @ Xx @ A ) ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(dsetconstrER_type,type,
% 1.71/1.38 dsetconstrER: $o ).
% 1.71/1.38
% 1.71/1.38 thf(dsetconstrER,definition,
% 1.71/1.38 ( dsetconstrER
% 1.71/1.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.71/1.38 ( ( in @ Xx
% 1.71/1.38 @ ( dsetconstr @ A
% 1.71/1.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.71/1.38 => ( Xphi @ Xx ) ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(exuE1_type,type,
% 1.71/1.38 exuE1: $o ).
% 1.71/1.38
% 1.71/1.38 thf(exuE1,definition,
% 1.71/1.38 ( exuE1
% 1.71/1.38 = ( ! [Xphi: $i > $o] :
% 1.71/1.38 ( ( exu
% 1.71/1.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.71/1.38 => ? [Xx: $i] :
% 1.71/1.38 ( ( Xphi @ Xx )
% 1.71/1.38 & ! [Xy: $i] :
% 1.71/1.38 ( ( Xphi @ Xy )
% 1.71/1.38 => ( Xx = Xy ) ) ) ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(prop2set_type,type,
% 1.71/1.38 prop2set: $o > $i ).
% 1.71/1.38
% 1.71/1.38 thf(prop2set,definition,
% 1.71/1.38 ( prop2set
% 1.71/1.38 = ( ^ [Xphi: $o] :
% 1.71/1.38 ( dsetconstr @ ( powerset @ emptyset )
% 1.71/1.38 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(prop2setE_type,type,
% 1.71/1.38 prop2setE: $o ).
% 1.71/1.38
% 1.71/1.38 thf(prop2setE,definition,
% 1.71/1.38 ( prop2setE
% 1.71/1.38 = ( ! [Xphi: $o,Xx: $i] :
% 1.71/1.38 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 1.71/1.38 => Xphi ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(emptysetE_type,type,
% 1.71/1.38 emptysetE: $o ).
% 1.71/1.38
% 1.71/1.38 thf(emptysetE,definition,
% 1.71/1.38 ( emptysetE
% 1.71/1.38 = ( ! [Xx: $i] :
% 1.71/1.38 ( ( in @ Xx @ emptyset )
% 1.71/1.38 => ! [Xphi: $o] : Xphi ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(emptysetimpfalse_type,type,
% 1.71/1.38 emptysetimpfalse: $o ).
% 1.71/1.38
% 1.71/1.38 thf(emptysetimpfalse,definition,
% 1.71/1.38 ( emptysetimpfalse
% 1.71/1.38 = ( ! [Xx: $i] :
% 1.71/1.38 ( ( in @ Xx @ emptyset )
% 1.71/1.38 => $false ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(notinemptyset_type,type,
% 1.71/1.38 notinemptyset: $o ).
% 1.71/1.38
% 1.71/1.38 thf(notinemptyset,definition,
% 1.71/1.38 ( notinemptyset
% 1.71/1.38 = ( ! [Xx: $i] :
% 1.71/1.38 ~ ( in @ Xx @ emptyset ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(exuE3e_type,type,
% 1.71/1.38 exuE3e: $o ).
% 1.71/1.38
% 1.71/1.38 thf(exuE3e,definition,
% 1.71/1.38 ( exuE3e
% 1.71/1.38 = ( ! [Xphi: $i > $o] :
% 1.71/1.38 ( ( exu
% 1.71/1.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.71/1.38 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(setext_type,type,
% 1.71/1.38 setext: $o ).
% 1.71/1.38
% 1.71/1.38 thf(setext,definition,
% 1.71/1.38 ( setext
% 1.71/1.38 = ( ! [A: $i,B: $i] :
% 1.71/1.38 ( ! [Xx: $i] :
% 1.71/1.38 ( ( in @ Xx @ A )
% 1.71/1.38 => ( in @ Xx @ B ) )
% 1.71/1.38 => ( ! [Xx: $i] :
% 1.71/1.38 ( ( in @ Xx @ B )
% 1.71/1.38 => ( in @ Xx @ A ) )
% 1.71/1.38 => ( A = B ) ) ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(emptyI_type,type,
% 1.71/1.38 emptyI: $o ).
% 1.71/1.38
% 1.71/1.38 thf(emptyI,definition,
% 1.71/1.38 ( emptyI
% 1.71/1.38 = ( ! [A: $i] :
% 1.71/1.38 ( ! [Xx: $i] :
% 1.71/1.38 ~ ( in @ Xx @ A )
% 1.71/1.38 => ( A = emptyset ) ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(noeltsimpempty_type,type,
% 1.71/1.38 noeltsimpempty: $o ).
% 1.71/1.38
% 1.71/1.38 thf(noeltsimpempty,definition,
% 1.71/1.38 ( noeltsimpempty
% 1.71/1.38 = ( ! [A: $i] :
% 1.71/1.38 ( ! [Xx: $i] :
% 1.71/1.38 ~ ( in @ Xx @ A )
% 1.71/1.38 => ( A = emptyset ) ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(setbeta_type,type,
% 1.71/1.38 setbeta: $o ).
% 1.71/1.38
% 1.71/1.38 thf(setbeta,definition,
% 1.71/1.38 ( setbeta
% 1.71/1.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.71/1.38 ( ( in @ Xx @ A )
% 1.71/1.38 => ( ( in @ Xx
% 1.71/1.38 @ ( dsetconstr @ A
% 1.71/1.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.71/1.38 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(nonempty_type,type,
% 1.71/1.38 nonempty: $i > $o ).
