%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU576^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n023.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 09:35:22 AM UTC 2026
% Result : Theorem 2.49s 1.68s
% Output : Refutation 2.49s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 4
% Syntax : Number of formulae : 25 ( 7 unt; 0 typ; 0 def)
% Number of atoms : 1581 ( 424 equ; 0 cnn)
% Maximal formula atoms : 94 ( 63 avg)
% Number of connectives : 842 ( 29 ~; 16 |; 289 &; 129 @)
% ( 2 <=>; 377 =>; 0 <=; 0 <~>)
% Maximal formula depth : 99 ( 30 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 206 ( 203 usr; 98 con; 0-4 aty)
% Number of variables : 40 ( 0 ^; 28 !; 12 ?; 40 :)
% 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_88,type,
subset: $i > $i > $o ).
thf(func_def_89,type,
disjoint: $i > $i > $o ).
thf(func_def_90,type,
setsmeet: $i > $i > $o ).
thf(func_def_120,type,
sP0: $i > $i > $o ).
thf(func_def_121,type,
sP1: $i > $o ).
thf(func_def_122,type,
sP2: $i > $i > $o ).
thf(func_def_123,type,
sP3: $i > $i > $o ).
thf(func_def_125,type,
sK5: $i > $o ).
thf(func_def_129,type,
sK9: $i > $i ).
thf(func_def_132,type,
sK12: ( $i > $o ) > $i ).
thf(func_def_133,type,
sK13: $i > $o ).
thf(func_def_134,type,
sK14: $i > $i ).
thf(func_def_135,type,
sK15: $i > $o ).
thf(func_def_140,type,
sK20: $i > $i > $i ).
thf(func_def_149,type,
sK29: $i > $o ).
thf(func_def_151,type,
sK31: ( $i > $o ) > $i > $i ).
thf(func_def_155,type,
sK35: $i > $o ).
thf(func_def_157,type,
sK37: ( $i > $o ) > $i > $i ).
thf(func_def_158,type,
sK38: $i > $i > $i ).
thf(func_def_159,type,
sK39: $i > $i > $i ).
thf(func_def_160,type,
sK40: $i > $i > $i ).
thf(func_def_161,type,
sK41: $i > $i > $i ).
thf(func_def_162,type,
sK42: $i > $i > $i ).
thf(func_def_163,type,
sK43: $i > $i > $i > $i ).
thf(func_def_164,type,
sK44: $i > $i ).
thf(func_def_165,type,
sK45: $i > $i ).
thf(func_def_166,type,
sK46: $i > $i ).
thf(func_def_167,type,
sK47: $i > $i ).
thf(func_def_168,type,
sK48: $i > $i > $i > $i ).
thf(func_def_169,type,
sK49: $i > $i > $i > $i ).
thf(func_def_170,type,
sK50: $i > $i > $i ).
thf(func_def_171,type,
sK51: $i > $i > $i ).
thf(func_def_172,type,
sK52: $i > $i > $i ).
thf(func_def_174,type,
sK54: $i > $i ).
thf(func_def_175,type,
sK55: $i > $i ).
thf(func_def_176,type,
sK56: $i > $i ).
thf(func_def_180,type,
sK60: $i > $i ).
thf(func_def_185,type,
sK65: $i > $o ).
thf(func_def_195,type,
sK75: $i > $i ).
thf(func_def_209,type,
sK89: $i > $i > $i ).
thf(func_def_213,type,
sK93: $i > $i ).
thf(func_def_215,type,
sK95: $i > $i ).
thf(func_def_220,type,
sK100: $i > $i > $i ).
thf(func_def_232,type,
sK112: $i > $i ).
thf(func_def_235,type,
sK115: $i > $o ).
thf(func_def_236,type,
sK116: ( $i > $o ) > $i ).
thf(func_def_243,type,
sK123: $i > $o ).
thf(func_def_246,type,
sK126: $i > $o ).
thf(func_def_250,type,
sK130: $i > $i > $i ).
thf(func_def_254,type,
sK134: $o > $i > $i > $i ).
thf(func_def_262,type,
sK142: $i > $o ).
thf(func_def_263,type,
sK143: $i > $o ).
thf(func_def_264,type,
sK144: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_265,type,
sK145: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_267,type,
sK147: $i > $o ).
thf(func_def_269,type,
sK149: ( $i > $o ) > $i > $i ).
thf(func_def_272,type,
sK152: $i > $i > $i ).
thf(func_def_276,type,
sK156: $i > $o ).
thf(func_def_279,type,
sK159: $i > $o ).
thf(func_def_285,type,
sK165: $i > $i > $o ).
thf(func_def_287,type,
sK167: $i > $i ).
thf(func_def_288,type,
sK168: $i > $i ).
thf(func_def_289,type,
sK169: $i > ( $i > $i > $o ) > $i ).
thf(func_def_290,type,
sK170: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_291,type,
sK171: $i > ( $i > $i > $o ) > $i ).
thf(func_def_292,type,
sK172: $i > $o ).
thf(func_def_294,type,
sK174: $i > ( $i > $o ) > $i ).
thf(func_def_295,type,
sK175: $i > $o ).
thf(func_def_308,type,
sK188: $i > $i > $i ).
thf(func_def_309,type,
sK189: $i > $i > $i ).
thf(func_def_311,type,
sK191: $i > $o ).
thf(func_def_313,type,
sK193: ( $i > $o ) > $i ).
thf(func_def_314,type,
sK194: $i > $o ).
thf(func_def_315,type,
sK195: $i > $i ).
thf(func_def_319,type,
sK199: $i > $i ).
thf(func_def_322,type,
sK202: $i > $i > $i ).
thf(func_def_323,type,
sK203: $i > $o ).
thf(func_def_324,type,
sK204: $i > $o ).
thf(func_def_325,type,
sK205: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_326,type,
sK206: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_327,type,
sK207: $i > ( $i > $o ) > $i ).
thf(func_def_328,type,
sK208: $i > $o ).
thf(func_def_336,type,
sK216: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_337,type,
sK217: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_340,type,
sK220: $i > $o ).
thf(func_def_341,type,
sK221: $i > $o ).
thf(func_def_346,type,
sK226: ( $i > $o ) > $i > $i ).
thf(func_def_348,type,
sK228: $i > $o ).
thf(func_def_350,type,
sK230: $i > ( $i > $o ) > $i ).
thf(func_def_351,type,
sK231: $i > $o ).
thf(func_def_356,type,
sK236: $i > $o ).
thf(func_def_358,type,
sK238: ( $i > $o ) > $i ).
thf(func_def_359,type,
sK239: ( $i > $o ) > $i ).
thf(func_def_361,type,
sK241: $i > $o ).
thf(func_def_363,type,
sK243: ( $i > $o ) > $i > $i ).
thf(func_def_375,type,
sK255: $i > $o ).
thf(func_def_381,type,
ph261:
!>[X0: $tType] : X0 ).
thf(f2336,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f2334,f985]) ).
thf(f985,plain,
( ( subset @ sK58 @ sK57 )
= $true ),
inference(cnf_transformation,[status(thm)],[f502]) ).
thf(f502,plain,
( ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( setadjoinSub2 = $true )
& ( emptysetsubset = $true )
& ( upairsetIL = $true )
& ( setunionE = $true )
& ( omegaSAx = $true )
& ( in__Cong = $true )
& ( subsetE2 = $true )
& ( exuE2 = $true )
& ( omegaIndAx = $true )
& ( notequalI1 = $true )
& ( exu__Cong = $true )
& ( emptyinunitempty = $true )
& ( notequalI2 = $true )
& ( eqinunit = $true )
& ( exuE1 = $true )
& ( notinsingleton = $true )
& ( notdallE = $true )
& ( setadjoin__Cong = $true )
& ( exuE3u = $true )
& ( wellorderingAx = $true )
& ( setunion__Cong = $true )
& ( uniqinunit = $true )
& ( quantDeMorgan2 = $true )
& ( eqimpsubset2 = $true )
& ( exuEu = $true )
& ( eqimpsubset1 = $true )
& ( subsetI2 = $true )
& ( notinemptyset = $true )
& ( foundationAx = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( subsetI1 = $true )
& ( nonemptyI1 = $true )
& ( setbeta = $true )
& ( emptysetimpfalse = $true )
& ( powerset__Cong = $true )
& ( quantDeMorgan1 = $true )
& ( subsetRefl = $true )
& ( exuE3e = $true )
& ( omega0Ax = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( powersetI = $true )
& ( dsetconstrEL = $true )
& ( upairsetE = $true )
& ( ( subset @ sK58 @ sK57 )
= $true )
& ( ( in @ sK58 @ ( powerset @ sK57 ) )
!= $true )
& ( setadjoinIL = $true )
& ( powersetAx = $true )
& ( setextsub = $true )
& ( singletonsswitch = $true )
& ( subsetemptysetimpeq = $true )
& ( emptyinPowerset = $true )
& ( nonemptyImpWitness = $true )
& ( replAx = $true )
& ( inCongP = $true )
& ( emptyInPowerset = $true )
& ( setadjoinAx = $true )
& ( subset2powerset = $true )
& ( upairsetIR = $true )
& ( setext = $true )
& ( notdexE = $true )
& ( exuI2 = $true )
& ( dsetconstrER = $true )
& ( exuI1 = $true )
& ( exuI3 = $true )
& ( setoftrueEq = $true )
& ( prop2set2propI = $true )
& ( notsubsetI = $true )
& ( setunionAx = $true )
& ( dsetconstrI = $true )
& ( nonemptyE1 = $true )
& ( emptysetE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( prop2setE = $true )
& ( setadjoinIR = $true )
& ( emptysetAx = $true )
& ( emptyset__Cong = $true )
& ( quantDeMorgan3 = $true )
& ( nonemptyI = $true )
& ( subsetE = $true )
& ( emptyI = $true )
& ( omega__Cong = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( descr__Cong = $true )
& ( dsetconstr__Cong = $true )
& ( setextAx = $true )
& ( descrp = $true )
& ( subPowSU = $true )
& ( noeltsimpempty = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK57,sK58])],[f329,f501]) ).
thf(f501,plain,
( ? [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
= $true )
& ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) ) )
=> ( ( ( subset @ sK58 @ sK57 )
= $true )
& ( ( in @ sK58 @ ( powerset @ sK57 ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f329,plain,
( ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( setadjoinSub2 = $true )
& ( emptysetsubset = $true )
& ( upairsetIL = $true )
& ( setunionE = $true )
& ( omegaSAx = $true )
& ( in__Cong = $true )
& ( subsetE2 = $true )
& ( exuE2 = $true )
& ( omegaIndAx = $true )
& ( notequalI1 = $true )
& ( exu__Cong = $true )
& ( emptyinunitempty = $true )
& ( notequalI2 = $true )
& ( eqinunit = $true )
& ( exuE1 = $true )
& ( notinsingleton = $true )
& ( notdallE = $true )
& ( setadjoin__Cong = $true )
& ( exuE3u = $true )
& ( wellorderingAx = $true )
& ( setunion__Cong = $true )
& ( uniqinunit = $true )
& ( quantDeMorgan2 = $true )
& ( eqimpsubset2 = $true )
& ( exuEu = $true )
& ( eqimpsubset1 = $true )
& ( subsetI2 = $true )
& ( notinemptyset = $true )
& ( foundationAx = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( subsetI1 = $true )
& ( nonemptyI1 = $true )
& ( setbeta = $true )
& ( emptysetimpfalse = $true )
& ( powerset__Cong = $true )
& ( quantDeMorgan1 = $true )
& ( subsetRefl = $true )
& ( exuE3e = $true )
& ( omega0Ax = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( powersetI = $true )
& ( dsetconstrEL = $true )
& ( upairsetE = $true )
& ? [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
= $true )
& ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) ) )
& ( setadjoinIL = $true )
& ( powersetAx = $true )
& ( setextsub = $true )
& ( singletonsswitch = $true )
& ( subsetemptysetimpeq = $true )
& ( emptyinPowerset = $true )
& ( nonemptyImpWitness = $true )
& ( replAx = $true )
& ( inCongP = $true )
& ( emptyInPowerset = $true )
& ( setadjoinAx = $true )
& ( subset2powerset = $true )
& ( upairsetIR = $true )
& ( setext = $true )
& ( notdexE = $true )
& ( exuI2 = $true )
& ( dsetconstrER = $true )
& ( exuI1 = $true )
& ( exuI3 = $true )
& ( setoftrueEq = $true )
& ( prop2set2propI = $true )
& ( notsubsetI = $true )
& ( setunionAx = $true )
& ( dsetconstrI = $true )
& ( nonemptyE1 = $true )
& ( emptysetE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( prop2setE = $true )
& ( setadjoinIR = $true )
& ( emptysetAx = $true )
& ( emptyset__Cong = $true )
& ( quantDeMorgan3 = $true )
& ( nonemptyI = $true )
& ( subsetE = $true )
& ( emptyI = $true )
& ( omega__Cong = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( descr__Cong = $true )
& ( dsetconstr__Cong = $true )
& ( setextAx = $true )
& ( descrp = $true )
& ( subPowSU = $true )
& ( noeltsimpempty = $true ) ),
inference(flattening,[status(thm)],[f328]) ).
thf(f328,plain,
( ? [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
= $true )
& ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) ) )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( 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)],[f277]) ).
thf(f277,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 )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ! [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
= $true )
=> ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f276]) ).
thf(f276,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
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ! [X0: $i,X1: $i] :
( ( subset @ X1 @ X0 )
=> ( in @ X1 @ ( powerset @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f96]) ).
thf(f96,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
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ! [X3: $i,X4: $i] :
( ( subset @ X4 @ X3 )
=> ( in @ X4 @ ( powerset @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f95]) ).
thf(f95,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
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ! [X3: $i,X4: $i] :
( ( subset @ X4 @ X3 )
=> ( in @ X4 @ ( powerset @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.ZrZtUQmNfW/DTF2THF_22788.p',powersetI1) ).
thf(f2334,plain,
( ( subset @ sK58 @ sK57 )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f2325]) ).
thf(f2325,plain,
( ( $true != $true )
| ( ( subset @ sK58 @ sK57 )
!= $true ) ),
inference(superposition,[status(thm)],[f984,f1794]) ).
thf(f1794,plain,
! [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
| ( ( subset @ X1 @ X0 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f1349]) ).
thf(f1349,plain,
! [X0: $i,X1: $i] :
( ( $true != $true )
| ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
| ( ( subset @ X1 @ X0 )
!= $true ) ),
inference(definition_unfolding,[status(thm)],[f877,f972]) ).
thf(f972,plain,
subset2powerset = $true,
inference(cnf_transformation,[status(thm)],[f502]) ).
thf(f877,plain,
! [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
!= $true )
| ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
| ( subset2powerset != $true ) ),
inference(cnf_transformation,[status(thm)],[f439]) ).
thf(f439,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
!= $true )
| ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) ) )
| ( subset2powerset != $true ) )
& ( ( subset2powerset = $true )
| ( ( $true
= ( subset @ sK19 @ sK18 ) )
& ( ( in @ sK19 @ ( powerset @ sK18 ) )
!= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK18,sK19])],[f437,f438]) ).
thf(f438,plain,
( ? [X2: $i,X3: $i] :
( ( $true
= ( subset @ X3 @ X2 ) )
& ( $true
!= ( in @ X3 @ ( powerset @ X2 ) ) ) )
=> ( ( $true
= ( subset @ sK19 @ sK18 ) )
& ( ( in @ sK19 @ ( powerset @ sK18 ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f437,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
!= $true )
| ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) ) )
| ( subset2powerset != $true ) )
& ( ( subset2powerset = $true )
| ? [X2: $i,X3: $i] :
( ( $true
= ( subset @ X3 @ X2 ) )
& ( $true
!= ( in @ X3 @ ( powerset @ X2 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f436]) ).
thf(f436,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
!= $true )
| ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) ) )
| ( subset2powerset != $true ) )
& ( ( subset2powerset = $true )
| ? [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
= $true )
& ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f319]) ).
thf(f319,plain,
( ! [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
!= $true )
| ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) ) )
<=> ( subset2powerset = $true ) ),
inference(ennf_transformation,[status(thm)],[f160]) ).
thf(f160,plain,
( ( subset2powerset = $true )
<=> ! [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
= $true )
=> ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f159]) ).
thf(f159,plain,
! [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
=> ( in @ X1 @ ( powerset @ X0 ) ) )
= subset2powerset ),
inference(rectify,[status(thm)],[f92]) ).
thf(f92,axiom,
! [X4: $i,X3: $i] :
( ( ( subset @ X3 @ X4 )
=> ( in @ X3 @ ( powerset @ X4 ) ) )
= subset2powerset ),
file('/export/starexec/sandbox/tmp/tmp.ZrZtUQmNfW/DTF2THF_22788.p',subset2powerset) ).
thf(f984,plain,
( ( in @ sK58 @ ( powerset @ sK57 ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f502]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : SEU576^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.09 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.11/0.29 % Computer : n023.cluster.edu
% 0.11/0.29 % Model : x86_64 x86_64
% 0.11/0.29 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.29 % Memory : 8042.1875MB
% 0.11/0.29 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.29 % CPULimit : 300
% 0.11/0.29 % WCLimit : 300
% 0.11/0.29 % DateTime : Tue Feb 24 12:28:04 EST 2026
% 0.11/0.29 % CPUTime :
% 0.11/0.29 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.38 ---- Original DTF file ---
% 0.19/0.38 thf(spec,logic,$$dhol).
% 0.19/0.38 %------------------------------------------------------------------------------
% 0.19/0.38 % File : SEU576^1 : TPTP v9.2.1. Released v3.7.0.
% 0.19/0.38 % Domain : Set Theory
% 0.19/0.38 % Problem : Preliminary Notions - Relations on Sets - Subsets
% 0.19/0.38 % Version : Especial.
