%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU520^1 : 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 : n029.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:35:03 AM UTC 2026
% Result : Theorem 2.06s 1.55s
% Output : Refutation 2.06s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 4
% Syntax : Number of formulae : 22 ( 7 unt; 0 typ; 0 def)
% Number of atoms : 669 ( 176 equ; 0 cnn)
% Maximal formula atoms : 38 ( 30 avg)
% Number of connectives : 358 ( 20 ~; 8 |; 111 &; 72 @)
% ( 1 <=>; 146 =>; 0 <=; 0 <~>)
% Maximal formula depth : 41 ( 14 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 101 ( 98 usr; 41 con; 0-3 aty)
% Number of variables : 19 ( 0 ^; 13 !; 6 ?; 19 :)
% 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_60,type,
sP0: $i > $i > $o ).
thf(func_def_61,type,
sP1: $i > $o ).
thf(func_def_62,type,
sP2: $i > $i > $o ).
thf(func_def_63,type,
sP3: $i > $i > $o ).
thf(func_def_69,type,
sK9: $i > $i ).
thf(func_def_70,type,
sK10: $i > $i > $i ).
thf(func_def_71,type,
sK11: $i > $i > $i ).
thf(func_def_72,type,
sK12: $i > $i > $i ).
thf(func_def_73,type,
sK13: $i > $i > $i ).
thf(func_def_74,type,
sK14: $i > $i > $i ).
thf(func_def_75,type,
sK15: $i > $i > $i > $i ).
thf(func_def_76,type,
sK16: $i > $i ).
thf(func_def_77,type,
sK17: $i > $i ).
thf(func_def_78,type,
sK18: $i > $i ).
thf(func_def_79,type,
sK19: $i > $i ).
thf(func_def_80,type,
sK20: $i > $i > $i > $i ).
thf(func_def_81,type,
sK21: $i > $i > $i > $i ).
thf(func_def_82,type,
sK22: $i > $i > $i ).
thf(func_def_83,type,
sK23: $i > $i > $i ).
thf(func_def_84,type,
sK24: $i > $i > $i ).
thf(func_def_86,type,
sK26: $i > $i ).
thf(func_def_87,type,
sK27: $i > $i ).
thf(func_def_88,type,
sK28: $i > $i ).
thf(func_def_100,type,
sK40: $i > $i > $i ).
thf(func_def_101,type,
sK41: $i > $i > $i ).
thf(func_def_102,type,
sK42: $i > $i ).
thf(func_def_105,type,
sK45: $i > $i > $o ).
thf(func_def_107,type,
sK47: $i > $i ).
thf(func_def_108,type,
sK48: $i > $i ).
thf(func_def_109,type,
sK49: $i > ( $i > $i > $o ) > $i ).
thf(func_def_110,type,
sK50: $i > ( $i > $i > $o ) > $i ).
thf(func_def_111,type,
sK51: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_117,type,
sK57: $i > $o ).
thf(func_def_127,type,
sK67: $i > $i ).
thf(func_def_131,type,
sK71: $i > $o ).
thf(func_def_136,type,
sK76: $i > $i ).
thf(func_def_137,type,
sK77: $i > $o ).
thf(func_def_138,type,
sK78: ( $i > $o ) > $i ).
thf(func_def_141,type,
sK81: $i > $i ).
thf(func_def_142,type,
sK82: $i > $i ).
thf(func_def_145,type,
sK85: $i > $o ).
thf(func_def_149,type,
sK89: $i > $i > $i ).
thf(func_def_152,type,
sK92: $i > $o ).
thf(func_def_155,type,
sK95: $i > $i > $i ).
thf(func_def_158,type,
sK98: $i > $o ).
thf(func_def_159,type,
sK99: $i > $i > $i ).
thf(func_def_163,type,
sK103: ( $i > $o ) > $i ).
thf(func_def_164,type,
sK104: $i > $o ).
thf(func_def_165,type,
sK105: $i > $i ).
thf(func_def_167,type,
ph107:
!>[X0: $tType] : X0 ).
thf(f798,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f797]) ).
thf(f797,plain,
$true != $true,
inference(superposition,[status(thm)],[f473,f700]) ).
thf(f700,plain,
! [X0: $i] :
( $true
= ( in @ emptyset @ ( powerset @ X0 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f624]) ).
thf(f624,plain,
! [X0: $i] :
( ( $true != $true )
| ( $true
= ( in @ emptyset @ ( powerset @ X0 ) ) ) ),
inference(definition_unfolding,[status(thm)],[f503,f455]) ).
thf(f455,plain,
emptyinPowerset = $true,
inference(cnf_transformation,[status(thm)],[f253]) ).
thf(f253,plain,
( ( $true
!= ( in @ emptyset @ ( powerset @ sK52 ) ) )
& ( dsetconstrI = $true )
& ( setadjoinIR = $true )
& ( setunionAx = $true )
& ( notinemptyset = $true )
& ( emptyinunitempty = $true )
& ( omegaIndAx = $true )
& ( emptyI = $true )
& ( setbeta = $true )
& ( prop2setE = $true )
& ( setadjoinIL = $true )
& ( exuE3e = $true )
& ( dsetconstrER = $true )
& ( nonemptyI = $true )
& ( setadjoinOr = $true )
& ( nonemptyI1 = $true )
& ( emptysetAx = $true )
& ( descrp = $true )
& ( emptyinPowerset = $true )
& ( setadjoinE = $true )
& ( noeltsimpempty = $true )
& ( setoftrueEq = $true )
& ( wellorderingAx = $true )
& ( powersetI = $true )
& ( nonemptyE1 = $true )
& ( omega0Ax = $true )
& ( dsetconstrEL = $true )
& ( setadjoinAx = $true )
& ( foundationAx = $true )
& ( exuE1 = $true )
& ( replAx = $true )
& ( omegaSAx = $true )
& ( emptysetE = $true )
& ( setext = $true )
& ( powersetAx = $true )
& ( setextAx = $true )
& ( emptysetimpfalse = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK52])],[f137,f252]) ).
thf(f252,plain,
( ? [X0: $i] :
( $true
!= ( in @ emptyset @ ( powerset @ X0 ) ) )
=> ( $true
!= ( in @ emptyset @ ( powerset @ sK52 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f137,plain,
( ? [X0: $i] :
( $true
!= ( in @ emptyset @ ( powerset @ X0 ) ) )
& ( dsetconstrI = $true )
& ( setadjoinIR = $true )
& ( setunionAx = $true )
& ( notinemptyset = $true )
& ( emptyinunitempty = $true )
& ( omegaIndAx = $true )
& ( emptyI = $true )
& ( setbeta = $true )
& ( prop2setE = $true )
& ( setadjoinIL = $true )
& ( exuE3e = $true )
& ( dsetconstrER = $true )
& ( nonemptyI = $true )
& ( setadjoinOr = $true )
& ( nonemptyI1 = $true )
& ( emptysetAx = $true )
& ( descrp = $true )
& ( emptyinPowerset = $true )
& ( setadjoinE = $true )
& ( noeltsimpempty = $true )
& ( setoftrueEq = $true )
& ( wellorderingAx = $true )
& ( powersetI = $true )
& ( nonemptyE1 = $true )
& ( omega0Ax = $true )
& ( dsetconstrEL = $true )
& ( setadjoinAx = $true )
& ( foundationAx = $true )
& ( exuE1 = $true )
& ( replAx = $true )
& ( omegaSAx = $true )
& ( emptysetE = $true )
& ( setext = $true )
& ( powersetAx = $true )
& ( setextAx = $true )
& ( emptysetimpfalse = $true ) ),
inference(flattening,[status(thm)],[f136]) ).
thf(f136,plain,
( ? [X0: $i] :
( $true
!= ( in @ emptyset @ ( powerset @ X0 ) ) )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[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 )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ! [X0: $i] :
( $true
= ( in @ emptyset @ ( powerset @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f76]) ).
thf(f76,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ! [X0: $i] : ( in @ emptyset @ ( powerset @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f41]) ).
thf(f41,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ! [X3: $i] : ( in @ emptyset @ ( powerset @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f40]) ).
thf(f40,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ! [X3: $i] : ( in @ emptyset @ ( powerset @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.iTElzqeOvB/DTF2THF_30638.p',emptyInPowerset) ).
thf(f503,plain,
! [X0: $i] :
( ( $true
= ( in @ emptyset @ ( powerset @ X0 ) ) )
| ( emptyinPowerset != $true ) ),
inference(cnf_transformation,[status(thm)],[f288]) ).
thf(f288,plain,
( ( ! [X0: $i] :
( $true
= ( in @ emptyset @ ( powerset @ X0 ) ) )
| ( emptyinPowerset != $true ) )
& ( ( emptyinPowerset = $true )
| ( ( in @ emptyset @ ( powerset @ sK70 ) )
!= $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK70])],[f286,f287]) ).
thf(f287,plain,
( ? [X1: $i] :
( $true
!= ( in @ emptyset @ ( powerset @ X1 ) ) )
=> ( ( in @ emptyset @ ( powerset @ sK70 ) )
!= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f286,plain,
( ( ! [X0: $i] :
( $true
= ( in @ emptyset @ ( powerset @ X0 ) ) )
| ( emptyinPowerset != $true ) )
& ( ( emptyinPowerset = $true )
| ? [X1: $i] :
( $true
!= ( in @ emptyset @ ( powerset @ X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f285]) ).
thf(f285,plain,
( ( ! [X0: $i] :
( $true
= ( in @ emptyset @ ( powerset @ X0 ) ) )
| ( emptyinPowerset != $true ) )
& ( ( emptyinPowerset = $true )
| ? [X0: $i] :
( $true
!= ( in @ emptyset @ ( powerset @ X0 ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f89]) ).
thf(f89,plain,
( ! [X0: $i] :
( $true
= ( in @ emptyset @ ( powerset @ X0 ) ) )
<=> ( emptyinPowerset = $true ) ),
inference(fool_elimination,[status(thm)],[f88]) ).
thf(f88,plain,
( emptyinPowerset
= ( ! [X0: $i] : ( in @ emptyset @ ( powerset @ X0 ) ) ) ),
inference(rectify,[status(thm)],[f39]) ).
thf(f39,axiom,
( emptyinPowerset
= ( ! [X3: $i] : ( in @ emptyset @ ( powerset @ X3 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.iTElzqeOvB/DTF2THF_30638.p',emptyinPowerset) ).
thf(f473,plain,
( $true
!= ( in @ emptyset @ ( powerset @ sK52 ) ) ),
inference(cnf_transformation,[status(thm)],[f253]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU520^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n029.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 12:25:44 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.44 ---- Original DTF file ---
% 0.19/0.44 thf(spec,logic,$$dhol).
