%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU513^1 : 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 : n011.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 09:35:01 AM UTC 2026
% Result : Theorem 1.95s 1.38s
% Output : Refutation 1.95s
% Verified :
% SZS Type : Refutation
% Derivation depth : 9
% Number of leaves : 3
% Syntax : Number of formulae : 19 ( 6 unt; 0 typ; 0 def)
% Number of atoms : 463 ( 115 equ; 0 cnn)
% Maximal formula atoms : 33 ( 24 avg)
% Number of connectives : 287 ( 16 ~; 8 |; 61 &; 84 @)
% ( 1 <=>; 117 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 12 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 93 ( 90 usr; 34 con; 0-3 aty)
% Number of variables : 24 ( 0 ^; 18 !; 6 ?; 24 :)
% 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_54,type,
sP1: $i > $i > $o ).
thf(func_def_55,type,
sP2: $i > $o ).
thf(func_def_56,type,
sP3: $i > $i > $o ).
thf(func_def_57,type,
sP4: $i > $i > $o ).
thf(func_def_62,type,
sK9: $i > $i > $i ).
thf(func_def_63,type,
sK10: $i > $i > $i ).
thf(func_def_65,type,
sK12: $i > $o ).
thf(func_def_70,type,
sK17: $i > $o ).
thf(func_def_71,type,
sK18: $i > $i ).
thf(func_def_73,type,
sK20: $i > $o ).
thf(func_def_74,type,
sK21: $i > $i ).
thf(func_def_75,type,
sK22: ( $i > $o ) > $i ).
thf(func_def_80,type,
sK27: $i > $i ).
thf(func_def_82,type,
sK29: $i > $i > $o ).
thf(func_def_84,type,
sK31: $i > $i ).
thf(func_def_85,type,
sK32: $i > $i ).
thf(func_def_86,type,
sK33: $i > ( $i > $i > $o ) > $i ).
thf(func_def_87,type,
sK34: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_88,type,
sK35: $i > ( $i > $i > $o ) > $i ).
thf(func_def_93,type,
sK40: $i > $i ).
thf(func_def_95,type,
sK42: $i > $i ).
thf(func_def_99,type,
sK46: $i > $i > $i ).
thf(func_def_102,type,
sK49: $i > $i ).
thf(func_def_104,type,
sK51: $i > $i ).
thf(func_def_107,type,
sK54: $i > $i > $i ).
thf(func_def_108,type,
sK55: $i > $i > $i ).
thf(func_def_109,type,
sK56: $i > $i > $i ).
thf(func_def_110,type,
sK57: $i > $i > $i ).
thf(func_def_111,type,
sK58: $i > $i > $i > $i ).
thf(func_def_112,type,
sK59: $i > $i ).
thf(func_def_113,type,
sK60: $i > $i ).
thf(func_def_114,type,
sK61: $i > $i ).
thf(func_def_115,type,
sK62: $i > $i ).
thf(func_def_116,type,
sK63: $i > $i > $i ).
thf(func_def_117,type,
sK64: $i > $i > $i > $i ).
thf(func_def_118,type,
sK65: $i > $i > $i > $i ).
thf(func_def_119,type,
sK66: $i > $i > $i ).
thf(func_def_120,type,
sK67: $i > $i > $i ).
thf(func_def_121,type,
sK68: $i > $i ).
thf(func_def_122,type,
sK69: $i > $i > $i ).
thf(func_def_124,type,
sK71: $i > $i ).
thf(func_def_125,type,
sK72: $i > $i ).
thf(func_def_127,type,
sK74: $i > $o ).
thf(func_def_129,type,
sK76: $i > $i > $i ).
thf(func_def_137,type,
sK84: $i > $o ).
thf(func_def_140,type,
sK87: $i > $o ).
thf(func_def_141,type,
sK88: ( $i > $o ) > $i ).
thf(func_def_142,type,
sK89: $i > $o ).
thf(func_def_144,type,
ph91:
!>[X0: $tType] : X0 ).
thf(f626,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f360,f609]) ).
thf(f609,plain,
! [X2: $i,X3: $i] :
( ( in @ X2 @ ( setadjoin @ X2 @ X3 ) )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f556]) ).
thf(f556,plain,
! [X2: $i,X3: $i] :
( ( $true != $true )
| ( ( in @ X2 @ ( setadjoin @ X2 @ X3 ) )
= $true ) ),
inference(definition_unfolding,[status(thm)],[f456,f364]) ).
thf(f364,plain,
setadjoinIL = $true,
inference(cnf_transformation,[status(thm)],[f110]) ).
thf(f110,plain,
( ( emptysetAx = $true )
& ( prop2setE = $true )
& ( setadjoinAx = $true )
& ( nonemptyI = $true )
& ( replAx = $true )
& ( setext = $true )
& ( setunionAx = $true )
& ( emptyI = $true )
& ( nonemptyE1 = $true )
& ( exuE3e = $true )
& ( noeltsimpempty = $true )
& ( descrp = $true )
& ( dsetconstrI = $true )
& ( powersetAx = $true )
& ( dsetconstrER = $true )
& ( omega0Ax = $true )
& ( nonemptyI1 = $true )
& ( setextAx = $true )
& ( exuE1 = $true )
& ( emptysetE = $true )
& ( notinemptyset = $true )
& ( setadjoinIL = $true )
& ( foundationAx = $true )
& ( setbeta = $true )
& ( wellorderingAx = $true )
& ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
!= $true )
& ( dsetconstrEL = $true )
& ( omegaSAx = $true )
& ( emptysetimpfalse = $true )
& ( omegaIndAx = $true ) ),
inference(flattening,[status(thm)],[f109]) ).
thf(f109,plain,
( ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
!= $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[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 )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
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
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f34]) ).
thf(f34,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f33]) ).
thf(f33,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.EScXwP1Jnj/DTF2THF_29057.p',emptyinunitempty) ).
thf(f456,plain,
! [X2: $i,X3: $i] :
( ( ( in @ X2 @ ( setadjoin @ X2 @ X3 ) )
= $true )
| ( setadjoinIL != $true ) ),
inference(cnf_transformation,[status(thm)],[f285]) ).
thf(f285,plain,
( ( ( setadjoinIL = $true )
| ( $true
!= ( in @ sK80 @ ( setadjoin @ sK80 @ sK81 ) ) ) )
& ( ! [X2: $i,X3: $i] :
( ( in @ X2 @ ( setadjoin @ X2 @ X3 ) )
= $true )
| ( setadjoinIL != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK80,sK81])],[f283,f284]) ).
thf(f284,plain,
( ? [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X0 @ X1 ) )
!= $true )
=> ( $true
!= ( in @ sK80 @ ( setadjoin @ sK80 @ sK81 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f283,plain,
( ( ( setadjoinIL = $true )
| ? [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X0 @ X1 ) )
!= $true ) )
& ( ! [X2: $i,X3: $i] :
( ( in @ X2 @ ( setadjoin @ X2 @ X3 ) )
= $true )
| ( setadjoinIL != $true ) ) ),
inference(rectify,[status(thm)],[f282]) ).
thf(f282,plain,
( ( ( setadjoinIL = $true )
| ? [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X0 @ X1 ) )
!= $true ) )
& ( ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X0 @ X1 ) )
= $true )
| ( setadjoinIL != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f73]) ).
thf(f73,plain,
( ( setadjoinIL = $true )
<=> ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X0 @ X1 ) )
= $true ) ),
inference(fool_elimination,[status(thm)],[f72]) ).
thf(f72,plain,
( setadjoinIL
= ( ! [X0: $i,X1: $i] : ( in @ X0 @ ( setadjoin @ X0 @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f32]) ).
thf(f32,axiom,
( setadjoinIL
= ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.EScXwP1Jnj/DTF2THF_29057.p',setadjoinIL) ).
thf(f360,plain,
( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f110]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU513^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n011.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 12:24:47 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.33/0.47 ---- Original DTF file ---
% 0.33/0.47 thf(spec,logic,$$dhol).