% 0.19/0.38 % English : (! A:i.! B:i.subset B A -> in B (powerset A))
% 0.19/0.38
% 0.19/0.38 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.19/0.38 % Source : [Bro08]
% 0.19/0.38 % Names : ZFC078g [Bro08]
% 0.19/0.38
% 0.19/0.38 % Status : Theorem
% 0.19/0.38 % Rating : 0.11 v9.1.0, 0.25 v9.0.0, 0.30 v8.2.0, 0.23 v8.1.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.67 v3.7.0
% 0.19/0.38 % Syntax : Number of formulae : 202 ( 94 unt; 107 typ; 94 def)
% 0.19/0.38 % Number of atoms : 513 ( 143 equ; 0 cnn)
% 0.19/0.38 % Maximal formula atoms : 94 ( 5 avg)
% 0.19/0.38 % Number of connectives : 805 ( 29 ~; 5 |; 27 &; 496 @)
% 0.19/0.38 % ( 14 <=>; 234 =>; 0 <=; 0 <~>)
% 0.19/0.38 % Maximal formula depth : 98 ( 2 avg)
% 0.19/0.38 % Number of types : 2 ( 0 usr)
% 0.19/0.38 % Number of type conns : 53 ( 53 >; 0 *; 0 +; 0 <<)
% 0.19/0.38 % Number of symbols : 110 ( 107 usr; 96 con; 0-2 aty)
% 0.19/0.38 % Number of variables : 283 ( 28 ^; 227 !; 28 ?; 283 :)
% 0.19/0.38 % SPC : TH0_THM_EQU_NAR
% 0.19/0.38
% 0.19/0.38 % Comments : http://mathgate.info/detsetitem.php?id=155
% 0.19/0.38 %------------------------------------------------------------------------------
% 0.19/0.38 thf(in_type,type,
% 0.19/0.38 in: $i > $i > $o ).
% 0.19/0.38
% 0.19/0.38 thf(exu_type,type,
% 0.19/0.38 exu: ( $i > $o ) > $o ).
% 0.19/0.38
% 0.19/0.38 thf(exu,definition,
% 0.19/0.38 ( exu
% 0.19/0.38 = ( ^ [Xphi: $i > $o] :
% 0.19/0.38 ? [Xx: $i] :
% 0.19/0.38 ( ( Xphi @ Xx )
% 0.19/0.38 & ! [Xy: $i] :
% 0.19/0.38 ( ( Xphi @ Xy )
% 0.19/0.38 => ( Xx = Xy ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setextAx_type,type,
% 0.19/0.38 setextAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setextAx,definition,
% 0.19/0.38 ( setextAx
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 <=> ( in @ Xx @ B ) )
% 0.19/0.38 => ( A = B ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(emptyset_type,type,
% 0.19/0.38 emptyset: $i ).
% 0.19/0.38
% 0.19/0.38 thf(emptysetAx_type,type,
% 0.19/0.38 emptysetAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(emptysetAx,definition,
% 0.19/0.38 ( emptysetAx
% 0.19/0.38 = ( ! [Xx: $i] :
% 0.19/0.38 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoin_type,type,
% 0.19/0.38 setadjoin: $i > $i > $i ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinAx_type,type,
% 0.19/0.38 setadjoinAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinAx,definition,
% 0.19/0.38 ( setadjoinAx
% 0.19/0.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.38 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.38 <=> ( ( Xy = Xx )
% 0.19/0.38 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(powerset_type,type,
% 0.19/0.38 powerset: $i > $i ).
% 0.19/0.38
% 0.19/0.38 thf(powersetAx_type,type,
% 0.19/0.38 powersetAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(powersetAx,definition,
% 0.19/0.38 ( powersetAx
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.38 <=> ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ B )
% 0.19/0.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setunion_type,type,
% 0.19/0.38 setunion: $i > $i ).
% 0.19/0.38
% 0.19/0.38 thf(setunionAx_type,type,
% 0.19/0.38 setunionAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setunionAx,definition,
% 0.19/0.38 ( setunionAx
% 0.19/0.38 = ( ! [A: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.38 <=> ? [B: $i] :
% 0.19/0.38 ( ( in @ Xx @ B )
% 0.19/0.38 & ( in @ B @ A ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(omega_type,type,
% 0.19/0.38 omega: $i ).
% 0.19/0.38
% 0.19/0.38 thf(omega0Ax_type,type,
% 0.19/0.38 omega0Ax: $o ).
% 0.19/0.38
% 0.19/0.38 thf(omega0Ax,definition,
% 0.19/0.38 ( omega0Ax
% 0.19/0.38 = ( in @ emptyset @ omega ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(omegaSAx_type,type,
% 0.19/0.38 omegaSAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(omegaSAx,definition,
% 0.19/0.38 ( omegaSAx
% 0.19/0.38 = ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ omega )
% 0.19/0.38 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(omegaIndAx_type,type,
% 0.19/0.38 omegaIndAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(omegaIndAx,definition,
% 0.19/0.38 ( omegaIndAx
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ( ( ( in @ emptyset @ A )
% 0.19/0.38 & ! [Xx: $i] :
% 0.19/0.38 ( ( ( in @ Xx @ omega )
% 0.19/0.38 & ( in @ Xx @ A ) )
% 0.19/0.38 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.19/0.38 => ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ omega )
% 0.19/0.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(replAx_type,type,
% 0.19/0.38 replAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(replAx,definition,
% 0.19/0.38 ( replAx
% 0.19/0.38 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.19/0.38 ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( exu
% 0.19/0.38 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.19/0.38 => ? [B: $i] :
% 0.19/0.38 ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ B )
% 0.19/0.38 <=> ? [Xy: $i] :
% 0.19/0.38 ( ( in @ Xy @ A )
% 0.19/0.38 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(foundationAx_type,type,
% 0.19/0.38 foundationAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(foundationAx,definition,
% 0.19/0.38 ( foundationAx
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.38 => ? [B: $i] :
% 0.19/0.38 ( ( in @ B @ A )
% 0.19/0.38 & ~ ? [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ B )
% 0.19/0.38 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(wellorderingAx_type,type,
% 0.19/0.38 wellorderingAx: $o ).
% 0.19/0.38
% 0.19/0.38 thf(wellorderingAx,definition,
% 0.19/0.38 ( wellorderingAx
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ? [B: $i] :
% 0.19/0.38 ( ! [C: $i] :
% 0.19/0.38 ( ( in @ C @ B )
% 0.19/0.38 => ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ C )
% 0.19/0.38 => ( in @ Xx @ A ) ) )
% 0.19/0.38 & ! [Xx: $i,Xy: $i] :
% 0.19/0.38 ( ( ( in @ Xx @ A )
% 0.19/0.38 & ( in @ Xy @ A ) )
% 0.19/0.38 => ( ! [C: $i] :
% 0.19/0.38 ( ( in @ C @ B )
% 0.19/0.38 => ( ( in @ Xx @ C )
% 0.19/0.38 <=> ( in @ Xy @ C ) ) )
% 0.19/0.38 => ( Xx = Xy ) ) )
% 0.19/0.38 & ! [C: $i,D: $i] :
% 0.19/0.38 ( ( ( in @ C @ B )
% 0.19/0.38 & ( in @ D @ B ) )
% 0.19/0.38 => ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ C )
% 0.19/0.38 => ( in @ Xx @ D ) )
% 0.19/0.38 | ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ D )
% 0.19/0.38 => ( in @ Xx @ C ) ) ) )
% 0.19/0.38 & ! [C: $i] :
% 0.19/0.38 ( ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ C )
% 0.19/0.38 => ( in @ Xx @ A ) )
% 0.19/0.38 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.19/0.38 => ? [D: $i,Xx: $i] :
% 0.19/0.38 ( ( in @ D @ B )
% 0.19/0.38 & ( in @ Xx @ C )
% 0.19/0.38 & ~ ? [Xy: $i] :
% 0.19/0.38 ( ( in @ Xy @ D )
% 0.19/0.38 & ( in @ Xy @ C ) )
% 0.19/0.38 & ! [E: $i] :
% 0.19/0.38 ( ( in @ E @ B )
% 0.19/0.38 => ( ! [Xy: $i] :
% 0.19/0.38 ( ( in @ Xy @ E )
% 0.19/0.38 => ( in @ Xy @ D ) )
% 0.19/0.38 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(descr_type,type,
% 0.19/0.38 descr: ( $i > $o ) > $i ).
% 0.19/0.38
% 0.19/0.38 thf(descrp_type,type,
% 0.19/0.38 descrp: $o ).
% 0.19/0.38
% 0.19/0.38 thf(descrp,definition,
% 0.19/0.38 ( descrp
% 0.19/0.38 = ( ! [Xphi: $i > $o] :
% 0.19/0.38 ( ( exu
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.38 => ( Xphi
% 0.19/0.38 @ ( descr
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(dsetconstr_type,type,
% 0.19/0.38 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.19/0.38
% 0.19/0.38 thf(dsetconstrI_type,type,
% 0.19/0.38 dsetconstrI: $o ).
% 0.19/0.38
% 0.19/0.38 thf(dsetconstrI,definition,
% 0.19/0.38 ( dsetconstrI
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( ( Xphi @ Xx )
% 0.19/0.38 => ( in @ Xx
% 0.19/0.38 @ ( dsetconstr @ A
% 0.19/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(dsetconstrEL_type,type,
% 0.19/0.38 dsetconstrEL: $o ).
% 0.19/0.38
% 0.19/0.38 thf(dsetconstrEL,definition,
% 0.19/0.38 ( dsetconstrEL
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx
% 0.19/0.38 @ ( dsetconstr @ A
% 0.19/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.38 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(dsetconstrER_type,type,
% 0.19/0.38 dsetconstrER: $o ).
% 0.19/0.38
% 0.19/0.38 thf(dsetconstrER,definition,
% 0.19/0.38 ( dsetconstrER
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx
% 0.19/0.38 @ ( dsetconstr @ A
% 0.19/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.38 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(exuE1_type,type,
% 0.19/0.38 exuE1: $o ).
% 0.19/0.38
% 0.19/0.38 thf(exuE1,definition,
% 0.19/0.38 ( exuE1
% 0.19/0.38 = ( ! [Xphi: $i > $o] :
% 0.19/0.38 ( ( exu
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.38 => ? [Xx: $i] :
% 0.19/0.38 ( ( Xphi @ Xx )
% 0.19/0.38 & ! [Xy: $i] :
% 0.19/0.38 ( ( Xphi @ Xy )
% 0.19/0.38 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(prop2set_type,type,
% 0.19/0.38 prop2set: $o > $i ).
% 0.19/0.38
% 0.19/0.38 thf(prop2setE_type,type,
% 0.19/0.38 prop2setE: $o ).
% 0.19/0.38
% 0.19/0.38 thf(prop2setE,definition,
% 0.19/0.38 ( prop2setE
% 0.19/0.38 = ( ! [Xphi: $o,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.19/0.38 => Xphi ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(emptysetE_type,type,
% 0.19/0.38 emptysetE: $o ).
% 0.19/0.38
% 0.19/0.38 thf(emptysetE,definition,
% 0.19/0.38 ( emptysetE
% 0.19/0.38 = ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ emptyset )
% 0.19/0.38 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(emptysetimpfalse_type,type,
% 0.19/0.38 emptysetimpfalse: $o ).
% 0.19/0.38
% 0.19/0.38 thf(emptysetimpfalse,definition,
% 0.19/0.38 ( emptysetimpfalse
% 0.19/0.38 = ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ emptyset )
% 0.19/0.38 => $false ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(notinemptyset_type,type,
% 0.19/0.38 notinemptyset: $o ).
% 0.19/0.38
% 0.19/0.38 thf(notinemptyset,definition,
% 0.19/0.38 ( notinemptyset
% 0.19/0.38 = ( ! [Xx: $i] :
% 0.19/0.38 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(exuE3e_type,type,
% 0.19/0.38 exuE3e: $o ).
% 0.19/0.38
% 0.19/0.38 thf(exuE3e,definition,
% 0.19/0.38 ( exuE3e
% 0.19/0.38 = ( ! [Xphi: $i > $o] :
% 0.19/0.38 ( ( exu
% 0.19/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.38 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setext_type,type,
% 0.19/0.38 setext: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setext,definition,
% 0.19/0.38 ( setext
% 0.19/0.38 = ( ! [A: $i,B: $i] :
% 0.19/0.38 ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( in @ Xx @ B ) )
% 0.19/0.38 => ( ! [Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ B )
% 0.19/0.38 => ( in @ Xx @ A ) )
% 0.19/0.38 => ( A = B ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(emptyI_type,type,
% 0.19/0.38 emptyI: $o ).
% 0.19/0.38
% 0.19/0.38 thf(emptyI,definition,
% 0.19/0.38 ( emptyI
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ( ! [Xx: $i] :
% 0.19/0.38 ~ ( in @ Xx @ A )
% 0.19/0.38 => ( A = emptyset ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(noeltsimpempty_type,type,
% 0.19/0.38 noeltsimpempty: $o ).
% 0.19/0.38
% 0.19/0.38 thf(noeltsimpempty,definition,
% 0.19/0.38 ( noeltsimpempty
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ( ! [Xx: $i] :
% 0.19/0.38 ~ ( in @ Xx @ A )
% 0.19/0.38 => ( A = emptyset ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setbeta_type,type,
% 0.19/0.38 setbeta: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setbeta,definition,
% 0.19/0.38 ( setbeta
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( ( in @ Xx
% 0.19/0.38 @ ( dsetconstr @ A
% 0.19/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.38 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(nonempty_type,type,
% 0.19/0.38 nonempty: $i > $o ).
% 0.19/0.38
% 0.19/0.38 thf(nonempty,definition,
% 0.19/0.38 ( nonempty
% 0.19/0.38 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(nonemptyE1_type,type,
% 0.19/0.38 nonemptyE1: $o ).
% 0.19/0.38
% 0.19/0.38 thf(nonemptyE1,definition,
% 0.19/0.38 ( nonemptyE1
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ( ( nonempty @ A )
% 0.19/0.38 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(nonemptyI_type,type,
% 0.19/0.38 nonemptyI: $o ).
% 0.19/0.38
% 0.19/0.38 thf(nonemptyI,definition,
% 0.19/0.38 ( nonemptyI
% 0.19/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.38 ( ( in @ Xx @ A )
% 0.19/0.38 => ( ( Xphi @ Xx )
% 0.19/0.38 => ( nonempty
% 0.19/0.38 @ ( dsetconstr @ A
% 0.19/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(nonemptyI1_type,type,
% 0.19/0.38 nonemptyI1: $o ).
% 0.19/0.38
% 0.19/0.38 thf(nonemptyI1,definition,
% 0.19/0.38 ( nonemptyI1
% 0.19/0.38 = ( ! [A: $i] :
% 0.19/0.38 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.38 => ( nonempty @ A ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinIL_type,type,
% 0.19/0.38 setadjoinIL: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinIL,definition,
% 0.19/0.38 ( setadjoinIL
% 0.19/0.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(emptyinunitempty_type,type,
% 0.19/0.38 emptyinunitempty: $o ).
% 0.19/0.38
% 0.19/0.38 thf(emptyinunitempty,definition,
% 0.19/0.38 ( emptyinunitempty
% 0.19/0.38 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinIR_type,type,
% 0.19/0.38 setadjoinIR: $o ).
% 0.19/0.38
% 0.19/0.38 thf(setadjoinIR,definition,
% 0.19/0.39 ( setadjoinIR
% 0.19/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ A )
% 0.19/0.39 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(setadjoinE_type,type,
% 0.19/0.39 setadjoinE: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setadjoinE,definition,
% 0.19/0.39 ( setadjoinE
% 0.19/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.39 => ! [Xphi: $o] :
% 0.19/0.39 ( ( ( Xy = Xx )
% 0.19/0.39 => Xphi )
% 0.19/0.39 => ( ( ( in @ Xy @ A )
% 0.19/0.39 => Xphi )
% 0.19/0.39 => Xphi ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(setadjoinOr_type,type,
% 0.19/0.39 setadjoinOr: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setadjoinOr,definition,
% 0.19/0.39 ( setadjoinOr
% 0.19/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.39 => ( ( Xy = Xx )
% 0.19/0.39 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(setoftrueEq_type,type,
% 0.19/0.39 setoftrueEq: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setoftrueEq,definition,
% 0.19/0.39 ( setoftrueEq
% 0.19/0.39 = ( ! [A: $i] :
% 0.19/0.39 ( ( dsetconstr @ A
% 0.19/0.39 @ ^ [Xx: $i] : $true )
% 0.19/0.39 = A ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(powersetI_type,type,
% 0.19/0.39 powersetI: $o ).
% 0.19/0.39
% 0.19/0.39 thf(powersetI,definition,
% 0.19/0.39 ( powersetI
% 0.19/0.39 = ( ! [A: $i,B: $i] :
% 0.19/0.39 ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ B )
% 0.19/0.39 => ( in @ Xx @ A ) )
% 0.19/0.39 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(emptyinPowerset_type,type,
% 0.19/0.39 emptyinPowerset: $o ).
% 0.19/0.39
% 0.19/0.39 thf(emptyinPowerset,definition,
% 0.19/0.39 ( emptyinPowerset
% 0.19/0.39 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(emptyInPowerset_type,type,
% 0.19/0.39 emptyInPowerset: $o ).
% 0.19/0.39
% 0.19/0.39 thf(emptyInPowerset,definition,
% 0.19/0.39 ( emptyInPowerset
% 0.19/0.39 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(powersetE_type,type,
% 0.19/0.39 powersetE: $o ).
% 0.19/0.39
% 0.19/0.39 thf(powersetE,definition,
% 0.19/0.39 ( powersetE
% 0.19/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.39 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.39 => ( ( in @ Xx @ B )
% 0.19/0.39 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(setunionI_type,type,
% 0.19/0.39 setunionI: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setunionI,definition,
% 0.19/0.39 ( setunionI
% 0.19/0.39 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.19/0.39 ( ( in @ Xx @ B )
% 0.19/0.39 => ( ( in @ B @ A )
% 0.19/0.39 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(setunionE_type,type,
% 0.19/0.39 setunionE: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setunionE,definition,
% 0.19/0.39 ( setunionE
% 0.19/0.39 = ( ! [A: $i,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.39 => ! [Xphi: $o] :
% 0.19/0.39 ( ! [B: $i] :
% 0.19/0.39 ( ( in @ Xx @ B )
% 0.19/0.39 => ( ( in @ B @ A )
% 0.19/0.39 => Xphi ) )
% 0.19/0.39 => Xphi ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(subPowSU_type,type,
% 0.19/0.39 subPowSU: $o ).
% 0.19/0.39
% 0.19/0.39 thf(subPowSU,definition,
% 0.19/0.39 ( subPowSU
% 0.19/0.39 = ( ! [A: $i,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(exuE2_type,type,
% 0.19/0.39 exuE2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(exuE2,definition,
% 0.19/0.39 ( exuE2
% 0.19/0.39 = ( ! [Xphi: $i > $o] :
% 0.19/0.39 ( ( exu
% 0.19/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.39 => ? [Xx: $i] :
% 0.19/0.39 ! [Xy: $i] :
% 0.19/0.39 ( ( Xphi @ Xy )
% 0.19/0.39 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(nonemptyImpWitness_type,type,
% 0.19/0.39 nonemptyImpWitness: $o ).
