%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU507^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n004.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:34:59 AM UTC 2026
% Result : Theorem 1.83s 1.38s
% Output : Refutation 1.83s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 5
% Syntax : Number of formulae : 28 ( 7 unt; 0 typ; 0 def)
% Number of atoms : 605 ( 183 equ; 0 cnn)
% Maximal formula atoms : 25 ( 21 avg)
% Number of connectives : 332 ( 41 ~; 16 |; 82 &; 63 @)
% ( 3 <=>; 127 =>; 0 <=; 0 <~>)
% Maximal formula depth : 31 ( 11 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 82 ( 79 usr; 28 con; 0-3 aty)
% Number of variables : 47 ( 0 ^; 37 !; 10 ?; 47 :)
% 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_45,type,
sP0: $i > $i > $o ).
thf(func_def_46,type,
sP1: $i > $o ).
thf(func_def_47,type,
sP2: $i > $i > $o ).
thf(func_def_48,type,
sP3: $i > $i > $o ).
thf(func_def_51,type,
sK6: $i > $i > $i ).
thf(func_def_52,type,
sK7: $i > $i > $i ).
thf(func_def_55,type,
sK10: $i > $o ).
thf(func_def_61,type,
sK16: $i > $i > $i ).
thf(func_def_62,type,
sK17: $i > $i > $i ).
thf(func_def_63,type,
sK18: $i > $i > $i ).
thf(func_def_64,type,
sK19: $i > $i > $i ).
thf(func_def_65,type,
sK20: $i > $i > $i > $i ).
thf(func_def_66,type,
sK21: $i > $i ).
thf(func_def_67,type,
sK22: $i > $i ).
thf(func_def_68,type,
sK23: $i > $i ).
thf(func_def_69,type,
sK24: $i > $i ).
thf(func_def_70,type,
sK25: $i > $i > $i ).
thf(func_def_71,type,
sK26: $i > $i > $i > $i ).
thf(func_def_72,type,
sK27: $i > $i > $i > $i ).
thf(func_def_73,type,
sK28: $i > $i > $i ).
thf(func_def_74,type,
sK29: $i > $i > $i ).
thf(func_def_75,type,
sK30: $i > $i ).
thf(func_def_76,type,
sK31: $i > $i > $i ).
thf(func_def_78,type,
sK33: $i > $i ).
thf(func_def_79,type,
sK34: $i > $i ).
thf(func_def_82,type,
sK37: $i > $i ).
thf(func_def_84,type,
sK39: $i > $i ).
thf(func_def_87,type,
sK42: $i > $i > $i ).
thf(func_def_88,type,
sK43: ( $i > $o ) > $i ).
thf(func_def_89,type,
sK44: $i > $o ).
thf(func_def_90,type,
sK45: $i > $i ).
thf(func_def_91,type,
sK46: $i > $i ).
thf(func_def_96,type,
sK51: $i > $i > $i ).
thf(func_def_102,type,
sK57: $i > $i ).
thf(func_def_109,type,
sK64: $i > $o ).
thf(func_def_110,type,
sK65: ( $i > $o ) > $i ).
thf(func_def_111,type,
sK66: $i > $o ).
thf(func_def_114,type,
sK69: $i > $o ).
thf(func_def_116,type,
sK71: $i > $i > $o ).
thf(func_def_117,type,
sK72: $i > $i ).
thf(func_def_118,type,
sK73: $i > $i ).
thf(func_def_119,type,
sK74: ( $i > $i > $o ) > $i > $i ).
thf(func_def_120,type,
sK75: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_121,type,
sK76: ( $i > $i > $o ) > $i > $i ).
thf(func_def_123,type,
ph78:
!>[X0: $tType] : X0 ).
thf(f621,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f619,f342]) ).
thf(f342,plain,
emptyset != sK58,
inference(cnf_transformation,[status(thm)],[f214]) ).
thf(f214,plain,
( ( descrp = $true )
& ( setadjoinAx = $true )
& ( omegaIndAx = $true )
& ( emptysetE = $true )
& ( dsetconstrEL = $true )
& ( exuE3e = $true )
& ( emptysetimpfalse = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( notinemptyset = $true )
& ( setextAx = $true )
& ( emptysetAx = $true )
& ( powersetAx = $true )
& ( foundationAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( replAx = $true )
& ( wellorderingAx = $true )
& ( dsetconstrI = $true )
& ( setunionAx = $true )
& ( emptyI = $true )
& ( setext = $true )
& ! [X1: $i] :
( $true
!= ( in @ X1 @ sK58 ) )
& ( emptyset != sK58 )
& ( prop2setE = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK58])],[f88,f213]) ).
thf(f213,plain,
( ? [X0: $i] :
( ! [X1: $i] :
( $true
!= ( in @ X1 @ X0 ) )
& ( emptyset != X0 ) )
=> ( ! [X1: $i] :
( $true
!= ( in @ X1 @ sK58 ) )
& ( emptyset != sK58 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f88,plain,
( ( descrp = $true )
& ( setadjoinAx = $true )
& ( omegaIndAx = $true )
& ( emptysetE = $true )
& ( dsetconstrEL = $true )
& ( exuE3e = $true )
& ( emptysetimpfalse = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( notinemptyset = $true )
& ( setextAx = $true )
& ( emptysetAx = $true )
& ( powersetAx = $true )
& ( foundationAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( replAx = $true )
& ( wellorderingAx = $true )
& ( dsetconstrI = $true )
& ( setunionAx = $true )
& ( emptyI = $true )
& ( setext = $true )
& ? [X0: $i] :
( ! [X1: $i] :
( $true
!= ( in @ X1 @ X0 ) )
& ( emptyset != X0 ) )
& ( prop2setE = $true ) ),
inference(flattening,[status(thm)],[f87]) ).
thf(f87,plain,
( ? [X0: $i] :
( ! [X1: $i] :
( $true
!= ( in @ X1 @ X0 ) )
& ( emptyset != X0 ) )
& ( 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)],[f78]) ).
thf(f78,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 )
=> ! [X0: $i] :
( ! [X1: $i] :
( $true
!= ( in @ X1 @ X0 ) )
=> ( emptyset = X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(flattening,[status(thm)],[f77]) ).
thf(f77,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 )
=> ! [X0: $i] :
( ! [X1: $i] :
( $true
!= ( in @ X1 @ X0 ) )
=> ( emptyset = X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f76]) ).
thf(f76,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ! [X0: $i] :
( ! [X1: $i] :
~ ( in @ X1 @ X0 )
=> ( emptyset = X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f27]) ).
thf(f27,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
=> ! [X3: $i] :
( ! [X1: $i] :
~ ( in @ X1 @ X3 )
=> ( emptyset = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f26]) ).
thf(f26,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ! [X3: $i] :
( ! [X1: $i] :
~ ( in @ X1 @ X3 )
=> ( emptyset = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.8JiX0Aov0I/DTF2THF_20125.p',noeltsimpempty) ).
thf(f619,plain,
emptyset = sK58,
inference(trivial_inequality_removal,[status(thm)],[f618]) ).
thf(f618,plain,
( ( emptyset = sK58 )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f343,f520]) ).
thf(f520,plain,
! [X0: $i] :
( ( $true
= ( in @ ( sK46 @ X0 ) @ X0 ) )
| ( emptyset = X0 ) ),
inference(trivial_inequality_removal,[status(thm)],[f436]) ).
thf(f436,plain,
! [X0: $i] :
( ( $true
= ( in @ ( sK46 @ X0 ) @ X0 ) )
| ( $true != $true )
| ( emptyset = X0 ) ),
inference(definition_unfolding,[status(thm)],[f322,f345]) ).
thf(f345,plain,
emptyI = $true,
inference(cnf_transformation,[status(thm)],[f214]) ).
thf(f322,plain,
! [X0: $i] :
( ( emptyset = X0 )
| ( $true
= ( in @ ( sK46 @ X0 ) @ X0 ) )
| ( emptyI != $true ) ),
inference(cnf_transformation,[status(thm)],[f192]) ).