% 0.33/0.47 %------------------------------------------------------------------------------
% 0.33/0.47 % File : SEU513^1 : TPTP v9.2.1. Released v3.7.0.
% 0.33/0.47 % Domain : Set Theory
% 0.33/0.47 % Problem : Preliminary Notions - Adjoining Elements to Sets
% 0.33/0.47 % Version : Especial.
% 0.33/0.47 % English : in emptyset (setadjoin emptyset emptyset)
% 0.33/0.47
% 0.33/0.47 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.33/0.47 % Source : [Bro08]
% 0.33/0.47 % Names : ZFC015g [Bro08]
% 0.33/0.47
% 0.33/0.47 % Status : Theorem
% 0.33/0.47 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.23 v8.1.0, 0.00 v7.1.0, 0.12 v7.0.0, 0.00 v6.2.0, 0.14 v6.0.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.2.0, 0.20 v5.1.0, 0.40 v5.0.0, 0.20 v4.1.0, 0.33 v3.7.0
% 0.33/0.47 % Syntax : Number of formulae : 73 ( 32 unt; 40 typ; 32 def)
% 0.33/0.47 % Number of atoms : 183 ( 41 equ; 0 cnn)
% 0.33/0.47 % Maximal formula atoms : 30 ( 5 avg)
% 0.33/0.47 % Number of connectives : 313 ( 7 ~; 3 |; 18 &; 205 @)
% 0.33/0.47 % ( 7 <=>; 73 =>; 0 <=; 0 <~>)
% 0.33/0.47 % Maximal formula depth : 33 ( 2 avg)
% 0.33/0.47 % Number of types : 2 ( 0 usr)
% 0.33/0.47 % Number of type conns : 26 ( 26 >; 0 *; 0 +; 0 <<)
% 0.33/0.47 % Number of symbols : 42 ( 40 usr; 32 con; 0-2 aty)
% 0.33/0.47 % Number of variables : 103 ( 14 ^; 73 !; 16 ?; 103 :)
% 0.33/0.47 % SPC : TH0_THM_EQU_NAR
% 0.33/0.47
% 0.33/0.47 % Comments : http://mathgate.info/detsetitem.php?id=390
% 0.33/0.47 %------------------------------------------------------------------------------
% 0.33/0.47 thf(in_type,type,
% 0.33/0.47 in: $i > $i > $o ).
% 0.33/0.47
% 0.33/0.47 thf(exu_type,type,
% 0.33/0.47 exu: ( $i > $o ) > $o ).
% 0.33/0.47
% 0.33/0.47 thf(exu,definition,
% 0.33/0.47 ( exu
% 0.33/0.47 = ( ^ [Xphi: $i > $o] :
% 0.33/0.47 ? [Xx: $i] :
% 0.33/0.47 ( ( Xphi @ Xx )
% 0.33/0.47 & ! [Xy: $i] :
% 0.33/0.47 ( ( Xphi @ Xy )
% 0.33/0.47 => ( Xx = Xy ) ) ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(setextAx_type,type,
% 0.33/0.47 setextAx: $o ).
% 0.33/0.47
% 0.33/0.47 thf(setextAx,definition,
% 0.33/0.47 ( setextAx
% 0.33/0.47 = ( ! [A: $i,B: $i] :
% 0.33/0.47 ( ! [Xx: $i] :
% 0.33/0.47 ( ( in @ Xx @ A )
% 0.33/0.47 <=> ( in @ Xx @ B ) )
% 0.33/0.47 => ( A = B ) ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(emptyset_type,type,
% 0.33/0.47 emptyset: $i ).
% 0.33/0.47
% 0.33/0.47 thf(emptysetAx_type,type,
% 0.33/0.47 emptysetAx: $o ).
% 0.33/0.47
% 0.33/0.47 thf(emptysetAx,definition,
% 0.33/0.47 ( emptysetAx
% 0.33/0.47 = ( ! [Xx: $i] :
% 0.33/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(setadjoin_type,type,
% 0.33/0.47 setadjoin: $i > $i > $i ).
% 0.33/0.47
% 0.33/0.47 thf(setadjoinAx_type,type,
% 0.33/0.47 setadjoinAx: $o ).
% 0.33/0.47
% 0.33/0.47 thf(setadjoinAx,definition,
% 0.33/0.47 ( setadjoinAx
% 0.33/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.33/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.33/0.47 <=> ( ( Xy = Xx )
% 0.33/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(powerset_type,type,
% 0.33/0.47 powerset: $i > $i ).
% 0.33/0.47
% 0.33/0.47 thf(powersetAx_type,type,
% 0.33/0.47 powersetAx: $o ).
% 0.33/0.47
% 0.33/0.47 thf(powersetAx,definition,
% 0.33/0.47 ( powersetAx
% 0.33/0.47 = ( ! [A: $i,B: $i] :
% 0.33/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.33/0.47 <=> ! [Xx: $i] :
% 0.33/0.47 ( ( in @ Xx @ B )
% 0.33/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(setunion_type,type,
% 0.33/0.47 setunion: $i > $i ).
% 0.33/0.47
% 0.33/0.47 thf(setunionAx_type,type,
% 0.33/0.47 setunionAx: $o ).
% 0.33/0.47
% 0.33/0.47 thf(setunionAx,definition,
% 0.33/0.47 ( setunionAx
% 0.33/0.47 = ( ! [A: $i,Xx: $i] :
% 0.33/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.33/0.47 <=> ? [B: $i] :
% 0.33/0.47 ( ( in @ Xx @ B )
% 0.33/0.47 & ( in @ B @ A ) ) ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(omega_type,type,
% 0.33/0.47 omega: $i ).
% 0.33/0.47
% 0.33/0.47 thf(omega0Ax_type,type,
% 0.33/0.47 omega0Ax: $o ).
% 0.33/0.47
% 0.33/0.47 thf(omega0Ax,definition,
% 0.33/0.47 ( omega0Ax
% 0.33/0.47 = ( in @ emptyset @ omega ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(omegaSAx_type,type,
% 0.33/0.47 omegaSAx: $o ).
% 0.33/0.47
% 0.33/0.47 thf(omegaSAx,definition,
% 0.33/0.47 ( omegaSAx
% 0.33/0.47 = ( ! [Xx: $i] :
% 0.33/0.47 ( ( in @ Xx @ omega )
% 0.33/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(omegaIndAx_type,type,
% 0.33/0.47 omegaIndAx: $o ).
% 0.33/0.47
% 0.33/0.47 thf(omegaIndAx,definition,
% 0.33/0.47 ( omegaIndAx
% 0.33/0.47 = ( ! [A: $i] :
% 0.33/0.47 ( ( ( in @ emptyset @ A )
% 0.33/0.47 & ! [Xx: $i] :
% 0.33/0.47 ( ( ( in @ Xx @ omega )
% 0.33/0.47 & ( in @ Xx @ A ) )
% 0.33/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.33/0.47 => ! [Xx: $i] :
% 0.33/0.47 ( ( in @ Xx @ omega )
% 0.33/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(replAx_type,type,
% 0.33/0.47 replAx: $o ).
% 0.33/0.47
% 0.33/0.47 thf(replAx,definition,
% 0.33/0.47 ( replAx
% 0.33/0.47 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.33/0.47 ( ! [Xx: $i] :
% 0.33/0.47 ( ( in @ Xx @ A )
% 0.33/0.47 => ( exu
% 0.33/0.47 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.33/0.47 => ? [B: $i] :
% 0.33/0.47 ! [Xx: $i] :
% 0.33/0.47 ( ( in @ Xx @ B )
% 0.33/0.47 <=> ? [Xy: $i] :
% 0.33/0.47 ( ( in @ Xy @ A )
% 0.33/0.47 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(foundationAx_type,type,
% 0.33/0.47 foundationAx: $o ).
% 0.33/0.47
% 0.33/0.47 thf(foundationAx,definition,
% 0.33/0.47 ( foundationAx
% 0.33/0.47 = ( ! [A: $i] :
% 0.33/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.33/0.47 => ? [B: $i] :
% 0.33/0.47 ( ( in @ B @ A )
% 0.33/0.47 & ~ ? [Xx: $i] :
% 0.33/0.47 ( ( in @ Xx @ B )
% 0.33/0.47 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(wellorderingAx_type,type,
% 0.33/0.47 wellorderingAx: $o ).