% 0.19/0.39
% 0.19/0.39 thf(nonemptyImpWitness,definition,
% 0.19/0.39 ( nonemptyImpWitness
% 0.19/0.39 = ( ! [A: $i] :
% 0.19/0.39 ( ( nonempty @ A )
% 0.19/0.39 => ? [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 & $true ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(uniqinunit_type,type,
% 0.19/0.39 uniqinunit: $o ).
% 0.19/0.39
% 0.19/0.39 thf(uniqinunit,definition,
% 0.19/0.39 ( uniqinunit
% 0.19/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.39 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.39 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(notinsingleton_type,type,
% 0.19/0.39 notinsingleton: $o ).
% 0.19/0.39
% 0.19/0.39 thf(notinsingleton,definition,
% 0.19/0.39 ( notinsingleton
% 0.19/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.39 ( ( Xx != Xy )
% 0.19/0.39 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(eqinunit_type,type,
% 0.19/0.39 eqinunit: $o ).
% 0.19/0.39
% 0.19/0.39 thf(eqinunit,definition,
% 0.19/0.39 ( eqinunit
% 0.19/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.39 ( ( Xx = Xy )
% 0.19/0.39 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(singletonsswitch_type,type,
% 0.19/0.39 singletonsswitch: $o ).
% 0.19/0.39
% 0.19/0.39 thf(singletonsswitch,definition,
% 0.19/0.39 ( singletonsswitch
% 0.19/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.39 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.39 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(upairsetE_type,type,
% 0.19/0.39 upairsetE: $o ).
% 0.19/0.39
% 0.19/0.39 thf(upairsetE,definition,
% 0.19/0.39 ( upairsetE
% 0.19/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.39 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.19/0.39 => ( ( Xz = Xx )
% 0.19/0.39 | ( Xz = Xy ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(upairsetIL_type,type,
% 0.19/0.39 upairsetIL: $o ).
% 0.19/0.39
% 0.19/0.39 thf(upairsetIL,definition,
% 0.19/0.39 ( upairsetIL
% 0.19/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(upairsetIR_type,type,
% 0.19/0.39 upairsetIR: $o ).
% 0.19/0.39
% 0.19/0.39 thf(upairsetIR,definition,
% 0.19/0.39 ( upairsetIR
% 0.19/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(emptyE1_type,type,
% 0.19/0.39 emptyE1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(emptyE1,definition,
% 0.19/0.39 ( emptyE1
% 0.19/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.39 ( ? [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 & ( Xphi @ Xx ) )
% 0.19/0.39 => ( ( ( dsetconstr @ A
% 0.19/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.39 = emptyset )
% 0.19/0.39 => $false ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(vacuousDall_type,type,
% 0.19/0.39 vacuousDall: $o ).
% 0.19/0.39
% 0.19/0.39 thf(vacuousDall,definition,
% 0.19/0.39 ( vacuousDall
% 0.19/0.39 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ emptyset )
% 0.19/0.39 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(quantDeMorgan1_type,type,
% 0.19/0.39 quantDeMorgan1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(quantDeMorgan1,definition,
% 0.19/0.39 ( quantDeMorgan1
% 0.19/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.39 ( ~ ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( Xphi @ Xx ) )
% 0.19/0.39 => ? [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(quantDeMorgan2_type,type,
% 0.19/0.39 quantDeMorgan2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(quantDeMorgan2,definition,
% 0.19/0.39 ( quantDeMorgan2
% 0.19/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.39 ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ~ ( Xphi @ Xx ) )
% 0.19/0.39 => ~ ? [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(quantDeMorgan3_type,type,
% 0.19/0.39 quantDeMorgan3: $o ).
% 0.19/0.39
% 0.19/0.39 thf(quantDeMorgan3,definition,
% 0.19/0.39 ( quantDeMorgan3
% 0.19/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.39 ( ~ ? [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 & ( Xphi @ Xx ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(quantDeMorgan4_type,type,
% 0.19/0.39 quantDeMorgan4: $o ).
% 0.19/0.39
% 0.19/0.39 thf(quantDeMorgan4,definition,
% 0.19/0.39 ( quantDeMorgan4
% 0.19/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.39 ( ? [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 & ~ ( Xphi @ Xx ) )
% 0.19/0.39 => ~ ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(prop2setI_type,type,
% 0.19/0.39 prop2setI: $o ).
% 0.19/0.39
% 0.19/0.39 thf(prop2setI,definition,
% 0.19/0.39 ( prop2setI
% 0.19/0.39 = ( ! [Xphi: $o] :
% 0.19/0.39 ( Xphi
% 0.19/0.39 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(set2prop_type,type,
% 0.19/0.39 set2prop: $i > $o ).
% 0.19/0.39
% 0.19/0.39 thf(prop2set2propI_type,type,
% 0.19/0.39 prop2set2propI: $o ).
% 0.19/0.39
% 0.19/0.39 thf(prop2set2propI,definition,
% 0.19/0.39 ( prop2set2propI
% 0.19/0.39 = ( ! [Xphi: $o] :
% 0.19/0.39 ( Xphi
% 0.19/0.39 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(notdexE_type,type,
% 0.19/0.39 notdexE: $o ).
% 0.19/0.39
% 0.19/0.39 thf(notdexE,definition,
% 0.19/0.39 ( notdexE
% 0.19/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.39 ( ~ ? [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 & ( Xphi @ Xx ) )
% 0.19/0.39 => ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(notdallE_type,type,
% 0.19/0.39 notdallE: $o ).
% 0.19/0.39
% 0.19/0.39 thf(notdallE,definition,
% 0.19/0.39 ( notdallE
% 0.19/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.39 ( ~ ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( Xphi @ Xx ) )
% 0.19/0.39 => ? [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(exuI1_type,type,
% 0.19/0.39 exuI1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(exuI1,definition,
% 0.19/0.39 ( exuI1
% 0.19/0.39 = ( ! [Xphi: $i > $o] :
% 0.19/0.39 ( ? [Xx: $i] :
% 0.19/0.39 ( ( Xphi @ Xx )
% 0.19/0.39 & ! [Xy: $i] :
% 0.19/0.39 ( ( Xphi @ Xy )
% 0.19/0.39 => ( Xx = Xy ) ) )
% 0.19/0.39 => ( exu
% 0.19/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(exuI3_type,type,
% 0.19/0.39 exuI3: $o ).
% 0.19/0.39
% 0.19/0.39 thf(exuI3,definition,
% 0.19/0.39 ( exuI3
% 0.19/0.39 = ( ! [Xphi: $i > $o] :
% 0.19/0.39 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.19/0.39 => ( ! [Xx: $i,Xy: $i] :
% 0.19/0.39 ( ( Xphi @ Xx )
% 0.19/0.39 => ( ( Xphi @ Xy )
% 0.19/0.39 => ( Xx = Xy ) ) )
% 0.19/0.39 => ( exu
% 0.19/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(exuI2_type,type,
% 0.19/0.39 exuI2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(exuI2,definition,
% 0.19/0.39 ( exuI2
% 0.19/0.39 = ( ! [Xphi: $i > $o] :
% 0.19/0.39 ( ? [Xx: $i] :
% 0.19/0.39 ! [Xy: $i] :
% 0.19/0.39 ( ( Xphi @ Xy )
% 0.19/0.39 <=> ( Xy = Xx ) )
% 0.19/0.39 => ( exu
% 0.19/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(inCongP_type,type,
% 0.19/0.39 inCongP: $o ).
% 0.19/0.39
% 0.19/0.39 thf(inCongP,definition,
% 0.19/0.39 ( inCongP
% 0.19/0.39 = ( ! [A: $i,B: $i] :
% 0.19/0.39 ( ( A = B )
% 0.19/0.39 => ! [Xx: $i,Xy: $i] :
% 0.19/0.39 ( ( Xx = Xy )
% 0.19/0.39 => ( ( in @ Xx @ A )
% 0.19/0.39 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(in__Cong_type,type,
% 0.19/0.39 in__Cong: $o ).
% 0.19/0.39
% 0.19/0.39 thf(in__Cong,definition,
% 0.19/0.39 ( in__Cong
% 0.19/0.39 = ( ! [A: $i,B: $i] :
% 0.19/0.39 ( ( A = B )
% 0.19/0.39 => ! [Xx: $i,Xy: $i] :
% 0.19/0.39 ( ( Xx = Xy )
% 0.19/0.39 => ( ( in @ Xx @ A )
% 0.19/0.39 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(exuE3u_type,type,
% 0.19/0.39 exuE3u: $o ).
% 0.19/0.39
% 0.19/0.39 thf(exuE3u,definition,
% 0.19/0.39 ( exuE3u
% 0.19/0.39 = ( ! [Xphi: $i > $o] :
% 0.19/0.39 ( ( exu
% 0.19/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.39 => ! [Xx: $i,Xy: $i] :
% 0.19/0.39 ( ( Xphi @ Xx )
% 0.19/0.39 => ( ( Xphi @ Xy )
% 0.19/0.39 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(exu__Cong_type,type,
% 0.19/0.39 exu__Cong: $o ).
% 0.19/0.39
% 0.19/0.39 thf(exu__Cong,definition,
% 0.19/0.39 ( exu__Cong
% 0.19/0.39 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.39 ( ! [Xx: $i,Xy: $i] :
% 0.19/0.39 ( ( Xx = Xy )
% 0.19/0.39 => ( ( Xphi @ Xx )
% 0.19/0.39 <=> ( Xpsi @ Xy ) ) )
% 0.19/0.39 => ( ( exu
% 0.19/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.39 <=> ( exu
% 0.19/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(emptyset__Cong_type,type,
% 0.19/0.39 emptyset__Cong: $o ).
% 0.19/0.39
% 0.19/0.39 thf(emptyset__Cong,definition,
% 0.19/0.39 ( emptyset__Cong
% 0.19/0.39 = ( emptyset = emptyset ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(setadjoin__Cong_type,type,
% 0.19/0.39 setadjoin__Cong: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setadjoin__Cong,definition,
% 0.19/0.39 ( setadjoin__Cong
% 0.19/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.39 ( ( Xx = Xy )
% 0.19/0.39 => ! [Xz: $i,Xu: $i] :
% 0.19/0.39 ( ( Xz = Xu )
% 0.19/0.39 => ( ( setadjoin @ Xx @ Xz )
% 0.19/0.39 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(powerset__Cong_type,type,
% 0.19/0.39 powerset__Cong: $o ).
% 0.19/0.39
% 0.19/0.39 thf(powerset__Cong,definition,
% 0.19/0.39 ( powerset__Cong
% 0.19/0.39 = ( ! [A: $i,B: $i] :
% 0.19/0.39 ( ( A = B )
% 0.19/0.39 => ( ( powerset @ A )
% 0.19/0.39 = ( powerset @ B ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(setunion__Cong_type,type,
% 0.19/0.39 setunion__Cong: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setunion__Cong,definition,
% 0.19/0.39 ( setunion__Cong
% 0.19/0.39 = ( ! [A: $i,B: $i] :
% 0.19/0.39 ( ( A = B )
% 0.19/0.39 => ( ( setunion @ A )
% 0.19/0.39 = ( setunion @ B ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(omega__Cong_type,type,
% 0.19/0.39 omega__Cong: $o ).
% 0.19/0.39
% 0.19/0.39 thf(omega__Cong,definition,
% 0.19/0.39 ( omega__Cong
% 0.19/0.39 = ( omega = omega ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(exuEu_type,type,
% 0.19/0.39 exuEu: $o ).
% 0.19/0.39
% 0.19/0.39 thf(exuEu,definition,
% 0.19/0.39 ( exuEu
% 0.19/0.39 = ( ! [Xphi: $i > $o] :
% 0.19/0.39 ( ( exu
% 0.19/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.39 => ! [Xx: $i,Xy: $i] :
% 0.19/0.39 ( ( Xphi @ Xx )
% 0.19/0.39 => ( ( Xphi @ Xy )
% 0.19/0.39 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(descr__Cong_type,type,
% 0.19/0.39 descr__Cong: $o ).
% 0.19/0.39
% 0.19/0.39 thf(descr__Cong,definition,
% 0.19/0.39 ( descr__Cong
% 0.19/0.39 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.39 ( ! [Xx: $i,Xy: $i] :
% 0.19/0.39 ( ( Xx = Xy )
% 0.19/0.39 => ( ( Xphi @ Xx )
% 0.19/0.39 <=> ( Xpsi @ Xy ) ) )
% 0.19/0.39 => ( ( exu
% 0.19/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.39 => ( ( exu
% 0.19/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.19/0.39 => ( ( descr
% 0.19/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.39 = ( descr
% 0.19/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(dsetconstr__Cong_type,type,
% 0.19/0.39 dsetconstr__Cong: $o ).
% 0.19/0.39
% 0.19/0.39 thf(dsetconstr__Cong,definition,
% 0.19/0.39 ( dsetconstr__Cong
% 0.19/0.39 = ( ! [A: $i,B: $i] :
% 0.19/0.39 ( ( A = B )
% 0.19/0.39 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.39 ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ! [Xy: $i] :
% 0.19/0.39 ( ( in @ Xy @ B )
% 0.19/0.39 => ( ( Xx = Xy )
% 0.19/0.39 => ( ( Xphi @ Xx )
% 0.19/0.39 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.19/0.39 => ( ( dsetconstr @ A
% 0.19/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.39 = ( dsetconstr @ B
% 0.19/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(subset_type,type,
% 0.19/0.39 subset: $i > $i > $o ).
% 0.19/0.39
% 0.19/0.39 thf(disjoint_type,type,
% 0.19/0.39 disjoint: $i > $i > $o ).
% 0.19/0.39
% 0.19/0.39 thf(setsmeet_type,type,
% 0.19/0.39 setsmeet: $i > $i > $o ).
% 0.19/0.39
% 0.19/0.39 thf(subsetI1_type,type,
% 0.19/0.39 subsetI1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(subsetI1,definition,
% 0.19/0.39 ( subsetI1
% 0.19/0.39 = ( ! [A: $i,B: $i] :
% 0.19/0.39 ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( in @ Xx @ B ) )
% 0.19/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(eqimpsubset2_type,type,
% 0.19/0.39 eqimpsubset2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(eqimpsubset2,definition,
% 0.19/0.39 ( eqimpsubset2
% 0.19/0.39 = ( ! [A: $i,B: $i] :
% 0.19/0.39 ( ( A = B )
% 0.19/0.39 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(eqimpsubset1_type,type,
% 0.19/0.39 eqimpsubset1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(eqimpsubset1,definition,
% 0.19/0.39 ( eqimpsubset1
% 0.19/0.39 = ( ! [A: $i,B: $i] :
% 0.19/0.39 ( ( A = B )
% 0.19/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(subsetI2_type,type,
% 0.19/0.39 subsetI2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(subsetI2,definition,
% 0.19/0.39 ( subsetI2
% 0.19/0.39 = ( ! [A: $i,B: $i] :
% 0.19/0.39 ( ! [Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( in @ Xx @ B ) )
% 0.19/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(emptysetsubset_type,type,
% 0.19/0.39 emptysetsubset: $o ).
% 0.19/0.39
% 0.19/0.39 thf(emptysetsubset,definition,
% 0.19/0.39 ( emptysetsubset
% 0.19/0.39 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(subsetE_type,type,
% 0.19/0.39 subsetE: $o ).
% 0.19/0.39
% 0.19/0.39 thf(subsetE,definition,
% 0.19/0.39 ( subsetE
% 0.19/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.39 ( ( subset @ A @ B )
% 0.19/0.39 => ( ( in @ Xx @ A )
% 0.19/0.39 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(subsetE2_type,type,
% 0.19/0.39 subsetE2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(subsetE2,definition,
% 0.19/0.39 ( subsetE2
% 0.19/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.39 ( ( subset @ A @ B )
% 0.19/0.39 => ( ~ ( in @ Xx @ B )
% 0.19/0.39 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(notsubsetI_type,type,
% 0.19/0.39 notsubsetI: $o ).
% 0.19/0.39
% 0.19/0.39 thf(notsubsetI,definition,
% 0.19/0.39 ( notsubsetI
% 0.19/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ~ ( in @ Xx @ B )
% 0.19/0.39 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(notequalI1_type,type,
% 0.19/0.39 notequalI1: $o ).
% 0.19/0.39
% 0.19/0.39 thf(notequalI1,definition,
% 0.19/0.39 ( notequalI1
% 0.19/0.39 = ( ! [A: $i,B: $i] :
% 0.19/0.39 ( ~ ( subset @ A @ B )
% 0.19/0.39 => ( A != B ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(notequalI2_type,type,
% 0.19/0.39 notequalI2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(notequalI2,definition,
% 0.19/0.39 ( notequalI2
% 0.19/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.39 ( ( in @ Xx @ A )
% 0.19/0.39 => ( ~ ( in @ Xx @ B )
% 0.19/0.39 => ( A != B ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(subsetRefl_type,type,
% 0.19/0.39 subsetRefl: $o ).
% 0.19/0.39
% 0.19/0.39 thf(subsetRefl,definition,
% 0.19/0.39 ( subsetRefl
% 0.19/0.39 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(subsetTrans_type,type,
% 0.19/0.39 subsetTrans: $o ).
% 0.19/0.39
% 0.19/0.39 thf(subsetTrans,definition,
% 0.19/0.39 ( subsetTrans
% 0.19/0.39 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.39 ( ( subset @ A @ B )
% 0.19/0.39 => ( ( subset @ B @ C )
% 0.19/0.39 => ( subset @ A @ C ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(setadjoinSub_type,type,
% 0.19/0.39 setadjoinSub: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setadjoinSub,definition,
% 0.19/0.39 ( setadjoinSub
% 0.19/0.39 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(setadjoinSub2_type,type,
% 0.19/0.39 setadjoinSub2: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setadjoinSub2,definition,
% 0.19/0.39 ( setadjoinSub2
% 0.19/0.39 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.19/0.39 ( ( subset @ A @ B )
% 0.19/0.39 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(subset2powerset_type,type,
% 0.19/0.39 subset2powerset: $o ).
