%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU503^1 : 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 : n026.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 1.76s 1.37s
% Output : Refutation 1.76s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 3
% Syntax : Number of formulae : 21 ( 12 unt; 0 typ; 0 def)
% Number of atoms : 425 ( 113 equ; 0 cnn)
% Maximal formula atoms : 21 ( 20 avg)
% Number of connectives : 217 ( 16 ~; 0 |; 57 &; 45 @)
% ( 0 <=>; 96 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 10 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 54 ( 50 usr; 24 con; 0-3 aty)
% ( 3 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 17 ( 4 ^; 10 !; 3 ?; 17 :)
% 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_48,type,
sK2: $i > $i ).
thf(func_def_49,type,
sK3: $i > $i ).
thf(func_def_50,type,
sK4: $i > $i > $i ).
thf(func_def_51,type,
sK5: $i > $i > $i ).
thf(func_def_52,type,
sK6: $i > $i > $i ).
thf(func_def_53,type,
sK7: $i > $i > $i ).
thf(func_def_54,type,
sK8: $i > $i > $i > $i ).
thf(func_def_55,type,
sK9: $i > $i > $i ).
thf(func_def_56,type,
sK10: ( $i > $o ) > $i > $i ).
thf(func_def_57,type,
sK11: ( $i > $o ) > $i ).
thf(func_def_58,type,
sK12: ( $i > $o ) > $i > $i ).
thf(func_def_59,type,
sK13: $i > ( $i > $o ) > $i ).
thf(func_def_60,type,
sK14: $i > $i ).
thf(func_def_61,type,
sK15: $i > $i > $i ).
thf(func_def_62,type,
sK16: $i > $i > $i ).
thf(func_def_63,type,
sK17: $i > ( $i > $i > $o ) > $i ).
thf(func_def_64,type,
sK18: $i > ( $i > $i > $o ) > $i ).
thf(func_def_65,type,
sK19: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_66,type,
sK20: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_67,type,
sK21: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_68,type,
sK22: $i > $i > ( $i > $i > $o ) > $i ).
thf(f448,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f447]) ).
thf(f447,plain,
$true = $false,
inference(superposition,[status(thm)],[f104,f382]) ).
thf(f382,plain,
! [X1: $i] :
( ( in @ X1 @ emptyset )
= $false ),
inference(not_proxy_clausification,[status(thm)],[f381]) ).
thf(f381,plain,
! [X1: $i] :
( ( ~ ( in @ X1 @ emptyset ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f380]) ).
thf(f380,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
~ ( in @ Y0 @ emptyset )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f127]) ).
thf(f127,plain,
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
~ ( in @ Y0 @ emptyset ) ) ),
inference(definition_unfolding,[status(thm)],[f86,f74]) ).
thf(f74,plain,
( emptysetAx
= ( !! @ $i
@ ^ [Y0: $i] :
~ ( in @ Y0 @ emptyset ) ) ),
inference(cnf_transformation,[status(thm)],[f41]) ).
thf(f41,plain,
( emptysetAx
= ( !! @ $i
@ ^ [Y0: $i] :
~ ( in @ Y0 @ emptyset ) ) ),
inference(fool_elimination,[status(thm)],[f40]) ).
thf(f40,plain,
! [X0: $i] :
( ( ~ ( in @ X0 @ emptyset ) )
= emptysetAx ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
! [X1: $i] :
( ( ~ ( in @ X1 @ emptyset ) )
= emptysetAx ),
file('/export/starexec/sandbox/tmp/tmp.S0r6rRSKS2/DTF2THF_24529.p',emptysetAx) ).
thf(f86,plain,
emptysetAx = $true,
inference(cnf_transformation,[status(thm)],[f70]) ).
thf(f70,plain,
( ( setunionAx = $true )
& ( ( in @ sK0 @ emptyset )
= $true )
& ( omegaSAx = $true )
& ( dsetconstrER = $true )
& ( replAx = $true )
& ( setextAx = $true )
& ( emptysetimpfalse = $true )
& ( descrp = $true )
& ( omega0Ax = $true )
& ( emptysetE = $true )
& ( powersetAx = $true )
& ( prop2setE = $true )
& ( omegaIndAx = $true )
& ( dsetconstrI = $true )
& ( exuE1 = $true )
& ( foundationAx = $true )
& ( wellorderingAx = $true )
& ( setadjoinAx = $true )
& ( dsetconstrEL = $true )
& ( emptysetAx = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0])],[f68,f69]) ).
thf(f69,plain,
( ? [X0: $i] :
( $true
= ( in @ X0 @ emptyset ) )
=> ( ( in @ sK0 @ emptyset )
= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f68,plain,
( ( setunionAx = $true )
& ? [X0: $i] :
( $true
= ( in @ X0 @ emptyset ) )
& ( omegaSAx = $true )
& ( dsetconstrER = $true )
& ( replAx = $true )
& ( setextAx = $true )
& ( emptysetimpfalse = $true )
& ( descrp = $true )
& ( omega0Ax = $true )
& ( emptysetE = $true )
& ( powersetAx = $true )
& ( prop2setE = $true )
& ( omegaIndAx = $true )
& ( dsetconstrI = $true )
& ( exuE1 = $true )
& ( foundationAx = $true )
& ( wellorderingAx = $true )
& ( setadjoinAx = $true )
& ( dsetconstrEL = $true )
& ( emptysetAx = $true ) ),
inference(flattening,[status(thm)],[f67]) ).
thf(f67,plain,
( ? [X0: $i] :
( $true
= ( in @ X0 @ emptyset ) )
& ( 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)],[f66]) ).
thf(f66,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 )
=> ! [X0: $i] :
( $true
!= ( in @ X0 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(flattening,[status(thm)],[f43]) ).
thf(f43,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 )
=> ! [X0: $i] :
( $true
!= ( in @ X0 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f42]) ).
thf(f42,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ! [X0: $i] :
~ ( in @ X0 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f23]) ).
thf(f23,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ! [X1: $i] :
~ ( in @ X1 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f22]) ).
thf(f22,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ! [X1: $i] :
~ ( in @ X1 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.S0r6rRSKS2/DTF2THF_24529.p',notinemptyset) ).
thf(f104,plain,
( ( in @ sK0 @ emptyset )
= $true ),
inference(cnf_transformation,[status(thm)],[f70]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU503^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n026.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 12:25:10 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.27/0.47 ---- Original DTF file ---
% 0.27/0.47 thf(spec,logic,$$dhol).
