%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU509^1 : 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 : n018.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 1.85s 1.41s
% Output : Refutation 1.85s
% Verified :
% SZS Type : Refutation
% Derivation depth : 18
% Number of leaves : 4
% Syntax : Number of formulae : 34 ( 20 unt; 0 typ; 0 def)
% Number of atoms : 548 ( 158 equ; 0 cnn)
% Maximal formula atoms : 27 ( 16 avg)
% Number of connectives : 316 ( 28 ~; 6 |; 80 &; 80 @)
% ( 0 <=>; 112 =>; 0 <=; 0 <~>)
% Maximal formula depth : 32 ( 7 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 68 ( 64 usr; 30 con; 0-3 aty)
% ( 10 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 45 ( 17 ^; 21 !; 7 ?; 45 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_55,type,
sK2: $i > $i > $i ).
thf(func_def_56,type,
sK3: $i > $i ).
thf(func_def_57,type,
sK4: $i > $i > $i ).
thf(func_def_58,type,
sK5: $i > $i > $i ).
thf(func_def_59,type,
sK6: $i > $i > $i ).
thf(func_def_60,type,
sK7: $i > $i ).
thf(func_def_61,type,
sK8: ( $i > $o ) > $i ).
thf(func_def_62,type,
sK9: ( $i > $o ) > $i > $i ).
thf(func_def_63,type,
sK10: ( $i > $o ) > $i > $i ).
thf(func_def_64,type,
sK11: $i > $i > $i ).
thf(func_def_65,type,
sK12: $i > ( $i > $i > $o ) > $i ).
thf(func_def_66,type,
sK13: $i > ( $i > $i > $o ) > $i ).
thf(func_def_67,type,
sK14: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_68,type,
sK15: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_69,type,
sK16: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_70,type,
sK17: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_71,type,
sK18: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_72,type,
sK19: $i > $i ).
thf(func_def_73,type,
sK20: $i > $i ).
thf(func_def_74,type,
sK21: ( $i > $o ) > $i ).
thf(func_def_75,type,
sK22: ( $i > $o ) > $i > $i ).
thf(func_def_76,type,
sK23: ( $i > $o ) > $i > $i ).
thf(func_def_77,type,
sK24: $i > $i ).
thf(func_def_78,type,
sK25: $i > $i > $i ).
thf(func_def_79,type,
sK26: $i > $i > $i ).
thf(func_def_80,type,
sK27: $i > $i > $i ).
thf(func_def_81,type,
sK28: $i > $i > $i > $i ).
thf(func_def_82,type,
sK29: $i > $i > $i ).
thf(f582,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f581,f284]) ).
thf(f284,plain,
emptyset != sK0,
inference(equality_proxy_clausification,[status(thm)],[f283]) ).
thf(f283,plain,
( ( emptyset = sK0 )
= $false ),
inference(not_proxy_clausification,[status(thm)],[f282]) ).
thf(f282,plain,
( ( emptyset != sK0 )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f161]) ).
thf(f161,plain,
( ( ^ [Y0: $i] : ( emptyset != Y0 )
@ sK0 )
= $true ),
inference(definition_unfolding,[status(thm)],[f118,f139]) ).
thf(f139,plain,
( nonempty
= ( ^ [Y0: $i] : ( emptyset != Y0 ) ) ),
inference(cnf_transformation,[status(thm)],[f85]) ).
thf(f85,plain,
( nonempty
= ( ^ [Y0: $i] : ( emptyset != Y0 ) ) ),
inference(fool_elimination,[status(thm)],[f84]) ).
thf(f84,plain,
( ( ^ [X0: $i] : ( emptyset != X0 ) )
= nonempty ),
inference(rectify,[status(thm)],[f28]) ).
thf(f28,axiom,
( ( ^ [X1: $i] : ( emptyset != X1 ) )
= nonempty ),
file('/export/starexec/sandbox/tmp/tmp.8shZsOoivs/DTF2THF_10582.p',nonempty) ).
thf(f118,plain,
( ( nonempty @ sK0 )
= $true ),
inference(cnf_transformation,[status(thm)],[f90]) ).
thf(f90,plain,
( ( exuE1 = $true )
& ( wellorderingAx = $true )
& ( descrp = $true )
& ( omega0Ax = $true )
& ( emptysetimpfalse = $true )
& ( prop2setE = $true )
& ( powersetAx = $true )
& ( exuE3e = $true )
& ( emptysetE = $true )
& ( noeltsimpempty = $true )
& ( dsetconstrER = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( setadjoinAx = $true )
& ( dsetconstrEL = $true )
& ( setextAx = $true )
& ! [X1: $i] :
( ( in @ X1 @ sK0 )
!= $true )
& ( ( nonempty @ sK0 )
= $true )
& ( setbeta = $true )
& ( emptysetAx = $true )
& ( emptyI = $true )
& ( setunionAx = $true )
& ( setext = $true )
& ( notinemptyset = $true )
& ( omegaSAx = $true )
& ( foundationAx = $true )
& ( dsetconstrI = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0])],[f88,f89]) ).
thf(f89,plain,
( ? [X0: $i] :
( ! [X1: $i] :
( ( in @ X1 @ X0 )
!= $true )
& ( ( nonempty @ X0 )
= $true ) )
=> ( ! [X1: $i] :
( ( in @ X1 @ sK0 )
!= $true )
& ( ( nonempty @ sK0 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f88,plain,
( ( exuE1 = $true )
& ( wellorderingAx = $true )
& ( descrp = $true )
& ( omega0Ax = $true )
& ( emptysetimpfalse = $true )
& ( prop2setE = $true )
& ( powersetAx = $true )
& ( exuE3e = $true )
& ( emptysetE = $true )
& ( noeltsimpempty = $true )
& ( dsetconstrER = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( setadjoinAx = $true )
& ( dsetconstrEL = $true )
& ( setextAx = $true )
& ? [X0: $i] :
( ! [X1: $i] :
( ( in @ X1 @ X0 )
!= $true )
& ( ( nonempty @ X0 )
= $true ) )
& ( setbeta = $true )
& ( emptysetAx = $true )
& ( emptyI = $true )
& ( setunionAx = $true )
& ( setext = $true )
& ( notinemptyset = $true )
& ( omegaSAx = $true )
& ( foundationAx = $true )
& ( dsetconstrI = $true ) ),
inference(flattening,[status(thm)],[f87]) ).
thf(f87,plain,
( ? [X0: $i] :
( ! [X1: $i] :
( ( in @ X1 @ X0 )
!= $true )
& ( ( nonempty @ X0 )
= $true ) )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[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 )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ! [X0: $i] :
( ( ( nonempty @ X0 )
= $true )
=> ? [X1: $i] :
( ( in @ X1 @ X0 )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
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
=> ( noeltsimpempty
=> ( setbeta
=> ! [X0: $i] :
( ( nonempty @ X0 )
=> ? [X1: $i] : ( in @ X1 @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f30]) ).
thf(f30,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ! [X3: $i] :
( ( nonempty @ X3 )
=> ? [X1: $i] : ( in @ X1 @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f29]) ).
thf(f29,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ! [X3: $i] :
( ( nonempty @ X3 )
=> ? [X1: $i] : ( in @ X1 @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.8shZsOoivs/DTF2THF_10582.p',nonemptyE1) ).
thf(f581,plain,
emptyset = sK0,
inference(trivial_inequality_removal,[status(thm)],[f579]) ).
thf(f579,plain,
( ( emptyset = sK0 )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f119,f206]) ).
thf(f206,plain,
! [X1: $i] :
( ( $true
= ( in @ ( sK3 @ X1 ) @ X1 ) )
| ( emptyset = X1 ) ),
inference(not_proxy_clausification,[status(thm)],[f205]) ).
thf(f205,plain,
! [X1: $i] :
( ( emptyset = X1 )
| ( ( ~ ( in @ ( sK3 @ X1 ) @ X1 ) )
= $false ) ),
inference(beta_eta_normalization,[status(thm)],[f204]) ).
thf(f204,plain,
! [X1: $i] :
( ( emptyset = X1 )
| ( ( ^ [Y0: $i] :
~ ( in @ Y0 @ X1 )
@ ( sK3 @ X1 ) )
= $false ) ),
inference(sigma_clausification,[status(thm)],[f203]) ).
thf(f203,plain,
! [X1: $i] :
( ( emptyset = X1 )
| ( ( !! @ $i
@ ^ [Y0: $i] :
~ ( in @ Y0 @ X1 ) )
= $false ) ),
inference(equality_proxy_clausification,[status(thm)],[f202]) ).