% 0.33/0.47
% 0.33/0.47 thf(wellorderingAx,definition,
% 0.33/0.47 ( wellorderingAx
% 0.33/0.47 = ( ! [A: $i] :
% 0.33/0.47 ? [B: $i] :
% 0.33/0.47 ( ! [C: $i] :
% 0.33/0.47 ( ( in @ C @ B )
% 0.33/0.47 => ! [Xx: $i] :
% 0.33/0.47 ( ( in @ Xx @ C )
% 0.33/0.47 => ( in @ Xx @ A ) ) )
% 0.33/0.47 & ! [Xx: $i,Xy: $i] :
% 0.33/0.47 ( ( ( in @ Xx @ A )
% 0.33/0.47 & ( in @ Xy @ A ) )
% 0.33/0.47 => ( ! [C: $i] :
% 0.33/0.47 ( ( in @ C @ B )
% 0.33/0.47 => ( ( in @ Xx @ C )
% 0.33/0.47 <=> ( in @ Xy @ C ) ) )
% 0.33/0.47 => ( Xx = Xy ) ) )
% 0.33/0.47 & ! [C: $i,D: $i] :
% 0.33/0.47 ( ( ( in @ C @ B )
% 0.33/0.47 & ( in @ D @ B ) )
% 0.33/0.47 => ( ! [Xx: $i] :
% 0.33/0.47 ( ( in @ Xx @ C )
% 0.33/0.47 => ( in @ Xx @ D ) )
% 0.33/0.47 | ! [Xx: $i] :
% 0.33/0.47 ( ( in @ Xx @ D )
% 0.33/0.47 => ( in @ Xx @ C ) ) ) )
% 0.33/0.47 & ! [C: $i] :
% 0.33/0.47 ( ( ! [Xx: $i] :
% 0.33/0.47 ( ( in @ Xx @ C )
% 0.33/0.47 => ( in @ Xx @ A ) )
% 0.33/0.47 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.33/0.47 => ? [D: $i,Xx: $i] :
% 0.33/0.47 ( ( in @ D @ B )
% 0.33/0.47 & ( in @ Xx @ C )
% 0.33/0.47 & ~ ? [Xy: $i] :
% 0.33/0.47 ( ( in @ Xy @ D )
% 0.33/0.47 & ( in @ Xy @ C ) )
% 0.33/0.47 & ! [E: $i] :
% 0.33/0.47 ( ( in @ E @ B )
% 0.33/0.47 => ( ! [Xy: $i] :
% 0.33/0.47 ( ( in @ Xy @ E )
% 0.33/0.47 => ( in @ Xy @ D ) )
% 0.33/0.47 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(descr_type,type,
% 0.33/0.47 descr: ( $i > $o ) > $i ).
% 0.33/0.47
% 0.33/0.47 thf(descrp_type,type,
% 0.33/0.47 descrp: $o ).
% 0.33/0.47
% 0.33/0.47 thf(descrp,definition,
% 0.33/0.47 ( descrp
% 0.33/0.47 = ( ! [Xphi: $i > $o] :
% 0.33/0.47 ( ( exu
% 0.33/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.33/0.47 => ( Xphi
% 0.33/0.47 @ ( descr
% 0.33/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(dsetconstr_type,type,
% 0.33/0.47 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.33/0.47
% 0.33/0.47 thf(dsetconstrI_type,type,
% 0.33/0.47 dsetconstrI: $o ).
% 0.33/0.47
% 0.33/0.47 thf(dsetconstrI,definition,
% 0.33/0.47 ( dsetconstrI
% 0.33/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.33/0.47 ( ( in @ Xx @ A )
% 0.33/0.47 => ( ( Xphi @ Xx )
% 0.33/0.47 => ( in @ Xx
% 0.33/0.47 @ ( dsetconstr @ A
% 0.33/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(dsetconstrEL_type,type,
% 0.33/0.47 dsetconstrEL: $o ).
% 0.33/0.47
% 0.33/0.47 thf(dsetconstrEL,definition,
% 0.33/0.47 ( dsetconstrEL
% 0.33/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.33/0.47 ( ( in @ Xx
% 0.33/0.47 @ ( dsetconstr @ A
% 0.33/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.33/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(dsetconstrER_type,type,
% 0.33/0.47 dsetconstrER: $o ).
% 0.33/0.47
% 0.33/0.47 thf(dsetconstrER,definition,
% 0.33/0.47 ( dsetconstrER
% 0.33/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.33/0.47 ( ( in @ Xx
% 0.33/0.47 @ ( dsetconstr @ A
% 0.33/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.33/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(exuE1_type,type,
% 0.33/0.47 exuE1: $o ).
% 0.33/0.47
% 0.33/0.47 thf(exuE1,definition,
% 0.33/0.47 ( exuE1
% 0.33/0.47 = ( ! [Xphi: $i > $o] :
% 0.33/0.47 ( ( exu
% 0.33/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.33/0.47 => ? [Xx: $i] :
% 0.33/0.47 ( ( Xphi @ Xx )
% 0.33/0.47 & ! [Xy: $i] :
% 0.33/0.47 ( ( Xphi @ Xy )
% 0.33/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(prop2set_type,type,
% 0.33/0.47 prop2set: $o > $i ).
% 0.33/0.47
% 0.33/0.47 thf(prop2set,definition,
% 0.33/0.47 ( prop2set
% 0.33/0.47 = ( ^ [Xphi: $o] :
% 0.33/0.47 ( dsetconstr @ ( powerset @ emptyset )
% 0.33/0.47 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(prop2setE_type,type,
% 0.33/0.47 prop2setE: $o ).
% 0.33/0.47
% 0.33/0.47 thf(prop2setE,definition,
% 0.33/0.47 ( prop2setE
% 0.33/0.47 = ( ! [Xphi: $o,Xx: $i] :
% 0.33/0.47 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.33/0.47 => Xphi ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(emptysetE_type,type,
% 0.33/0.47 emptysetE: $o ).
% 0.33/0.47
% 0.33/0.47 thf(emptysetE,definition,
% 0.33/0.47 ( emptysetE
% 0.33/0.47 = ( ! [Xx: $i] :
% 0.33/0.47 ( ( in @ Xx @ emptyset )
% 0.33/0.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(emptysetimpfalse_type,type,
% 0.33/0.47 emptysetimpfalse: $o ).
% 0.33/0.47
% 0.33/0.47 thf(emptysetimpfalse,definition,
% 0.33/0.47 ( emptysetimpfalse
% 0.33/0.47 = ( ! [Xx: $i] :
% 0.33/0.47 ( ( in @ Xx @ emptyset )
% 0.33/0.47 => $false ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(notinemptyset_type,type,
% 0.33/0.47 notinemptyset: $o ).
% 0.33/0.47
% 0.33/0.47 thf(notinemptyset,definition,
% 0.33/0.47 ( notinemptyset
% 0.33/0.47 = ( ! [Xx: $i] :
% 0.33/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(exuE3e_type,type,
% 0.33/0.47 exuE3e: $o ).
% 0.33/0.47
% 0.33/0.47 thf(exuE3e,definition,
% 0.33/0.47 ( exuE3e
% 0.33/0.47 = ( ! [Xphi: $i > $o] :
% 0.33/0.47 ( ( exu
% 0.33/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.33/0.47 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(setext_type,type,
% 0.33/0.47 setext: $o ).
% 0.33/0.47
% 0.33/0.47 thf(setext,definition,
% 0.33/0.47 ( setext
% 0.33/0.47 = ( ! [A: $i,B: $i] :
% 0.33/0.47 ( ! [Xx: $i] :
% 0.33/0.47 ( ( in @ Xx @ A )
% 0.33/0.47 => ( in @ Xx @ B ) )
% 0.33/0.47 => ( ! [Xx: $i] :
% 0.33/0.47 ( ( in @ Xx @ B )
% 0.33/0.47 => ( in @ Xx @ A ) )
% 0.33/0.47 => ( A = B ) ) ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(emptyI_type,type,
% 0.33/0.47 emptyI: $o ).