% 0.19/0.44 %------------------------------------------------------------------------------
% 0.19/0.44 % File : SEU520^1 : TPTP v9.2.1. Released v3.7.0.
% 0.19/0.44 % Domain : Set Theory
% 0.19/0.44 % Problem : Preliminary Notions - Power Sets and Unions
% 0.19/0.44 % Version : Especial.
% 0.19/0.44 % English : (! A:i.in emptyset (powerset A))
% 0.19/0.44
% 0.19/0.44 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.19/0.44 % Source : [Bro08]
% 0.19/0.44 % Names : ZFC022g [Bro08]
% 0.19/0.44
% 0.19/0.44 % Status : Theorem
% 0.19/0.44 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.23 v8.1.0, 0.00 v6.2.0, 0.14 v6.0.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.2.0, 0.20 v5.1.0, 0.40 v5.0.0, 0.20 v4.1.0, 0.33 v3.7.0
% 0.19/0.44 % Syntax : Number of formulae : 87 ( 39 unt; 47 typ; 39 def)
% 0.19/0.44 % Number of atoms : 219 ( 51 equ; 0 cnn)
% 0.19/0.44 % Maximal formula atoms : 37 ( 5 avg)
% 0.19/0.44 % Number of connectives : 363 ( 7 ~; 4 |; 18 &; 238 @)
% 0.19/0.44 % ( 7 <=>; 89 =>; 0 <=; 0 <~>)
% 0.19/0.44 % Maximal formula depth : 40 ( 2 avg)
% 0.19/0.44 % Number of types : 2 ( 0 usr)
% 0.19/0.44 % Number of type conns : 26 ( 26 >; 0 *; 0 +; 0 <<)
% 0.19/0.44 % Number of symbols : 50 ( 47 usr; 40 con; 0-2 aty)
% 0.19/0.44 % Number of variables : 120 ( 15 ^; 89 !; 16 ?; 120 :)
% 0.19/0.44 % SPC : TH0_THM_EQU_NAR
% 0.19/0.44
% 0.19/0.44 % Comments : http://mathgate.info/detsetitem.php?id=388
% 0.19/0.44 %------------------------------------------------------------------------------
% 0.19/0.44 thf(in_type,type,
% 0.19/0.44 in: $i > $i > $o ).
% 0.19/0.44
% 0.19/0.44 thf(exu_type,type,
% 0.19/0.44 exu: ( $i > $o ) > $o ).
% 0.19/0.44
% 0.19/0.44 thf(exu,definition,
% 0.19/0.44 ( exu
% 0.19/0.44 = ( ^ [Xphi: $i > $o] :
% 0.19/0.44 ? [Xx: $i] :
% 0.19/0.44 ( ( Xphi @ Xx )
% 0.19/0.44 & ! [Xy: $i] :
% 0.19/0.44 ( ( Xphi @ Xy )
% 0.19/0.44 => ( Xx = Xy ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(setextAx_type,type,
% 0.19/0.44 setextAx: $o ).
% 0.19/0.44
% 0.19/0.44 thf(setextAx,definition,
% 0.19/0.44 ( setextAx
% 0.19/0.44 = ( ! [A: $i,B: $i] :
% 0.19/0.44 ( ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 <=> ( in @ Xx @ B ) )
% 0.19/0.44 => ( A = B ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(emptyset_type,type,
% 0.19/0.44 emptyset: $i ).
% 0.19/0.44
% 0.19/0.44 thf(emptysetAx_type,type,
% 0.19/0.44 emptysetAx: $o ).
% 0.19/0.44
% 0.19/0.44 thf(emptysetAx,definition,
% 0.19/0.44 ( emptysetAx
% 0.19/0.44 = ( ! [Xx: $i] :
% 0.19/0.44 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(setadjoin_type,type,
% 0.19/0.44 setadjoin: $i > $i > $i ).
% 0.19/0.44
% 0.19/0.44 thf(setadjoinAx_type,type,
% 0.19/0.44 setadjoinAx: $o ).
% 0.19/0.44
% 0.19/0.44 thf(setadjoinAx,definition,
% 0.19/0.44 ( setadjoinAx
% 0.19/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.44 <=> ( ( Xy = Xx )
% 0.19/0.44 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(powerset_type,type,
% 0.19/0.44 powerset: $i > $i ).
% 0.19/0.44
% 0.19/0.44 thf(powersetAx_type,type,
% 0.19/0.44 powersetAx: $o ).
% 0.19/0.44
% 0.19/0.44 thf(powersetAx,definition,
% 0.19/0.44 ( powersetAx
% 0.19/0.44 = ( ! [A: $i,B: $i] :
% 0.19/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.44 <=> ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ B )
% 0.19/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(setunion_type,type,
% 0.19/0.44 setunion: $i > $i ).
% 0.19/0.44
% 0.19/0.44 thf(setunionAx_type,type,
% 0.19/0.44 setunionAx: $o ).
% 0.19/0.44
% 0.19/0.44 thf(setunionAx,definition,
% 0.19/0.44 ( setunionAx
% 0.19/0.44 = ( ! [A: $i,Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.44 <=> ? [B: $i] :
% 0.19/0.44 ( ( in @ Xx @ B )
% 0.19/0.44 & ( in @ B @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(omega_type,type,
% 0.19/0.44 omega: $i ).
% 0.19/0.44
% 0.19/0.44 thf(omega0Ax_type,type,
% 0.19/0.44 omega0Ax: $o ).
% 0.19/0.44
% 0.19/0.44 thf(omega0Ax,definition,
% 0.19/0.44 ( omega0Ax
% 0.19/0.44 = ( in @ emptyset @ omega ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(omegaSAx_type,type,
% 0.19/0.44 omegaSAx: $o ).
% 0.19/0.44
% 0.19/0.44 thf(omegaSAx,definition,
% 0.19/0.44 ( omegaSAx
% 0.19/0.44 = ( ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ omega )
% 0.19/0.44 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(omegaIndAx_type,type,
% 0.19/0.44 omegaIndAx: $o ).
% 0.19/0.44
% 0.19/0.44 thf(omegaIndAx,definition,
% 0.19/0.44 ( omegaIndAx
% 0.19/0.44 = ( ! [A: $i] :
% 0.19/0.44 ( ( ( in @ emptyset @ A )
% 0.19/0.44 & ! [Xx: $i] :
% 0.19/0.44 ( ( ( in @ Xx @ omega )
% 0.19/0.44 & ( in @ Xx @ A ) )
% 0.19/0.44 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.19/0.44 => ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ omega )
% 0.19/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(replAx_type,type,
% 0.19/0.44 replAx: $o ).
% 0.19/0.44
% 0.19/0.44 thf(replAx,definition,
% 0.19/0.44 ( replAx
% 0.19/0.44 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.19/0.44 ( ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 => ( exu
% 0.19/0.44 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.19/0.44 => ? [B: $i] :
% 0.19/0.44 ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ B )
% 0.19/0.44 <=> ? [Xy: $i] :
% 0.19/0.44 ( ( in @ Xy @ A )
% 0.19/0.44 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(foundationAx_type,type,
% 0.19/0.44 foundationAx: $o ).
% 0.19/0.44
% 0.19/0.44 thf(foundationAx,definition,
% 0.19/0.44 ( foundationAx
% 0.19/0.44 = ( ! [A: $i] :
% 0.19/0.44 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.44 => ? [B: $i] :
% 0.19/0.44 ( ( in @ B @ A )
% 0.19/0.44 & ~ ? [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ B )
% 0.19/0.44 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(wellorderingAx_type,type,
% 0.19/0.44 wellorderingAx: $o ).
