%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU554^1 : TPTP v9.2.1. Bugfixed v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n027.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 09:35:14 AM UTC 2026
% Result : Theorem 2.17s 1.48s
% Output : Refutation 2.17s
% Verified :
% SZS Type : Refutation
% Derivation depth : 9
% Number of leaves : 2
% Syntax : Number of formulae : 12 ( 4 unt; 0 typ; 0 def)
% Number of atoms : 1086 ( 301 equ; 0 cnn)
% Maximal formula atoms : 72 ( 90 avg)
% Number of connectives : 532 ( 10 ~; 0 |; 215 &; 22 @)
% ( 0 <=>; 285 =>; 0 <=; 0 <~>)
% Maximal formula depth : 75 ( 44 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 104 ( 101 usr; 74 con; 0-2 aty)
% Number of variables : 14 ( 0 ^; 8 !; 6 ?; 14 :)
% 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_69,type,
set2prop: $i > $o ).
thf(func_def_102,type,
sK3: $i > $i ).
thf(func_def_103,type,
sK4: $i > ( $i > $o ) > $i ).
thf(func_def_104,type,
sK5: $i > ( $i > $o ) > $i ).
thf(func_def_105,type,
sK6: $i > $i > $i ).
thf(func_def_106,type,
sK7: ( $i > $o ) > $i ).
thf(func_def_107,type,
sK8: ( $i > $o ) > $i > $i ).
thf(func_def_108,type,
sK9: $i > $i > $i ).
thf(func_def_109,type,
sK10: ( $i > $o ) > $i ).
thf(func_def_110,type,
sK11: ( $i > $o ) > $i ).
thf(func_def_111,type,
sK12: ( $i > $o ) > $i ).
thf(func_def_112,type,
sK13: ( $i > $o ) > $i ).
thf(func_def_113,type,
sK14: ( $i > $o ) > $i > $i ).
thf(func_def_114,type,
sK15: $i > $i ).
thf(func_def_115,type,
sK16: $i > $i ).
thf(func_def_116,type,
sK17: ( $i > $o ) > $i > $i ).
thf(func_def_117,type,
sK18: $i > $i ).
thf(func_def_118,type,
sK19: ( $i > $o ) > $i ).
thf(func_def_119,type,
sK20: ( $i > $o ) > $i > $i ).
thf(func_def_120,type,
sK21: $i > $i ).
thf(f730,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f366]) ).
thf(f366,plain,
( ( setunion @ sK0 )
!= ( setunion @ sK0 ) ),
inference(definition_unfolding,[status(thm)],[f265,f264]) ).
thf(f264,plain,
sK1 = sK0,
inference(cnf_transformation,[status(thm)],[f220]) ).
thf(f220,plain,
( ( notinsingleton = $true )
& ( powersetE = $true )
& ( exu__Cong = $true )
& ( setbeta = $true )
& ( descrp = $true )
& ( powerset__Cong = $true )
& ( dsetconstrEL = $true )
& ( emptysetE = $true )
& ( setunionE = $true )
& ( emptysetAx = $true )
& ( upairsetIL = $true )
& ( noeltsimpempty = $true )
& ( prop2setI = $true )
& ( emptyE1 = $true )
& ( inCongP = $true )
& ( exuI3 = $true )
& ( dsetconstrI = $true )
& ( exuE3u = $true )
& ( foundationAx = $true )
& ( setunionAx = $true )
& ( exuE2 = $true )
& ( notdexE = $true )
& ( omega0Ax = $true )
& ( exuE1 = $true )
& ( nonemptyI = $true )
& ( quantDeMorgan1 = $true )
& ( setadjoinIL = $true )
& ( upairsetE = $true )
& ( vacuousDall = $true )
& ( setadjoinOr = $true )
& ( ( setunion @ sK1 )
!= ( setunion @ sK0 ) )
& ( sK1 = sK0 )
& ( emptyInPowerset = $true )
& ( subPowSU = $true )
& ( exuI1 = $true )
& ( setoftrueEq = $true )
& ( setadjoinIR = $true )
& ( uniqinunit = $true )
& ( setadjoinAx = $true )
& ( quantDeMorgan4 = $true )
& ( nonemptyE1 = $true )
& ( wellorderingAx = $true )
& ( emptyset__Cong = $true )
& ( nonemptyImpWitness = $true )
& ( setextAx = $true )
& ( replAx = $true )
& ( prop2set2propI = $true )
& ( omegaSAx = $true )
& ( prop2setE = $true )
& ( upairsetIR = $true )
& ( powersetI = $true )
& ( nonemptyI1 = $true )
& ( dsetconstrER = $true )
& ( notinemptyset = $true )
& ( emptyI = $true )
& ( quantDeMorgan2 = $true )
& ( exuE3e = $true )
& ( setext = $true )
& ( setadjoin__Cong = $true )
& ( singletonsswitch = $true )
& ( in__Cong = $true )
& ( exuI2 = $true )
& ( emptyinunitempty = $true )
& ( emptyinPowerset = $true )
& ( quantDeMorgan3 = $true )
& ( notdallE = $true )
& ( omegaIndAx = $true )
& ( powersetAx = $true )
& ( emptysetimpfalse = $true )
& ( eqinunit = $true )
& ( setunionI = $true )
& ( setadjoinE = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f218,f219]) ).
thf(f219,plain,
( ? [X0: $i,X1: $i] :
( ( ( setunion @ X1 )
!= ( setunion @ X0 ) )
& ( X0 = X1 ) )
=> ( ( ( setunion @ sK1 )
!= ( setunion @ sK0 ) )
& ( sK1 = sK0 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f218,plain,
( ( notinsingleton = $true )
& ( powersetE = $true )
& ( exu__Cong = $true )
& ( setbeta = $true )
& ( descrp = $true )
& ( powerset__Cong = $true )
& ( dsetconstrEL = $true )
& ( emptysetE = $true )
& ( setunionE = $true )
& ( emptysetAx = $true )
& ( upairsetIL = $true )
& ( noeltsimpempty = $true )
& ( prop2setI = $true )
& ( emptyE1 = $true )
& ( inCongP = $true )
& ( exuI3 = $true )
& ( dsetconstrI = $true )
& ( exuE3u = $true )
& ( foundationAx = $true )
& ( setunionAx = $true )
& ( exuE2 = $true )
& ( notdexE = $true )
& ( omega0Ax = $true )
& ( exuE1 = $true )
& ( nonemptyI = $true )
& ( quantDeMorgan1 = $true )
& ( setadjoinIL = $true )
& ( upairsetE = $true )
& ( vacuousDall = $true )
& ( setadjoinOr = $true )
& ? [X0: $i,X1: $i] :
( ( ( setunion @ X1 )
!= ( setunion @ X0 ) )
& ( X0 = X1 ) )
& ( emptyInPowerset = $true )
& ( subPowSU = $true )
& ( exuI1 = $true )
& ( setoftrueEq = $true )
& ( setadjoinIR = $true )
& ( uniqinunit = $true )
& ( setadjoinAx = $true )
& ( quantDeMorgan4 = $true )
& ( nonemptyE1 = $true )
& ( wellorderingAx = $true )
& ( emptyset__Cong = $true )
& ( nonemptyImpWitness = $true )
& ( setextAx = $true )
& ( replAx = $true )
& ( prop2set2propI = $true )
& ( omegaSAx = $true )
& ( prop2setE = $true )
& ( upairsetIR = $true )
& ( powersetI = $true )
& ( nonemptyI1 = $true )
& ( dsetconstrER = $true )
& ( notinemptyset = $true )
& ( emptyI = $true )
& ( quantDeMorgan2 = $true )
& ( exuE3e = $true )
& ( setext = $true )
& ( setadjoin__Cong = $true )
& ( singletonsswitch = $true )
& ( in__Cong = $true )
& ( exuI2 = $true )
& ( emptyinunitempty = $true )
& ( emptyinPowerset = $true )
& ( quantDeMorgan3 = $true )
& ( notdallE = $true )
& ( omegaIndAx = $true )
& ( powersetAx = $true )
& ( emptysetimpfalse = $true )
& ( eqinunit = $true )
& ( setunionI = $true )
& ( setadjoinE = $true ) ),
inference(flattening,[status(thm)],[f217]) ).
thf(f217,plain,
( ? [X0: $i,X1: $i] :
( ( ( setunion @ X1 )
!= ( setunion @ X0 ) )
& ( X0 = X1 ) )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( 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)],[f91]) ).
thf(f91,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 )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ! [X0: $i,X1: $i] :
( ( X0 = X1 )
=> ( ( setunion @ X1 )
= ( setunion @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f90]) ).
thf(f90,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
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ! [X0: $i,X1: $i] :
( ( X0 = X1 )
=> ( ( setunion @ X1 )
= ( setunion @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f74]) ).
thf(f74,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
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setunion @ X3 )
= ( setunion @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f73]) ).
thf(f73,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
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setunion @ X3 )
= ( setunion @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.BmuO3nUN66/DTF2THF_29642.p',setunion__Cong) ).
thf(f265,plain,
( ( setunion @ sK1 )
!= ( setunion @ sK0 ) ),
inference(cnf_transformation,[status(thm)],[f220]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU554^1 : TPTP v9.2.1. Bugfixed v4.0.0.
% 0.13/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n027.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 12:26:37 EST 2026
% 0.16/0.33 % CPUTime :
% 0.16/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.30/0.45 ---- Original DTF file ---
% 0.30/0.45 thf(spec,logic,$$dhol).
% 0.30/0.45 %------------------------------------------------------------------------------
% 0.30/0.45 % File : SEU554^1 : TPTP v9.2.1. Bugfixed v4.0.0.
% 0.30/0.45 % Domain : Set Theory
% 0.30/0.45 % Problem : A simple congruence property of setunion
% 0.30/0.45 % Version : Especial.
% 0.30/0.45 % English : (forall A:i.forall B:i.A = B -> setunion A = setunion B)
% 0.30/0.45
% 0.30/0.45 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.30/0.45 % Source : [Bro08]
% 0.30/0.45 % Names : ZFC056g [Bro08]
% 0.30/0.45
% 0.30/0.45 % Status : Theorem
% 0.30/0.45 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.15 v8.1.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.17 v5.4.0, 0.20 v5.3.0, 0.40 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.33 v4.0.1, 0.67 v4.0.0
% 0.30/0.45 % Syntax : Number of formulae : 155 ( 72 unt; 82 typ; 72 def)
% 0.30/0.45 % Number of atoms : 394 ( 108 equ; 0 cnn)
% 0.30/0.45 % Maximal formula atoms : 72 ( 5 avg)
% 0.30/0.45 % Number of connectives : 636 ( 21 ~; 5 |; 27 &; 394 @)
% 0.30/0.45 % ( 12 <=>; 177 =>; 0 <=; 0 <~>)
% 0.30/0.45 % Maximal formula depth : 74 ( 2 avg)
% 0.30/0.45 % Number of types : 2 ( 0 usr)
% 0.30/0.45 % Number of type conns : 42 ( 42 >; 0 *; 0 +; 0 <<)
% 0.30/0.45 % Number of symbols : 85 ( 82 usr; 74 con; 0-2 aty)
% 0.30/0.45 % Number of variables : 222 ( 21 ^; 173 !; 28 ?; 222 :)
% 0.30/0.45 % SPC : TH0_THM_EQU_NAR
% 0.30/0.45
% 0.30/0.45 % Comments : http://mathgate.info/detsetitem.php?id=34
% 0.30/0.45 % Bugfixes : v4.0.0 - Adding missing type delaration for in
% 0.30/0.45 %------------------------------------------------------------------------------
% 0.30/0.45 thf(in_type,type,
% 0.30/0.45 in: $i > $i > $o ).