thf(f202,plain,
! [X1: $i] :
( ( $true
= ( emptyset = X1 ) )
| ( ( !! @ $i
@ ^ [Y0: $i] :
~ ( in @ Y0 @ X1 ) )
= $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f201]) ).
thf(f201,plain,
! [X1: $i] :
( $true
= ( ( !! @ $i
@ ^ [Y0: $i] :
~ ( in @ Y0 @ X1 ) )
=> ( emptyset = X1 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f200]) ).
thf(f200,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
( ( !! @ $i
@ ^ [Y1: $i] :
~ ( in @ Y1 @ Y0 ) )
=> ( emptyset = Y0 ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f154]) ).
thf(f154,plain,
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( ( !! @ $i
@ ^ [Y1: $i] :
~ ( in @ Y1 @ Y0 ) )
=> ( emptyset = Y0 ) ) ) ),
inference(definition_unfolding,[status(thm)],[f126,f107]) ).
thf(f107,plain,
( noeltsimpempty
= ( !! @ $i
@ ^ [Y0: $i] :
( ( !! @ $i
@ ^ [Y1: $i] :
~ ( in @ Y1 @ Y0 ) )
=> ( emptyset = Y0 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f51]) ).
thf(f51,plain,
( noeltsimpempty
= ( !! @ $i
@ ^ [Y0: $i] :
( ( !! @ $i
@ ^ [Y1: $i] :
~ ( in @ Y1 @ Y0 ) )
=> ( emptyset = Y0 ) ) ) ),
inference(fool_elimination,[status(thm)],[f50]) ).
thf(f50,plain,
( noeltsimpempty
= ( ! [X0: $i] :
( ! [X1: $i] :
~ ( in @ X1 @ X0 )
=> ( emptyset = X0 ) ) ) ),
inference(rectify,[status(thm)],[f26]) ).
thf(f26,axiom,
( noeltsimpempty
= ( ! [X3: $i] :
( ! [X1: $i] :
~ ( in @ X1 @ X3 )
=> ( emptyset = X3 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.8shZsOoivs/DTF2THF_10582.p',noeltsimpempty) ).
thf(f126,plain,
noeltsimpempty = $true,
inference(cnf_transformation,[status(thm)],[f90]) ).
thf(f119,plain,
! [X1: $i] :
( ( in @ X1 @ sK0 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f90]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.08 % Problem : SEU509^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.09 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.11/0.29 % Computer : n018.cluster.edu
% 0.11/0.29 % Model : x86_64 x86_64
% 0.11/0.29 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.29 % Memory : 8042.1875MB
% 0.11/0.29 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.29 % CPULimit : 300
% 0.11/0.29 % WCLimit : 300
% 0.11/0.29 % DateTime : Tue Feb 24 12:25:00 EST 2026
% 0.11/0.29 % CPUTime :
% 0.11/0.29 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.18/0.38 ---- Original DTF file ---
% 0.18/0.38 thf(spec,logic,$$dhol).
% 0.18/0.38 %------------------------------------------------------------------------------
% 0.18/0.38 % File : SEU509^1 : TPTP v9.2.1. Released v3.7.0.
% 0.18/0.38 % Domain : Set Theory
% 0.18/0.38 % Problem : Preliminary Notions - Basic Laws of Logic
% 0.18/0.38 % Version : Especial.
% 0.18/0.38 % English : (! A:i.nonempty A -> (? x:i.in x A))
% 0.18/0.38
% 0.18/0.38 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.18/0.38 % Source : [Bro08]
% 0.18/0.38 % Names : ZFC011g [Bro08]
% 0.18/0.38
% 0.18/0.38 % Status : Theorem
% 0.18/0.38 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.23 v8.1.0, 0.09 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.43 v5.5.0, 0.33 v5.4.0, 0.40 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.67 v3.7.0
% 0.18/0.38 % Syntax : Number of formulae : 65 ( 28 unt; 36 typ; 28 def)
% 0.18/0.38 % Number of atoms : 165 ( 37 equ; 0 cnn)
% 0.18/0.38 % Maximal formula atoms : 27 ( 5 avg)
% 0.18/0.38 % Number of connectives : 288 ( 7 ~; 3 |; 18 &; 187 @)
% 0.18/0.38 % ( 7 <=>; 66 =>; 0 <=; 0 <~>)
% 0.18/0.38 % Maximal formula depth : 31 ( 2 avg)
% 0.18/0.38 % Number of types : 2 ( 0 usr)
% 0.18/0.38 % Number of type conns : 25 ( 25 >; 0 *; 0 +; 0 <<)
% 0.18/0.38 % Number of symbols : 38 ( 36 usr; 28 con; 0-2 aty)
% 0.18/0.38 % Number of variables : 95 ( 13 ^; 67 !; 15 ?; 95 :)
% 0.18/0.38 % SPC : TH0_THM_EQU_NAR
% 0.18/0.38
% 0.18/0.38 % Comments : http://mathgate.info/detsetitem.php?id=428
% 0.18/0.38 %------------------------------------------------------------------------------
% 0.18/0.38 thf(in_type,type,
% 0.18/0.38 in: $i > $i > $o ).
% 0.18/0.38
% 0.18/0.38 thf(exu_type,type,
% 0.18/0.38 exu: ( $i > $o ) > $o ).
% 0.18/0.38
% 0.18/0.38 thf(exu,definition,
% 0.18/0.38 ( exu
% 0.18/0.38 = ( ^ [Xphi: $i > $o] :
% 0.18/0.38 ? [Xx: $i] :
% 0.18/0.38 ( ( Xphi @ Xx )
% 0.18/0.38 & ! [Xy: $i] :
% 0.18/0.38 ( ( Xphi @ Xy )
% 0.18/0.38 => ( Xx = Xy ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(setextAx_type,type,
% 0.18/0.38 setextAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(setextAx,definition,
% 0.18/0.38 ( setextAx
% 0.18/0.38 = ( ! [A: $i,B: $i] :
% 0.18/0.38 ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 <=> ( in @ Xx @ B ) )
% 0.18/0.38 => ( A = B ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(emptyset_type,type,
% 0.18/0.38 emptyset: $i ).
% 0.18/0.38
% 0.18/0.38 thf(emptysetAx_type,type,
% 0.18/0.38 emptysetAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(emptysetAx,definition,
% 0.18/0.38 ( emptysetAx
% 0.18/0.38 = ( ! [Xx: $i] :
% 0.18/0.38 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(setadjoin_type,type,
% 0.18/0.38 setadjoin: $i > $i > $i ).
% 0.18/0.38
% 0.18/0.38 thf(setadjoinAx_type,type,
% 0.18/0.38 setadjoinAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(setadjoinAx,definition,
% 0.18/0.38 ( setadjoinAx
% 0.18/0.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.18/0.38 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.18/0.38 <=> ( ( Xy = Xx )
% 0.18/0.38 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(powerset_type,type,
% 0.18/0.38 powerset: $i > $i ).
% 0.18/0.38
% 0.18/0.38 thf(powersetAx_type,type,
% 0.18/0.38 powersetAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(powersetAx,definition,
% 0.18/0.38 ( powersetAx
% 0.18/0.38 = ( ! [A: $i,B: $i] :
% 0.18/0.38 ( ( in @ B @ ( powerset @ A ) )
% 0.18/0.38 <=> ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ B )
% 0.18/0.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(setunion_type,type,
% 0.18/0.38 setunion: $i > $i ).
% 0.18/0.38
% 0.18/0.38 thf(setunionAx_type,type,
% 0.18/0.38 setunionAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(setunionAx,definition,
% 0.18/0.38 ( setunionAx
% 0.18/0.38 = ( ! [A: $i,Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ ( setunion @ A ) )
% 0.18/0.38 <=> ? [B: $i] :
% 0.18/0.38 ( ( in @ Xx @ B )
% 0.18/0.38 & ( in @ B @ A ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(omega_type,type,
% 0.18/0.38 omega: $i ).
% 0.18/0.38
% 0.18/0.38 thf(omega0Ax_type,type,
% 0.18/0.38 omega0Ax: $o ).