thf(f192,plain,
( ( ! [X0: $i] :
( ( emptyset = X0 )
| ( $true
= ( in @ ( sK46 @ X0 ) @ X0 ) ) )
| ( emptyI != $true ) )
& ( ( emptyI = $true )
| ( ( emptyset != sK47 )
& ! [X3: $i] :
( $true
!= ( in @ X3 @ sK47 ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK46,sK47])],[f189,f191,f190]) ).
thf(f190,plain,
! [X0: $i] :
( ? [X1: $i] :
( $true
= ( in @ X1 @ X0 ) )
=> ( $true
= ( in @ ( sK46 @ X0 ) @ X0 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f191,plain,
( ? [X2: $i] :
( ( emptyset != X2 )
& ! [X3: $i] :
( $true
!= ( in @ X3 @ X2 ) ) )
=> ( ( emptyset != sK47 )
& ! [X3: $i] :
( $true
!= ( in @ X3 @ sK47 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f189,plain,
( ( ! [X0: $i] :
( ( emptyset = X0 )
| ? [X1: $i] :
( $true
= ( in @ X1 @ X0 ) ) )
| ( emptyI != $true ) )
& ( ( emptyI = $true )
| ? [X2: $i] :
( ( emptyset != X2 )
& ! [X3: $i] :
( $true
!= ( in @ X3 @ X2 ) ) ) ) ),
inference(rectify,[status(thm)],[f188]) ).
thf(f188,plain,
( ( ! [X0: $i] :
( ( emptyset = X0 )
| ? [X1: $i] :
( $true
= ( in @ X1 @ X0 ) ) )
| ( emptyI != $true ) )
& ( ( emptyI = $true )
| ? [X0: $i] :
( ( emptyset != X0 )
& ! [X1: $i] :
( $true
!= ( in @ X1 @ X0 ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f97]) ).
thf(f97,plain,
( ! [X0: $i] :
( ( emptyset = X0 )
| ? [X1: $i] :
( $true
= ( in @ X1 @ X0 ) ) )
<=> ( emptyI = $true ) ),
inference(ennf_transformation,[status(thm)],[f86]) ).
thf(f86,plain,
( ! [X0: $i] :
( ! [X1: $i] :
( $true
!= ( in @ X1 @ X0 ) )
=> ( emptyset = X0 ) )
<=> ( emptyI = $true ) ),
inference(flattening,[status(thm)],[f41]) ).
thf(f41,plain,
( ( emptyI = $true )
<=> ! [X0: $i] :
( ! [X1: $i] :
( $true
!= ( in @ X1 @ X0 ) )
=> ( emptyset = X0 ) ) ),
inference(fool_elimination,[status(thm)],[f40]) ).
thf(f40,plain,
( emptyI
= ( ! [X0: $i] :
( ! [X1: $i] :
~ ( in @ X1 @ X0 )
=> ( emptyset = X0 ) ) ) ),
inference(rectify,[status(thm)],[f25]) ).
thf(f25,axiom,
( emptyI
= ( ! [X3: $i] :
( ! [X1: $i] :
~ ( in @ X1 @ X3 )
=> ( emptyset = X3 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.8JiX0Aov0I/DTF2THF_20125.p',emptyI) ).
thf(f343,plain,
! [X1: $i] :
( $true
!= ( in @ X1 @ sK58 ) ),
inference(cnf_transformation,[status(thm)],[f214]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU507^1 : TPTP v9.2.1. Released v3.7.0.
% 0.11/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n004.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 12:24:26 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.27/0.46 ---- Original DTF file ---
% 0.27/0.46 thf(spec,logic,$$dhol).
% 0.27/0.46 %------------------------------------------------------------------------------
% 0.27/0.46 % File : SEU507^1 : TPTP v9.2.1. Released v3.7.0.
% 0.27/0.46 % Domain : Set Theory
% 0.27/0.46 % Problem : Preliminary Notions - Basic Laws of Logic
% 0.27/0.46 % Version : Especial.
% 0.27/0.46 % English : (! A:i.(! x:i.~(in x A)) -> A = emptyset)
% 0.27/0.46
% 0.27/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.27/0.46 % Source : [Bro08]
% 0.27/0.46 % Names : ZFC009g [Bro08]
% 0.27/0.46
% 0.27/0.46 % Status : Theorem
% 0.27/0.46 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.23 v8.1.0, 0.00 v6.0.0, 0.14 v5.5.0, 0.17 v5.4.0, 0.20 v5.3.0, 0.40 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.67 v4.0.0, 0.33 v3.7.0
% 0.27/0.46 % Syntax : Number of formulae : 59 ( 25 unt; 33 typ; 25 def)
% 0.27/0.46 % Number of atoms : 152 ( 33 equ; 0 cnn)
% 0.27/0.46 % Maximal formula atoms : 25 ( 5 avg)
% 0.27/0.46 % Number of connectives : 271 ( 6 ~; 3 |; 18 &; 176 @)
% 0.27/0.46 % ( 6 <=>; 62 =>; 0 <=; 0 <~>)
% 0.27/0.46 % Maximal formula depth : 30 ( 2 avg)
% 0.27/0.46 % Number of types : 2 ( 0 usr)
% 0.27/0.46 % Number of type conns : 23 ( 23 >; 0 *; 0 +; 0 <<)
% 0.27/0.46 % Number of symbols : 35 ( 33 usr; 26 con; 0-2 aty)
% 0.27/0.46 % Number of variables : 88 ( 11 ^; 63 !; 14 ?; 88 :)
% 0.27/0.46 % SPC : TH0_THM_EQU_NAR
% 0.27/0.46
% 0.27/0.46 % Comments : http://mathgate.info/detsetitem.php?id=480
% 0.27/0.46 %------------------------------------------------------------------------------
% 0.27/0.46 thf(in_type,type,
% 0.27/0.46 in: $i > $i > $o ).
% 0.27/0.46
% 0.27/0.46 thf(exu_type,type,
% 0.27/0.46 exu: ( $i > $o ) > $o ).
% 0.27/0.46
% 0.27/0.46 thf(exu,definition,
% 0.27/0.46 ( exu
% 0.27/0.46 = ( ^ [Xphi: $i > $o] :
% 0.27/0.46 ? [Xx: $i] :
% 0.27/0.46 ( ( Xphi @ Xx )
% 0.27/0.46 & ! [Xy: $i] :
% 0.27/0.46 ( ( Xphi @ Xy )
% 0.27/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setextAx_type,type,
% 0.27/0.46 setextAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setextAx,definition,
% 0.27/0.46 ( setextAx
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 <=> ( in @ Xx @ B ) )
% 0.27/0.46 => ( A = B ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptyset_type,type,
% 0.27/0.46 emptyset: $i ).
% 0.27/0.46
% 0.27/0.46 thf(emptysetAx_type,type,
% 0.27/0.46 emptysetAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptysetAx,definition,
% 0.27/0.46 ( emptysetAx
% 0.27/0.46 = ( ! [Xx: $i] :
% 0.27/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoin_type,type,
% 0.27/0.46 setadjoin: $i > $i > $i ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinAx_type,type,
% 0.27/0.46 setadjoinAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setadjoinAx,definition,
% 0.27/0.46 ( setadjoinAx
% 0.27/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.27/0.46 <=> ( ( Xy = Xx )
% 0.27/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(powerset_type,type,
% 0.27/0.46 powerset: $i > $i ).
% 0.27/0.46
% 0.27/0.46 thf(powersetAx_type,type,
% 0.27/0.46 powersetAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(powersetAx,definition,
% 0.27/0.46 ( powersetAx
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.27/0.46 <=> ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setunion_type,type,
% 0.27/0.46 setunion: $i > $i ).