% 0.30/0.45
% 0.30/0.45 thf(exu_type,type,
% 0.30/0.45 exu: ( $i > $o ) > $o ).
% 0.30/0.45
% 0.30/0.45 thf(exu,definition,
% 0.30/0.45 ( exu
% 0.30/0.45 = ( ^ [Xphi: $i > $o] :
% 0.30/0.45 ? [Xx: $i] :
% 0.30/0.45 ( ( Xphi @ Xx )
% 0.30/0.45 & ! [Xy: $i] :
% 0.30/0.45 ( ( Xphi @ Xy )
% 0.30/0.45 => ( Xx = Xy ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setextAx_type,type,
% 0.30/0.45 setextAx: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setextAx,definition,
% 0.30/0.45 ( setextAx
% 0.30/0.45 = ( ! [A: $i,B: $i] :
% 0.30/0.45 ( ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 <=> ( in @ Xx @ B ) )
% 0.30/0.45 => ( A = B ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(emptyset_type,type,
% 0.30/0.45 emptyset: $i ).
% 0.30/0.45
% 0.30/0.45 thf(emptysetAx_type,type,
% 0.30/0.45 emptysetAx: $o ).
% 0.30/0.45
% 0.30/0.45 thf(emptysetAx,definition,
% 0.30/0.45 ( emptysetAx
% 0.30/0.45 = ( ! [Xx: $i] :
% 0.30/0.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setadjoin_type,type,
% 0.30/0.45 setadjoin: $i > $i > $i ).
% 0.30/0.45
% 0.30/0.45 thf(setadjoinAx_type,type,
% 0.30/0.45 setadjoinAx: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setadjoinAx,definition,
% 0.30/0.45 ( setadjoinAx
% 0.30/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.45 <=> ( ( Xy = Xx )
% 0.30/0.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(powerset_type,type,
% 0.30/0.45 powerset: $i > $i ).
% 0.30/0.45
% 0.30/0.45 thf(powersetAx_type,type,
% 0.30/0.45 powersetAx: $o ).
% 0.30/0.45
% 0.30/0.45 thf(powersetAx,definition,
% 0.30/0.45 ( powersetAx
% 0.30/0.45 = ( ! [A: $i,B: $i] :
% 0.30/0.45 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.45 <=> ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ B )
% 0.30/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setunion_type,type,
% 0.30/0.45 setunion: $i > $i ).
% 0.30/0.45
% 0.30/0.45 thf(setunionAx_type,type,
% 0.30/0.45 setunionAx: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setunionAx,definition,
% 0.30/0.45 ( setunionAx
% 0.30/0.45 = ( ! [A: $i,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.45 <=> ? [B: $i] :
% 0.30/0.45 ( ( in @ Xx @ B )
% 0.30/0.45 & ( in @ B @ A ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(omega_type,type,
% 0.30/0.45 omega: $i ).
% 0.30/0.45
% 0.30/0.45 thf(omega0Ax_type,type,
% 0.30/0.45 omega0Ax: $o ).
% 0.30/0.45
% 0.30/0.45 thf(omega0Ax,definition,
% 0.30/0.45 ( omega0Ax
% 0.30/0.45 = ( in @ emptyset @ omega ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(omegaSAx_type,type,
% 0.30/0.45 omegaSAx: $o ).
% 0.30/0.45
% 0.30/0.45 thf(omegaSAx,definition,
% 0.30/0.45 ( omegaSAx
% 0.30/0.45 = ( ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ omega )
% 0.30/0.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(omegaIndAx_type,type,
% 0.30/0.45 omegaIndAx: $o ).
% 0.30/0.45
% 0.30/0.45 thf(omegaIndAx,definition,
% 0.30/0.45 ( omegaIndAx
% 0.30/0.45 = ( ! [A: $i] :
% 0.30/0.45 ( ( ( in @ emptyset @ A )
% 0.30/0.45 & ! [Xx: $i] :
% 0.30/0.45 ( ( ( in @ Xx @ omega )
% 0.30/0.45 & ( in @ Xx @ A ) )
% 0.30/0.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.30/0.45 => ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ omega )
% 0.30/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(replAx_type,type,
% 0.30/0.45 replAx: $o ).
% 0.30/0.45
% 0.30/0.45 thf(replAx,definition,
% 0.30/0.45 ( replAx
% 0.30/0.45 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.30/0.45 ( ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ( exu
% 0.30/0.45 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.30/0.45 => ? [B: $i] :
% 0.30/0.45 ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ B )
% 0.30/0.45 <=> ? [Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ A )
% 0.30/0.45 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(foundationAx_type,type,
% 0.30/0.45 foundationAx: $o ).
% 0.30/0.45
% 0.30/0.45 thf(foundationAx,definition,
% 0.30/0.45 ( foundationAx
% 0.30/0.45 = ( ! [A: $i] :
% 0.30/0.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/0.45 => ? [B: $i] :
% 0.30/0.45 ( ( in @ B @ A )
% 0.30/0.45 & ~ ? [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ B )
% 0.30/0.45 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(wellorderingAx_type,type,
% 0.30/0.45 wellorderingAx: $o ).
% 0.30/0.45
% 0.30/0.45 thf(wellorderingAx,definition,
% 0.30/0.45 ( wellorderingAx
% 0.30/0.45 = ( ! [A: $i] :
% 0.30/0.45 ? [B: $i] :
% 0.30/0.45 ( ! [C: $i] :
% 0.30/0.45 ( ( in @ C @ B )
% 0.30/0.45 => ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ C )
% 0.30/0.45 => ( in @ Xx @ A ) ) )
% 0.30/0.45 & ! [Xx: $i,Xy: $i] :
% 0.30/0.45 ( ( ( in @ Xx @ A )
% 0.30/0.45 & ( in @ Xy @ A ) )
% 0.30/0.45 => ( ! [C: $i] :
% 0.30/0.45 ( ( in @ C @ B )
% 0.30/0.45 => ( ( in @ Xx @ C )
% 0.30/0.45 <=> ( in @ Xy @ C ) ) )
% 0.30/0.45 => ( Xx = Xy ) ) )
% 0.30/0.45 & ! [C: $i,D: $i] :
% 0.30/0.45 ( ( ( in @ C @ B )
% 0.30/0.45 & ( in @ D @ B ) )
% 0.30/0.45 => ( ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ C )
% 0.30/0.45 => ( in @ Xx @ D ) )
% 0.30/0.45 | ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ D )
% 0.30/0.45 => ( in @ Xx @ C ) ) ) )
% 0.30/0.45 & ! [C: $i] :
% 0.30/0.45 ( ( ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ C )
% 0.30/0.45 => ( in @ Xx @ A ) )
% 0.30/0.45 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.30/0.45 => ? [D: $i,Xx: $i] :
% 0.30/0.45 ( ( in @ D @ B )
% 0.30/0.45 & ( in @ Xx @ C )
% 0.30/0.45 & ~ ? [Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ D )
% 0.30/0.45 & ( in @ Xy @ C ) )
% 0.30/0.45 & ! [E: $i] :
% 0.30/0.45 ( ( in @ E @ B )
% 0.30/0.45 => ( ! [Xy: $i] :
% 0.30/0.45 ( ( in @ Xy @ E )
% 0.30/0.45 => ( in @ Xy @ D ) )
% 0.30/0.45 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(descr_type,type,
% 0.30/0.45 descr: ( $i > $o ) > $i ).
% 0.30/0.45
% 0.30/0.45 thf(descrp_type,type,
% 0.30/0.45 descrp: $o ).
% 0.30/0.45
% 0.30/0.45 thf(descrp,definition,
% 0.30/0.45 ( descrp
% 0.30/0.45 = ( ! [Xphi: $i > $o] :
% 0.30/0.45 ( ( exu
% 0.30/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.45 => ( Xphi
% 0.30/0.45 @ ( descr
% 0.30/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(dsetconstr_type,type,
% 0.30/0.45 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.30/0.45
% 0.30/0.45 thf(dsetconstrI_type,type,
% 0.30/0.45 dsetconstrI: $o ).
% 0.30/0.45
% 0.30/0.45 thf(dsetconstrI,definition,
% 0.30/0.45 ( dsetconstrI
% 0.30/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ( ( Xphi @ Xx )
% 0.30/0.45 => ( in @ Xx
% 0.30/0.45 @ ( dsetconstr @ A
% 0.30/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(dsetconstrEL_type,type,
% 0.30/0.45 dsetconstrEL: $o ).
% 0.30/0.45
% 0.30/0.45 thf(dsetconstrEL,definition,
% 0.30/0.45 ( dsetconstrEL
% 0.30/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx
% 0.30/0.45 @ ( dsetconstr @ A
% 0.30/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(dsetconstrER_type,type,
% 0.30/0.45 dsetconstrER: $o ).
% 0.30/0.45
% 0.30/0.45 thf(dsetconstrER,definition,
% 0.30/0.45 ( dsetconstrER
% 0.30/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx
% 0.30/0.45 @ ( dsetconstr @ A
% 0.30/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.45 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(exuE1_type,type,
% 0.30/0.45 exuE1: $o ).
% 0.30/0.45
% 0.30/0.45 thf(exuE1,definition,
% 0.30/0.45 ( exuE1
% 0.30/0.45 = ( ! [Xphi: $i > $o] :
% 0.30/0.45 ( ( exu
% 0.30/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.45 => ? [Xx: $i] :
% 0.30/0.45 ( ( Xphi @ Xx )
% 0.30/0.45 & ! [Xy: $i] :
% 0.30/0.45 ( ( Xphi @ Xy )
% 0.30/0.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(prop2set_type,type,
% 0.30/0.45 prop2set: $o > $i ).
% 0.30/0.45
% 0.30/0.45 thf(prop2setE_type,type,
% 0.30/0.45 prop2setE: $o ).
