%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU501^1 : 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 09:34:57 AM UTC 2026
% Result : Theorem 1.97s 1.54s
% Output : Refutation 1.97s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 5
% Syntax : Number of formulae : 24 ( 7 unt; 0 typ; 0 def)
% Number of atoms : 409 ( 110 equ; 0 cnn)
% Maximal formula atoms : 19 ( 17 avg)
% Number of connectives : 220 ( 26 ~; 8 |; 59 &; 50 @)
% ( 2 <=>; 75 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 8 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 72 ( 69 usr; 23 con; 0-3 aty)
% Number of variables : 29 ( 0 ^; 18 !; 11 ?; 29 :)
% 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_40,type,
sP1: $i > $i > $o ).
thf(func_def_41,type,
sP2: $i > $o ).
thf(func_def_42,type,
sP3: $i > $i > $o ).
thf(func_def_43,type,
sP4: $i > $i > $o ).
thf(func_def_44,type,
sK5: $i > $o ).
thf(func_def_47,type,
sK8: ( $i > $o ) > $i ).
thf(func_def_48,type,
sK9: $i > $o ).
thf(func_def_49,type,
sK10: $i > $i ).
thf(func_def_51,type,
sK12: $i > $i > $o ).
thf(func_def_52,type,
sK13: $i > $i ).
thf(func_def_53,type,
sK14: $i > $i ).
thf(func_def_54,type,
sK15: ( $i > $i > $o ) > $i > $i ).
thf(func_def_55,type,
sK16: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_56,type,
sK17: ( $i > $i > $o ) > $i > $i ).
thf(func_def_59,type,
sK20: $i > $i > $i ).
thf(func_def_60,type,
sK21: $i > $i > $i ).
thf(func_def_61,type,
sK22: $i > $i > $i ).
thf(func_def_62,type,
sK23: $i > $i > $i ).
thf(func_def_63,type,
sK24: $i > $i > $i ).
thf(func_def_64,type,
sK25: $i > $i > $i > $i ).
thf(func_def_65,type,
sK26: $i > $i ).
thf(func_def_66,type,
sK27: $i > $i ).
thf(func_def_67,type,
sK28: $i > $i ).
thf(func_def_68,type,
sK29: $i > $i ).
thf(func_def_69,type,
sK30: $i > $i > $i > $i ).
thf(func_def_70,type,
sK31: $i > $i > $i > $i ).
thf(func_def_71,type,
sK32: $i > $i > $i ).
thf(func_def_72,type,
sK33: $i > $i > $i ).
thf(func_def_73,type,
sK34: $i > $i > $i ).
thf(func_def_75,type,
sK36: $i > $i ).
thf(func_def_76,type,
sK37: $i > $i ).
thf(func_def_77,type,
sK38: $i > $i ).
thf(func_def_83,type,
sK44: $i > $i > $i ).
thf(func_def_89,type,
sK50: $i > $i ).
thf(func_def_90,type,
sK51: $i > $i ).
thf(func_def_93,type,
sK54: $i > $o ).
thf(func_def_94,type,
sK55: $i > $o ).
thf(func_def_98,type,
sK59: $i > $i > $i ).
thf(func_def_104,type,
sK65: $i > $i ).
thf(func_def_106,type,
ph67:
!>[X0: $tType] : X0 ).
thf(f438,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f425]) ).
thf(f425,plain,
$true != $true,
inference(superposition,[status(thm)],[f379,f264]) ).
thf(f264,plain,
( ( in @ sK45 @ emptyset )
= $true ),
inference(cnf_transformation,[status(thm)],[f154]) ).
thf(f154,plain,
( ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( prop2setE = $true )
& ( wellorderingAx = $true )
& ( setadjoinAx = $true )
& ( exuE1 = $true )
& ( replAx = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrER = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( foundationAx = $true )
& ( powersetAx = $true )
& ( sK46 != $true )
& ( ( in @ sK45 @ emptyset )
= $true )
& ( setextAx = $true )
& ( dsetconstrI = $true )
& ( emptysetAx = $true )
& ( descrp = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK45,sK46])],[f69,f153,f152]) ).
thf(f152,plain,
( ? [X0: $i] :
( ? [X1: $o] : ( $true != X1 )
& ( $true
= ( in @ X0 @ emptyset ) ) )
=> ( ? [X1: $o] : ( $true != X1 )
& ( ( in @ sK45 @ emptyset )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f153,plain,
( ? [X1: $o] : ( $true != X1 )
=> ( sK46 != $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f69,plain,
( ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( prop2setE = $true )
& ( wellorderingAx = $true )
& ( setadjoinAx = $true )
& ( exuE1 = $true )
& ( replAx = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrER = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( foundationAx = $true )
& ( powersetAx = $true )
& ? [X0: $i] :
( ? [X1: $o] : ( $true != X1 )
& ( $true
= ( in @ X0 @ emptyset ) ) )
& ( setextAx = $true )
& ( dsetconstrI = $true )
& ( emptysetAx = $true )
& ( descrp = $true ) ),
inference(flattening,[status(thm)],[f68]) ).
thf(f68,plain,
( ? [X0: $i] :
( ? [X1: $o] : ( $true != X1 )
& ( $true
= ( in @ X0 @ emptyset ) ) )
& ( 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)],[f56]) ).
thf(f56,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 )
=> ! [X0: $i] :
( ( $true
= ( in @ X0 @ emptyset ) )
=> ! [X1: $o] : ( $true = X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f55]) ).
thf(f55,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ! [X0: $i] :
( ( in @ X0 @ emptyset )
=> ! [X1: $o] : X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f21]) ).
thf(f21,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X0: $o] : X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f20]) ).
thf(f20,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X0: $o] : X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.YoXTo2yw5Z/DTF2THF_26364.p',emptysetE) ).
thf(f379,plain,
! [X1: $i] :
( ( in @ X1 @ emptyset )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f341]) ).
thf(f341,plain,
! [X1: $i] :
( ( $true != $true )
| ( ( in @ X1 @ emptyset )
!= $true ) ),
inference(definition_unfolding,[status(thm)],[f279,f261]) ).
thf(f261,plain,
emptysetAx = $true,
inference(cnf_transformation,[status(thm)],[f154]) ).
thf(f279,plain,
! [X1: $i] :
( ( ( in @ X1 @ emptyset )
!= $true )
| ( emptysetAx != $true ) ),
inference(cnf_transformation,[status(thm)],[f158]) ).
thf(f158,plain,
( ( ( emptysetAx = $true )
| ( $true
= ( in @ sK47 @ emptyset ) ) )
& ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
!= $true )
| ( emptysetAx != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK47])],[f156,f157]) ).