% 0.27/0.46
% 0.27/0.46 thf(setunionAx_type,type,
% 0.27/0.46 setunionAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setunionAx,definition,
% 0.27/0.46 ( setunionAx
% 0.27/0.46 = ( ! [A: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.27/0.46 <=> ? [B: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(omega_type,type,
% 0.27/0.46 omega: $i ).
% 0.27/0.46
% 0.27/0.46 thf(omega0Ax_type,type,
% 0.27/0.46 omega0Ax: $o ).
% 0.27/0.46
% 0.27/0.46 thf(omega0Ax,definition,
% 0.27/0.46 ( omega0Ax
% 0.27/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(omegaSAx_type,type,
% 0.27/0.46 omegaSAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(omegaSAx,definition,
% 0.27/0.46 ( omegaSAx
% 0.27/0.46 = ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ omega )
% 0.27/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(omegaIndAx_type,type,
% 0.27/0.46 omegaIndAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(omegaIndAx,definition,
% 0.27/0.46 ( omegaIndAx
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ( ( in @ emptyset @ A )
% 0.27/0.46 & ! [Xx: $i] :
% 0.27/0.46 ( ( ( in @ Xx @ omega )
% 0.27/0.46 & ( in @ Xx @ A ) )
% 0.27/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ omega )
% 0.27/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(replAx_type,type,
% 0.27/0.46 replAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(replAx,definition,
% 0.27/0.46 ( replAx
% 0.27/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( exu
% 0.27/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.27/0.46 => ? [B: $i] :
% 0.27/0.46 ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 <=> ? [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ A )
% 0.27/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(foundationAx_type,type,
% 0.27/0.46 foundationAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(foundationAx,definition,
% 0.27/0.46 ( foundationAx
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.27/0.46 => ? [B: $i] :
% 0.27/0.46 ( ( in @ B @ A )
% 0.27/0.46 & ~ ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(wellorderingAx_type,type,
% 0.27/0.46 wellorderingAx: $o ).
% 0.27/0.46
% 0.27/0.46 thf(wellorderingAx,definition,
% 0.27/0.46 ( wellorderingAx
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ? [B: $i] :
% 0.27/0.46 ( ! [C: $i] :
% 0.27/0.46 ( ( in @ C @ B )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ C )
% 0.27/0.46 => ( in @ Xx @ A ) ) )
% 0.27/0.46 & ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( ( in @ Xx @ A )
% 0.27/0.46 & ( in @ Xy @ A ) )
% 0.27/0.46 => ( ! [C: $i] :
% 0.27/0.46 ( ( in @ C @ B )
% 0.27/0.46 => ( ( in @ Xx @ C )
% 0.27/0.46 <=> ( in @ Xy @ C ) ) )
% 0.27/0.46 => ( Xx = Xy ) ) )
% 0.27/0.46 & ! [C: $i,D: $i] :
% 0.27/0.46 ( ( ( in @ C @ B )
% 0.27/0.46 & ( in @ D @ B ) )
% 0.27/0.46 => ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ C )
% 0.27/0.46 => ( in @ Xx @ D ) )
% 0.27/0.46 | ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ D )
% 0.27/0.46 => ( in @ Xx @ C ) ) ) )
% 0.27/0.46 & ! [C: $i] :
% 0.27/0.46 ( ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ C )
% 0.27/0.46 => ( in @ Xx @ A ) )
% 0.27/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.27/0.46 => ? [D: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ D @ B )
% 0.27/0.46 & ( in @ Xx @ C )
% 0.27/0.46 & ~ ? [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ D )
% 0.27/0.46 & ( in @ Xy @ C ) )
% 0.27/0.46 & ! [E: $i] :
% 0.27/0.46 ( ( in @ E @ B )
% 0.27/0.46 => ( ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ E )
% 0.27/0.46 => ( in @ Xy @ D ) )
% 0.27/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(descr_type,type,
% 0.27/0.46 descr: ( $i > $o ) > $i ).
% 0.27/0.46
% 0.27/0.46 thf(descrp_type,type,
% 0.27/0.46 descrp: $o ).
% 0.27/0.46
% 0.27/0.46 thf(descrp,definition,
% 0.27/0.46 ( descrp
% 0.27/0.46 = ( ! [Xphi: $i > $o] :
% 0.27/0.46 ( ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 => ( Xphi
% 0.27/0.46 @ ( descr
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstr_type,type,
% 0.27/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstrI_type,type,
% 0.27/0.46 dsetconstrI: $o ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstrI,definition,
% 0.27/0.46 ( dsetconstrI
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( Xphi @ Xx )
% 0.27/0.46 => ( in @ Xx
% 0.27/0.46 @ ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstrEL_type,type,
% 0.27/0.46 dsetconstrEL: $o ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstrEL,definition,
% 0.27/0.46 ( dsetconstrEL
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx
% 0.27/0.46 @ ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.27/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstrER_type,type,
% 0.27/0.46 dsetconstrER: $o ).
% 0.27/0.46
% 0.27/0.46 thf(dsetconstrER,definition,
% 0.27/0.46 ( dsetconstrER
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx
% 0.27/0.46 @ ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.27/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(exuE1_type,type,
% 0.27/0.46 exuE1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(exuE1,definition,
% 0.27/0.46 ( exuE1
% 0.27/0.46 = ( ! [Xphi: $i > $o] :
% 0.27/0.46 ( ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 => ? [Xx: $i] :
% 0.27/0.46 ( ( Xphi @ Xx )
% 0.27/0.46 & ! [Xy: $i] :
% 0.27/0.46 ( ( Xphi @ Xy )
% 0.27/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(prop2set_type,type,
% 0.27/0.46 prop2set: $o > $i ).
% 0.27/0.46
% 0.27/0.46 thf(prop2set,definition,
% 0.27/0.46 ( prop2set
% 0.27/0.46 = ( ^ [Xphi: $o] :
% 0.27/0.46 ( dsetconstr @ ( powerset @ emptyset )
% 0.27/0.46 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(prop2setE_type,type,
% 0.27/0.46 prop2setE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(prop2setE,definition,
% 0.27/0.46 ( prop2setE
% 0.27/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.27/0.46 => Xphi ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptysetE_type,type,
% 0.27/0.46 emptysetE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptysetE,definition,
% 0.27/0.46 ( emptysetE
% 0.27/0.46 = ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ emptyset )
% 0.27/0.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptysetimpfalse_type,type,
% 0.27/0.46 emptysetimpfalse: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptysetimpfalse,definition,
% 0.27/0.46 ( emptysetimpfalse
% 0.27/0.46 = ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ emptyset )
% 0.27/0.46 => $false ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(notinemptyset_type,type,
% 0.27/0.46 notinemptyset: $o ).
% 0.27/0.46
% 0.27/0.46 thf(notinemptyset,definition,
% 0.27/0.46 ( notinemptyset
% 0.27/0.46 = ( ! [Xx: $i] :
% 0.27/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(exuE3e_type,type,
% 0.27/0.46 exuE3e: $o ).
% 0.27/0.46
% 0.27/0.46 thf(exuE3e,definition,
% 0.27/0.46 ( exuE3e
% 0.27/0.46 = ( ! [Xphi: $i > $o] :
% 0.27/0.46 ( ( exu
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setext_type,type,
% 0.27/0.46 setext: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setext,definition,
% 0.27/0.46 ( setext
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( in @ Xx @ B ) )
% 0.27/0.46 => ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 => ( in @ Xx @ A ) )
% 0.27/0.46 => ( A = B ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(emptyI_type,type,
% 0.27/0.46 emptyI: $o ).