% 0.27/0.47 %------------------------------------------------------------------------------
% 0.27/0.47 % File : SEU503^1 : TPTP v9.2.1. Released v3.7.0.
% 0.27/0.47 % Domain : Set Theory
% 0.27/0.47 % Problem : Preliminary Notions - Basic Laws of Logic
% 0.27/0.47 % Version : Especial.
% 0.27/0.47 % English : (! x:i.~(in x emptyset))
% 0.27/0.47
% 0.27/0.47 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.27/0.47 % Source : [Bro08]
% 0.27/0.47 % Names : ZFC005g [Bro08]
% 0.27/0.47
% 0.27/0.47 % Status : Theorem
% 0.27/0.47 % 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 v5.2.0, 0.20 v4.1.0, 0.33 v3.7.0
% 0.27/0.47 % Syntax : Number of formulae : 51 ( 21 unt; 29 typ; 21 def)
% 0.27/0.47 % Number of atoms : 130 ( 26 equ; 0 cnn)
% 0.27/0.47 % Maximal formula atoms : 20 ( 5 avg)
% 0.27/0.47 % Number of connectives : 243 ( 4 ~; 3 |; 18 &; 161 @)
% 0.27/0.47 % ( 6 <=>; 51 =>; 0 <=; 0 <~>)
% 0.27/0.47 % Maximal formula depth : 24 ( 2 avg)
% 0.27/0.47 % Number of types : 2 ( 0 usr)
% 0.27/0.47 % Number of type conns : 22 ( 22 >; 0 *; 0 +; 0 <<)
% 0.27/0.47 % Number of symbols : 31 ( 29 usr; 22 con; 0-2 aty)
% 0.27/0.47 % Number of variables : 77 ( 10 ^; 54 !; 13 ?; 77 :)
% 0.27/0.47 % SPC : TH0_THM_EQU_NAR
% 0.27/0.47
% 0.27/0.47 % Comments : http://mathgate.info/detsetitem.php?id=434
% 0.27/0.47 %------------------------------------------------------------------------------
% 0.27/0.47 thf(in_type,type,
% 0.27/0.47 in: $i > $i > $o ).
% 0.27/0.47
% 0.27/0.47 thf(exu_type,type,
% 0.27/0.47 exu: ( $i > $o ) > $o ).
% 0.27/0.47
% 0.27/0.47 thf(exu,definition,
% 0.27/0.47 ( exu
% 0.27/0.47 = ( ^ [Xphi: $i > $o] :
% 0.27/0.47 ? [Xx: $i] :
% 0.27/0.47 ( ( Xphi @ Xx )
% 0.27/0.47 & ! [Xy: $i] :
% 0.27/0.47 ( ( Xphi @ Xy )
% 0.27/0.47 => ( Xx = Xy ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setextAx_type,type,
% 0.27/0.47 setextAx: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setextAx,definition,
% 0.27/0.47 ( setextAx
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 <=> ( in @ Xx @ B ) )
% 0.27/0.47 => ( A = B ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(emptyset_type,type,
% 0.27/0.47 emptyset: $i ).
% 0.27/0.47
% 0.27/0.47 thf(emptysetAx_type,type,
% 0.27/0.47 emptysetAx: $o ).
% 0.27/0.47
% 0.27/0.47 thf(emptysetAx,definition,
% 0.27/0.47 ( emptysetAx
% 0.27/0.47 = ( ! [Xx: $i] :
% 0.27/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoin_type,type,
% 0.27/0.47 setadjoin: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoinAx_type,type,
% 0.27/0.47 setadjoinAx: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setadjoinAx,definition,
% 0.27/0.47 ( setadjoinAx
% 0.27/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.27/0.47 <=> ( ( Xy = Xx )
% 0.27/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(powerset_type,type,
% 0.27/0.47 powerset: $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(powersetAx_type,type,
% 0.27/0.47 powersetAx: $o ).
% 0.27/0.47
% 0.27/0.47 thf(powersetAx,definition,
% 0.27/0.47 ( powersetAx
% 0.27/0.47 = ( ! [A: $i,B: $i] :
% 0.27/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.27/0.47 <=> ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ B )
% 0.27/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(setunion_type,type,
% 0.27/0.47 setunion: $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(setunionAx_type,type,
% 0.27/0.47 setunionAx: $o ).
% 0.27/0.47
% 0.27/0.47 thf(setunionAx,definition,
% 0.27/0.47 ( setunionAx
% 0.27/0.47 = ( ! [A: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.27/0.47 <=> ? [B: $i] :
% 0.27/0.47 ( ( in @ Xx @ B )
% 0.27/0.47 & ( in @ B @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(omega_type,type,
% 0.27/0.47 omega: $i ).
% 0.27/0.47
% 0.27/0.47 thf(omega0Ax_type,type,
% 0.27/0.47 omega0Ax: $o ).
% 0.27/0.47
% 0.27/0.47 thf(omega0Ax,definition,
% 0.27/0.47 ( omega0Ax
% 0.27/0.47 = ( in @ emptyset @ omega ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(omegaSAx_type,type,
% 0.27/0.47 omegaSAx: $o ).
% 0.27/0.47
% 0.27/0.47 thf(omegaSAx,definition,
% 0.27/0.47 ( omegaSAx
% 0.27/0.47 = ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ omega )
% 0.27/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(omegaIndAx_type,type,
% 0.27/0.47 omegaIndAx: $o ).
% 0.27/0.47
% 0.27/0.47 thf(omegaIndAx,definition,
% 0.27/0.47 ( omegaIndAx
% 0.27/0.47 = ( ! [A: $i] :
% 0.27/0.47 ( ( ( in @ emptyset @ A )
% 0.27/0.47 & ! [Xx: $i] :
% 0.27/0.47 ( ( ( in @ Xx @ omega )
% 0.27/0.47 & ( in @ Xx @ A ) )
% 0.27/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ omega )
% 0.27/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(replAx_type,type,
% 0.27/0.47 replAx: $o ).