thf(f157,plain,
( ? [X0: $i] :
( $true
= ( in @ X0 @ emptyset ) )
=> ( $true
= ( in @ sK47 @ emptyset ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f156,plain,
( ( ( emptysetAx = $true )
| ? [X0: $i] :
( $true
= ( in @ X0 @ emptyset ) ) )
& ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
!= $true )
| ( emptysetAx != $true ) ) ),
inference(rectify,[status(thm)],[f155]) ).
thf(f155,plain,
( ( ( emptysetAx = $true )
| ? [X0: $i] :
( $true
= ( in @ X0 @ emptyset ) ) )
& ( ! [X0: $i] :
( $true
!= ( in @ X0 @ emptyset ) )
| ( emptysetAx != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f60]) ).
thf(f60,plain,
( ( emptysetAx = $true )
<=> ! [X0: $i] :
( $true
!= ( in @ X0 @ emptyset ) ) ),
inference(flattening,[status(thm)],[f32]) ).
thf(f32,plain,
( ( emptysetAx = $true )
<=> ! [X0: $i] :
( $true
!= ( in @ X0 @ emptyset ) ) ),
inference(fool_elimination,[status(thm)],[f31]) ).
thf(f31,plain,
( emptysetAx
= ( ! [X0: $i] :
~ ( in @ X0 @ emptyset ) ) ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
( emptysetAx
= ( ! [X1: $i] :
~ ( in @ X1 @ emptyset ) ) ),
file('/export/starexec/sandbox/tmp/tmp.YoXTo2yw5Z/DTF2THF_26364.p',emptysetAx) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU501^1 : 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
% 0.16/0.34 % Computer : n011.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 12:24:02 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.48 ---- Original DTF file ---
% 0.19/0.48 thf(spec,logic,$$dhol).
% 0.19/0.48 %------------------------------------------------------------------------------
% 0.19/0.48 % File : SEU501^1 : TPTP v9.2.1. Released v3.7.0.
% 0.19/0.48 % Domain : Set Theory
% 0.19/0.48 % Problem : Preliminary Notions - Basic Laws of Logic
% 0.19/0.48 % Version : Especial.
% 0.19/0.48 % English : (! x:i.in x emptyset -> (! phi:o.phi))
% 0.19/0.48
% 0.19/0.48 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.19/0.48 % Source : [Bro08]
% 0.19/0.48 % Names : ZFC003g [Bro08]
% 0.19/0.48
% 0.19/0.48 % Status : Theorem
% 0.19/0.48 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.23 v8.1.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.0.0, 0.29 v5.5.0, 0.17 v5.4.0, 0.20 v5.3.0, 0.40 v5.2.0, 0.20 v4.1.0, 0.00 v4.0.1, 0.33 v3.7.0
% 0.19/0.48 % Syntax : Number of formulae : 47 ( 19 unt; 27 typ; 19 def)
% 0.19/0.48 % Number of atoms : 121 ( 24 equ; 0 cnn)
% 0.19/0.48 % Maximal formula atoms : 18 ( 6 avg)
% 0.19/0.48 % Number of connectives : 235 ( 3 ~; 3 |; 18 &; 157 @)
% 0.19/0.48 % ( 6 <=>; 48 =>; 0 <=; 0 <~>)
% 0.19/0.48 % Maximal formula depth : 22 ( 2 avg)
% 0.19/0.48 % Number of types : 2 ( 0 usr)
% 0.19/0.48 % Number of type conns : 22 ( 22 >; 0 *; 0 +; 0 <<)
% 0.19/0.48 % Number of symbols : 28 ( 27 usr; 19 con; 0-2 aty)
% 0.19/0.48 % Number of variables : 75 ( 10 ^; 52 !; 13 ?; 75 :)
% 0.19/0.48 % SPC : TH0_THM_EQU_NAR
% 0.19/0.48
% 0.19/0.48 % Comments : http://mathgate.info/detsetitem.php?id=57
% 0.19/0.48 %------------------------------------------------------------------------------
% 0.19/0.48 thf(in_type,type,
% 0.19/0.48 in: $i > $i > $o ).
% 0.19/0.48
% 0.19/0.48 thf(exu_type,type,
% 0.19/0.48 exu: ( $i > $o ) > $o ).
% 0.19/0.48
% 0.19/0.48 thf(exu,definition,
% 0.19/0.48 ( exu
% 0.19/0.48 = ( ^ [Xphi: $i > $o] :
% 0.19/0.48 ? [Xx: $i] :
% 0.19/0.48 ( ( Xphi @ Xx )
% 0.19/0.48 & ! [Xy: $i] :
% 0.19/0.48 ( ( Xphi @ Xy )
% 0.19/0.48 => ( Xx = Xy ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(setextAx_type,type,
% 0.19/0.48 setextAx: $o ).
% 0.19/0.48
% 0.19/0.48 thf(setextAx,definition,
% 0.19/0.48 ( setextAx
% 0.19/0.48 = ( ! [A: $i,B: $i] :
% 0.19/0.48 ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 <=> ( in @ Xx @ B ) )
% 0.19/0.48 => ( A = B ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(emptyset_type,type,
% 0.19/0.48 emptyset: $i ).
% 0.19/0.48
% 0.19/0.48 thf(emptysetAx_type,type,
% 0.19/0.48 emptysetAx: $o ).
% 0.19/0.48
% 0.19/0.48 thf(emptysetAx,definition,
% 0.19/0.48 ( emptysetAx
% 0.19/0.48 = ( ! [Xx: $i] :
% 0.19/0.48 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(setadjoin_type,type,
% 0.19/0.48 setadjoin: $i > $i > $i ).
% 0.19/0.48
% 0.19/0.48 thf(setadjoinAx_type,type,
% 0.19/0.48 setadjoinAx: $o ).
% 0.19/0.48
% 0.19/0.48 thf(setadjoinAx,definition,
% 0.19/0.48 ( setadjoinAx
% 0.19/0.48 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.48 <=> ( ( Xy = Xx )
% 0.19/0.48 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(powerset_type,type,
% 0.19/0.48 powerset: $i > $i ).
% 0.19/0.48
% 0.19/0.48 thf(powersetAx_type,type,
% 0.19/0.48 powersetAx: $o ).
% 0.19/0.48
% 0.19/0.48 thf(powersetAx,definition,
% 0.19/0.48 ( powersetAx
% 0.19/0.48 = ( ! [A: $i,B: $i] :
% 0.19/0.48 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.48 <=> ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ B )
% 0.19/0.48 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(setunion_type,type,
% 0.19/0.48 setunion: $i > $i ).
% 0.19/0.48
% 0.19/0.48 thf(setunionAx_type,type,
% 0.19/0.48 setunionAx: $o ).
% 0.19/0.48
% 0.19/0.48 thf(setunionAx,definition,
% 0.19/0.48 ( setunionAx
% 0.19/0.48 = ( ! [A: $i,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.48 <=> ? [B: $i] :
% 0.19/0.48 ( ( in @ Xx @ B )
% 0.19/0.48 & ( in @ B @ A ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(omega_type,type,
% 0.19/0.48 omega: $i ).