% 0.27/0.46
% 0.27/0.46 thf(emptyI,definition,
% 0.27/0.46 ( emptyI
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ~ ( in @ Xx @ A )
% 0.27/0.46 => ( A = emptyset ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(noeltsimpempty,conjecture,
% 0.27/0.46 ( setextAx
% 0.27/0.46 => ( emptysetAx
% 0.27/0.46 => ( setadjoinAx
% 0.27/0.46 => ( powersetAx
% 0.27/0.46 => ( setunionAx
% 0.27/0.46 => ( omega0Ax
% 0.27/0.46 => ( omegaSAx
% 0.27/0.46 => ( omegaIndAx
% 0.27/0.46 => ( replAx
% 0.27/0.46 => ( foundationAx
% 0.27/0.46 => ( wellorderingAx
% 0.27/0.46 => ( descrp
% 0.27/0.46 => ( dsetconstrI
% 0.27/0.46 => ( dsetconstrEL
% 0.27/0.46 => ( dsetconstrER
% 0.27/0.46 => ( exuE1
% 0.27/0.46 => ( prop2setE
% 0.27/0.46 => ( emptysetE
% 0.27/0.46 => ( emptysetimpfalse
% 0.27/0.46 => ( notinemptyset
% 0.27/0.46 => ( exuE3e
% 0.27/0.46 => ( setext
% 0.27/0.46 => ( emptyI
% 0.27/0.46 => ! [A: $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ~ ( in @ Xx @ A )
% 0.27/0.46 => ( A = emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 %------------------------------------------------------------------------------
% 0.27/0.46 ------------------------
% 1.83/1.32 ---- Embedded in THF ---
% 1.83/1.33 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.83/1.33 %%% Generated on Tue Feb 24 12:24:27 EST 2026
% 1.83/1.33 %%% using '$$dhol' embedding, version 1.3.0.
% 1.83/1.33 %%% Logic specification used:
% 1.83/1.33 %%% thf(spec, logic, $$dhol).
% 1.83/1.33
% 1.83/1.33 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.8JiX0Aov0I/DTF2DTF20125.p
% See solution above
% 1.83/1.33 ------------------------
% 1.83/1.34 ---- Cleaned THF ---
% 1.83/1.34 thf(in_type,type,
% 1.83/1.34 in: $i > $i > $o ).
% 1.83/1.34
% 1.83/1.34 thf(exu_type,type,
% 1.83/1.34 exu: ( $i > $o ) > $o ).
% 1.83/1.34
% 1.83/1.34 thf(exu,definition,
% 1.83/1.34 ( exu
% 1.83/1.34 = ( ^ [Xphi: $i > $o] :
% 1.83/1.34 ? [Xx: $i] :
% 1.83/1.34 ( ( Xphi @ Xx )
% 1.83/1.34 & ! [Xy: $i] :
% 1.83/1.34 ( ( Xphi @ Xy )
% 1.83/1.34 => ( Xx = Xy ) ) ) ) ) ).
% 1.83/1.34
% 1.83/1.34 thf(setextAx_type,type,
% 1.83/1.34 setextAx: $o ).
% 1.83/1.34
% 1.83/1.34 thf(setextAx,definition,
% 1.83/1.34 ( setextAx
% 1.83/1.34 = ( ! [A: $i,B: $i] :
% 1.83/1.34 ( ! [Xx: $i] :
% 1.83/1.34 ( ( in @ Xx @ A )
% 1.83/1.34 <=> ( in @ Xx @ B ) )
% 1.83/1.34 => ( A = B ) ) ) ) ).
% 1.83/1.34
% 1.83/1.34 thf(emptyset_type,type,
% 1.83/1.34 emptyset: $i ).
% 1.83/1.34
% 1.83/1.34 thf(emptysetAx_type,type,
% 1.83/1.34 emptysetAx: $o ).
% 1.83/1.34
% 1.83/1.34 thf(emptysetAx,definition,
% 1.83/1.34 ( emptysetAx
% 1.83/1.34 = ( ! [Xx: $i] :
% 1.83/1.34 ~ ( in @ Xx @ emptyset ) ) ) ).
% 1.83/1.34
% 1.83/1.34 thf(setadjoin_type,type,
% 1.83/1.34 setadjoin: $i > $i > $i ).
% 1.83/1.34
% 1.83/1.34 thf(setadjoinAx_type,type,
% 1.83/1.34 setadjoinAx: $o ).
% 1.83/1.34
% 1.83/1.34 thf(setadjoinAx,definition,
% 1.83/1.34 ( setadjoinAx
% 1.83/1.34 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.83/1.34 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.83/1.34 <=> ( ( Xy = Xx )
% 1.83/1.34 | ( in @ Xy @ A ) ) ) ) ) ).
% 1.83/1.34
% 1.83/1.34 thf(powerset_type,type,
% 1.83/1.34 powerset: $i > $i ).
% 1.83/1.34
% 1.83/1.34 thf(powersetAx_type,type,
% 1.83/1.34 powersetAx: $o ).
% 1.83/1.34
% 1.83/1.34 thf(powersetAx,definition,
% 1.83/1.34 ( powersetAx
% 1.83/1.34 = ( ! [A: $i,B: $i] :
% 1.83/1.34 ( ( in @ B @ ( powerset @ A ) )
% 1.83/1.34 <=> ! [Xx: $i] :
% 1.83/1.34 ( ( in @ Xx @ B )
% 1.83/1.34 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.83/1.34
% 1.83/1.34 thf(setunion_type,type,
% 1.83/1.34 setunion: $i > $i ).
% 1.83/1.34
% 1.83/1.34 thf(setunionAx_type,type,
% 1.83/1.34 setunionAx: $o ).
% 1.83/1.34
% 1.83/1.34 thf(setunionAx,definition,
% 1.83/1.34 ( setunionAx
% 1.83/1.34 = ( ! [A: $i,Xx: $i] :
% 1.83/1.34 ( ( in @ Xx @ ( setunion @ A ) )
% 1.83/1.34 <=> ? [B: $i] :
% 1.83/1.34 ( ( in @ Xx @ B )
% 1.83/1.34 & ( in @ B @ A ) ) ) ) ) ).
% 1.83/1.34
% 1.83/1.34 thf(omega_type,type,
% 1.83/1.34 omega: $i ).
% 1.83/1.34
% 1.83/1.34 thf(omega0Ax_type,type,
% 1.83/1.34 omega0Ax: $o ).
% 1.83/1.34
% 1.83/1.34 thf(omega0Ax,definition,
% 1.83/1.34 ( omega0Ax
% 1.83/1.34 = ( in @ emptyset @ omega ) ) ).
% 1.83/1.34
% 1.83/1.34 thf(omegaSAx_type,type,
% 1.83/1.34 omegaSAx: $o ).
% 1.83/1.34
% 1.83/1.34 thf(omegaSAx,definition,
% 1.83/1.34 ( omegaSAx
% 1.83/1.34 = ( ! [Xx: $i] :
% 1.83/1.34 ( ( in @ Xx @ omega )
% 1.83/1.34 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 1.83/1.34
% 1.83/1.34 thf(omegaIndAx_type,type,
% 1.83/1.34 omegaIndAx: $o ).
% 1.83/1.34
% 1.83/1.34 thf(omegaIndAx,definition,
% 1.83/1.34 ( omegaIndAx
% 1.83/1.34 = ( ! [A: $i] :
% 1.83/1.34 ( ( ( in @ emptyset @ A )
% 1.83/1.34 & ! [Xx: $i] :
% 1.83/1.34 ( ( ( in @ Xx @ omega )
% 1.83/1.34 & ( in @ Xx @ A ) )
% 1.83/1.34 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 1.83/1.34 => ! [Xx: $i] :
% 1.83/1.34 ( ( in @ Xx @ omega )
% 1.83/1.34 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.83/1.34
% 1.83/1.34 thf(replAx_type,type,
% 1.83/1.34 replAx: $o ).