% 0.19/0.39
% 0.19/0.39 thf(subset2powerset,definition,
% 0.19/0.39 ( subset2powerset
% 0.19/0.39 = ( ! [A: $i,B: $i] :
% 0.19/0.39 ( ( subset @ A @ B )
% 0.19/0.39 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(setextsub_type,type,
% 0.19/0.39 setextsub: $o ).
% 0.19/0.39
% 0.19/0.39 thf(setextsub,definition,
% 0.19/0.39 ( setextsub
% 0.19/0.39 = ( ! [A: $i,B: $i] :
% 0.19/0.39 ( ( subset @ A @ B )
% 0.19/0.39 => ( ( subset @ B @ A )
% 0.19/0.39 => ( A = B ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(subsetemptysetimpeq_type,type,
% 0.19/0.39 subsetemptysetimpeq: $o ).
% 0.19/0.39
% 0.19/0.39 thf(subsetemptysetimpeq,definition,
% 0.19/0.39 ( subsetemptysetimpeq
% 0.19/0.39 = ( ! [A: $i] :
% 0.19/0.39 ( ( subset @ A @ emptyset )
% 0.19/0.39 => ( A = emptyset ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 thf(powersetI1,conjecture,
% 0.19/0.39 ( setextAx
% 0.19/0.39 => ( emptysetAx
% 0.19/0.39 => ( setadjoinAx
% 0.19/0.39 => ( powersetAx
% 0.19/0.39 => ( setunionAx
% 0.19/0.39 => ( omega0Ax
% 0.19/0.39 => ( omegaSAx
% 0.19/0.39 => ( omegaIndAx
% 0.19/0.39 => ( replAx
% 0.19/0.39 => ( foundationAx
% 0.19/0.39 => ( wellorderingAx
% 0.19/0.39 => ( descrp
% 0.19/0.39 => ( dsetconstrI
% 0.19/0.39 => ( dsetconstrEL
% 0.19/0.39 => ( dsetconstrER
% 0.19/0.39 => ( exuE1
% 0.19/0.39 => ( prop2setE
% 0.19/0.39 => ( emptysetE
% 0.19/0.39 => ( emptysetimpfalse
% 0.19/0.39 => ( notinemptyset
% 0.19/0.39 => ( exuE3e
% 0.19/0.39 => ( setext
% 0.19/0.39 => ( emptyI
% 0.19/0.39 => ( noeltsimpempty
% 0.19/0.39 => ( setbeta
% 0.19/0.39 => ( nonemptyE1
% 0.19/0.39 => ( nonemptyI
% 0.19/0.39 => ( nonemptyI1
% 0.19/0.39 => ( setadjoinIL
% 0.19/0.39 => ( emptyinunitempty
% 0.19/0.39 => ( setadjoinIR
% 0.19/0.39 => ( setadjoinE
% 0.19/0.39 => ( setadjoinOr
% 0.19/0.39 => ( setoftrueEq
% 0.19/0.39 => ( powersetI
% 0.19/0.39 => ( emptyinPowerset
% 0.19/0.39 => ( emptyInPowerset
% 0.19/0.39 => ( powersetE
% 0.19/0.39 => ( setunionI
% 0.19/0.39 => ( setunionE
% 0.19/0.39 => ( subPowSU
% 0.19/0.39 => ( exuE2
% 0.19/0.39 => ( nonemptyImpWitness
% 0.19/0.39 => ( uniqinunit
% 0.19/0.39 => ( notinsingleton
% 0.19/0.39 => ( eqinunit
% 0.19/0.39 => ( singletonsswitch
% 0.19/0.39 => ( upairsetE
% 0.19/0.39 => ( upairsetIL
% 0.19/0.39 => ( upairsetIR
% 0.19/0.39 => ( emptyE1
% 0.19/0.39 => ( vacuousDall
% 0.19/0.39 => ( quantDeMorgan1
% 0.19/0.39 => ( quantDeMorgan2
% 0.19/0.39 => ( quantDeMorgan3
% 0.19/0.39 => ( quantDeMorgan4
% 0.19/0.39 => ( prop2setI
% 0.19/0.39 => ( prop2set2propI
% 0.19/0.39 => ( notdexE
% 0.19/0.39 => ( notdallE
% 0.19/0.39 => ( exuI1
% 0.19/0.39 => ( exuI3
% 0.19/0.39 => ( exuI2
% 0.19/0.39 => ( inCongP
% 0.19/0.39 => ( in__Cong
% 0.19/0.39 => ( exuE3u
% 0.19/0.39 => ( exu__Cong
% 0.19/0.39 => ( emptyset__Cong
% 0.19/0.39 => ( setadjoin__Cong
% 0.19/0.39 => ( powerset__Cong
% 0.19/0.39 => ( setunion__Cong
% 0.19/0.39 => ( omega__Cong
% 0.19/0.39 => ( exuEu
% 0.19/0.39 => ( descr__Cong
% 0.19/0.39 => ( dsetconstr__Cong
% 0.19/0.39 => ( subsetI1
% 0.19/0.39 => ( eqimpsubset2
% 0.19/0.39 => ( eqimpsubset1
% 0.19/0.39 => ( subsetI2
% 0.19/0.39 => ( emptysetsubset
% 0.19/0.39 => ( subsetE
% 0.19/0.39 => ( subsetE2
% 0.19/0.39 => ( notsubsetI
% 0.19/0.39 => ( notequalI1
% 0.19/0.39 => ( notequalI2
% 0.19/0.39 => ( subsetRefl
% 0.19/0.39 => ( subsetTrans
% 0.19/0.39 => ( setadjoinSub
% 0.19/0.39 => ( setadjoinSub2
% 0.19/0.39 => ( subset2powerset
% 0.19/0.39 => ( setextsub
% 0.19/0.39 => ( subsetemptysetimpeq
% 0.19/0.39 => ! [A: $i,B: $i] :
% 0.19/0.39 ( ( subset @ B @ A )
% 0.19/0.39 => ( in @ B @ ( powerset @ A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.39
% 0.19/0.39 %------------------------------------------------------------------------------
% 0.19/0.39 ------------------------
% 2.44/1.51 ---- Embedded in THF ---
% 2.44/1.51 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.44/1.51 %%% Generated on Tue Feb 24 12:28:05 EST 2026
% 2.44/1.51 %%% using '$$dhol' embedding, version 1.3.0.
% 2.44/1.51 %%% Logic specification used:
% 2.44/1.51 %%% thf(spec, logic, $$dhol).
% 2.44/1.51
% 2.44/1.51 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.ZrZtUQmNfW/DTF2DTF22788.p
% See solution above
% 2.44/1.51 ------------------------
% 2.44/1.53 ---- Cleaned THF ---
% 2.44/1.53 thf(in_type,type,
% 2.44/1.53 in: $i > $i > $o ).
% 2.44/1.53
% 2.44/1.53 thf(exu_type,type,
% 2.44/1.53 exu: ( $i > $o ) > $o ).
% 2.44/1.53
% 2.44/1.53 thf(exu,definition,
% 2.44/1.53 ( exu
% 2.44/1.53 = ( ^ [Xphi: $i > $o] :
% 2.44/1.53 ? [Xx: $i] :
% 2.44/1.53 ( ( Xphi @ Xx )
% 2.44/1.53 & ! [Xy: $i] :
% 2.44/1.53 ( ( Xphi @ Xy )
% 2.44/1.53 => ( Xx = Xy ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(setextAx_type,type,
% 2.44/1.53 setextAx: $o ).
% 2.44/1.53
% 2.44/1.53 thf(setextAx,definition,
% 2.44/1.53 ( setextAx
% 2.44/1.53 = ( ! [A: $i,B: $i] :
% 2.44/1.53 ( ! [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 <=> ( in @ Xx @ B ) )
% 2.44/1.53 => ( A = B ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(emptyset_type,type,
% 2.44/1.53 emptyset: $i ).
% 2.44/1.53
% 2.44/1.53 thf(emptysetAx_type,type,
% 2.44/1.53 emptysetAx: $o ).
% 2.44/1.53
% 2.44/1.53 thf(emptysetAx,definition,
% 2.44/1.53 ( emptysetAx
% 2.44/1.53 = ( ! [Xx: $i] :
% 2.44/1.53 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(setadjoin_type,type,
% 2.44/1.53 setadjoin: $i > $i > $i ).
% 2.44/1.53
% 2.44/1.53 thf(setadjoinAx_type,type,
% 2.44/1.53 setadjoinAx: $o ).
% 2.44/1.53
% 2.44/1.53 thf(setadjoinAx,definition,
% 2.44/1.53 ( setadjoinAx
% 2.44/1.53 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.44/1.53 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.44/1.53 <=> ( ( Xy = Xx )
% 2.44/1.53 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(powerset_type,type,
% 2.44/1.53 powerset: $i > $i ).
% 2.44/1.53
% 2.44/1.53 thf(powersetAx_type,type,
% 2.44/1.53 powersetAx: $o ).
% 2.44/1.53
% 2.44/1.53 thf(powersetAx,definition,
% 2.44/1.53 ( powersetAx
% 2.44/1.53 = ( ! [A: $i,B: $i] :
% 2.44/1.53 ( ( in @ B @ ( powerset @ A ) )
% 2.44/1.53 <=> ! [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ B )
% 2.44/1.53 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(setunion_type,type,
% 2.44/1.53 setunion: $i > $i ).
% 2.44/1.53
% 2.44/1.53 thf(setunionAx_type,type,
% 2.44/1.53 setunionAx: $o ).
% 2.44/1.53
% 2.44/1.53 thf(setunionAx,definition,
% 2.44/1.53 ( setunionAx
% 2.44/1.53 = ( ! [A: $i,Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ ( setunion @ A ) )
% 2.44/1.53 <=> ? [B: $i] :
% 2.44/1.53 ( ( in @ Xx @ B )
% 2.44/1.53 & ( in @ B @ A ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(omega_type,type,
% 2.44/1.53 omega: $i ).
% 2.44/1.53
% 2.44/1.53 thf(omega0Ax_type,type,
% 2.44/1.53 omega0Ax: $o ).
% 2.44/1.53
% 2.44/1.53 thf(omega0Ax,definition,
% 2.44/1.53 ( omega0Ax
% 2.44/1.53 = ( in @ emptyset @ omega ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(omegaSAx_type,type,
% 2.44/1.53 omegaSAx: $o ).
% 2.44/1.53
% 2.44/1.53 thf(omegaSAx,definition,
% 2.44/1.53 ( omegaSAx
% 2.44/1.53 = ( ! [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ omega )
% 2.44/1.53 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(omegaIndAx_type,type,
% 2.44/1.53 omegaIndAx: $o ).
% 2.44/1.53
% 2.44/1.53 thf(omegaIndAx,definition,
% 2.44/1.53 ( omegaIndAx
% 2.44/1.53 = ( ! [A: $i] :
% 2.44/1.53 ( ( ( in @ emptyset @ A )
% 2.44/1.53 & ! [Xx: $i] :
% 2.44/1.53 ( ( ( in @ Xx @ omega )
% 2.44/1.53 & ( in @ Xx @ A ) )
% 2.44/1.53 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.44/1.53 => ! [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ omega )
% 2.44/1.53 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(replAx_type,type,
% 2.44/1.53 replAx: $o ).
% 2.44/1.53
% 2.44/1.53 thf(replAx,definition,
% 2.44/1.53 ( replAx
% 2.44/1.53 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.44/1.53 ( ! [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 => ( exu
% 2.44/1.53 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.44/1.53 => ? [B: $i] :
% 2.44/1.53 ! [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ B )
% 2.44/1.53 <=> ? [Xy: $i] :
% 2.44/1.53 ( ( in @ Xy @ A )
% 2.44/1.53 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(foundationAx_type,type,
% 2.44/1.53 foundationAx: $o ).
% 2.44/1.53
% 2.44/1.53 thf(foundationAx,definition,
% 2.44/1.53 ( foundationAx
% 2.44/1.53 = ( ! [A: $i] :
% 2.44/1.53 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.44/1.53 => ? [B: $i] :
% 2.44/1.53 ( ( in @ B @ A )
% 2.44/1.53 & ~ ? [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ B )
% 2.44/1.53 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(wellorderingAx_type,type,
% 2.44/1.53 wellorderingAx: $o ).
% 2.44/1.53
% 2.44/1.53 thf(wellorderingAx,definition,
% 2.44/1.53 ( wellorderingAx
% 2.44/1.53 = ( ! [A: $i] :
% 2.44/1.53 ? [B: $i] :
% 2.44/1.53 ( ! [C: $i] :
% 2.44/1.53 ( ( in @ C @ B )
% 2.44/1.53 => ! [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ C )
% 2.44/1.53 => ( in @ Xx @ A ) ) )
% 2.44/1.53 & ! [Xx: $i,Xy: $i] :
% 2.44/1.53 ( ( ( in @ Xx @ A )
% 2.44/1.53 & ( in @ Xy @ A ) )
% 2.44/1.53 => ( ! [C: $i] :
% 2.44/1.53 ( ( in @ C @ B )
% 2.44/1.53 => ( ( in @ Xx @ C )
% 2.44/1.53 <=> ( in @ Xy @ C ) ) )
% 2.44/1.53 => ( Xx = Xy ) ) )
% 2.44/1.53 & ! [C: $i,D: $i] :
% 2.44/1.53 ( ( ( in @ C @ B )
% 2.44/1.53 & ( in @ D @ B ) )
% 2.44/1.53 => ( ! [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ C )
% 2.44/1.53 => ( in @ Xx @ D ) )
% 2.44/1.53 | ! [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ D )
% 2.44/1.53 => ( in @ Xx @ C ) ) ) )
% 2.44/1.53 & ! [C: $i] :
% 2.44/1.53 ( ( ! [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ C )
% 2.44/1.53 => ( in @ Xx @ A ) )
% 2.44/1.53 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.44/1.53 => ? [D: $i,Xx: $i] :
% 2.44/1.53 ( ( in @ D @ B )
% 2.44/1.53 & ( in @ Xx @ C )
% 2.44/1.53 & ~ ? [Xy: $i] :
% 2.44/1.53 ( ( in @ Xy @ D )
% 2.44/1.53 & ( in @ Xy @ C ) )
% 2.44/1.53 & ! [E: $i] :
% 2.44/1.53 ( ( in @ E @ B )
% 2.44/1.53 => ( ! [Xy: $i] :
% 2.44/1.53 ( ( in @ Xy @ E )
% 2.44/1.53 => ( in @ Xy @ D ) )
% 2.44/1.53 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(descr_type,type,
% 2.44/1.53 descr: ( $i > $o ) > $i ).
% 2.44/1.53
% 2.44/1.53 thf(descrp_type,type,
% 2.44/1.53 descrp: $o ).
% 2.44/1.53
% 2.44/1.53 thf(descrp,definition,
% 2.44/1.53 ( descrp
% 2.44/1.53 = ( ! [Xphi: $i > $o] :
% 2.44/1.53 ( ( exu
% 2.44/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.44/1.53 => ( Xphi
% 2.44/1.53 @ ( descr
% 2.44/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(dsetconstr_type,type,
% 2.44/1.53 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.44/1.53
% 2.44/1.53 thf(dsetconstrI_type,type,
% 2.44/1.53 dsetconstrI: $o ).
% 2.44/1.53
% 2.44/1.53 thf(dsetconstrI,definition,
% 2.44/1.53 ( dsetconstrI
% 2.44/1.53 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 => ( ( Xphi @ Xx )
% 2.44/1.53 => ( in @ Xx
% 2.44/1.53 @ ( dsetconstr @ A
% 2.44/1.53 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(dsetconstrEL_type,type,
% 2.44/1.53 dsetconstrEL: $o ).
% 2.44/1.53
% 2.44/1.53 thf(dsetconstrEL,definition,
% 2.44/1.53 ( dsetconstrEL
% 2.44/1.53 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.44/1.53 ( ( in @ Xx
% 2.44/1.53 @ ( dsetconstr @ A
% 2.44/1.53 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.44/1.53 => ( in @ Xx @ A ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(dsetconstrER_type,type,
% 2.44/1.53 dsetconstrER: $o ).
% 2.44/1.53
% 2.44/1.53 thf(dsetconstrER,definition,
% 2.44/1.53 ( dsetconstrER
% 2.44/1.53 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.44/1.53 ( ( in @ Xx
% 2.44/1.53 @ ( dsetconstr @ A
% 2.44/1.53 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.44/1.53 => ( Xphi @ Xx ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(exuE1_type,type,
% 2.44/1.53 exuE1: $o ).
% 2.44/1.53
% 2.44/1.53 thf(exuE1,definition,
% 2.44/1.53 ( exuE1
% 2.44/1.53 = ( ! [Xphi: $i > $o] :
% 2.44/1.53 ( ( exu
% 2.44/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.44/1.53 => ? [Xx: $i] :
% 2.44/1.53 ( ( Xphi @ Xx )
% 2.44/1.53 & ! [Xy: $i] :
% 2.44/1.53 ( ( Xphi @ Xy )
% 2.44/1.53 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(prop2set_type,type,
% 2.44/1.53 prop2set: $o > $i ).
% 2.44/1.53
% 2.44/1.53 thf(prop2setE_type,type,
% 2.44/1.53 prop2setE: $o ).
% 2.44/1.53
% 2.44/1.53 thf(prop2setE,definition,
% 2.44/1.53 ( prop2setE
% 2.44/1.53 = ( ! [Xphi: $o,Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.44/1.53 => Xphi ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(emptysetE_type,type,
% 2.44/1.53 emptysetE: $o ).
% 2.44/1.53
% 2.44/1.53 thf(emptysetE,definition,
% 2.44/1.53 ( emptysetE
% 2.44/1.53 = ( ! [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ emptyset )
% 2.44/1.53 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(emptysetimpfalse_type,type,
% 2.44/1.53 emptysetimpfalse: $o ).
% 2.44/1.53
% 2.44/1.53 thf(emptysetimpfalse,definition,
% 2.44/1.53 ( emptysetimpfalse
% 2.44/1.53 = ( ! [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ emptyset )
% 2.44/1.53 => $false ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(notinemptyset_type,type,
% 2.44/1.53 notinemptyset: $o ).
% 2.44/1.53
% 2.44/1.53 thf(notinemptyset,definition,
% 2.44/1.53 ( notinemptyset
% 2.44/1.53 = ( ! [Xx: $i] :
% 2.44/1.53 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(exuE3e_type,type,
% 2.44/1.53 exuE3e: $o ).