% 0.19/0.48
% 0.19/0.48 thf(omega0Ax_type,type,
% 0.19/0.48 omega0Ax: $o ).
% 0.19/0.48
% 0.19/0.48 thf(omega0Ax,definition,
% 0.19/0.48 ( omega0Ax
% 0.19/0.48 = ( in @ emptyset @ omega ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(omegaSAx_type,type,
% 0.19/0.48 omegaSAx: $o ).
% 0.19/0.48
% 0.19/0.48 thf(omegaSAx,definition,
% 0.19/0.48 ( omegaSAx
% 0.19/0.48 = ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ omega )
% 0.19/0.48 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(omegaIndAx_type,type,
% 0.19/0.48 omegaIndAx: $o ).
% 0.19/0.48
% 0.19/0.48 thf(omegaIndAx,definition,
% 0.19/0.48 ( omegaIndAx
% 0.19/0.48 = ( ! [A: $i] :
% 0.19/0.48 ( ( ( in @ emptyset @ A )
% 0.19/0.48 & ! [Xx: $i] :
% 0.19/0.48 ( ( ( in @ Xx @ omega )
% 0.19/0.48 & ( in @ Xx @ A ) )
% 0.19/0.48 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ omega )
% 0.19/0.48 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(replAx_type,type,
% 0.19/0.48 replAx: $o ).
% 0.19/0.48
% 0.19/0.48 thf(replAx,definition,
% 0.19/0.48 ( replAx
% 0.19/0.48 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.19/0.48 ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( exu
% 0.19/0.48 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.19/0.48 => ? [B: $i] :
% 0.19/0.48 ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ B )
% 0.19/0.48 <=> ? [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ A )
% 0.19/0.48 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(foundationAx_type,type,
% 0.19/0.48 foundationAx: $o ).
% 0.19/0.48
% 0.19/0.48 thf(foundationAx,definition,
% 0.19/0.48 ( foundationAx
% 0.19/0.48 = ( ! [A: $i] :
% 0.19/0.48 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.48 => ? [B: $i] :
% 0.19/0.48 ( ( in @ B @ A )
% 0.19/0.48 & ~ ? [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ B )
% 0.19/0.48 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(wellorderingAx_type,type,
% 0.19/0.48 wellorderingAx: $o ).
% 0.19/0.48
% 0.19/0.48 thf(wellorderingAx,definition,
% 0.19/0.48 ( wellorderingAx
% 0.19/0.48 = ( ! [A: $i] :
% 0.19/0.48 ? [B: $i] :
% 0.19/0.48 ( ! [C: $i] :
% 0.19/0.48 ( ( in @ C @ B )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ C )
% 0.19/0.48 => ( in @ Xx @ A ) ) )
% 0.19/0.48 & ! [Xx: $i,Xy: $i] :
% 0.19/0.48 ( ( ( in @ Xx @ A )
% 0.19/0.48 & ( in @ Xy @ A ) )
% 0.19/0.48 => ( ! [C: $i] :
% 0.19/0.48 ( ( in @ C @ B )
% 0.19/0.48 => ( ( in @ Xx @ C )
% 0.19/0.48 <=> ( in @ Xy @ C ) ) )
% 0.19/0.48 => ( Xx = Xy ) ) )
% 0.19/0.48 & ! [C: $i,D: $i] :
% 0.19/0.48 ( ( ( in @ C @ B )
% 0.19/0.48 & ( in @ D @ B ) )
% 0.19/0.48 => ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ C )
% 0.19/0.48 => ( in @ Xx @ D ) )
% 0.19/0.48 | ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ D )
% 0.19/0.48 => ( in @ Xx @ C ) ) ) )
% 0.19/0.48 & ! [C: $i] :
% 0.19/0.48 ( ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ C )
% 0.19/0.48 => ( in @ Xx @ A ) )
% 0.19/0.48 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.19/0.48 => ? [D: $i,Xx: $i] :
% 0.19/0.48 ( ( in @ D @ B )
% 0.19/0.48 & ( in @ Xx @ C )
% 0.19/0.48 & ~ ? [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ D )
% 0.19/0.48 & ( in @ Xy @ C ) )
% 0.19/0.48 & ! [E: $i] :
% 0.19/0.48 ( ( in @ E @ B )
% 0.19/0.48 => ( ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ E )
% 0.19/0.48 => ( in @ Xy @ D ) )
% 0.19/0.48 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(descr_type,type,
% 0.19/0.48 descr: ( $i > $o ) > $i ).
% 0.19/0.48
% 0.19/0.48 thf(descrp_type,type,
% 0.19/0.48 descrp: $o ).
% 0.19/0.48
% 0.19/0.48 thf(descrp,definition,
% 0.19/0.48 ( descrp
% 0.19/0.48 = ( ! [Xphi: $i > $o] :
% 0.19/0.48 ( ( exu
% 0.19/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.48 => ( Xphi
% 0.19/0.48 @ ( descr
% 0.19/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(dsetconstr_type,type,
% 0.19/0.48 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.19/0.48
% 0.19/0.48 thf(dsetconstrI_type,type,
% 0.19/0.48 dsetconstrI: $o ).
% 0.19/0.48
% 0.19/0.48 thf(dsetconstrI,definition,
% 0.19/0.48 ( dsetconstrI
% 0.19/0.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( Xphi @ Xx )
% 0.19/0.48 => ( in @ Xx
% 0.19/0.48 @ ( dsetconstr @ A
% 0.19/0.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(dsetconstrEL_type,type,
% 0.19/0.48 dsetconstrEL: $o ).
% 0.19/0.48
% 0.19/0.48 thf(dsetconstrEL,definition,
% 0.19/0.48 ( dsetconstrEL
% 0.19/0.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx
% 0.19/0.48 @ ( dsetconstr @ A
% 0.19/0.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(dsetconstrER_type,type,
% 0.19/0.48 dsetconstrER: $o ).
% 0.19/0.48
% 0.19/0.48 thf(dsetconstrER,definition,
% 0.19/0.48 ( dsetconstrER
% 0.19/0.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx
% 0.19/0.48 @ ( dsetconstr @ A
% 0.19/0.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.48 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(exuE1_type,type,
% 0.19/0.48 exuE1: $o ).
% 0.19/0.48
% 0.19/0.48 thf(exuE1,definition,
% 0.19/0.48 ( exuE1
% 0.19/0.48 = ( ! [Xphi: $i > $o] :
% 0.19/0.48 ( ( exu
% 0.19/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.48 => ? [Xx: $i] :
% 0.19/0.48 ( ( Xphi @ Xx )
% 0.19/0.48 & ! [Xy: $i] :
% 0.19/0.48 ( ( Xphi @ Xy )
% 0.19/0.48 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(prop2set_type,type,
% 0.19/0.48 prop2set: $o > $i ).