% 0.18/0.38
% 0.18/0.38 thf(omega0Ax,definition,
% 0.18/0.38 ( omega0Ax
% 0.18/0.38 = ( in @ emptyset @ omega ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(omegaSAx_type,type,
% 0.18/0.38 omegaSAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(omegaSAx,definition,
% 0.18/0.38 ( omegaSAx
% 0.18/0.38 = ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ omega )
% 0.18/0.38 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(omegaIndAx_type,type,
% 0.18/0.38 omegaIndAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(omegaIndAx,definition,
% 0.18/0.38 ( omegaIndAx
% 0.18/0.38 = ( ! [A: $i] :
% 0.18/0.38 ( ( ( in @ emptyset @ A )
% 0.18/0.38 & ! [Xx: $i] :
% 0.18/0.38 ( ( ( in @ Xx @ omega )
% 0.18/0.38 & ( in @ Xx @ A ) )
% 0.18/0.38 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.18/0.38 => ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ omega )
% 0.18/0.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(replAx_type,type,
% 0.18/0.38 replAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(replAx,definition,
% 0.18/0.38 ( replAx
% 0.18/0.38 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.18/0.38 ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 => ( exu
% 0.18/0.38 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.18/0.38 => ? [B: $i] :
% 0.18/0.38 ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ B )
% 0.18/0.38 <=> ? [Xy: $i] :
% 0.18/0.38 ( ( in @ Xy @ A )
% 0.18/0.38 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(foundationAx_type,type,
% 0.18/0.38 foundationAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(foundationAx,definition,
% 0.18/0.38 ( foundationAx
% 0.18/0.38 = ( ! [A: $i] :
% 0.18/0.38 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.18/0.38 => ? [B: $i] :
% 0.18/0.38 ( ( in @ B @ A )
% 0.18/0.38 & ~ ? [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ B )
% 0.18/0.38 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(wellorderingAx_type,type,
% 0.18/0.38 wellorderingAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(wellorderingAx,definition,
% 0.18/0.38 ( wellorderingAx
% 0.18/0.38 = ( ! [A: $i] :
% 0.18/0.38 ? [B: $i] :
% 0.18/0.38 ( ! [C: $i] :
% 0.18/0.38 ( ( in @ C @ B )
% 0.18/0.38 => ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ C )
% 0.18/0.38 => ( in @ Xx @ A ) ) )
% 0.18/0.38 & ! [Xx: $i,Xy: $i] :
% 0.18/0.38 ( ( ( in @ Xx @ A )
% 0.18/0.38 & ( in @ Xy @ A ) )
% 0.18/0.38 => ( ! [C: $i] :
% 0.18/0.38 ( ( in @ C @ B )
% 0.18/0.38 => ( ( in @ Xx @ C )
% 0.18/0.38 <=> ( in @ Xy @ C ) ) )
% 0.18/0.38 => ( Xx = Xy ) ) )
% 0.18/0.38 & ! [C: $i,D: $i] :
% 0.18/0.38 ( ( ( in @ C @ B )
% 0.18/0.38 & ( in @ D @ B ) )
% 0.18/0.38 => ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ C )
% 0.18/0.38 => ( in @ Xx @ D ) )
% 0.18/0.38 | ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ D )
% 0.18/0.38 => ( in @ Xx @ C ) ) ) )
% 0.18/0.38 & ! [C: $i] :
% 0.18/0.38 ( ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ C )
% 0.18/0.38 => ( in @ Xx @ A ) )
% 0.18/0.38 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.18/0.38 => ? [D: $i,Xx: $i] :
% 0.18/0.38 ( ( in @ D @ B )
% 0.18/0.38 & ( in @ Xx @ C )
% 0.18/0.38 & ~ ? [Xy: $i] :
% 0.18/0.38 ( ( in @ Xy @ D )
% 0.18/0.38 & ( in @ Xy @ C ) )
% 0.18/0.38 & ! [E: $i] :
% 0.18/0.38 ( ( in @ E @ B )
% 0.18/0.38 => ( ! [Xy: $i] :
% 0.18/0.38 ( ( in @ Xy @ E )
% 0.18/0.38 => ( in @ Xy @ D ) )
% 0.18/0.38 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(descr_type,type,
% 0.18/0.38 descr: ( $i > $o ) > $i ).
% 0.18/0.38
% 0.18/0.38 thf(descrp_type,type,
% 0.18/0.38 descrp: $o ).
% 0.18/0.38
% 0.18/0.38 thf(descrp,definition,
% 0.18/0.38 ( descrp
% 0.18/0.38 = ( ! [Xphi: $i > $o] :
% 0.18/0.38 ( ( exu
% 0.18/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.38 => ( Xphi
% 0.18/0.38 @ ( descr
% 0.18/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(dsetconstr_type,type,
% 0.18/0.38 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.18/0.38
% 0.18/0.38 thf(dsetconstrI_type,type,
% 0.18/0.38 dsetconstrI: $o ).
% 0.18/0.38
% 0.18/0.38 thf(dsetconstrI,definition,
% 0.18/0.38 ( dsetconstrI
% 0.18/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 => ( ( Xphi @ Xx )
% 0.18/0.38 => ( in @ Xx
% 0.18/0.38 @ ( dsetconstr @ A
% 0.18/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(dsetconstrEL_type,type,
% 0.18/0.38 dsetconstrEL: $o ).
% 0.18/0.38
% 0.18/0.38 thf(dsetconstrEL,definition,
% 0.18/0.38 ( dsetconstrEL
% 0.18/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.38 ( ( in @ Xx
% 0.18/0.38 @ ( dsetconstr @ A
% 0.18/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.18/0.38 => ( in @ Xx @ A ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(dsetconstrER_type,type,
% 0.18/0.38 dsetconstrER: $o ).
% 0.18/0.38
% 0.18/0.38 thf(dsetconstrER,definition,
% 0.18/0.38 ( dsetconstrER
% 0.18/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.38 ( ( in @ Xx
% 0.18/0.38 @ ( dsetconstr @ A
% 0.18/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.18/0.38 => ( Xphi @ Xx ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(exuE1_type,type,
% 0.18/0.38 exuE1: $o ).
% 0.18/0.38
% 0.18/0.38 thf(exuE1,definition,
% 0.18/0.38 ( exuE1
% 0.18/0.38 = ( ! [Xphi: $i > $o] :
% 0.18/0.38 ( ( exu
% 0.18/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.38 => ? [Xx: $i] :
% 0.18/0.38 ( ( Xphi @ Xx )
% 0.18/0.38 & ! [Xy: $i] :
% 0.18/0.38 ( ( Xphi @ Xy )
% 0.18/0.38 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(prop2set_type,type,
% 0.18/0.38 prop2set: $o > $i ).
% 0.18/0.38
% 0.18/0.38 thf(prop2set,definition,
% 0.18/0.38 ( prop2set
% 0.18/0.38 = ( ^ [Xphi: $o] :
% 0.18/0.38 ( dsetconstr @ ( powerset @ emptyset )
% 0.18/0.38 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(prop2setE_type,type,
% 0.18/0.38 prop2setE: $o ).
% 0.18/0.38
% 0.18/0.38 thf(prop2setE,definition,
% 0.18/0.38 ( prop2setE
% 0.18/0.38 = ( ! [Xphi: $o,Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.18/0.38 => Xphi ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(emptysetE_type,type,
% 0.18/0.38 emptysetE: $o ).
% 0.18/0.38
% 0.18/0.38 thf(emptysetE,definition,
% 0.18/0.38 ( emptysetE
% 0.18/0.38 = ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ emptyset )
% 0.18/0.38 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(emptysetimpfalse_type,type,
% 0.18/0.38 emptysetimpfalse: $o ).
% 0.18/0.38
% 0.18/0.38 thf(emptysetimpfalse,definition,
% 0.18/0.38 ( emptysetimpfalse
% 0.18/0.38 = ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ emptyset )
% 0.18/0.38 => $false ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(notinemptyset_type,type,
% 0.18/0.38 notinemptyset: $o ).
% 0.18/0.38
% 0.18/0.38 thf(notinemptyset,definition,
% 0.18/0.38 ( notinemptyset
% 0.18/0.38 = ( ! [Xx: $i] :
% 0.18/0.38 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(exuE3e_type,type,
% 0.18/0.38 exuE3e: $o ).
% 0.18/0.38
% 0.18/0.38 thf(exuE3e,definition,
% 0.18/0.38 ( exuE3e
% 0.18/0.38 = ( ! [Xphi: $i > $o] :
% 0.18/0.38 ( ( exu
% 0.18/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.38 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(setext_type,type,
% 0.18/0.38 setext: $o ).
% 0.18/0.38
% 0.18/0.38 thf(setext,definition,
% 0.18/0.38 ( setext
% 0.18/0.38 = ( ! [A: $i,B: $i] :
% 0.18/0.38 ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 => ( in @ Xx @ B ) )
% 0.18/0.38 => ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ B )
% 0.18/0.38 => ( in @ Xx @ A ) )
% 0.18/0.38 => ( A = B ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(emptyI_type,type,
% 0.18/0.38 emptyI: $o ).