% 0.19/0.44
% 0.19/0.44 thf(wellorderingAx,definition,
% 0.19/0.44 ( wellorderingAx
% 0.19/0.44 = ( ! [A: $i] :
% 0.19/0.44 ? [B: $i] :
% 0.19/0.44 ( ! [C: $i] :
% 0.19/0.44 ( ( in @ C @ B )
% 0.19/0.44 => ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ C )
% 0.19/0.44 => ( in @ Xx @ A ) ) )
% 0.19/0.44 & ! [Xx: $i,Xy: $i] :
% 0.19/0.44 ( ( ( in @ Xx @ A )
% 0.19/0.44 & ( in @ Xy @ A ) )
% 0.19/0.44 => ( ! [C: $i] :
% 0.19/0.44 ( ( in @ C @ B )
% 0.19/0.44 => ( ( in @ Xx @ C )
% 0.19/0.44 <=> ( in @ Xy @ C ) ) )
% 0.19/0.44 => ( Xx = Xy ) ) )
% 0.19/0.44 & ! [C: $i,D: $i] :
% 0.19/0.44 ( ( ( in @ C @ B )
% 0.19/0.44 & ( in @ D @ B ) )
% 0.19/0.44 => ( ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ C )
% 0.19/0.44 => ( in @ Xx @ D ) )
% 0.19/0.44 | ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ D )
% 0.19/0.44 => ( in @ Xx @ C ) ) ) )
% 0.19/0.44 & ! [C: $i] :
% 0.19/0.44 ( ( ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ C )
% 0.19/0.44 => ( in @ Xx @ A ) )
% 0.19/0.44 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.19/0.44 => ? [D: $i,Xx: $i] :
% 0.19/0.44 ( ( in @ D @ B )
% 0.19/0.44 & ( in @ Xx @ C )
% 0.19/0.44 & ~ ? [Xy: $i] :
% 0.19/0.44 ( ( in @ Xy @ D )
% 0.19/0.44 & ( in @ Xy @ C ) )
% 0.19/0.44 & ! [E: $i] :
% 0.19/0.44 ( ( in @ E @ B )
% 0.19/0.44 => ( ! [Xy: $i] :
% 0.19/0.44 ( ( in @ Xy @ E )
% 0.19/0.44 => ( in @ Xy @ D ) )
% 0.19/0.44 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(descr_type,type,
% 0.19/0.44 descr: ( $i > $o ) > $i ).
% 0.19/0.44
% 0.19/0.44 thf(descrp_type,type,
% 0.19/0.44 descrp: $o ).
% 0.19/0.44
% 0.19/0.44 thf(descrp,definition,
% 0.19/0.44 ( descrp
% 0.19/0.44 = ( ! [Xphi: $i > $o] :
% 0.19/0.44 ( ( exu
% 0.19/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.44 => ( Xphi
% 0.19/0.44 @ ( descr
% 0.19/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(dsetconstr_type,type,
% 0.19/0.44 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.19/0.44
% 0.19/0.44 thf(dsetconstrI_type,type,
% 0.19/0.44 dsetconstrI: $o ).
% 0.19/0.44
% 0.19/0.44 thf(dsetconstrI,definition,
% 0.19/0.44 ( dsetconstrI
% 0.19/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 => ( ( Xphi @ Xx )
% 0.19/0.44 => ( in @ Xx
% 0.19/0.44 @ ( dsetconstr @ A
% 0.19/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(dsetconstrEL_type,type,
% 0.19/0.44 dsetconstrEL: $o ).
% 0.19/0.44
% 0.19/0.44 thf(dsetconstrEL,definition,
% 0.19/0.44 ( dsetconstrEL
% 0.19/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.44 ( ( in @ Xx
% 0.19/0.44 @ ( dsetconstr @ A
% 0.19/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(dsetconstrER_type,type,
% 0.19/0.44 dsetconstrER: $o ).
% 0.19/0.44
% 0.19/0.44 thf(dsetconstrER,definition,
% 0.19/0.44 ( dsetconstrER
% 0.19/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.44 ( ( in @ Xx
% 0.19/0.44 @ ( dsetconstr @ A
% 0.19/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.44 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(exuE1_type,type,
% 0.19/0.44 exuE1: $o ).
% 0.19/0.44
% 0.19/0.44 thf(exuE1,definition,
% 0.19/0.44 ( exuE1
% 0.19/0.44 = ( ! [Xphi: $i > $o] :
% 0.19/0.44 ( ( exu
% 0.19/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.44 => ? [Xx: $i] :
% 0.19/0.44 ( ( Xphi @ Xx )
% 0.19/0.44 & ! [Xy: $i] :
% 0.19/0.44 ( ( Xphi @ Xy )
% 0.19/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(prop2set_type,type,
% 0.19/0.44 prop2set: $o > $i ).
% 0.19/0.44
% 0.19/0.44 thf(prop2set,definition,
% 0.19/0.44 ( prop2set
% 0.19/0.44 = ( ^ [Xphi: $o] :
% 0.19/0.44 ( dsetconstr @ ( powerset @ emptyset )
% 0.19/0.44 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(prop2setE_type,type,
% 0.19/0.44 prop2setE: $o ).
% 0.19/0.44
% 0.19/0.44 thf(prop2setE,definition,
% 0.19/0.44 ( prop2setE
% 0.19/0.44 = ( ! [Xphi: $o,Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.19/0.44 => Xphi ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(emptysetE_type,type,
% 0.19/0.44 emptysetE: $o ).
% 0.19/0.44
% 0.19/0.44 thf(emptysetE,definition,
% 0.19/0.44 ( emptysetE
% 0.19/0.44 = ( ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ emptyset )
% 0.19/0.44 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(emptysetimpfalse_type,type,
% 0.19/0.44 emptysetimpfalse: $o ).
% 0.19/0.44
% 0.19/0.44 thf(emptysetimpfalse,definition,
% 0.19/0.44 ( emptysetimpfalse
% 0.19/0.44 = ( ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ emptyset )
% 0.19/0.44 => $false ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(notinemptyset_type,type,
% 0.19/0.44 notinemptyset: $o ).
% 0.19/0.44
% 0.19/0.44 thf(notinemptyset,definition,
% 0.19/0.44 ( notinemptyset
% 0.19/0.44 = ( ! [Xx: $i] :
% 0.19/0.44 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(exuE3e_type,type,
% 0.19/0.44 exuE3e: $o ).
% 0.19/0.44
% 0.19/0.44 thf(exuE3e,definition,
% 0.19/0.44 ( exuE3e
% 0.19/0.44 = ( ! [Xphi: $i > $o] :
% 0.19/0.44 ( ( exu
% 0.19/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.44 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(setext_type,type,
% 0.19/0.44 setext: $o ).
% 0.19/0.44
% 0.19/0.44 thf(setext,definition,
% 0.19/0.44 ( setext
% 0.19/0.44 = ( ! [A: $i,B: $i] :
% 0.19/0.44 ( ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 => ( in @ Xx @ B ) )
% 0.19/0.44 => ( ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ B )
% 0.19/0.44 => ( in @ Xx @ A ) )
% 0.19/0.44 => ( A = B ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(emptyI_type,type,
% 0.19/0.44 emptyI: $o ).
% 0.19/0.44
% 0.19/0.44 thf(emptyI,definition,
% 0.19/0.44 ( emptyI
% 0.19/0.44 = ( ! [A: $i] :
% 0.19/0.44 ( ! [Xx: $i] :
% 0.19/0.44 ~ ( in @ Xx @ A )
% 0.19/0.44 => ( A = emptyset ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(noeltsimpempty_type,type,
% 0.19/0.44 noeltsimpempty: $o ).
% 0.19/0.44
% 0.19/0.44 thf(noeltsimpempty,definition,
% 0.19/0.44 ( noeltsimpempty
% 0.19/0.44 = ( ! [A: $i] :
% 0.19/0.44 ( ! [Xx: $i] :
% 0.19/0.44 ~ ( in @ Xx @ A )
% 0.19/0.44 => ( A = emptyset ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(setbeta_type,type,
% 0.19/0.44 setbeta: $o ).
% 0.19/0.44
% 0.19/0.44 thf(setbeta,definition,
% 0.19/0.44 ( setbeta
% 0.19/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 => ( ( in @ Xx
% 0.19/0.44 @ ( dsetconstr @ A
% 0.19/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.44 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(nonempty_type,type,
% 0.19/0.44 nonempty: $i > $o ).
% 0.19/0.44
% 0.19/0.44 thf(nonempty,definition,
% 0.19/0.44 ( nonempty
% 0.19/0.44 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(nonemptyE1_type,type,
% 0.19/0.44 nonemptyE1: $o ).
% 0.19/0.44
% 0.19/0.44 thf(nonemptyE1,definition,
% 0.19/0.44 ( nonemptyE1
% 0.19/0.44 = ( ! [A: $i] :
% 0.19/0.44 ( ( nonempty @ A )
% 0.19/0.44 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(nonemptyI_type,type,
% 0.19/0.44 nonemptyI: $o ).
% 0.19/0.44
% 0.19/0.44 thf(nonemptyI,definition,
% 0.19/0.44 ( nonemptyI
% 0.19/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 => ( ( Xphi @ Xx )
% 0.19/0.44 => ( nonempty
% 0.19/0.44 @ ( dsetconstr @ A
% 0.19/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(nonemptyI1_type,type,
% 0.19/0.44 nonemptyI1: $o ).
% 0.19/0.44
% 0.19/0.44 thf(nonemptyI1,definition,
% 0.19/0.44 ( nonemptyI1
% 0.19/0.44 = ( ! [A: $i] :
% 0.19/0.44 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.44 => ( nonempty @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(setadjoinIL_type,type,
% 0.19/0.44 setadjoinIL: $o ).
% 0.19/0.44
% 0.19/0.44 thf(setadjoinIL,definition,
% 0.19/0.44 ( setadjoinIL
% 0.19/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(emptyinunitempty_type,type,
% 0.19/0.44 emptyinunitempty: $o ).