% 0.27/0.47
% 0.27/0.47 thf(replAx,definition,
% 0.27/0.47 ( replAx
% 0.27/0.47 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.27/0.47 ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( exu
% 0.27/0.47 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.27/0.47 => ? [B: $i] :
% 0.27/0.47 ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ B )
% 0.27/0.47 <=> ? [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ A )
% 0.27/0.47 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(foundationAx_type,type,
% 0.27/0.47 foundationAx: $o ).
% 0.27/0.47
% 0.27/0.47 thf(foundationAx,definition,
% 0.27/0.47 ( foundationAx
% 0.27/0.47 = ( ! [A: $i] :
% 0.27/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.27/0.47 => ? [B: $i] :
% 0.27/0.47 ( ( in @ B @ A )
% 0.27/0.47 & ~ ? [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ B )
% 0.27/0.47 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(wellorderingAx_type,type,
% 0.27/0.47 wellorderingAx: $o ).
% 0.27/0.47
% 0.27/0.47 thf(wellorderingAx,definition,
% 0.27/0.47 ( wellorderingAx
% 0.27/0.47 = ( ! [A: $i] :
% 0.27/0.47 ? [B: $i] :
% 0.27/0.47 ( ! [C: $i] :
% 0.27/0.47 ( ( in @ C @ B )
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ C )
% 0.27/0.47 => ( in @ Xx @ A ) ) )
% 0.27/0.47 & ! [Xx: $i,Xy: $i] :
% 0.27/0.47 ( ( ( in @ Xx @ A )
% 0.27/0.47 & ( in @ Xy @ A ) )
% 0.27/0.47 => ( ! [C: $i] :
% 0.27/0.47 ( ( in @ C @ B )
% 0.27/0.47 => ( ( in @ Xx @ C )
% 0.27/0.47 <=> ( in @ Xy @ C ) ) )
% 0.27/0.47 => ( Xx = Xy ) ) )
% 0.27/0.47 & ! [C: $i,D: $i] :
% 0.27/0.47 ( ( ( in @ C @ B )
% 0.27/0.47 & ( in @ D @ B ) )
% 0.27/0.47 => ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ C )
% 0.27/0.47 => ( in @ Xx @ D ) )
% 0.27/0.47 | ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ D )
% 0.27/0.47 => ( in @ Xx @ C ) ) ) )
% 0.27/0.47 & ! [C: $i] :
% 0.27/0.47 ( ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ C )
% 0.27/0.47 => ( in @ Xx @ A ) )
% 0.27/0.47 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.27/0.47 => ? [D: $i,Xx: $i] :
% 0.27/0.47 ( ( in @ D @ B )
% 0.27/0.47 & ( in @ Xx @ C )
% 0.27/0.47 & ~ ? [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ D )
% 0.27/0.47 & ( in @ Xy @ C ) )
% 0.27/0.47 & ! [E: $i] :
% 0.27/0.47 ( ( in @ E @ B )
% 0.27/0.47 => ( ! [Xy: $i] :
% 0.27/0.47 ( ( in @ Xy @ E )
% 0.27/0.47 => ( in @ Xy @ D ) )
% 0.27/0.47 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(descr_type,type,
% 0.27/0.47 descr: ( $i > $o ) > $i ).
% 0.27/0.47
% 0.27/0.47 thf(descrp_type,type,
% 0.27/0.47 descrp: $o ).
% 0.27/0.47
% 0.27/0.47 thf(descrp,definition,
% 0.27/0.47 ( descrp
% 0.27/0.47 = ( ! [Xphi: $i > $o] :
% 0.27/0.47 ( ( exu
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 => ( Xphi
% 0.27/0.47 @ ( descr
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dsetconstr_type,type,
% 0.27/0.47 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.27/0.47
% 0.27/0.47 thf(dsetconstrI_type,type,
% 0.27/0.47 dsetconstrI: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dsetconstrI,definition,
% 0.27/0.47 ( dsetconstrI
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ A )
% 0.27/0.47 => ( ( Xphi @ Xx )
% 0.27/0.47 => ( in @ Xx
% 0.27/0.47 @ ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dsetconstrEL_type,type,
% 0.27/0.47 dsetconstrEL: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dsetconstrEL,definition,
% 0.27/0.47 ( dsetconstrEL
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx
% 0.27/0.47 @ ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.27/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(dsetconstrER_type,type,
% 0.27/0.47 dsetconstrER: $o ).
% 0.27/0.47
% 0.27/0.47 thf(dsetconstrER,definition,
% 0.27/0.47 ( dsetconstrER
% 0.27/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx
% 0.27/0.47 @ ( dsetconstr @ A
% 0.27/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.27/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(exuE1_type,type,
% 0.27/0.47 exuE1: $o ).
% 0.27/0.47
% 0.27/0.47 thf(exuE1,definition,
% 0.27/0.47 ( exuE1
% 0.27/0.47 = ( ! [Xphi: $i > $o] :
% 0.27/0.47 ( ( exu
% 0.27/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.47 => ? [Xx: $i] :
% 0.27/0.47 ( ( Xphi @ Xx )
% 0.27/0.47 & ! [Xy: $i] :
% 0.27/0.47 ( ( Xphi @ Xy )
% 0.27/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(prop2set_type,type,
% 0.27/0.47 prop2set: $o > $i ).
% 0.27/0.47
% 0.27/0.47 thf(prop2set,definition,
% 0.27/0.47 ( prop2set
% 0.27/0.47 = ( ^ [Xphi: $o] :
% 0.27/0.47 ( dsetconstr @ ( powerset @ emptyset )
% 0.27/0.47 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(prop2setE_type,type,
% 0.27/0.47 prop2setE: $o ).
% 0.27/0.47
% 0.27/0.47 thf(prop2setE,definition,
% 0.27/0.47 ( prop2setE
% 0.27/0.47 = ( ! [Xphi: $o,Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.27/0.47 => Xphi ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(emptysetE_type,type,
% 0.27/0.47 emptysetE: $o ).
% 0.27/0.47
% 0.27/0.47 thf(emptysetE,definition,
% 0.27/0.47 ( emptysetE
% 0.27/0.47 = ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ emptyset )
% 0.27/0.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(emptysetimpfalse_type,type,
% 0.27/0.47 emptysetimpfalse: $o ).