% 1.83/1.34
% 1.83/1.34 thf(replAx,definition,
% 1.83/1.34 ( replAx
% 1.83/1.34 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 1.83/1.34 ( ! [Xx: $i] :
% 1.83/1.34 ( ( in @ Xx @ A )
% 1.83/1.34 => ( exu
% 1.83/1.34 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 1.83/1.34 => ? [B: $i] :
% 1.83/1.34 ! [Xx: $i] :
% 1.83/1.34 ( ( in @ Xx @ B )
% 1.83/1.34 <=> ? [Xy: $i] :
% 1.83/1.34 ( ( in @ Xy @ A )
% 1.83/1.34 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 1.83/1.34
% 1.83/1.34 thf(foundationAx_type,type,
% 1.83/1.34 foundationAx: $o ).
% 1.83/1.34
% 1.83/1.34 thf(foundationAx,definition,
% 1.83/1.34 ( foundationAx
% 1.83/1.34 = ( ! [A: $i] :
% 1.83/1.34 ( ? [Xx: $i] : ( in @ Xx @ A )
% 1.83/1.34 => ? [B: $i] :
% 1.83/1.34 ( ( in @ B @ A )
% 1.83/1.34 & ~ ? [Xx: $i] :
% 1.83/1.34 ( ( in @ Xx @ B )
% 1.83/1.34 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 1.83/1.34
% 1.83/1.34 thf(wellorderingAx_type,type,
% 1.83/1.34 wellorderingAx: $o ).
% 1.83/1.34
% 1.83/1.34 thf(wellorderingAx,definition,
% 1.83/1.34 ( wellorderingAx
% 1.83/1.34 = ( ! [A: $i] :
% 1.83/1.34 ? [B: $i] :
% 1.83/1.34 ( ! [C: $i] :
% 1.83/1.34 ( ( in @ C @ B )
% 1.83/1.34 => ! [Xx: $i] :
% 1.83/1.34 ( ( in @ Xx @ C )
% 1.83/1.34 => ( in @ Xx @ A ) ) )
% 1.83/1.34 & ! [Xx: $i,Xy: $i] :
% 1.83/1.34 ( ( ( in @ Xx @ A )
% 1.83/1.34 & ( in @ Xy @ A ) )
% 1.83/1.34 => ( ! [C: $i] :
% 1.83/1.34 ( ( in @ C @ B )
% 1.83/1.34 => ( ( in @ Xx @ C )
% 1.83/1.34 <=> ( in @ Xy @ C ) ) )
% 1.83/1.34 => ( Xx = Xy ) ) )
% 1.83/1.34 & ! [C: $i,D: $i] :
% 1.83/1.34 ( ( ( in @ C @ B )
% 1.83/1.34 & ( in @ D @ B ) )
% 1.83/1.34 => ( ! [Xx: $i] :
% 1.83/1.34 ( ( in @ Xx @ C )
% 1.83/1.34 => ( in @ Xx @ D ) )
% 1.83/1.34 | ! [Xx: $i] :
% 1.83/1.34 ( ( in @ Xx @ D )
% 1.83/1.34 => ( in @ Xx @ C ) ) ) )
% 1.83/1.34 & ! [C: $i] :
% 1.83/1.34 ( ( ! [Xx: $i] :
% 1.83/1.34 ( ( in @ Xx @ C )
% 1.83/1.34 => ( in @ Xx @ A ) )
% 1.83/1.34 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 1.83/1.34 => ? [D: $i,Xx: $i] :
% 1.83/1.34 ( ( in @ D @ B )
% 1.83/1.34 & ( in @ Xx @ C )
% 1.83/1.34 & ~ ? [Xy: $i] :
% 1.83/1.34 ( ( in @ Xy @ D )
% 1.83/1.34 & ( in @ Xy @ C ) )
% 1.83/1.34 & ! [E: $i] :
% 1.83/1.34 ( ( in @ E @ B )
% 1.83/1.34 => ( ! [Xy: $i] :
% 1.83/1.34 ( ( in @ Xy @ E )
% 1.83/1.34 => ( in @ Xy @ D ) )
% 1.83/1.34 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 1.83/1.34
% 1.83/1.34 thf(descr_type,type,
% 1.83/1.34 descr: ( $i > $o ) > $i ).
% 1.83/1.34
% 1.83/1.34 thf(descrp_type,type,
% 1.83/1.34 descrp: $o ).
% 1.83/1.34
% 1.83/1.34 thf(descrp,definition,
% 1.83/1.34 ( descrp
% 1.83/1.34 = ( ! [Xphi: $i > $o] :
% 1.83/1.34 ( ( exu
% 1.83/1.34 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.83/1.34 => ( Xphi
% 1.83/1.34 @ ( descr
% 1.83/1.34 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 1.83/1.34
% 1.83/1.34 thf(dsetconstr_type,type,
% 1.83/1.34 dsetconstr: $i > ( $i > $o ) > $i ).
% 1.83/1.34
% 1.83/1.34 thf(dsetconstrI_type,type,
% 1.83/1.34 dsetconstrI: $o ).
% 1.83/1.34
% 1.83/1.34 thf(dsetconstrI,definition,
% 1.83/1.34 ( dsetconstrI
% 1.83/1.34 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.83/1.34 ( ( in @ Xx @ A )
% 1.83/1.34 => ( ( Xphi @ Xx )
% 1.83/1.34 => ( in @ Xx
% 1.83/1.34 @ ( dsetconstr @ A
% 1.83/1.34 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 1.83/1.34
% 1.83/1.34 thf(dsetconstrEL_type,type,
% 1.83/1.34 dsetconstrEL: $o ).
% 1.83/1.34
% 1.83/1.34 thf(dsetconstrEL,definition,
% 1.83/1.34 ( dsetconstrEL
% 1.83/1.34 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.83/1.34 ( ( in @ Xx
% 1.83/1.34 @ ( dsetconstr @ A
% 1.83/1.34 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.83/1.34 => ( in @ Xx @ A ) ) ) ) ).
% 1.83/1.34
% 1.83/1.34 thf(dsetconstrER_type,type,
% 1.83/1.34 dsetconstrER: $o ).
% 1.83/1.34
% 1.83/1.34 thf(dsetconstrER,definition,
% 1.83/1.34 ( dsetconstrER
% 1.83/1.34 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.83/1.34 ( ( in @ Xx
% 1.83/1.34 @ ( dsetconstr @ A
% 1.83/1.34 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.83/1.34 => ( Xphi @ Xx ) ) ) ) ).
% 1.83/1.34
% 1.83/1.34 thf(exuE1_type,type,
% 1.83/1.34 exuE1: $o ).
% 1.83/1.34
% 1.83/1.34 thf(exuE1,definition,
% 1.83/1.34 ( exuE1
% 1.83/1.34 = ( ! [Xphi: $i > $o] :
% 1.83/1.34 ( ( exu
% 1.83/1.34 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.83/1.34 => ? [Xx: $i] :
% 1.83/1.34 ( ( Xphi @ Xx )
% 1.83/1.34 & ! [Xy: $i] :
% 1.83/1.34 ( ( Xphi @ Xy )
% 1.83/1.34 => ( Xx = Xy ) ) ) ) ) ) ).
% 1.83/1.34
% 1.83/1.34 thf(prop2set_type,type,
% 1.83/1.34 prop2set: $o > $i ).
% 1.83/1.34
% 1.83/1.34 thf(prop2set,definition,
% 1.83/1.34 ( prop2set
% 1.83/1.34 = ( ^ [Xphi: $o] :
% 1.83/1.34 ( dsetconstr @ ( powerset @ emptyset )
% 1.83/1.34 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 1.83/1.34
% 1.83/1.34 thf(prop2setE_type,type,
% 1.83/1.34 prop2setE: $o ).
% 1.83/1.34
% 1.83/1.34 thf(prop2setE,definition,
% 1.83/1.34 ( prop2setE
% 1.83/1.34 = ( ! [Xphi: $o,Xx: $i] :
% 1.83/1.34 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 1.83/1.34 => Xphi ) ) ) ).
% 1.83/1.34
% 1.83/1.34 thf(emptysetE_type,type,
% 1.83/1.34 emptysetE: $o ).