% 0.30/0.45
% 0.30/0.45 thf(prop2setE,definition,
% 0.30/0.45 ( prop2setE
% 0.30/0.45 = ( ! [Xphi: $o,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.30/0.45 => Xphi ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(emptysetE_type,type,
% 0.30/0.45 emptysetE: $o ).
% 0.30/0.45
% 0.30/0.45 thf(emptysetE,definition,
% 0.30/0.45 ( emptysetE
% 0.30/0.45 = ( ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ emptyset )
% 0.30/0.45 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(emptysetimpfalse_type,type,
% 0.30/0.45 emptysetimpfalse: $o ).
% 0.30/0.45
% 0.30/0.45 thf(emptysetimpfalse,definition,
% 0.30/0.45 ( emptysetimpfalse
% 0.30/0.45 = ( ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ emptyset )
% 0.30/0.45 => $false ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(notinemptyset_type,type,
% 0.30/0.45 notinemptyset: $o ).
% 0.30/0.45
% 0.30/0.45 thf(notinemptyset,definition,
% 0.30/0.45 ( notinemptyset
% 0.30/0.45 = ( ! [Xx: $i] :
% 0.30/0.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(exuE3e_type,type,
% 0.30/0.45 exuE3e: $o ).
% 0.30/0.45
% 0.30/0.45 thf(exuE3e,definition,
% 0.30/0.45 ( exuE3e
% 0.30/0.45 = ( ! [Xphi: $i > $o] :
% 0.30/0.45 ( ( exu
% 0.30/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.45 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setext_type,type,
% 0.30/0.45 setext: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setext,definition,
% 0.30/0.45 ( setext
% 0.30/0.45 = ( ! [A: $i,B: $i] :
% 0.30/0.45 ( ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ( in @ Xx @ B ) )
% 0.30/0.45 => ( ! [Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ B )
% 0.30/0.45 => ( in @ Xx @ A ) )
% 0.30/0.45 => ( A = B ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(emptyI_type,type,
% 0.30/0.45 emptyI: $o ).
% 0.30/0.45
% 0.30/0.45 thf(emptyI,definition,
% 0.30/0.45 ( emptyI
% 0.30/0.45 = ( ! [A: $i] :
% 0.30/0.45 ( ! [Xx: $i] :
% 0.30/0.45 ~ ( in @ Xx @ A )
% 0.30/0.45 => ( A = emptyset ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(noeltsimpempty_type,type,
% 0.30/0.45 noeltsimpempty: $o ).
% 0.30/0.45
% 0.30/0.45 thf(noeltsimpempty,definition,
% 0.30/0.45 ( noeltsimpempty
% 0.30/0.45 = ( ! [A: $i] :
% 0.30/0.45 ( ! [Xx: $i] :
% 0.30/0.45 ~ ( in @ Xx @ A )
% 0.30/0.45 => ( A = emptyset ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(setbeta_type,type,
% 0.30/0.45 setbeta: $o ).
% 0.30/0.45
% 0.30/0.45 thf(setbeta,definition,
% 0.30/0.45 ( setbeta
% 0.30/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.45 ( ( in @ Xx @ A )
% 0.30/0.45 => ( ( in @ Xx
% 0.30/0.45 @ ( dsetconstr @ A
% 0.30/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.45 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(nonempty_type,type,
% 0.30/0.45 nonempty: $i > $o ).
% 0.30/0.45
% 0.30/0.45 thf(nonempty,definition,
% 0.30/0.45 ( nonempty
% 0.30/0.45 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(nonemptyE1_type,type,
% 0.30/0.46 nonemptyE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyE1,definition,
% 0.30/0.46 ( nonemptyE1
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( nonempty @ A )
% 0.30/0.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI_type,type,
% 0.30/0.46 nonemptyI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI,definition,
% 0.30/0.46 ( nonemptyI
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( nonempty
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI1_type,type,
% 0.30/0.46 nonemptyI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI1,definition,
% 0.30/0.46 ( nonemptyI1
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/0.46 => ( nonempty @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIL_type,type,
% 0.30/0.46 setadjoinIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIL,definition,
% 0.30/0.46 ( setadjoinIL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinunitempty_type,type,
% 0.30/0.46 emptyinunitempty: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinunitempty,definition,
% 0.30/0.46 ( emptyinunitempty
% 0.30/0.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIR_type,type,
% 0.30/0.46 setadjoinIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIR,definition,
% 0.30/0.46 ( setadjoinIR
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinE_type,type,
% 0.30/0.46 setadjoinE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinE,definition,
% 0.30/0.46 ( setadjoinE
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ( ( Xy = Xx )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => ( ( ( in @ Xy @ A )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinOr_type,type,
% 0.30/0.46 setadjoinOr: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinOr,definition,
% 0.30/0.46 ( setadjoinOr
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 => ( ( Xy = Xx )
% 0.30/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setoftrueEq_type,type,
% 0.30/0.46 setoftrueEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setoftrueEq,definition,
% 0.30/0.46 ( setoftrueEq
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : $true )
% 0.30/0.46 = A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI_type,type,
% 0.30/0.46 powersetI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI,definition,
% 0.30/0.46 ( powersetI
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) )
% 0.30/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinPowerset_type,type,
% 0.30/0.46 emptyinPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinPowerset,definition,
% 0.30/0.46 ( emptyinPowerset
% 0.30/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyInPowerset_type,type,
% 0.30/0.46 emptyInPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyInPowerset,definition,
% 0.30/0.46 ( emptyInPowerset
% 0.30/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE_type,type,
% 0.30/0.46 powersetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE,definition,
% 0.30/0.46 ( powersetE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionI_type,type,
% 0.30/0.46 setunionI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionI,definition,
% 0.30/0.46 ( setunionI
% 0.30/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( ( in @ B @ A )
% 0.30/0.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE_type,type,
% 0.30/0.46 setunionE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE,definition,
% 0.30/0.46 ( setunionE
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ! [B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( ( in @ B @ A )
% 0.30/0.46 => Xphi ) )
% 0.30/0.46 => Xphi ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subPowSU_type,type,
% 0.30/0.46 subPowSU: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subPowSU,definition,
% 0.30/0.46 ( subPowSU
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE2_type,type,
% 0.30/0.46 exuE2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE2,definition,
% 0.30/0.46 ( exuE2
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyImpWitness_type,type,
% 0.30/0.46 nonemptyImpWitness: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyImpWitness,definition,
% 0.30/0.46 ( nonemptyImpWitness
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( nonempty @ A )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & $true ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(uniqinunit_type,type,
% 0.30/0.46 uniqinunit: $o ).
% 0.30/0.46
% 0.30/0.46 thf(uniqinunit,definition,
% 0.30/0.46 ( uniqinunit
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notinsingleton_type,type,
% 0.30/0.46 notinsingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notinsingleton,definition,
% 0.30/0.46 ( notinsingleton
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx != Xy )
% 0.30/0.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqinunit_type,type,
% 0.30/0.46 eqinunit: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqinunit,definition,
% 0.30/0.46 ( eqinunit
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsswitch_type,type,
% 0.30/0.46 singletonsswitch: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsswitch,definition,
% 0.30/0.46 ( singletonsswitch
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetE_type,type,
% 0.30/0.46 upairsetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetE,definition,
% 0.30/0.46 ( upairsetE
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.30/0.46 => ( ( Xz = Xx )
% 0.30/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIL_type,type,
% 0.30/0.46 upairsetIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIL,definition,
% 0.30/0.46 ( upairsetIL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIR_type,type,
% 0.30/0.46 upairsetIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIR,definition,
% 0.30/0.46 ( upairsetIR
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyE1_type,type,
% 0.30/0.46 emptyE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyE1,definition,
% 0.30/0.46 ( emptyE1
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ( ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 = emptyset )
% 0.30/0.46 => $false ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(vacuousDall_type,type,
% 0.30/0.46 vacuousDall: $o ).