% 0.18/0.38
% 0.18/0.38 thf(emptyI,definition,
% 0.18/0.38 ( emptyI
% 0.18/0.38 = ( ! [A: $i] :
% 0.18/0.38 ( ! [Xx: $i] :
% 0.18/0.38 ~ ( in @ Xx @ A )
% 0.18/0.38 => ( A = emptyset ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(noeltsimpempty_type,type,
% 0.18/0.38 noeltsimpempty: $o ).
% 0.18/0.38
% 0.18/0.38 thf(noeltsimpempty,definition,
% 0.18/0.38 ( noeltsimpempty
% 0.18/0.38 = ( ! [A: $i] :
% 0.18/0.38 ( ! [Xx: $i] :
% 0.18/0.38 ~ ( in @ Xx @ A )
% 0.18/0.38 => ( A = emptyset ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(setbeta_type,type,
% 0.18/0.38 setbeta: $o ).
% 0.18/0.38
% 0.18/0.38 thf(setbeta,definition,
% 0.18/0.38 ( setbeta
% 0.18/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 => ( ( in @ Xx
% 0.18/0.38 @ ( dsetconstr @ A
% 0.18/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.18/0.38 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(nonempty_type,type,
% 0.18/0.38 nonempty: $i > $o ).
% 0.18/0.38
% 0.18/0.38 thf(nonempty,definition,
% 0.18/0.38 ( nonempty
% 0.18/0.38 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(nonemptyE1,conjecture,
% 0.18/0.38 ( setextAx
% 0.18/0.38 => ( emptysetAx
% 0.18/0.38 => ( setadjoinAx
% 0.18/0.38 => ( powersetAx
% 0.18/0.38 => ( setunionAx
% 0.18/0.38 => ( omega0Ax
% 0.18/0.38 => ( omegaSAx
% 0.18/0.38 => ( omegaIndAx
% 0.18/0.38 => ( replAx
% 0.18/0.38 => ( foundationAx
% 0.18/0.38 => ( wellorderingAx
% 0.18/0.38 => ( descrp
% 0.18/0.38 => ( dsetconstrI
% 0.18/0.38 => ( dsetconstrEL
% 0.18/0.38 => ( dsetconstrER
% 0.18/0.38 => ( exuE1
% 0.18/0.38 => ( prop2setE
% 0.18/0.38 => ( emptysetE
% 0.18/0.38 => ( emptysetimpfalse
% 0.18/0.38 => ( notinemptyset
% 0.18/0.38 => ( exuE3e
% 0.18/0.38 => ( setext
% 0.18/0.38 => ( emptyI
% 0.18/0.38 => ( noeltsimpempty
% 0.18/0.38 => ( setbeta
% 0.18/0.38 => ! [A: $i] :
% 0.18/0.38 ( ( nonempty @ A )
% 0.18/0.38 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 %------------------------------------------------------------------------------
% 0.18/0.38 ------------------------
% 1.72/1.35 ---- Embedded in THF ---
% 1.72/1.35 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.72/1.35 %%% Generated on Tue Feb 24 12:25:01 EST 2026
% 1.72/1.35 %%% using '$$dhol' embedding, version 1.3.0.
% 1.72/1.35 %%% Logic specification used:
% 1.72/1.35 %%% thf(spec, logic, $$dhol).
% 1.72/1.35
% 1.72/1.35 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.8shZsOoivs/DTF2DTF10582.p
% See solution above
% 1.72/1.35 ------------------------
% 1.72/1.36 ---- Cleaned THF ---
% 1.72/1.37 thf(in_type,type,
% 1.72/1.37 in: $i > $i > $o ).
% 1.72/1.37
% 1.72/1.37 thf(exu_type,type,
% 1.72/1.37 exu: ( $i > $o ) > $o ).
% 1.72/1.37
% 1.72/1.37 thf(exu,definition,
% 1.72/1.37 ( exu
% 1.72/1.37 = ( ^ [Xphi: $i > $o] :
% 1.72/1.37 ? [Xx: $i] :
% 1.72/1.37 ( ( Xphi @ Xx )
% 1.72/1.37 & ! [Xy: $i] :
% 1.72/1.37 ( ( Xphi @ Xy )
% 1.72/1.37 => ( Xx = Xy ) ) ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(setextAx_type,type,
% 1.72/1.37 setextAx: $o ).
% 1.72/1.37
% 1.72/1.37 thf(setextAx,definition,
% 1.72/1.37 ( setextAx
% 1.72/1.37 = ( ! [A: $i,B: $i] :
% 1.72/1.37 ( ! [Xx: $i] :
% 1.72/1.37 ( ( in @ Xx @ A )
% 1.72/1.37 <=> ( in @ Xx @ B ) )
% 1.72/1.37 => ( A = B ) ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(emptyset_type,type,
% 1.72/1.37 emptyset: $i ).
% 1.72/1.37
% 1.72/1.37 thf(emptysetAx_type,type,
% 1.72/1.37 emptysetAx: $o ).
% 1.72/1.37
% 1.72/1.37 thf(emptysetAx,definition,
% 1.72/1.37 ( emptysetAx
% 1.72/1.37 = ( ! [Xx: $i] :
% 1.72/1.37 ~ ( in @ Xx @ emptyset ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(setadjoin_type,type,
% 1.72/1.37 setadjoin: $i > $i > $i ).
% 1.72/1.37
% 1.72/1.37 thf(setadjoinAx_type,type,
% 1.72/1.37 setadjoinAx: $o ).
% 1.72/1.37
% 1.72/1.37 thf(setadjoinAx,definition,
% 1.72/1.37 ( setadjoinAx
% 1.72/1.37 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.72/1.37 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.72/1.37 <=> ( ( Xy = Xx )
% 1.72/1.37 | ( in @ Xy @ A ) ) ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(powerset_type,type,
% 1.72/1.37 powerset: $i > $i ).
% 1.72/1.37
% 1.72/1.37 thf(powersetAx_type,type,
% 1.72/1.37 powersetAx: $o ).
% 1.72/1.37
% 1.72/1.37 thf(powersetAx,definition,
% 1.72/1.37 ( powersetAx
% 1.72/1.37 = ( ! [A: $i,B: $i] :
% 1.72/1.37 ( ( in @ B @ ( powerset @ A ) )
% 1.72/1.37 <=> ! [Xx: $i] :
% 1.72/1.37 ( ( in @ Xx @ B )
% 1.72/1.37 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(setunion_type,type,
% 1.72/1.37 setunion: $i > $i ).
% 1.72/1.37
% 1.72/1.37 thf(setunionAx_type,type,
% 1.72/1.37 setunionAx: $o ).
% 1.72/1.37
% 1.72/1.37 thf(setunionAx,definition,
% 1.72/1.37 ( setunionAx
% 1.72/1.37 = ( ! [A: $i,Xx: $i] :
% 1.72/1.37 ( ( in @ Xx @ ( setunion @ A ) )
% 1.72/1.37 <=> ? [B: $i] :
% 1.72/1.37 ( ( in @ Xx @ B )
% 1.72/1.37 & ( in @ B @ A ) ) ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(omega_type,type,
% 1.72/1.37 omega: $i ).
% 1.72/1.37
% 1.72/1.37 thf(omega0Ax_type,type,
% 1.72/1.37 omega0Ax: $o ).
% 1.72/1.37
% 1.72/1.37 thf(omega0Ax,definition,
% 1.72/1.37 ( omega0Ax
% 1.72/1.37 = ( in @ emptyset @ omega ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(omegaSAx_type,type,
% 1.72/1.37 omegaSAx: $o ).
% 1.72/1.37
% 1.72/1.37 thf(omegaSAx,definition,
% 1.72/1.37 ( omegaSAx
% 1.72/1.37 = ( ! [Xx: $i] :
% 1.72/1.37 ( ( in @ Xx @ omega )
% 1.72/1.37 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(omegaIndAx_type,type,
% 1.72/1.37 omegaIndAx: $o ).
% 1.72/1.37
% 1.72/1.37 thf(omegaIndAx,definition,
% 1.72/1.37 ( omegaIndAx
% 1.72/1.37 = ( ! [A: $i] :
% 1.72/1.37 ( ( ( in @ emptyset @ A )
% 1.72/1.37 & ! [Xx: $i] :
% 1.72/1.37 ( ( ( in @ Xx @ omega )
% 1.72/1.37 & ( in @ Xx @ A ) )
% 1.72/1.37 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 1.72/1.37 => ! [Xx: $i] :
% 1.72/1.37 ( ( in @ Xx @ omega )
% 1.72/1.37 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(replAx_type,type,
% 1.72/1.37 replAx: $o ).