% 1.71/1.38
% 1.71/1.38 thf(nonempty,definition,
% 1.71/1.38 ( nonempty
% 1.71/1.38 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(nonemptyE1_type,type,
% 1.71/1.38 nonemptyE1: $o ).
% 1.71/1.38
% 1.71/1.38 thf(nonemptyE1,definition,
% 1.71/1.38 ( nonemptyE1
% 1.71/1.38 = ( ! [A: $i] :
% 1.71/1.38 ( ( nonempty @ A )
% 1.71/1.38 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 1.71/1.38
% 1.71/1.38 thf(nonemptyI,conjecture,
% 1.71/1.38 ( setextAx
% 1.71/1.38 => ( emptysetAx
% 1.71/1.38 => ( setadjoinAx
% 1.71/1.38 => ( powersetAx
% 1.71/1.38 => ( setunionAx
% 1.71/1.38 => ( omega0Ax
% 1.71/1.38 => ( omegaSAx
% 1.71/1.38 => ( omegaIndAx
% 1.71/1.38 => ( replAx
% 1.71/1.38 => ( foundationAx
% 1.71/1.38 => ( wellorderingAx
% 1.71/1.38 => ( descrp
% 1.71/1.38 => ( dsetconstrI
% 1.71/1.38 => ( dsetconstrEL
% 1.71/1.38 => ( dsetconstrER
% 1.71/1.38 => ( exuE1
% 1.71/1.38 => ( prop2setE
% 1.71/1.38 => ( emptysetE
% 1.71/1.38 => ( emptysetimpfalse
% 1.71/1.38 => ( notinemptyset
% 1.71/1.38 => ( exuE3e
% 1.71/1.38 => ( setext
% 1.71/1.38 => ( emptyI
% 1.71/1.38 => ( noeltsimpempty
% 1.71/1.38 => ( setbeta
% 1.71/1.38 => ( nonemptyE1
% 1.71/1.38 => ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.71/1.38 ( ( in @ Xx @ A )
% 1.71/1.38 => ( ( Xphi @ Xx )
% 1.71/1.38 => ( nonempty
% 1.71/1.38 @ ( dsetconstr @ A
% 1.71/1.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 1.71/1.38 ------------------------
% 1.89/1.40 % (17849)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_17746 for (2999ds/275Mi)
% 1.89/1.40 % (17847)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_17746 for (2999ds/2Mi)
% 1.89/1.40 % (17845)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_17746 for (2999ds/4Mi)
% 1.89/1.40 % (17846)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_17746 for (2999ds/27Mi)
% 1.89/1.40 % (17848)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_17746 for (2999ds/2Mi)
% 1.89/1.40 % (17844)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_17746 for (2999ds/183Mi)
% 1.89/1.40 % (17850)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_17746 for (2999ds/18Mi)
% 1.89/1.41 % (17847)Instruction limit reached!
% 1.89/1.41 % (17847)------------------------------
% 1.89/1.41 % (17847)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.89/1.41 % (17847)Termination reason: Unknown
% 1.89/1.41 % (17847)Termination phase: shuffling
% 1.89/1.41 % (17848)Instruction limit reached!
% 1.89/1.41 % (17848)------------------------------
% 1.89/1.41 % (17848)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.89/1.41 % (17848)Termination reason: Unknown
% 1.89/1.41 % (17848)Termination phase: shuffling
% 1.89/1.41
% 1.89/1.41 % (17848)Memory used [KB]: 1023
% 1.89/1.41 % (17848)Time elapsed: 0.003 s
% 1.89/1.41 % (17848)Instructions burned: 2 (million)
% 1.89/1.41 % (17848)------------------------------
% 1.89/1.41 % (17848)------------------------------
% 1.89/1.41
% 1.89/1.41 % (17847)Memory used [KB]: 1023
% 1.89/1.41 % (17847)Time elapsed: 0.003 s
% 1.89/1.41 % (17847)Instructions burned: 2 (million)
% 1.89/1.41 % (17847)------------------------------
% 1.89/1.41 % (17847)------------------------------
% 1.89/1.41 % (17845)Instruction limit reached!
% 1.89/1.41 % (17845)------------------------------
% 1.89/1.41 % (17845)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.89/1.41 % (17845)Termination reason: Unknown
% 1.89/1.41 % (17845)Termination phase: shuffling
% 1.89/1.41
% 1.89/1.41 % (17845)Memory used [KB]: 1023
% 1.89/1.41 % (17845)Time elapsed: 0.004 s
% 1.89/1.41 % (17845)Instructions burned: 4 (million)
% 1.89/1.41 % (17845)------------------------------
% 1.89/1.41 % (17845)------------------------------
% 1.89/1.41 % (17850)Instruction limit reached!
% 1.89/1.41 % (17850)------------------------------
% 1.89/1.41 % (17850)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.89/1.41 % (17850)Termination reason: Unknown
% 1.89/1.41 % (17850)Termination phase: Saturation
% 1.89/1.41
% 1.89/1.41 % (17850)Memory used [KB]: 1407
% 1.89/1.41 % (17850)Time elapsed: 0.011 s
% 1.89/1.41 % (17850)Instructions burned: 19 (million)
% 1.89/1.41 % (17850)------------------------------
% 1.89/1.41 % (17850)------------------------------
% 1.89/1.42 % (17846)Instruction limit reached!