% 0.19/0.44
% 0.19/0.44 thf(emptyinunitempty,definition,
% 0.19/0.44 ( emptyinunitempty
% 0.19/0.44 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(setadjoinIR_type,type,
% 0.19/0.44 setadjoinIR: $o ).
% 0.19/0.44
% 0.19/0.44 thf(setadjoinIR,definition,
% 0.19/0.44 ( setadjoinIR
% 0.19/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.44 ( ( in @ Xy @ A )
% 0.19/0.44 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(setadjoinE_type,type,
% 0.19/0.44 setadjoinE: $o ).
% 0.19/0.44
% 0.19/0.44 thf(setadjoinE,definition,
% 0.19/0.44 ( setadjoinE
% 0.19/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.44 => ! [Xphi: $o] :
% 0.19/0.44 ( ( ( Xy = Xx )
% 0.19/0.44 => Xphi )
% 0.19/0.44 => ( ( ( in @ Xy @ A )
% 0.19/0.44 => Xphi )
% 0.19/0.44 => Xphi ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(setadjoinOr_type,type,
% 0.19/0.44 setadjoinOr: $o ).
% 0.19/0.44
% 0.19/0.44 thf(setadjoinOr,definition,
% 0.19/0.44 ( setadjoinOr
% 0.19/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.44 => ( ( Xy = Xx )
% 0.19/0.44 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(setoftrueEq_type,type,
% 0.19/0.44 setoftrueEq: $o ).
% 0.19/0.44
% 0.19/0.44 thf(setoftrueEq,definition,
% 0.19/0.44 ( setoftrueEq
% 0.19/0.44 = ( ! [A: $i] :
% 0.19/0.44 ( ( dsetconstr @ A
% 0.19/0.44 @ ^ [Xx: $i] : $true )
% 0.19/0.44 = A ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(powersetI_type,type,
% 0.19/0.44 powersetI: $o ).
% 0.19/0.44
% 0.19/0.44 thf(powersetI,definition,
% 0.19/0.44 ( powersetI
% 0.19/0.44 = ( ! [A: $i,B: $i] :
% 0.19/0.44 ( ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ B )
% 0.19/0.44 => ( in @ Xx @ A ) )
% 0.19/0.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(emptyinPowerset_type,type,
% 0.19/0.44 emptyinPowerset: $o ).
% 0.19/0.44
% 0.19/0.44 thf(emptyinPowerset,definition,
% 0.19/0.44 ( emptyinPowerset
% 0.19/0.44 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(emptyInPowerset,conjecture,
% 0.19/0.44 ( setextAx
% 0.19/0.44 => ( emptysetAx
% 0.19/0.44 => ( setadjoinAx
% 0.19/0.44 => ( powersetAx
% 0.19/0.44 => ( setunionAx
% 0.19/0.44 => ( omega0Ax
% 0.19/0.44 => ( omegaSAx
% 0.19/0.44 => ( omegaIndAx
% 0.19/0.44 => ( replAx
% 0.19/0.44 => ( foundationAx
% 0.19/0.44 => ( wellorderingAx
% 0.19/0.44 => ( descrp
% 0.19/0.44 => ( dsetconstrI
% 0.19/0.44 => ( dsetconstrEL
% 0.19/0.44 => ( dsetconstrER
% 0.19/0.44 => ( exuE1
% 0.19/0.44 => ( prop2setE
% 0.19/0.44 => ( emptysetE
% 0.19/0.44 => ( emptysetimpfalse
% 0.19/0.44 => ( notinemptyset
% 0.19/0.44 => ( exuE3e
% 0.19/0.44 => ( setext
% 0.19/0.44 => ( emptyI
% 0.19/0.44 => ( noeltsimpempty
% 0.19/0.44 => ( setbeta
% 0.19/0.44 => ( nonemptyE1
% 0.19/0.44 => ( nonemptyI
% 0.19/0.44 => ( nonemptyI1
% 0.19/0.44 => ( setadjoinIL
% 0.19/0.44 => ( emptyinunitempty
% 0.19/0.44 => ( setadjoinIR
% 0.19/0.44 => ( setadjoinE
% 0.19/0.44 => ( setadjoinOr
% 0.19/0.44 => ( setoftrueEq
% 0.19/0.44 => ( powersetI
% 0.19/0.44 => ( emptyinPowerset
% 0.19/0.44 => ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 %------------------------------------------------------------------------------
% 0.19/0.44 ------------------------
% 1.86/1.49 ---- Embedded in THF ---
% 1.86/1.49 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.86/1.49 %%% Generated on Tue Feb 24 12:25:45 EST 2026
% 1.86/1.49 %%% using '$$dhol' embedding, version 1.3.0.
% 1.86/1.49 %%% Logic specification used:
% 1.86/1.49 %%% thf(spec, logic, $$dhol).
% 1.86/1.49
% 1.86/1.49 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.iTElzqeOvB/DTF2DTF30638.p
% See solution above
% 1.86/1.49 ------------------------
% 2.06/1.50 ---- Cleaned THF ---
% 2.06/1.50 thf(in_type,type,
% 2.06/1.50 in: $i > $i > $o ).
% 2.06/1.50
% 2.06/1.50 thf(exu_type,type,
% 2.06/1.50 exu: ( $i > $o ) > $o ).
% 2.06/1.50
% 2.06/1.50 thf(exu,definition,
% 2.06/1.50 ( exu
% 2.06/1.50 = ( ^ [Xphi: $i > $o] :
% 2.06/1.50 ? [Xx: $i] :
% 2.06/1.50 ( ( Xphi @ Xx )
% 2.06/1.50 & ! [Xy: $i] :
% 2.06/1.50 ( ( Xphi @ Xy )
% 2.06/1.50 => ( Xx = Xy ) ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(setextAx_type,type,
% 2.06/1.50 setextAx: $o ).
% 2.06/1.50
% 2.06/1.50 thf(setextAx,definition,
% 2.06/1.50 ( setextAx
% 2.06/1.50 = ( ! [A: $i,B: $i] :
% 2.06/1.50 ( ! [Xx: $i] :
% 2.06/1.50 ( ( in @ Xx @ A )
% 2.06/1.50 <=> ( in @ Xx @ B ) )
% 2.06/1.50 => ( A = B ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(emptyset_type,type,
% 2.06/1.50 emptyset: $i ).
% 2.06/1.50
% 2.06/1.50 thf(emptysetAx_type,type,
% 2.06/1.50 emptysetAx: $o ).
% 2.06/1.50
% 2.06/1.50 thf(emptysetAx,definition,
% 2.06/1.50 ( emptysetAx
% 2.06/1.50 = ( ! [Xx: $i] :
% 2.06/1.50 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(setadjoin_type,type,
% 2.06/1.50 setadjoin: $i > $i > $i ).
% 2.06/1.50
% 2.06/1.50 thf(setadjoinAx_type,type,
% 2.06/1.50 setadjoinAx: $o ).
% 2.06/1.50
% 2.06/1.50 thf(setadjoinAx,definition,
% 2.06/1.50 ( setadjoinAx
% 2.06/1.50 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.06/1.50 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.06/1.50 <=> ( ( Xy = Xx )
% 2.06/1.50 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(powerset_type,type,
% 2.06/1.50 powerset: $i > $i ).
% 2.06/1.50
% 2.06/1.50 thf(powersetAx_type,type,
% 2.06/1.50 powersetAx: $o ).
% 2.06/1.50
% 2.06/1.50 thf(powersetAx,definition,
% 2.06/1.50 ( powersetAx
% 2.06/1.50 = ( ! [A: $i,B: $i] :
% 2.06/1.50 ( ( in @ B @ ( powerset @ A ) )
% 2.06/1.50 <=> ! [Xx: $i] :
% 2.06/1.50 ( ( in @ Xx @ B )
% 2.06/1.50 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(setunion_type,type,
% 2.06/1.50 setunion: $i > $i ).
% 2.06/1.50
% 2.06/1.50 thf(setunionAx_type,type,
% 2.06/1.50 setunionAx: $o ).
% 2.06/1.50
% 2.06/1.50 thf(setunionAx,definition,
% 2.06/1.50 ( setunionAx
% 2.06/1.50 = ( ! [A: $i,Xx: $i] :
% 2.06/1.50 ( ( in @ Xx @ ( setunion @ A ) )
% 2.06/1.50 <=> ? [B: $i] :
% 2.06/1.50 ( ( in @ Xx @ B )
% 2.06/1.50 & ( in @ B @ A ) ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(omega_type,type,
% 2.06/1.50 omega: $i ).
% 2.06/1.50
% 2.06/1.50 thf(omega0Ax_type,type,
% 2.06/1.50 omega0Ax: $o ).
% 2.06/1.50
% 2.06/1.50 thf(omega0Ax,definition,
% 2.06/1.50 ( omega0Ax
% 2.06/1.50 = ( in @ emptyset @ omega ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(omegaSAx_type,type,
% 2.06/1.50 omegaSAx: $o ).
% 2.06/1.50
% 2.06/1.50 thf(omegaSAx,definition,
% 2.06/1.50 ( omegaSAx
% 2.06/1.50 = ( ! [Xx: $i] :
% 2.06/1.50 ( ( in @ Xx @ omega )
% 2.06/1.50 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(omegaIndAx_type,type,
% 2.06/1.50 omegaIndAx: $o ).