% 0.27/0.47
% 0.27/0.47 thf(emptysetimpfalse,definition,
% 0.27/0.47 ( emptysetimpfalse
% 0.27/0.47 = ( ! [Xx: $i] :
% 0.27/0.47 ( ( in @ Xx @ emptyset )
% 0.27/0.47 => $false ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(notinemptyset,conjecture,
% 0.27/0.47 ( setextAx
% 0.27/0.47 => ( emptysetAx
% 0.27/0.47 => ( setadjoinAx
% 0.27/0.47 => ( powersetAx
% 0.27/0.47 => ( setunionAx
% 0.27/0.47 => ( omega0Ax
% 0.27/0.47 => ( omegaSAx
% 0.27/0.47 => ( omegaIndAx
% 0.27/0.47 => ( replAx
% 0.27/0.47 => ( foundationAx
% 0.27/0.47 => ( wellorderingAx
% 0.27/0.47 => ( descrp
% 0.27/0.47 => ( dsetconstrI
% 0.27/0.47 => ( dsetconstrEL
% 0.27/0.47 => ( dsetconstrER
% 0.27/0.47 => ( exuE1
% 0.27/0.47 => ( prop2setE
% 0.27/0.47 => ( emptysetE
% 0.27/0.47 => ( emptysetimpfalse
% 0.27/0.47 => ! [Xx: $i] :
% 0.27/0.47 ~ ( in @ Xx @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 %------------------------------------------------------------------------------
% 0.27/0.47 ------------------------
% 1.76/1.33 ---- Embedded in THF ---
% 1.76/1.33 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.76/1.33 %%% Generated on Tue Feb 24 12:25:11 EST 2026
% 1.76/1.33 %%% using '$$dhol' embedding, version 1.3.0.
% 1.76/1.33 %%% Logic specification used:
% 1.76/1.33 %%% thf(spec, logic, $$dhol).
% 1.76/1.33
% 1.76/1.33 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.S0r6rRSKS2/DTF2DTF24529.p
% See solution above
% 1.76/1.33 ------------------------
% 1.76/1.34 ---- Cleaned THF ---
% 1.76/1.34 thf(in_type,type,
% 1.76/1.34 in: $i > $i > $o ).
% 1.76/1.34
% 1.76/1.34 thf(exu_type,type,
% 1.76/1.34 exu: ( $i > $o ) > $o ).
% 1.76/1.34
% 1.76/1.34 thf(exu,definition,
% 1.76/1.34 ( exu
% 1.76/1.34 = ( ^ [Xphi: $i > $o] :
% 1.76/1.34 ? [Xx: $i] :
% 1.76/1.34 ( ( Xphi @ Xx )
% 1.76/1.34 & ! [Xy: $i] :
% 1.76/1.34 ( ( Xphi @ Xy )
% 1.76/1.34 => ( Xx = Xy ) ) ) ) ) ).
% 1.76/1.34
% 1.76/1.34 thf(setextAx_type,type,
% 1.76/1.34 setextAx: $o ).
% 1.76/1.34
% 1.76/1.34 thf(setextAx,definition,
% 1.76/1.34 ( setextAx
% 1.76/1.34 = ( ! [A: $i,B: $i] :
% 1.76/1.34 ( ! [Xx: $i] :
% 1.76/1.34 ( ( in @ Xx @ A )
% 1.76/1.34 <=> ( in @ Xx @ B ) )
% 1.76/1.34 => ( A = B ) ) ) ) ).
% 1.76/1.34
% 1.76/1.34 thf(emptyset_type,type,
% 1.76/1.34 emptyset: $i ).
% 1.76/1.34
% 1.76/1.34 thf(emptysetAx_type,type,
% 1.76/1.34 emptysetAx: $o ).
% 1.76/1.34
% 1.76/1.34 thf(emptysetAx,definition,
% 1.76/1.34 ( emptysetAx
% 1.76/1.34 = ( ! [Xx: $i] :
% 1.76/1.34 ~ ( in @ Xx @ emptyset ) ) ) ).
% 1.76/1.34
% 1.76/1.34 thf(setadjoin_type,type,
% 1.76/1.34 setadjoin: $i > $i > $i ).
% 1.76/1.34
% 1.76/1.34 thf(setadjoinAx_type,type,
% 1.76/1.34 setadjoinAx: $o ).
% 1.76/1.34
% 1.76/1.34 thf(setadjoinAx,definition,
% 1.76/1.34 ( setadjoinAx
% 1.76/1.34 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.76/1.34 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.76/1.34 <=> ( ( Xy = Xx )
% 1.76/1.34 | ( in @ Xy @ A ) ) ) ) ) ).
% 1.76/1.34
% 1.76/1.34 thf(powerset_type,type,
% 1.76/1.34 powerset: $i > $i ).
% 1.76/1.34
% 1.76/1.34 thf(powersetAx_type,type,
% 1.76/1.34 powersetAx: $o ).
% 1.76/1.34
% 1.76/1.34 thf(powersetAx,definition,
% 1.76/1.34 ( powersetAx
% 1.76/1.34 = ( ! [A: $i,B: $i] :
% 1.76/1.34 ( ( in @ B @ ( powerset @ A ) )
% 1.76/1.34 <=> ! [Xx: $i] :
% 1.76/1.34 ( ( in @ Xx @ B )
% 1.76/1.34 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.76/1.34
% 1.76/1.34 thf(setunion_type,type,
% 1.76/1.34 setunion: $i > $i ).
% 1.76/1.34
% 1.76/1.34 thf(setunionAx_type,type,
% 1.76/1.34 setunionAx: $o ).
% 1.76/1.34
% 1.76/1.34 thf(setunionAx,definition,
% 1.76/1.34 ( setunionAx
% 1.76/1.34 = ( ! [A: $i,Xx: $i] :
% 1.76/1.34 ( ( in @ Xx @ ( setunion @ A ) )
% 1.76/1.34 <=> ? [B: $i] :
% 1.76/1.34 ( ( in @ Xx @ B )
% 1.76/1.34 & ( in @ B @ A ) ) ) ) ) ).
% 1.76/1.34
% 1.76/1.34 thf(omega_type,type,
% 1.76/1.34 omega: $i ).
% 1.76/1.34
% 1.76/1.34 thf(omega0Ax_type,type,
% 1.76/1.34 omega0Ax: $o ).
