%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU586^1 : 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 : n012.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 09:35:26 AM UTC 2026
% Result : Theorem 2.62s 1.79s
% Output : Refutation 2.62s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 7
% Syntax : Number of formulae : 43 ( 10 unt; 0 typ; 3 def)
% Number of atoms : 1977 ( 537 equ; 0 cnn)
% Maximal formula atoms : 105 ( 45 avg)
% Number of connectives : 1126 ( 69 ~; 62 |; 344 &; 216 @)
% ( 6 <=>; 429 =>; 0 <=; 0 <~>)
% Maximal formula depth : 111 ( 21 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 225 ( 222 usr; 114 con; 0-4 aty)
% Number of variables : 75 ( 0 ^; 54 !; 21 ?; 75 :)
% 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_114,type,
binunion: $i > $i > $i ).
thf(func_def_132,type,
sP1: $i > $i > $o ).
thf(func_def_133,type,
sP2: $i > $o ).
thf(func_def_134,type,
sP3: $i > $i > $o ).
thf(func_def_135,type,
sP4: $i > $i > $o ).
thf(func_def_137,type,
sK6: $i > $i ).
thf(func_def_138,type,
sK7: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_139,type,
sK8: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_140,type,
sK9: $i > $o ).
thf(func_def_141,type,
sK10: $i > $o ).
thf(func_def_149,type,
sK18: $i > $o ).
thf(func_def_153,type,
sK22: $i > $o ).
thf(func_def_157,type,
sK26: $i > $o ).
thf(func_def_159,type,
sK28: $i > $o ).
thf(func_def_161,type,
sK30: ( $i > $o ) > $i > $i ).
thf(func_def_162,type,
sK31: $i > $o ).
thf(func_def_163,type,
sK32: $i > $i > $i ).
thf(func_def_175,type,
sK44: $i > $i > $o ).
thf(func_def_177,type,
sK46: $i > $i ).
thf(func_def_178,type,
sK47: $i > $i ).
thf(func_def_179,type,
sK48: $i > ( $i > $i > $o ) > $i ).
thf(func_def_180,type,
sK49: $i > ( $i > $i > $o ) > $i ).
thf(func_def_181,type,
sK50: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_187,type,
sK56: $i > $i ).
thf(func_def_189,type,
sK58: $i > $i ).
thf(func_def_191,type,
sK60: $i > $o ).
thf(func_def_192,type,
sK61: $i > $i ).
thf(func_def_193,type,
sK62: ( $i > $o ) > $i ).
thf(func_def_198,type,
sK67: $i > ( $i > $o ) > $i ).
thf(func_def_199,type,
sK68: $i > $o ).
thf(func_def_205,type,
sK74: $i > $o ).
thf(func_def_208,type,
sK77: $i > $i ).
thf(func_def_210,type,
sK79: $i > $i ).
thf(func_def_212,type,
sK81: $i > $o ).
thf(func_def_225,type,
sK94: $i > $i ).
thf(func_def_227,type,
sK96: $i > $i > $i ).
thf(func_def_231,type,
sK100: $i > $i > $i ).
thf(func_def_232,type,
sK101: $i > $i > $i ).
thf(func_def_238,type,
sK107: $i > $o ).
thf(func_def_244,type,
sK113: $i > $o ).
thf(func_def_247,type,
sK116: $i > ( $i > $o ) > $i ).
thf(func_def_248,type,
sK117: $i > $o ).
thf(func_def_253,type,
sK122: $i > $o ).
thf(func_def_255,type,
sK124: ( $i > $o ) > $i ).
thf(func_def_256,type,
sK125: ( $i > $o ) > $i ).
thf(func_def_267,type,
sK136: ( $i > $o ) > $i > $i ).
thf(func_def_269,type,
sK138: $i > $o ).
thf(func_def_273,type,
sK142: $i > $o ).
thf(func_def_292,type,
sK161: $i > $o ).
thf(func_def_295,type,
sK164: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_296,type,
sK165: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_297,type,
sK166: $i > $o ).
thf(func_def_298,type,
sK167: $i > $o ).
thf(func_def_305,type,
sK174: $i > ( $i > $o ) > $i ).
thf(func_def_306,type,
sK175: $i > $o ).
thf(func_def_322,type,
sK191: $o > $i > $i > $i ).
thf(func_def_329,type,
sK198: $i > $o ).
thf(func_def_330,type,
sK199: $i > $o ).
thf(func_def_331,type,
sK200: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_332,type,
sK201: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_336,type,
sK205: $i > $i > $i ).
thf(func_def_337,type,
sK206: $i > $i > $i ).
thf(func_def_338,type,
sK207: $i > $i > $i ).
thf(func_def_339,type,
sK208: $i > $i > $i ).
thf(func_def_340,type,
sK209: $i > $i > $i ).
thf(func_def_341,type,
sK210: $i > $i > $i > $i ).
thf(func_def_342,type,
sK211: $i > $i ).
thf(func_def_343,type,
sK212: $i > $i ).
thf(func_def_344,type,
sK213: $i > $i ).
thf(func_def_345,type,
sK214: $i > $i ).
thf(func_def_346,type,
sK215: $i > $i > $i ).
thf(func_def_347,type,
sK216: $i > $i > $i > $i ).
thf(func_def_348,type,
sK217: $i > $i > $i > $i ).
thf(func_def_349,type,
sK218: $i > $i > $i ).
thf(func_def_350,type,
sK219: $i > $i > $i ).
thf(func_def_352,type,
sK221: $i > $i ).
thf(func_def_353,type,
sK222: $i > $i ).
thf(func_def_354,type,
sK223: $i > $i ).
thf(func_def_360,type,
sK229: ( $i > $o ) > $i ).
thf(func_def_361,type,
sK230: $i > $o ).
thf(func_def_362,type,
sK231: $i > $i ).
thf(func_def_364,type,
sK233: $i > $i > $i ).
thf(func_def_370,type,
sK239: $i > $i > $i ).
thf(func_def_371,type,
sK240: $i > $i > $i ).
thf(func_def_374,type,
sK243: $i > $o ).
thf(func_def_375,type,
sK244: ( $i > $o ) > $i ).
thf(func_def_378,type,
sK247: $i > $o ).
thf(func_def_380,type,
sK249: ( $i > $o ) > $i > $i ).
thf(func_def_381,type,
sK250: $i > $o ).
thf(func_def_383,type,
sK252: $i > ( $i > $o ) > $i ).
thf(func_def_386,type,
sK255: $i > $o ).