% 2.06/1.50
% 2.06/1.50 thf(omegaIndAx,definition,
% 2.06/1.50 ( omegaIndAx
% 2.06/1.50 = ( ! [A: $i] :
% 2.06/1.50 ( ( ( in @ emptyset @ A )
% 2.06/1.50 & ! [Xx: $i] :
% 2.06/1.50 ( ( ( in @ Xx @ omega )
% 2.06/1.50 & ( in @ Xx @ A ) )
% 2.06/1.50 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.06/1.50 => ! [Xx: $i] :
% 2.06/1.50 ( ( in @ Xx @ omega )
% 2.06/1.50 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(replAx_type,type,
% 2.06/1.50 replAx: $o ).
% 2.06/1.50
% 2.06/1.50 thf(replAx,definition,
% 2.06/1.50 ( replAx
% 2.06/1.50 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.06/1.50 ( ! [Xx: $i] :
% 2.06/1.50 ( ( in @ Xx @ A )
% 2.06/1.50 => ( exu
% 2.06/1.50 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.06/1.50 => ? [B: $i] :
% 2.06/1.50 ! [Xx: $i] :
% 2.06/1.50 ( ( in @ Xx @ B )
% 2.06/1.50 <=> ? [Xy: $i] :
% 2.06/1.50 ( ( in @ Xy @ A )
% 2.06/1.50 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(foundationAx_type,type,
% 2.06/1.50 foundationAx: $o ).
% 2.06/1.50
% 2.06/1.50 thf(foundationAx,definition,
% 2.06/1.50 ( foundationAx
% 2.06/1.50 = ( ! [A: $i] :
% 2.06/1.50 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.06/1.50 => ? [B: $i] :
% 2.06/1.50 ( ( in @ B @ A )
% 2.06/1.50 & ~ ? [Xx: $i] :
% 2.06/1.50 ( ( in @ Xx @ B )
% 2.06/1.50 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(wellorderingAx_type,type,
% 2.06/1.50 wellorderingAx: $o ).
% 2.06/1.50
% 2.06/1.50 thf(wellorderingAx,definition,
% 2.06/1.50 ( wellorderingAx
% 2.06/1.50 = ( ! [A: $i] :
% 2.06/1.50 ? [B: $i] :
% 2.06/1.50 ( ! [C: $i] :
% 2.06/1.50 ( ( in @ C @ B )
% 2.06/1.50 => ! [Xx: $i] :
% 2.06/1.50 ( ( in @ Xx @ C )
% 2.06/1.50 => ( in @ Xx @ A ) ) )
% 2.06/1.50 & ! [Xx: $i,Xy: $i] :
% 2.06/1.50 ( ( ( in @ Xx @ A )
% 2.06/1.50 & ( in @ Xy @ A ) )
% 2.06/1.50 => ( ! [C: $i] :
% 2.06/1.50 ( ( in @ C @ B )
% 2.06/1.50 => ( ( in @ Xx @ C )
% 2.06/1.50 <=> ( in @ Xy @ C ) ) )
% 2.06/1.50 => ( Xx = Xy ) ) )
% 2.06/1.50 & ! [C: $i,D: $i] :
% 2.06/1.50 ( ( ( in @ C @ B )
% 2.06/1.50 & ( in @ D @ B ) )
% 2.06/1.50 => ( ! [Xx: $i] :
% 2.06/1.50 ( ( in @ Xx @ C )
% 2.06/1.50 => ( in @ Xx @ D ) )
% 2.06/1.50 | ! [Xx: $i] :
% 2.06/1.50 ( ( in @ Xx @ D )
% 2.06/1.50 => ( in @ Xx @ C ) ) ) )
% 2.06/1.50 & ! [C: $i] :
% 2.06/1.50 ( ( ! [Xx: $i] :
% 2.06/1.50 ( ( in @ Xx @ C )
% 2.06/1.50 => ( in @ Xx @ A ) )
% 2.06/1.50 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.06/1.50 => ? [D: $i,Xx: $i] :
% 2.06/1.50 ( ( in @ D @ B )
% 2.06/1.50 & ( in @ Xx @ C )
% 2.06/1.50 & ~ ? [Xy: $i] :
% 2.06/1.50 ( ( in @ Xy @ D )
% 2.06/1.50 & ( in @ Xy @ C ) )
% 2.06/1.50 & ! [E: $i] :
% 2.06/1.50 ( ( in @ E @ B )
% 2.06/1.50 => ( ! [Xy: $i] :
% 2.06/1.50 ( ( in @ Xy @ E )
% 2.06/1.50 => ( in @ Xy @ D ) )
% 2.06/1.50 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(descr_type,type,
% 2.06/1.50 descr: ( $i > $o ) > $i ).
% 2.06/1.50
% 2.06/1.50 thf(descrp_type,type,
% 2.06/1.50 descrp: $o ).
% 2.06/1.50
% 2.06/1.50 thf(descrp,definition,
% 2.06/1.50 ( descrp
% 2.06/1.50 = ( ! [Xphi: $i > $o] :
% 2.06/1.50 ( ( exu
% 2.06/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.06/1.50 => ( Xphi
% 2.06/1.50 @ ( descr
% 2.06/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(dsetconstr_type,type,
% 2.06/1.50 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.06/1.50
% 2.06/1.50 thf(dsetconstrI_type,type,
% 2.06/1.50 dsetconstrI: $o ).
% 2.06/1.50
% 2.06/1.50 thf(dsetconstrI,definition,
% 2.06/1.50 ( dsetconstrI
% 2.06/1.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.06/1.50 ( ( in @ Xx @ A )
% 2.06/1.50 => ( ( Xphi @ Xx )
% 2.06/1.50 => ( in @ Xx
% 2.06/1.50 @ ( dsetconstr @ A
% 2.06/1.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(dsetconstrEL_type,type,
% 2.06/1.50 dsetconstrEL: $o ).
% 2.06/1.50
% 2.06/1.50 thf(dsetconstrEL,definition,
% 2.06/1.50 ( dsetconstrEL
% 2.06/1.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.06/1.50 ( ( in @ Xx
% 2.06/1.50 @ ( dsetconstr @ A
% 2.06/1.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.06/1.50 => ( in @ Xx @ A ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(dsetconstrER_type,type,
% 2.06/1.50 dsetconstrER: $o ).
% 2.06/1.50
% 2.06/1.50 thf(dsetconstrER,definition,
% 2.06/1.50 ( dsetconstrER
% 2.06/1.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.06/1.50 ( ( in @ Xx
% 2.06/1.50 @ ( dsetconstr @ A
% 2.06/1.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.06/1.50 => ( Xphi @ Xx ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(exuE1_type,type,
% 2.06/1.50 exuE1: $o ).
% 2.06/1.50
% 2.06/1.50 thf(exuE1,definition,
% 2.06/1.50 ( exuE1
% 2.06/1.50 = ( ! [Xphi: $i > $o] :
% 2.06/1.50 ( ( exu
% 2.06/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.06/1.50 => ? [Xx: $i] :
% 2.06/1.50 ( ( Xphi @ Xx )
% 2.06/1.50 & ! [Xy: $i] :
% 2.06/1.50 ( ( Xphi @ Xy )
% 2.06/1.50 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(prop2set_type,type,
% 2.06/1.50 prop2set: $o > $i ).
% 2.06/1.50
% 2.06/1.50 thf(prop2set,definition,
% 2.06/1.50 ( prop2set
% 2.06/1.50 = ( ^ [Xphi: $o] :
% 2.06/1.50 ( dsetconstr @ ( powerset @ emptyset )
% 2.06/1.50 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(prop2setE_type,type,
% 2.06/1.50 prop2setE: $o ).
% 2.06/1.50
% 2.06/1.50 thf(prop2setE,definition,
% 2.06/1.50 ( prop2setE
% 2.06/1.50 = ( ! [Xphi: $o,Xx: $i] :
% 2.06/1.50 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.06/1.50 => Xphi ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(emptysetE_type,type,
% 2.06/1.50 emptysetE: $o ).
% 2.06/1.50
% 2.06/1.50 thf(emptysetE,definition,
% 2.06/1.50 ( emptysetE
% 2.06/1.50 = ( ! [Xx: $i] :
% 2.06/1.50 ( ( in @ Xx @ emptyset )
% 2.06/1.50 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(emptysetimpfalse_type,type,
% 2.06/1.50 emptysetimpfalse: $o ).
% 2.06/1.50
% 2.06/1.50 thf(emptysetimpfalse,definition,
% 2.06/1.50 ( emptysetimpfalse
% 2.06/1.50 = ( ! [Xx: $i] :
% 2.06/1.50 ( ( in @ Xx @ emptyset )
% 2.06/1.50 => $false ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(notinemptyset_type,type,
% 2.06/1.50 notinemptyset: $o ).
% 2.06/1.50
% 2.06/1.50 thf(notinemptyset,definition,
% 2.06/1.50 ( notinemptyset
% 2.06/1.50 = ( ! [Xx: $i] :
% 2.06/1.50 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(exuE3e_type,type,
% 2.06/1.50 exuE3e: $o ).
% 2.06/1.50
% 2.06/1.50 thf(exuE3e,definition,
% 2.06/1.50 ( exuE3e
% 2.06/1.50 = ( ! [Xphi: $i > $o] :
% 2.06/1.50 ( ( exu
% 2.06/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.06/1.50 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(setext_type,type,
% 2.06/1.50 setext: $o ).