% 0.30/0.46
% 0.30/0.46 thf(vacuousDall,definition,
% 0.30/0.46 ( vacuousDall
% 0.30/0.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ emptyset )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan1_type,type,
% 0.30/0.46 quantDeMorgan1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan1,definition,
% 0.30/0.46 ( quantDeMorgan1
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan2_type,type,
% 0.30/0.46 quantDeMorgan2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan2,definition,
% 0.30/0.46 ( quantDeMorgan2
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ~ ( Xphi @ Xx ) )
% 0.30/0.46 => ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan3_type,type,
% 0.30/0.46 quantDeMorgan3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan3,definition,
% 0.30/0.46 ( quantDeMorgan3
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan4_type,type,
% 0.30/0.46 quantDeMorgan4: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan4,definition,
% 0.30/0.46 ( quantDeMorgan4
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ~ ( Xphi @ Xx ) )
% 0.30/0.46 => ~ ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setI_type,type,
% 0.30/0.46 prop2setI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setI,definition,
% 0.30/0.46 ( prop2setI
% 0.30/0.46 = ( ! [Xphi: $o] :
% 0.30/0.46 ( Xphi
% 0.30/0.46 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(set2prop_type,type,
% 0.30/0.46 set2prop: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2set2propI_type,type,
% 0.30/0.46 prop2set2propI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2set2propI,definition,
% 0.30/0.46 ( prop2set2propI
% 0.30/0.46 = ( ! [Xphi: $o] :
% 0.30/0.46 ( Xphi
% 0.30/0.46 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notdexE_type,type,
% 0.30/0.46 notdexE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notdexE,definition,
% 0.30/0.46 ( notdexE
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notdallE_type,type,
% 0.30/0.46 notdallE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notdallE,definition,
% 0.30/0.46 ( notdallE
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuI1_type,type,
% 0.30/0.46 exuI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuI1,definition,
% 0.30/0.46 ( exuI1
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 & ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuI3_type,type,
% 0.30/0.46 exuI3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuI3,definition,
% 0.30/0.46 ( exuI3
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.30/0.46 => ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuI2_type,type,
% 0.30/0.46 exuI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuI2,definition,
% 0.30/0.46 ( exuI2
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 <=> ( Xy = Xx ) )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inCongP_type,type,
% 0.30/0.46 inCongP: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inCongP,definition,
% 0.30/0.46 ( inCongP
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(in__Cong_type,type,
% 0.30/0.46 in__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(in__Cong,definition,
% 0.30/0.46 ( in__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3u_type,type,
% 0.30/0.46 exuE3u: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3u,definition,
% 0.30/0.46 ( exuE3u
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exu__Cong_type,type,
% 0.30/0.46 exu__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exu__Cong,definition,
% 0.30/0.46 ( exu__Cong
% 0.30/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.30/0.46 => ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 <=> ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyset__Cong_type,type,
% 0.30/0.46 emptyset__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyset__Cong,definition,
% 0.30/0.46 ( emptyset__Cong
% 0.30/0.46 = ( emptyset = emptyset ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoin__Cong_type,type,
% 0.30/0.46 setadjoin__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoin__Cong,definition,
% 0.30/0.46 ( setadjoin__Cong
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ! [Xz: $i,Xu: $i] :
% 0.30/0.46 ( ( Xz = Xu )
% 0.30/0.46 => ( ( setadjoin @ Xx @ Xz )
% 0.30/0.46 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powerset__Cong_type,type,
% 0.30/0.46 powerset__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powerset__Cong,definition,
% 0.30/0.46 ( powerset__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( ( powerset @ A )
% 0.30/0.46 = ( powerset @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunion__Cong,conjecture,
% 0.30/0.46 ( setextAx
% 0.30/0.46 => ( emptysetAx
% 0.30/0.46 => ( setadjoinAx
% 0.30/0.46 => ( powersetAx
% 0.30/0.46 => ( setunionAx
% 0.30/0.46 => ( omega0Ax
% 0.30/0.46 => ( omegaSAx
% 0.30/0.46 => ( omegaIndAx
% 0.30/0.46 => ( replAx
% 0.30/0.46 => ( foundationAx
% 0.30/0.46 => ( wellorderingAx
% 0.30/0.46 => ( descrp
% 0.30/0.46 => ( dsetconstrI
% 0.30/0.46 => ( dsetconstrEL
% 0.30/0.46 => ( dsetconstrER
% 0.30/0.46 => ( exuE1
% 0.30/0.46 => ( prop2setE
% 0.30/0.46 => ( emptysetE
% 0.30/0.46 => ( emptysetimpfalse
% 0.30/0.46 => ( notinemptyset
% 0.30/0.46 => ( exuE3e
% 0.30/0.46 => ( setext
% 0.30/0.46 => ( emptyI
% 0.30/0.46 => ( noeltsimpempty
% 0.30/0.46 => ( setbeta
% 0.30/0.46 => ( nonemptyE1
% 0.30/0.46 => ( nonemptyI
% 0.30/0.46 => ( nonemptyI1
% 0.30/0.46 => ( setadjoinIL
% 0.30/0.46 => ( emptyinunitempty
% 0.30/0.46 => ( setadjoinIR
% 0.30/0.46 => ( setadjoinE
% 0.30/0.46 => ( setadjoinOr
% 0.30/0.46 => ( setoftrueEq
% 0.30/0.46 => ( powersetI
% 0.30/0.46 => ( emptyinPowerset
% 0.30/0.46 => ( emptyInPowerset
% 0.30/0.46 => ( powersetE
% 0.30/0.46 => ( setunionI
% 0.30/0.46 => ( setunionE
% 0.30/0.46 => ( subPowSU
% 0.30/0.46 => ( exuE2
% 0.30/0.46 => ( nonemptyImpWitness
% 0.30/0.46 => ( uniqinunit
% 0.30/0.46 => ( notinsingleton
% 0.30/0.46 => ( eqinunit
% 0.30/0.46 => ( singletonsswitch
% 0.30/0.46 => ( upairsetE
% 0.30/0.46 => ( upairsetIL
% 0.30/0.46 => ( upairsetIR
% 0.30/0.46 => ( emptyE1
% 0.30/0.46 => ( vacuousDall
% 0.30/0.46 => ( quantDeMorgan1
% 0.30/0.46 => ( quantDeMorgan2
% 0.30/0.46 => ( quantDeMorgan3
% 0.30/0.46 => ( quantDeMorgan4
% 0.30/0.46 => ( prop2setI
% 0.30/0.46 => ( prop2set2propI
% 0.30/0.46 => ( notdexE
% 0.30/0.46 => ( notdallE
% 0.30/0.46 => ( exuI1
% 0.30/0.46 => ( exuI3
% 0.30/0.46 => ( exuI2
% 0.30/0.46 => ( inCongP
% 0.30/0.46 => ( in__Cong
% 0.30/0.46 => ( exuE3u
% 0.30/0.46 => ( exu__Cong
% 0.30/0.46 => ( emptyset__Cong
% 0.30/0.46 => ( setadjoin__Cong
% 0.30/0.46 => ( powerset__Cong
% 0.30/0.46 => ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( ( setunion @ A )
% 0.30/0.46 = ( setunion @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 ------------------------
% 2.17/1.43 ---- Embedded in THF ---
% 2.17/1.43 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.17/1.43 %%% Generated on Tue Feb 24 12:26:38 EST 2026
% 2.17/1.43 %%% using '$$dhol' embedding, version 1.3.0.
% 2.17/1.43 %%% Logic specification used:
% 2.17/1.43 %%% thf(spec, logic, $$dhol).
% 2.17/1.43
% 2.17/1.43 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.BmuO3nUN66/DTF2DTF29642.p
% See solution above
% 2.17/1.43 ------------------------
% 2.17/1.45 ---- Cleaned THF ---
% 2.17/1.45 thf(in_type,type,
% 2.17/1.45 in: $i > $i > $o ).
% 2.17/1.45
% 2.17/1.45 thf(exu_type,type,
% 2.17/1.45 exu: ( $i > $o ) > $o ).
% 2.17/1.45
% 2.17/1.45 thf(exu,definition,
% 2.17/1.45 ( exu
% 2.17/1.45 = ( ^ [Xphi: $i > $o] :
% 2.17/1.45 ? [Xx: $i] :
% 2.17/1.45 ( ( Xphi @ Xx )
% 2.17/1.45 & ! [Xy: $i] :
% 2.17/1.45 ( ( Xphi @ Xy )
% 2.17/1.45 => ( Xx = Xy ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setextAx_type,type,
% 2.17/1.45 setextAx: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setextAx,definition,
% 2.17/1.45 ( setextAx
% 2.17/1.45 = ( ! [A: $i,B: $i] :
% 2.17/1.45 ( ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 <=> ( in @ Xx @ B ) )
% 2.17/1.45 => ( A = B ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(emptyset_type,type,
% 2.17/1.45 emptyset: $i ).
% 2.17/1.45
% 2.17/1.45 thf(emptysetAx_type,type,
% 2.17/1.45 emptysetAx: $o ).
% 2.17/1.45
% 2.17/1.45 thf(emptysetAx,definition,
% 2.17/1.45 ( emptysetAx
% 2.17/1.45 = ( ! [Xx: $i] :
% 2.17/1.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoin_type,type,
% 2.17/1.45 setadjoin: $i > $i > $i ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoinAx_type,type,
% 2.17/1.45 setadjoinAx: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoinAx,definition,
% 2.17/1.45 ( setadjoinAx
% 2.17/1.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.17/1.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.17/1.45 <=> ( ( Xy = Xx )
% 2.17/1.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(powerset_type,type,
% 2.17/1.45 powerset: $i > $i ).
% 2.17/1.45
% 2.17/1.45 thf(powersetAx_type,type,
% 2.17/1.45 powersetAx: $o ).
% 2.17/1.45
% 2.17/1.45 thf(powersetAx,definition,
% 2.17/1.45 ( powersetAx
% 2.17/1.45 = ( ! [A: $i,B: $i] :
% 2.17/1.45 ( ( in @ B @ ( powerset @ A ) )
% 2.17/1.45 <=> ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ B )
% 2.17/1.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setunion_type,type,
% 2.17/1.45 setunion: $i > $i ).
% 2.17/1.45
% 2.17/1.45 thf(setunionAx_type,type,
% 2.17/1.45 setunionAx: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setunionAx,definition,
% 2.17/1.45 ( setunionAx
% 2.17/1.45 = ( ! [A: $i,Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ ( setunion @ A ) )
% 2.17/1.45 <=> ? [B: $i] :
% 2.17/1.45 ( ( in @ Xx @ B )
% 2.17/1.45 & ( in @ B @ A ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(omega_type,type,
% 2.17/1.45 omega: $i ).
% 2.17/1.45
% 2.17/1.45 thf(omega0Ax_type,type,
% 2.17/1.45 omega0Ax: $o ).
% 2.17/1.45
% 2.17/1.45 thf(omega0Ax,definition,
% 2.17/1.45 ( omega0Ax
% 2.17/1.45 = ( in @ emptyset @ omega ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(omegaSAx_type,type,
% 2.17/1.45 omegaSAx: $o ).
% 2.17/1.45
% 2.17/1.45 thf(omegaSAx,definition,
% 2.17/1.45 ( omegaSAx
% 2.17/1.45 = ( ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ omega )
% 2.17/1.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(omegaIndAx_type,type,
% 2.17/1.45 omegaIndAx: $o ).
% 2.17/1.45
% 2.17/1.45 thf(omegaIndAx,definition,
% 2.17/1.45 ( omegaIndAx
% 2.17/1.45 = ( ! [A: $i] :
% 2.17/1.45 ( ( ( in @ emptyset @ A )
% 2.17/1.45 & ! [Xx: $i] :
% 2.17/1.45 ( ( ( in @ Xx @ omega )
% 2.17/1.45 & ( in @ Xx @ A ) )
% 2.17/1.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.17/1.45 => ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ omega )
% 2.17/1.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(replAx_type,type,
% 2.17/1.45 replAx: $o ).
% 2.17/1.45
% 2.17/1.45 thf(replAx,definition,
% 2.17/1.45 ( replAx
% 2.17/1.45 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.17/1.45 ( ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ( exu
% 2.17/1.45 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.17/1.45 => ? [B: $i] :
% 2.17/1.45 ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ B )
% 2.17/1.45 <=> ? [Xy: $i] :
% 2.17/1.45 ( ( in @ Xy @ A )
% 2.17/1.45 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(foundationAx_type,type,
% 2.17/1.45 foundationAx: $o ).
% 2.17/1.45
% 2.17/1.45 thf(foundationAx,definition,
% 2.17/1.45 ( foundationAx
% 2.17/1.45 = ( ! [A: $i] :
% 2.17/1.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.17/1.45 => ? [B: $i] :
% 2.17/1.45 ( ( in @ B @ A )
% 2.17/1.45 & ~ ? [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ B )
% 2.17/1.45 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(wellorderingAx_type,type,
% 2.17/1.45 wellorderingAx: $o ).