% 2.44/1.53
% 2.44/1.53 thf(exuE3e,definition,
% 2.44/1.53 ( exuE3e
% 2.44/1.53 = ( ! [Xphi: $i > $o] :
% 2.44/1.53 ( ( exu
% 2.44/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.44/1.53 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(setext_type,type,
% 2.44/1.53 setext: $o ).
% 2.44/1.53
% 2.44/1.53 thf(setext,definition,
% 2.44/1.53 ( setext
% 2.44/1.53 = ( ! [A: $i,B: $i] :
% 2.44/1.53 ( ! [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 => ( in @ Xx @ B ) )
% 2.44/1.53 => ( ! [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ B )
% 2.44/1.53 => ( in @ Xx @ A ) )
% 2.44/1.53 => ( A = B ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(emptyI_type,type,
% 2.44/1.53 emptyI: $o ).
% 2.44/1.53
% 2.44/1.53 thf(emptyI,definition,
% 2.44/1.53 ( emptyI
% 2.44/1.53 = ( ! [A: $i] :
% 2.44/1.53 ( ! [Xx: $i] :
% 2.44/1.53 ~ ( in @ Xx @ A )
% 2.44/1.53 => ( A = emptyset ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(noeltsimpempty_type,type,
% 2.44/1.53 noeltsimpempty: $o ).
% 2.44/1.53
% 2.44/1.53 thf(noeltsimpempty,definition,
% 2.44/1.53 ( noeltsimpempty
% 2.44/1.53 = ( ! [A: $i] :
% 2.44/1.53 ( ! [Xx: $i] :
% 2.44/1.53 ~ ( in @ Xx @ A )
% 2.44/1.53 => ( A = emptyset ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(setbeta_type,type,
% 2.44/1.53 setbeta: $o ).
% 2.44/1.53
% 2.44/1.53 thf(setbeta,definition,
% 2.44/1.53 ( setbeta
% 2.44/1.53 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 => ( ( in @ Xx
% 2.44/1.53 @ ( dsetconstr @ A
% 2.44/1.53 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.44/1.53 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(nonempty_type,type,
% 2.44/1.53 nonempty: $i > $o ).
% 2.44/1.53
% 2.44/1.53 thf(nonempty,definition,
% 2.44/1.53 ( nonempty
% 2.44/1.53 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(nonemptyE1_type,type,
% 2.44/1.53 nonemptyE1: $o ).
% 2.44/1.53
% 2.44/1.53 thf(nonemptyE1,definition,
% 2.44/1.53 ( nonemptyE1
% 2.44/1.53 = ( ! [A: $i] :
% 2.44/1.53 ( ( nonempty @ A )
% 2.44/1.53 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(nonemptyI_type,type,
% 2.44/1.53 nonemptyI: $o ).
% 2.44/1.53
% 2.44/1.53 thf(nonemptyI,definition,
% 2.44/1.53 ( nonemptyI
% 2.44/1.53 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 => ( ( Xphi @ Xx )
% 2.44/1.53 => ( nonempty
% 2.44/1.53 @ ( dsetconstr @ A
% 2.44/1.53 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(nonemptyI1_type,type,
% 2.44/1.53 nonemptyI1: $o ).
% 2.44/1.53
% 2.44/1.53 thf(nonemptyI1,definition,
% 2.44/1.53 ( nonemptyI1
% 2.44/1.53 = ( ! [A: $i] :
% 2.44/1.53 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.44/1.53 => ( nonempty @ A ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(setadjoinIL_type,type,
% 2.44/1.53 setadjoinIL: $o ).
% 2.44/1.53
% 2.44/1.53 thf(setadjoinIL,definition,
% 2.44/1.53 ( setadjoinIL
% 2.44/1.53 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(emptyinunitempty_type,type,
% 2.44/1.53 emptyinunitempty: $o ).
% 2.44/1.53
% 2.44/1.53 thf(emptyinunitempty,definition,
% 2.44/1.53 ( emptyinunitempty
% 2.44/1.53 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(setadjoinIR_type,type,
% 2.44/1.53 setadjoinIR: $o ).
% 2.44/1.53
% 2.44/1.53 thf(setadjoinIR,definition,
% 2.44/1.53 ( setadjoinIR
% 2.44/1.53 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.44/1.53 ( ( in @ Xy @ A )
% 2.44/1.53 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(setadjoinE_type,type,
% 2.44/1.53 setadjoinE: $o ).
% 2.44/1.53
% 2.44/1.53 thf(setadjoinE,definition,
% 2.44/1.53 ( setadjoinE
% 2.44/1.53 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.44/1.53 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.44/1.53 => ! [Xphi: $o] :
% 2.44/1.53 ( ( ( Xy = Xx )
% 2.44/1.53 => Xphi )
% 2.44/1.53 => ( ( ( in @ Xy @ A )
% 2.44/1.53 => Xphi )
% 2.44/1.53 => Xphi ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(setadjoinOr_type,type,
% 2.44/1.53 setadjoinOr: $o ).
% 2.44/1.53
% 2.44/1.53 thf(setadjoinOr,definition,
% 2.44/1.53 ( setadjoinOr
% 2.44/1.53 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.44/1.53 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.44/1.53 => ( ( Xy = Xx )
% 2.44/1.53 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(setoftrueEq_type,type,
% 2.44/1.53 setoftrueEq: $o ).
% 2.44/1.53
% 2.44/1.53 thf(setoftrueEq,definition,
% 2.44/1.53 ( setoftrueEq
% 2.44/1.53 = ( ! [A: $i] :
% 2.44/1.53 ( ( dsetconstr @ A
% 2.44/1.53 @ ^ [Xx: $i] : $true )
% 2.44/1.53 = A ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(powersetI_type,type,
% 2.44/1.53 powersetI: $o ).
% 2.44/1.53
% 2.44/1.53 thf(powersetI,definition,
% 2.44/1.53 ( powersetI
% 2.44/1.53 = ( ! [A: $i,B: $i] :
% 2.44/1.53 ( ! [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ B )
% 2.44/1.53 => ( in @ Xx @ A ) )
% 2.44/1.53 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(emptyinPowerset_type,type,
% 2.44/1.53 emptyinPowerset: $o ).
% 2.44/1.53
% 2.44/1.53 thf(emptyinPowerset,definition,
% 2.44/1.53 ( emptyinPowerset
% 2.44/1.53 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(emptyInPowerset_type,type,
% 2.44/1.53 emptyInPowerset: $o ).
% 2.44/1.53
% 2.44/1.53 thf(emptyInPowerset,definition,
% 2.44/1.53 ( emptyInPowerset
% 2.44/1.53 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(powersetE_type,type,
% 2.44/1.53 powersetE: $o ).
% 2.44/1.53
% 2.44/1.53 thf(powersetE,definition,
% 2.44/1.53 ( powersetE
% 2.44/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.44/1.53 ( ( in @ B @ ( powerset @ A ) )
% 2.44/1.53 => ( ( in @ Xx @ B )
% 2.44/1.53 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(setunionI_type,type,
% 2.44/1.53 setunionI: $o ).
% 2.44/1.53
% 2.44/1.53 thf(setunionI,definition,
% 2.44/1.53 ( setunionI
% 2.44/1.53 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.44/1.53 ( ( in @ Xx @ B )
% 2.44/1.53 => ( ( in @ B @ A )
% 2.44/1.53 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(setunionE_type,type,
% 2.44/1.53 setunionE: $o ).
% 2.44/1.53
% 2.44/1.53 thf(setunionE,definition,
% 2.44/1.53 ( setunionE
% 2.44/1.53 = ( ! [A: $i,Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ ( setunion @ A ) )
% 2.44/1.53 => ! [Xphi: $o] :
% 2.44/1.53 ( ! [B: $i] :
% 2.44/1.53 ( ( in @ Xx @ B )
% 2.44/1.53 => ( ( in @ B @ A )
% 2.44/1.53 => Xphi ) )
% 2.44/1.53 => Xphi ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(subPowSU_type,type,
% 2.44/1.53 subPowSU: $o ).
% 2.44/1.53
% 2.44/1.53 thf(subPowSU,definition,
% 2.44/1.53 ( subPowSU
% 2.44/1.53 = ( ! [A: $i,Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(exuE2_type,type,
% 2.44/1.53 exuE2: $o ).
% 2.44/1.53
% 2.44/1.53 thf(exuE2,definition,
% 2.44/1.53 ( exuE2
% 2.44/1.53 = ( ! [Xphi: $i > $o] :
% 2.44/1.53 ( ( exu
% 2.44/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.44/1.53 => ? [Xx: $i] :
% 2.44/1.53 ! [Xy: $i] :
% 2.44/1.53 ( ( Xphi @ Xy )
% 2.44/1.53 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(nonemptyImpWitness_type,type,
% 2.44/1.53 nonemptyImpWitness: $o ).
% 2.44/1.53
% 2.44/1.53 thf(nonemptyImpWitness,definition,
% 2.44/1.53 ( nonemptyImpWitness
% 2.44/1.53 = ( ! [A: $i] :
% 2.44/1.53 ( ( nonempty @ A )
% 2.44/1.53 => ? [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 & $true ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(uniqinunit_type,type,
% 2.44/1.53 uniqinunit: $o ).
% 2.44/1.53
% 2.44/1.53 thf(uniqinunit,definition,
% 2.44/1.53 ( uniqinunit
% 2.44/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.44/1.53 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.44/1.53 => ( Xx = Xy ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(notinsingleton_type,type,
% 2.44/1.53 notinsingleton: $o ).
% 2.44/1.53
% 2.44/1.53 thf(notinsingleton,definition,
% 2.44/1.53 ( notinsingleton
% 2.44/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.44/1.53 ( ( Xx != Xy )
% 2.44/1.53 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(eqinunit_type,type,
% 2.44/1.53 eqinunit: $o ).
% 2.44/1.53
% 2.44/1.53 thf(eqinunit,definition,
% 2.44/1.53 ( eqinunit
% 2.44/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.44/1.53 ( ( Xx = Xy )
% 2.44/1.53 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(singletonsswitch_type,type,
% 2.44/1.53 singletonsswitch: $o ).
% 2.44/1.53
% 2.44/1.53 thf(singletonsswitch,definition,
% 2.44/1.53 ( singletonsswitch
% 2.44/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.44/1.53 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.44/1.53 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(upairsetE_type,type,
% 2.44/1.53 upairsetE: $o ).
% 2.44/1.53
% 2.44/1.53 thf(upairsetE,definition,
% 2.44/1.53 ( upairsetE
% 2.44/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.44/1.53 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.44/1.53 => ( ( Xz = Xx )
% 2.44/1.53 | ( Xz = Xy ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(upairsetIL_type,type,
% 2.44/1.53 upairsetIL: $o ).
% 2.44/1.53
% 2.44/1.53 thf(upairsetIL,definition,
% 2.44/1.53 ( upairsetIL
% 2.44/1.53 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(upairsetIR_type,type,
% 2.44/1.53 upairsetIR: $o ).
% 2.44/1.53
% 2.44/1.53 thf(upairsetIR,definition,
% 2.44/1.53 ( upairsetIR
% 2.44/1.53 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(emptyE1_type,type,
% 2.44/1.53 emptyE1: $o ).
% 2.44/1.53
% 2.44/1.53 thf(emptyE1,definition,
% 2.44/1.53 ( emptyE1
% 2.44/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.44/1.53 ( ? [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 & ( Xphi @ Xx ) )
% 2.44/1.53 => ( ( ( dsetconstr @ A
% 2.44/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.44/1.53 = emptyset )
% 2.44/1.53 => $false ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(vacuousDall_type,type,
% 2.44/1.53 vacuousDall: $o ).
% 2.44/1.53
% 2.44/1.53 thf(vacuousDall,definition,
% 2.44/1.53 ( vacuousDall
% 2.44/1.53 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ emptyset )
% 2.44/1.53 => ( Xphi @ Xx ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(quantDeMorgan1_type,type,
% 2.44/1.53 quantDeMorgan1: $o ).
% 2.44/1.53
% 2.44/1.53 thf(quantDeMorgan1,definition,
% 2.44/1.53 ( quantDeMorgan1
% 2.44/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.44/1.53 ( ~ ! [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 => ( Xphi @ Xx ) )
% 2.44/1.53 => ? [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(quantDeMorgan2_type,type,
% 2.44/1.53 quantDeMorgan2: $o ).
% 2.44/1.53
% 2.44/1.53 thf(quantDeMorgan2,definition,
% 2.44/1.53 ( quantDeMorgan2
% 2.44/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.44/1.53 ( ! [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 => ~ ( Xphi @ Xx ) )
% 2.44/1.53 => ~ ? [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(quantDeMorgan3_type,type,
% 2.44/1.53 quantDeMorgan3: $o ).
% 2.44/1.53
% 2.44/1.53 thf(quantDeMorgan3,definition,
% 2.44/1.53 ( quantDeMorgan3
% 2.44/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.44/1.53 ( ~ ? [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 & ( Xphi @ Xx ) )
% 2.44/1.53 => ! [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(quantDeMorgan4_type,type,
% 2.44/1.53 quantDeMorgan4: $o ).
% 2.44/1.53
% 2.44/1.53 thf(quantDeMorgan4,definition,
% 2.44/1.53 ( quantDeMorgan4
% 2.44/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.44/1.53 ( ? [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 & ~ ( Xphi @ Xx ) )
% 2.44/1.53 => ~ ! [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(prop2setI_type,type,
% 2.44/1.53 prop2setI: $o ).
% 2.44/1.53
% 2.44/1.53 thf(prop2setI,definition,
% 2.44/1.53 ( prop2setI
% 2.44/1.53 = ( ! [Xphi: $o] :
% 2.44/1.53 ( Xphi
% 2.44/1.53 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(set2prop_type,type,
% 2.44/1.53 set2prop: $i > $o ).
% 2.44/1.53
% 2.44/1.53 thf(prop2set2propI_type,type,
% 2.44/1.53 prop2set2propI: $o ).
% 2.44/1.53
% 2.44/1.53 thf(prop2set2propI,definition,
% 2.44/1.53 ( prop2set2propI
% 2.44/1.53 = ( ! [Xphi: $o] :
% 2.44/1.53 ( Xphi
% 2.44/1.53 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(notdexE_type,type,
% 2.44/1.53 notdexE: $o ).
% 2.44/1.53
% 2.44/1.53 thf(notdexE,definition,
% 2.44/1.53 ( notdexE
% 2.44/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.44/1.53 ( ~ ? [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 & ( Xphi @ Xx ) )
% 2.44/1.53 => ! [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(notdallE_type,type,
% 2.44/1.53 notdallE: $o ).
% 2.44/1.53
% 2.44/1.53 thf(notdallE,definition,
% 2.44/1.53 ( notdallE
% 2.44/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.44/1.53 ( ~ ! [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 => ( Xphi @ Xx ) )
% 2.44/1.53 => ? [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(exuI1_type,type,
% 2.44/1.53 exuI1: $o ).
% 2.44/1.53
% 2.44/1.53 thf(exuI1,definition,
% 2.44/1.53 ( exuI1
% 2.44/1.53 = ( ! [Xphi: $i > $o] :
% 2.44/1.53 ( ? [Xx: $i] :
% 2.44/1.53 ( ( Xphi @ Xx )
% 2.44/1.53 & ! [Xy: $i] :
% 2.44/1.53 ( ( Xphi @ Xy )
% 2.44/1.53 => ( Xx = Xy ) ) )
% 2.44/1.53 => ( exu
% 2.44/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(exuI3_type,type,
% 2.44/1.53 exuI3: $o ).
% 2.44/1.53
% 2.44/1.53 thf(exuI3,definition,
% 2.44/1.53 ( exuI3
% 2.44/1.53 = ( ! [Xphi: $i > $o] :
% 2.44/1.53 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.44/1.53 => ( ! [Xx: $i,Xy: $i] :
% 2.44/1.53 ( ( Xphi @ Xx )
% 2.44/1.53 => ( ( Xphi @ Xy )
% 2.44/1.53 => ( Xx = Xy ) ) )
% 2.44/1.53 => ( exu
% 2.44/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(exuI2_type,type,
% 2.44/1.53 exuI2: $o ).
% 2.44/1.53
% 2.44/1.53 thf(exuI2,definition,
% 2.44/1.53 ( exuI2
% 2.44/1.53 = ( ! [Xphi: $i > $o] :
% 2.44/1.53 ( ? [Xx: $i] :
% 2.44/1.53 ! [Xy: $i] :
% 2.44/1.53 ( ( Xphi @ Xy )
% 2.44/1.53 <=> ( Xy = Xx ) )
% 2.44/1.53 => ( exu
% 2.44/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(inCongP_type,type,
% 2.44/1.53 inCongP: $o ).
% 2.44/1.53
% 2.44/1.53 thf(inCongP,definition,
% 2.44/1.53 ( inCongP
% 2.44/1.53 = ( ! [A: $i,B: $i] :
% 2.44/1.53 ( ( A = B )
% 2.44/1.53 => ! [Xx: $i,Xy: $i] :
% 2.44/1.53 ( ( Xx = Xy )
% 2.44/1.53 => ( ( in @ Xx @ A )
% 2.44/1.53 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(in__Cong_type,type,
% 2.44/1.53 in__Cong: $o ).
% 2.44/1.53
% 2.44/1.53 thf(in__Cong,definition,
% 2.44/1.53 ( in__Cong
% 2.44/1.53 = ( ! [A: $i,B: $i] :
% 2.44/1.53 ( ( A = B )
% 2.44/1.53 => ! [Xx: $i,Xy: $i] :
% 2.44/1.53 ( ( Xx = Xy )
% 2.44/1.53 => ( ( in @ Xx @ A )
% 2.44/1.53 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(exuE3u_type,type,
% 2.44/1.53 exuE3u: $o ).
% 2.44/1.53
% 2.44/1.53 thf(exuE3u,definition,
% 2.44/1.53 ( exuE3u
% 2.44/1.53 = ( ! [Xphi: $i > $o] :
% 2.44/1.53 ( ( exu
% 2.44/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.44/1.53 => ! [Xx: $i,Xy: $i] :
% 2.44/1.53 ( ( Xphi @ Xx )
% 2.44/1.53 => ( ( Xphi @ Xy )
% 2.44/1.53 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(exu__Cong_type,type,
% 2.44/1.53 exu__Cong: $o ).