% 2.06/1.50
% 2.06/1.50 thf(setext,definition,
% 2.06/1.50 ( setext
% 2.06/1.50 = ( ! [A: $i,B: $i] :
% 2.06/1.50 ( ! [Xx: $i] :
% 2.06/1.50 ( ( in @ Xx @ A )
% 2.06/1.50 => ( in @ Xx @ B ) )
% 2.06/1.50 => ( ! [Xx: $i] :
% 2.06/1.50 ( ( in @ Xx @ B )
% 2.06/1.50 => ( in @ Xx @ A ) )
% 2.06/1.50 => ( A = B ) ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(emptyI_type,type,
% 2.06/1.50 emptyI: $o ).
% 2.06/1.50
% 2.06/1.50 thf(emptyI,definition,
% 2.06/1.50 ( emptyI
% 2.06/1.50 = ( ! [A: $i] :
% 2.06/1.50 ( ! [Xx: $i] :
% 2.06/1.50 ~ ( in @ Xx @ A )
% 2.06/1.50 => ( A = emptyset ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(noeltsimpempty_type,type,
% 2.06/1.50 noeltsimpempty: $o ).
% 2.06/1.50
% 2.06/1.50 thf(noeltsimpempty,definition,
% 2.06/1.50 ( noeltsimpempty
% 2.06/1.50 = ( ! [A: $i] :
% 2.06/1.50 ( ! [Xx: $i] :
% 2.06/1.50 ~ ( in @ Xx @ A )
% 2.06/1.50 => ( A = emptyset ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(setbeta_type,type,
% 2.06/1.50 setbeta: $o ).
% 2.06/1.50
% 2.06/1.50 thf(setbeta,definition,
% 2.06/1.50 ( setbeta
% 2.06/1.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.06/1.50 ( ( in @ Xx @ A )
% 2.06/1.50 => ( ( in @ Xx
% 2.06/1.50 @ ( dsetconstr @ A
% 2.06/1.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.06/1.50 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(nonempty_type,type,
% 2.06/1.50 nonempty: $i > $o ).
% 2.06/1.50
% 2.06/1.50 thf(nonempty,definition,
% 2.06/1.50 ( nonempty
% 2.06/1.50 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(nonemptyE1_type,type,
% 2.06/1.50 nonemptyE1: $o ).
% 2.06/1.50
% 2.06/1.50 thf(nonemptyE1,definition,
% 2.06/1.50 ( nonemptyE1
% 2.06/1.50 = ( ! [A: $i] :
% 2.06/1.50 ( ( nonempty @ A )
% 2.06/1.50 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(nonemptyI_type,type,
% 2.06/1.50 nonemptyI: $o ).
% 2.06/1.50
% 2.06/1.50 thf(nonemptyI,definition,
% 2.06/1.50 ( nonemptyI
% 2.06/1.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.06/1.50 ( ( in @ Xx @ A )
% 2.06/1.50 => ( ( Xphi @ Xx )
% 2.06/1.50 => ( nonempty
% 2.06/1.50 @ ( dsetconstr @ A
% 2.06/1.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(nonemptyI1_type,type,
% 2.06/1.50 nonemptyI1: $o ).
% 2.06/1.50
% 2.06/1.50 thf(nonemptyI1,definition,
% 2.06/1.50 ( nonemptyI1
% 2.06/1.50 = ( ! [A: $i] :
% 2.06/1.50 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.06/1.50 => ( nonempty @ A ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(setadjoinIL_type,type,
% 2.06/1.50 setadjoinIL: $o ).
% 2.06/1.50
% 2.06/1.50 thf(setadjoinIL,definition,
% 2.06/1.50 ( setadjoinIL
% 2.06/1.50 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(emptyinunitempty_type,type,
% 2.06/1.50 emptyinunitempty: $o ).
% 2.06/1.50
% 2.06/1.50 thf(emptyinunitempty,definition,
% 2.06/1.50 ( emptyinunitempty
% 2.06/1.50 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(setadjoinIR_type,type,
% 2.06/1.50 setadjoinIR: $o ).
% 2.06/1.50
% 2.06/1.50 thf(setadjoinIR,definition,
% 2.06/1.50 ( setadjoinIR
% 2.06/1.50 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.06/1.50 ( ( in @ Xy @ A )
% 2.06/1.50 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(setadjoinE_type,type,
% 2.06/1.50 setadjoinE: $o ).
% 2.06/1.50
% 2.06/1.50 thf(setadjoinE,definition,
% 2.06/1.50 ( setadjoinE
% 2.06/1.50 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.06/1.50 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.06/1.50 => ! [Xphi: $o] :
% 2.06/1.50 ( ( ( Xy = Xx )
% 2.06/1.50 => Xphi )
% 2.06/1.50 => ( ( ( in @ Xy @ A )
% 2.06/1.50 => Xphi )
% 2.06/1.50 => Xphi ) ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(setadjoinOr_type,type,
% 2.06/1.50 setadjoinOr: $o ).
% 2.06/1.50
% 2.06/1.50 thf(setadjoinOr,definition,
% 2.06/1.50 ( setadjoinOr
% 2.06/1.50 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.06/1.50 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.06/1.50 => ( ( Xy = Xx )
% 2.06/1.50 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(setoftrueEq_type,type,
% 2.06/1.50 setoftrueEq: $o ).
% 2.06/1.50
% 2.06/1.50 thf(setoftrueEq,definition,
% 2.06/1.50 ( setoftrueEq
% 2.06/1.50 = ( ! [A: $i] :
% 2.06/1.50 ( ( dsetconstr @ A
% 2.06/1.50 @ ^ [Xx: $i] : $true )
% 2.06/1.50 = A ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(powersetI_type,type,
% 2.06/1.50 powersetI: $o ).
% 2.06/1.50
% 2.06/1.50 thf(powersetI,definition,
% 2.06/1.50 ( powersetI
% 2.06/1.50 = ( ! [A: $i,B: $i] :
% 2.06/1.50 ( ! [Xx: $i] :
% 2.06/1.50 ( ( in @ Xx @ B )
% 2.06/1.50 => ( in @ Xx @ A ) )
% 2.06/1.50 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(emptyinPowerset_type,type,
% 2.06/1.50 emptyinPowerset: $o ).
% 2.06/1.50
% 2.06/1.50 thf(emptyinPowerset,definition,
% 2.06/1.50 ( emptyinPowerset
% 2.06/1.50 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.06/1.50
% 2.06/1.50 thf(emptyInPowerset,conjecture,
% 2.06/1.50 ( setextAx
% 2.06/1.50 => ( emptysetAx
% 2.06/1.50 => ( setadjoinAx
% 2.06/1.50 => ( powersetAx
% 2.06/1.50 => ( setunionAx
% 2.06/1.50 => ( omega0Ax
% 2.06/1.50 => ( omegaSAx
% 2.06/1.50 => ( omegaIndAx
% 2.06/1.50 => ( replAx
% 2.06/1.50 => ( foundationAx
% 2.06/1.50 => ( wellorderingAx
% 2.06/1.50 => ( descrp
% 2.06/1.50 => ( dsetconstrI
% 2.06/1.50 => ( dsetconstrEL
% 2.06/1.50 => ( dsetconstrER
% 2.06/1.50 => ( exuE1
% 2.06/1.50 => ( prop2setE
% 2.06/1.50 => ( emptysetE
% 2.06/1.50 => ( emptysetimpfalse
% 2.06/1.50 => ( notinemptyset
% 2.06/1.50 => ( exuE3e
% 2.06/1.50 => ( setext
% 2.06/1.50 => ( emptyI
% 2.06/1.50 => ( noeltsimpempty
% 2.06/1.50 => ( setbeta
% 2.06/1.50 => ( nonemptyE1
% 2.06/1.50 => ( nonemptyI
% 2.06/1.50 => ( nonemptyI1
% 2.06/1.50 => ( setadjoinIL
% 2.06/1.50 => ( emptyinunitempty
% 2.06/1.50 => ( setadjoinIR
% 2.06/1.50 => ( setadjoinE
% 2.06/1.50 => ( setadjoinOr
% 2.06/1.50 => ( setoftrueEq
% 2.06/1.50 => ( powersetI
% 2.06/1.50 => ( emptyinPowerset
% 2.06/1.50 => ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.06/1.50 ------------------------
% 2.06/1.52 % (30743)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_30638 for (2999ds/18Mi)
% 2.06/1.52 % (30737)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_30638 for (2999ds/183Mi)
% 2.06/1.52 % (30740)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_30638 for (2999ds/2Mi)
% 2.06/1.52 % (30739)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_30638 for (2999ds/27Mi)
% 2.06/1.52 % (30738)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_30638 for (2999ds/4Mi)
% 2.06/1.52 % (30741)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_30638 for (2999ds/2Mi)
% 2.06/1.52 % (30742)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_30638 for (2999ds/275Mi)
% 2.06/1.52 % (30740)Instruction limit reached!
% 2.06/1.52 % (30740)------------------------------
% 2.06/1.52 % (30740)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.06/1.52 % (30740)Termination reason: Unknown
% 2.06/1.52 % (30740)Termination phase: shuffling
% 2.06/1.52 % (30741)Instruction limit reached!