% 2.17/1.45
% 2.17/1.45 thf(wellorderingAx,definition,
% 2.17/1.45 ( wellorderingAx
% 2.17/1.45 = ( ! [A: $i] :
% 2.17/1.45 ? [B: $i] :
% 2.17/1.45 ( ! [C: $i] :
% 2.17/1.45 ( ( in @ C @ B )
% 2.17/1.45 => ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ C )
% 2.17/1.45 => ( in @ Xx @ A ) ) )
% 2.17/1.45 & ! [Xx: $i,Xy: $i] :
% 2.17/1.45 ( ( ( in @ Xx @ A )
% 2.17/1.45 & ( in @ Xy @ A ) )
% 2.17/1.45 => ( ! [C: $i] :
% 2.17/1.45 ( ( in @ C @ B )
% 2.17/1.45 => ( ( in @ Xx @ C )
% 2.17/1.45 <=> ( in @ Xy @ C ) ) )
% 2.17/1.45 => ( Xx = Xy ) ) )
% 2.17/1.45 & ! [C: $i,D: $i] :
% 2.17/1.45 ( ( ( in @ C @ B )
% 2.17/1.45 & ( in @ D @ B ) )
% 2.17/1.45 => ( ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ C )
% 2.17/1.45 => ( in @ Xx @ D ) )
% 2.17/1.45 | ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ D )
% 2.17/1.45 => ( in @ Xx @ C ) ) ) )
% 2.17/1.45 & ! [C: $i] :
% 2.17/1.45 ( ( ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ C )
% 2.17/1.45 => ( in @ Xx @ A ) )
% 2.17/1.45 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.17/1.45 => ? [D: $i,Xx: $i] :
% 2.17/1.45 ( ( in @ D @ B )
% 2.17/1.45 & ( in @ Xx @ C )
% 2.17/1.45 & ~ ? [Xy: $i] :
% 2.17/1.45 ( ( in @ Xy @ D )
% 2.17/1.45 & ( in @ Xy @ C ) )
% 2.17/1.45 & ! [E: $i] :
% 2.17/1.45 ( ( in @ E @ B )
% 2.17/1.45 => ( ! [Xy: $i] :
% 2.17/1.45 ( ( in @ Xy @ E )
% 2.17/1.45 => ( in @ Xy @ D ) )
% 2.17/1.45 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(descr_type,type,
% 2.17/1.45 descr: ( $i > $o ) > $i ).
% 2.17/1.45
% 2.17/1.45 thf(descrp_type,type,
% 2.17/1.45 descrp: $o ).
% 2.17/1.45
% 2.17/1.45 thf(descrp,definition,
% 2.17/1.45 ( descrp
% 2.17/1.45 = ( ! [Xphi: $i > $o] :
% 2.17/1.45 ( ( exu
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.17/1.45 => ( Xphi
% 2.17/1.45 @ ( descr
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(dsetconstr_type,type,
% 2.17/1.45 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.17/1.45
% 2.17/1.45 thf(dsetconstrI_type,type,
% 2.17/1.45 dsetconstrI: $o ).
% 2.17/1.45
% 2.17/1.45 thf(dsetconstrI,definition,
% 2.17/1.45 ( dsetconstrI
% 2.17/1.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ( ( Xphi @ Xx )
% 2.17/1.45 => ( in @ Xx
% 2.17/1.45 @ ( dsetconstr @ A
% 2.17/1.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(dsetconstrEL_type,type,
% 2.17/1.45 dsetconstrEL: $o ).
% 2.17/1.45
% 2.17/1.45 thf(dsetconstrEL,definition,
% 2.17/1.45 ( dsetconstrEL
% 2.17/1.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.17/1.45 ( ( in @ Xx
% 2.17/1.45 @ ( dsetconstr @ A
% 2.17/1.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.17/1.45 => ( in @ Xx @ A ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(dsetconstrER_type,type,
% 2.17/1.45 dsetconstrER: $o ).
% 2.17/1.45
% 2.17/1.45 thf(dsetconstrER,definition,
% 2.17/1.45 ( dsetconstrER
% 2.17/1.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.17/1.45 ( ( in @ Xx
% 2.17/1.45 @ ( dsetconstr @ A
% 2.17/1.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.17/1.45 => ( Xphi @ Xx ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(exuE1_type,type,
% 2.17/1.45 exuE1: $o ).
% 2.17/1.45
% 2.17/1.45 thf(exuE1,definition,
% 2.17/1.45 ( exuE1
% 2.17/1.45 = ( ! [Xphi: $i > $o] :
% 2.17/1.45 ( ( exu
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.17/1.45 => ? [Xx: $i] :
% 2.17/1.45 ( ( Xphi @ Xx )
% 2.17/1.45 & ! [Xy: $i] :
% 2.17/1.45 ( ( Xphi @ Xy )
% 2.17/1.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(prop2set_type,type,
% 2.17/1.45 prop2set: $o > $i ).
% 2.17/1.45
% 2.17/1.45 thf(prop2setE_type,type,
% 2.17/1.45 prop2setE: $o ).
% 2.17/1.45
% 2.17/1.45 thf(prop2setE,definition,
% 2.17/1.45 ( prop2setE
% 2.17/1.45 = ( ! [Xphi: $o,Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.17/1.45 => Xphi ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(emptysetE_type,type,
% 2.17/1.45 emptysetE: $o ).
% 2.17/1.45
% 2.17/1.45 thf(emptysetE,definition,
% 2.17/1.45 ( emptysetE
% 2.17/1.45 = ( ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ emptyset )
% 2.17/1.45 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(emptysetimpfalse_type,type,
% 2.17/1.45 emptysetimpfalse: $o ).
% 2.17/1.45
% 2.17/1.45 thf(emptysetimpfalse,definition,
% 2.17/1.45 ( emptysetimpfalse
% 2.17/1.45 = ( ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ emptyset )
% 2.17/1.45 => $false ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(notinemptyset_type,type,
% 2.17/1.45 notinemptyset: $o ).
% 2.17/1.45
% 2.17/1.45 thf(notinemptyset,definition,
% 2.17/1.45 ( notinemptyset
% 2.17/1.45 = ( ! [Xx: $i] :
% 2.17/1.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(exuE3e_type,type,
% 2.17/1.45 exuE3e: $o ).
% 2.17/1.45
% 2.17/1.45 thf(exuE3e,definition,
% 2.17/1.45 ( exuE3e
% 2.17/1.45 = ( ! [Xphi: $i > $o] :
% 2.17/1.45 ( ( exu
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.17/1.45 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setext_type,type,
% 2.17/1.45 setext: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setext,definition,
% 2.17/1.45 ( setext
% 2.17/1.45 = ( ! [A: $i,B: $i] :
% 2.17/1.45 ( ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ( in @ Xx @ B ) )
% 2.17/1.45 => ( ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ B )
% 2.17/1.45 => ( in @ Xx @ A ) )
% 2.17/1.45 => ( A = B ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(emptyI_type,type,
% 2.17/1.45 emptyI: $o ).
% 2.17/1.45
% 2.17/1.45 thf(emptyI,definition,
% 2.17/1.45 ( emptyI
% 2.17/1.45 = ( ! [A: $i] :
% 2.17/1.45 ( ! [Xx: $i] :
% 2.17/1.45 ~ ( in @ Xx @ A )
% 2.17/1.45 => ( A = emptyset ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(noeltsimpempty_type,type,
% 2.17/1.45 noeltsimpempty: $o ).
% 2.17/1.45
% 2.17/1.45 thf(noeltsimpempty,definition,
% 2.17/1.45 ( noeltsimpempty
% 2.17/1.45 = ( ! [A: $i] :
% 2.17/1.45 ( ! [Xx: $i] :
% 2.17/1.45 ~ ( in @ Xx @ A )
% 2.17/1.45 => ( A = emptyset ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setbeta_type,type,
% 2.17/1.45 setbeta: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setbeta,definition,
% 2.17/1.45 ( setbeta
% 2.17/1.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ( ( in @ Xx
% 2.17/1.45 @ ( dsetconstr @ A
% 2.17/1.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.17/1.45 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(nonempty_type,type,
% 2.17/1.45 nonempty: $i > $o ).
% 2.17/1.45
% 2.17/1.45 thf(nonempty,definition,
% 2.17/1.45 ( nonempty
% 2.17/1.45 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(nonemptyE1_type,type,
% 2.17/1.45 nonemptyE1: $o ).
% 2.17/1.45
% 2.17/1.45 thf(nonemptyE1,definition,
% 2.17/1.45 ( nonemptyE1
% 2.17/1.45 = ( ! [A: $i] :
% 2.17/1.45 ( ( nonempty @ A )
% 2.17/1.45 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(nonemptyI_type,type,
% 2.17/1.45 nonemptyI: $o ).
% 2.17/1.45
% 2.17/1.45 thf(nonemptyI,definition,
% 2.17/1.45 ( nonemptyI
% 2.17/1.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ( ( Xphi @ Xx )
% 2.17/1.45 => ( nonempty
% 2.17/1.45 @ ( dsetconstr @ A
% 2.17/1.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(nonemptyI1_type,type,
% 2.17/1.45 nonemptyI1: $o ).
% 2.17/1.45
% 2.17/1.45 thf(nonemptyI1,definition,
% 2.17/1.45 ( nonemptyI1
% 2.17/1.45 = ( ! [A: $i] :
% 2.17/1.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.17/1.45 => ( nonempty @ A ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoinIL_type,type,
% 2.17/1.45 setadjoinIL: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoinIL,definition,
% 2.17/1.45 ( setadjoinIL
% 2.17/1.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(emptyinunitempty_type,type,
% 2.17/1.45 emptyinunitempty: $o ).
% 2.17/1.45
% 2.17/1.45 thf(emptyinunitempty,definition,
% 2.17/1.45 ( emptyinunitempty
% 2.17/1.45 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoinIR_type,type,
% 2.17/1.45 setadjoinIR: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoinIR,definition,
% 2.17/1.45 ( setadjoinIR
% 2.17/1.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.17/1.45 ( ( in @ Xy @ A )
% 2.17/1.45 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoinE_type,type,
% 2.17/1.45 setadjoinE: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoinE,definition,
% 2.17/1.45 ( setadjoinE
% 2.17/1.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.17/1.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.17/1.45 => ! [Xphi: $o] :
% 2.17/1.45 ( ( ( Xy = Xx )
% 2.17/1.45 => Xphi )
% 2.17/1.45 => ( ( ( in @ Xy @ A )
% 2.17/1.45 => Xphi )
% 2.17/1.45 => Xphi ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoinOr_type,type,
% 2.17/1.45 setadjoinOr: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoinOr,definition,
% 2.17/1.45 ( setadjoinOr
% 2.17/1.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.17/1.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.17/1.45 => ( ( Xy = Xx )
% 2.17/1.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setoftrueEq_type,type,
% 2.17/1.45 setoftrueEq: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setoftrueEq,definition,
% 2.17/1.45 ( setoftrueEq
% 2.17/1.45 = ( ! [A: $i] :
% 2.17/1.45 ( ( dsetconstr @ A
% 2.17/1.45 @ ^ [Xx: $i] : $true )
% 2.17/1.45 = A ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(powersetI_type,type,
% 2.17/1.45 powersetI: $o ).
% 2.17/1.45
% 2.17/1.45 thf(powersetI,definition,
% 2.17/1.45 ( powersetI
% 2.17/1.45 = ( ! [A: $i,B: $i] :
% 2.17/1.45 ( ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ B )
% 2.17/1.45 => ( in @ Xx @ A ) )
% 2.17/1.45 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(emptyinPowerset_type,type,
% 2.17/1.45 emptyinPowerset: $o ).