% 1.83/1.34
% 1.83/1.34 thf(emptysetE,definition,
% 1.83/1.34 ( emptysetE
% 1.83/1.34 = ( ! [Xx: $i] :
% 1.83/1.34 ( ( in @ Xx @ emptyset )
% 1.83/1.34 => ! [Xphi: $o] : Xphi ) ) ) ).
% 1.83/1.34
% 1.83/1.34 thf(emptysetimpfalse_type,type,
% 1.83/1.34 emptysetimpfalse: $o ).
% 1.83/1.34
% 1.83/1.34 thf(emptysetimpfalse,definition,
% 1.83/1.34 ( emptysetimpfalse
% 1.83/1.34 = ( ! [Xx: $i] :
% 1.83/1.34 ( ( in @ Xx @ emptyset )
% 1.83/1.34 => $false ) ) ) ).
% 1.83/1.34
% 1.83/1.34 thf(notinemptyset_type,type,
% 1.83/1.34 notinemptyset: $o ).
% 1.83/1.34
% 1.83/1.34 thf(notinemptyset,definition,
% 1.83/1.34 ( notinemptyset
% 1.83/1.34 = ( ! [Xx: $i] :
% 1.83/1.34 ~ ( in @ Xx @ emptyset ) ) ) ).
% 1.83/1.34
% 1.83/1.34 thf(exuE3e_type,type,
% 1.83/1.34 exuE3e: $o ).
% 1.83/1.34
% 1.83/1.34 thf(exuE3e,definition,
% 1.83/1.34 ( exuE3e
% 1.83/1.34 = ( ! [Xphi: $i > $o] :
% 1.83/1.34 ( ( exu
% 1.83/1.34 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.83/1.34 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 1.83/1.34
% 1.83/1.34 thf(setext_type,type,
% 1.83/1.34 setext: $o ).
% 1.83/1.34
% 1.83/1.34 thf(setext,definition,
% 1.83/1.34 ( setext
% 1.83/1.34 = ( ! [A: $i,B: $i] :
% 1.83/1.34 ( ! [Xx: $i] :
% 1.83/1.34 ( ( in @ Xx @ A )
% 1.83/1.34 => ( in @ Xx @ B ) )
% 1.83/1.34 => ( ! [Xx: $i] :
% 1.83/1.34 ( ( in @ Xx @ B )
% 1.83/1.34 => ( in @ Xx @ A ) )
% 1.83/1.34 => ( A = B ) ) ) ) ) ).
% 1.83/1.34
% 1.83/1.34 thf(emptyI_type,type,
% 1.83/1.34 emptyI: $o ).
% 1.83/1.34
% 1.83/1.34 thf(emptyI,definition,
% 1.83/1.34 ( emptyI
% 1.83/1.34 = ( ! [A: $i] :
% 1.83/1.34 ( ! [Xx: $i] :
% 1.83/1.34 ~ ( in @ Xx @ A )
% 1.83/1.34 => ( A = emptyset ) ) ) ) ).
% 1.83/1.34
% 1.83/1.34 thf(noeltsimpempty,conjecture,
% 1.83/1.34 ( setextAx
% 1.83/1.34 => ( emptysetAx
% 1.83/1.34 => ( setadjoinAx
% 1.83/1.34 => ( powersetAx
% 1.83/1.34 => ( setunionAx
% 1.83/1.34 => ( omega0Ax
% 1.83/1.34 => ( omegaSAx
% 1.83/1.34 => ( omegaIndAx
% 1.83/1.34 => ( replAx
% 1.83/1.34 => ( foundationAx
% 1.83/1.34 => ( wellorderingAx
% 1.83/1.34 => ( descrp
% 1.83/1.34 => ( dsetconstrI
% 1.83/1.34 => ( dsetconstrEL
% 1.83/1.34 => ( dsetconstrER
% 1.83/1.34 => ( exuE1
% 1.83/1.34 => ( prop2setE
% 1.83/1.34 => ( emptysetE
% 1.83/1.34 => ( emptysetimpfalse
% 1.83/1.34 => ( notinemptyset
% 1.83/1.34 => ( exuE3e
% 1.83/1.34 => ( setext
% 1.83/1.34 => ( emptyI
% 1.83/1.34 => ! [A: $i] :
% 1.83/1.34 ( ! [Xx: $i] :
% 1.83/1.34 ~ ( in @ Xx @ A )
% 1.83/1.34 => ( A = emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 1.83/1.34 ------------------------
% 1.83/1.36 % (20339)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_20125 for (2999ds/2Mi)
% 1.83/1.36 % (20339)Instruction limit reached!
% 1.83/1.36 % (20339)------------------------------
% 1.83/1.36 % (20339)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.83/1.36 % (20339)Termination reason: Unknown
% 1.83/1.36 % (20339)Termination phase: shuffling
% 1.83/1.36
% 1.83/1.36 % (20339)Memory used [KB]: 1023
% 1.83/1.36 % (20339)Time elapsed: 0.002 s
% 1.83/1.36 % (20339)Instructions burned: 3 (million)
% 1.83/1.36 % (20339)------------------------------
% 1.83/1.36 % (20339)------------------------------
% 1.83/1.36 % (20340)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_20125 for (2999ds/275Mi)
% 1.83/1.36 % (20335)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_20125 for (2999ds/183Mi)
% 1.83/1.36 % (20336)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_20125 for (2999ds/4Mi)
% 1.83/1.36 % (20337)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_20125 for (2999ds/27Mi)
% 1.83/1.36 % (20338)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_20125 for (2999ds/2Mi)
% 1.83/1.36 % (20341)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_20125 for (2999ds/18Mi)
% 1.83/1.37 % (20338)Instruction limit reached!
% 1.83/1.37 % (20338)------------------------------
% 1.83/1.37 % (20338)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.83/1.37 % (20338)Termination reason: Unknown
% 1.83/1.37 % (20338)Termination phase: shuffling
% 1.83/1.37
% 1.83/1.37 % (20338)Memory used [KB]: 1023
% 1.83/1.37 % (20338)Time elapsed: 0.003 s
% 1.83/1.37 % (20338)Instructions burned: 3 (million)
% 1.83/1.37 % (20338)------------------------------
% 1.83/1.37 % (20338)------------------------------
% 1.83/1.37 % (20336)Instruction limit reached!
% 1.83/1.37 % (20336)------------------------------
% 1.83/1.37 % (20336)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.83/1.37 % (20336)Termination reason: Unknown
% 1.83/1.37 % (20336)Termination phase: Property scanning
% 1.83/1.37
% 1.83/1.37 % (20336)Memory used [KB]: 1023
% 1.83/1.37 % (20336)Time elapsed: 0.005 s
% 1.83/1.37 % (20336)Instructions burned: 5 (million)
% 1.83/1.37 % (20336)------------------------------
% 1.83/1.37 % (20336)------------------------------
% 1.83/1.37 % (20342)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_20125 for (2999ds/3Mi)
% 1.83/1.37 % (20342)Instruction limit reached!
% 1.83/1.37 % (20342)------------------------------
% 1.83/1.37 % (20342)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.83/1.37 % (20342)Termination reason: Unknown
% 1.83/1.37 % (20342)Termination phase: shuffling
% 1.83/1.37
% 1.83/1.37 % (20342)Memory used [KB]: 1023
% 1.83/1.37 % (20342)Time elapsed: 0.002 s
% 1.83/1.37 % (20342)Instructions burned: 4 (million)
% 1.83/1.37 % (20342)------------------------------
% 1.83/1.37 % (20342)------------------------------
% 1.83/1.37 % (20341)Instruction limit reached!
% 1.83/1.37 % (20341)------------------------------
% 1.83/1.37 % (20341)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.83/1.37 % (20341)Termination reason: Unknown
% 1.83/1.37 % (20341)Termination phase: Function definition elimination
% 1.83/1.37
% 1.83/1.37 % (20341)Memory used [KB]: 1407
% 1.83/1.37 % (20341)Time elapsed: 0.012 s
% 1.83/1.37 % (20341)Instructions burned: 18 (million)
% 1.83/1.37 % (20341)------------------------------
% 1.83/1.37 % (20341)------------------------------
% 1.83/1.37 % (20340)First to succeed.