% 0.33/0.47
% 0.33/0.47 thf(emptyI,definition,
% 0.33/0.47 ( emptyI
% 0.33/0.47 = ( ! [A: $i] :
% 0.33/0.47 ( ! [Xx: $i] :
% 0.33/0.47 ~ ( in @ Xx @ A )
% 0.33/0.47 => ( A = emptyset ) ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(noeltsimpempty_type,type,
% 0.33/0.47 noeltsimpempty: $o ).
% 0.33/0.47
% 0.33/0.47 thf(noeltsimpempty,definition,
% 0.33/0.47 ( noeltsimpempty
% 0.33/0.47 = ( ! [A: $i] :
% 0.33/0.47 ( ! [Xx: $i] :
% 0.33/0.47 ~ ( in @ Xx @ A )
% 0.33/0.47 => ( A = emptyset ) ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(setbeta_type,type,
% 0.33/0.47 setbeta: $o ).
% 0.33/0.47
% 0.33/0.47 thf(setbeta,definition,
% 0.33/0.47 ( setbeta
% 0.33/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.33/0.47 ( ( in @ Xx @ A )
% 0.33/0.47 => ( ( in @ Xx
% 0.33/0.47 @ ( dsetconstr @ A
% 0.33/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.33/0.47 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(nonempty_type,type,
% 0.33/0.47 nonempty: $i > $o ).
% 0.33/0.47
% 0.33/0.47 thf(nonempty,definition,
% 0.33/0.47 ( nonempty
% 0.33/0.47 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(nonemptyE1_type,type,
% 0.33/0.47 nonemptyE1: $o ).
% 0.33/0.47
% 0.33/0.47 thf(nonemptyE1,definition,
% 0.33/0.47 ( nonemptyE1
% 0.33/0.47 = ( ! [A: $i] :
% 0.33/0.47 ( ( nonempty @ A )
% 0.33/0.47 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(nonemptyI_type,type,
% 0.33/0.47 nonemptyI: $o ).
% 0.33/0.47
% 0.33/0.47 thf(nonemptyI,definition,
% 0.33/0.47 ( nonemptyI
% 0.33/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.33/0.47 ( ( in @ Xx @ A )
% 0.33/0.47 => ( ( Xphi @ Xx )
% 0.33/0.47 => ( nonempty
% 0.33/0.47 @ ( dsetconstr @ A
% 0.33/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(nonemptyI1_type,type,
% 0.33/0.47 nonemptyI1: $o ).
% 0.33/0.47
% 0.33/0.47 thf(nonemptyI1,definition,
% 0.33/0.47 ( nonemptyI1
% 0.33/0.47 = ( ! [A: $i] :
% 0.33/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.33/0.47 => ( nonempty @ A ) ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(setadjoinIL_type,type,
% 0.33/0.47 setadjoinIL: $o ).
% 0.33/0.47
% 0.33/0.47 thf(setadjoinIL,definition,
% 0.33/0.47 ( setadjoinIL
% 0.33/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.33/0.47
% 0.33/0.47 thf(emptyinunitempty,conjecture,
% 0.33/0.47 ( setextAx
% 0.33/0.47 => ( emptysetAx
% 0.33/0.47 => ( setadjoinAx
% 0.33/0.47 => ( powersetAx
% 0.33/0.47 => ( setunionAx
% 0.33/0.47 => ( omega0Ax
% 0.33/0.47 => ( omegaSAx
% 0.33/0.47 => ( omegaIndAx
% 0.33/0.47 => ( replAx
% 0.33/0.47 => ( foundationAx
% 0.33/0.47 => ( wellorderingAx
% 0.33/0.47 => ( descrp
% 0.33/0.47 => ( dsetconstrI
% 0.33/0.47 => ( dsetconstrEL
% 0.33/0.47 => ( dsetconstrER
% 0.33/0.47 => ( exuE1
% 0.33/0.47 => ( prop2setE
% 0.33/0.47 => ( emptysetE
% 0.33/0.47 => ( emptysetimpfalse
% 0.33/0.47 => ( notinemptyset
% 0.33/0.47 => ( exuE3e
% 0.33/0.47 => ( setext
% 0.33/0.47 => ( emptyI
% 0.33/0.47 => ( noeltsimpempty
% 0.33/0.47 => ( setbeta
% 0.33/0.47 => ( nonemptyE1
% 0.33/0.47 => ( nonemptyI
% 0.33/0.47 => ( nonemptyI1
% 0.33/0.47 => ( setadjoinIL
% 0.33/0.47 => ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.47
% 0.33/0.47 %------------------------------------------------------------------------------
% 0.33/0.47 ------------------------
% 1.95/1.34 ---- Embedded in THF ---
% 1.95/1.34 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.95/1.34 %%% Generated on Tue Feb 24 12:24:48 EST 2026
% 1.95/1.34 %%% using '$$dhol' embedding, version 1.3.0.
% 1.95/1.34 %%% Logic specification used:
% 1.95/1.34 %%% thf(spec, logic, $$dhol).
% 1.95/1.34
% 1.95/1.34 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.EScXwP1Jnj/DTF2DTF29057.p
% See solution above
% 1.95/1.34 ------------------------
% 1.95/1.35 ---- Cleaned THF ---
% 1.95/1.35 thf(in_type,type,
% 1.95/1.35 in: $i > $i > $o ).
% 1.95/1.35
% 1.95/1.35 thf(exu_type,type,
% 1.95/1.35 exu: ( $i > $o ) > $o ).
% 1.95/1.35
% 1.95/1.35 thf(exu,definition,
% 1.95/1.35 ( exu
% 1.95/1.35 = ( ^ [Xphi: $i > $o] :
% 1.95/1.35 ? [Xx: $i] :
% 1.95/1.35 ( ( Xphi @ Xx )
% 1.95/1.35 & ! [Xy: $i] :
% 1.95/1.35 ( ( Xphi @ Xy )
% 1.95/1.35 => ( Xx = Xy ) ) ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(setextAx_type,type,
% 1.95/1.35 setextAx: $o ).
% 1.95/1.35
% 1.95/1.35 thf(setextAx,definition,
% 1.95/1.35 ( setextAx
% 1.95/1.35 = ( ! [A: $i,B: $i] :
% 1.95/1.35 ( ! [Xx: $i] :
% 1.95/1.35 ( ( in @ Xx @ A )
% 1.95/1.35 <=> ( in @ Xx @ B ) )
% 1.95/1.35 => ( A = B ) ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(emptyset_type,type,
% 1.95/1.35 emptyset: $i ).
% 1.95/1.35
% 1.95/1.35 thf(emptysetAx_type,type,
% 1.95/1.35 emptysetAx: $o ).
% 1.95/1.35
% 1.95/1.35 thf(emptysetAx,definition,
% 1.95/1.35 ( emptysetAx
% 1.95/1.35 = ( ! [Xx: $i] :
% 1.95/1.35 ~ ( in @ Xx @ emptyset ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(setadjoin_type,type,
% 1.95/1.35 setadjoin: $i > $i > $i ).
% 1.95/1.35
% 1.95/1.35 thf(setadjoinAx_type,type,
% 1.95/1.35 setadjoinAx: $o ).
% 1.95/1.35
% 1.95/1.35 thf(setadjoinAx,definition,
% 1.95/1.35 ( setadjoinAx
% 1.95/1.35 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.95/1.35 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.95/1.35 <=> ( ( Xy = Xx )
% 1.95/1.35 | ( in @ Xy @ A ) ) ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(powerset_type,type,
% 1.95/1.35 powerset: $i > $i ).
% 1.95/1.35
% 1.95/1.35 thf(powersetAx_type,type,
% 1.95/1.35 powersetAx: $o ).