% 1.76/1.34
% 1.76/1.34 thf(omega0Ax,definition,
% 1.76/1.34 ( omega0Ax
% 1.76/1.34 = ( in @ emptyset @ omega ) ) ).
% 1.76/1.34
% 1.76/1.34 thf(omegaSAx_type,type,
% 1.76/1.34 omegaSAx: $o ).
% 1.76/1.34
% 1.76/1.34 thf(omegaSAx,definition,
% 1.76/1.34 ( omegaSAx
% 1.76/1.34 = ( ! [Xx: $i] :
% 1.76/1.34 ( ( in @ Xx @ omega )
% 1.76/1.34 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 1.76/1.34
% 1.76/1.34 thf(omegaIndAx_type,type,
% 1.76/1.34 omegaIndAx: $o ).
% 1.76/1.34
% 1.76/1.34 thf(omegaIndAx,definition,
% 1.76/1.34 ( omegaIndAx
% 1.76/1.34 = ( ! [A: $i] :
% 1.76/1.34 ( ( ( in @ emptyset @ A )
% 1.76/1.34 & ! [Xx: $i] :
% 1.76/1.34 ( ( ( in @ Xx @ omega )
% 1.76/1.34 & ( in @ Xx @ A ) )
% 1.76/1.34 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 1.76/1.34 => ! [Xx: $i] :
% 1.76/1.34 ( ( in @ Xx @ omega )
% 1.76/1.34 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.76/1.34
% 1.76/1.34 thf(replAx_type,type,
% 1.76/1.34 replAx: $o ).
% 1.76/1.34
% 1.76/1.34 thf(replAx,definition,
% 1.76/1.34 ( replAx
% 1.76/1.34 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 1.76/1.34 ( ! [Xx: $i] :
% 1.76/1.34 ( ( in @ Xx @ A )
% 1.76/1.34 => ( exu
% 1.76/1.34 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 1.76/1.34 => ? [B: $i] :
% 1.76/1.34 ! [Xx: $i] :
% 1.76/1.34 ( ( in @ Xx @ B )
% 1.76/1.34 <=> ? [Xy: $i] :
% 1.76/1.34 ( ( in @ Xy @ A )
% 1.76/1.34 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 1.76/1.34
% 1.76/1.34 thf(foundationAx_type,type,
% 1.76/1.34 foundationAx: $o ).
% 1.76/1.34
% 1.76/1.34 thf(foundationAx,definition,
% 1.76/1.34 ( foundationAx
% 1.76/1.34 = ( ! [A: $i] :
% 1.76/1.34 ( ? [Xx: $i] : ( in @ Xx @ A )
% 1.76/1.34 => ? [B: $i] :
% 1.76/1.34 ( ( in @ B @ A )
% 1.76/1.34 & ~ ? [Xx: $i] :
% 1.76/1.34 ( ( in @ Xx @ B )
% 1.76/1.34 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 1.76/1.34
% 1.76/1.34 thf(wellorderingAx_type,type,
% 1.76/1.34 wellorderingAx: $o ).
% 1.76/1.34
% 1.76/1.34 thf(wellorderingAx,definition,
% 1.76/1.34 ( wellorderingAx
% 1.76/1.34 = ( ! [A: $i] :
% 1.76/1.34 ? [B: $i] :
% 1.76/1.34 ( ! [C: $i] :
% 1.76/1.34 ( ( in @ C @ B )
% 1.76/1.34 => ! [Xx: $i] :
% 1.76/1.34 ( ( in @ Xx @ C )
% 1.76/1.34 => ( in @ Xx @ A ) ) )
% 1.76/1.34 & ! [Xx: $i,Xy: $i] :
% 1.76/1.34 ( ( ( in @ Xx @ A )
% 1.76/1.34 & ( in @ Xy @ A ) )
% 1.76/1.34 => ( ! [C: $i] :
% 1.76/1.34 ( ( in @ C @ B )
% 1.76/1.34 => ( ( in @ Xx @ C )
% 1.76/1.34 <=> ( in @ Xy @ C ) ) )
% 1.76/1.34 => ( Xx = Xy ) ) )
% 1.76/1.34 & ! [C: $i,D: $i] :
% 1.76/1.34 ( ( ( in @ C @ B )
% 1.76/1.34 & ( in @ D @ B ) )
% 1.76/1.34 => ( ! [Xx: $i] :
% 1.76/1.34 ( ( in @ Xx @ C )
% 1.76/1.34 => ( in @ Xx @ D ) )
% 1.76/1.34 | ! [Xx: $i] :
% 1.76/1.34 ( ( in @ Xx @ D )
% 1.76/1.34 => ( in @ Xx @ C ) ) ) )
% 1.76/1.34 & ! [C: $i] :
% 1.76/1.34 ( ( ! [Xx: $i] :
% 1.76/1.34 ( ( in @ Xx @ C )
% 1.76/1.34 => ( in @ Xx @ A ) )
% 1.76/1.34 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 1.76/1.34 => ? [D: $i,Xx: $i] :
% 1.76/1.34 ( ( in @ D @ B )
% 1.76/1.34 & ( in @ Xx @ C )
% 1.76/1.34 & ~ ? [Xy: $i] :
% 1.76/1.34 ( ( in @ Xy @ D )
% 1.76/1.34 & ( in @ Xy @ C ) )
% 1.76/1.34 & ! [E: $i] :
% 1.76/1.34 ( ( in @ E @ B )
% 1.76/1.34 => ( ! [Xy: $i] :
% 1.76/1.34 ( ( in @ Xy @ E )
% 1.76/1.34 => ( in @ Xy @ D ) )
% 1.76/1.34 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 1.76/1.34
% 1.76/1.34 thf(descr_type,type,
% 1.76/1.34 descr: ( $i > $o ) > $i ).
% 1.76/1.34
% 1.76/1.34 thf(descrp_type,type,
% 1.76/1.34 descrp: $o ).
% 1.76/1.34
% 1.76/1.34 thf(descrp,definition,
% 1.76/1.34 ( descrp
% 1.76/1.34 = ( ! [Xphi: $i > $o] :
% 1.76/1.34 ( ( exu
% 1.76/1.34 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.76/1.34 => ( Xphi
% 1.76/1.34 @ ( descr
% 1.76/1.34 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 1.76/1.34
% 1.76/1.34 thf(dsetconstr_type,type,
% 1.76/1.34 dsetconstr: $i > ( $i > $o ) > $i ).