% 2.06/1.52 % (30741)------------------------------
% 2.06/1.52 % (30741)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.06/1.52
% 2.06/1.52 % (30740)Memory used [KB]: 1023
% 2.06/1.52 % (30740)Time elapsed: 0.003 s
% 2.06/1.52 % (30740)Instructions burned: 3 (million)
% 2.06/1.52 % (30740)------------------------------
% 2.06/1.52 % (30740)------------------------------
% 2.06/1.52 % (30741)Termination reason: Unknown
% 2.06/1.52 % (30741)Termination phase: shuffling
% 2.06/1.52
% 2.06/1.52 % (30741)Memory used [KB]: 1023
% 2.06/1.52 % (30741)Time elapsed: 0.003 s
% 2.06/1.52 % (30741)Instructions burned: 3 (million)
% 2.06/1.52 % (30741)------------------------------
% 2.06/1.52 % (30741)------------------------------
% 2.06/1.53 % (30738)Instruction limit reached!
% 2.06/1.53 % (30738)------------------------------
% 2.06/1.53 % (30738)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.06/1.53 % (30738)Termination reason: Unknown
% 2.06/1.53 % (30738)Termination phase: shuffling
% 2.06/1.53
% 2.06/1.53 % (30738)Memory used [KB]: 1151
% 2.06/1.53 % (30738)Time elapsed: 0.004 s
% 2.06/1.53 % (30738)Instructions burned: 5 (million)
% 2.06/1.53 % (30738)------------------------------
% 2.06/1.53 % (30738)------------------------------
% 2.06/1.53 % (30743)Instruction limit reached!
% 2.06/1.53 % (30743)------------------------------
% 2.06/1.53 % (30743)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.06/1.53 % (30743)Termination reason: Unknown
% 2.06/1.53 % (30743)Termination phase: Property scanning
% 2.06/1.53
% 2.06/1.53 % (30743)Memory used [KB]: 1535
% 2.06/1.53 % (30743)Time elapsed: 0.011 s
% 2.06/1.53 % (30743)Instructions burned: 19 (million)
% 2.06/1.53 % (30743)------------------------------
% 2.06/1.53 % (30743)------------------------------
% 2.06/1.54 % (30739)Instruction limit reached!
% 2.06/1.54 % (30739)------------------------------
% 2.06/1.54 % (30739)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.06/1.54 % (30739)Termination reason: Unknown
% 2.06/1.54 % (30739)Termination phase: Saturation
% 2.06/1.54
% 2.06/1.54 % (30739)Memory used [KB]: 5756
% 2.06/1.54 % (30739)Time elapsed: 0.014 s
% 2.06/1.54 % (30739)Instructions burned: 27 (million)
% 2.06/1.54 % (30739)------------------------------
% 2.06/1.54 % (30739)------------------------------
% 2.06/1.54 % (30744)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_30638 for (2999ds/3Mi)
% 2.06/1.54 % (30745)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_30638 for (2999ds/37Mi)
% 2.06/1.54 % (30746)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_30638 for (2999ds/15Mi)
% 2.06/1.54 % (30744)Instruction limit reached!
% 2.06/1.54 % (30744)------------------------------
% 2.06/1.54 % (30744)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.06/1.54 % (30744)Termination reason: Unknown
% 2.06/1.54 % (30744)Termination phase: shuffling
% 2.06/1.54
% 2.06/1.54 % (30744)Memory used [KB]: 1023
% 2.06/1.54 % (30744)Time elapsed: 0.003 s
% 2.06/1.54 % (30744)Instructions burned: 4 (million)
% 2.06/1.54 % (30744)------------------------------
% 2.06/1.54 % (30744)------------------------------
% 2.06/1.55 % (30747)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_30638 for (2999ds/3Mi)
% 2.06/1.55 % (30746)Instruction limit reached!
% 2.06/1.55 % (30746)------------------------------
% 2.06/1.55 % (30746)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.06/1.55 % (30746)Termination reason: Unknown
% 2.06/1.55 % (30746)Termination phase: Preprocessing 3
% 2.06/1.55
% 2.06/1.55 % (30746)Memory used [KB]: 1407
% 2.06/1.55 % (30746)Time elapsed: 0.009 s
% 2.06/1.55 % (30746)Instructions burned: 16 (million)
% 2.06/1.55 % (30746)------------------------------
% 2.06/1.55 % (30746)------------------------------
% 2.06/1.55 % (30747)Instruction limit reached!
% 2.06/1.55 % (30747)------------------------------
% 2.06/1.55 % (30747)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.06/1.55 % (30747)Termination reason: Unknown
% 2.06/1.55 % (30747)Termination phase: shuffling
% 2.06/1.55
% 2.06/1.55 % (30747)Memory used [KB]: 1151
% 2.06/1.55 % (30747)Time elapsed: 0.004 s
% 2.06/1.55 % (30747)Instructions burned: 5 (million)
% 2.06/1.55 % (30747)------------------------------
% 2.06/1.55 % (30747)------------------------------
% 2.06/1.55 % (30742)First to succeed.
% 2.06/1.55 % (30748)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_30638 for (2999ds/1041Mi)
% 2.06/1.55 % (30742)Refutation found. Thanks to Tanya!
% 2.06/1.55 % SZS status Theorem for DTF2THF_30638
% 2.06/1.55 % SZS output start Proof for DTF2THF_30638
% 2.06/1.55 thf(func_def_0, type, in: $i > $i > $o).
% 2.06/1.55 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.06/1.55 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.06/1.55 thf(func_def_8, type, powerset: $i > $i).
% 2.06/1.55 thf(func_def_10, type, setunion: $i > $i).
% 2.06/1.55 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.06/1.55 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.06/1.55 thf(func_def_26, type, prop2set: $o > $i).
% 2.06/1.55 thf(func_def_36, type, nonempty: $i > $o).
% 2.06/1.55 thf(func_def_60, type, sP0: $i > $i > $o).
% 2.06/1.55 thf(func_def_61, type, sP1: $i > $o).
% 2.06/1.55 thf(func_def_62, type, sP2: $i > $i > $o).
% 2.06/1.55 thf(func_def_63, type, sP3: $i > $i > $o).
% 2.06/1.55 thf(func_def_69, type, sK9: $i > $i).
% 2.06/1.55 thf(func_def_70, type, sK10: $i > $i > $i).
% 2.06/1.55 thf(func_def_71, type, sK11: $i > $i > $i).
% 2.06/1.55 thf(func_def_72, type, sK12: $i > $i > $i).
% 2.06/1.55 thf(func_def_73, type, sK13: $i > $i > $i).
% 2.06/1.55 thf(func_def_74, type, sK14: $i > $i > $i).
% 2.06/1.55 thf(func_def_75, type, sK15: $i > $i > $i > $i).
% 2.06/1.55 thf(func_def_76, type, sK16: $i > $i).
% 2.06/1.55 thf(func_def_77, type, sK17: $i > $i).
% 2.06/1.55 thf(func_def_78, type, sK18: $i > $i).
% 2.06/1.55 thf(func_def_79, type, sK19: $i > $i).
% 2.06/1.55 thf(func_def_80, type, sK20: $i > $i > $i > $i).
% 2.06/1.55 thf(func_def_81, type, sK21: $i > $i > $i > $i).
% 2.06/1.55 thf(func_def_82, type, sK22: $i > $i > $i).
% 2.06/1.55 thf(func_def_83, type, sK23: $i > $i > $i).
% 2.06/1.55 thf(func_def_84, type, sK24: $i > $i > $i).
% 2.06/1.55 thf(func_def_86, type, sK26: $i > $i).
% 2.06/1.55 thf(func_def_87, type, sK27: $i > $i).
% 2.06/1.55 thf(func_def_88, type, sK28: $i > $i).
% 2.06/1.55 thf(func_def_100, type, sK40: $i > $i > $i).
% 2.06/1.55 thf(func_def_101, type, sK41: $i > $i > $i).
% 2.06/1.55 thf(func_def_102, type, sK42: $i > $i).
% 2.06/1.55 thf(func_def_105, type, sK45: $i > $i > $o).
% 2.06/1.55 thf(func_def_107, type, sK47: $i > $i).
% 2.06/1.55 thf(func_def_108, type, sK48: $i > $i).
% 2.06/1.55 thf(func_def_109, type, sK49: $i > ($i > $i > $o) > $i).
% 2.06/1.55 thf(func_def_110, type, sK50: $i > ($i > $i > $o) > $i).
% 2.06/1.55 thf(func_def_111, type, sK51: $i > $i > ($i > $i > $o) > $i).
% 2.06/1.55 thf(func_def_117, type, sK57: $i > $o).
% 2.06/1.55 thf(func_def_127, type, sK67: $i > $i).
% 2.06/1.55 thf(func_def_131, type, sK71: $i > $o).
% 2.06/1.55 thf(func_def_136, type, sK76: $i > $i).
% 2.06/1.55 thf(func_def_137, type, sK77: $i > $o).
% 2.06/1.55 thf(func_def_138, type, sK78: ($i > $o) > $i).
% 2.06/1.55 thf(func_def_141, type, sK81: $i > $i).
% 2.06/1.55 thf(func_def_142, type, sK82: $i > $i).
% 2.06/1.55 thf(func_def_145, type, sK85: $i > $o).
% 2.06/1.55 thf(func_def_149, type, sK89: $i > $i > $i).
% 2.06/1.55 thf(func_def_152, type, sK92: $i > $o).
% 2.06/1.55 thf(func_def_155, type, sK95: $i > $i > $i).
% 2.06/1.55 thf(func_def_158, type, sK98: $i > $o).
% 2.06/1.55 thf(func_def_159, type, sK99: $i > $i > $i).
% 2.06/1.55 thf(func_def_163, type, sK103: ($i > $o) > $i).
% 2.06/1.55 thf(func_def_164, type, sK104: $i > $o).