% 1.89/1.42 % (17846)------------------------------
% 1.89/1.42 % (17846)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.89/1.42 % (17846)Termination reason: Unknown
% 1.89/1.42 % (17846)Termination phase: Saturation
% 1.89/1.42
% 1.89/1.42 % (17846)Memory used [KB]: 5756
% 1.89/1.42 % (17846)Time elapsed: 0.015 s
% 1.89/1.42 % (17846)Instructions burned: 28 (million)
% 1.89/1.42 % (17846)------------------------------
% 1.89/1.42 % (17846)------------------------------
% 1.89/1.42 % (17851)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_17746 for (2999ds/3Mi)
% 1.89/1.42 % (17852)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_17746 for (2999ds/37Mi)
% 1.89/1.42 % (17853)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_17746 for (2999ds/15Mi)
% 1.89/1.42 % (17851)Instruction limit reached!
% 1.89/1.42 % (17851)------------------------------
% 1.89/1.42 % (17851)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.89/1.42 % (17851)Termination reason: Unknown
% 1.89/1.42 % (17851)Termination phase: shuffling
% 1.89/1.42
% 1.89/1.42 % (17851)Memory used [KB]: 1023
% 1.89/1.42 % (17851)Time elapsed: 0.003 s
% 1.89/1.42 % (17851)Instructions burned: 3 (million)
% 1.89/1.42 % (17851)------------------------------
% 1.89/1.42 % (17851)------------------------------
% 1.89/1.43 % (17854)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_17746 for (2999ds/3Mi)
% 1.89/1.43 % (17853)Instruction limit reached!
% 1.89/1.43 % (17853)------------------------------
% 1.89/1.43 % (17853)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.89/1.43 % (17853)Termination reason: Unknown
% 1.89/1.43 % (17853)Termination phase: Property scanning
% 1.89/1.43
% 1.89/1.43 % (17853)Memory used [KB]: 1407
% 1.89/1.43 % (17853)Time elapsed: 0.009 s
% 1.89/1.43 % (17853)Instructions burned: 16 (million)
% 1.89/1.43 % (17853)------------------------------
% 1.89/1.43 % (17853)------------------------------
% 1.89/1.43 % (17854)Instruction limit reached!
% 1.89/1.43 % (17854)------------------------------
% 1.89/1.43 % (17854)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.89/1.43 % (17854)Termination reason: Unknown
% 1.89/1.43 % (17854)Termination phase: shuffling
% 1.89/1.43
% 1.89/1.43 % (17854)Memory used [KB]: 1023
% 1.89/1.43 % (17854)Time elapsed: 0.003 s
% 1.89/1.43 % (17854)Instructions burned: 4 (million)
% 1.89/1.43 % (17854)------------------------------
% 1.89/1.43 % (17854)------------------------------
% 1.89/1.43 % (17855)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_17746 for (2999ds/1041Mi)
% 1.89/1.44 % (17852)Instruction limit reached!
% 1.89/1.44 % (17852)------------------------------
% 1.89/1.44 % (17852)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.89/1.44 % (17852)Termination reason: Unknown
% 1.89/1.44 % (17852)Termination phase: Saturation
% 1.89/1.44
% 1.89/1.44 % (17852)Memory used [KB]: 5756
% 1.89/1.44 % (17852)Time elapsed: 0.018 s
% 1.89/1.44 % (17852)Instructions burned: 38 (million)
% 1.89/1.44 % (17852)------------------------------
% 1.89/1.44 % (17852)------------------------------
% 1.89/1.44 % (17856)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_17746 for (2999ds/7Mi)
% 1.89/1.44 % (17856)Instruction limit reached!
% 1.89/1.44 % (17856)------------------------------
% 1.89/1.44 % (17856)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.89/1.44 % (17856)Termination reason: Unknown
% 1.89/1.44 % (17856)Termination phase: Property scanning
% 1.89/1.44
% 1.89/1.44 % (17856)Memory used [KB]: 1151
% 1.89/1.44 % (17856)Time elapsed: 0.005 s
% 1.89/1.44 % (17856)Instructions burned: 8 (million)
% 1.89/1.44 % (17856)------------------------------
% 1.89/1.44 % (17856)------------------------------
% 1.89/1.45 % (17857)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_17746 for (2999ds/16Mi)
% 1.89/1.45 % (17858)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_17746 for (2999ds/3Mi)
% 1.89/1.45 % (17858)Instruction limit reached!
% 1.89/1.45 % (17858)------------------------------
% 1.89/1.45 % (17858)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.89/1.45 % (17858)Termination reason: Unknown
% 1.89/1.45 % (17858)Termination phase: shuffling
% 1.89/1.45
% 1.89/1.45 % (17858)Memory used [KB]: 1023
% 1.89/1.45 % (17858)Time elapsed: 0.003 s
% 1.89/1.45 % (17858)Instructions burned: 4 (million)
% 1.89/1.45 % (17858)------------------------------
% 1.89/1.45 % (17858)------------------------------
% 1.89/1.45 % (17857)Instruction limit reached!
% 1.89/1.45 % (17857)------------------------------
% 1.89/1.45 % (17857)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.89/1.45 % (17857)Termination reason: Unknown
% 1.89/1.45 % (17857)Termination phase: Property scanning
% 1.89/1.45
% 1.89/1.45 % (17857)Memory used [KB]: 1279
% 1.89/1.45 % (17857)Time elapsed: 0.009 s
% 1.89/1.45 % (17857)Instructions burned: 17 (million)
% 1.89/1.45 % (17857)------------------------------
% 1.89/1.45 % (17857)------------------------------
% 1.89/1.45 % (17859)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_17746 for (2999ds/3Mi)
% 1.89/1.46 % (17859)Instruction limit reached!