thf(func_def_388,type,
sK257: $i > $o ).
thf(func_def_390,type,
sK259: ( $i > $o ) > $i > $i ).
thf(func_def_397,type,
sK266: $i > $i > $i ).
thf(func_def_401,type,
sK270: $i > $o ).
thf(func_def_404,type,
sK273: $i > $i ).
thf(func_def_417,type,
ph286:
!>[X0: $tType] : X0 ).
thf(f2054,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f1599,f1604,f1915,f2051]) ).
thf(f2051,plain,
~ spl285_1,
inference(avatar_contradiction_clause,[status(thm)],[f2050]) ).
thf(f2050,plain,
( $false
| ~ spl285_1 ),
inference(trivial_inequality_removal,[status(thm)],[f2047]) ).
thf(f2047,plain,
( ( $true != $true )
| ~ spl285_1 ),
inference(superposition,[status(thm)],[f1011,f1591]) ).
thf(f1591,plain,
( ( $true
= ( in @ sK37 @ sK36 ) )
| ~ spl285_1 ),
inference(avatar_component_clause,[status(thm)],[f1589]) ).
thf(f1589,definition,
( spl285_1
<=> ( $true
= ( in @ sK37 @ sK36 ) ) ),
introduced(definition,[new_symbols(naming,[spl285_1])],[avatar_definition]) ).
thf(f1011,plain,
( $true
!= ( in @ sK37 @ sK36 ) ),
inference(cnf_transformation,[status(thm)],[f502]) ).
thf(f502,plain,
( ( exuI1 = $true )
& ( descrp = $true )
& ( binunionIL = $true )
& ( setadjoinE = $true )
& ( subsetE = $true )
& ( in__Cong = $true )
& ( setadjoinSub2 = $true )
& ( emptysetAx = $true )
& ( setadjoinOr = $true )
& ( eqimpsubset1 = $true )
& ( exuEu = $true )
& ( nonemptyI = $true )
& ( exuE1 = $true )
& ( emptyinunitempty = $true )
& ( notequalI1 = $true )
& ( nonemptyE1 = $true )
& ( emptyinPowerset = $true )
& ( setadjoinSub = $true )
& ( exuE3e = $true )
& ( replAx = $true )
& ( sepInPowerset = $true )
& ( emptyset__Cong = $true )
& ( setbeta = $true )
& ( emptyInPowerset = $true )
& ( subPowSU = $true )
& ( inPowerset = $true )
& ( subset2powerset = $true )
& ( exu__Cong = $true )
& ( subsetI1 = $true )
& ( eqimpsubset2 = $true )
& ( upairset2IR = $true )
& ( emptysetsubset = $true )
& ( powersetsubset = $true )
& ( setadjoin__Cong = $true )
& ( setextAx = $true )
& ( wellorderingAx = $true )
& ( exuI3 = $true )
& ( powersetAx = $true )
& ( eqinunit = $true )
& ( setoftrueEq = $true )
& ( singletonsswitch = $true )
& ( setext = $true )
& ( subsetI2 = $true )
& ( notdexE = $true )
& ( descr__Cong = $true )
& ( emptysetE = $true )
& ( subsetemptysetimpeq = $true )
& ( setunionE = $true )
& ( powersetI1 = $true )
& ( dsetconstrER = $true )
& ( sepSubset = $true )
& ( notequalI2 = $true )
& ( powersetE = $true )
& ( upairsetE = $true )
& ( subsetE2 = $true )
& ( quantDeMorgan2 = $true )
& ( binunionIR = $true )
& ( dsetconstr__Cong = $true )
& ( omega0Ax = $true )
& ( noeltsimpempty = $true )
& ( setadjoinAx = $true )
& ( quantDeMorgan3 = $true )
& ( upairsetIR = $true )
& ( prop2setI = $true )
& ( upairsetIL = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( subsetRefl = $true )
& ( exuE2 = $true )
& ( quantDeMorgan1 = $true )
& ( emptysetimpfalse = $true )
& ( nonemptyImpWitness = $true )
& ( $true
!= ( in @ sK37 @ sK35 ) )
& ( $true
= ( in @ sK37 @ ( binunion @ sK36 @ sK35 ) ) )
& ( $true
!= ( in @ sK37 @ sK36 ) )
& ( prop2set2propI = $true )
& ( inCongP = $true )
& ( powerset__Cong = $true )
& ( omegaIndAx = $true )
& ( notsubsetI = $true )
& ( foundationAx = $true )
& ( exuE3u = $true )
& ( setunionI = $true )
& ( binunionEcases = $true )
& ( notdallE = $true )
& ( vacuousDall = $true )
& ( setunion__Cong = $true )
& ( subsetTrans = $true )
& ( emptyI = $true )
& ( nonemptyI1 = $true )
& ( dsetconstrEL = $true )
& ( exuI2 = $true )
& ( omegaSAx = $true )
& ( dsetconstrI = $true )
& ( setadjoinIL = $true )
& ( omega__Cong = $true )
& ( setextsub = $true )
& ( setadjoinIR = $true )
& ( powersetI = $true )
& ( quantDeMorgan4 = $true )
& ( powersetE1 = $true )
& ( notinemptyset = $true )
& ( prop2setE = $true )
& ( emptyE1 = $true )
& ( setunionAx = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK35,sK36,sK37])],[f355,f501]) ).