% 0.19/0.48
% 0.19/0.48 thf(prop2set,definition,
% 0.19/0.48 ( prop2set
% 0.19/0.48 = ( ^ [Xphi: $o] :
% 0.19/0.48 ( dsetconstr @ ( powerset @ emptyset )
% 0.19/0.48 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(prop2setE_type,type,
% 0.19/0.48 prop2setE: $o ).
% 0.19/0.48
% 0.19/0.48 thf(prop2setE,definition,
% 0.19/0.48 ( prop2setE
% 0.19/0.48 = ( ! [Xphi: $o,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.19/0.48 => Xphi ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(emptysetE,conjecture,
% 0.19/0.48 ( setextAx
% 0.19/0.48 => ( emptysetAx
% 0.19/0.48 => ( setadjoinAx
% 0.19/0.48 => ( powersetAx
% 0.19/0.48 => ( setunionAx
% 0.19/0.48 => ( omega0Ax
% 0.19/0.48 => ( omegaSAx
% 0.19/0.48 => ( omegaIndAx
% 0.19/0.48 => ( replAx
% 0.19/0.48 => ( foundationAx
% 0.19/0.48 => ( wellorderingAx
% 0.19/0.48 => ( descrp
% 0.19/0.48 => ( dsetconstrI
% 0.19/0.48 => ( dsetconstrEL
% 0.19/0.48 => ( dsetconstrER
% 0.19/0.48 => ( exuE1
% 0.19/0.48 => ( prop2setE
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ emptyset )
% 0.19/0.48 => ! [Xphi: $o] : Xphi ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 %------------------------------------------------------------------------------
% 0.19/0.48 ------------------------
% 1.74/1.49 ---- Embedded in THF ---
% 1.74/1.49 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.74/1.49 %%% Generated on Tue Feb 24 12:24:03 EST 2026
% 1.74/1.49 %%% using '$$dhol' embedding, version 1.3.0.
% 1.74/1.49 %%% Logic specification used:
% 1.74/1.49 %%% thf(spec, logic, $$dhol).
% 1.74/1.49
% 1.74/1.49 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.YoXTo2yw5Z/DTF2DTF26364.p
% See solution above
% 1.74/1.49 ------------------------
% 1.97/1.50 ---- Cleaned THF ---
% 1.97/1.50 thf(in_type,type,
% 1.97/1.50 in: $i > $i > $o ).
% 1.97/1.50
% 1.97/1.50 thf(exu_type,type,
% 1.97/1.50 exu: ( $i > $o ) > $o ).
% 1.97/1.50
% 1.97/1.50 thf(exu,definition,
% 1.97/1.50 ( exu
% 1.97/1.50 = ( ^ [Xphi: $i > $o] :
% 1.97/1.50 ? [Xx: $i] :
% 1.97/1.50 ( ( Xphi @ Xx )
% 1.97/1.50 & ! [Xy: $i] :
% 1.97/1.50 ( ( Xphi @ Xy )
% 1.97/1.50 => ( Xx = Xy ) ) ) ) ) ).
% 1.97/1.50
% 1.97/1.50 thf(setextAx_type,type,
% 1.97/1.50 setextAx: $o ).
% 1.97/1.50
% 1.97/1.50 thf(setextAx,definition,
% 1.97/1.50 ( setextAx
% 1.97/1.50 = ( ! [A: $i,B: $i] :
% 1.97/1.50 ( ! [Xx: $i] :
% 1.97/1.50 ( ( in @ Xx @ A )
% 1.97/1.50 <=> ( in @ Xx @ B ) )
% 1.97/1.50 => ( A = B ) ) ) ) ).
% 1.97/1.50
% 1.97/1.50 thf(emptyset_type,type,
% 1.97/1.50 emptyset: $i ).
% 1.97/1.50
% 1.97/1.50 thf(emptysetAx_type,type,
% 1.97/1.50 emptysetAx: $o ).
% 1.97/1.50
% 1.97/1.50 thf(emptysetAx,definition,
% 1.97/1.50 ( emptysetAx
% 1.97/1.50 = ( ! [Xx: $i] :
% 1.97/1.50 ~ ( in @ Xx @ emptyset ) ) ) ).
% 1.97/1.50
% 1.97/1.50 thf(setadjoin_type,type,
% 1.97/1.50 setadjoin: $i > $i > $i ).
% 1.97/1.50
% 1.97/1.50 thf(setadjoinAx_type,type,
% 1.97/1.50 setadjoinAx: $o ).
% 1.97/1.50
% 1.97/1.50 thf(setadjoinAx,definition,
% 1.97/1.50 ( setadjoinAx
% 1.97/1.50 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.97/1.50 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.97/1.50 <=> ( ( Xy = Xx )
% 1.97/1.50 | ( in @ Xy @ A ) ) ) ) ) ).
% 1.97/1.50
% 1.97/1.50 thf(powerset_type,type,
% 1.97/1.50 powerset: $i > $i ).
% 1.97/1.50
% 1.97/1.50 thf(powersetAx_type,type,
% 1.97/1.50 powersetAx: $o ).
% 1.97/1.50
% 1.97/1.50 thf(powersetAx,definition,
% 1.97/1.50 ( powersetAx
% 1.97/1.50 = ( ! [A: $i,B: $i] :
% 1.97/1.50 ( ( in @ B @ ( powerset @ A ) )
% 1.97/1.50 <=> ! [Xx: $i] :
% 1.97/1.50 ( ( in @ Xx @ B )
% 1.97/1.50 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.97/1.50
% 1.97/1.50 thf(setunion_type,type,
% 1.97/1.50 setunion: $i > $i ).
% 1.97/1.50
% 1.97/1.50 thf(setunionAx_type,type,
% 1.97/1.50 setunionAx: $o ).
% 1.97/1.50
% 1.97/1.50 thf(setunionAx,definition,
% 1.97/1.50 ( setunionAx
% 1.97/1.50 = ( ! [A: $i,Xx: $i] :
% 1.97/1.50 ( ( in @ Xx @ ( setunion @ A ) )
% 1.97/1.50 <=> ? [B: $i] :
% 1.97/1.50 ( ( in @ Xx @ B )
% 1.97/1.50 & ( in @ B @ A ) ) ) ) ) ).
% 1.97/1.50
% 1.97/1.50 thf(omega_type,type,
% 1.97/1.50 omega: $i ).
% 1.97/1.50
% 1.97/1.50 thf(omega0Ax_type,type,
% 1.97/1.50 omega0Ax: $o ).
% 1.97/1.50
% 1.97/1.50 thf(omega0Ax,definition,
% 1.97/1.50 ( omega0Ax
% 1.97/1.50 = ( in @ emptyset @ omega ) ) ).