% 1.76/1.34
% 1.76/1.34 thf(dsetconstrI_type,type,
% 1.76/1.34 dsetconstrI: $o ).
% 1.76/1.34
% 1.76/1.34 thf(dsetconstrI,definition,
% 1.76/1.34 ( dsetconstrI
% 1.76/1.34 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.76/1.34 ( ( in @ Xx @ A )
% 1.76/1.34 => ( ( Xphi @ Xx )
% 1.76/1.34 => ( in @ Xx
% 1.76/1.34 @ ( dsetconstr @ A
% 1.76/1.34 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 1.76/1.34
% 1.76/1.34 thf(dsetconstrEL_type,type,
% 1.76/1.34 dsetconstrEL: $o ).
% 1.76/1.34
% 1.76/1.34 thf(dsetconstrEL,definition,
% 1.76/1.34 ( dsetconstrEL
% 1.76/1.34 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.76/1.34 ( ( in @ Xx
% 1.76/1.34 @ ( dsetconstr @ A
% 1.76/1.34 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.76/1.34 => ( in @ Xx @ A ) ) ) ) ).
% 1.76/1.34
% 1.76/1.34 thf(dsetconstrER_type,type,
% 1.76/1.34 dsetconstrER: $o ).
% 1.76/1.34
% 1.76/1.34 thf(dsetconstrER,definition,
% 1.76/1.34 ( dsetconstrER
% 1.76/1.34 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.76/1.34 ( ( in @ Xx
% 1.76/1.34 @ ( dsetconstr @ A
% 1.76/1.34 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.76/1.34 => ( Xphi @ Xx ) ) ) ) ).
% 1.76/1.34
% 1.76/1.34 thf(exuE1_type,type,
% 1.76/1.34 exuE1: $o ).
% 1.76/1.34
% 1.76/1.34 thf(exuE1,definition,
% 1.76/1.34 ( exuE1
% 1.76/1.34 = ( ! [Xphi: $i > $o] :
% 1.76/1.34 ( ( exu
% 1.76/1.34 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.76/1.34 => ? [Xx: $i] :
% 1.76/1.34 ( ( Xphi @ Xx )
% 1.76/1.34 & ! [Xy: $i] :
% 1.76/1.34 ( ( Xphi @ Xy )
% 1.76/1.34 => ( Xx = Xy ) ) ) ) ) ) ).
% 1.76/1.34
% 1.76/1.34 thf(prop2set_type,type,
% 1.76/1.34 prop2set: $o > $i ).
% 1.76/1.34
% 1.76/1.34 thf(prop2set,definition,
% 1.76/1.34 ( prop2set
% 1.76/1.34 = ( ^ [Xphi: $o] :
% 1.76/1.34 ( dsetconstr @ ( powerset @ emptyset )
% 1.76/1.34 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 1.76/1.34
% 1.76/1.34 thf(prop2setE_type,type,
% 1.76/1.34 prop2setE: $o ).
% 1.76/1.34
% 1.76/1.34 thf(prop2setE,definition,
% 1.76/1.34 ( prop2setE
% 1.76/1.34 = ( ! [Xphi: $o,Xx: $i] :
% 1.76/1.34 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 1.76/1.34 => Xphi ) ) ) ).
% 1.76/1.34
% 1.76/1.34 thf(emptysetE_type,type,
% 1.76/1.34 emptysetE: $o ).
% 1.76/1.34
% 1.76/1.34 thf(emptysetE,definition,
% 1.76/1.34 ( emptysetE
% 1.76/1.34 = ( ! [Xx: $i] :
% 1.76/1.34 ( ( in @ Xx @ emptyset )
% 1.76/1.34 => ! [Xphi: $o] : Xphi ) ) ) ).
% 1.76/1.34
% 1.76/1.34 thf(emptysetimpfalse_type,type,
% 1.76/1.34 emptysetimpfalse: $o ).
% 1.76/1.34
% 1.76/1.34 thf(emptysetimpfalse,definition,
% 1.76/1.34 ( emptysetimpfalse
% 1.76/1.34 = ( ! [Xx: $i] :
% 1.76/1.34 ( ( in @ Xx @ emptyset )
% 1.76/1.34 => $false ) ) ) ).
% 1.76/1.34
% 1.76/1.34 thf(notinemptyset,conjecture,
% 1.76/1.34 ( setextAx
% 1.76/1.34 => ( emptysetAx
% 1.76/1.34 => ( setadjoinAx
% 1.76/1.34 => ( powersetAx
% 1.76/1.34 => ( setunionAx
% 1.76/1.34 => ( omega0Ax
% 1.76/1.34 => ( omegaSAx
% 1.76/1.34 => ( omegaIndAx
% 1.76/1.34 => ( replAx
% 1.76/1.34 => ( foundationAx
% 1.76/1.34 => ( wellorderingAx
% 1.76/1.34 => ( descrp
% 1.76/1.34 => ( dsetconstrI
% 1.76/1.34 => ( dsetconstrEL
% 1.76/1.34 => ( dsetconstrER
% 1.76/1.34 => ( exuE1
% 1.76/1.34 => ( prop2setE
% 1.76/1.34 => ( emptysetE
% 1.76/1.34 => ( emptysetimpfalse
% 1.76/1.34 => ! [Xx: $i] :
% 1.76/1.34 ~ ( in @ Xx @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 1.76/1.34 ------------------------
% 1.76/1.35 % (24813)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_24529 for (2999ds/4Mi)
% 1.76/1.36 % (24812)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_24529 for (2999ds/183Mi)
% 1.76/1.36 % (24813)Instruction limit reached!