% 1.89/1.46 % (17859)------------------------------
% 1.89/1.46 % (17859)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.89/1.46 % (17859)Termination reason: Unknown
% 1.89/1.46 % (17859)Termination phase: shuffling
% 1.89/1.46
% 1.89/1.46 % (17859)Memory used [KB]: 1023
% 1.89/1.46 % (17859)Time elapsed: 0.003 s
% 1.89/1.46 % (17859)Instructions burned: 4 (million)
% 1.89/1.46 % (17859)------------------------------
% 1.89/1.46 % (17859)------------------------------
% 1.89/1.46 % (17860)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_17746 for (2999ds/7Mi)
% 1.89/1.46 % (17860)Instruction limit reached!
% 1.89/1.46 % (17860)------------------------------
% 1.89/1.46 % (17860)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.89/1.46 % (17860)Termination reason: Unknown
% 1.89/1.46 % (17860)Termination phase: Property scanning
% 1.89/1.46
% 1.89/1.46 % (17860)Memory used [KB]: 1151
% 1.89/1.46 % (17860)Time elapsed: 0.005 s
% 1.89/1.46 % (17860)Instructions burned: 8 (million)
% 1.89/1.46 % (17860)------------------------------
% 1.89/1.46 % (17860)------------------------------
% 1.89/1.46 % (17861)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_17746 for (2999ds/3Mi)
% 1.89/1.47 % (17861)Instruction limit reached!
% 1.89/1.47 % (17861)------------------------------
% 1.89/1.47 % (17861)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.89/1.47 % (17861)Termination reason: Unknown
% 1.89/1.47 % (17861)Termination phase: shuffling
% 1.89/1.47
% 1.89/1.47 % (17861)Memory used [KB]: 1023
% 1.89/1.47 % (17861)Time elapsed: 0.003 s
% 1.89/1.47 % (17861)Instructions burned: 4 (million)
% 1.89/1.47 % (17861)------------------------------
% 1.89/1.47 % (17861)------------------------------
% 1.89/1.47 % (17862)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_17746 for (2999ds/4Mi)
% 1.89/1.47 % (17862)Instruction limit reached!
% 1.89/1.47 % (17862)------------------------------
% 1.89/1.47 % (17862)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.89/1.47 % (17862)Termination reason: Unknown
% 1.89/1.47 % (17862)Termination phase: shuffling
% 1.89/1.47
% 1.89/1.47 % (17862)Memory used [KB]: 1023
% 1.89/1.47 % (17862)Time elapsed: 0.003 s
% 1.89/1.47 % (17862)Instructions burned: 4 (million)
% 1.89/1.47 % (17862)------------------------------
% 1.89/1.47 % (17862)------------------------------
% 1.89/1.47 % (17863)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_17746 for (2999ds/18Mi)
% 1.89/1.48 % (17864)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_17746 for (2999ds/710Mi)
% 1.89/1.48 % (17863)Instruction limit reached!
% 1.89/1.48 % (17863)------------------------------
% 1.89/1.48 % (17863)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.89/1.48 % (17863)Termination reason: Unknown
% 1.89/1.48 % (17863)Termination phase: Saturation
% 1.89/1.48
% 1.89/1.48 % (17863)Memory used [KB]: 5628
% 1.89/1.48 % (17863)Time elapsed: 0.010 s
% 1.89/1.48 % (17863)Instructions burned: 19 (million)
% 1.89/1.48 % (17863)------------------------------
% 1.89/1.48 % (17863)------------------------------
% 1.89/1.48 % (17865)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_17746 for (2999ds/6Mi)
% 1.89/1.49 % (17865)Instruction limit reached!
% 1.89/1.49 % (17865)------------------------------
% 1.89/1.49 % (17865)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.89/1.49 % (17865)Termination reason: Unknown
% 1.89/1.49 % (17865)Termination phase: Property scanning
% 1.89/1.49
% 1.89/1.49 % (17865)Memory used [KB]: 1151
% 1.89/1.49 % (17865)Time elapsed: 0.004 s
% 1.89/1.49 % (17865)Instructions burned: 6 (million)
% 1.89/1.49 % (17865)------------------------------
% 1.89/1.49 % (17865)------------------------------
% 1.89/1.49 % (17866)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_17746 for (2999ds/902Mi)
% 1.89/1.49 % (17844)First to succeed.
% 1.89/1.49 % (17844)Refutation found. Thanks to Tanya!
% 1.89/1.49 % SZS status Theorem for DTF2THF_17746
% 1.89/1.49 % SZS output start Proof for DTF2THF_17746
% 1.89/1.49 thf(func_def_0, type, in: $i > $i > $o).
% 1.89/1.49 thf(func_def_1, type, exu: ($i > $o) > $o).
% 1.89/1.49 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 1.89/1.49 thf(func_def_8, type, powerset: $i > $i).
% 1.89/1.49 thf(func_def_10, type, setunion: $i > $i).
% 1.89/1.49 thf(func_def_19, type, descr: ($i > $o) > $i).
% 1.89/1.49 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 1.89/1.49 thf(func_def_26, type, prop2set: $o > $i).
% 1.89/1.49 thf(func_def_36, type, nonempty: $i > $o).
% 1.89/1.49 thf(func_def_55, type, sK1: $i > $o).
% 1.89/1.49 thf(func_def_58, type, sK4: $i > $i > $i).
% 1.89/1.49 thf(func_def_60, type, sK5: ($i > $i > $o) > $i > $i).
% 1.89/1.49 thf(func_def_61, type, sK6: ($i > $i > $o) > $i > $i).
% 1.89/1.49 thf(func_def_62, type, sK7: $i > ($i > $i > $o) > $i > $i).
% 1.89/1.49 thf(func_def_63, type, sK8: $i > ($i > $i > $o) > $i > $i).