% 2.44/1.53
% 2.44/1.53 thf(exu__Cong,definition,
% 2.44/1.53 ( exu__Cong
% 2.44/1.53 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.44/1.53 ( ! [Xx: $i,Xy: $i] :
% 2.44/1.53 ( ( Xx = Xy )
% 2.44/1.53 => ( ( Xphi @ Xx )
% 2.44/1.53 <=> ( Xpsi @ Xy ) ) )
% 2.44/1.53 => ( ( exu
% 2.44/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.44/1.53 <=> ( exu
% 2.44/1.53 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(emptyset__Cong_type,type,
% 2.44/1.53 emptyset__Cong: $o ).
% 2.44/1.53
% 2.44/1.53 thf(emptyset__Cong,definition,
% 2.44/1.53 ( emptyset__Cong
% 2.44/1.53 = ( emptyset = emptyset ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(setadjoin__Cong_type,type,
% 2.44/1.53 setadjoin__Cong: $o ).
% 2.44/1.53
% 2.44/1.53 thf(setadjoin__Cong,definition,
% 2.44/1.53 ( setadjoin__Cong
% 2.44/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.44/1.53 ( ( Xx = Xy )
% 2.44/1.53 => ! [Xz: $i,Xu: $i] :
% 2.44/1.53 ( ( Xz = Xu )
% 2.44/1.53 => ( ( setadjoin @ Xx @ Xz )
% 2.44/1.53 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(powerset__Cong_type,type,
% 2.44/1.53 powerset__Cong: $o ).
% 2.44/1.53
% 2.44/1.53 thf(powerset__Cong,definition,
% 2.44/1.53 ( powerset__Cong
% 2.44/1.53 = ( ! [A: $i,B: $i] :
% 2.44/1.53 ( ( A = B )
% 2.44/1.53 => ( ( powerset @ A )
% 2.44/1.53 = ( powerset @ B ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(setunion__Cong_type,type,
% 2.44/1.53 setunion__Cong: $o ).
% 2.44/1.53
% 2.44/1.53 thf(setunion__Cong,definition,
% 2.44/1.53 ( setunion__Cong
% 2.44/1.53 = ( ! [A: $i,B: $i] :
% 2.44/1.53 ( ( A = B )
% 2.44/1.53 => ( ( setunion @ A )
% 2.44/1.53 = ( setunion @ B ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(omega__Cong_type,type,
% 2.44/1.53 omega__Cong: $o ).
% 2.44/1.53
% 2.44/1.53 thf(omega__Cong,definition,
% 2.44/1.53 ( omega__Cong
% 2.44/1.53 = ( omega = omega ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(exuEu_type,type,
% 2.44/1.53 exuEu: $o ).
% 2.44/1.53
% 2.44/1.53 thf(exuEu,definition,
% 2.44/1.53 ( exuEu
% 2.44/1.53 = ( ! [Xphi: $i > $o] :
% 2.44/1.53 ( ( exu
% 2.44/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.44/1.53 => ! [Xx: $i,Xy: $i] :
% 2.44/1.53 ( ( Xphi @ Xx )
% 2.44/1.53 => ( ( Xphi @ Xy )
% 2.44/1.53 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(descr__Cong_type,type,
% 2.44/1.53 descr__Cong: $o ).
% 2.44/1.53
% 2.44/1.53 thf(descr__Cong,definition,
% 2.44/1.53 ( descr__Cong
% 2.44/1.53 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.44/1.53 ( ! [Xx: $i,Xy: $i] :
% 2.44/1.53 ( ( Xx = Xy )
% 2.44/1.53 => ( ( Xphi @ Xx )
% 2.44/1.53 <=> ( Xpsi @ Xy ) ) )
% 2.44/1.53 => ( ( exu
% 2.44/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.44/1.53 => ( ( exu
% 2.44/1.53 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.44/1.53 => ( ( descr
% 2.44/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.44/1.53 = ( descr
% 2.44/1.53 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(dsetconstr__Cong_type,type,
% 2.44/1.53 dsetconstr__Cong: $o ).
% 2.44/1.53
% 2.44/1.53 thf(dsetconstr__Cong,definition,
% 2.44/1.53 ( dsetconstr__Cong
% 2.44/1.53 = ( ! [A: $i,B: $i] :
% 2.44/1.53 ( ( A = B )
% 2.44/1.53 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.44/1.53 ( ! [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 => ! [Xy: $i] :
% 2.44/1.53 ( ( in @ Xy @ B )
% 2.44/1.53 => ( ( Xx = Xy )
% 2.44/1.53 => ( ( Xphi @ Xx )
% 2.44/1.53 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.44/1.53 => ( ( dsetconstr @ A
% 2.44/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.44/1.53 = ( dsetconstr @ B
% 2.44/1.53 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(subset_type,type,
% 2.44/1.53 subset: $i > $i > $o ).
% 2.44/1.53
% 2.44/1.53 thf(disjoint_type,type,
% 2.44/1.53 disjoint: $i > $i > $o ).
% 2.44/1.53
% 2.44/1.53 thf(setsmeet_type,type,
% 2.44/1.53 setsmeet: $i > $i > $o ).
% 2.44/1.53
% 2.44/1.53 thf(subsetI1_type,type,
% 2.44/1.53 subsetI1: $o ).
% 2.44/1.53
% 2.44/1.53 thf(subsetI1,definition,
% 2.44/1.53 ( subsetI1
% 2.44/1.53 = ( ! [A: $i,B: $i] :
% 2.44/1.53 ( ! [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 => ( in @ Xx @ B ) )
% 2.44/1.53 => ( subset @ A @ B ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(eqimpsubset2_type,type,
% 2.44/1.53 eqimpsubset2: $o ).
% 2.44/1.53
% 2.44/1.53 thf(eqimpsubset2,definition,
% 2.44/1.53 ( eqimpsubset2
% 2.44/1.53 = ( ! [A: $i,B: $i] :
% 2.44/1.53 ( ( A = B )
% 2.44/1.53 => ( subset @ B @ A ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(eqimpsubset1_type,type,
% 2.44/1.53 eqimpsubset1: $o ).
% 2.44/1.53
% 2.44/1.53 thf(eqimpsubset1,definition,
% 2.44/1.53 ( eqimpsubset1
% 2.44/1.53 = ( ! [A: $i,B: $i] :
% 2.44/1.53 ( ( A = B )
% 2.44/1.53 => ( subset @ A @ B ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(subsetI2_type,type,
% 2.44/1.53 subsetI2: $o ).
% 2.44/1.53
% 2.44/1.53 thf(subsetI2,definition,
% 2.44/1.53 ( subsetI2
% 2.44/1.53 = ( ! [A: $i,B: $i] :
% 2.44/1.53 ( ! [Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 => ( in @ Xx @ B ) )
% 2.44/1.53 => ( subset @ A @ B ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(emptysetsubset_type,type,
% 2.44/1.53 emptysetsubset: $o ).
% 2.44/1.53
% 2.44/1.53 thf(emptysetsubset,definition,
% 2.44/1.53 ( emptysetsubset
% 2.44/1.53 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(subsetE_type,type,
% 2.44/1.53 subsetE: $o ).
% 2.44/1.53
% 2.44/1.53 thf(subsetE,definition,
% 2.44/1.53 ( subsetE
% 2.44/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.44/1.53 ( ( subset @ A @ B )
% 2.44/1.53 => ( ( in @ Xx @ A )
% 2.44/1.53 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(subsetE2_type,type,
% 2.44/1.53 subsetE2: $o ).
% 2.44/1.53
% 2.44/1.53 thf(subsetE2,definition,
% 2.44/1.53 ( subsetE2
% 2.44/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.44/1.53 ( ( subset @ A @ B )
% 2.44/1.53 => ( ~ ( in @ Xx @ B )
% 2.44/1.53 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(notsubsetI_type,type,
% 2.44/1.53 notsubsetI: $o ).
% 2.44/1.53
% 2.44/1.53 thf(notsubsetI,definition,
% 2.44/1.53 ( notsubsetI
% 2.44/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 => ( ~ ( in @ Xx @ B )
% 2.44/1.53 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(notequalI1_type,type,
% 2.44/1.53 notequalI1: $o ).
% 2.44/1.53
% 2.44/1.53 thf(notequalI1,definition,
% 2.44/1.53 ( notequalI1
% 2.44/1.53 = ( ! [A: $i,B: $i] :
% 2.44/1.53 ( ~ ( subset @ A @ B )
% 2.44/1.53 => ( A != B ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(notequalI2_type,type,
% 2.44/1.53 notequalI2: $o ).
% 2.44/1.53
% 2.44/1.53 thf(notequalI2,definition,
% 2.44/1.53 ( notequalI2
% 2.44/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.44/1.53 ( ( in @ Xx @ A )
% 2.44/1.53 => ( ~ ( in @ Xx @ B )
% 2.44/1.53 => ( A != B ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(subsetRefl_type,type,
% 2.44/1.53 subsetRefl: $o ).
% 2.44/1.53
% 2.44/1.53 thf(subsetRefl,definition,
% 2.44/1.53 ( subsetRefl
% 2.44/1.53 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(subsetTrans_type,type,
% 2.44/1.53 subsetTrans: $o ).
% 2.44/1.53
% 2.44/1.53 thf(subsetTrans,definition,
% 2.44/1.53 ( subsetTrans
% 2.44/1.53 = ( ! [A: $i,B: $i,C: $i] :
% 2.44/1.53 ( ( subset @ A @ B )
% 2.44/1.53 => ( ( subset @ B @ C )
% 2.44/1.53 => ( subset @ A @ C ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(setadjoinSub_type,type,
% 2.44/1.53 setadjoinSub: $o ).
% 2.44/1.53
% 2.44/1.53 thf(setadjoinSub,definition,
% 2.44/1.53 ( setadjoinSub
% 2.44/1.53 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(setadjoinSub2_type,type,
% 2.44/1.53 setadjoinSub2: $o ).
% 2.44/1.53
% 2.44/1.53 thf(setadjoinSub2,definition,
% 2.44/1.53 ( setadjoinSub2
% 2.44/1.53 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.44/1.53 ( ( subset @ A @ B )
% 2.44/1.53 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(subset2powerset_type,type,
% 2.44/1.53 subset2powerset: $o ).
% 2.44/1.53
% 2.44/1.53 thf(subset2powerset,definition,
% 2.44/1.53 ( subset2powerset
% 2.44/1.53 = ( ! [A: $i,B: $i] :
% 2.44/1.53 ( ( subset @ A @ B )
% 2.44/1.53 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(setextsub_type,type,
% 2.44/1.53 setextsub: $o ).
% 2.44/1.53
% 2.44/1.53 thf(setextsub,definition,
% 2.44/1.53 ( setextsub
% 2.44/1.53 = ( ! [A: $i,B: $i] :
% 2.44/1.53 ( ( subset @ A @ B )
% 2.44/1.53 => ( ( subset @ B @ A )
% 2.44/1.53 => ( A = B ) ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(subsetemptysetimpeq_type,type,
% 2.44/1.53 subsetemptysetimpeq: $o ).
% 2.44/1.53
% 2.44/1.53 thf(subsetemptysetimpeq,definition,
% 2.44/1.53 ( subsetemptysetimpeq
% 2.44/1.53 = ( ! [A: $i] :
% 2.44/1.53 ( ( subset @ A @ emptyset )
% 2.44/1.53 => ( A = emptyset ) ) ) ) ).
% 2.44/1.53
% 2.44/1.53 thf(powersetI1,conjecture,
% 2.44/1.53 ( setextAx
% 2.44/1.53 => ( emptysetAx
% 2.44/1.53 => ( setadjoinAx
% 2.44/1.53 => ( powersetAx
% 2.44/1.53 => ( setunionAx
% 2.44/1.53 => ( omega0Ax
% 2.44/1.53 => ( omegaSAx
% 2.44/1.53 => ( omegaIndAx
% 2.44/1.53 => ( replAx
% 2.44/1.53 => ( foundationAx
% 2.44/1.53 => ( wellorderingAx
% 2.44/1.53 => ( descrp
% 2.44/1.53 => ( dsetconstrI
% 2.44/1.53 => ( dsetconstrEL
% 2.44/1.53 => ( dsetconstrER
% 2.44/1.53 => ( exuE1
% 2.44/1.53 => ( prop2setE
% 2.44/1.53 => ( emptysetE
% 2.44/1.53 => ( emptysetimpfalse
% 2.44/1.53 => ( notinemptyset
% 2.44/1.53 => ( exuE3e
% 2.44/1.53 => ( setext
% 2.44/1.53 => ( emptyI
% 2.44/1.53 => ( noeltsimpempty
% 2.44/1.53 => ( setbeta
% 2.44/1.53 => ( nonemptyE1
% 2.44/1.53 => ( nonemptyI
% 2.44/1.53 => ( nonemptyI1
% 2.44/1.53 => ( setadjoinIL
% 2.44/1.53 => ( emptyinunitempty
% 2.44/1.53 => ( setadjoinIR
% 2.44/1.53 => ( setadjoinE
% 2.44/1.53 => ( setadjoinOr
% 2.44/1.53 => ( setoftrueEq
% 2.44/1.53 => ( powersetI
% 2.44/1.53 => ( emptyinPowerset
% 2.44/1.53 => ( emptyInPowerset
% 2.44/1.53 => ( powersetE
% 2.44/1.53 => ( setunionI
% 2.44/1.53 => ( setunionE
% 2.44/1.53 => ( subPowSU
% 2.44/1.53 => ( exuE2
% 2.44/1.53 => ( nonemptyImpWitness
% 2.44/1.53 => ( uniqinunit
% 2.44/1.53 => ( notinsingleton
% 2.44/1.53 => ( eqinunit
% 2.44/1.53 => ( singletonsswitch
% 2.44/1.53 => ( upairsetE
% 2.44/1.53 => ( upairsetIL
% 2.44/1.53 => ( upairsetIR
% 2.44/1.53 => ( emptyE1
% 2.44/1.53 => ( vacuousDall
% 2.44/1.53 => ( quantDeMorgan1
% 2.44/1.53 => ( quantDeMorgan2
% 2.44/1.53 => ( quantDeMorgan3
% 2.44/1.53 => ( quantDeMorgan4
% 2.44/1.53 => ( prop2setI
% 2.44/1.53 => ( prop2set2propI
% 2.44/1.53 => ( notdexE
% 2.44/1.53 => ( notdallE
% 2.44/1.53 => ( exuI1
% 2.44/1.53 => ( exuI3
% 2.44/1.53 => ( exuI2
% 2.44/1.53 => ( inCongP
% 2.44/1.53 => ( in__Cong
% 2.44/1.53 => ( exuE3u
% 2.44/1.53 => ( exu__Cong
% 2.44/1.53 => ( emptyset__Cong
% 2.44/1.53 => ( setadjoin__Cong
% 2.44/1.53 => ( powerset__Cong
% 2.44/1.53 => ( setunion__Cong
% 2.44/1.53 => ( omega__Cong
% 2.44/1.53 => ( exuEu
% 2.44/1.53 => ( descr__Cong
% 2.44/1.53 => ( dsetconstr__Cong
% 2.44/1.53 => ( subsetI1
% 2.44/1.53 => ( eqimpsubset2
% 2.44/1.53 => ( eqimpsubset1
% 2.44/1.53 => ( subsetI2
% 2.44/1.53 => ( emptysetsubset
% 2.44/1.53 => ( subsetE
% 2.44/1.53 => ( subsetE2
% 2.44/1.53 => ( notsubsetI
% 2.44/1.53 => ( notequalI1
% 2.44/1.53 => ( notequalI2
% 2.44/1.53 => ( subsetRefl
% 2.44/1.53 => ( subsetTrans
% 2.44/1.53 => ( setadjoinSub
% 2.44/1.53 => ( setadjoinSub2
% 2.44/1.53 => ( subset2powerset
% 2.44/1.53 => ( setextsub
% 2.44/1.53 => ( subsetemptysetimpeq
% 2.44/1.53 => ! [A: $i,B: $i] :
% 2.44/1.53 ( ( subset @ B @ A )
% 2.44/1.53 => ( in @ B @ ( powerset @ A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.44/1.53 ------------------------
% 2.44/1.55 % (22888)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_22788 for (2999ds/27Mi)
% 2.44/1.55 % (22886)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_22788 for (2999ds/183Mi)
% 2.44/1.55 % (22890)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_22788 for (2999ds/2Mi)
% 2.44/1.55 % (22891)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_22788 for (2999ds/275Mi)
% 2.44/1.55 % (22887)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_22788 for (2999ds/4Mi)
% 2.44/1.55 % (22892)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_22788 for (2999ds/18Mi)
% 2.44/1.55 % (22889)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_22788 for (2999ds/2Mi)
% 2.44/1.55 % (22889)Instruction limit reached!
% 2.44/1.55 % (22889)------------------------------
% 2.44/1.55 % (22889)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.44/1.55 % (22889)Termination reason: Unknown
% 2.44/1.55 % (22889)Termination phase: shuffling
% 2.44/1.55
% 2.44/1.55 % (22889)Memory used [KB]: 1151
% 2.44/1.55 % (22889)Time elapsed: 0.003 s
% 2.44/1.55 % (22889)Instructions burned: 2 (million)
% 2.44/1.55 % (22889)------------------------------
% 2.44/1.55 % (22889)------------------------------
% 2.44/1.55 % (22890)Instruction limit reached!
% 2.44/1.55 % (22890)------------------------------
% 2.44/1.55 % (22890)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.44/1.55 % (22890)Termination reason: Unknown
% 2.44/1.55 % (22890)Termination phase: shuffling
% 2.44/1.55
% 2.44/1.55 % (22890)Memory used [KB]: 1151
% 2.44/1.55 % (22890)Time elapsed: 0.003 s
% 2.44/1.55 % (22890)Instructions burned: 3 (million)
% 2.44/1.55 % (22890)------------------------------
% 2.44/1.55 % (22890)------------------------------
% 2.44/1.56 % (22887)Instruction limit reached!
% 2.44/1.56 % (22887)------------------------------
% 2.44/1.56 % (22887)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.44/1.56 % (22887)Termination reason: Unknown
% 2.44/1.56 % (22887)Termination phase: shuffling
% 2.44/1.56
% 2.44/1.56 % (22887)Memory used [KB]: 1151
% 2.44/1.56 % (22887)Time elapsed: 0.004 s
% 2.44/1.56 % (22887)Instructions burned: 4 (million)
% 2.44/1.56 % (22887)------------------------------
% 2.44/1.56 % (22887)------------------------------
% 2.44/1.56 % (22892)Instruction limit reached!