thf(f501,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
!= $true )
& ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
= $true )
& ( $true
!= ( in @ X2 @ X1 ) ) )
=> ( ( $true
!= ( in @ sK37 @ sK35 ) )
& ( $true
= ( in @ sK37 @ ( binunion @ sK36 @ sK35 ) ) )
& ( $true
!= ( in @ sK37 @ sK36 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f355,plain,
( ( exuI1 = $true )
& ( descrp = $true )
& ( binunionIL = $true )
& ( setadjoinE = $true )
& ( subsetE = $true )
& ( in__Cong = $true )
& ( setadjoinSub2 = $true )
& ( emptysetAx = $true )
& ( setadjoinOr = $true )
& ( eqimpsubset1 = $true )
& ( exuEu = $true )
& ( nonemptyI = $true )
& ( exuE1 = $true )
& ( emptyinunitempty = $true )
& ( notequalI1 = $true )
& ( nonemptyE1 = $true )
& ( emptyinPowerset = $true )
& ( setadjoinSub = $true )
& ( exuE3e = $true )
& ( replAx = $true )
& ( sepInPowerset = $true )
& ( emptyset__Cong = $true )
& ( setbeta = $true )
& ( emptyInPowerset = $true )
& ( subPowSU = $true )
& ( inPowerset = $true )
& ( subset2powerset = $true )
& ( exu__Cong = $true )
& ( subsetI1 = $true )
& ( eqimpsubset2 = $true )
& ( upairset2IR = $true )
& ( emptysetsubset = $true )
& ( powersetsubset = $true )
& ( setadjoin__Cong = $true )
& ( setextAx = $true )
& ( wellorderingAx = $true )
& ( exuI3 = $true )
& ( powersetAx = $true )
& ( eqinunit = $true )
& ( setoftrueEq = $true )
& ( singletonsswitch = $true )
& ( setext = $true )
& ( subsetI2 = $true )
& ( notdexE = $true )
& ( descr__Cong = $true )
& ( emptysetE = $true )
& ( subsetemptysetimpeq = $true )
& ( setunionE = $true )
& ( powersetI1 = $true )
& ( dsetconstrER = $true )
& ( sepSubset = $true )
& ( notequalI2 = $true )
& ( powersetE = $true )
& ( upairsetE = $true )
& ( subsetE2 = $true )
& ( quantDeMorgan2 = $true )
& ( binunionIR = $true )
& ( dsetconstr__Cong = $true )
& ( omega0Ax = $true )
& ( noeltsimpempty = $true )
& ( setadjoinAx = $true )
& ( quantDeMorgan3 = $true )
& ( upairsetIR = $true )
& ( prop2setI = $true )
& ( upairsetIL = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( subsetRefl = $true )
& ( exuE2 = $true )
& ( quantDeMorgan1 = $true )
& ( emptysetimpfalse = $true )
& ( nonemptyImpWitness = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ X0 )
!= $true )
& ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
= $true )
& ( $true
!= ( in @ X2 @ X1 ) ) )
& ( prop2set2propI = $true )
& ( inCongP = $true )
& ( powerset__Cong = $true )
& ( omegaIndAx = $true )
& ( notsubsetI = $true )
& ( foundationAx = $true )
& ( exuE3u = $true )
& ( setunionI = $true )
& ( binunionEcases = $true )
& ( notdallE = $true )
& ( vacuousDall = $true )
& ( setunion__Cong = $true )
& ( subsetTrans = $true )
& ( emptyI = $true )
& ( nonemptyI1 = $true )
& ( dsetconstrEL = $true )
& ( exuI2 = $true )
& ( omegaSAx = $true )
& ( dsetconstrI = $true )
& ( setadjoinIL = $true )
& ( omega__Cong = $true )
& ( setextsub = $true )
& ( setadjoinIR = $true )
& ( powersetI = $true )
& ( quantDeMorgan4 = $true )
& ( powersetE1 = $true )
& ( notinemptyset = $true )
& ( prop2setE = $true )
& ( emptyE1 = $true )
& ( setunionAx = $true ) ),
inference(flattening,[status(thm)],[f354]) ).
thf(f354,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X2 @ X1 ) )
& ( ( in @ X2 @ X0 )
!= $true )
& ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
= $true ) )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( 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)],[f261]) ).
thf(f261,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 )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
= $true )
=> ( ( $true
= ( in @ X2 @ X1 ) )
| ( ( in @ X2 @ X0 )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f260]) ).
thf(f260,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
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ ( binunion @ X1 @ X0 ) )
=> ( ( in @ X2 @ X1 )
| ( in @ X2 @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f106]) ).
thf(f106,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
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ! [X4: $i,X3: $i,X1: $i] :
( ( in @ X1 @ ( binunion @ X3 @ X4 ) )
=> ( ( in @ X1 @ X3 )
| ( in @ X1 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f105]) ).
thf(f105,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
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ! [X4: $i,X3: $i,X1: $i] :
( ( in @ X1 @ ( binunion @ X3 @ X4 ) )
=> ( ( in @ X1 @ X3 )
| ( in @ X1 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.9nobNTBT08/DTF2THF_21348.p',binunionE) ).
thf(f1915,plain,
~ spl285_3,
inference(avatar_contradiction_clause,[status(thm)],[f1914]) ).
thf(f1914,plain,
( $false
| ~ spl285_3 ),
inference(trivial_inequality_removal,[status(thm)],[f1691]) ).
thf(f1691,plain,
( ( $true != $true )
| ~ spl285_3 ),
inference(superposition,[status(thm)],[f1011,f1598]) ).
thf(f1598,plain,
( ! [X0: $o] : ( $true = X0 )
| ~ spl285_3 ),
inference(avatar_component_clause,[status(thm)],[f1597]) ).
thf(f1597,definition,
( spl285_3
<=> ! [X0: $o] : ( $true = X0 ) ),
introduced(definition,[new_symbols(naming,[spl285_3])],[avatar_definition]) ).
thf(f1604,plain,
~ spl285_2,
inference(avatar_contradiction_clause,[status(thm)],[f1603]) ).
thf(f1603,plain,
( $false
| ~ spl285_2 ),
inference(trivial_inequality_removal,[status(thm)],[f1600]) ).
thf(f1600,plain,
( ( $true != $true )
| ~ spl285_2 ),
inference(superposition,[status(thm)],[f1013,f1595]) ).
thf(f1595,plain,
( ( $true
= ( in @ sK37 @ sK35 ) )
| ~ spl285_2 ),
inference(avatar_component_clause,[status(thm)],[f1593]) ).
thf(f1593,definition,
( spl285_2
<=> ( $true
= ( in @ sK37 @ sK35 ) ) ),
introduced(definition,[new_symbols(naming,[spl285_2])],[avatar_definition]) ).
thf(f1013,plain,
( $true
!= ( in @ sK37 @ sK35 ) ),
inference(cnf_transformation,[status(thm)],[f502]) ).
thf(f1599,plain,
( spl285_1
| spl285_2
| spl285_3 ),
inference(avatar_split_clause,[status(thm)],[f1587,f1597,f1593,f1589]) ).
thf(f1587,plain,
! [X0: $o] :
( ( $true = X0 )
| ( $true
= ( in @ sK37 @ sK36 ) )
| ( $true
= ( in @ sK37 @ sK35 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f1586]) ).
thf(f1586,plain,
! [X0: $o] :
( ( $true != $true )
| ( $true = X0 )
| ( $true
= ( in @ sK37 @ sK35 ) )
| ( $true
= ( in @ sK37 @ sK36 ) ) ),
inference(forward_demodulation,[status(thm)],[f1582,f1002]) ).
thf(f1002,plain,
binunionEcases = $true,
inference(cnf_transformation,[status(thm)],[f502]) ).