% 1.89/1.49 thf(func_def_64, type, sK9: $i > ($i > $i > $o) > $i > $i).
% 1.89/1.49 thf(func_def_65, type, sK10: $i > $i > $i).
% 1.89/1.49 thf(func_def_66, type, sK11: ($i > $o) > $i > $i).
% 1.89/1.49 thf(func_def_67, type, sK12: $i > $i).
% 1.89/1.49 thf(func_def_68, type, sK13: $i > $i).
% 1.89/1.49 thf(func_def_69, type, sK14: $i > $i).
% 1.89/1.49 thf(func_def_70, type, sK15: $i > $i).
% 1.89/1.49 thf(func_def_71, type, sK16: $i > $i).
% 1.89/1.49 thf(func_def_72, type, sK17: $i > $i > $i > $i).
% 1.89/1.49 thf(func_def_73, type, sK18: $i > $i > $i).
% 1.89/1.49 thf(func_def_74, type, sK19: $i > $i > $i).
% 1.89/1.49 thf(func_def_75, type, sK20: $i > $i > $i).
% 1.89/1.49 thf(func_def_76, type, sK21: $i > $i > $i).
% 1.89/1.49 thf(func_def_77, type, sK22: $i > $i > $i).
% 1.89/1.49 thf(func_def_78, type, sK23: $i > $i > $i).
% 1.89/1.49 thf(func_def_79, type, sK24: ($i > $o) > $i).
% 1.89/1.49 thf(func_def_80, type, sK25: ($i > $o) > $i > $i).
% 1.89/1.49 thf(func_def_81, type, sK26: $i > ($i > $o) > $i).
% 1.89/1.49 thf(func_def_82, type, sK27: $i > $i > $i).
% 1.89/1.49 thf(func_def_83, type, sK28: ($i > $o) > $i).
% 1.89/1.49 thf(func_def_84, type, sK29: ($i > $o) > $i > $i).
% 1.89/1.49 thf(func_def_85, type, sK30: $i > $i).
% 1.89/1.49 thf(func_def_86, type, ph31: !>[X0: $tType]:(X0)).
% 1.89/1.49 thf(func_def_87, type, sK32: $i > $i > $i).
% 1.89/1.49 thf(f1138,plain,(
% 1.89/1.49 $false),
% 1.89/1.49 inference(avatar_sat_refutation,[status(thm)],[f1054,f1099,f1137])).
% 1.89/1.49 thf(f1137,plain,(
% 1.89/1.49 ~spl3_1),
% 1.89/1.49 inference(avatar_contradiction_clause,[status(thm)],[f1136])).
% 1.89/1.49 thf(f1136,plain,(
% 1.89/1.49 $false | ~spl3_1),
% 1.89/1.49 inference(trivial_inequality_removal,[status(thm)],[f1102])).
% 1.89/1.49 thf(f1102,plain,(
% 1.89/1.49 ($false = $true) | ~spl3_1),
% 1.89/1.49 inference(superposition,[status(thm)],[f1005,f133])).
% 1.89/1.49 thf(f133,plain,(
% 1.89/1.49 ((sK1 @ sK0) = $true)),
% 1.89/1.49 inference(cnf_transformation,[status(thm)],[f93])).
% 1.89/1.49 thf(f93,plain,(
% 1.89/1.49 (noeltsimpempty = $true) & (notinemptyset = $true) & (emptysetE = $true) & (setext = $true) & (foundationAx = $true) & (((nonempty @ (dsetconstr @ sK2 @ (^[Y0 : $i]: (sK1 @ Y0)))) != $true) & ((sK1 @ sK0) = $true) & ((in @ sK0 @ sK2) = $true)) & (emptysetimpfalse = $true) & (setadjoinAx = $true) & (setbeta = $true) & (exuE1 = $true) & (dsetconstrER = $true) & (dsetconstrEL = $true) & (prop2setE = $true) & (setunionAx = $true) & (omega0Ax = $true) & (emptysetAx = $true) & (dsetconstrI = $true) & (setextAx = $true) & (powersetAx = $true) & (descrp = $true) & (replAx = $true) & (omegaSAx = $true) & (exuE3e = $true) & (emptyI = $true) & (nonemptyE1 = $true) & (omegaIndAx = $true) & (wellorderingAx = $true)),
% 1.89/1.49 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f91,f92])).
% 1.89/1.49 thf(f92,plain,(
% 1.89/1.49 ? [X0 : $i,X1 : $i > $o,X2 : $i] : (((nonempty @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X1 @ Y0)))) != $true) & ((X1 @ X0) = $true) & ((in @ X0 @ X2) = $true)) => (((nonempty @ (dsetconstr @ sK2 @ (^[Y0 : $i]: (sK1 @ Y0)))) != $true) & ((sK1 @ sK0) = $true) & ((in @ sK0 @ sK2) = $true))),
% 1.89/1.49 introduced(definition,[],[choice_axiom])).
% 1.89/1.49 thf(f91,plain,(
% 1.89/1.49 (noeltsimpempty = $true) & (notinemptyset = $true) & (emptysetE = $true) & (setext = $true) & (foundationAx = $true) & ? [X0 : $i,X1 : $i > $o,X2 : $i] : (((nonempty @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X1 @ Y0)))) != $true) & ((X1 @ X0) = $true) & ((in @ X0 @ X2) = $true)) & (emptysetimpfalse = $true) & (setadjoinAx = $true) & (setbeta = $true) & (exuE1 = $true) & (dsetconstrER = $true) & (dsetconstrEL = $true) & (prop2setE = $true) & (setunionAx = $true) & (omega0Ax = $true) & (emptysetAx = $true) & (dsetconstrI = $true) & (setextAx = $true) & (powersetAx = $true) & (descrp = $true) & (replAx = $true) & (omegaSAx = $true) & (exuE3e = $true) & (emptyI = $true) & (nonemptyE1 = $true) & (omegaIndAx = $true) & (wellorderingAx = $true)),
% 1.89/1.49 inference(flattening,[status(thm)],[f90])).