% 1.72/1.37
% 1.72/1.37 thf(replAx,definition,
% 1.72/1.37 ( replAx
% 1.72/1.37 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 1.72/1.37 ( ! [Xx: $i] :
% 1.72/1.37 ( ( in @ Xx @ A )
% 1.72/1.37 => ( exu
% 1.72/1.37 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 1.72/1.37 => ? [B: $i] :
% 1.72/1.37 ! [Xx: $i] :
% 1.72/1.37 ( ( in @ Xx @ B )
% 1.72/1.37 <=> ? [Xy: $i] :
% 1.72/1.37 ( ( in @ Xy @ A )
% 1.72/1.37 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(foundationAx_type,type,
% 1.72/1.37 foundationAx: $o ).
% 1.72/1.37
% 1.72/1.37 thf(foundationAx,definition,
% 1.72/1.37 ( foundationAx
% 1.72/1.37 = ( ! [A: $i] :
% 1.72/1.37 ( ? [Xx: $i] : ( in @ Xx @ A )
% 1.72/1.37 => ? [B: $i] :
% 1.72/1.37 ( ( in @ B @ A )
% 1.72/1.37 & ~ ? [Xx: $i] :
% 1.72/1.37 ( ( in @ Xx @ B )
% 1.72/1.37 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(wellorderingAx_type,type,
% 1.72/1.37 wellorderingAx: $o ).
% 1.72/1.37
% 1.72/1.37 thf(wellorderingAx,definition,
% 1.72/1.37 ( wellorderingAx
% 1.72/1.37 = ( ! [A: $i] :
% 1.72/1.37 ? [B: $i] :
% 1.72/1.37 ( ! [C: $i] :
% 1.72/1.37 ( ( in @ C @ B )
% 1.72/1.37 => ! [Xx: $i] :
% 1.72/1.37 ( ( in @ Xx @ C )
% 1.72/1.37 => ( in @ Xx @ A ) ) )
% 1.72/1.37 & ! [Xx: $i,Xy: $i] :
% 1.72/1.37 ( ( ( in @ Xx @ A )
% 1.72/1.37 & ( in @ Xy @ A ) )
% 1.72/1.37 => ( ! [C: $i] :
% 1.72/1.37 ( ( in @ C @ B )
% 1.72/1.37 => ( ( in @ Xx @ C )
% 1.72/1.37 <=> ( in @ Xy @ C ) ) )
% 1.72/1.37 => ( Xx = Xy ) ) )
% 1.72/1.37 & ! [C: $i,D: $i] :
% 1.72/1.37 ( ( ( in @ C @ B )
% 1.72/1.37 & ( in @ D @ B ) )
% 1.72/1.37 => ( ! [Xx: $i] :
% 1.72/1.37 ( ( in @ Xx @ C )
% 1.72/1.37 => ( in @ Xx @ D ) )
% 1.72/1.37 | ! [Xx: $i] :
% 1.72/1.37 ( ( in @ Xx @ D )
% 1.72/1.37 => ( in @ Xx @ C ) ) ) )
% 1.72/1.37 & ! [C: $i] :
% 1.72/1.37 ( ( ! [Xx: $i] :
% 1.72/1.37 ( ( in @ Xx @ C )
% 1.72/1.37 => ( in @ Xx @ A ) )
% 1.72/1.37 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 1.72/1.37 => ? [D: $i,Xx: $i] :
% 1.72/1.37 ( ( in @ D @ B )
% 1.72/1.37 & ( in @ Xx @ C )
% 1.72/1.37 & ~ ? [Xy: $i] :
% 1.72/1.37 ( ( in @ Xy @ D )
% 1.72/1.37 & ( in @ Xy @ C ) )
% 1.72/1.37 & ! [E: $i] :
% 1.72/1.37 ( ( in @ E @ B )
% 1.72/1.37 => ( ! [Xy: $i] :
% 1.72/1.37 ( ( in @ Xy @ E )
% 1.72/1.37 => ( in @ Xy @ D ) )
% 1.72/1.37 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(descr_type,type,
% 1.72/1.37 descr: ( $i > $o ) > $i ).
% 1.72/1.37
% 1.72/1.37 thf(descrp_type,type,
% 1.72/1.37 descrp: $o ).
% 1.72/1.37
% 1.72/1.37 thf(descrp,definition,
% 1.72/1.37 ( descrp
% 1.72/1.37 = ( ! [Xphi: $i > $o] :
% 1.72/1.37 ( ( exu
% 1.72/1.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.72/1.37 => ( Xphi
% 1.72/1.37 @ ( descr
% 1.72/1.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(dsetconstr_type,type,
% 1.72/1.37 dsetconstr: $i > ( $i > $o ) > $i ).
% 1.72/1.37
% 1.72/1.37 thf(dsetconstrI_type,type,
% 1.72/1.37 dsetconstrI: $o ).
% 1.72/1.37
% 1.72/1.37 thf(dsetconstrI,definition,
% 1.72/1.37 ( dsetconstrI
% 1.72/1.37 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.72/1.37 ( ( in @ Xx @ A )
% 1.72/1.37 => ( ( Xphi @ Xx )
% 1.72/1.37 => ( in @ Xx
% 1.72/1.37 @ ( dsetconstr @ A
% 1.72/1.37 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(dsetconstrEL_type,type,
% 1.72/1.37 dsetconstrEL: $o ).
% 1.72/1.37
% 1.72/1.37 thf(dsetconstrEL,definition,
% 1.72/1.37 ( dsetconstrEL
% 1.72/1.37 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.72/1.37 ( ( in @ Xx
% 1.72/1.37 @ ( dsetconstr @ A
% 1.72/1.37 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.72/1.37 => ( in @ Xx @ A ) ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(dsetconstrER_type,type,
% 1.72/1.37 dsetconstrER: $o ).
% 1.72/1.37
% 1.72/1.37 thf(dsetconstrER,definition,
% 1.72/1.37 ( dsetconstrER
% 1.72/1.37 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.72/1.37 ( ( in @ Xx
% 1.72/1.37 @ ( dsetconstr @ A
% 1.72/1.37 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.72/1.37 => ( Xphi @ Xx ) ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(exuE1_type,type,
% 1.72/1.37 exuE1: $o ).
% 1.72/1.37
% 1.72/1.37 thf(exuE1,definition,
% 1.72/1.37 ( exuE1
% 1.72/1.37 = ( ! [Xphi: $i > $o] :
% 1.72/1.37 ( ( exu
% 1.72/1.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.72/1.37 => ? [Xx: $i] :
% 1.72/1.37 ( ( Xphi @ Xx )
% 1.72/1.37 & ! [Xy: $i] :
% 1.72/1.37 ( ( Xphi @ Xy )
% 1.72/1.37 => ( Xx = Xy ) ) ) ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(prop2set_type,type,
% 1.72/1.37 prop2set: $o > $i ).
% 1.72/1.37
% 1.72/1.37 thf(prop2set,definition,
% 1.72/1.37 ( prop2set
% 1.72/1.37 = ( ^ [Xphi: $o] :
% 1.72/1.37 ( dsetconstr @ ( powerset @ emptyset )
% 1.72/1.37 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(prop2setE_type,type,
% 1.72/1.37 prop2setE: $o ).
% 1.72/1.37
% 1.72/1.37 thf(prop2setE,definition,
% 1.72/1.37 ( prop2setE
% 1.72/1.37 = ( ! [Xphi: $o,Xx: $i] :
% 1.72/1.37 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 1.72/1.37 => Xphi ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(emptysetE_type,type,
% 1.72/1.37 emptysetE: $o ).
% 1.72/1.37
% 1.72/1.37 thf(emptysetE,definition,
% 1.72/1.37 ( emptysetE
% 1.72/1.37 = ( ! [Xx: $i] :
% 1.72/1.37 ( ( in @ Xx @ emptyset )
% 1.72/1.37 => ! [Xphi: $o] : Xphi ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(emptysetimpfalse_type,type,
% 1.72/1.37 emptysetimpfalse: $o ).
% 1.72/1.37
% 1.72/1.37 thf(emptysetimpfalse,definition,
% 1.72/1.37 ( emptysetimpfalse
% 1.72/1.37 = ( ! [Xx: $i] :
% 1.72/1.37 ( ( in @ Xx @ emptyset )
% 1.72/1.37 => $false ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(notinemptyset_type,type,
% 1.72/1.37 notinemptyset: $o ).