% 1.83/1.38 % (20340)Refutation found. Thanks to Tanya!
% 1.83/1.38 % SZS status Theorem for DTF2THF_20125
% 1.83/1.38 % SZS output start Proof for DTF2THF_20125
% 1.83/1.38 thf(func_def_0, type, in: $i > $i > $o).
% 1.83/1.38 thf(func_def_1, type, exu: ($i > $o) > $o).
% 1.83/1.38 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 1.83/1.38 thf(func_def_8, type, powerset: $i > $i).
% 1.83/1.38 thf(func_def_10, type, setunion: $i > $i).
% 1.83/1.38 thf(func_def_19, type, descr: ($i > $o) > $i).
% 1.83/1.38 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 1.83/1.38 thf(func_def_26, type, prop2set: $o > $i).
% 1.83/1.38 thf(func_def_45, type, sP0: $i > $i > $o).
% 1.83/1.38 thf(func_def_46, type, sP1: $i > $o).
% 1.83/1.38 thf(func_def_47, type, sP2: $i > $i > $o).
% 1.83/1.38 thf(func_def_48, type, sP3: $i > $i > $o).
% 1.83/1.38 thf(func_def_51, type, sK6: $i > $i > $i).
% 1.83/1.38 thf(func_def_52, type, sK7: $i > $i > $i).
% 1.83/1.38 thf(func_def_55, type, sK10: $i > $o).
% 1.83/1.38 thf(func_def_61, type, sK16: $i > $i > $i).
% 1.83/1.38 thf(func_def_62, type, sK17: $i > $i > $i).
% 1.83/1.38 thf(func_def_63, type, sK18: $i > $i > $i).
% 1.83/1.38 thf(func_def_64, type, sK19: $i > $i > $i).
% 1.83/1.38 thf(func_def_65, type, sK20: $i > $i > $i > $i).
% 1.83/1.38 thf(func_def_66, type, sK21: $i > $i).
% 1.83/1.38 thf(func_def_67, type, sK22: $i > $i).
% 1.83/1.38 thf(func_def_68, type, sK23: $i > $i).
% 1.83/1.38 thf(func_def_69, type, sK24: $i > $i).
% 1.83/1.38 thf(func_def_70, type, sK25: $i > $i > $i).
% 1.83/1.38 thf(func_def_71, type, sK26: $i > $i > $i > $i).
% 1.83/1.38 thf(func_def_72, type, sK27: $i > $i > $i > $i).
% 1.83/1.38 thf(func_def_73, type, sK28: $i > $i > $i).
% 1.83/1.38 thf(func_def_74, type, sK29: $i > $i > $i).
% 1.83/1.38 thf(func_def_75, type, sK30: $i > $i).
% 1.83/1.38 thf(func_def_76, type, sK31: $i > $i > $i).
% 1.83/1.38 thf(func_def_78, type, sK33: $i > $i).
% 1.83/1.38 thf(func_def_79, type, sK34: $i > $i).
% 1.83/1.38 thf(func_def_82, type, sK37: $i > $i).
% 1.83/1.38 thf(func_def_84, type, sK39: $i > $i).
% 1.83/1.38 thf(func_def_87, type, sK42: $i > $i > $i).
% 1.83/1.38 thf(func_def_88, type, sK43: ($i > $o) > $i).
% 1.83/1.38 thf(func_def_89, type, sK44: $i > $o).
% 1.83/1.38 thf(func_def_90, type, sK45: $i > $i).
% 1.83/1.38 thf(func_def_91, type, sK46: $i > $i).
% 1.83/1.38 thf(func_def_96, type, sK51: $i > $i > $i).
% 1.83/1.38 thf(func_def_102, type, sK57: $i > $i).
% 1.83/1.38 thf(func_def_109, type, sK64: $i > $o).
% 1.83/1.38 thf(func_def_110, type, sK65: ($i > $o) > $i).
% 1.83/1.38 thf(func_def_111, type, sK66: $i > $o).
% 1.83/1.38 thf(func_def_114, type, sK69: $i > $o).
% 1.83/1.38 thf(func_def_116, type, sK71: $i > $i > $o).
% 1.83/1.38 thf(func_def_117, type, sK72: $i > $i).
% 1.83/1.38 thf(func_def_118, type, sK73: $i > $i).
% 1.83/1.38 thf(func_def_119, type, sK74: ($i > $i > $o) > $i > $i).
% 1.83/1.38 thf(func_def_120, type, sK75: $i > ($i > $i > $o) > $i > $i).
% 1.83/1.38 thf(func_def_121, type, sK76: ($i > $i > $o) > $i > $i).
% 1.83/1.38 thf(func_def_123, type, ph78: !>[X0: $tType]:(X0)).
% 1.83/1.38 thf(f621,plain,(
% 1.83/1.38 $false),
% 1.83/1.38 inference(subsumption_resolution,[status(thm)],[f619,f342])).
% 1.83/1.38 thf(f342,plain,(
% 1.83/1.38 (emptyset != sK58)),
% 1.83/1.38 inference(cnf_transformation,[status(thm)],[f214])).
% 1.83/1.38 thf(f214,plain,(
% 1.83/1.38 (descrp = $true) & (setadjoinAx = $true) & (omegaIndAx = $true) & (emptysetE = $true) & (dsetconstrEL = $true) & (exuE3e = $true) & (emptysetimpfalse = $true) & (exuE1 = $true) & (dsetconstrER = $true) & (notinemptyset = $true) & (setextAx = $true) & (emptysetAx = $true) & (powersetAx = $true) & (foundationAx = $true) & (omegaSAx = $true) & (omega0Ax = $true) & (replAx = $true) & (wellorderingAx = $true) & (dsetconstrI = $true) & (setunionAx = $true) & (emptyI = $true) & (setext = $true) & (! [X1 : $i] : ($true != (in @ X1 @ sK58)) & (emptyset != sK58)) & (prop2setE = $true)),
% 1.83/1.38 inference(skolemisation,[status(esa),new_symbols(skolem,[sK58])],[f88,f213])).
% 1.83/1.38 thf(f213,plain,(
% 1.83/1.38 ? [X0 : $i] : (! [X1 : $i] : ($true != (in @ X1 @ X0)) & (emptyset != X0)) => (! [X1 : $i] : ($true != (in @ X1 @ sK58)) & (emptyset != sK58))),
% 1.83/1.38 introduced(definition,[],[choice_axiom])).
% 1.83/1.38 thf(f88,plain,(
% 1.83/1.38 (descrp = $true) & (setadjoinAx = $true) & (omegaIndAx = $true) & (emptysetE = $true) & (dsetconstrEL = $true) & (exuE3e = $true) & (emptysetimpfalse = $true) & (exuE1 = $true) & (dsetconstrER = $true) & (notinemptyset = $true) & (setextAx = $true) & (emptysetAx = $true) & (powersetAx = $true) & (foundationAx = $true) & (omegaSAx = $true) & (omega0Ax = $true) & (replAx = $true) & (wellorderingAx = $true) & (dsetconstrI = $true) & (setunionAx = $true) & (emptyI = $true) & (setext = $true) & ? [X0 : $i] : (! [X1 : $i] : ($true != (in @ X1 @ X0)) & (emptyset != X0)) & (prop2setE = $true)),
% 1.83/1.38 inference(flattening,[status(thm)],[f87])).
% 1.83/1.38 thf(f87,plain,(
% 1.83/1.38 ((((((((((((((((((((((? [X0 : $i] : (! [X1 : $i] : ($true != (in @ X1 @ X0)) & (emptyset != X0)) & (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)),
% 1.83/1.38 inference(ennf_transformation,[status(thm)],[f78])).