thf(f1582,plain,
! [X0: $o] :
( ( $true
= ( in @ sK37 @ sK36 ) )
| ( $true = X0 )
| ( binunionEcases != $true )
| ( $true
= ( in @ sK37 @ sK35 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f1580]) ).
thf(f1580,plain,
! [X0: $o] :
( ( $true != $true )
| ( $true
= ( in @ sK37 @ sK35 ) )
| ( $true = X0 )
| ( $true
= ( in @ sK37 @ sK36 ) )
| ( binunionEcases != $true ) ),
inference(superposition,[status(thm)],[f1260,f1012]) ).
thf(f1012,plain,
( $true
= ( in @ sK37 @ ( binunion @ sK36 @ sK35 ) ) ),
inference(cnf_transformation,[status(thm)],[f502]) ).
thf(f1260,plain,
! [X6: $i,X7: $i,X4: $i,X5: $o] :
( ( $true
!= ( in @ X7 @ ( binunion @ X4 @ X6 ) ) )
| ( $true = X5 )
| ( $true
= ( in @ X7 @ X6 ) )
| ( ( in @ X7 @ X4 )
= $true )
| ( binunionEcases != $true ) ),
inference(cnf_transformation,[status(thm)],[f711]) ).
thf(f711,plain,
( ( ( binunionEcases = $true )
| ( ( ( $true = sK154 )
| ( $true
!= ( in @ sK156 @ sK153 ) ) )
& ( ( $true = sK154 )
| ( $true
!= ( in @ sK156 @ sK155 ) ) )
& ( ( in @ sK156 @ ( binunion @ sK153 @ sK155 ) )
= $true )
& ( $true != sK154 ) ) )
& ( ! [X4: $i,X5: $o,X6: $i,X7: $i] :
( ( ( $true != X5 )
& ( ( in @ X7 @ X4 )
= $true ) )
| ( ( $true != X5 )
& ( $true
= ( in @ X7 @ X6 ) ) )
| ( $true
!= ( in @ X7 @ ( binunion @ X4 @ X6 ) ) )
| ( $true = X5 ) )
| ( binunionEcases != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK153,sK154,sK155,sK156])],[f709,f710]) ).
thf(f710,plain,
( ? [X0: $i,X1: $o,X2: $i,X3: $i] :
( ( ( $true = X1 )
| ( $true
!= ( in @ X3 @ X0 ) ) )
& ( ( $true = X1 )
| ( $true
!= ( in @ X3 @ X2 ) ) )
& ( ( in @ X3 @ ( binunion @ X0 @ X2 ) )
= $true )
& ( $true != X1 ) )
=> ( ( ( $true = sK154 )
| ( $true
!= ( in @ sK156 @ sK153 ) ) )
& ( ( $true = sK154 )
| ( $true
!= ( in @ sK156 @ sK155 ) ) )
& ( ( in @ sK156 @ ( binunion @ sK153 @ sK155 ) )
= $true )
& ( $true != sK154 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f709,plain,
( ( ( binunionEcases = $true )
| ? [X0: $i,X1: $o,X2: $i,X3: $i] :
( ( ( $true = X1 )
| ( $true
!= ( in @ X3 @ X0 ) ) )
& ( ( $true = X1 )
| ( $true
!= ( in @ X3 @ X2 ) ) )
& ( ( in @ X3 @ ( binunion @ X0 @ X2 ) )
= $true )
& ( $true != X1 ) ) )
& ( ! [X4: $i,X5: $o,X6: $i,X7: $i] :
( ( ( $true != X5 )
& ( ( in @ X7 @ X4 )
= $true ) )
| ( ( $true != X5 )
& ( $true
= ( in @ X7 @ X6 ) ) )
| ( $true
!= ( in @ X7 @ ( binunion @ X4 @ X6 ) ) )
| ( $true = X5 ) )
| ( binunionEcases != $true ) ) ),
inference(rectify,[status(thm)],[f708]) ).
thf(f708,plain,
( ( ( binunionEcases = $true )
| ? [X0: $i,X1: $o,X2: $i,X3: $i] :
( ( ( $true = X1 )
| ( $true
!= ( in @ X3 @ X0 ) ) )
& ( ( $true = X1 )
| ( $true
!= ( in @ X3 @ X2 ) ) )
& ( ( in @ X3 @ ( binunion @ X0 @ X2 ) )
= $true )
& ( $true != X1 ) ) )
& ( ! [X0: $i,X1: $o,X2: $i,X3: $i] :
( ( ( $true != X1 )
& ( $true
= ( in @ X3 @ X0 ) ) )
| ( ( $true != X1 )
& ( $true
= ( in @ X3 @ X2 ) ) )
| ( ( in @ X3 @ ( binunion @ X0 @ X2 ) )
!= $true )
| ( $true = X1 ) )
| ( binunionEcases != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f429]) ).
thf(f429,plain,
( ( binunionEcases = $true )
<=> ! [X0: $i,X1: $o,X2: $i,X3: $i] :
( ( ( $true != X1 )
& ( $true
= ( in @ X3 @ X0 ) ) )
| ( ( $true != X1 )
& ( $true
= ( in @ X3 @ X2 ) ) )
| ( ( in @ X3 @ ( binunion @ X0 @ X2 ) )
!= $true )
| ( $true = X1 ) ) ),
inference(flattening,[status(thm)],[f428]) ).
thf(f428,plain,
( ! [X0: $i,X1: $o,X2: $i,X3: $i] :
( ( $true = X1 )
| ( ( $true != X1 )
& ( $true
= ( in @ X3 @ X2 ) ) )
| ( ( $true != X1 )
& ( $true
= ( in @ X3 @ X0 ) ) )
| ( ( in @ X3 @ ( binunion @ X0 @ X2 ) )
!= $true ) )
<=> ( binunionEcases = $true ) ),
inference(ennf_transformation,[status(thm)],[f133]) ).
thf(f133,plain,
( ! [X0: $i,X1: $o,X2: $i,X3: $i] :
( ( ( in @ X3 @ ( binunion @ X0 @ X2 ) )
= $true )
=> ( ( ( $true
= ( in @ X3 @ X0 ) )
=> ( $true = X1 ) )
=> ( ( ( $true
= ( in @ X3 @ X2 ) )
=> ( $true = X1 ) )
=> ( $true = X1 ) ) ) )
<=> ( binunionEcases = $true ) ),
inference(fool_elimination,[status(thm)],[f132]) ).