% 1.72/1.37
% 1.72/1.37 thf(notinemptyset,definition,
% 1.72/1.37 ( notinemptyset
% 1.72/1.37 = ( ! [Xx: $i] :
% 1.72/1.37 ~ ( in @ Xx @ emptyset ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(exuE3e_type,type,
% 1.72/1.37 exuE3e: $o ).
% 1.72/1.37
% 1.72/1.37 thf(exuE3e,definition,
% 1.72/1.37 ( exuE3e
% 1.72/1.37 = ( ! [Xphi: $i > $o] :
% 1.72/1.37 ( ( exu
% 1.72/1.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.72/1.37 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(setext_type,type,
% 1.72/1.37 setext: $o ).
% 1.72/1.37
% 1.72/1.37 thf(setext,definition,
% 1.72/1.37 ( setext
% 1.72/1.37 = ( ! [A: $i,B: $i] :
% 1.72/1.37 ( ! [Xx: $i] :
% 1.72/1.37 ( ( in @ Xx @ A )
% 1.72/1.37 => ( in @ Xx @ B ) )
% 1.72/1.37 => ( ! [Xx: $i] :
% 1.72/1.37 ( ( in @ Xx @ B )
% 1.72/1.37 => ( in @ Xx @ A ) )
% 1.72/1.37 => ( A = B ) ) ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(emptyI_type,type,
% 1.72/1.37 emptyI: $o ).
% 1.72/1.37
% 1.72/1.37 thf(emptyI,definition,
% 1.72/1.37 ( emptyI
% 1.72/1.37 = ( ! [A: $i] :
% 1.72/1.37 ( ! [Xx: $i] :
% 1.72/1.37 ~ ( in @ Xx @ A )
% 1.72/1.37 => ( A = emptyset ) ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(noeltsimpempty_type,type,
% 1.72/1.37 noeltsimpempty: $o ).
% 1.72/1.37
% 1.72/1.37 thf(noeltsimpempty,definition,
% 1.72/1.37 ( noeltsimpempty
% 1.72/1.37 = ( ! [A: $i] :
% 1.72/1.37 ( ! [Xx: $i] :
% 1.72/1.37 ~ ( in @ Xx @ A )
% 1.72/1.37 => ( A = emptyset ) ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(setbeta_type,type,
% 1.72/1.37 setbeta: $o ).
% 1.72/1.37
% 1.72/1.37 thf(setbeta,definition,
% 1.72/1.37 ( setbeta
% 1.72/1.37 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.72/1.37 ( ( in @ Xx @ A )
% 1.72/1.37 => ( ( in @ Xx
% 1.72/1.37 @ ( dsetconstr @ A
% 1.72/1.37 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.72/1.37 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(nonempty_type,type,
% 1.72/1.37 nonempty: $i > $o ).
% 1.72/1.37
% 1.72/1.37 thf(nonempty,definition,
% 1.72/1.37 ( nonempty
% 1.72/1.37 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 1.72/1.37
% 1.72/1.37 thf(nonemptyE1,conjecture,
% 1.72/1.37 ( setextAx
% 1.72/1.37 => ( emptysetAx
% 1.72/1.37 => ( setadjoinAx
% 1.72/1.37 => ( powersetAx
% 1.72/1.37 => ( setunionAx
% 1.72/1.37 => ( omega0Ax
% 1.72/1.37 => ( omegaSAx
% 1.72/1.37 => ( omegaIndAx
% 1.72/1.37 => ( replAx
% 1.72/1.37 => ( foundationAx
% 1.72/1.37 => ( wellorderingAx
% 1.72/1.37 => ( descrp
% 1.72/1.37 => ( dsetconstrI
% 1.72/1.37 => ( dsetconstrEL
% 1.72/1.37 => ( dsetconstrER
% 1.72/1.37 => ( exuE1
% 1.72/1.37 => ( prop2setE
% 1.72/1.37 => ( emptysetE
% 1.72/1.37 => ( emptysetimpfalse
% 1.72/1.37 => ( notinemptyset
% 1.72/1.37 => ( exuE3e
% 1.72/1.37 => ( setext
% 1.72/1.37 => ( emptyI
% 1.72/1.37 => ( noeltsimpempty
% 1.72/1.37 => ( setbeta
% 1.72/1.37 => ! [A: $i] :
% 1.72/1.37 ( ( nonempty @ A )
% 1.72/1.37 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 1.72/1.37 ------------------------
% 1.72/1.39 % (10687)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_10582 for (2999ds/275Mi)
% 1.72/1.39 % (10686)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_10582 for (2999ds/2Mi)
% 1.72/1.39 % (10684)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_10582 for (2999ds/27Mi)
% 1.72/1.39 % (10683)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_10582 for (2999ds/4Mi)
% 1.72/1.39 % (10685)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_10582 for (2999ds/2Mi)
% 1.72/1.39 % (10688)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_10582 for (2999ds/18Mi)
% 1.72/1.39 % (10682)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_10582 for (2999ds/183Mi)
% 1.72/1.39 % (10685)Instruction limit reached!
% 1.72/1.39 % (10685)------------------------------
% 1.72/1.39 % (10685)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.72/1.39 % (10685)Termination reason: Unknown
% 1.72/1.39 % (10685)Termination phase: shuffling
% 1.72/1.39 % (10686)Instruction limit reached!
% 1.72/1.39 % (10686)------------------------------
% 1.72/1.39 % (10686)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.72/1.39 % (10686)Termination reason: Unknown
% 1.72/1.39 % (10686)Termination phase: shuffling
% 1.72/1.39
% 1.72/1.39 % (10686)Memory used [KB]: 1023
% 1.72/1.39 % (10686)Time elapsed: 0.003 s
% 1.72/1.39 % (10686)Instructions burned: 3 (million)
% 1.72/1.39 % (10686)------------------------------
% 1.72/1.39 % (10686)------------------------------
% 1.72/1.39
% 1.72/1.39 % (10685)Memory used [KB]: 1023
% 1.72/1.39 % (10685)Time elapsed: 0.003 s
% 1.72/1.39 % (10685)Instructions burned: 3 (million)
% 1.72/1.39 % (10685)------------------------------
% 1.72/1.39 % (10685)------------------------------
% 1.72/1.39 % (10683)Instruction limit reached!
% 1.72/1.39 % (10683)------------------------------
% 1.72/1.39 % (10683)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.72/1.39 % (10683)Termination reason: Unknown
% 1.72/1.39 % (10683)Termination phase: shuffling
% 1.72/1.39
% 1.72/1.39 % (10683)Memory used [KB]: 1151
% 1.72/1.39 % (10683)Time elapsed: 0.004 s
% 1.72/1.39 % (10683)Instructions burned: 5 (million)
% 1.72/1.39 % (10683)------------------------------
% 1.72/1.39 % (10683)------------------------------
% 1.72/1.40 % (10688)Instruction limit reached!
% 1.72/1.40 % (10688)------------------------------
% 1.72/1.40 % (10688)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.72/1.40 % (10688)Termination reason: Unknown
% 1.72/1.40 % (10688)Termination phase: Function definition elimination
% 1.72/1.40
% 1.72/1.40 % (10688)Memory used [KB]: 1407
% 1.72/1.40 % (10688)Time elapsed: 0.010 s
% 1.72/1.40 % (10688)Instructions burned: 18 (million)
% 1.72/1.40 % (10688)------------------------------
% 1.72/1.40 % (10688)------------------------------
% 1.85/1.40 % (10684)Instruction limit reached!
% 1.85/1.40 % (10684)------------------------------
% 1.85/1.40 % (10684)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.85/1.40 % (10684)Termination reason: Unknown
% 1.85/1.40 % (10684)Termination phase: Saturation
% 1.85/1.40
% 1.85/1.40 % (10684)Memory used [KB]: 5756
% 1.85/1.40 % (10684)Time elapsed: 0.014 s
% 1.85/1.40 % (10684)Instructions burned: 27 (million)
% 1.85/1.40 % (10684)------------------------------
% 1.85/1.40 % (10684)------------------------------
% 1.85/1.40 % (10689)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_10582 for (2999ds/3Mi)
% 1.85/1.40 % (10690)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_10582 for (2999ds/37Mi)
% 1.85/1.40 % (10691)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_10582 for (2999ds/15Mi)
% 1.85/1.41 % (10689)Instruction limit reached!
% 1.85/1.41 % (10689)------------------------------
% 1.85/1.41 % (10689)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.85/1.41 % (10689)Termination reason: Unknown
% 1.85/1.41 % (10689)Termination phase: shuffling
% 1.85/1.41
% 1.85/1.41 % (10689)Memory used [KB]: 1023
% 1.85/1.41 % (10689)Time elapsed: 0.003 s
% 1.85/1.41 % (10689)Instructions burned: 4 (million)
% 1.85/1.41 % (10689)------------------------------
% 1.85/1.41 % (10689)------------------------------
% 1.85/1.41 % (10692)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_10582 for (2999ds/3Mi)
% 1.85/1.41 % (10682)First to succeed.