% 1.76/1.36 % (24813)------------------------------
% 1.76/1.36 % (24813)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.76/1.36 % (24813)Termination reason: Unknown
% 1.76/1.36 % (24813)Termination phase: Property scanning
% 1.76/1.36
% 1.76/1.36 % (24813)Memory used [KB]: 1023
% 1.76/1.36 % (24813)Time elapsed: 0.003 s
% 1.76/1.36 % (24813)Instructions burned: 5 (million)
% 1.76/1.36 % (24813)------------------------------
% 1.76/1.36 % (24813)------------------------------
% 1.76/1.36 % (24814)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_24529 for (2999ds/27Mi)
% 1.76/1.36 % (24815)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_24529 for (2999ds/2Mi)
% 1.76/1.36 % (24816)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_24529 for (2999ds/2Mi)
% 1.76/1.36 % (24817)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_24529 for (2999ds/275Mi)
% 1.76/1.36 % (24818)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_24529 for (2999ds/18Mi)
% 1.76/1.36 % (24815)Instruction limit reached!
% 1.76/1.36 % (24815)------------------------------
% 1.76/1.36 % (24815)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.76/1.36 % (24815)Termination reason: Unknown
% 1.76/1.36 % (24815)Termination phase: shuffling
% 1.76/1.36
% 1.76/1.36 % (24815)Memory used [KB]: 1023
% 1.76/1.36 % (24815)Time elapsed: 0.003 s
% 1.76/1.36 % (24815)Instructions burned: 2 (million)
% 1.76/1.36 % (24815)------------------------------
% 1.76/1.36 % (24815)------------------------------
% 1.76/1.36 % (24816)Instruction limit reached!
% 1.76/1.36 % (24816)------------------------------
% 1.76/1.36 % (24816)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.76/1.36 % (24816)Termination reason: Unknown
% 1.76/1.36 % (24816)Termination phase: shuffling
% 1.76/1.36
% 1.76/1.36 % (24816)Memory used [KB]: 1023
% 1.76/1.36 % (24816)Time elapsed: 0.003 s
% 1.76/1.36 % (24816)Instructions burned: 2 (million)
% 1.76/1.36 % (24816)------------------------------
% 1.76/1.36 % (24816)------------------------------
% 1.76/1.37 % (24819)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_24529 for (2999ds/3Mi)
% 1.76/1.37 % (24819)Instruction limit reached!
% 1.76/1.37 % (24819)------------------------------
% 1.76/1.37 % (24819)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.76/1.37 % (24819)Termination reason: Unknown
% 1.76/1.37 % (24819)Termination phase: shuffling
% 1.76/1.37
% 1.76/1.37 % (24819)Memory used [KB]: 1023
% 1.76/1.37 % (24819)Time elapsed: 0.002 s
% 1.76/1.37 % (24819)Instructions burned: 3 (million)
% 1.76/1.37 % (24819)------------------------------
% 1.76/1.37 % (24819)------------------------------
% 1.76/1.37 % (24812)First to succeed.
% 1.76/1.37 % (24812)Refutation found. Thanks to Tanya!
% 1.76/1.37 % SZS status Theorem for DTF2THF_24529
% 1.76/1.37 % SZS output start Proof for DTF2THF_24529
% 1.76/1.37 thf(func_def_0, type, in: $i > $i > $o).
% 1.76/1.37 thf(func_def_1, type, exu: ($i > $o) > $o).
% 1.76/1.37 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 1.76/1.37 thf(func_def_8, type, powerset: $i > $i).
% 1.76/1.37 thf(func_def_10, type, setunion: $i > $i).
% 1.76/1.37 thf(func_def_19, type, descr: ($i > $o) > $i).
% 1.76/1.37 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 1.76/1.37 thf(func_def_26, type, prop2set: $o > $i).
% 1.76/1.37 thf(func_def_48, type, sK2: $i > $i).
% 1.76/1.37 thf(func_def_49, type, sK3: $i > $i).
% 1.76/1.37 thf(func_def_50, type, sK4: $i > $i > $i).
% 1.76/1.37 thf(func_def_51, type, sK5: $i > $i > $i).
% 1.76/1.37 thf(func_def_52, type, sK6: $i > $i > $i).
% 1.76/1.37 thf(func_def_53, type, sK7: $i > $i > $i).
% 1.76/1.37 thf(func_def_54, type, sK8: $i > $i > $i > $i).
% 1.76/1.37 thf(func_def_55, type, sK9: $i > $i > $i).
% 1.76/1.37 thf(func_def_56, type, sK10: ($i > $o) > $i > $i).
% 1.76/1.37 thf(func_def_57, type, sK11: ($i > $o) > $i).
% 1.76/1.37 thf(func_def_58, type, sK12: ($i > $o) > $i > $i).
% 1.76/1.37 thf(func_def_59, type, sK13: $i > ($i > $o) > $i).
% 1.76/1.37 thf(func_def_60, type, sK14: $i > $i).
% 1.76/1.37 thf(func_def_61, type, sK15: $i > $i > $i).
% 1.76/1.37 thf(func_def_62, type, sK16: $i > $i > $i).
% 1.76/1.37 thf(func_def_63, type, sK17: $i > ($i > $i > $o) > $i).
% 1.76/1.37 thf(func_def_64, type, sK18: $i > ($i > $i > $o) > $i).
% 1.76/1.37 thf(func_def_65, type, sK19: $i > $i > ($i > $i > $o) > $i).
% 1.76/1.37 thf(func_def_66, type, sK20: $i > $i > ($i > $i > $o) > $i).
% 1.76/1.37 thf(func_def_67, type, sK21: $i > $i > ($i > $i > $o) > $i).
% 1.76/1.37 thf(func_def_68, type, sK22: $i > $i > ($i > $i > $o) > $i).
% 1.76/1.37 thf(f448,plain,(
% 1.76/1.37 $false),
% 1.76/1.37 inference(trivial_inequality_removal,[status(thm)],[f447])).
% 1.76/1.37 thf(f447,plain,(
% 1.76/1.37 ($true = $false)),
% 1.76/1.37 inference(superposition,[status(thm)],[f104,f382])).
% 1.76/1.37 thf(f382,plain,(
% 1.76/1.37 ( ! [X1 : $i] : (((in @ X1 @ emptyset) = $false)) )),
% 1.76/1.37 inference(not_proxy_clausification,[status(thm)],[f381])).
% 1.76/1.37 thf(f381,plain,(
% 1.76/1.37 ( ! [X1 : $i] : (((~ (in @ X1 @ emptyset)) = $true)) )),
% 1.76/1.37 inference(beta_eta_normalization,[status(thm)],[f380])).