% 1.83/1.38 thf(f78,plain,(
% 1.83/1.38 ~((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) => ! [X0 : $i] : (! [X1 : $i] : ($true != (in @ X1 @ X0)) => (emptyset = X0)))))))))))))))))))))))))),
% 1.83/1.38 inference(flattening,[status(thm)],[f77])).
% 1.83/1.38 thf(f77,plain,(
% 1.83/1.38 ~((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) => ! [X0 : $i] : (! [X1 : $i] : ~($true = (in @ X1 @ X0)) => (emptyset = X0)))))))))))))))))))))))))),
% 1.83/1.38 inference(fool_elimination,[status(thm)],[f76])).
% 1.83/1.38 thf(f76,plain,(
% 1.83/1.38 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => ! [X0 : $i] : (! [X1 : $i] : ~(in @ X1 @ X0) => (emptyset = X0)))))))))))))))))))))))))),
% 1.83/1.38 inference(rectify,[status(thm)],[f27])).
% 1.83/1.38 thf(f27,negated_conjecture,(
% 1.83/1.38 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => ! [X3 : $i] : (! [X1 : $i] : ~(in @ X1 @ X3) => (emptyset = X3)))))))))))))))))))))))))),
% 1.83/1.38 inference(negated_conjecture,[status(cth)],[f26])).
% 1.83/1.38 thf(f26,conjecture,(
% 1.83/1.38 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => ! [X3 : $i] : (! [X1 : $i] : ~(in @ X1 @ X3) => (emptyset = X3))))))))))))))))))))))))),
% 1.83/1.38 file('/export/starexec/sandbox2/tmp/tmp.8JiX0Aov0I/DTF2THF_20125.p',noeltsimpempty)).
% 1.83/1.38 thf(f619,plain,(
% 1.83/1.38 (emptyset = sK58)),
% 1.83/1.38 inference(trivial_inequality_removal,[status(thm)],[f618])).
% 1.83/1.38 thf(f618,plain,(
% 1.83/1.38 (emptyset = sK58) | ($true != $true)),
% 1.83/1.38 inference(superposition,[status(thm)],[f343,f520])).
% 1.83/1.38 thf(f520,plain,(
% 1.83/1.38 ( ! [X0 : $i] : (($true = (in @ (sK46 @ X0) @ X0)) | (emptyset = X0)) )),
% 1.83/1.38 inference(trivial_inequality_removal,[status(thm)],[f436])).
% 1.83/1.38 thf(f436,plain,(
% 1.83/1.38 ( ! [X0 : $i] : (($true = (in @ (sK46 @ X0) @ X0)) | ($true != $true) | (emptyset = X0)) )),
% 1.83/1.38 inference(definition_unfolding,[status(thm)],[f322,f345])).
% 1.83/1.38 thf(f345,plain,(
% 1.83/1.38 (emptyI = $true)),
% 1.83/1.38 inference(cnf_transformation,[status(thm)],[f214])).
% 1.83/1.38 thf(f322,plain,(
% 1.83/1.38 ( ! [X0 : $i] : ((emptyset = X0) | ($true = (in @ (sK46 @ X0) @ X0)) | (emptyI != $true)) )),
% 1.83/1.38 inference(cnf_transformation,[status(thm)],[f192])).
% 1.83/1.38 thf(f192,plain,(
% 1.83/1.38 (! [X0 : $i] : ((emptyset = X0) | ($true = (in @ (sK46 @ X0) @ X0))) | (emptyI != $true)) & ((emptyI = $true) | ((emptyset != sK47) & ! [X3 : $i] : ($true != (in @ X3 @ sK47))))),
% 1.83/1.38 inference(skolemisation,[status(esa),new_symbols(skolem,[sK46,sK47])],[f189,f191,f190])).
% 1.83/1.38 thf(f190,plain,(
% 1.83/1.38 ! [X0 : $i] : (? [X1 : $i] : ($true = (in @ X1 @ X0)) => ($true = (in @ (sK46 @ X0) @ X0)))),
% 1.83/1.38 introduced(definition,[],[choice_axiom])).
% 1.83/1.38 thf(f191,plain,(
% 1.83/1.38 ? [X2 : $i] : ((emptyset != X2) & ! [X3 : $i] : ($true != (in @ X3 @ X2))) => ((emptyset != sK47) & ! [X3 : $i] : ($true != (in @ X3 @ sK47)))),
% 1.83/1.38 introduced(definition,[],[choice_axiom])).
% 1.83/1.38 thf(f189,plain,(
% 1.83/1.38 (! [X0 : $i] : ((emptyset = X0) | ? [X1 : $i] : ($true = (in @ X1 @ X0))) | (emptyI != $true)) & ((emptyI = $true) | ? [X2 : $i] : ((emptyset != X2) & ! [X3 : $i] : ($true != (in @ X3 @ X2))))),
% 1.83/1.38 inference(rectify,[status(thm)],[f188])).
% 1.83/1.38 thf(f188,plain,(
% 1.83/1.38 (! [X0 : $i] : ((emptyset = X0) | ? [X1 : $i] : ($true = (in @ X1 @ X0))) | (emptyI != $true)) & ((emptyI = $true) | ? [X0 : $i] : ((emptyset != X0) & ! [X1 : $i] : ($true != (in @ X1 @ X0))))),
% 1.83/1.38 inference(nnf_transformation,[status(thm)],[f97])).
% 1.83/1.38 thf(f97,plain,(
% 1.83/1.38 ! [X0 : $i] : ((emptyset = X0) | ? [X1 : $i] : ($true = (in @ X1 @ X0))) <=> (emptyI = $true)),
% 1.83/1.38 inference(ennf_transformation,[status(thm)],[f86])).
% 1.83/1.38 thf(f86,plain,(
% 1.83/1.38 ! [X0 : $i] : (! [X1 : $i] : ($true != (in @ X1 @ X0)) => (emptyset = X0)) <=> (emptyI = $true)),
% 1.83/1.38 inference(flattening,[status(thm)],[f41])).
% 1.83/1.38 thf(f41,plain,(
% 1.83/1.38 (emptyI = $true) <=> ! [X0 : $i] : (! [X1 : $i] : ~($true = (in @ X1 @ X0)) => (emptyset = X0))),
% 1.83/1.38 inference(fool_elimination,[status(thm)],[f40])).
% 1.83/1.38 thf(f40,plain,(
% 1.83/1.38 (emptyI = ! [X0 : $i] : (! [X1 : $i] : ~(in @ X1 @ X0) => (emptyset = X0)))),
% 1.83/1.38 inference(rectify,[status(thm)],[f25])).
% 1.83/1.38 thf(f25,axiom,(
% 1.83/1.38 (emptyI = ! [X3 : $i] : (! [X1 : $i] : ~(in @ X1 @ X3) => (emptyset = X3)))),
% 1.83/1.38 file('/export/starexec/sandbox2/tmp/tmp.8JiX0Aov0I/DTF2THF_20125.p',emptyI)).
% 1.83/1.38 thf(f343,plain,(
% 1.83/1.38 ( ! [X1 : $i] : (($true != (in @ X1 @ sK58))) )),
% 1.83/1.38 inference(cnf_transformation,[status(thm)],[f214])).
% 1.83/1.38 % SZS output end Proof for DTF2THF_20125
% 1.83/1.38 % (20340)------------------------------
% 1.83/1.38 % (20340)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.83/1.38 % (20340)Termination reason: Refutation
% 1.83/1.38
% 1.83/1.38 % (20340)Memory used [KB]: 6140
% 1.83/1.38 % (20340)Time elapsed: 0.018 s
% 1.83/1.38 % (20340)Instructions burned: 50 (million)
% 1.83/1.38 % (20340)------------------------------
% 1.83/1.38 % (20340)------------------------------
% 1.83/1.38 % (20334)Success in time 0.027 s
%------------------------------------------------------------------------------