% 1.85/1.41 % (10691)Instruction limit reached!
% 1.85/1.41 % (10691)------------------------------
% 1.85/1.41 % (10691)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.85/1.41 % (10691)Termination reason: Unknown
% 1.85/1.41 % (10691)Termination phase: Property scanning
% 1.85/1.41
% 1.85/1.41 % (10691)Memory used [KB]: 1407
% 1.85/1.41 % (10691)Time elapsed: 0.009 s
% 1.85/1.41 % (10691)Instructions burned: 16 (million)
% 1.85/1.41 % (10691)------------------------------
% 1.85/1.41 % (10691)------------------------------
% 1.85/1.41 % (10692)Instruction limit reached!
% 1.85/1.41 % (10692)------------------------------
% 1.85/1.41 % (10692)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.85/1.41 % (10692)Termination reason: Unknown
% 1.85/1.41 % (10692)Termination phase: shuffling
% 1.85/1.41
% 1.85/1.41 % (10692)Memory used [KB]: 1023
% 1.85/1.41 % (10692)Time elapsed: 0.003 s
% 1.85/1.41 % (10692)Instructions burned: 4 (million)
% 1.85/1.41 % (10692)------------------------------
% 1.85/1.41 % (10692)------------------------------
% 1.85/1.41 % (10682)Refutation found. Thanks to Tanya!
% 1.85/1.41 % SZS status Theorem for DTF2THF_10582
% 1.85/1.41 % SZS output start Proof for DTF2THF_10582
% 1.85/1.41 thf(func_def_0, type, in: $i > $i > $o).
% 1.85/1.41 thf(func_def_1, type, exu: ($i > $o) > $o).
% 1.85/1.41 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 1.85/1.41 thf(func_def_8, type, powerset: $i > $i).
% 1.85/1.41 thf(func_def_10, type, setunion: $i > $i).
% 1.85/1.41 thf(func_def_19, type, descr: ($i > $o) > $i).
% 1.85/1.41 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 1.85/1.41 thf(func_def_26, type, prop2set: $o > $i).
% 1.85/1.41 thf(func_def_36, type, nonempty: $i > $o).
% 1.85/1.41 thf(func_def_55, type, sK2: $i > $i > $i).
% 1.85/1.41 thf(func_def_56, type, sK3: $i > $i).
% 1.85/1.41 thf(func_def_57, type, sK4: $i > $i > $i).
% 1.85/1.41 thf(func_def_58, type, sK5: $i > $i > $i).
% 1.85/1.41 thf(func_def_59, type, sK6: $i > $i > $i).
% 1.85/1.41 thf(func_def_60, type, sK7: $i > $i).
% 1.85/1.41 thf(func_def_61, type, sK8: ($i > $o) > $i).
% 1.85/1.41 thf(func_def_62, type, sK9: ($i > $o) > $i > $i).
% 1.85/1.41 thf(func_def_63, type, sK10: ($i > $o) > $i > $i).
% 1.85/1.41 thf(func_def_64, type, sK11: $i > $i > $i).
% 1.85/1.41 thf(func_def_65, type, sK12: $i > ($i > $i > $o) > $i).
% 1.85/1.41 thf(func_def_66, type, sK13: $i > ($i > $i > $o) > $i).
% 1.85/1.41 thf(func_def_67, type, sK14: $i > $i > ($i > $i > $o) > $i).
% 1.85/1.41 thf(func_def_68, type, sK15: $i > $i > ($i > $i > $o) > $i).
% 1.85/1.41 thf(func_def_69, type, sK16: $i > $i > ($i > $i > $o) > $i).
% 1.85/1.41 thf(func_def_70, type, sK17: $i > $i > ($i > $i > $o) > $i).
% 1.85/1.41 thf(func_def_71, type, sK18: $i > $i > ($i > $i > $o) > $i).
% 1.85/1.41 thf(func_def_72, type, sK19: $i > $i).
% 1.85/1.41 thf(func_def_73, type, sK20: $i > $i).
% 1.85/1.41 thf(func_def_74, type, sK21: ($i > $o) > $i).
% 1.85/1.41 thf(func_def_75, type, sK22: ($i > $o) > $i > $i).
% 1.85/1.41 thf(func_def_76, type, sK23: ($i > $o) > $i > $i).
% 1.85/1.41 thf(func_def_77, type, sK24: $i > $i).
% 1.85/1.41 thf(func_def_78, type, sK25: $i > $i > $i).
% 1.85/1.41 thf(func_def_79, type, sK26: $i > $i > $i).
% 1.85/1.41 thf(func_def_80, type, sK27: $i > $i > $i).
% 1.85/1.41 thf(func_def_81, type, sK28: $i > $i > $i > $i).
% 1.85/1.41 thf(func_def_82, type, sK29: $i > $i > $i).
% 1.85/1.41 thf(f582,plain,(
% 1.85/1.41 $false),
% 1.85/1.41 inference(subsumption_resolution,[status(thm)],[f581,f284])).
% 1.85/1.41 thf(f284,plain,(
% 1.85/1.41 (emptyset != sK0)),
% 1.85/1.41 inference(equality_proxy_clausification,[status(thm)],[f283])).
% 1.85/1.41 thf(f283,plain,(
% 1.85/1.41 ((emptyset = sK0) = $false)),
% 1.85/1.41 inference(not_proxy_clausification,[status(thm)],[f282])).
% 1.85/1.41 thf(f282,plain,(
% 1.85/1.41 ((~ (emptyset = sK0)) = $true)),
% 1.85/1.41 inference(beta_eta_normalization,[status(thm)],[f161])).
% 1.85/1.41 thf(f161,plain,(
% 1.85/1.41 (((^[Y0 : $i]: (~ (emptyset = Y0))) @ sK0) = $true)),
% 1.85/1.41 inference(definition_unfolding,[status(thm)],[f118,f139])).
% 1.85/1.41 thf(f139,plain,(
% 1.85/1.41 (nonempty = (^[Y0 : $i]: (~ (emptyset = Y0))))),
% 1.85/1.41 inference(cnf_transformation,[status(thm)],[f85])).
% 1.85/1.41 thf(f85,plain,(
% 1.85/1.41 (nonempty = (^[Y0 : $i]: (~ (emptyset = Y0))))),
% 1.85/1.41 inference(fool_elimination,[status(thm)],[f84])).
% 1.85/1.41 thf(f84,plain,(
% 1.85/1.41 ((^[X0 : $i] : ((emptyset != X0))) = nonempty)),
% 1.85/1.41 inference(rectify,[status(thm)],[f28])).
% 1.85/1.41 thf(f28,axiom,(
% 1.85/1.41 ((^[X1 : $i] : ((emptyset != X1))) = nonempty)),
% 1.85/1.41 file('/export/starexec/sandbox/tmp/tmp.8shZsOoivs/DTF2THF_10582.p',nonempty)).
% 1.85/1.41 thf(f118,plain,(
% 1.85/1.41 ((nonempty @ sK0) = $true)),
% 1.85/1.41 inference(cnf_transformation,[status(thm)],[f90])).
% 1.85/1.41 thf(f90,plain,(
% 1.85/1.41 (exuE1 = $true) & (wellorderingAx = $true) & (descrp = $true) & (omega0Ax = $true) & (emptysetimpfalse = $true) & (prop2setE = $true) & (powersetAx = $true) & (exuE3e = $true) & (emptysetE = $true) & (noeltsimpempty = $true) & (dsetconstrER = $true) & (replAx = $true) & (omegaIndAx = $true) & (setadjoinAx = $true) & (dsetconstrEL = $true) & (setextAx = $true) & (! [X1 : $i] : ((in @ X1 @ sK0) != $true) & ((nonempty @ sK0) = $true)) & (setbeta = $true) & (emptysetAx = $true) & (emptyI = $true) & (setunionAx = $true) & (setext = $true) & (notinemptyset = $true) & (omegaSAx = $true) & (foundationAx = $true) & (dsetconstrI = $true)),
% 1.85/1.41 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0])],[f88,f89])).
% 1.85/1.41 thf(f89,plain,(
% 1.85/1.41 ? [X0 : $i] : (! [X1 : $i] : ((in @ X1 @ X0) != $true) & ((nonempty @ X0) = $true)) => (! [X1 : $i] : ((in @ X1 @ sK0) != $true) & ((nonempty @ sK0) = $true))),
% 1.85/1.41 introduced(definition,[],[choice_axiom])).