thf(f132,plain,
( binunionEcases
= ( ! [X0: $i,X1: $o,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X0 @ X2 ) )
=> ( ( ( in @ X3 @ X0 )
=> X1 )
=> ( ( ( in @ X3 @ X2 )
=> X1 )
=> X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f104]) ).
thf(f104,axiom,
( binunionEcases
= ( ! [X3: $i,X0: $o,X4: $i,X1: $i] :
( ( in @ X1 @ ( binunion @ X3 @ X4 ) )
=> ( ( ( in @ X1 @ X3 )
=> X0 )
=> ( ( ( in @ X1 @ X4 )
=> X0 )
=> X0 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.9nobNTBT08/DTF2THF_21348.p',binunionEcases) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.06 % Problem : SEU586^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.07 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.08/0.25 % Computer : n012.cluster.edu
% 0.08/0.25 % Model : x86_64 x86_64
% 0.08/0.25 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.25 % Memory : 8042.1875MB
% 0.08/0.25 % OS : Linux 3.10.0-693.el7.x86_64
% 0.08/0.25 % CPULimit : 300
% 0.08/0.25 % WCLimit : 300
% 0.08/0.25 % DateTime : Tue Feb 24 12:33:45 EST 2026
% 0.08/0.25 % CPUTime :
% 0.08/0.25 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.14/0.36 ---- Original DTF file ---
% 0.14/0.36 thf(spec,logic,$$dhol).
% 0.14/0.36 %------------------------------------------------------------------------------
% 0.14/0.36 % File : SEU586^1 : TPTP v9.2.1. Released v3.7.0.
% 0.14/0.36 % Domain : Set Theory
% 0.14/0.36 % Problem : Preliminary Notions - Ops on Sets - Unions and Intersections
% 0.14/0.36 % Version : Especial.
% 0.14/0.36 % English : (! A:i.! B:i.! x:i.in x (binunion A B) -> in x A | in x B)
% 0.14/0.36
% 0.14/0.36 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.14/0.36 % Source : [Bro08]
% 0.14/0.36 % Names : ZFC088g [Bro08]
% 0.14/0.36
% 0.14/0.36 % Status : Theorem
% 0.14/0.36 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.23 v8.1.0, 0.00 v7.4.0, 0.22 v7.2.0, 0.12 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, 0.60 v5.0.0, 0.40 v4.1.0, 0.67 v3.7.0
% 0.14/0.36 % Syntax : Number of formulae : 223 ( 104 unt; 118 typ; 104 def)
% 0.14/0.36 % Number of atoms : 561 ( 153 equ; 0 cnn)
% 0.14/0.36 % Maximal formula atoms : 105 ( 5 avg)
% 0.14/0.36 % Number of connectives : 885 ( 29 ~; 6 |; 27 &; 555 @)
% 0.14/0.36 % ( 14 <=>; 254 =>; 0 <=; 0 <~>)
% 0.14/0.36 % Maximal formula depth : 110 ( 2 avg)
% 0.14/0.36 % Number of types : 2 ( 0 usr)
% 0.14/0.36 % Number of type conns : 57 ( 57 >; 0 *; 0 +; 0 <<)
% 0.14/0.36 % Number of symbols : 121 ( 118 usr; 106 con; 0-2 aty)
% 0.14/0.36 % Number of variables : 309 ( 30 ^; 251 !; 28 ?; 309 :)
% 0.14/0.36 % SPC : TH0_THM_EQU_NAR
% 0.14/0.36
% 0.14/0.36 % Comments : http://mathgate.info/detsetitem.php?id=162
% 0.14/0.36 %------------------------------------------------------------------------------
% 0.14/0.36 thf(in_type,type,
% 0.14/0.36 in: $i > $i > $o ).
% 0.14/0.36
% 0.14/0.36 thf(exu_type,type,
% 0.14/0.36 exu: ( $i > $o ) > $o ).
% 0.14/0.36
% 0.14/0.36 thf(exu,definition,
% 0.14/0.36 ( exu
% 0.14/0.36 = ( ^ [Xphi: $i > $o] :
% 0.14/0.36 ? [Xx: $i] :
% 0.14/0.36 ( ( Xphi @ Xx )
% 0.14/0.36 & ! [Xy: $i] :
% 0.14/0.36 ( ( Xphi @ Xy )
% 0.14/0.36 => ( Xx = Xy ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(setextAx_type,type,
% 0.14/0.36 setextAx: $o ).
% 0.14/0.36
% 0.14/0.36 thf(setextAx,definition,
% 0.14/0.36 ( setextAx
% 0.14/0.36 = ( ! [A: $i,B: $i] :
% 0.14/0.36 ( ! [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 <=> ( in @ Xx @ B ) )
% 0.14/0.36 => ( A = B ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(emptyset_type,type,
% 0.14/0.36 emptyset: $i ).
% 0.14/0.36
% 0.14/0.36 thf(emptysetAx_type,type,
% 0.14/0.36 emptysetAx: $o ).
% 0.14/0.36
% 0.14/0.36 thf(emptysetAx,definition,
% 0.14/0.36 ( emptysetAx
% 0.14/0.36 = ( ! [Xx: $i] :
% 0.14/0.36 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(setadjoin_type,type,
% 0.14/0.36 setadjoin: $i > $i > $i ).
% 0.14/0.36
% 0.14/0.36 thf(setadjoinAx_type,type,
% 0.14/0.36 setadjoinAx: $o ).
% 0.14/0.36
% 0.14/0.36 thf(setadjoinAx,definition,
% 0.14/0.36 ( setadjoinAx
% 0.14/0.36 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.14/0.36 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.14/0.36 <=> ( ( Xy = Xx )
% 0.14/0.36 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(powerset_type,type,
% 0.14/0.36 powerset: $i > $i ).
% 0.14/0.36
% 0.14/0.36 thf(powersetAx_type,type,
% 0.14/0.36 powersetAx: $o ).
% 0.14/0.36
% 0.14/0.36 thf(powersetAx,definition,
% 0.14/0.36 ( powersetAx
% 0.14/0.36 = ( ! [A: $i,B: $i] :
% 0.14/0.36 ( ( in @ B @ ( powerset @ A ) )
% 0.14/0.36 <=> ! [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ B )
% 0.14/0.36 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(setunion_type,type,
% 0.14/0.36 setunion: $i > $i ).
% 0.14/0.36
% 0.14/0.36 thf(setunionAx_type,type,
% 0.14/0.36 setunionAx: $o ).