% 2.44/1.56 % (22892)------------------------------
% 2.44/1.56 % (22892)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.44/1.56 % (22892)Termination reason: Unknown
% 2.44/1.56 % (22892)Termination phase: Preprocessing 3
% 2.44/1.56
% 2.44/1.56 % (22892)Memory used [KB]: 1663
% 2.44/1.56 % (22892)Time elapsed: 0.011 s
% 2.44/1.56 % (22892)Instructions burned: 19 (million)
% 2.44/1.56 % (22892)------------------------------
% 2.44/1.56 % (22892)------------------------------
% 2.44/1.57 % (22888)Instruction limit reached!
% 2.44/1.57 % (22888)------------------------------
% 2.44/1.57 % (22888)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.44/1.57 % (22888)Termination reason: Unknown
% 2.44/1.57 % (22888)Termination phase: Property scanning
% 2.44/1.57
% 2.44/1.57 % (22888)Memory used [KB]: 1407
% 2.44/1.57 % (22888)Time elapsed: 0.014 s
% 2.44/1.57 % (22888)Instructions burned: 27 (million)
% 2.44/1.57 % (22888)------------------------------
% 2.44/1.57 % (22888)------------------------------
% 2.44/1.57 % (22893)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_22788 for (2999ds/3Mi)
% 2.44/1.57 % (22894)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_22788 for (2999ds/37Mi)
% 2.44/1.57 % (22895)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_22788 for (2999ds/15Mi)
% 2.44/1.57 % (22893)Instruction limit reached!
% 2.44/1.57 % (22893)------------------------------
% 2.44/1.57 % (22893)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.44/1.57 % (22893)Termination reason: Unknown
% 2.44/1.57 % (22893)Termination phase: shuffling
% 2.44/1.57
% 2.44/1.57 % (22893)Memory used [KB]: 1151
% 2.44/1.57 % (22893)Time elapsed: 0.003 s
% 2.44/1.57 % (22893)Instructions burned: 3 (million)
% 2.44/1.57 % (22893)------------------------------
% 2.44/1.57 % (22893)------------------------------
% 2.44/1.58 % (22896)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_22788 for (2999ds/3Mi)
% 2.44/1.58 % (22895)Instruction limit reached!
% 2.44/1.58 % (22895)------------------------------
% 2.44/1.58 % (22895)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.44/1.58 % (22895)Termination reason: Unknown
% 2.44/1.58 % (22895)Termination phase: Preprocessing 3
% 2.44/1.58
% 2.44/1.58 % (22895)Memory used [KB]: 1407
% 2.44/1.58 % (22895)Time elapsed: 0.009 s
% 2.44/1.58 % (22895)Instructions burned: 15 (million)
% 2.44/1.58 % (22895)------------------------------
% 2.44/1.58 % (22895)------------------------------
% 2.44/1.58 % (22896)Instruction limit reached!
% 2.44/1.58 % (22896)------------------------------
% 2.44/1.58 % (22896)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.44/1.58 % (22896)Termination reason: Unknown
% 2.44/1.58 % (22896)Termination phase: shuffling
% 2.44/1.58
% 2.44/1.58 % (22896)Memory used [KB]: 1151
% 2.44/1.58 % (22896)Time elapsed: 0.003 s
% 2.44/1.58 % (22896)Instructions burned: 4 (million)
% 2.44/1.58 % (22896)------------------------------
% 2.44/1.58 % (22896)------------------------------
% 2.44/1.58 % (22897)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_22788 for (2999ds/1041Mi)
% 2.44/1.59 % (22894)Instruction limit reached!
% 2.44/1.59 % (22894)------------------------------
% 2.44/1.59 % (22894)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.44/1.59 % (22894)Termination reason: Unknown
% 2.44/1.59 % (22894)Termination phase: Property scanning
% 2.44/1.59
% 2.44/1.59 % (22894)Memory used [KB]: 1535
% 2.44/1.59 % (22894)Time elapsed: 0.018 s
% 2.44/1.59 % (22894)Instructions burned: 38 (million)
% 2.44/1.59 % (22894)------------------------------
% 2.44/1.59 % (22894)------------------------------
% 2.44/1.59 % (22898)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_22788 for (2999ds/7Mi)
% 2.44/1.59 % (22898)Instruction limit reached!
% 2.44/1.59 % (22898)------------------------------
% 2.44/1.59 % (22898)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.44/1.59 % (22898)Termination reason: Unknown
% 2.44/1.59 % (22898)Termination phase: shuffling
% 2.44/1.59
% 2.44/1.59 % (22898)Memory used [KB]: 1279
% 2.44/1.59 % (22898)Time elapsed: 0.005 s
% 2.44/1.59 % (22898)Instructions burned: 8 (million)
% 2.44/1.59 % (22898)------------------------------
% 2.44/1.59 % (22898)------------------------------
% 2.44/1.60 % (22899)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_22788 for (2999ds/16Mi)
% 2.49/1.60 % (22900)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_22788 for (2999ds/3Mi)
% 2.49/1.60 % (22900)Instruction limit reached!
% 2.49/1.60 % (22900)------------------------------
% 2.49/1.60 % (22900)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.49/1.60 % (22900)Termination reason: Unknown
% 2.49/1.60 % (22900)Termination phase: shuffling
% 2.49/1.60
% 2.49/1.60 % (22900)Memory used [KB]: 1151
% 2.49/1.60 % (22900)Time elapsed: 0.003 s
% 2.49/1.60 % (22900)Instructions burned: 3 (million)
% 2.49/1.60 % (22900)------------------------------
% 2.49/1.60 % (22900)------------------------------
% 2.49/1.60 % (22899)Instruction limit reached!
% 2.49/1.60 % (22899)------------------------------
% 2.49/1.60 % (22899)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.49/1.60 % (22899)Termination reason: Unknown
% 2.49/1.60 % (22899)Termination phase: Property scanning
% 2.49/1.60
% 2.49/1.60 % (22899)Memory used [KB]: 1407
% 2.49/1.60 % (22899)Time elapsed: 0.009 s
% 2.49/1.60 % (22899)Instructions burned: 17 (million)
% 2.49/1.60 % (22899)------------------------------
% 2.49/1.60 % (22899)------------------------------
% 2.49/1.60 % (22901)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_22788 for (2999ds/3Mi)
% 2.49/1.61 % (22901)Instruction limit reached!
% 2.49/1.61 % (22901)------------------------------
% 2.49/1.61 % (22901)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.49/1.61 % (22901)Termination reason: Unknown
% 2.49/1.61 % (22901)Termination phase: shuffling
% 2.49/1.61
% 2.49/1.61 % (22901)Memory used [KB]: 1151
% 2.49/1.61 % (22901)Time elapsed: 0.003 s
% 2.49/1.61 % (22901)Instructions burned: 3 (million)
% 2.49/1.61 % (22901)------------------------------
% 2.49/1.61 % (22901)------------------------------
% 2.49/1.61 % (22902)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_22788 for (2999ds/7Mi)
% 2.49/1.61 % (22902)Instruction limit reached!
% 2.49/1.61 % (22902)------------------------------
% 2.49/1.61 % (22902)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.49/1.61 % (22902)Termination reason: Unknown
% 2.49/1.61 % (22902)Termination phase: shuffling
% 2.49/1.61
% 2.49/1.61 % (22902)Memory used [KB]: 1279
% 2.49/1.61 % (22902)Time elapsed: 0.005 s
% 2.49/1.61 % (22902)Instructions burned: 7 (million)
% 2.49/1.61 % (22902)------------------------------
% 2.49/1.61 % (22902)------------------------------
% 2.49/1.61 % (22903)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_22788 for (2999ds/3Mi)
% 2.49/1.62 % (22903)Instruction limit reached!
% 2.49/1.62 % (22903)------------------------------
% 2.49/1.62 % (22903)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.49/1.62 % (22903)Termination reason: Unknown
% 2.49/1.62 % (22903)Termination phase: shuffling
% 2.49/1.62
% 2.49/1.62 % (22903)Memory used [KB]: 1151
% 2.49/1.62 % (22903)Time elapsed: 0.003 s
% 2.49/1.62 % (22903)Instructions burned: 3 (million)
% 2.49/1.62 % (22903)------------------------------
% 2.49/1.62 % (22903)------------------------------
% 2.49/1.62 % (22904)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_22788 for (2999ds/4Mi)
% 2.49/1.62 % (22904)Instruction limit reached!
% 2.49/1.62 % (22904)------------------------------
% 2.49/1.62 % (22904)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.49/1.62 % (22904)Termination reason: Unknown
% 2.49/1.62 % (22904)Termination phase: shuffling
% 2.49/1.62
% 2.49/1.62 % (22904)Memory used [KB]: 1151
% 2.49/1.62 % (22904)Time elapsed: 0.003 s
% 2.49/1.62 % (22904)Instructions burned: 4 (million)
% 2.49/1.62 % (22904)------------------------------
% 2.49/1.62 % (22904)------------------------------
% 2.49/1.62 % (22905)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_22788 for (2999ds/18Mi)
% 2.49/1.63 % (22906)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_22788 for (2999ds/710Mi)
% 2.49/1.63 % (22905)Instruction limit reached!
% 2.49/1.63 % (22905)------------------------------
% 2.49/1.63 % (22905)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.49/1.63 % (22905)Termination reason: Unknown
% 2.49/1.63 % (22905)Termination phase: Property scanning
% 2.49/1.63
% 2.49/1.63 % (22905)Memory used [KB]: 1407
% 2.49/1.63 % (22905)Time elapsed: 0.010 s
% 2.49/1.63 % (22905)Instructions burned: 19 (million)
% 2.49/1.63 % (22905)------------------------------
% 2.49/1.63 % (22905)------------------------------
% 2.49/1.63 % (22907)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_22788 for (2999ds/6Mi)
% 2.49/1.63 % (22907)Instruction limit reached!
% 2.49/1.63 % (22907)------------------------------
% 2.49/1.63 % (22907)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.49/1.63 % (22907)Termination reason: Unknown
% 2.49/1.63 % (22907)Termination phase: shuffling
% 2.49/1.63
% 2.49/1.63 % (22907)Memory used [KB]: 1279
% 2.49/1.63 % (22907)Time elapsed: 0.004 s
% 2.49/1.63 % (22907)Instructions burned: 6 (million)
% 2.49/1.63 % (22907)------------------------------
% 2.49/1.63 % (22907)------------------------------
% 2.49/1.64 % (22908)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_22788 for (2999ds/902Mi)
% 2.49/1.64 % (22886)Instruction limit reached!
% 2.49/1.64 % (22886)------------------------------
% 2.49/1.64 % (22886)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.49/1.64 % (22886)Termination reason: Unknown
% 2.49/1.64 % (22886)Termination phase: Saturation
% 2.49/1.64
% 2.49/1.64 % (22886)Memory used [KB]: 6908
% 2.49/1.64 % (22886)Time elapsed: 0.090 s
% 2.49/1.64 % (22886)Instructions burned: 183 (million)
% 2.49/1.64 % (22886)------------------------------
% 2.49/1.64 % (22886)------------------------------
% 2.49/1.65 % (22909)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_22788 for (2999ds/21Mi)
% 2.49/1.65 % (22910)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_22788 for (2999ds/5Mi)
% 2.49/1.65 % (22910)Instruction limit reached!
% 2.49/1.65 % (22910)------------------------------
% 2.49/1.65 % (22910)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.49/1.65 % (22910)Termination reason: Unknown
% 2.49/1.65 % (22910)Termination phase: shuffling
% 2.49/1.65
% 2.49/1.65 % (22910)Memory used [KB]: 1279
% 2.49/1.65 % (22910)Time elapsed: 0.004 s
% 2.49/1.65 % (22910)Instructions burned: 6 (million)
% 2.49/1.65 % (22910)------------------------------
% 2.49/1.65 % (22910)------------------------------
% 2.49/1.66 % (22909)Instruction limit reached!
% 2.49/1.66 % (22909)------------------------------
% 2.49/1.66 % (22909)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.49/1.66 % (22909)Termination reason: Unknown
% 2.49/1.66 % (22909)Termination phase: Property scanning
% 2.49/1.66
% 2.49/1.66 % (22909)Memory used [KB]: 1407
% 2.49/1.66 % (22909)Time elapsed: 0.011 s
% 2.49/1.66 % (22909)Instructions burned: 21 (million)
% 2.49/1.66 % (22909)------------------------------
% 2.49/1.66 % (22909)------------------------------
% 2.49/1.66 % (22911)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_22788 for (2999ds/6Mi)
% 2.49/1.66 % (22911)Instruction limit reached!
% 2.49/1.66 % (22911)------------------------------
% 2.49/1.66 % (22911)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.49/1.66 % (22911)Termination reason: Unknown
% 2.49/1.66 % (22911)Termination phase: shuffling
% 2.49/1.66
% 2.49/1.66 % (22911)Memory used [KB]: 1279
% 2.49/1.66 % (22911)Time elapsed: 0.005 s
% 2.49/1.66 % (22911)Instructions burned: 7 (million)
% 2.49/1.66 % (22911)------------------------------
% 2.49/1.66 % (22911)------------------------------
% 2.49/1.67 % (22912)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_22788 for (2998ds/377Mi)
% 2.49/1.67 % (22913)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_22788 for (2998ds/779Mi)
% 2.49/1.67 % (22891)First to succeed.
% 2.49/1.68 % (22914)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_22788 for (2998ds/19Mi)
% 2.49/1.68 % (22908)Also succeeded, but the first one will report.
% 2.49/1.68 % (22891)Refutation found. Thanks to Tanya!
% 2.49/1.68 % SZS status Theorem for DTF2THF_22788
% 2.49/1.68 % SZS output start Proof for DTF2THF_22788
% 2.49/1.68 thf(func_def_0, type, in: $i > $i > $o).
% 2.49/1.68 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.49/1.68 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.49/1.68 thf(func_def_8, type, powerset: $i > $i).
% 2.49/1.68 thf(func_def_10, type, setunion: $i > $i).
% 2.49/1.68 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.49/1.68 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.49/1.68 thf(func_def_26, type, prop2set: $o > $i).
% 2.49/1.68 thf(func_def_36, type, nonempty: $i > $o).
% 2.49/1.68 thf(func_def_69, type, set2prop: $i > $o).
% 2.49/1.68 thf(func_def_88, type, subset: $i > $i > $o).
% 2.49/1.68 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.49/1.68 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.49/1.68 thf(func_def_120, type, sP0: $i > $i > $o).
% 2.49/1.68 thf(func_def_121, type, sP1: $i > $o).
% 2.49/1.68 thf(func_def_122, type, sP2: $i > $i > $o).
% 2.49/1.68 thf(func_def_123, type, sP3: $i > $i > $o).
% 2.49/1.68 thf(func_def_125, type, sK5: $i > $o).
% 2.49/1.68 thf(func_def_129, type, sK9: $i > $i).
% 2.49/1.68 thf(func_def_132, type, sK12: ($i > $o) > $i).
% 2.49/1.68 thf(func_def_133, type, sK13: $i > $o).
% 2.49/1.68 thf(func_def_134, type, sK14: $i > $i).
% 2.49/1.68 thf(func_def_135, type, sK15: $i > $o).
% 2.49/1.68 thf(func_def_140, type, sK20: $i > $i > $i).
% 2.49/1.68 thf(func_def_149, type, sK29: $i > $o).
% 2.49/1.68 thf(func_def_151, type, sK31: ($i > $o) > $i > $i).
% 2.49/1.68 thf(func_def_155, type, sK35: $i > $o).
% 2.49/1.68 thf(func_def_157, type, sK37: ($i > $o) > $i > $i).
% 2.49/1.68 thf(func_def_158, type, sK38: $i > $i > $i).
% 2.49/1.68 thf(func_def_159, type, sK39: $i > $i > $i).
% 2.49/1.68 thf(func_def_160, type, sK40: $i > $i > $i).
% 2.49/1.68 thf(func_def_161, type, sK41: $i > $i > $i).
% 2.49/1.68 thf(func_def_162, type, sK42: $i > $i > $i).
% 2.49/1.68 thf(func_def_163, type, sK43: $i > $i > $i > $i).
% 2.49/1.68 thf(func_def_164, type, sK44: $i > $i).
% 2.49/1.68 thf(func_def_165, type, sK45: $i > $i).
% 2.49/1.68 thf(func_def_166, type, sK46: $i > $i).
% 2.49/1.68 thf(func_def_167, type, sK47: $i > $i).
% 2.49/1.68 thf(func_def_168, type, sK48: $i > $i > $i > $i).
% 2.49/1.68 thf(func_def_169, type, sK49: $i > $i > $i > $i).
% 2.49/1.68 thf(func_def_170, type, sK50: $i > $i > $i).
% 2.49/1.68 thf(func_def_171, type, sK51: $i > $i > $i).
% 2.49/1.68 thf(func_def_172, type, sK52: $i > $i > $i).
% 2.49/1.68 thf(func_def_174, type, sK54: $i > $i).
% 2.49/1.68 thf(func_def_175, type, sK55: $i > $i).
% 2.49/1.68 thf(func_def_176, type, sK56: $i > $i).
% 2.49/1.68 thf(func_def_180, type, sK60: $i > $i).
% 2.49/1.68 thf(func_def_185, type, sK65: $i > $o).
% 2.49/1.68 thf(func_def_195, type, sK75: $i > $i).
% 2.49/1.68 thf(func_def_209, type, sK89: $i > $i > $i).
% 2.49/1.68 thf(func_def_213, type, sK93: $i > $i).
% 2.49/1.68 thf(func_def_215, type, sK95: $i > $i).
% 2.49/1.68 thf(func_def_220, type, sK100: $i > $i > $i).
% 2.49/1.68 thf(func_def_232, type, sK112: $i > $i).
% 2.49/1.68 thf(func_def_235, type, sK115: $i > $o).
% 2.49/1.68 thf(func_def_236, type, sK116: ($i > $o) > $i).
% 2.49/1.68 thf(func_def_243, type, sK123: $i > $o).
% 2.49/1.68 thf(func_def_246, type, sK126: $i > $o).
% 2.49/1.68 thf(func_def_250, type, sK130: $i > $i > $i).
% 2.49/1.68 thf(func_def_254, type, sK134: $o > $i > $i > $i).