% 1.89/1.49 thf(f90,plain,(
% 1.89/1.49 (((((((((((((((((((((((((? [X0 : $i,X1 : $i > $o,X2 : $i] : ((((nonempty @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X1 @ Y0)))) != $true) & ((X1 @ X0) = $true)) & ((in @ X0 @ X2) = $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)),
% 1.89/1.49 inference(ennf_transformation,[status(thm)],[f71])).
% 1.89/1.49 thf(f71,plain,(
% 1.89/1.49 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ! [X0 : $i,X1 : $i > $o,X2 : $i] : (((in @ X0 @ X2) = $true) => (((X1 @ X0) = $true) => ((nonempty @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X1 @ Y0)))) = $true)))))))))))))))))))))))))))))),
% 1.89/1.49 inference(fool_elimination,[status(thm)],[f70])).
% 1.89/1.49 thf(f70,plain,(
% 1.89/1.49 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => ! [X0 : $i,X1 : $i > $o,X2 : $i] : ((in @ X0 @ X2) => ((X1 @ X0) => (nonempty @ (dsetconstr @ X2 @ (^[X3 : $i] : (X1 @ X3))))))))))))))))))))))))))))))))),
% 1.89/1.49 inference(rectify,[status(thm)],[f31])).
% 1.89/1.49 thf(f31,negated_conjecture,(
% 1.89/1.49 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => ! [X1 : $i,X0 : $i > $o,X3 : $i] : ((in @ X1 @ X3) => ((X0 @ X1) => (nonempty @ (dsetconstr @ X3 @ (^[X2 : $i] : (X0 @ X2))))))))))))))))))))))))))))))))),
% 1.89/1.49 inference(negated_conjecture,[status(cth)],[f30])).
% 1.89/1.49 thf(f30,conjecture,(
% 1.89/1.49 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => ! [X1 : $i,X0 : $i > $o,X3 : $i] : ((in @ X1 @ X3) => ((X0 @ X1) => (nonempty @ (dsetconstr @ X3 @ (^[X2 : $i] : (X0 @ X2)))))))))))))))))))))))))))))))),
% 1.89/1.49 file('/export/starexec/sandbox2/tmp/tmp.vEp0jiah3y/DTF2THF_17746.p',nonemptyI)).
% 1.89/1.49 thf(f1005,plain,(
% 1.89/1.49 ((sK1 @ sK0) = $false) | ~spl3_1),
% 1.89/1.49 inference(avatar_component_clause,[status(thm)],[f1003])).
% 1.89/1.49 thf(f1003,definition,(
% 1.89/1.49 spl3_1 <=> ((sK1 @ sK0) = $false)),
% 1.89/1.49 introduced(definition,[new_symbols(naming,[spl3_1])],[avatar_definition])).
% 1.89/1.49 thf(f1099,plain,(
% 1.89/1.49 ~spl3_2),
% 1.89/1.49 inference(avatar_contradiction_clause,[status(thm)],[f1098])).
% 1.89/1.49 thf(f1098,plain,(
% 1.89/1.49 $false | ~spl3_2),
% 1.89/1.49 inference(trivial_inequality_removal,[status(thm)],[f1090])).
% 1.89/1.49 thf(f1090,plain,(
% 1.89/1.49 ($false = $true) | ~spl3_2),
% 1.89/1.49 inference(superposition,[status(thm)],[f1009,f558])).
% 1.89/1.49 thf(f558,plain,(
% 1.89/1.49 ( ! [X1 : $i] : (((in @ X1 @ emptyset) = $false)) )),
% 1.89/1.49 inference(not_proxy_clausification,[status(thm)],[f557])).
% 1.89/1.49 thf(f557,plain,(
% 1.89/1.49 ( ! [X1 : $i] : (((~ (in @ X1 @ emptyset)) = $true)) )),
% 1.89/1.49 inference(beta_eta_normalization,[status(thm)],[f556])).
% 1.89/1.49 thf(f556,plain,(
% 1.89/1.49 ( ! [X1 : $i] : ((((^[Y0 : $i]: (~ (in @ Y0 @ emptyset))) @ X1) = $true)) )),
% 1.89/1.49 inference(pi_clausification,[status(thm)],[f158])).
% 1.89/1.49 thf(f158,plain,(
% 1.89/1.49 ($true = (!! @ $i @ (^[Y0 : $i]: (~ (in @ Y0 @ emptyset)))))),
% 1.89/1.49 inference(definition_unfolding,[status(thm)],[f138,f100])).
% 1.89/1.49 thf(f100,plain,(
% 1.89/1.49 (notinemptyset = (!! @ $i @ (^[Y0 : $i]: (~ (in @ Y0 @ emptyset)))))),
% 1.89/1.49 inference(cnf_transformation,[status(thm)],[f56])).
% 1.89/1.49 thf(f56,plain,(
% 1.89/1.49 (notinemptyset = (!! @ $i @ (^[Y0 : $i]: (~ (in @ Y0 @ emptyset)))))),
% 1.89/1.49 inference(fool_elimination,[status(thm)],[f55])).
% 1.89/1.49 thf(f55,plain,(
% 1.89/1.49 (notinemptyset = ! [X0 : $i] : ~(in @ X0 @ emptyset))),
% 1.89/1.49 inference(rectify,[status(thm)],[f22])).