% 2.17/1.45
% 2.17/1.45 thf(emptyinPowerset,definition,
% 2.17/1.45 ( emptyinPowerset
% 2.17/1.45 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(emptyInPowerset_type,type,
% 2.17/1.45 emptyInPowerset: $o ).
% 2.17/1.45
% 2.17/1.45 thf(emptyInPowerset,definition,
% 2.17/1.45 ( emptyInPowerset
% 2.17/1.45 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(powersetE_type,type,
% 2.17/1.45 powersetE: $o ).
% 2.17/1.45
% 2.17/1.45 thf(powersetE,definition,
% 2.17/1.45 ( powersetE
% 2.17/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.17/1.45 ( ( in @ B @ ( powerset @ A ) )
% 2.17/1.45 => ( ( in @ Xx @ B )
% 2.17/1.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setunionI_type,type,
% 2.17/1.45 setunionI: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setunionI,definition,
% 2.17/1.45 ( setunionI
% 2.17/1.45 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.17/1.45 ( ( in @ Xx @ B )
% 2.17/1.45 => ( ( in @ B @ A )
% 2.17/1.45 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setunionE_type,type,
% 2.17/1.45 setunionE: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setunionE,definition,
% 2.17/1.45 ( setunionE
% 2.17/1.45 = ( ! [A: $i,Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ ( setunion @ A ) )
% 2.17/1.45 => ! [Xphi: $o] :
% 2.17/1.45 ( ! [B: $i] :
% 2.17/1.45 ( ( in @ Xx @ B )
% 2.17/1.45 => ( ( in @ B @ A )
% 2.17/1.45 => Xphi ) )
% 2.17/1.45 => Xphi ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(subPowSU_type,type,
% 2.17/1.45 subPowSU: $o ).
% 2.17/1.45
% 2.17/1.45 thf(subPowSU,definition,
% 2.17/1.45 ( subPowSU
% 2.17/1.45 = ( ! [A: $i,Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(exuE2_type,type,
% 2.17/1.45 exuE2: $o ).
% 2.17/1.45
% 2.17/1.45 thf(exuE2,definition,
% 2.17/1.45 ( exuE2
% 2.17/1.45 = ( ! [Xphi: $i > $o] :
% 2.17/1.45 ( ( exu
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.17/1.45 => ? [Xx: $i] :
% 2.17/1.45 ! [Xy: $i] :
% 2.17/1.45 ( ( Xphi @ Xy )
% 2.17/1.45 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(nonemptyImpWitness_type,type,
% 2.17/1.45 nonemptyImpWitness: $o ).
% 2.17/1.45
% 2.17/1.45 thf(nonemptyImpWitness,definition,
% 2.17/1.45 ( nonemptyImpWitness
% 2.17/1.45 = ( ! [A: $i] :
% 2.17/1.45 ( ( nonempty @ A )
% 2.17/1.45 => ? [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 & $true ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(uniqinunit_type,type,
% 2.17/1.45 uniqinunit: $o ).
% 2.17/1.45
% 2.17/1.45 thf(uniqinunit,definition,
% 2.17/1.45 ( uniqinunit
% 2.17/1.45 = ( ! [Xx: $i,Xy: $i] :
% 2.17/1.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.17/1.45 => ( Xx = Xy ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(notinsingleton_type,type,
% 2.17/1.45 notinsingleton: $o ).
% 2.17/1.45
% 2.17/1.45 thf(notinsingleton,definition,
% 2.17/1.45 ( notinsingleton
% 2.17/1.45 = ( ! [Xx: $i,Xy: $i] :
% 2.17/1.45 ( ( Xx != Xy )
% 2.17/1.45 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(eqinunit_type,type,
% 2.17/1.45 eqinunit: $o ).
% 2.17/1.45
% 2.17/1.45 thf(eqinunit,definition,
% 2.17/1.45 ( eqinunit
% 2.17/1.45 = ( ! [Xx: $i,Xy: $i] :
% 2.17/1.45 ( ( Xx = Xy )
% 2.17/1.45 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(singletonsswitch_type,type,
% 2.17/1.45 singletonsswitch: $o ).
% 2.17/1.45
% 2.17/1.45 thf(singletonsswitch,definition,
% 2.17/1.45 ( singletonsswitch
% 2.17/1.45 = ( ! [Xx: $i,Xy: $i] :
% 2.17/1.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.17/1.45 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(upairsetE_type,type,
% 2.17/1.45 upairsetE: $o ).
% 2.17/1.45
% 2.17/1.45 thf(upairsetE,definition,
% 2.17/1.45 ( upairsetE
% 2.17/1.45 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.17/1.45 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.17/1.45 => ( ( Xz = Xx )
% 2.17/1.45 | ( Xz = Xy ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(upairsetIL_type,type,
% 2.17/1.45 upairsetIL: $o ).
% 2.17/1.45
% 2.17/1.45 thf(upairsetIL,definition,
% 2.17/1.45 ( upairsetIL
% 2.17/1.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(upairsetIR_type,type,
% 2.17/1.45 upairsetIR: $o ).
% 2.17/1.45
% 2.17/1.45 thf(upairsetIR,definition,
% 2.17/1.45 ( upairsetIR
% 2.17/1.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(emptyE1_type,type,
% 2.17/1.45 emptyE1: $o ).
% 2.17/1.45
% 2.17/1.45 thf(emptyE1,definition,
% 2.17/1.45 ( emptyE1
% 2.17/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 2.17/1.45 ( ? [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 & ( Xphi @ Xx ) )
% 2.17/1.45 => ( ( ( dsetconstr @ A
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.17/1.45 = emptyset )
% 2.17/1.45 => $false ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(vacuousDall_type,type,
% 2.17/1.45 vacuousDall: $o ).
% 2.17/1.45
% 2.17/1.45 thf(vacuousDall,definition,
% 2.17/1.45 ( vacuousDall
% 2.17/1.45 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ emptyset )
% 2.17/1.45 => ( Xphi @ Xx ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(quantDeMorgan1_type,type,
% 2.17/1.45 quantDeMorgan1: $o ).
% 2.17/1.45
% 2.17/1.45 thf(quantDeMorgan1,definition,
% 2.17/1.45 ( quantDeMorgan1
% 2.17/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 2.17/1.45 ( ~ ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ( Xphi @ Xx ) )
% 2.17/1.45 => ? [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(quantDeMorgan2_type,type,
% 2.17/1.45 quantDeMorgan2: $o ).
% 2.17/1.45
% 2.17/1.45 thf(quantDeMorgan2,definition,
% 2.17/1.45 ( quantDeMorgan2
% 2.17/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 2.17/1.45 ( ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ~ ( Xphi @ Xx ) )
% 2.17/1.45 => ~ ? [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(quantDeMorgan3_type,type,
% 2.17/1.45 quantDeMorgan3: $o ).
% 2.17/1.45
% 2.17/1.45 thf(quantDeMorgan3,definition,
% 2.17/1.45 ( quantDeMorgan3
% 2.17/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 2.17/1.45 ( ~ ? [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 & ( Xphi @ Xx ) )
% 2.17/1.45 => ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(quantDeMorgan4_type,type,
% 2.17/1.45 quantDeMorgan4: $o ).
% 2.17/1.45
% 2.17/1.45 thf(quantDeMorgan4,definition,
% 2.17/1.45 ( quantDeMorgan4
% 2.17/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 2.17/1.45 ( ? [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 & ~ ( Xphi @ Xx ) )
% 2.17/1.45 => ~ ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(prop2setI_type,type,
% 2.17/1.45 prop2setI: $o ).
% 2.17/1.45
% 2.17/1.45 thf(prop2setI,definition,
% 2.17/1.45 ( prop2setI
% 2.17/1.45 = ( ! [Xphi: $o] :
% 2.17/1.45 ( Xphi
% 2.17/1.45 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(set2prop_type,type,
% 2.17/1.45 set2prop: $i > $o ).
% 2.17/1.45
% 2.17/1.45 thf(prop2set2propI_type,type,
% 2.17/1.45 prop2set2propI: $o ).
% 2.17/1.45
% 2.17/1.45 thf(prop2set2propI,definition,
% 2.17/1.45 ( prop2set2propI
% 2.17/1.45 = ( ! [Xphi: $o] :
% 2.17/1.45 ( Xphi
% 2.17/1.45 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(notdexE_type,type,
% 2.17/1.45 notdexE: $o ).
% 2.17/1.45
% 2.17/1.45 thf(notdexE,definition,
% 2.17/1.45 ( notdexE
% 2.17/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 2.17/1.45 ( ~ ? [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 & ( Xphi @ Xx ) )
% 2.17/1.45 => ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(notdallE_type,type,
% 2.17/1.45 notdallE: $o ).
% 2.17/1.45
% 2.17/1.45 thf(notdallE,definition,
% 2.17/1.45 ( notdallE
% 2.17/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 2.17/1.45 ( ~ ! [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 => ( Xphi @ Xx ) )
% 2.17/1.45 => ? [Xx: $i] :
% 2.17/1.45 ( ( in @ Xx @ A )
% 2.17/1.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(exuI1_type,type,
% 2.17/1.45 exuI1: $o ).
% 2.17/1.45
% 2.17/1.45 thf(exuI1,definition,
% 2.17/1.45 ( exuI1
% 2.17/1.45 = ( ! [Xphi: $i > $o] :
% 2.17/1.45 ( ? [Xx: $i] :
% 2.17/1.45 ( ( Xphi @ Xx )
% 2.17/1.45 & ! [Xy: $i] :
% 2.17/1.45 ( ( Xphi @ Xy )
% 2.17/1.45 => ( Xx = Xy ) ) )
% 2.17/1.45 => ( exu
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(exuI3_type,type,
% 2.17/1.45 exuI3: $o ).
% 2.17/1.45
% 2.17/1.45 thf(exuI3,definition,
% 2.17/1.45 ( exuI3
% 2.17/1.45 = ( ! [Xphi: $i > $o] :
% 2.17/1.45 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.17/1.45 => ( ! [Xx: $i,Xy: $i] :
% 2.17/1.45 ( ( Xphi @ Xx )
% 2.17/1.45 => ( ( Xphi @ Xy )
% 2.17/1.45 => ( Xx = Xy ) ) )
% 2.17/1.45 => ( exu
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(exuI2_type,type,
% 2.17/1.45 exuI2: $o ).
% 2.17/1.45
% 2.17/1.45 thf(exuI2,definition,
% 2.17/1.45 ( exuI2
% 2.17/1.45 = ( ! [Xphi: $i > $o] :
% 2.17/1.45 ( ? [Xx: $i] :
% 2.17/1.45 ! [Xy: $i] :
% 2.17/1.45 ( ( Xphi @ Xy )
% 2.17/1.45 <=> ( Xy = Xx ) )
% 2.17/1.45 => ( exu
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(inCongP_type,type,
% 2.17/1.45 inCongP: $o ).