% 2.49/1.68 thf(func_def_262, type, sK142: $i > $o).
% 2.49/1.68 thf(func_def_263, type, sK143: $i > $o).
% 2.49/1.68 thf(func_def_264, type, sK144: ($i > $o) > ($i > $o) > $i).
% 2.49/1.68 thf(func_def_265, type, sK145: ($i > $o) > ($i > $o) > $i).
% 2.49/1.68 thf(func_def_267, type, sK147: $i > $o).
% 2.49/1.68 thf(func_def_269, type, sK149: ($i > $o) > $i > $i).
% 2.49/1.68 thf(func_def_272, type, sK152: $i > $i > $i).
% 2.49/1.68 thf(func_def_276, type, sK156: $i > $o).
% 2.49/1.68 thf(func_def_279, type, sK159: $i > $o).
% 2.49/1.68 thf(func_def_285, type, sK165: $i > $i > $o).
% 2.49/1.68 thf(func_def_287, type, sK167: $i > $i).
% 2.49/1.68 thf(func_def_288, type, sK168: $i > $i).
% 2.49/1.68 thf(func_def_289, type, sK169: $i > ($i > $i > $o) > $i).
% 2.49/1.68 thf(func_def_290, type, sK170: $i > $i > ($i > $i > $o) > $i).
% 2.49/1.68 thf(func_def_291, type, sK171: $i > ($i > $i > $o) > $i).
% 2.49/1.68 thf(func_def_292, type, sK172: $i > $o).
% 2.49/1.68 thf(func_def_294, type, sK174: $i > ($i > $o) > $i).
% 2.49/1.68 thf(func_def_295, type, sK175: $i > $o).
% 2.49/1.68 thf(func_def_308, type, sK188: $i > $i > $i).
% 2.49/1.68 thf(func_def_309, type, sK189: $i > $i > $i).
% 2.49/1.68 thf(func_def_311, type, sK191: $i > $o).
% 2.49/1.68 thf(func_def_313, type, sK193: ($i > $o) > $i).
% 2.49/1.68 thf(func_def_314, type, sK194: $i > $o).
% 2.49/1.68 thf(func_def_315, type, sK195: $i > $i).
% 2.49/1.68 thf(func_def_319, type, sK199: $i > $i).
% 2.49/1.68 thf(func_def_322, type, sK202: $i > $i > $i).
% 2.49/1.68 thf(func_def_323, type, sK203: $i > $o).
% 2.49/1.68 thf(func_def_324, type, sK204: $i > $o).
% 2.49/1.68 thf(func_def_325, type, sK205: ($i > $o) > ($i > $o) > $i).
% 2.49/1.68 thf(func_def_326, type, sK206: ($i > $o) > ($i > $o) > $i).
% 2.49/1.68 thf(func_def_327, type, sK207: $i > ($i > $o) > $i).
% 2.49/1.68 thf(func_def_328, type, sK208: $i > $o).
% 2.49/1.68 thf(func_def_336, type, sK216: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.49/1.68 thf(func_def_337, type, sK217: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.49/1.68 thf(func_def_340, type, sK220: $i > $o).
% 2.49/1.68 thf(func_def_341, type, sK221: $i > $o).
% 2.49/1.68 thf(func_def_346, type, sK226: ($i > $o) > $i > $i).
% 2.49/1.68 thf(func_def_348, type, sK228: $i > $o).
% 2.49/1.68 thf(func_def_350, type, sK230: $i > ($i > $o) > $i).
% 2.49/1.68 thf(func_def_351, type, sK231: $i > $o).
% 2.49/1.68 thf(func_def_356, type, sK236: $i > $o).
% 2.49/1.68 thf(func_def_358, type, sK238: ($i > $o) > $i).
% 2.49/1.68 thf(func_def_359, type, sK239: ($i > $o) > $i).
% 2.49/1.68 thf(func_def_361, type, sK241: $i > $o).
% 2.49/1.68 thf(func_def_363, type, sK243: ($i > $o) > $i > $i).
% 2.49/1.68 thf(func_def_375, type, sK255: $i > $o).
% 2.49/1.68 thf(func_def_381, type, ph261: !>[X0: $tType]:(X0)).
% 2.49/1.68 thf(f2336,plain,(
% 2.49/1.68 $false),
% 2.49/1.68 inference(subsumption_resolution,[status(thm)],[f2334,f985])).
% 2.49/1.68 thf(f985,plain,(
% 2.49/1.68 ((subset @ sK58 @ sK57) = $true)),
% 2.49/1.68 inference(cnf_transformation,[status(thm)],[f502])).
% 2.49/1.68 thf(f502,plain,(
% 2.49/1.68 (prop2setI = $true) & (quantDeMorgan4 = $true) & (setadjoinSub2 = $true) & (emptysetsubset = $true) & (upairsetIL = $true) & (setunionE = $true) & (omegaSAx = $true) & (in__Cong = $true) & (subsetE2 = $true) & (exuE2 = $true) & (omegaIndAx = $true) & (notequalI1 = $true) & (exu__Cong = $true) & (emptyinunitempty = $true) & (notequalI2 = $true) & (eqinunit = $true) & (exuE1 = $true) & (notinsingleton = $true) & (notdallE = $true) & (setadjoin__Cong = $true) & (exuE3u = $true) & (wellorderingAx = $true) & (setunion__Cong = $true) & (uniqinunit = $true) & (quantDeMorgan2 = $true) & (eqimpsubset2 = $true) & (exuEu = $true) & (eqimpsubset1 = $true) & (subsetI2 = $true) & (notinemptyset = $true) & (foundationAx = $true) & (setadjoinOr = $true) & (setadjoinE = $true) & (subsetI1 = $true) & (nonemptyI1 = $true) & (setbeta = $true) & (emptysetimpfalse = $true) & (powerset__Cong = $true) & (quantDeMorgan1 = $true) & (subsetRefl = $true) & (exuE3e = $true) & (omega0Ax = $true) & (setadjoinSub = $true) & (subsetTrans = $true) & (powersetI = $true) & (dsetconstrEL = $true) & (upairsetE = $true) & (((subset @ sK58 @ sK57) = $true) & ((in @ sK58 @ (powerset @ sK57)) != $true)) & (setadjoinIL = $true) & (powersetAx = $true) & (setextsub = $true) & (singletonsswitch = $true) & (subsetemptysetimpeq = $true) & (emptyinPowerset = $true) & (nonemptyImpWitness = $true) & (replAx = $true) & (inCongP = $true) & (emptyInPowerset = $true) & (setadjoinAx = $true) & (subset2powerset = $true) & (upairsetIR = $true) & (setext = $true) & (notdexE = $true) & (exuI2 = $true) & (dsetconstrER = $true) & (exuI1 = $true) & (exuI3 = $true) & (setoftrueEq = $true) & (prop2set2propI = $true) & (notsubsetI = $true) & (setunionAx = $true) & (dsetconstrI = $true) & (nonemptyE1 = $true) & (emptysetE = $true) & (setunionI = $true) & (powersetE = $true) & (prop2setE = $true) & (setadjoinIR = $true) & (emptysetAx = $true) & (emptyset__Cong = $true) & (quantDeMorgan3 = $true) & (nonemptyI = $true) & (subsetE = $true) & (emptyI = $true) & (omega__Cong = $true) & (vacuousDall = $true) & (emptyE1 = $true) & (descr__Cong = $true) & (dsetconstr__Cong = $true) & (setextAx = $true) & (descrp = $true) & (subPowSU = $true) & (noeltsimpempty = $true)),
% 2.49/1.68 inference(skolemisation,[status(esa),new_symbols(skolem,[sK57,sK58])],[f329,f501])).
% 2.49/1.68 thf(f501,plain,(
% 2.49/1.68 ? [X0 : $i,X1 : $i] : (((subset @ X1 @ X0) = $true) & ($true != (in @ X1 @ (powerset @ X0)))) => (((subset @ sK58 @ sK57) = $true) & ((in @ sK58 @ (powerset @ sK57)) != $true))),
% 2.49/1.68 introduced(definition,[],[choice_axiom])).
% 2.49/1.68 thf(f329,plain,(
% 2.49/1.68 (prop2setI = $true) & (quantDeMorgan4 = $true) & (setadjoinSub2 = $true) & (emptysetsubset = $true) & (upairsetIL = $true) & (setunionE = $true) & (omegaSAx = $true) & (in__Cong = $true) & (subsetE2 = $true) & (exuE2 = $true) & (omegaIndAx = $true) & (notequalI1 = $true) & (exu__Cong = $true) & (emptyinunitempty = $true) & (notequalI2 = $true) & (eqinunit = $true) & (exuE1 = $true) & (notinsingleton = $true) & (notdallE = $true) & (setadjoin__Cong = $true) & (exuE3u = $true) & (wellorderingAx = $true) & (setunion__Cong = $true) & (uniqinunit = $true) & (quantDeMorgan2 = $true) & (eqimpsubset2 = $true) & (exuEu = $true) & (eqimpsubset1 = $true) & (subsetI2 = $true) & (notinemptyset = $true) & (foundationAx = $true) & (setadjoinOr = $true) & (setadjoinE = $true) & (subsetI1 = $true) & (nonemptyI1 = $true) & (setbeta = $true) & (emptysetimpfalse = $true) & (powerset__Cong = $true) & (quantDeMorgan1 = $true) & (subsetRefl = $true) & (exuE3e = $true) & (omega0Ax = $true) & (setadjoinSub = $true) & (subsetTrans = $true) & (powersetI = $true) & (dsetconstrEL = $true) & (upairsetE = $true) & ? [X0 : $i,X1 : $i] : (((subset @ X1 @ X0) = $true) & ($true != (in @ X1 @ (powerset @ X0)))) & (setadjoinIL = $true) & (powersetAx = $true) & (setextsub = $true) & (singletonsswitch = $true) & (subsetemptysetimpeq = $true) & (emptyinPowerset = $true) & (nonemptyImpWitness = $true) & (replAx = $true) & (inCongP = $true) & (emptyInPowerset = $true) & (setadjoinAx = $true) & (subset2powerset = $true) & (upairsetIR = $true) & (setext = $true) & (notdexE = $true) & (exuI2 = $true) & (dsetconstrER = $true) & (exuI1 = $true) & (exuI3 = $true) & (setoftrueEq = $true) & (prop2set2propI = $true) & (notsubsetI = $true) & (setunionAx = $true) & (dsetconstrI = $true) & (nonemptyE1 = $true) & (emptysetE = $true) & (setunionI = $true) & (powersetE = $true) & (prop2setE = $true) & (setadjoinIR = $true) & (emptysetAx = $true) & (emptyset__Cong = $true) & (quantDeMorgan3 = $true) & (nonemptyI = $true) & (subsetE = $true) & (emptyI = $true) & (omega__Cong = $true) & (vacuousDall = $true) & (emptyE1 = $true) & (descr__Cong = $true) & (dsetconstr__Cong = $true) & (setextAx = $true) & (descrp = $true) & (subPowSU = $true) & (noeltsimpempty = $true)),
% 2.49/1.68 inference(flattening,[status(thm)],[f328])).
% 2.49/1.68 thf(f328,plain,(
% 2.49/1.68 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (((subset @ X1 @ X0) = $true) & ($true != (in @ X1 @ (powerset @ X0)))) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (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)),
% 2.49/1.68 inference(ennf_transformation,[status(thm)],[f277])).
% 2.49/1.68 thf(f277,plain,(
% 2.49/1.68 ~((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) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ! [X0 : $i,X1 : $i] : (((subset @ X1 @ X0) = $true) => ($true = (in @ X1 @ (powerset @ X0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.49/1.68 inference(fool_elimination,[status(thm)],[f276])).
% 2.49/1.68 thf(f276,plain,(
% 2.49/1.68 ~(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 => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => ! [X0 : $i,X1 : $i] : ((subset @ X1 @ X0) => (in @ X1 @ (powerset @ X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.49/1.68 inference(rectify,[status(thm)],[f96])).
% 2.49/1.68 thf(f96,negated_conjecture,(
% 2.49/1.68 ~(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 => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => ! [X3 : $i,X4 : $i] : ((subset @ X4 @ X3) => (in @ X4 @ (powerset @ X3)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.49/1.68 inference(negated_conjecture,[status(cth)],[f95])).
% 2.49/1.68 thf(f95,conjecture,(
% 2.49/1.68 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 => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => ! [X3 : $i,X4 : $i] : ((subset @ X4 @ X3) => (in @ X4 @ (powerset @ X3))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.49/1.68 file('/export/starexec/sandbox/tmp/tmp.ZrZtUQmNfW/DTF2THF_22788.p',powersetI1)).
% 2.49/1.68 thf(f2334,plain,(
% 2.49/1.68 ((subset @ sK58 @ sK57) != $true)),
% 2.49/1.68 inference(trivial_inequality_removal,[status(thm)],[f2325])).
% 2.49/1.68 thf(f2325,plain,(
% 2.49/1.68 ($true != $true) | ((subset @ sK58 @ sK57) != $true)),
% 2.49/1.68 inference(superposition,[status(thm)],[f984,f1794])).
% 2.49/1.68 thf(f1794,plain,(
% 2.49/1.68 ( ! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) | ((subset @ X1 @ X0) != $true)) )),
% 2.49/1.68 inference(trivial_inequality_removal,[status(thm)],[f1349])).
% 2.49/1.68 thf(f1349,plain,(
% 2.49/1.68 ( ! [X0 : $i,X1 : $i] : (($true != $true) | ($true = (in @ X1 @ (powerset @ X0))) | ((subset @ X1 @ X0) != $true)) )),
% 2.49/1.68 inference(definition_unfolding,[status(thm)],[f877,f972])).
% 2.49/1.68 thf(f972,plain,(
% 2.49/1.68 (subset2powerset = $true)),
% 2.49/1.68 inference(cnf_transformation,[status(thm)],[f502])).
% 2.49/1.68 thf(f877,plain,(
% 2.49/1.68 ( ! [X0 : $i,X1 : $i] : (((subset @ X1 @ X0) != $true) | ($true = (in @ X1 @ (powerset @ X0))) | (subset2powerset != $true)) )),
% 2.49/1.68 inference(cnf_transformation,[status(thm)],[f439])).
% 2.49/1.68 thf(f439,plain,(
% 2.49/1.68 (! [X0 : $i,X1 : $i] : (((subset @ X1 @ X0) != $true) | ($true = (in @ X1 @ (powerset @ X0)))) | (subset2powerset != $true)) & ((subset2powerset = $true) | (($true = (subset @ sK19 @ sK18)) & ((in @ sK19 @ (powerset @ sK18)) != $true)))),
% 2.49/1.68 inference(skolemisation,[status(esa),new_symbols(skolem,[sK18,sK19])],[f437,f438])).
% 2.49/1.68 thf(f438,plain,(
% 2.49/1.68 ? [X2 : $i,X3 : $i] : (($true = (subset @ X3 @ X2)) & ($true != (in @ X3 @ (powerset @ X2)))) => (($true = (subset @ sK19 @ sK18)) & ((in @ sK19 @ (powerset @ sK18)) != $true))),
% 2.49/1.68 introduced(definition,[],[choice_axiom])).
% 2.49/1.68 thf(f437,plain,(
% 2.49/1.68 (! [X0 : $i,X1 : $i] : (((subset @ X1 @ X0) != $true) | ($true = (in @ X1 @ (powerset @ X0)))) | (subset2powerset != $true)) & ((subset2powerset = $true) | ? [X2 : $i,X3 : $i] : (($true = (subset @ X3 @ X2)) & ($true != (in @ X3 @ (powerset @ X2)))))),
% 2.49/1.68 inference(rectify,[status(thm)],[f436])).
% 2.49/1.68 thf(f436,plain,(
% 2.49/1.68 (! [X0 : $i,X1 : $i] : (((subset @ X1 @ X0) != $true) | ($true = (in @ X1 @ (powerset @ X0)))) | (subset2powerset != $true)) & ((subset2powerset = $true) | ? [X0 : $i,X1 : $i] : (((subset @ X1 @ X0) = $true) & ($true != (in @ X1 @ (powerset @ X0)))))),
% 2.49/1.68 inference(nnf_transformation,[status(thm)],[f319])).
% 2.49/1.68 thf(f319,plain,(
% 2.49/1.68 ! [X0 : $i,X1 : $i] : (((subset @ X1 @ X0) != $true) | ($true = (in @ X1 @ (powerset @ X0)))) <=> (subset2powerset = $true)),
% 2.49/1.68 inference(ennf_transformation,[status(thm)],[f160])).
% 2.49/1.68 thf(f160,plain,(
% 2.49/1.68 (subset2powerset = $true) <=> ! [X0 : $i,X1 : $i] : (((subset @ X1 @ X0) = $true) => ($true = (in @ X1 @ (powerset @ X0))))),
% 2.49/1.68 inference(fool_elimination,[status(thm)],[f159])).
% 2.49/1.68 thf(f159,plain,(
% 2.49/1.68 (! [X0 : $i,X1 : $i] : ((subset @ X1 @ X0) => (in @ X1 @ (powerset @ X0))) = subset2powerset)),
% 2.49/1.68 inference(rectify,[status(thm)],[f92])).
% 2.49/1.68 thf(f92,axiom,(
% 2.49/1.68 (! [X4 : $i,X3 : $i] : ((subset @ X3 @ X4) => (in @ X3 @ (powerset @ X4))) = subset2powerset)),
% 2.49/1.68 file('/export/starexec/sandbox/tmp/tmp.ZrZtUQmNfW/DTF2THF_22788.p',subset2powerset)).
% 2.49/1.68 thf(f984,plain,(
% 2.49/1.68 ((in @ sK58 @ (powerset @ sK57)) != $true)),
% 2.49/1.68 inference(cnf_transformation,[status(thm)],[f502])).
% 2.49/1.68 % SZS output end Proof for DTF2THF_22788
% 2.49/1.68 % (22891)------------------------------
% 2.49/1.68 % (22891)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.49/1.68 % (22891)Termination reason: Refutation
% 2.49/1.68
% 2.49/1.68 % (22891)Memory used [KB]: 7547
% 2.49/1.68 % (22891)Time elapsed: 0.123 s
% 2.49/1.68 % (22891)Instructions burned: 236 (million)
% 2.49/1.68 % (22891)------------------------------
% 2.49/1.68 % (22891)------------------------------
% 2.49/1.68 % (22885)Success in time 0.128 s
%------------------------------------------------------------------------------