% 0.14/0.36
% 0.14/0.36 thf(setunionAx,definition,
% 0.14/0.36 ( setunionAx
% 0.14/0.36 = ( ! [A: $i,Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ ( setunion @ A ) )
% 0.14/0.36 <=> ? [B: $i] :
% 0.14/0.36 ( ( in @ Xx @ B )
% 0.14/0.36 & ( in @ B @ A ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(omega_type,type,
% 0.14/0.36 omega: $i ).
% 0.14/0.36
% 0.14/0.36 thf(omega0Ax_type,type,
% 0.14/0.36 omega0Ax: $o ).
% 0.14/0.36
% 0.14/0.36 thf(omega0Ax,definition,
% 0.14/0.36 ( omega0Ax
% 0.14/0.36 = ( in @ emptyset @ omega ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(omegaSAx_type,type,
% 0.14/0.36 omegaSAx: $o ).
% 0.14/0.36
% 0.14/0.36 thf(omegaSAx,definition,
% 0.14/0.36 ( omegaSAx
% 0.14/0.36 = ( ! [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ omega )
% 0.14/0.36 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(omegaIndAx_type,type,
% 0.14/0.36 omegaIndAx: $o ).
% 0.14/0.36
% 0.14/0.36 thf(omegaIndAx,definition,
% 0.14/0.36 ( omegaIndAx
% 0.14/0.36 = ( ! [A: $i] :
% 0.14/0.36 ( ( ( in @ emptyset @ A )
% 0.14/0.36 & ! [Xx: $i] :
% 0.14/0.36 ( ( ( in @ Xx @ omega )
% 0.14/0.36 & ( in @ Xx @ A ) )
% 0.14/0.36 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.14/0.36 => ! [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ omega )
% 0.14/0.36 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(replAx_type,type,
% 0.14/0.36 replAx: $o ).
% 0.14/0.36
% 0.14/0.36 thf(replAx,definition,
% 0.14/0.36 ( replAx
% 0.14/0.36 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.14/0.36 ( ! [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 => ( exu
% 0.14/0.36 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.14/0.36 => ? [B: $i] :
% 0.14/0.36 ! [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ B )
% 0.14/0.36 <=> ? [Xy: $i] :
% 0.14/0.36 ( ( in @ Xy @ A )
% 0.14/0.36 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(foundationAx_type,type,
% 0.14/0.36 foundationAx: $o ).
% 0.14/0.36
% 0.14/0.36 thf(foundationAx,definition,
% 0.14/0.36 ( foundationAx
% 0.14/0.36 = ( ! [A: $i] :
% 0.14/0.36 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.14/0.36 => ? [B: $i] :
% 0.14/0.36 ( ( in @ B @ A )
% 0.14/0.36 & ~ ? [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ B )
% 0.14/0.36 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(wellorderingAx_type,type,
% 0.14/0.36 wellorderingAx: $o ).
% 0.14/0.36
% 0.14/0.36 thf(wellorderingAx,definition,
% 0.14/0.36 ( wellorderingAx
% 0.14/0.36 = ( ! [A: $i] :
% 0.14/0.36 ? [B: $i] :
% 0.14/0.36 ( ! [C: $i] :
% 0.14/0.36 ( ( in @ C @ B )
% 0.14/0.36 => ! [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ C )
% 0.14/0.36 => ( in @ Xx @ A ) ) )
% 0.14/0.36 & ! [Xx: $i,Xy: $i] :
% 0.14/0.36 ( ( ( in @ Xx @ A )
% 0.14/0.36 & ( in @ Xy @ A ) )
% 0.14/0.36 => ( ! [C: $i] :
% 0.14/0.36 ( ( in @ C @ B )
% 0.14/0.36 => ( ( in @ Xx @ C )
% 0.14/0.36 <=> ( in @ Xy @ C ) ) )
% 0.14/0.36 => ( Xx = Xy ) ) )
% 0.14/0.36 & ! [C: $i,D: $i] :
% 0.14/0.36 ( ( ( in @ C @ B )
% 0.14/0.36 & ( in @ D @ B ) )
% 0.14/0.36 => ( ! [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ C )
% 0.14/0.36 => ( in @ Xx @ D ) )
% 0.14/0.36 | ! [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ D )
% 0.14/0.36 => ( in @ Xx @ C ) ) ) )
% 0.14/0.36 & ! [C: $i] :
% 0.14/0.36 ( ( ! [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ C )
% 0.14/0.36 => ( in @ Xx @ A ) )
% 0.14/0.36 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.14/0.36 => ? [D: $i,Xx: $i] :
% 0.14/0.36 ( ( in @ D @ B )
% 0.14/0.36 & ( in @ Xx @ C )
% 0.14/0.36 & ~ ? [Xy: $i] :
% 0.14/0.36 ( ( in @ Xy @ D )
% 0.14/0.36 & ( in @ Xy @ C ) )
% 0.14/0.36 & ! [E: $i] :
% 0.14/0.36 ( ( in @ E @ B )
% 0.14/0.36 => ( ! [Xy: $i] :
% 0.14/0.36 ( ( in @ Xy @ E )
% 0.14/0.36 => ( in @ Xy @ D ) )
% 0.14/0.36 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(descr_type,type,
% 0.14/0.36 descr: ( $i > $o ) > $i ).
% 0.14/0.36
% 0.14/0.36 thf(descrp_type,type,
% 0.14/0.36 descrp: $o ).
% 0.14/0.36
% 0.14/0.36 thf(descrp,definition,
% 0.14/0.36 ( descrp
% 0.14/0.36 = ( ! [Xphi: $i > $o] :
% 0.14/0.36 ( ( exu
% 0.14/0.36 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.14/0.36 => ( Xphi
% 0.14/0.36 @ ( descr
% 0.14/0.36 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(dsetconstr_type,type,
% 0.14/0.36 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.14/0.36
% 0.14/0.36 thf(dsetconstrI_type,type,
% 0.14/0.36 dsetconstrI: $o ).
% 0.14/0.36
% 0.14/0.36 thf(dsetconstrI,definition,
% 0.14/0.36 ( dsetconstrI
% 0.14/0.36 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 => ( ( Xphi @ Xx )
% 0.14/0.36 => ( in @ Xx
% 0.14/0.36 @ ( dsetconstr @ A
% 0.14/0.36 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(dsetconstrEL_type,type,
% 0.14/0.36 dsetconstrEL: $o ).
% 0.14/0.36
% 0.14/0.36 thf(dsetconstrEL,definition,
% 0.14/0.36 ( dsetconstrEL
% 0.14/0.36 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.14/0.36 ( ( in @ Xx
% 0.14/0.36 @ ( dsetconstr @ A
% 0.14/0.36 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.14/0.36 => ( in @ Xx @ A ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(dsetconstrER_type,type,
% 0.14/0.36 dsetconstrER: $o ).