% 1.85/1.41 thf(f88,plain,(
% 1.85/1.41 (exuE1 = $true) & (wellorderingAx = $true) & (descrp = $true) & (omega0Ax = $true) & (emptysetimpfalse = $true) & (prop2setE = $true) & (powersetAx = $true) & (exuE3e = $true) & (emptysetE = $true) & (noeltsimpempty = $true) & (dsetconstrER = $true) & (replAx = $true) & (omegaIndAx = $true) & (setadjoinAx = $true) & (dsetconstrEL = $true) & (setextAx = $true) & ? [X0 : $i] : (! [X1 : $i] : ((in @ X1 @ X0) != $true) & ((nonempty @ X0) = $true)) & (setbeta = $true) & (emptysetAx = $true) & (emptyI = $true) & (setunionAx = $true) & (setext = $true) & (notinemptyset = $true) & (omegaSAx = $true) & (foundationAx = $true) & (dsetconstrI = $true)),
% 1.85/1.41 inference(flattening,[status(thm)],[f87])).
% 1.85/1.41 thf(f87,plain,(
% 1.85/1.41 ((((((((((((((((((((((((? [X0 : $i] : (! [X1 : $i] : ((in @ X1 @ X0) != $true) & ((nonempty @ X0) = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 1.85/1.41 inference(ennf_transformation,[status(thm)],[f77])).
% 1.85/1.41 thf(f77,plain,(
% 1.85/1.41 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ! [X0 : $i] : (((nonempty @ X0) = $true) => ? [X1 : $i] : ((in @ X1 @ X0) = $true)))))))))))))))))))))))))))),
% 1.85/1.41 inference(fool_elimination,[status(thm)],[f76])).
% 1.85/1.41 thf(f76,plain,(
% 1.85/1.41 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => ! [X0 : $i] : ((nonempty @ X0) => ? [X1 : $i] : (in @ X1 @ X0)))))))))))))))))))))))))))),
% 1.85/1.41 inference(rectify,[status(thm)],[f30])).
% 1.85/1.41 thf(f30,negated_conjecture,(
% 1.85/1.41 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => ! [X3 : $i] : ((nonempty @ X3) => ? [X1 : $i] : (in @ X1 @ X3)))))))))))))))))))))))))))),
% 1.85/1.41 inference(negated_conjecture,[status(cth)],[f29])).
% 1.85/1.41 thf(f29,conjecture,(
% 1.85/1.41 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => ! [X3 : $i] : ((nonempty @ X3) => ? [X1 : $i] : (in @ X1 @ X3))))))))))))))))))))))))))),
% 1.85/1.41 file('/export/starexec/sandbox/tmp/tmp.8shZsOoivs/DTF2THF_10582.p',nonemptyE1)).
% 1.85/1.41 thf(f581,plain,(
% 1.85/1.41 (emptyset = sK0)),
% 1.85/1.41 inference(trivial_inequality_removal,[status(thm)],[f579])).
% 1.85/1.41 thf(f579,plain,(
% 1.85/1.41 (emptyset = sK0) | ($true != $true)),
% 1.85/1.41 inference(superposition,[status(thm)],[f119,f206])).
% 1.85/1.41 thf(f206,plain,(
% 1.85/1.41 ( ! [X1 : $i] : (($true = (in @ (sK3 @ X1) @ X1)) | (emptyset = X1)) )),
% 1.85/1.41 inference(not_proxy_clausification,[status(thm)],[f205])).
% 1.85/1.41 thf(f205,plain,(
% 1.85/1.41 ( ! [X1 : $i] : ((emptyset = X1) | ((~ (in @ (sK3 @ X1) @ X1)) = $false)) )),
% 1.85/1.41 inference(beta_eta_normalization,[status(thm)],[f204])).
% 1.85/1.41 thf(f204,plain,(
% 1.85/1.41 ( ! [X1 : $i] : ((emptyset = X1) | (((^[Y0 : $i]: (~ (in @ Y0 @ X1))) @ (sK3 @ X1)) = $false)) )),
% 1.85/1.41 inference(sigma_clausification,[status(thm)],[f203])).
% 1.85/1.41 thf(f203,plain,(
% 1.85/1.41 ( ! [X1 : $i] : ((emptyset = X1) | ((!! @ $i @ (^[Y0 : $i]: (~ (in @ Y0 @ X1)))) = $false)) )),
% 1.85/1.41 inference(equality_proxy_clausification,[status(thm)],[f202])).
% 1.85/1.41 thf(f202,plain,(
% 1.85/1.41 ( ! [X1 : $i] : (($true = (emptyset = X1)) | ((!! @ $i @ (^[Y0 : $i]: (~ (in @ Y0 @ X1)))) = $false)) )),
% 1.85/1.41 inference(binary_proxy_clausification,[status(thm)],[f201])).
% 1.85/1.41 thf(f201,plain,(
% 1.85/1.41 ( ! [X1 : $i] : (($true = ((!! @ $i @ (^[Y0 : $i]: (~ (in @ Y0 @ X1)))) => (emptyset = X1)))) )),
% 1.85/1.41 inference(beta_eta_normalization,[status(thm)],[f200])).
% 1.85/1.41 thf(f200,plain,(
% 1.85/1.41 ( ! [X1 : $i] : ((((^[Y0 : $i]: ((!! @ $i @ (^[Y1 : $i]: (~ (in @ Y1 @ Y0)))) => (emptyset = Y0))) @ X1) = $true)) )),
% 1.85/1.41 inference(pi_clausification,[status(thm)],[f154])).
% 1.85/1.41 thf(f154,plain,(
% 1.85/1.41 ($true = (!! @ $i @ (^[Y0 : $i]: ((!! @ $i @ (^[Y1 : $i]: (~ (in @ Y1 @ Y0)))) => (emptyset = Y0)))))),
% 1.85/1.41 inference(definition_unfolding,[status(thm)],[f126,f107])).
% 1.85/1.41 thf(f107,plain,(
% 1.85/1.41 (noeltsimpempty = (!! @ $i @ (^[Y0 : $i]: ((!! @ $i @ (^[Y1 : $i]: (~ (in @ Y1 @ Y0)))) => (emptyset = Y0)))))),
% 1.85/1.41 inference(cnf_transformation,[status(thm)],[f51])).
% 1.85/1.41 thf(f51,plain,(
% 1.85/1.41 (noeltsimpempty = (!! @ $i @ (^[Y0 : $i]: ((!! @ $i @ (^[Y1 : $i]: (~ (in @ Y1 @ Y0)))) => (emptyset = Y0)))))),
% 1.85/1.41 inference(fool_elimination,[status(thm)],[f50])).
% 1.85/1.41 thf(f50,plain,(
% 1.85/1.41 (noeltsimpempty = ! [X0 : $i] : (! [X1 : $i] : ~(in @ X1 @ X0) => (emptyset = X0)))),
% 1.85/1.41 inference(rectify,[status(thm)],[f26])).
% 1.85/1.41 thf(f26,axiom,(
% 1.85/1.41 (noeltsimpempty = ! [X3 : $i] : (! [X1 : $i] : ~(in @ X1 @ X3) => (emptyset = X3)))),
% 1.85/1.41 file('/export/starexec/sandbox/tmp/tmp.8shZsOoivs/DTF2THF_10582.p',noeltsimpempty)).
% 1.85/1.41 thf(f126,plain,(
% 1.85/1.41 (noeltsimpempty = $true)),
% 1.85/1.41 inference(cnf_transformation,[status(thm)],[f90])).
% 1.85/1.41 thf(f119,plain,(
% 1.85/1.41 ( ! [X1 : $i] : (((in @ X1 @ sK0) != $true)) )),
% 1.85/1.41 inference(cnf_transformation,[status(thm)],[f90])).
% 1.85/1.41 % SZS output end Proof for DTF2THF_10582
% 1.85/1.41 % (10682)------------------------------
% 1.85/1.41 % (10682)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.85/1.41 % (10682)Termination reason: Refutation
% 1.85/1.41
% 1.85/1.41 % (10682)Memory used [KB]: 6012
% 1.85/1.41 % (10682)Time elapsed: 0.027 s
% 1.85/1.41 % (10682)Instructions burned: 54 (million)
% 1.85/1.41 % (10682)------------------------------
% 1.85/1.41 % (10682)------------------------------
% 1.85/1.41 % (10681)Success in time 0.041 s
%------------------------------------------------------------------------------