% 1.89/1.49 thf(f22,axiom,(
% 1.89/1.49 (notinemptyset = ! [X1 : $i] : ~(in @ X1 @ emptyset))),
% 1.89/1.49 file('/export/starexec/sandbox2/tmp/tmp.vEp0jiah3y/DTF2THF_17746.p',notinemptyset)).
% 1.89/1.49 thf(f138,plain,(
% 1.89/1.49 (notinemptyset = $true)),
% 1.89/1.49 inference(cnf_transformation,[status(thm)],[f93])).
% 1.89/1.49 thf(f1009,plain,(
% 1.89/1.49 ((in @ sK0 @ emptyset) = $true) | ~spl3_2),
% 1.89/1.49 inference(avatar_component_clause,[status(thm)],[f1007])).
% 1.89/1.49 thf(f1007,definition,(
% 1.89/1.49 spl3_2 <=> ((in @ sK0 @ emptyset) = $true)),
% 1.89/1.49 introduced(definition,[new_symbols(naming,[spl3_2])],[avatar_definition])).
% 1.89/1.49 thf(f1054,plain,(
% 1.89/1.49 spl3_2 | spl3_1),
% 1.89/1.49 inference(avatar_split_clause,[status(thm)],[f983,f1003,f1007])).
% 1.89/1.49 thf(f983,plain,(
% 1.89/1.49 ((in @ sK0 @ emptyset) = $true) | ((sK1 @ sK0) = $false)),
% 1.89/1.49 inference(trivial_inequality_removal,[status(thm)],[f952])).
% 1.89/1.49 thf(f952,plain,(
% 1.89/1.49 ($false = $true) | ((sK1 @ sK0) = $false) | ((in @ sK0 @ emptyset) = $true)),
% 1.89/1.49 inference(superposition,[status(thm)],[f697,f132])).
% 1.89/1.49 thf(f132,plain,(
% 1.89/1.49 ((in @ sK0 @ sK2) = $true)),
% 1.89/1.49 inference(cnf_transformation,[status(thm)],[f93])).
% 1.89/1.49 thf(f697,plain,(
% 1.89/1.49 ( ! [X0 : $i] : (((in @ X0 @ sK2) = $false) | ((in @ X0 @ emptyset) = $true) | ((sK1 @ X0) = $false)) )),
% 1.89/1.49 inference(superposition,[status(thm)],[f538,f507])).
% 1.89/1.49 thf(f507,plain,(
% 1.89/1.49 (emptyset = (dsetconstr @ sK2 @ sK1))),
% 1.89/1.49 inference(equality_proxy_clausification,[status(thm)],[f506])).
% 1.89/1.49 thf(f506,plain,(
% 1.89/1.49 ((emptyset = (dsetconstr @ sK2 @ sK1)) = $true)),
% 1.89/1.49 inference(not_proxy_clausification,[status(thm)],[f505])).
% 1.89/1.49 thf(f505,plain,(
% 1.89/1.49 ((~ (emptyset = (dsetconstr @ sK2 @ sK1))) != $true)),
% 1.89/1.49 inference(beta_eta_normalization,[status(thm)],[f159])).
% 1.89/1.49 thf(f159,plain,(
% 1.89/1.49 (((^[Y0 : $i]: (~ (emptyset = Y0))) @ (dsetconstr @ sK2 @ (^[Y0 : $i]: (sK1 @ Y0)))) != $true)),
% 1.89/1.49 inference(definition_unfolding,[status(thm)],[f134,f99])).
% 1.89/1.49 thf(f99,plain,(
% 1.89/1.49 (nonempty = (^[Y0 : $i]: (~ (emptyset = Y0))))),
% 1.89/1.49 inference(cnf_transformation,[status(thm)],[f40])).
% 1.89/1.49 thf(f40,plain,(
% 1.89/1.49 (nonempty = (^[Y0 : $i]: (~ (emptyset = Y0))))),
% 1.89/1.49 inference(fool_elimination,[status(thm)],[f39])).
% 1.89/1.49 thf(f39,plain,(
% 1.89/1.49 ((^[X0 : $i] : ((emptyset != X0))) = nonempty)),
% 1.89/1.49 inference(rectify,[status(thm)],[f28])).
% 1.89/1.49 thf(f28,axiom,(
% 1.89/1.49 ((^[X1 : $i] : ((emptyset != X1))) = nonempty)),
% 1.89/1.49 file('/export/starexec/sandbox2/tmp/tmp.vEp0jiah3y/DTF2THF_17746.p',nonempty)).
% 1.89/1.49 thf(f134,plain,(
% 1.89/1.49 ((nonempty @ (dsetconstr @ sK2 @ (^[Y0 : $i]: (sK1 @ Y0)))) != $true)),
% 1.89/1.49 inference(cnf_transformation,[status(thm)],[f93])).
% 1.89/1.49 thf(f538,plain,(
% 1.89/1.49 ( ! [X2 : $i,X3 : $i > $o,X1 : $i] : (((in @ X2 @ (dsetconstr @ X1 @ X3)) = $true) | ((X3 @ X2) = $false) | ((in @ X2 @ X1) = $false)) )),
% 1.89/1.49 inference(binary_proxy_clausification,[status(thm)],[f537])).
% 1.89/1.49 thf(f537,plain,(
% 1.89/1.49 ( ! [X2 : $i,X3 : $i > $o,X1 : $i] : (((in @ X2 @ X1) = $false) | (((X3 @ X2) => (in @ X2 @ (dsetconstr @ X1 @ X3))) = $true)) )),
% 1.89/1.49 inference(binary_proxy_clausification,[status(thm)],[f536])).