% 1.97/1.50
% 1.97/1.50 thf(omegaSAx_type,type,
% 1.97/1.50 omegaSAx: $o ).
% 1.97/1.50
% 1.97/1.50 thf(omegaSAx,definition,
% 1.97/1.50 ( omegaSAx
% 1.97/1.50 = ( ! [Xx: $i] :
% 1.97/1.50 ( ( in @ Xx @ omega )
% 1.97/1.50 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 1.97/1.50
% 1.97/1.50 thf(omegaIndAx_type,type,
% 1.97/1.50 omegaIndAx: $o ).
% 1.97/1.50
% 1.97/1.50 thf(omegaIndAx,definition,
% 1.97/1.50 ( omegaIndAx
% 1.97/1.50 = ( ! [A: $i] :
% 1.97/1.50 ( ( ( in @ emptyset @ A )
% 1.97/1.50 & ! [Xx: $i] :
% 1.97/1.50 ( ( ( in @ Xx @ omega )
% 1.97/1.50 & ( in @ Xx @ A ) )
% 1.97/1.50 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 1.97/1.50 => ! [Xx: $i] :
% 1.97/1.50 ( ( in @ Xx @ omega )
% 1.97/1.50 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.97/1.50
% 1.97/1.50 thf(replAx_type,type,
% 1.97/1.50 replAx: $o ).
% 1.97/1.50
% 1.97/1.50 thf(replAx,definition,
% 1.97/1.50 ( replAx
% 1.97/1.50 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 1.97/1.50 ( ! [Xx: $i] :
% 1.97/1.50 ( ( in @ Xx @ A )
% 1.97/1.50 => ( exu
% 1.97/1.50 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 1.97/1.50 => ? [B: $i] :
% 1.97/1.50 ! [Xx: $i] :
% 1.97/1.50 ( ( in @ Xx @ B )
% 1.97/1.50 <=> ? [Xy: $i] :
% 1.97/1.50 ( ( in @ Xy @ A )
% 1.97/1.50 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 1.97/1.50
% 1.97/1.50 thf(foundationAx_type,type,
% 1.97/1.50 foundationAx: $o ).
% 1.97/1.50
% 1.97/1.50 thf(foundationAx,definition,
% 1.97/1.50 ( foundationAx
% 1.97/1.50 = ( ! [A: $i] :
% 1.97/1.50 ( ? [Xx: $i] : ( in @ Xx @ A )
% 1.97/1.50 => ? [B: $i] :
% 1.97/1.50 ( ( in @ B @ A )
% 1.97/1.50 & ~ ? [Xx: $i] :
% 1.97/1.50 ( ( in @ Xx @ B )
% 1.97/1.50 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 1.97/1.50
% 1.97/1.50 thf(wellorderingAx_type,type,
% 1.97/1.50 wellorderingAx: $o ).
% 1.97/1.50
% 1.97/1.50 thf(wellorderingAx,definition,
% 1.97/1.50 ( wellorderingAx
% 1.97/1.50 = ( ! [A: $i] :
% 1.97/1.50 ? [B: $i] :
% 1.97/1.50 ( ! [C: $i] :
% 1.97/1.50 ( ( in @ C @ B )
% 1.97/1.50 => ! [Xx: $i] :
% 1.97/1.50 ( ( in @ Xx @ C )
% 1.97/1.50 => ( in @ Xx @ A ) ) )
% 1.97/1.50 & ! [Xx: $i,Xy: $i] :
% 1.97/1.50 ( ( ( in @ Xx @ A )
% 1.97/1.50 & ( in @ Xy @ A ) )
% 1.97/1.50 => ( ! [C: $i] :
% 1.97/1.50 ( ( in @ C @ B )
% 1.97/1.50 => ( ( in @ Xx @ C )
% 1.97/1.50 <=> ( in @ Xy @ C ) ) )
% 1.97/1.50 => ( Xx = Xy ) ) )
% 1.97/1.50 & ! [C: $i,D: $i] :
% 1.97/1.50 ( ( ( in @ C @ B )
% 1.97/1.50 & ( in @ D @ B ) )
% 1.97/1.50 => ( ! [Xx: $i] :
% 1.97/1.50 ( ( in @ Xx @ C )
% 1.97/1.50 => ( in @ Xx @ D ) )
% 1.97/1.50 | ! [Xx: $i] :
% 1.97/1.50 ( ( in @ Xx @ D )
% 1.97/1.50 => ( in @ Xx @ C ) ) ) )
% 1.97/1.50 & ! [C: $i] :
% 1.97/1.50 ( ( ! [Xx: $i] :
% 1.97/1.50 ( ( in @ Xx @ C )
% 1.97/1.50 => ( in @ Xx @ A ) )
% 1.97/1.50 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 1.97/1.50 => ? [D: $i,Xx: $i] :
% 1.97/1.50 ( ( in @ D @ B )
% 1.97/1.50 & ( in @ Xx @ C )
% 1.97/1.50 & ~ ? [Xy: $i] :
% 1.97/1.50 ( ( in @ Xy @ D )
% 1.97/1.50 & ( in @ Xy @ C ) )
% 1.97/1.50 & ! [E: $i] :
% 1.97/1.50 ( ( in @ E @ B )
% 1.97/1.50 => ( ! [Xy: $i] :
% 1.97/1.50 ( ( in @ Xy @ E )
% 1.97/1.50 => ( in @ Xy @ D ) )
% 1.97/1.50 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 1.97/1.50
% 1.97/1.50 thf(descr_type,type,
% 1.97/1.50 descr: ( $i > $o ) > $i ).
% 1.97/1.50
% 1.97/1.50 thf(descrp_type,type,
% 1.97/1.50 descrp: $o ).
% 1.97/1.50
% 1.97/1.50 thf(descrp,definition,
% 1.97/1.50 ( descrp
% 1.97/1.50 = ( ! [Xphi: $i > $o] :
% 1.97/1.50 ( ( exu
% 1.97/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.97/1.50 => ( Xphi
% 1.97/1.50 @ ( descr
% 1.97/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 1.97/1.50
% 1.97/1.50 thf(dsetconstr_type,type,
% 1.97/1.50 dsetconstr: $i > ( $i > $o ) > $i ).
% 1.97/1.50
% 1.97/1.50 thf(dsetconstrI_type,type,
% 1.97/1.50 dsetconstrI: $o ).
% 1.97/1.50
% 1.97/1.50 thf(dsetconstrI,definition,
% 1.97/1.50 ( dsetconstrI
% 1.97/1.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.97/1.50 ( ( in @ Xx @ A )
% 1.97/1.50 => ( ( Xphi @ Xx )
% 1.97/1.50 => ( in @ Xx
% 1.97/1.50 @ ( dsetconstr @ A
% 1.97/1.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 1.97/1.50
% 1.97/1.50 thf(dsetconstrEL_type,type,
% 1.97/1.50 dsetconstrEL: $o ).