% 1.95/1.35
% 1.95/1.35 thf(powersetAx,definition,
% 1.95/1.35 ( powersetAx
% 1.95/1.35 = ( ! [A: $i,B: $i] :
% 1.95/1.35 ( ( in @ B @ ( powerset @ A ) )
% 1.95/1.35 <=> ! [Xx: $i] :
% 1.95/1.35 ( ( in @ Xx @ B )
% 1.95/1.35 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(setunion_type,type,
% 1.95/1.35 setunion: $i > $i ).
% 1.95/1.35
% 1.95/1.35 thf(setunionAx_type,type,
% 1.95/1.35 setunionAx: $o ).
% 1.95/1.35
% 1.95/1.35 thf(setunionAx,definition,
% 1.95/1.35 ( setunionAx
% 1.95/1.35 = ( ! [A: $i,Xx: $i] :
% 1.95/1.35 ( ( in @ Xx @ ( setunion @ A ) )
% 1.95/1.35 <=> ? [B: $i] :
% 1.95/1.35 ( ( in @ Xx @ B )
% 1.95/1.35 & ( in @ B @ A ) ) ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(omega_type,type,
% 1.95/1.35 omega: $i ).
% 1.95/1.35
% 1.95/1.35 thf(omega0Ax_type,type,
% 1.95/1.35 omega0Ax: $o ).
% 1.95/1.35
% 1.95/1.35 thf(omega0Ax,definition,
% 1.95/1.35 ( omega0Ax
% 1.95/1.35 = ( in @ emptyset @ omega ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(omegaSAx_type,type,
% 1.95/1.35 omegaSAx: $o ).
% 1.95/1.35
% 1.95/1.35 thf(omegaSAx,definition,
% 1.95/1.35 ( omegaSAx
% 1.95/1.35 = ( ! [Xx: $i] :
% 1.95/1.35 ( ( in @ Xx @ omega )
% 1.95/1.35 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(omegaIndAx_type,type,
% 1.95/1.35 omegaIndAx: $o ).
% 1.95/1.35
% 1.95/1.35 thf(omegaIndAx,definition,
% 1.95/1.35 ( omegaIndAx
% 1.95/1.35 = ( ! [A: $i] :
% 1.95/1.35 ( ( ( in @ emptyset @ A )
% 1.95/1.35 & ! [Xx: $i] :
% 1.95/1.35 ( ( ( in @ Xx @ omega )
% 1.95/1.35 & ( in @ Xx @ A ) )
% 1.95/1.35 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 1.95/1.35 => ! [Xx: $i] :
% 1.95/1.35 ( ( in @ Xx @ omega )
% 1.95/1.35 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(replAx_type,type,
% 1.95/1.35 replAx: $o ).
% 1.95/1.35
% 1.95/1.35 thf(replAx,definition,
% 1.95/1.35 ( replAx
% 1.95/1.35 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 1.95/1.35 ( ! [Xx: $i] :
% 1.95/1.35 ( ( in @ Xx @ A )
% 1.95/1.35 => ( exu
% 1.95/1.35 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 1.95/1.35 => ? [B: $i] :
% 1.95/1.35 ! [Xx: $i] :
% 1.95/1.35 ( ( in @ Xx @ B )
% 1.95/1.35 <=> ? [Xy: $i] :
% 1.95/1.35 ( ( in @ Xy @ A )
% 1.95/1.35 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(foundationAx_type,type,
% 1.95/1.35 foundationAx: $o ).
% 1.95/1.35
% 1.95/1.35 thf(foundationAx,definition,
% 1.95/1.35 ( foundationAx
% 1.95/1.35 = ( ! [A: $i] :
% 1.95/1.35 ( ? [Xx: $i] : ( in @ Xx @ A )
% 1.95/1.35 => ? [B: $i] :
% 1.95/1.35 ( ( in @ B @ A )
% 1.95/1.35 & ~ ? [Xx: $i] :
% 1.95/1.35 ( ( in @ Xx @ B )
% 1.95/1.35 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(wellorderingAx_type,type,
% 1.95/1.35 wellorderingAx: $o ).
% 1.95/1.35
% 1.95/1.35 thf(wellorderingAx,definition,
% 1.95/1.35 ( wellorderingAx
% 1.95/1.35 = ( ! [A: $i] :
% 1.95/1.35 ? [B: $i] :
% 1.95/1.35 ( ! [C: $i] :
% 1.95/1.35 ( ( in @ C @ B )
% 1.95/1.35 => ! [Xx: $i] :
% 1.95/1.35 ( ( in @ Xx @ C )
% 1.95/1.35 => ( in @ Xx @ A ) ) )
% 1.95/1.35 & ! [Xx: $i,Xy: $i] :
% 1.95/1.35 ( ( ( in @ Xx @ A )
% 1.95/1.35 & ( in @ Xy @ A ) )
% 1.95/1.35 => ( ! [C: $i] :
% 1.95/1.35 ( ( in @ C @ B )
% 1.95/1.35 => ( ( in @ Xx @ C )
% 1.95/1.35 <=> ( in @ Xy @ C ) ) )
% 1.95/1.35 => ( Xx = Xy ) ) )
% 1.95/1.35 & ! [C: $i,D: $i] :
% 1.95/1.35 ( ( ( in @ C @ B )
% 1.95/1.35 & ( in @ D @ B ) )
% 1.95/1.35 => ( ! [Xx: $i] :
% 1.95/1.35 ( ( in @ Xx @ C )
% 1.95/1.35 => ( in @ Xx @ D ) )
% 1.95/1.35 | ! [Xx: $i] :
% 1.95/1.35 ( ( in @ Xx @ D )
% 1.95/1.35 => ( in @ Xx @ C ) ) ) )
% 1.95/1.35 & ! [C: $i] :
% 1.95/1.35 ( ( ! [Xx: $i] :
% 1.95/1.35 ( ( in @ Xx @ C )
% 1.95/1.35 => ( in @ Xx @ A ) )
% 1.95/1.35 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 1.95/1.35 => ? [D: $i,Xx: $i] :
% 1.95/1.35 ( ( in @ D @ B )
% 1.95/1.35 & ( in @ Xx @ C )
% 1.95/1.35 & ~ ? [Xy: $i] :
% 1.95/1.35 ( ( in @ Xy @ D )
% 1.95/1.35 & ( in @ Xy @ C ) )
% 1.95/1.35 & ! [E: $i] :
% 1.95/1.35 ( ( in @ E @ B )
% 1.95/1.35 => ( ! [Xy: $i] :
% 1.95/1.35 ( ( in @ Xy @ E )
% 1.95/1.35 => ( in @ Xy @ D ) )
% 1.95/1.35 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(descr_type,type,
% 1.95/1.35 descr: ( $i > $o ) > $i ).
% 1.95/1.35
% 1.95/1.35 thf(descrp_type,type,
% 1.95/1.35 descrp: $o ).
% 1.95/1.35
% 1.95/1.35 thf(descrp,definition,
% 1.95/1.35 ( descrp
% 1.95/1.35 = ( ! [Xphi: $i > $o] :
% 1.95/1.35 ( ( exu
% 1.95/1.35 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.95/1.35 => ( Xphi
% 1.95/1.35 @ ( descr
% 1.95/1.35 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(dsetconstr_type,type,
% 1.95/1.35 dsetconstr: $i > ( $i > $o ) > $i ).
% 1.95/1.35
% 1.95/1.35 thf(dsetconstrI_type,type,
% 1.95/1.35 dsetconstrI: $o ).
% 1.95/1.35
% 1.95/1.35 thf(dsetconstrI,definition,
% 1.95/1.35 ( dsetconstrI
% 1.95/1.35 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.95/1.35 ( ( in @ Xx @ A )
% 1.95/1.35 => ( ( Xphi @ Xx )
% 1.95/1.35 => ( in @ Xx
% 1.95/1.35 @ ( dsetconstr @ A
% 1.95/1.35 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(dsetconstrEL_type,type,
% 1.95/1.35 dsetconstrEL: $o ).
% 1.95/1.35
% 1.95/1.35 thf(dsetconstrEL,definition,
% 1.95/1.35 ( dsetconstrEL
% 1.95/1.35 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.95/1.35 ( ( in @ Xx
% 1.95/1.35 @ ( dsetconstr @ A
% 1.95/1.35 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.95/1.35 => ( in @ Xx @ A ) ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(dsetconstrER_type,type,
% 1.95/1.35 dsetconstrER: $o ).