% 2.17/1.45
% 2.17/1.45 thf(inCongP,definition,
% 2.17/1.45 ( inCongP
% 2.17/1.45 = ( ! [A: $i,B: $i] :
% 2.17/1.45 ( ( A = B )
% 2.17/1.45 => ! [Xx: $i,Xy: $i] :
% 2.17/1.45 ( ( Xx = Xy )
% 2.17/1.45 => ( ( in @ Xx @ A )
% 2.17/1.45 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(in__Cong_type,type,
% 2.17/1.45 in__Cong: $o ).
% 2.17/1.45
% 2.17/1.45 thf(in__Cong,definition,
% 2.17/1.45 ( in__Cong
% 2.17/1.45 = ( ! [A: $i,B: $i] :
% 2.17/1.45 ( ( A = B )
% 2.17/1.45 => ! [Xx: $i,Xy: $i] :
% 2.17/1.45 ( ( Xx = Xy )
% 2.17/1.45 => ( ( in @ Xx @ A )
% 2.17/1.45 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(exuE3u_type,type,
% 2.17/1.45 exuE3u: $o ).
% 2.17/1.45
% 2.17/1.45 thf(exuE3u,definition,
% 2.17/1.45 ( exuE3u
% 2.17/1.45 = ( ! [Xphi: $i > $o] :
% 2.17/1.45 ( ( exu
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.17/1.45 => ! [Xx: $i,Xy: $i] :
% 2.17/1.45 ( ( Xphi @ Xx )
% 2.17/1.45 => ( ( Xphi @ Xy )
% 2.17/1.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(exu__Cong_type,type,
% 2.17/1.45 exu__Cong: $o ).
% 2.17/1.45
% 2.17/1.45 thf(exu__Cong,definition,
% 2.17/1.45 ( exu__Cong
% 2.17/1.45 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.17/1.45 ( ! [Xx: $i,Xy: $i] :
% 2.17/1.45 ( ( Xx = Xy )
% 2.17/1.45 => ( ( Xphi @ Xx )
% 2.17/1.45 <=> ( Xpsi @ Xy ) ) )
% 2.17/1.45 => ( ( exu
% 2.17/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.17/1.45 <=> ( exu
% 2.17/1.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(emptyset__Cong_type,type,
% 2.17/1.45 emptyset__Cong: $o ).
% 2.17/1.45
% 2.17/1.45 thf(emptyset__Cong,definition,
% 2.17/1.45 ( emptyset__Cong
% 2.17/1.45 = ( emptyset = emptyset ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoin__Cong_type,type,
% 2.17/1.45 setadjoin__Cong: $o ).
% 2.17/1.45
% 2.17/1.45 thf(setadjoin__Cong,definition,
% 2.17/1.45 ( setadjoin__Cong
% 2.17/1.45 = ( ! [Xx: $i,Xy: $i] :
% 2.17/1.45 ( ( Xx = Xy )
% 2.17/1.45 => ! [Xz: $i,Xu: $i] :
% 2.17/1.45 ( ( Xz = Xu )
% 2.17/1.45 => ( ( setadjoin @ Xx @ Xz )
% 2.17/1.45 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(powerset__Cong_type,type,
% 2.17/1.45 powerset__Cong: $o ).
% 2.17/1.45
% 2.17/1.45 thf(powerset__Cong,definition,
% 2.17/1.45 ( powerset__Cong
% 2.17/1.45 = ( ! [A: $i,B: $i] :
% 2.17/1.45 ( ( A = B )
% 2.17/1.45 => ( ( powerset @ A )
% 2.17/1.45 = ( powerset @ B ) ) ) ) ) ).
% 2.17/1.45
% 2.17/1.45 thf(setunion__Cong,conjecture,
% 2.17/1.45 ( setextAx
% 2.17/1.45 => ( emptysetAx
% 2.17/1.45 => ( setadjoinAx
% 2.17/1.45 => ( powersetAx
% 2.17/1.45 => ( setunionAx
% 2.17/1.45 => ( omega0Ax
% 2.17/1.45 => ( omegaSAx
% 2.17/1.45 => ( omegaIndAx
% 2.17/1.45 => ( replAx
% 2.17/1.45 => ( foundationAx
% 2.17/1.45 => ( wellorderingAx
% 2.17/1.45 => ( descrp
% 2.17/1.45 => ( dsetconstrI
% 2.17/1.45 => ( dsetconstrEL
% 2.17/1.45 => ( dsetconstrER
% 2.17/1.45 => ( exuE1
% 2.17/1.45 => ( prop2setE
% 2.17/1.45 => ( emptysetE
% 2.17/1.45 => ( emptysetimpfalse
% 2.17/1.45 => ( notinemptyset
% 2.17/1.45 => ( exuE3e
% 2.17/1.45 => ( setext
% 2.17/1.45 => ( emptyI
% 2.17/1.45 => ( noeltsimpempty
% 2.17/1.45 => ( setbeta
% 2.17/1.45 => ( nonemptyE1
% 2.17/1.45 => ( nonemptyI
% 2.17/1.45 => ( nonemptyI1
% 2.17/1.45 => ( setadjoinIL
% 2.17/1.45 => ( emptyinunitempty
% 2.17/1.45 => ( setadjoinIR
% 2.17/1.45 => ( setadjoinE
% 2.17/1.45 => ( setadjoinOr
% 2.17/1.45 => ( setoftrueEq
% 2.17/1.45 => ( powersetI
% 2.17/1.45 => ( emptyinPowerset
% 2.17/1.45 => ( emptyInPowerset
% 2.17/1.45 => ( powersetE
% 2.17/1.45 => ( setunionI
% 2.17/1.45 => ( setunionE
% 2.17/1.45 => ( subPowSU
% 2.17/1.45 => ( exuE2
% 2.17/1.45 => ( nonemptyImpWitness
% 2.17/1.45 => ( uniqinunit
% 2.17/1.45 => ( notinsingleton
% 2.17/1.45 => ( eqinunit
% 2.17/1.45 => ( singletonsswitch
% 2.17/1.45 => ( upairsetE
% 2.17/1.45 => ( upairsetIL
% 2.17/1.45 => ( upairsetIR
% 2.17/1.45 => ( emptyE1
% 2.17/1.45 => ( vacuousDall
% 2.17/1.45 => ( quantDeMorgan1
% 2.17/1.45 => ( quantDeMorgan2
% 2.17/1.45 => ( quantDeMorgan3
% 2.17/1.45 => ( quantDeMorgan4
% 2.17/1.45 => ( prop2setI
% 2.17/1.45 => ( prop2set2propI
% 2.17/1.45 => ( notdexE
% 2.17/1.45 => ( notdallE
% 2.17/1.45 => ( exuI1
% 2.17/1.45 => ( exuI3
% 2.17/1.45 => ( exuI2
% 2.17/1.45 => ( inCongP
% 2.17/1.45 => ( in__Cong
% 2.17/1.45 => ( exuE3u
% 2.17/1.45 => ( exu__Cong
% 2.17/1.45 => ( emptyset__Cong
% 2.17/1.45 => ( setadjoin__Cong
% 2.17/1.45 => ( powerset__Cong
% 2.17/1.45 => ! [A: $i,B: $i] :
% 2.17/1.45 ( ( A = B )
% 2.17/1.45 => ( ( setunion @ A )
% 2.17/1.45 = ( setunion @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.17/1.45 ------------------------
% 2.17/1.46 % (29966)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_29642 for (2999ds/183Mi)
% 2.17/1.46 % (29967)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_29642 for (2999ds/4Mi)
% 2.17/1.46 % (29968)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_29642 for (2999ds/27Mi)
% 2.17/1.47 % (29967)Instruction limit reached!
% 2.17/1.47 % (29967)------------------------------
% 2.17/1.47 % (29967)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.47 % (29967)Termination reason: Unknown
% 2.17/1.47 % (29967)Termination phase: shuffling
% 2.17/1.47
% 2.17/1.47 % (29967)Memory used [KB]: 1151
% 2.17/1.47 % (29967)Time elapsed: 0.003 s
% 2.17/1.47 % (29967)Instructions burned: 6 (million)
% 2.17/1.47 % (29967)------------------------------
% 2.17/1.47 % (29967)------------------------------
% 2.17/1.47 % (29969)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_29642 for (2999ds/2Mi)
% 2.17/1.47 % (29970)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_29642 for (2999ds/2Mi)
% 2.17/1.47 % (29971)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_29642 for (2999ds/275Mi)
% 2.17/1.47 % (29972)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_29642 for (2999ds/18Mi)
% 2.17/1.47 % (29968)Instruction limit reached!
% 2.17/1.47 % (29968)------------------------------
% 2.17/1.47 % (29968)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.47 % (29968)Termination reason: Unknown
% 2.17/1.47 % (29968)Termination phase: Property scanning
% 2.17/1.47
% 2.17/1.47 % (29968)Memory used [KB]: 1407
% 2.17/1.47 % (29968)Time elapsed: 0.009 s
% 2.17/1.47 % (29968)Instructions burned: 28 (million)
% 2.17/1.47 % (29968)------------------------------
% 2.17/1.47 % (29968)------------------------------
% 2.17/1.47 % (29969)Instruction limit reached!
% 2.17/1.47 % (29969)------------------------------
% 2.17/1.47 % (29969)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.47 % (29969)Termination reason: Unknown
% 2.17/1.47 % (29969)Termination phase: shuffling
% 2.17/1.47
% 2.17/1.47 % (29969)Memory used [KB]: 1023
% 2.17/1.47 % (29969)Time elapsed: 0.003 s
% 2.17/1.47 % (29969)Instructions burned: 2 (million)
% 2.17/1.47 % (29969)------------------------------
% 2.17/1.47 % (29969)------------------------------
% 2.17/1.47 % (29970)Instruction limit reached!
% 2.17/1.47 % (29970)------------------------------
% 2.17/1.47 % (29970)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.47 % (29970)Termination reason: Unknown
% 2.17/1.47 % (29970)Termination phase: shuffling
% 2.17/1.47
% 2.17/1.47 % (29970)Memory used [KB]: 1023
% 2.17/1.47 % (29970)Time elapsed: 0.003 s
% 2.17/1.47 % (29970)Instructions burned: 2 (million)
% 2.17/1.47 % (29970)------------------------------
% 2.17/1.47 % (29970)------------------------------
% 2.17/1.48 % (29973)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_29642 for (2999ds/3Mi)
% 2.17/1.48 % (29966)First to succeed.
% 2.17/1.48 % (29973)Instruction limit reached!