% 1.97/1.50
% 1.97/1.50 thf(dsetconstrEL,definition,
% 1.97/1.50 ( dsetconstrEL
% 1.97/1.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.97/1.50 ( ( in @ Xx
% 1.97/1.50 @ ( dsetconstr @ A
% 1.97/1.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.97/1.50 => ( in @ Xx @ A ) ) ) ) ).
% 1.97/1.50
% 1.97/1.50 thf(dsetconstrER_type,type,
% 1.97/1.50 dsetconstrER: $o ).
% 1.97/1.50
% 1.97/1.50 thf(dsetconstrER,definition,
% 1.97/1.50 ( dsetconstrER
% 1.97/1.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.97/1.50 ( ( in @ Xx
% 1.97/1.50 @ ( dsetconstr @ A
% 1.97/1.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.97/1.50 => ( Xphi @ Xx ) ) ) ) ).
% 1.97/1.50
% 1.97/1.50 thf(exuE1_type,type,
% 1.97/1.50 exuE1: $o ).
% 1.97/1.50
% 1.97/1.50 thf(exuE1,definition,
% 1.97/1.50 ( exuE1
% 1.97/1.50 = ( ! [Xphi: $i > $o] :
% 1.97/1.50 ( ( exu
% 1.97/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.97/1.50 => ? [Xx: $i] :
% 1.97/1.50 ( ( Xphi @ Xx )
% 1.97/1.50 & ! [Xy: $i] :
% 1.97/1.50 ( ( Xphi @ Xy )
% 1.97/1.50 => ( Xx = Xy ) ) ) ) ) ) ).
% 1.97/1.50
% 1.97/1.50 thf(prop2set_type,type,
% 1.97/1.50 prop2set: $o > $i ).
% 1.97/1.50
% 1.97/1.50 thf(prop2set,definition,
% 1.97/1.50 ( prop2set
% 1.97/1.50 = ( ^ [Xphi: $o] :
% 1.97/1.50 ( dsetconstr @ ( powerset @ emptyset )
% 1.97/1.50 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 1.97/1.50
% 1.97/1.50 thf(prop2setE_type,type,
% 1.97/1.50 prop2setE: $o ).
% 1.97/1.50
% 1.97/1.50 thf(prop2setE,definition,
% 1.97/1.50 ( prop2setE
% 1.97/1.50 = ( ! [Xphi: $o,Xx: $i] :
% 1.97/1.50 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 1.97/1.50 => Xphi ) ) ) ).
% 1.97/1.50
% 1.97/1.50 thf(emptysetE,conjecture,
% 1.97/1.50 ( setextAx
% 1.97/1.50 => ( emptysetAx
% 1.97/1.50 => ( setadjoinAx
% 1.97/1.50 => ( powersetAx
% 1.97/1.50 => ( setunionAx
% 1.97/1.50 => ( omega0Ax
% 1.97/1.50 => ( omegaSAx
% 1.97/1.50 => ( omegaIndAx
% 1.97/1.50 => ( replAx
% 1.97/1.50 => ( foundationAx
% 1.97/1.50 => ( wellorderingAx
% 1.97/1.50 => ( descrp
% 1.97/1.50 => ( dsetconstrI
% 1.97/1.50 => ( dsetconstrEL
% 1.97/1.50 => ( dsetconstrER
% 1.97/1.50 => ( exuE1
% 1.97/1.50 => ( prop2setE
% 1.97/1.50 => ! [Xx: $i] :
% 1.97/1.50 ( ( in @ Xx @ emptyset )
% 1.97/1.50 => ! [Xphi: $o] : Xphi ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 1.97/1.50 ------------------------
% 1.97/1.52 % (26465)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_26364 for (2999ds/27Mi)
% 1.97/1.52 % (26466)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_26364 for (2999ds/2Mi)
% 1.97/1.52 % (26467)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_26364 for (2999ds/2Mi)
% 1.97/1.52 % (26463)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_26364 for (2999ds/183Mi)
% 1.97/1.52 % (26464)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_26364 for (2999ds/4Mi)
% 1.97/1.52 % (26468)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_26364 for (2999ds/275Mi)
% 1.97/1.52 % (26469)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_26364 for (2999ds/18Mi)
% 1.97/1.53 % (26466)Instruction limit reached!
% 1.97/1.53 % (26466)------------------------------
% 1.97/1.53 % (26466)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.97/1.53 % (26466)Termination reason: Unknown
% 1.97/1.53 % (26466)Termination phase: shuffling
% 1.97/1.53
% 1.97/1.53 % (26466)Memory used [KB]: 1023
% 1.97/1.53 % (26466)Time elapsed: 0.003 s
% 1.97/1.53 % (26466)Instructions burned: 3 (million)
% 1.97/1.53 % (26466)------------------------------
% 1.97/1.53 % (26466)------------------------------
% 1.97/1.53 % (26467)Instruction limit reached!
% 1.97/1.53 % (26467)------------------------------
% 1.97/1.53 % (26467)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.97/1.53 % (26467)Termination reason: Unknown
% 1.97/1.53 % (26467)Termination phase: shuffling
% 1.97/1.53
% 1.97/1.53 % (26467)Memory used [KB]: 1023
% 1.97/1.53 % (26467)Time elapsed: 0.003 s
% 1.97/1.53 % (26467)Instructions burned: 3 (million)
% 1.97/1.53 % (26467)------------------------------
% 1.97/1.53 % (26467)------------------------------
% 1.97/1.53 % (26464)Instruction limit reached!
% 1.97/1.53 % (26464)------------------------------
% 1.97/1.53 % (26464)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.97/1.53 % (26464)Termination reason: Unknown
% 1.97/1.53 % (26464)Termination phase: Property scanning
% 1.97/1.53
% 1.97/1.53 % (26464)Memory used [KB]: 1023
% 1.97/1.53 % (26464)Time elapsed: 0.004 s
% 1.97/1.53 % (26464)Instructions burned: 5 (million)
% 1.97/1.53 % (26464)------------------------------
% 1.97/1.53 % (26464)------------------------------
% 1.97/1.53 % (26469)Instruction limit reached!
% 1.97/1.53 % (26469)------------------------------
% 1.97/1.53 % (26469)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.97/1.53 % (26469)Termination reason: Unknown
% 1.97/1.53 % (26469)Termination phase: Property scanning
% 1.97/1.53
% 1.97/1.53 % (26469)Memory used [KB]: 1279
% 1.97/1.53 % (26469)Time elapsed: 0.010 s
% 1.97/1.53 % (26469)Instructions burned: 18 (million)
% 1.97/1.53 % (26469)------------------------------
% 1.97/1.53 % (26469)------------------------------
% 1.97/1.54 % (26465)Instruction limit reached!