% 0.14/0.36
% 0.14/0.36 thf(dsetconstrER,definition,
% 0.14/0.36 ( dsetconstrER
% 0.14/0.36 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.14/0.36 ( ( in @ Xx
% 0.14/0.36 @ ( dsetconstr @ A
% 0.14/0.36 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.14/0.36 => ( Xphi @ Xx ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(exuE1_type,type,
% 0.14/0.36 exuE1: $o ).
% 0.14/0.36
% 0.14/0.36 thf(exuE1,definition,
% 0.14/0.36 ( exuE1
% 0.14/0.36 = ( ! [Xphi: $i > $o] :
% 0.14/0.36 ( ( exu
% 0.14/0.36 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.14/0.36 => ? [Xx: $i] :
% 0.14/0.36 ( ( Xphi @ Xx )
% 0.14/0.36 & ! [Xy: $i] :
% 0.14/0.36 ( ( Xphi @ Xy )
% 0.14/0.36 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(prop2set_type,type,
% 0.14/0.36 prop2set: $o > $i ).
% 0.14/0.36
% 0.14/0.36 thf(prop2setE_type,type,
% 0.14/0.36 prop2setE: $o ).
% 0.14/0.36
% 0.14/0.36 thf(prop2setE,definition,
% 0.14/0.36 ( prop2setE
% 0.14/0.36 = ( ! [Xphi: $o,Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.14/0.36 => Xphi ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(emptysetE_type,type,
% 0.14/0.36 emptysetE: $o ).
% 0.14/0.36
% 0.14/0.36 thf(emptysetE,definition,
% 0.14/0.36 ( emptysetE
% 0.14/0.36 = ( ! [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ emptyset )
% 0.14/0.36 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(emptysetimpfalse_type,type,
% 0.14/0.36 emptysetimpfalse: $o ).
% 0.14/0.36
% 0.14/0.36 thf(emptysetimpfalse,definition,
% 0.14/0.36 ( emptysetimpfalse
% 0.14/0.36 = ( ! [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ emptyset )
% 0.14/0.36 => $false ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(notinemptyset_type,type,
% 0.14/0.36 notinemptyset: $o ).
% 0.14/0.36
% 0.14/0.36 thf(notinemptyset,definition,
% 0.14/0.36 ( notinemptyset
% 0.14/0.36 = ( ! [Xx: $i] :
% 0.14/0.36 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(exuE3e_type,type,
% 0.14/0.36 exuE3e: $o ).
% 0.14/0.36
% 0.14/0.36 thf(exuE3e,definition,
% 0.14/0.36 ( exuE3e
% 0.14/0.36 = ( ! [Xphi: $i > $o] :
% 0.14/0.36 ( ( exu
% 0.14/0.36 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.14/0.36 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(setext_type,type,
% 0.14/0.36 setext: $o ).
% 0.14/0.36
% 0.14/0.36 thf(setext,definition,
% 0.14/0.36 ( setext
% 0.14/0.36 = ( ! [A: $i,B: $i] :
% 0.14/0.36 ( ! [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 => ( in @ Xx @ B ) )
% 0.14/0.36 => ( ! [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ B )
% 0.14/0.36 => ( in @ Xx @ A ) )
% 0.14/0.36 => ( A = B ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(emptyI_type,type,
% 0.14/0.36 emptyI: $o ).
% 0.14/0.36
% 0.14/0.36 thf(emptyI,definition,
% 0.14/0.36 ( emptyI
% 0.14/0.36 = ( ! [A: $i] :
% 0.14/0.36 ( ! [Xx: $i] :
% 0.14/0.36 ~ ( in @ Xx @ A )
% 0.14/0.36 => ( A = emptyset ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(noeltsimpempty_type,type,
% 0.14/0.36 noeltsimpempty: $o ).
% 0.14/0.36
% 0.14/0.36 thf(noeltsimpempty,definition,
% 0.14/0.36 ( noeltsimpempty
% 0.14/0.36 = ( ! [A: $i] :
% 0.14/0.36 ( ! [Xx: $i] :
% 0.14/0.36 ~ ( in @ Xx @ A )
% 0.14/0.36 => ( A = emptyset ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(setbeta_type,type,
% 0.14/0.36 setbeta: $o ).
% 0.14/0.36
% 0.14/0.36 thf(setbeta,definition,
% 0.14/0.36 ( setbeta
% 0.14/0.36 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 => ( ( in @ Xx
% 0.14/0.36 @ ( dsetconstr @ A
% 0.14/0.36 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.14/0.36 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(nonempty_type,type,
% 0.14/0.36 nonempty: $i > $o ).
% 0.14/0.36
% 0.14/0.36 thf(nonempty,definition,
% 0.14/0.36 ( nonempty
% 0.14/0.36 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(nonemptyE1_type,type,
% 0.14/0.36 nonemptyE1: $o ).
% 0.14/0.36
% 0.14/0.36 thf(nonemptyE1,definition,
% 0.14/0.36 ( nonemptyE1
% 0.14/0.36 = ( ! [A: $i] :
% 0.14/0.36 ( ( nonempty @ A )
% 0.14/0.36 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(nonemptyI_type,type,
% 0.14/0.36 nonemptyI: $o ).
% 0.14/0.36
% 0.14/0.36 thf(nonemptyI,definition,
% 0.14/0.36 ( nonemptyI
% 0.14/0.36 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 => ( ( Xphi @ Xx )
% 0.14/0.36 => ( nonempty
% 0.14/0.36 @ ( dsetconstr @ A
% 0.14/0.36 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(nonemptyI1_type,type,
% 0.14/0.36 nonemptyI1: $o ).
% 0.14/0.36
% 0.14/0.36 thf(nonemptyI1,definition,
% 0.14/0.36 ( nonemptyI1
% 0.14/0.36 = ( ! [A: $i] :
% 0.14/0.36 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.14/0.36 => ( nonempty @ A ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(setadjoinIL_type,type,
% 0.14/0.36 setadjoinIL: $o ).
% 0.14/0.36
% 0.14/0.36 thf(setadjoinIL,definition,
% 0.14/0.36 ( setadjoinIL
% 0.14/0.36 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(emptyinunitempty_type,type,
% 0.14/0.36 emptyinunitempty: $o ).