% 2.06/1.55 thf(func_def_165, type, sK105: $i > $i).
% 2.06/1.55 thf(func_def_167, type, ph107: !>[X0: $tType]:(X0)).
% 2.06/1.55 thf(f798,plain,(
% 2.06/1.55 $false),
% 2.06/1.55 inference(trivial_inequality_removal,[status(thm)],[f797])).
% 2.06/1.55 thf(f797,plain,(
% 2.06/1.55 ($true != $true)),
% 2.06/1.55 inference(superposition,[status(thm)],[f473,f700])).
% 2.06/1.55 thf(f700,plain,(
% 2.06/1.55 ( ! [X0 : $i] : (($true = (in @ emptyset @ (powerset @ X0)))) )),
% 2.06/1.55 inference(trivial_inequality_removal,[status(thm)],[f624])).
% 2.06/1.55 thf(f624,plain,(
% 2.06/1.55 ( ! [X0 : $i] : (($true != $true) | ($true = (in @ emptyset @ (powerset @ X0)))) )),
% 2.06/1.55 inference(definition_unfolding,[status(thm)],[f503,f455])).
% 2.06/1.55 thf(f455,plain,(
% 2.06/1.55 (emptyinPowerset = $true)),
% 2.06/1.55 inference(cnf_transformation,[status(thm)],[f253])).
% 2.06/1.55 thf(f253,plain,(
% 2.06/1.55 ($true != (in @ emptyset @ (powerset @ sK52))) & (dsetconstrI = $true) & (setadjoinIR = $true) & (setunionAx = $true) & (notinemptyset = $true) & (emptyinunitempty = $true) & (omegaIndAx = $true) & (emptyI = $true) & (setbeta = $true) & (prop2setE = $true) & (setadjoinIL = $true) & (exuE3e = $true) & (dsetconstrER = $true) & (nonemptyI = $true) & (setadjoinOr = $true) & (nonemptyI1 = $true) & (emptysetAx = $true) & (descrp = $true) & (emptyinPowerset = $true) & (setadjoinE = $true) & (noeltsimpempty = $true) & (setoftrueEq = $true) & (wellorderingAx = $true) & (powersetI = $true) & (nonemptyE1 = $true) & (omega0Ax = $true) & (dsetconstrEL = $true) & (setadjoinAx = $true) & (foundationAx = $true) & (exuE1 = $true) & (replAx = $true) & (omegaSAx = $true) & (emptysetE = $true) & (setext = $true) & (powersetAx = $true) & (setextAx = $true) & (emptysetimpfalse = $true)),
% 2.06/1.55 inference(skolemisation,[status(esa),new_symbols(skolem,[sK52])],[f137,f252])).
% 2.06/1.55 thf(f252,plain,(
% 2.06/1.55 ? [X0 : $i] : ($true != (in @ emptyset @ (powerset @ X0))) => ($true != (in @ emptyset @ (powerset @ sK52)))),
% 2.06/1.55 introduced(definition,[],[choice_axiom])).
% 2.06/1.55 thf(f137,plain,(
% 2.06/1.55 ? [X0 : $i] : ($true != (in @ emptyset @ (powerset @ X0))) & (dsetconstrI = $true) & (setadjoinIR = $true) & (setunionAx = $true) & (notinemptyset = $true) & (emptyinunitempty = $true) & (omegaIndAx = $true) & (emptyI = $true) & (setbeta = $true) & (prop2setE = $true) & (setadjoinIL = $true) & (exuE3e = $true) & (dsetconstrER = $true) & (nonemptyI = $true) & (setadjoinOr = $true) & (nonemptyI1 = $true) & (emptysetAx = $true) & (descrp = $true) & (emptyinPowerset = $true) & (setadjoinE = $true) & (noeltsimpempty = $true) & (setoftrueEq = $true) & (wellorderingAx = $true) & (powersetI = $true) & (nonemptyE1 = $true) & (omega0Ax = $true) & (dsetconstrEL = $true) & (setadjoinAx = $true) & (foundationAx = $true) & (exuE1 = $true) & (replAx = $true) & (omegaSAx = $true) & (emptysetE = $true) & (setext = $true) & (powersetAx = $true) & (setextAx = $true) & (emptysetimpfalse = $true)),
% 2.06/1.55 inference(flattening,[status(thm)],[f136])).
% 2.06/1.55 thf(f136,plain,(
% 2.06/1.55 (((((((((((((((((((((((((((((((((((? [X0 : $i] : ($true != (in @ emptyset @ (powerset @ X0))) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.06/1.55 inference(ennf_transformation,[status(thm)],[f77])).
% 2.06/1.55 thf(f77,plain,(
% 2.06/1.55 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ! [X0 : $i] : ($true = (in @ emptyset @ (powerset @ X0)))))))))))))))))))))))))))))))))))))))),
% 2.06/1.55 inference(fool_elimination,[status(thm)],[f76])).
% 2.06/1.55 thf(f76,plain,(
% 2.06/1.55 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => ! [X0 : $i] : (in @ emptyset @ (powerset @ X0))))))))))))))))))))))))))))))))))))))),
% 2.06/1.55 inference(rectify,[status(thm)],[f41])).
% 2.06/1.55 thf(f41,negated_conjecture,(
% 2.06/1.55 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => ! [X3 : $i] : (in @ emptyset @ (powerset @ X3))))))))))))))))))))))))))))))))))))))),
% 2.06/1.55 inference(negated_conjecture,[status(cth)],[f40])).
% 2.06/1.55 thf(f40,conjecture,(
% 2.06/1.55 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => ! [X3 : $i] : (in @ emptyset @ (powerset @ X3)))))))))))))))))))))))))))))))))))))),
% 2.06/1.55 file('/export/starexec/sandbox/tmp/tmp.iTElzqeOvB/DTF2THF_30638.p',emptyInPowerset)).
% 2.06/1.55 thf(f503,plain,(
% 2.06/1.55 ( ! [X0 : $i] : (($true = (in @ emptyset @ (powerset @ X0))) | (emptyinPowerset != $true)) )),
% 2.06/1.55 inference(cnf_transformation,[status(thm)],[f288])).
% 2.06/1.55 thf(f288,plain,(
% 2.06/1.55 (! [X0 : $i] : ($true = (in @ emptyset @ (powerset @ X0))) | (emptyinPowerset != $true)) & ((emptyinPowerset = $true) | ((in @ emptyset @ (powerset @ sK70)) != $true))),
% 2.06/1.55 inference(skolemisation,[status(esa),new_symbols(skolem,[sK70])],[f286,f287])).
% 2.06/1.55 thf(f287,plain,(
% 2.06/1.55 ? [X1 : $i] : ($true != (in @ emptyset @ (powerset @ X1))) => ((in @ emptyset @ (powerset @ sK70)) != $true)),
% 2.06/1.55 introduced(definition,[],[choice_axiom])).
% 2.06/1.55 thf(f286,plain,(
% 2.06/1.55 (! [X0 : $i] : ($true = (in @ emptyset @ (powerset @ X0))) | (emptyinPowerset != $true)) & ((emptyinPowerset = $true) | ? [X1 : $i] : ($true != (in @ emptyset @ (powerset @ X1))))),
% 2.06/1.55 inference(rectify,[status(thm)],[f285])).
% 2.06/1.55 thf(f285,plain,(
% 2.06/1.55 (! [X0 : $i] : ($true = (in @ emptyset @ (powerset @ X0))) | (emptyinPowerset != $true)) & ((emptyinPowerset = $true) | ? [X0 : $i] : ($true != (in @ emptyset @ (powerset @ X0))))),
% 2.06/1.55 inference(nnf_transformation,[status(thm)],[f89])).
% 2.06/1.55 thf(f89,plain,(
% 2.06/1.55 ! [X0 : $i] : ($true = (in @ emptyset @ (powerset @ X0))) <=> (emptyinPowerset = $true)),
% 2.06/1.55 inference(fool_elimination,[status(thm)],[f88])).
% 2.06/1.55 thf(f88,plain,(
% 2.06/1.55 (emptyinPowerset = ! [X0 : $i] : (in @ emptyset @ (powerset @ X0)))),
% 2.06/1.55 inference(rectify,[status(thm)],[f39])).
% 2.06/1.55 thf(f39,axiom,(
% 2.06/1.55 (emptyinPowerset = ! [X3 : $i] : (in @ emptyset @ (powerset @ X3)))),
% 2.06/1.55 file('/export/starexec/sandbox/tmp/tmp.iTElzqeOvB/DTF2THF_30638.p',emptyinPowerset)).
% 2.06/1.55 thf(f473,plain,(
% 2.06/1.55 ($true != (in @ emptyset @ (powerset @ sK52)))),
% 2.06/1.55 inference(cnf_transformation,[status(thm)],[f253])).
% 2.06/1.55 % SZS output end Proof for DTF2THF_30638
% 2.06/1.55 % (30742)------------------------------
% 2.06/1.55 % (30742)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.06/1.55 % (30742)Termination reason: Refutation
% 2.06/1.55
% 2.06/1.55 % (30742)Memory used [KB]: 6140
% 2.06/1.55 % (30742)Time elapsed: 0.030 s
% 2.06/1.55 % (30742)Instructions burned: 56 (million)
% 2.06/1.55 % (30742)------------------------------
% 2.06/1.55 % (30742)------------------------------
% 2.06/1.55 % (30736)Success in time 0.044 s
%------------------------------------------------------------------------------