% 1.97/1.54 % (26465)------------------------------
% 1.97/1.54 % (26465)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.97/1.54 % (26465)Termination reason: Unknown
% 1.97/1.54 % (26465)Termination phase: Saturation
% 1.97/1.54
% 1.97/1.54 % (26465)Memory used [KB]: 5756
% 1.97/1.54 % (26465)Time elapsed: 0.015 s
% 1.97/1.54 % (26465)Instructions burned: 29 (million)
% 1.97/1.54 % (26465)------------------------------
% 1.97/1.54 % (26465)------------------------------
% 1.97/1.54 % (26468)First to succeed.
% 1.97/1.54 % (26470)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_26364 for (2999ds/3Mi)
% 1.97/1.54 % (26471)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_26364 for (2999ds/37Mi)
% 1.97/1.54 % (26472)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_26364 for (2999ds/15Mi)
% 1.97/1.54 % (26468)Refutation found. Thanks to Tanya!
% 1.97/1.54 % SZS status Theorem for DTF2THF_26364
% 1.97/1.54 % SZS output start Proof for DTF2THF_26364
% 1.97/1.54 thf(func_def_0, type, in: $i > $i > $o).
% 1.97/1.54 thf(func_def_1, type, exu: ($i > $o) > $o).
% 1.97/1.54 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 1.97/1.54 thf(func_def_8, type, powerset: $i > $i).
% 1.97/1.54 thf(func_def_10, type, setunion: $i > $i).
% 1.97/1.54 thf(func_def_19, type, descr: ($i > $o) > $i).
% 1.97/1.54 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 1.97/1.54 thf(func_def_26, type, prop2set: $o > $i).
% 1.97/1.54 thf(func_def_40, type, sP1: $i > $i > $o).
% 1.97/1.54 thf(func_def_41, type, sP2: $i > $o).
% 1.97/1.54 thf(func_def_42, type, sP3: $i > $i > $o).
% 1.97/1.54 thf(func_def_43, type, sP4: $i > $i > $o).
% 1.97/1.54 thf(func_def_44, type, sK5: $i > $o).
% 1.97/1.54 thf(func_def_47, type, sK8: ($i > $o) > $i).
% 1.97/1.54 thf(func_def_48, type, sK9: $i > $o).
% 1.97/1.54 thf(func_def_49, type, sK10: $i > $i).
% 1.97/1.54 thf(func_def_51, type, sK12: $i > $i > $o).
% 1.97/1.54 thf(func_def_52, type, sK13: $i > $i).
% 1.97/1.54 thf(func_def_53, type, sK14: $i > $i).
% 1.97/1.54 thf(func_def_54, type, sK15: ($i > $i > $o) > $i > $i).
% 1.97/1.54 thf(func_def_55, type, sK16: $i > ($i > $i > $o) > $i > $i).
% 1.97/1.54 thf(func_def_56, type, sK17: ($i > $i > $o) > $i > $i).
% 1.97/1.54 thf(func_def_59, type, sK20: $i > $i > $i).
% 1.97/1.54 thf(func_def_60, type, sK21: $i > $i > $i).
% 1.97/1.54 thf(func_def_61, type, sK22: $i > $i > $i).
% 1.97/1.54 thf(func_def_62, type, sK23: $i > $i > $i).
% 1.97/1.54 thf(func_def_63, type, sK24: $i > $i > $i).
% 1.97/1.54 thf(func_def_64, type, sK25: $i > $i > $i > $i).
% 1.97/1.54 thf(func_def_65, type, sK26: $i > $i).
% 1.97/1.54 thf(func_def_66, type, sK27: $i > $i).
% 1.97/1.54 thf(func_def_67, type, sK28: $i > $i).
% 1.97/1.54 thf(func_def_68, type, sK29: $i > $i).
% 1.97/1.54 thf(func_def_69, type, sK30: $i > $i > $i > $i).
% 1.97/1.54 thf(func_def_70, type, sK31: $i > $i > $i > $i).
% 1.97/1.54 thf(func_def_71, type, sK32: $i > $i > $i).
% 1.97/1.54 thf(func_def_72, type, sK33: $i > $i > $i).
% 1.97/1.54 thf(func_def_73, type, sK34: $i > $i > $i).
% 1.97/1.54 thf(func_def_75, type, sK36: $i > $i).
% 1.97/1.54 thf(func_def_76, type, sK37: $i > $i).
% 1.97/1.54 thf(func_def_77, type, sK38: $i > $i).
% 1.97/1.54 thf(func_def_83, type, sK44: $i > $i > $i).
% 1.97/1.54 thf(func_def_89, type, sK50: $i > $i).
% 1.97/1.54 thf(func_def_90, type, sK51: $i > $i).
% 1.97/1.54 thf(func_def_93, type, sK54: $i > $o).
% 1.97/1.54 thf(func_def_94, type, sK55: $i > $o).
% 1.97/1.54 thf(func_def_98, type, sK59: $i > $i > $i).
% 1.97/1.54 thf(func_def_104, type, sK65: $i > $i).
% 1.97/1.54 thf(func_def_106, type, ph67: !>[X0: $tType]:(X0)).
% 1.97/1.54 thf(f438,plain,(
% 1.97/1.54 $false),
% 1.97/1.54 inference(trivial_inequality_removal,[status(thm)],[f425])).
% 1.97/1.54 thf(f425,plain,(
% 1.97/1.54 ($true != $true)),
% 1.97/1.54 inference(superposition,[status(thm)],[f379,f264])).
% 1.97/1.54 thf(f264,plain,(
% 1.97/1.54 ((in @ sK45 @ emptyset) = $true)),
% 1.97/1.54 inference(cnf_transformation,[status(thm)],[f154])).
% 1.97/1.54 thf(f154,plain,(
% 1.97/1.54 (omegaIndAx = $true) & (omegaSAx = $true) & (prop2setE = $true) & (wellorderingAx = $true) & (setadjoinAx = $true) & (exuE1 = $true) & (replAx = $true) & (dsetconstrEL = $true) & (dsetconstrER = $true) & (omega0Ax = $true) & (setunionAx = $true) & (foundationAx = $true) & (powersetAx = $true) & ((sK46 != $true) & ((in @ sK45 @ emptyset) = $true)) & (setextAx = $true) & (dsetconstrI = $true) & (emptysetAx = $true) & (descrp = $true)),
% 1.97/1.54 inference(skolemisation,[status(esa),new_symbols(skolem,[sK45,sK46])],[f69,f153,f152])).
% 1.97/1.54 thf(f152,plain,(
% 1.97/1.54 ? [X0 : $i] : (? [X1 : $o] : ($true != X1) & ($true = (in @ X0 @ emptyset))) => (? [X1 : $o] : ($true != X1) & ((in @ sK45 @ emptyset) = $true))),
% 1.97/1.54 introduced(definition,[],[choice_axiom])).