% 1.95/1.35
% 1.95/1.35 thf(dsetconstrER,definition,
% 1.95/1.35 ( dsetconstrER
% 1.95/1.35 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.95/1.35 ( ( in @ Xx
% 1.95/1.35 @ ( dsetconstr @ A
% 1.95/1.35 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.95/1.35 => ( Xphi @ Xx ) ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(exuE1_type,type,
% 1.95/1.35 exuE1: $o ).
% 1.95/1.35
% 1.95/1.35 thf(exuE1,definition,
% 1.95/1.35 ( exuE1
% 1.95/1.35 = ( ! [Xphi: $i > $o] :
% 1.95/1.35 ( ( exu
% 1.95/1.35 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.95/1.35 => ? [Xx: $i] :
% 1.95/1.35 ( ( Xphi @ Xx )
% 1.95/1.35 & ! [Xy: $i] :
% 1.95/1.35 ( ( Xphi @ Xy )
% 1.95/1.35 => ( Xx = Xy ) ) ) ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(prop2set_type,type,
% 1.95/1.35 prop2set: $o > $i ).
% 1.95/1.35
% 1.95/1.35 thf(prop2set,definition,
% 1.95/1.35 ( prop2set
% 1.95/1.35 = ( ^ [Xphi: $o] :
% 1.95/1.35 ( dsetconstr @ ( powerset @ emptyset )
% 1.95/1.35 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(prop2setE_type,type,
% 1.95/1.35 prop2setE: $o ).
% 1.95/1.35
% 1.95/1.35 thf(prop2setE,definition,
% 1.95/1.35 ( prop2setE
% 1.95/1.35 = ( ! [Xphi: $o,Xx: $i] :
% 1.95/1.35 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 1.95/1.35 => Xphi ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(emptysetE_type,type,
% 1.95/1.35 emptysetE: $o ).
% 1.95/1.35
% 1.95/1.35 thf(emptysetE,definition,
% 1.95/1.35 ( emptysetE
% 1.95/1.35 = ( ! [Xx: $i] :
% 1.95/1.35 ( ( in @ Xx @ emptyset )
% 1.95/1.35 => ! [Xphi: $o] : Xphi ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(emptysetimpfalse_type,type,
% 1.95/1.35 emptysetimpfalse: $o ).
% 1.95/1.35
% 1.95/1.35 thf(emptysetimpfalse,definition,
% 1.95/1.35 ( emptysetimpfalse
% 1.95/1.35 = ( ! [Xx: $i] :
% 1.95/1.35 ( ( in @ Xx @ emptyset )
% 1.95/1.35 => $false ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(notinemptyset_type,type,
% 1.95/1.35 notinemptyset: $o ).
% 1.95/1.35
% 1.95/1.35 thf(notinemptyset,definition,
% 1.95/1.35 ( notinemptyset
% 1.95/1.35 = ( ! [Xx: $i] :
% 1.95/1.35 ~ ( in @ Xx @ emptyset ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(exuE3e_type,type,
% 1.95/1.35 exuE3e: $o ).
% 1.95/1.35
% 1.95/1.35 thf(exuE3e,definition,
% 1.95/1.35 ( exuE3e
% 1.95/1.35 = ( ! [Xphi: $i > $o] :
% 1.95/1.35 ( ( exu
% 1.95/1.35 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.95/1.35 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(setext_type,type,
% 1.95/1.35 setext: $o ).
% 1.95/1.35
% 1.95/1.35 thf(setext,definition,
% 1.95/1.35 ( setext
% 1.95/1.35 = ( ! [A: $i,B: $i] :
% 1.95/1.35 ( ! [Xx: $i] :
% 1.95/1.35 ( ( in @ Xx @ A )
% 1.95/1.35 => ( in @ Xx @ B ) )
% 1.95/1.35 => ( ! [Xx: $i] :
% 1.95/1.35 ( ( in @ Xx @ B )
% 1.95/1.35 => ( in @ Xx @ A ) )
% 1.95/1.35 => ( A = B ) ) ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(emptyI_type,type,
% 1.95/1.35 emptyI: $o ).
% 1.95/1.35
% 1.95/1.35 thf(emptyI,definition,
% 1.95/1.35 ( emptyI
% 1.95/1.35 = ( ! [A: $i] :
% 1.95/1.35 ( ! [Xx: $i] :
% 1.95/1.35 ~ ( in @ Xx @ A )
% 1.95/1.35 => ( A = emptyset ) ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(noeltsimpempty_type,type,
% 1.95/1.35 noeltsimpempty: $o ).
% 1.95/1.35
% 1.95/1.35 thf(noeltsimpempty,definition,
% 1.95/1.35 ( noeltsimpempty
% 1.95/1.35 = ( ! [A: $i] :
% 1.95/1.35 ( ! [Xx: $i] :
% 1.95/1.35 ~ ( in @ Xx @ A )
% 1.95/1.35 => ( A = emptyset ) ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(setbeta_type,type,
% 1.95/1.35 setbeta: $o ).
% 1.95/1.35
% 1.95/1.35 thf(setbeta,definition,
% 1.95/1.35 ( setbeta
% 1.95/1.35 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.95/1.35 ( ( in @ Xx @ A )
% 1.95/1.35 => ( ( in @ Xx
% 1.95/1.35 @ ( dsetconstr @ A
% 1.95/1.35 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.95/1.35 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(nonempty_type,type,
% 1.95/1.35 nonempty: $i > $o ).
% 1.95/1.35
% 1.95/1.35 thf(nonempty,definition,
% 1.95/1.35 ( nonempty
% 1.95/1.35 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(nonemptyE1_type,type,
% 1.95/1.35 nonemptyE1: $o ).
% 1.95/1.35
% 1.95/1.35 thf(nonemptyE1,definition,
% 1.95/1.35 ( nonemptyE1
% 1.95/1.35 = ( ! [A: $i] :
% 1.95/1.35 ( ( nonempty @ A )
% 1.95/1.35 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(nonemptyI_type,type,
% 1.95/1.35 nonemptyI: $o ).
% 1.95/1.35
% 1.95/1.35 thf(nonemptyI,definition,
% 1.95/1.35 ( nonemptyI
% 1.95/1.35 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.95/1.35 ( ( in @ Xx @ A )
% 1.95/1.35 => ( ( Xphi @ Xx )
% 1.95/1.35 => ( nonempty
% 1.95/1.35 @ ( dsetconstr @ A
% 1.95/1.35 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(nonemptyI1_type,type,
% 1.95/1.35 nonemptyI1: $o ).
% 1.95/1.35
% 1.95/1.35 thf(nonemptyI1,definition,
% 1.95/1.35 ( nonemptyI1
% 1.95/1.35 = ( ! [A: $i] :
% 1.95/1.35 ( ? [Xx: $i] : ( in @ Xx @ A )
% 1.95/1.35 => ( nonempty @ A ) ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(setadjoinIL_type,type,
% 1.95/1.35 setadjoinIL: $o ).