% 2.17/1.48 % (29973)------------------------------
% 2.17/1.48 % (29973)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.48 % (29973)Termination reason: Unknown
% 2.17/1.48 % (29973)Termination phase: shuffling
% 2.17/1.48
% 2.17/1.48 % (29973)Memory used [KB]: 1151
% 2.17/1.48 % (29973)Time elapsed: 0.003 s
% 2.17/1.48 % (29973)Instructions burned: 5 (million)
% 2.17/1.48 % (29973)------------------------------
% 2.17/1.48 % (29973)------------------------------
% 2.17/1.48 % (29966)Refutation found. Thanks to Tanya!
% 2.17/1.48 % SZS status Theorem for DTF2THF_29642
% 2.17/1.48 % SZS output start Proof for DTF2THF_29642
% 2.17/1.48 thf(func_def_0, type, in: $i > $i > $o).
% 2.17/1.48 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.17/1.48 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.17/1.48 thf(func_def_8, type, powerset: $i > $i).
% 2.17/1.48 thf(func_def_10, type, setunion: $i > $i).
% 2.17/1.48 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.17/1.48 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.17/1.48 thf(func_def_26, type, prop2set: $o > $i).
% 2.17/1.48 thf(func_def_36, type, nonempty: $i > $o).
% 2.17/1.48 thf(func_def_69, type, set2prop: $i > $o).
% 2.17/1.48 thf(func_def_102, type, sK3: $i > $i).
% 2.17/1.48 thf(func_def_103, type, sK4: $i > ($i > $o) > $i).
% 2.17/1.48 thf(func_def_104, type, sK5: $i > ($i > $o) > $i).
% 2.17/1.48 thf(func_def_105, type, sK6: $i > $i > $i).
% 2.17/1.48 thf(func_def_106, type, sK7: ($i > $o) > $i).
% 2.17/1.48 thf(func_def_107, type, sK8: ($i > $o) > $i > $i).
% 2.17/1.48 thf(func_def_108, type, sK9: $i > $i > $i).
% 2.17/1.48 thf(func_def_109, type, sK10: ($i > $o) > $i).
% 2.17/1.48 thf(func_def_110, type, sK11: ($i > $o) > $i).
% 2.17/1.48 thf(func_def_111, type, sK12: ($i > $o) > $i).
% 2.17/1.48 thf(func_def_112, type, sK13: ($i > $o) > $i).
% 2.17/1.48 thf(func_def_113, type, sK14: ($i > $o) > $i > $i).
% 2.17/1.48 thf(func_def_114, type, sK15: $i > $i).
% 2.17/1.48 thf(func_def_115, type, sK16: $i > $i).
% 2.17/1.48 thf(func_def_116, type, sK17: ($i > $o) > $i > $i).
% 2.17/1.48 thf(func_def_117, type, sK18: $i > $i).
% 2.17/1.48 thf(func_def_118, type, sK19: ($i > $o) > $i).
% 2.17/1.48 thf(func_def_119, type, sK20: ($i > $o) > $i > $i).
% 2.17/1.48 thf(func_def_120, type, sK21: $i > $i).
% 2.17/1.48 thf(f730,plain,(
% 2.17/1.48 $false),
% 2.17/1.48 inference(trivial_inequality_removal,[status(thm)],[f366])).
% 2.17/1.48 thf(f366,plain,(
% 2.17/1.48 ((setunion @ sK0) != (setunion @ sK0))),
% 2.17/1.48 inference(definition_unfolding,[status(thm)],[f265,f264])).
% 2.17/1.48 thf(f264,plain,(
% 2.17/1.48 (sK1 = sK0)),
% 2.17/1.48 inference(cnf_transformation,[status(thm)],[f220])).
% 2.17/1.48 thf(f220,plain,(
% 2.17/1.48 (notinsingleton = $true) & (powersetE = $true) & (exu__Cong = $true) & (setbeta = $true) & (descrp = $true) & (powerset__Cong = $true) & (dsetconstrEL = $true) & (emptysetE = $true) & (setunionE = $true) & (emptysetAx = $true) & (upairsetIL = $true) & (noeltsimpempty = $true) & (prop2setI = $true) & (emptyE1 = $true) & (inCongP = $true) & (exuI3 = $true) & (dsetconstrI = $true) & (exuE3u = $true) & (foundationAx = $true) & (setunionAx = $true) & (exuE2 = $true) & (notdexE = $true) & (omega0Ax = $true) & (exuE1 = $true) & (nonemptyI = $true) & (quantDeMorgan1 = $true) & (setadjoinIL = $true) & (upairsetE = $true) & (vacuousDall = $true) & (setadjoinOr = $true) & (((setunion @ sK1) != (setunion @ sK0)) & (sK1 = sK0)) & (emptyInPowerset = $true) & (subPowSU = $true) & (exuI1 = $true) & (setoftrueEq = $true) & (setadjoinIR = $true) & (uniqinunit = $true) & (setadjoinAx = $true) & (quantDeMorgan4 = $true) & (nonemptyE1 = $true) & (wellorderingAx = $true) & (emptyset__Cong = $true) & (nonemptyImpWitness = $true) & (setextAx = $true) & (replAx = $true) & (prop2set2propI = $true) & (omegaSAx = $true) & (prop2setE = $true) & (upairsetIR = $true) & (powersetI = $true) & (nonemptyI1 = $true) & (dsetconstrER = $true) & (notinemptyset = $true) & (emptyI = $true) & (quantDeMorgan2 = $true) & (exuE3e = $true) & (setext = $true) & (setadjoin__Cong = $true) & (singletonsswitch = $true) & (in__Cong = $true) & (exuI2 = $true) & (emptyinunitempty = $true) & (emptyinPowerset = $true) & (quantDeMorgan3 = $true) & (notdallE = $true) & (omegaIndAx = $true) & (powersetAx = $true) & (emptysetimpfalse = $true) & (eqinunit = $true) & (setunionI = $true) & (setadjoinE = $true)),
% 2.17/1.48 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f218,f219])).
% 2.17/1.48 thf(f219,plain,(
% 2.17/1.48 ? [X0 : $i,X1 : $i] : (((setunion @ X1) != (setunion @ X0)) & (X0 = X1)) => (((setunion @ sK1) != (setunion @ sK0)) & (sK1 = sK0))),
% 2.17/1.48 introduced(definition,[],[choice_axiom])).
% 2.17/1.48 thf(f218,plain,(
% 2.17/1.48 (notinsingleton = $true) & (powersetE = $true) & (exu__Cong = $true) & (setbeta = $true) & (descrp = $true) & (powerset__Cong = $true) & (dsetconstrEL = $true) & (emptysetE = $true) & (setunionE = $true) & (emptysetAx = $true) & (upairsetIL = $true) & (noeltsimpempty = $true) & (prop2setI = $true) & (emptyE1 = $true) & (inCongP = $true) & (exuI3 = $true) & (dsetconstrI = $true) & (exuE3u = $true) & (foundationAx = $true) & (setunionAx = $true) & (exuE2 = $true) & (notdexE = $true) & (omega0Ax = $true) & (exuE1 = $true) & (nonemptyI = $true) & (quantDeMorgan1 = $true) & (setadjoinIL = $true) & (upairsetE = $true) & (vacuousDall = $true) & (setadjoinOr = $true) & ? [X0 : $i,X1 : $i] : (((setunion @ X1) != (setunion @ X0)) & (X0 = X1)) & (emptyInPowerset = $true) & (subPowSU = $true) & (exuI1 = $true) & (setoftrueEq = $true) & (setadjoinIR = $true) & (uniqinunit = $true) & (setadjoinAx = $true) & (quantDeMorgan4 = $true) & (nonemptyE1 = $true) & (wellorderingAx = $true) & (emptyset__Cong = $true) & (nonemptyImpWitness = $true) & (setextAx = $true) & (replAx = $true) & (prop2set2propI = $true) & (omegaSAx = $true) & (prop2setE = $true) & (upairsetIR = $true) & (powersetI = $true) & (nonemptyI1 = $true) & (dsetconstrER = $true) & (notinemptyset = $true) & (emptyI = $true) & (quantDeMorgan2 = $true) & (exuE3e = $true) & (setext = $true) & (setadjoin__Cong = $true) & (singletonsswitch = $true) & (in__Cong = $true) & (exuI2 = $true) & (emptyinunitempty = $true) & (emptyinPowerset = $true) & (quantDeMorgan3 = $true) & (notdallE = $true) & (omegaIndAx = $true) & (powersetAx = $true) & (emptysetimpfalse = $true) & (eqinunit = $true) & (setunionI = $true) & (setadjoinE = $true)),
% 2.17/1.48 inference(flattening,[status(thm)],[f217])).
% 2.17/1.48 thf(f217,plain,(
% 2.17/1.48 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (((setunion @ X1) != (setunion @ X0)) & (X0 = X1)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (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)),
% 2.17/1.48 inference(ennf_transformation,[status(thm)],[f91])).
% 2.17/1.48 thf(f91,plain,(
% 2.17/1.48 ~((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) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ! [X0 : $i,X1 : $i] : ((X0 = X1) => ((setunion @ X1) = (setunion @ X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.17/1.48 inference(fool_elimination,[status(thm)],[f90])).
% 2.17/1.48 thf(f90,plain,(
% 2.17/1.48 ~(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 => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => ! [X0 : $i,X1 : $i] : ((X0 = X1) => ((setunion @ X1) = (setunion @ X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.17/1.48 inference(rectify,[status(thm)],[f74])).
% 2.17/1.48 thf(f74,negated_conjecture,(
% 2.17/1.48 ~(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 => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => ! [X3 : $i,X4 : $i] : ((X3 = X4) => ((setunion @ X3) = (setunion @ X4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.17/1.48 inference(negated_conjecture,[status(cth)],[f73])).
% 2.17/1.48 thf(f73,conjecture,(
% 2.17/1.48 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 => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => ! [X3 : $i,X4 : $i] : ((X3 = X4) => ((setunion @ X3) = (setunion @ X4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.17/1.48 file('/export/starexec/sandbox/tmp/tmp.BmuO3nUN66/DTF2THF_29642.p',setunion__Cong)).
% 2.17/1.48 thf(f265,plain,(
% 2.17/1.48 ((setunion @ sK1) != (setunion @ sK0))),
% 2.17/1.48 inference(cnf_transformation,[status(thm)],[f220])).
% 2.17/1.48 % SZS output end Proof for DTF2THF_29642
% 2.17/1.48 % (29966)------------------------------
% 2.17/1.48 % (29966)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.17/1.48 % (29966)Termination reason: Refutation
% 2.17/1.48
% 2.17/1.48 % (29966)Memory used [KB]: 6140
% 2.17/1.48 % (29966)Time elapsed: 0.014 s
% 2.17/1.48 % (29966)Instructions burned: 45 (million)
% 2.17/1.48 % (29966)------------------------------
% 2.17/1.48 % (29966)------------------------------
% 2.17/1.48 % (29965)Success in time 0.024 s
%------------------------------------------------------------------------------