% 1.76/1.37 thf(f380,plain,(
% 1.76/1.37 ( ! [X1 : $i] : ((((^[Y0 : $i]: (~ (in @ Y0 @ emptyset))) @ X1) = $true)) )),
% 1.76/1.37 inference(pi_clausification,[status(thm)],[f127])).
% 1.76/1.37 thf(f127,plain,(
% 1.76/1.37 ($true = (!! @ $i @ (^[Y0 : $i]: (~ (in @ Y0 @ emptyset)))))),
% 1.76/1.37 inference(definition_unfolding,[status(thm)],[f86,f74])).
% 1.76/1.37 thf(f74,plain,(
% 1.76/1.37 (emptysetAx = (!! @ $i @ (^[Y0 : $i]: (~ (in @ Y0 @ emptyset)))))),
% 1.76/1.37 inference(cnf_transformation,[status(thm)],[f41])).
% 1.76/1.37 thf(f41,plain,(
% 1.76/1.37 (emptysetAx = (!! @ $i @ (^[Y0 : $i]: (~ (in @ Y0 @ emptyset)))))),
% 1.76/1.37 inference(fool_elimination,[status(thm)],[f40])).
% 1.76/1.37 thf(f40,plain,(
% 1.76/1.37 (! [X0 : $i] : ~(in @ X0 @ emptyset) = emptysetAx)),
% 1.76/1.37 inference(rectify,[status(thm)],[f3])).
% 1.76/1.37 thf(f3,axiom,(
% 1.76/1.37 (! [X1 : $i] : ~(in @ X1 @ emptyset) = emptysetAx)),
% 1.76/1.37 file('/export/starexec/sandbox/tmp/tmp.S0r6rRSKS2/DTF2THF_24529.p',emptysetAx)).
% 1.76/1.37 thf(f86,plain,(
% 1.76/1.37 (emptysetAx = $true)),
% 1.76/1.37 inference(cnf_transformation,[status(thm)],[f70])).
% 1.76/1.37 thf(f70,plain,(
% 1.76/1.37 (setunionAx = $true) & ((in @ sK0 @ emptyset) = $true) & (omegaSAx = $true) & (dsetconstrER = $true) & (replAx = $true) & (setextAx = $true) & (emptysetimpfalse = $true) & (descrp = $true) & (omega0Ax = $true) & (emptysetE = $true) & (powersetAx = $true) & (prop2setE = $true) & (omegaIndAx = $true) & (dsetconstrI = $true) & (exuE1 = $true) & (foundationAx = $true) & (wellorderingAx = $true) & (setadjoinAx = $true) & (dsetconstrEL = $true) & (emptysetAx = $true)),
% 1.76/1.37 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0])],[f68,f69])).
% 1.76/1.37 thf(f69,plain,(
% 1.76/1.37 ? [X0 : $i] : ($true = (in @ X0 @ emptyset)) => ((in @ sK0 @ emptyset) = $true)),
% 1.76/1.37 introduced(definition,[],[choice_axiom])).
% 1.76/1.37 thf(f68,plain,(
% 1.76/1.37 (setunionAx = $true) & ? [X0 : $i] : ($true = (in @ X0 @ emptyset)) & (omegaSAx = $true) & (dsetconstrER = $true) & (replAx = $true) & (setextAx = $true) & (emptysetimpfalse = $true) & (descrp = $true) & (omega0Ax = $true) & (emptysetE = $true) & (powersetAx = $true) & (prop2setE = $true) & (omegaIndAx = $true) & (dsetconstrI = $true) & (exuE1 = $true) & (foundationAx = $true) & (wellorderingAx = $true) & (setadjoinAx = $true) & (dsetconstrEL = $true) & (emptysetAx = $true)),
% 1.76/1.37 inference(flattening,[status(thm)],[f67])).
% 1.76/1.37 thf(f67,plain,(
% 1.76/1.37 ((((((((((((((((((? [X0 : $i] : ($true = (in @ X0 @ emptyset)) & (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)),
% 1.76/1.37 inference(ennf_transformation,[status(thm)],[f66])).
% 1.76/1.37 thf(f66,plain,(
% 1.76/1.37 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ! [X0 : $i] : ($true != (in @ X0 @ emptyset)))))))))))))))))))))),
% 1.76/1.37 inference(flattening,[status(thm)],[f43])).
% 1.76/1.37 thf(f43,plain,(
% 1.76/1.37 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ! [X0 : $i] : ~($true = (in @ X0 @ emptyset)))))))))))))))))))))),
% 1.76/1.37 inference(fool_elimination,[status(thm)],[f42])).
% 1.76/1.37 thf(f42,plain,(
% 1.76/1.37 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => ! [X0 : $i] : ~(in @ X0 @ emptyset))))))))))))))))))))),
% 1.76/1.37 inference(rectify,[status(thm)],[f23])).
% 1.76/1.37 thf(f23,negated_conjecture,(
% 1.76/1.37 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => ! [X1 : $i] : ~(in @ X1 @ emptyset))))))))))))))))))))),
% 1.76/1.37 inference(negated_conjecture,[status(cth)],[f22])).
% 1.76/1.37 thf(f22,conjecture,(
% 1.76/1.37 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => ! [X1 : $i] : ~(in @ X1 @ emptyset)))))))))))))))))))),
% 1.76/1.37 file('/export/starexec/sandbox/tmp/tmp.S0r6rRSKS2/DTF2THF_24529.p',notinemptyset)).
% 1.76/1.37 thf(f104,plain,(
% 1.76/1.37 ((in @ sK0 @ emptyset) = $true)),
% 1.76/1.37 inference(cnf_transformation,[status(thm)],[f70])).
% 1.76/1.37 % SZS output end Proof for DTF2THF_24529
% 1.76/1.37 % (24812)------------------------------
% 1.76/1.37 % (24812)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.76/1.37 % (24812)Termination reason: Refutation
% 1.76/1.37
% 1.76/1.37 % (24812)Memory used [KB]: 5884
% 1.76/1.37 % (24812)Time elapsed: 0.016 s
% 1.76/1.37 % (24812)Instructions burned: 47 (million)
% 1.76/1.37 % (24812)------------------------------
% 1.76/1.37 % (24812)------------------------------
% 1.76/1.37 % (24811)Success in time 0.025 s
%------------------------------------------------------------------------------