% 1.97/1.54 thf(f153,plain,(
% 1.97/1.54 ? [X1 : $o] : ($true != X1) => (sK46 != $true)),
% 1.97/1.54 introduced(definition,[],[choice_axiom])).
% 1.97/1.54 thf(f69,plain,(
% 1.97/1.54 (omegaIndAx = $true) & (omegaSAx = $true) & (prop2setE = $true) & (wellorderingAx = $true) & (setadjoinAx = $true) & (exuE1 = $true) & (replAx = $true) & (dsetconstrEL = $true) & (dsetconstrER = $true) & (omega0Ax = $true) & (setunionAx = $true) & (foundationAx = $true) & (powersetAx = $true) & ? [X0 : $i] : (? [X1 : $o] : ($true != X1) & ($true = (in @ X0 @ emptyset))) & (setextAx = $true) & (dsetconstrI = $true) & (emptysetAx = $true) & (descrp = $true)),
% 1.97/1.54 inference(flattening,[status(thm)],[f68])).
% 1.97/1.54 thf(f68,plain,(
% 1.97/1.54 ((((((((((((((((? [X0 : $i] : (? [X1 : $o] : ($true != X1) & ($true = (in @ X0 @ emptyset))) & (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)),
% 1.97/1.54 inference(ennf_transformation,[status(thm)],[f56])).
% 1.97/1.54 thf(f56,plain,(
% 1.97/1.54 ~((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) => ! [X0 : $i] : (($true = (in @ X0 @ emptyset)) => ! [X1 : $o] : ($true = X1)))))))))))))))))))),
% 1.97/1.54 inference(fool_elimination,[status(thm)],[f55])).
% 1.97/1.54 thf(f55,plain,(
% 1.97/1.54 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => ! [X0 : $i] : ((in @ X0 @ emptyset) => ! [X1 : $o] : X1))))))))))))))))))),
% 1.97/1.54 inference(rectify,[status(thm)],[f21])).
% 1.97/1.54 thf(f21,negated_conjecture,(
% 1.97/1.54 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => ! [X1 : $i] : ((in @ X1 @ emptyset) => ! [X0 : $o] : X0))))))))))))))))))),
% 1.97/1.54 inference(negated_conjecture,[status(cth)],[f20])).
% 1.97/1.54 thf(f20,conjecture,(
% 1.97/1.54 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => ! [X1 : $i] : ((in @ X1 @ emptyset) => ! [X0 : $o] : X0)))))))))))))))))),
% 1.97/1.54 file('/export/starexec/sandbox/tmp/tmp.YoXTo2yw5Z/DTF2THF_26364.p',emptysetE)).
% 1.97/1.54 thf(f379,plain,(
% 1.97/1.54 ( ! [X1 : $i] : (((in @ X1 @ emptyset) != $true)) )),
% 1.97/1.54 inference(trivial_inequality_removal,[status(thm)],[f341])).
% 1.97/1.54 thf(f341,plain,(
% 1.97/1.54 ( ! [X1 : $i] : (($true != $true) | ((in @ X1 @ emptyset) != $true)) )),
% 1.97/1.54 inference(definition_unfolding,[status(thm)],[f279,f261])).
% 1.97/1.54 thf(f261,plain,(
% 1.97/1.54 (emptysetAx = $true)),
% 1.97/1.54 inference(cnf_transformation,[status(thm)],[f154])).
% 1.97/1.54 thf(f279,plain,(
% 1.97/1.54 ( ! [X1 : $i] : (((in @ X1 @ emptyset) != $true) | (emptysetAx != $true)) )),
% 1.97/1.54 inference(cnf_transformation,[status(thm)],[f158])).
% 1.97/1.54 thf(f158,plain,(
% 1.97/1.54 ((emptysetAx = $true) | ($true = (in @ sK47 @ emptyset))) & (! [X1 : $i] : ((in @ X1 @ emptyset) != $true) | (emptysetAx != $true))),
% 1.97/1.54 inference(skolemisation,[status(esa),new_symbols(skolem,[sK47])],[f156,f157])).
% 1.97/1.54 thf(f157,plain,(
% 1.97/1.54 ? [X0 : $i] : ($true = (in @ X0 @ emptyset)) => ($true = (in @ sK47 @ emptyset))),
% 1.97/1.54 introduced(definition,[],[choice_axiom])).
% 1.97/1.54 thf(f156,plain,(
% 1.97/1.54 ((emptysetAx = $true) | ? [X0 : $i] : ($true = (in @ X0 @ emptyset))) & (! [X1 : $i] : ((in @ X1 @ emptyset) != $true) | (emptysetAx != $true))),
% 1.97/1.54 inference(rectify,[status(thm)],[f155])).
% 1.97/1.54 thf(f155,plain,(
% 1.97/1.54 ((emptysetAx = $true) | ? [X0 : $i] : ($true = (in @ X0 @ emptyset))) & (! [X0 : $i] : ($true != (in @ X0 @ emptyset)) | (emptysetAx != $true))),
% 1.97/1.54 inference(nnf_transformation,[status(thm)],[f60])).
% 1.97/1.54 thf(f60,plain,(
% 1.97/1.54 (emptysetAx = $true) <=> ! [X0 : $i] : ($true != (in @ X0 @ emptyset))),
% 1.97/1.54 inference(flattening,[status(thm)],[f32])).
% 1.97/1.54 thf(f32,plain,(
% 1.97/1.54 (emptysetAx = $true) <=> ! [X0 : $i] : ~($true = (in @ X0 @ emptyset))),
% 1.97/1.54 inference(fool_elimination,[status(thm)],[f31])).
% 1.97/1.54 thf(f31,plain,(
% 1.97/1.54 (emptysetAx = ! [X0 : $i] : ~(in @ X0 @ emptyset))),
% 1.97/1.54 inference(rectify,[status(thm)],[f3])).
% 1.97/1.54 thf(f3,axiom,(
% 1.97/1.54 (emptysetAx = ! [X1 : $i] : ~(in @ X1 @ emptyset))),
% 1.97/1.54 file('/export/starexec/sandbox/tmp/tmp.YoXTo2yw5Z/DTF2THF_26364.p',emptysetAx)).
% 1.97/1.54 % SZS output end Proof for DTF2THF_26364
% 1.97/1.54 % (26468)------------------------------
% 1.97/1.54 % (26468)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.97/1.54 % (26468)Termination reason: Refutation
% 1.97/1.54
% 1.97/1.54 % (26468)Memory used [KB]: 6012
% 1.97/1.54 % (26468)Time elapsed: 0.019 s
% 1.97/1.54 % (26468)Instructions burned: 33 (million)
% 1.97/1.54 % (26468)------------------------------
% 1.97/1.54 % (26468)------------------------------
% 1.97/1.54 % (26462)Success in time 0.034 s
%------------------------------------------------------------------------------