% 1.95/1.35
% 1.95/1.35 thf(setadjoinIL,definition,
% 1.95/1.35 ( setadjoinIL
% 1.95/1.35 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 1.95/1.35
% 1.95/1.35 thf(emptyinunitempty,conjecture,
% 1.95/1.35 ( setextAx
% 1.95/1.35 => ( emptysetAx
% 1.95/1.35 => ( setadjoinAx
% 1.95/1.35 => ( powersetAx
% 1.95/1.35 => ( setunionAx
% 1.95/1.35 => ( omega0Ax
% 1.95/1.35 => ( omegaSAx
% 1.95/1.35 => ( omegaIndAx
% 1.95/1.35 => ( replAx
% 1.95/1.35 => ( foundationAx
% 1.95/1.35 => ( wellorderingAx
% 1.95/1.35 => ( descrp
% 1.95/1.35 => ( dsetconstrI
% 1.95/1.35 => ( dsetconstrEL
% 1.95/1.35 => ( dsetconstrER
% 1.95/1.35 => ( exuE1
% 1.95/1.35 => ( prop2setE
% 1.95/1.35 => ( emptysetE
% 1.95/1.35 => ( emptysetimpfalse
% 1.95/1.35 => ( notinemptyset
% 1.95/1.35 => ( exuE3e
% 1.95/1.35 => ( setext
% 1.95/1.35 => ( emptyI
% 1.95/1.35 => ( noeltsimpempty
% 1.95/1.35 => ( setbeta
% 1.95/1.35 => ( nonemptyE1
% 1.95/1.35 => ( nonemptyI
% 1.95/1.35 => ( nonemptyI1
% 1.95/1.35 => ( setadjoinIL
% 1.95/1.35 => ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 1.95/1.35 ------------------------
% 1.95/1.37 % (29254)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_29057 for (2999ds/18Mi)
% 1.95/1.37 % (29249)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_29057 for (2999ds/4Mi)
% 1.95/1.37 % (29253)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_29057 for (2999ds/275Mi)
% 1.95/1.37 % (29249)Instruction limit reached!
% 1.95/1.37 % (29249)------------------------------
% 1.95/1.37 % (29249)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.95/1.37 % (29249)Termination reason: Unknown
% 1.95/1.37 % (29249)Termination phase: Property scanning
% 1.95/1.37
% 1.95/1.37 % (29249)Memory used [KB]: 1151
% 1.95/1.37 % (29249)Time elapsed: 0.003 s
% 1.95/1.37 % (29249)Instructions burned: 6 (million)
% 1.95/1.37 % (29249)------------------------------
% 1.95/1.37 % (29249)------------------------------
% 1.95/1.37 % (29250)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_29057 for (2999ds/27Mi)
% 1.95/1.37 % (29254)Instruction limit reached!
% 1.95/1.37 % (29254)------------------------------
% 1.95/1.37 % (29254)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.95/1.37 % (29254)Termination reason: Unknown
% 1.95/1.37 % (29254)Termination phase: Property scanning
% 1.95/1.37
% 1.95/1.37 % (29254)Memory used [KB]: 1407
% 1.95/1.37 % (29254)Time elapsed: 0.007 s
% 1.95/1.37 % (29254)Instructions burned: 20 (million)
% 1.95/1.37 % (29254)------------------------------
% 1.95/1.37 % (29254)------------------------------
% 1.95/1.37 % (29248)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_29057 for (2999ds/183Mi)
% 1.95/1.37 % (29252)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_29057 for (2999ds/2Mi)
% 1.95/1.38 % (29252)Instruction limit reached!
% 1.95/1.38 % (29252)------------------------------
% 1.95/1.38 % (29252)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.95/1.38 % (29252)Termination reason: Unknown
% 1.95/1.38 % (29252)Termination phase: shuffling
% 1.95/1.38
% 1.95/1.38 % (29252)Memory used [KB]: 1023
% 1.95/1.38 % (29252)Time elapsed: 0.004 s
% 1.95/1.38 % (29252)Instructions burned: 3 (million)
% 1.95/1.38 % (29252)------------------------------
% 1.95/1.38 % (29252)------------------------------
% 1.95/1.38 % (29250)Instruction limit reached!
% 1.95/1.38 % (29250)------------------------------
% 1.95/1.38 % (29250)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.95/1.38 % (29250)Termination reason: Unknown
% 1.95/1.38 % (29250)Termination phase: Saturation
% 1.95/1.38
% 1.95/1.38 % (29250)Memory used [KB]: 5756
% 1.95/1.38 % (29250)Time elapsed: 0.009 s
% 1.95/1.38 % (29250)Instructions burned: 27 (million)
% 1.95/1.38 % (29250)------------------------------
% 1.95/1.38 % (29250)------------------------------
% 1.95/1.38 % (29253)First to succeed.
% 1.95/1.38 % (29253)Refutation found. Thanks to Tanya!
% 1.95/1.38 % SZS status Theorem for DTF2THF_29057
% 1.95/1.38 % SZS output start Proof for DTF2THF_29057
% 1.95/1.38 thf(func_def_0, type, in: $i > $i > $o).
% 1.95/1.38 thf(func_def_1, type, exu: ($i > $o) > $o).
% 1.95/1.38 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 1.95/1.38 thf(func_def_8, type, powerset: $i > $i).
% 1.95/1.38 thf(func_def_10, type, setunion: $i > $i).
% 1.95/1.38 thf(func_def_19, type, descr: ($i > $o) > $i).
% 1.95/1.38 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 1.95/1.38 thf(func_def_26, type, prop2set: $o > $i).
% 1.95/1.38 thf(func_def_36, type, nonempty: $i > $o).
% 1.95/1.38 thf(func_def_54, type, sP1: $i > $i > $o).
% 1.95/1.38 thf(func_def_55, type, sP2: $i > $o).
% 1.95/1.38 thf(func_def_56, type, sP3: $i > $i > $o).
% 1.95/1.38 thf(func_def_57, type, sP4: $i > $i > $o).
% 1.95/1.38 thf(func_def_62, type, sK9: $i > $i > $i).
% 1.95/1.38 thf(func_def_63, type, sK10: $i > $i > $i).
% 1.95/1.38 thf(func_def_65, type, sK12: $i > $o).
% 1.95/1.38 thf(func_def_70, type, sK17: $i > $o).
% 1.95/1.38 thf(func_def_71, type, sK18: $i > $i).
% 1.95/1.38 thf(func_def_73, type, sK20: $i > $o).
% 1.95/1.38 thf(func_def_74, type, sK21: $i > $i).
% 1.95/1.38 thf(func_def_75, type, sK22: ($i > $o) > $i).
% 1.95/1.38 thf(func_def_80, type, sK27: $i > $i).
% 1.95/1.38 thf(func_def_82, type, sK29: $i > $i > $o).
% 1.95/1.38 thf(func_def_84, type, sK31: $i > $i).
% 1.95/1.38 thf(func_def_85, type, sK32: $i > $i).
% 1.95/1.38 thf(func_def_86, type, sK33: $i > ($i > $i > $o) > $i).
% 1.95/1.38 thf(func_def_87, type, sK34: $i > $i > ($i > $i > $o) > $i).
% 1.95/1.38 thf(func_def_88, type, sK35: $i > ($i > $i > $o) > $i).
% 1.95/1.38 thf(func_def_93, type, sK40: $i > $i).
% 1.95/1.38 thf(func_def_95, type, sK42: $i > $i).
% 1.95/1.38 thf(func_def_99, type, sK46: $i > $i > $i).
% 1.95/1.38 thf(func_def_102, type, sK49: $i > $i).
% 1.95/1.38 thf(func_def_104, type, sK51: $i > $i).
% 1.95/1.38 thf(func_def_107, type, sK54: $i > $i > $i).
% 1.95/1.38 thf(func_def_108, type, sK55: $i > $i > $i).
% 1.95/1.38 thf(func_def_109, type, sK56: $i > $i > $i).
% 1.95/1.38 thf(func_def_110, type, sK57: $i > $i > $i).
% 1.95/1.38 thf(func_def_111, type, sK58: $i > $i > $i > $i).
% 1.95/1.38 thf(func_def_112, type, sK59: $i > $i).
% 1.95/1.38 thf(func_def_113, type, sK60: $i > $i).
% 1.95/1.38 thf(func_def_114, type, sK61: $i > $i).
% 1.95/1.38 thf(func_def_115, type, sK62: $i > $i).
% 1.95/1.38 thf(func_def_116, type, sK63: $i > $i > $i).
% 1.95/1.38 thf(func_def_117, type, sK64: $i > $i > $i > $i).
% 1.95/1.38 thf(func_def_118, type, sK65: $i > $i > $i > $i).
% 1.95/1.38 thf(func_def_119, type, sK66: $i > $i > $i).
% 1.95/1.38 thf(func_def_120, type, sK67: $i > $i > $i).
% 1.95/1.38 thf(func_def_121, type, sK68: $i > $i).