% 1.89/1.49 thf(f536,plain,(
% 1.89/1.49 ( ! [X2 : $i,X3 : $i > $o,X1 : $i] : ((((in @ X2 @ X1) => ((X3 @ X2) => (in @ X2 @ (dsetconstr @ X1 @ X3)))) = $true)) )),
% 1.89/1.49 inference(beta_eta_normalization,[status(thm)],[f535])).
% 1.89/1.49 thf(f535,plain,(
% 1.89/1.49 ( ! [X2 : $i,X3 : $i > $o,X1 : $i] : ((((^[Y0 : $i > $o]: ((in @ X2 @ X1) => ((Y0 @ X2) => (in @ X2 @ (dsetconstr @ X1 @ Y0))))) @ X3) = $true)) )),
% 1.89/1.49 inference(pi_clausification,[status(thm)],[f534])).
% 1.89/1.49 thf(f534,plain,(
% 1.89/1.49 ( ! [X2 : $i,X1 : $i] : (((!! @ ($i > $o) @ (^[Y0 : $i > $o]: ((in @ X2 @ X1) => ((Y0 @ X2) => (in @ X2 @ (dsetconstr @ X1 @ Y0)))))) = $true)) )),
% 1.89/1.49 inference(beta_eta_normalization,[status(thm)],[f533])).
% 1.89/1.49 thf(f533,plain,(
% 1.89/1.49 ( ! [X2 : $i,X1 : $i] : ((((^[Y0 : $i]: (!! @ ($i > $o) @ (^[Y1 : $i > $o]: ((in @ Y0 @ X1) => ((Y1 @ Y0) => (in @ Y0 @ (dsetconstr @ X1 @ Y1))))))) @ X2) = $true)) )),
% 1.89/1.49 inference(pi_clausification,[status(thm)],[f532])).
% 1.89/1.49 thf(f532,plain,(
% 1.89/1.49 ( ! [X1 : $i] : (((!! @ $i @ (^[Y0 : $i]: (!! @ ($i > $o) @ (^[Y1 : $i > $o]: ((in @ Y0 @ X1) => ((Y1 @ Y0) => (in @ Y0 @ (dsetconstr @ X1 @ Y1)))))))) = $true)) )),
% 1.89/1.49 inference(beta_eta_normalization,[status(thm)],[f531])).
% 1.89/1.49 thf(f531,plain,(
% 1.89/1.49 ( ! [X1 : $i] : ((((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ ($i > $o) @ (^[Y2 : $i > $o]: ((in @ Y1 @ Y0) => ((Y2 @ Y1) => (in @ Y1 @ (dsetconstr @ Y0 @ Y2))))))))) @ X1) = $true)) )),
% 1.89/1.49 inference(pi_clausification,[status(thm)],[f530])).
% 1.89/1.49 thf(f530,plain,(
% 1.89/1.49 ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ ($i > $o) @ (^[Y2 : $i > $o]: ((in @ Y1 @ Y0) => ((Y2 @ Y1) => (in @ Y1 @ (dsetconstr @ Y0 @ Y2)))))))))) = $true)),
% 1.89/1.49 inference(beta_eta_normalization,[status(thm)],[f175])).
% 1.89/1.49 thf(f175,plain,(
% 1.89/1.49 ((!! @ $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)),
% 1.89/1.49 inference(definition_unfolding,[status(thm)],[f143,f121])).
% 1.89/1.49 thf(f121,plain,(
% 1.89/1.49 (dsetconstrI = $true)),
% 1.89/1.49 inference(cnf_transformation,[status(thm)],[f93])).
% 1.89/1.49 thf(f143,plain,(
% 1.89/1.49 (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)))))))))))))),
% 1.89/1.49 inference(cnf_transformation,[status(thm)],[f38])).
% 1.89/1.49 thf(f38,plain,(
% 1.89/1.49 (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)))))))))))))),
% 1.89/1.49 inference(fool_elimination,[status(thm)],[f37])).
% 1.89/1.49 thf(f37,plain,(
% 1.89/1.49 (! [X0 : $i > $o,X1 : $i,X2 : $i] : ((in @ X1 @ X2) => ((X0 @ X1) => (in @ X1 @ (dsetconstr @ X2 @ (^[X3 : $i] : (X0 @ X3)))))) = dsetconstrI)),
% 1.89/1.49 inference(rectify,[status(thm)],[f14])).
% 1.89/1.49 thf(f14,axiom,(
% 1.89/1.49 (! [X0 : $i > $o,X1 : $i,X3 : $i] : ((in @ X1 @ X3) => ((X0 @ X1) => (in @ X1 @ (dsetconstr @ X3 @ (^[X2 : $i] : (X0 @ X2)))))) = dsetconstrI)),
% 1.89/1.49 file('/export/starexec/sandbox2/tmp/tmp.vEp0jiah3y/DTF2THF_17746.p',dsetconstrI)).
% 1.89/1.49 % SZS output end Proof for DTF2THF_17746
% 1.89/1.49 % (17844)------------------------------
% 1.89/1.49 % (17844)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.89/1.49 % (17844)Termination reason: Refutation
% 1.89/1.49
% 1.89/1.49 % (17844)Memory used [KB]: 6908
% 1.89/1.49 % (17844)Time elapsed: 0.087 s
% 1.89/1.49 % (17844)Instructions burned: 169 (million)
% 1.89/1.49 % (17844)------------------------------
% 1.89/1.49 % (17844)------------------------------
% 1.89/1.49 % (17843)Success in time 0.088 s
%------------------------------------------------------------------------------