% 1.95/1.38 thf(func_def_122, type, sK69: $i > $i > $i).
% 1.95/1.38 thf(func_def_124, type, sK71: $i > $i).
% 1.95/1.38 thf(func_def_125, type, sK72: $i > $i).
% 1.95/1.38 thf(func_def_127, type, sK74: $i > $o).
% 1.95/1.38 thf(func_def_129, type, sK76: $i > $i > $i).
% 1.95/1.38 thf(func_def_137, type, sK84: $i > $o).
% 1.95/1.38 thf(func_def_140, type, sK87: $i > $o).
% 1.95/1.38 thf(func_def_141, type, sK88: ($i > $o) > $i).
% 1.95/1.38 thf(func_def_142, type, sK89: $i > $o).
% 1.95/1.38 thf(func_def_144, type, ph91: !>[X0: $tType]:(X0)).
% 1.95/1.38 thf(f626,plain,(
% 1.95/1.38 $false),
% 1.95/1.38 inference(subsumption_resolution,[status(thm)],[f360,f609])).
% 1.95/1.38 thf(f609,plain,(
% 1.95/1.38 ( ! [X2 : $i,X3 : $i] : (((in @ X2 @ (setadjoin @ X2 @ X3)) = $true)) )),
% 1.95/1.38 inference(trivial_inequality_removal,[status(thm)],[f556])).
% 1.95/1.38 thf(f556,plain,(
% 1.95/1.38 ( ! [X2 : $i,X3 : $i] : (($true != $true) | ((in @ X2 @ (setadjoin @ X2 @ X3)) = $true)) )),
% 1.95/1.38 inference(definition_unfolding,[status(thm)],[f456,f364])).
% 1.95/1.38 thf(f364,plain,(
% 1.95/1.38 (setadjoinIL = $true)),
% 1.95/1.38 inference(cnf_transformation,[status(thm)],[f110])).
% 1.95/1.38 thf(f110,plain,(
% 1.95/1.38 (emptysetAx = $true) & (prop2setE = $true) & (setadjoinAx = $true) & (nonemptyI = $true) & (replAx = $true) & (setext = $true) & (setunionAx = $true) & (emptyI = $true) & (nonemptyE1 = $true) & (exuE3e = $true) & (noeltsimpempty = $true) & (descrp = $true) & (dsetconstrI = $true) & (powersetAx = $true) & (dsetconstrER = $true) & (omega0Ax = $true) & (nonemptyI1 = $true) & (setextAx = $true) & (exuE1 = $true) & (emptysetE = $true) & (notinemptyset = $true) & (setadjoinIL = $true) & (foundationAx = $true) & (setbeta = $true) & (wellorderingAx = $true) & ((in @ emptyset @ (setadjoin @ emptyset @ emptyset)) != $true) & (dsetconstrEL = $true) & (omegaSAx = $true) & (emptysetimpfalse = $true) & (omegaIndAx = $true)),
% 1.95/1.38 inference(flattening,[status(thm)],[f109])).
% 1.95/1.38 thf(f109,plain,(
% 1.95/1.38 ((((((((((((((((((((((((((((((in @ emptyset @ (setadjoin @ emptyset @ emptyset)) != $true) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 1.95/1.38 inference(ennf_transformation,[status(thm)],[f41])).
% 1.95/1.38 thf(f41,plain,(
% 1.95/1.38 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((in @ emptyset @ (setadjoin @ emptyset @ emptyset)) = $true))))))))))))))))))))))))))))))),
% 1.95/1.38 inference(fool_elimination,[status(thm)],[f40])).
% 1.95/1.38 thf(f40,plain,(
% 1.95/1.38 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (in @ emptyset @ (setadjoin @ emptyset @ emptyset)))))))))))))))))))))))))))))))),
% 1.95/1.38 inference(rectify,[status(thm)],[f34])).
% 1.95/1.38 thf(f34,negated_conjecture,(
% 1.95/1.38 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (in @ emptyset @ (setadjoin @ emptyset @ emptyset)))))))))))))))))))))))))))))))),
% 1.95/1.38 inference(negated_conjecture,[status(cth)],[f33])).
% 1.95/1.38 thf(f33,conjecture,(
% 1.95/1.38 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (in @ emptyset @ (setadjoin @ emptyset @ emptyset))))))))))))))))))))))))))))))),
% 1.95/1.38 file('/export/starexec/sandbox/tmp/tmp.EScXwP1Jnj/DTF2THF_29057.p',emptyinunitempty)).
% 1.95/1.38 thf(f456,plain,(
% 1.95/1.38 ( ! [X2 : $i,X3 : $i] : (((in @ X2 @ (setadjoin @ X2 @ X3)) = $true) | (setadjoinIL != $true)) )),
% 1.95/1.38 inference(cnf_transformation,[status(thm)],[f285])).
% 1.95/1.38 thf(f285,plain,(
% 1.95/1.38 ((setadjoinIL = $true) | ($true != (in @ sK80 @ (setadjoin @ sK80 @ sK81)))) & (! [X2 : $i,X3 : $i] : ((in @ X2 @ (setadjoin @ X2 @ X3)) = $true) | (setadjoinIL != $true))),
% 1.95/1.38 inference(skolemisation,[status(esa),new_symbols(skolem,[sK80,sK81])],[f283,f284])).
% 1.95/1.38 thf(f284,plain,(
% 1.95/1.38 ? [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X0 @ X1)) != $true) => ($true != (in @ sK80 @ (setadjoin @ sK80 @ sK81)))),
% 1.95/1.38 introduced(definition,[],[choice_axiom])).
% 1.95/1.38 thf(f283,plain,(
% 1.95/1.38 ((setadjoinIL = $true) | ? [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X0 @ X1)) != $true)) & (! [X2 : $i,X3 : $i] : ((in @ X2 @ (setadjoin @ X2 @ X3)) = $true) | (setadjoinIL != $true))),
% 1.95/1.38 inference(rectify,[status(thm)],[f282])).
% 1.95/1.38 thf(f282,plain,(
% 1.95/1.38 ((setadjoinIL = $true) | ? [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X0 @ X1)) != $true)) & (! [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X0 @ X1)) = $true) | (setadjoinIL != $true))),
% 1.95/1.38 inference(nnf_transformation,[status(thm)],[f73])).
% 1.95/1.38 thf(f73,plain,(
% 1.95/1.38 (setadjoinIL = $true) <=> ! [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X0 @ X1)) = $true)),
% 1.95/1.38 inference(fool_elimination,[status(thm)],[f72])).
% 1.95/1.38 thf(f72,plain,(
% 1.95/1.38 (setadjoinIL = ! [X0 : $i,X1 : $i] : (in @ X0 @ (setadjoin @ X0 @ X1)))),
% 1.95/1.38 inference(rectify,[status(thm)],[f32])).
% 1.95/1.38 thf(f32,axiom,(
% 1.95/1.38 (setadjoinIL = ! [X1 : $i,X2 : $i] : (in @ X1 @ (setadjoin @ X1 @ X2)))),
% 1.95/1.38 file('/export/starexec/sandbox/tmp/tmp.EScXwP1Jnj/DTF2THF_29057.p',setadjoinIL)).
% 1.95/1.38 thf(f360,plain,(
% 1.95/1.38 ((in @ emptyset @ (setadjoin @ emptyset @ emptyset)) != $true)),
% 1.95/1.38 inference(cnf_transformation,[status(thm)],[f110])).
% 1.95/1.38 % SZS output end Proof for DTF2THF_29057
% 1.95/1.38 % (29253)------------------------------
% 1.95/1.38 % (29253)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.95/1.38 % (29253)Termination reason: Refutation
% 1.95/1.38
% 1.95/1.38 % (29253)Memory used [KB]: 6012
% 1.95/1.38 % (29253)Time elapsed: 0.012 s
% 1.95/1.38 % (29253)Instructions burned: 33 (million)
% 1.95/1.38 % (29253)------------------------------
% 1.95/1.38 % (29253)------------------------------
% 1.95/1.38 % (29247)Success in time 0.012 s
%------------------------------------------------------------------------------