% 0.14/0.36
% 0.14/0.36 thf(emptyinunitempty,definition,
% 0.14/0.36 ( emptyinunitempty
% 0.14/0.36 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(setadjoinIR_type,type,
% 0.14/0.36 setadjoinIR: $o ).
% 0.14/0.36
% 0.14/0.36 thf(setadjoinIR,definition,
% 0.14/0.36 ( setadjoinIR
% 0.14/0.36 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.14/0.36 ( ( in @ Xy @ A )
% 0.14/0.36 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(setadjoinE_type,type,
% 0.14/0.36 setadjoinE: $o ).
% 0.14/0.36
% 0.14/0.36 thf(setadjoinE,definition,
% 0.14/0.36 ( setadjoinE
% 0.14/0.36 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.14/0.36 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.14/0.36 => ! [Xphi: $o] :
% 0.14/0.36 ( ( ( Xy = Xx )
% 0.14/0.36 => Xphi )
% 0.14/0.36 => ( ( ( in @ Xy @ A )
% 0.14/0.36 => Xphi )
% 0.14/0.36 => Xphi ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(setadjoinOr_type,type,
% 0.14/0.36 setadjoinOr: $o ).
% 0.14/0.36
% 0.14/0.36 thf(setadjoinOr,definition,
% 0.14/0.36 ( setadjoinOr
% 0.14/0.36 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.14/0.36 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.14/0.36 => ( ( Xy = Xx )
% 0.14/0.36 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(setoftrueEq_type,type,
% 0.14/0.36 setoftrueEq: $o ).
% 0.14/0.36
% 0.14/0.36 thf(setoftrueEq,definition,
% 0.14/0.36 ( setoftrueEq
% 0.14/0.36 = ( ! [A: $i] :
% 0.14/0.36 ( ( dsetconstr @ A
% 0.14/0.36 @ ^ [Xx: $i] : $true )
% 0.14/0.36 = A ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(powersetI_type,type,
% 0.14/0.36 powersetI: $o ).
% 0.14/0.36
% 0.14/0.36 thf(powersetI,definition,
% 0.14/0.36 ( powersetI
% 0.14/0.36 = ( ! [A: $i,B: $i] :
% 0.14/0.36 ( ! [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ B )
% 0.14/0.36 => ( in @ Xx @ A ) )
% 0.14/0.36 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(emptyinPowerset_type,type,
% 0.14/0.36 emptyinPowerset: $o ).
% 0.14/0.36
% 0.14/0.36 thf(emptyinPowerset,definition,
% 0.14/0.36 ( emptyinPowerset
% 0.14/0.36 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(emptyInPowerset_type,type,
% 0.14/0.36 emptyInPowerset: $o ).
% 0.14/0.36
% 0.14/0.36 thf(emptyInPowerset,definition,
% 0.14/0.36 ( emptyInPowerset
% 0.14/0.36 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(powersetE_type,type,
% 0.14/0.36 powersetE: $o ).
% 0.14/0.36
% 0.14/0.36 thf(powersetE,definition,
% 0.14/0.36 ( powersetE
% 0.14/0.36 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.14/0.36 ( ( in @ B @ ( powerset @ A ) )
% 0.14/0.36 => ( ( in @ Xx @ B )
% 0.14/0.36 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(setunionI_type,type,
% 0.14/0.36 setunionI: $o ).
% 0.14/0.36
% 0.14/0.36 thf(setunionI,definition,
% 0.14/0.36 ( setunionI
% 0.14/0.36 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.14/0.36 ( ( in @ Xx @ B )
% 0.14/0.36 => ( ( in @ B @ A )
% 0.14/0.36 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(setunionE_type,type,
% 0.14/0.36 setunionE: $o ).
% 0.14/0.36
% 0.14/0.36 thf(setunionE,definition,
% 0.14/0.36 ( setunionE
% 0.14/0.36 = ( ! [A: $i,Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ ( setunion @ A ) )
% 0.14/0.36 => ! [Xphi: $o] :
% 0.14/0.36 ( ! [B: $i] :
% 0.14/0.36 ( ( in @ Xx @ B )
% 0.14/0.36 => ( ( in @ B @ A )
% 0.14/0.36 => Xphi ) )
% 0.14/0.36 => Xphi ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(subPowSU_type,type,
% 0.14/0.36 subPowSU: $o ).
% 0.14/0.36
% 0.14/0.36 thf(subPowSU,definition,
% 0.14/0.36 ( subPowSU
% 0.14/0.36 = ( ! [A: $i,Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(exuE2_type,type,
% 0.14/0.36 exuE2: $o ).
% 0.14/0.36
% 0.14/0.36 thf(exuE2,definition,
% 0.14/0.36 ( exuE2
% 0.14/0.36 = ( ! [Xphi: $i > $o] :
% 0.14/0.36 ( ( exu
% 0.14/0.36 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.14/0.36 => ? [Xx: $i] :
% 0.14/0.36 ! [Xy: $i] :
% 0.14/0.36 ( ( Xphi @ Xy )
% 0.14/0.36 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(nonemptyImpWitness_type,type,
% 0.14/0.36 nonemptyImpWitness: $o ).
% 0.14/0.36
% 0.14/0.36 thf(nonemptyImpWitness,definition,
% 0.14/0.36 ( nonemptyImpWitness
% 0.14/0.36 = ( ! [A: $i] :
% 0.14/0.36 ( ( nonempty @ A )
% 0.14/0.36 => ? [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 & $true ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(uniqinunit_type,type,
% 0.14/0.36 uniqinunit: $o ).
% 0.14/0.36
% 0.14/0.36 thf(uniqinunit,definition,
% 0.14/0.36 ( uniqinunit
% 0.14/0.36 = ( ! [Xx: $i,Xy: $i] :
% 0.14/0.36 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.14/0.36 => ( Xx = Xy ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(notinsingleton_type,type,
% 0.14/0.36 notinsingleton: $o ).
% 0.14/0.36
% 0.14/0.36 thf(notinsingleton,definition,
% 0.14/0.36 ( notinsingleton
% 0.14/0.36 = ( ! [Xx: $i,Xy: $i] :
% 0.14/0.36 ( ( Xx != Xy )
% 0.14/0.36 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(eqinunit_type,type,
% 0.14/0.36 eqinunit: $o ).
% 0.14/0.36
% 0.14/0.36 thf(eqinunit,definition,
% 0.14/0.36 ( eqinunit
% 0.14/0.36 = ( ! [Xx: $i,Xy: $i] :
% 0.14/0.36 ( ( Xx = Xy )
% 0.14/0.36 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(singletonsswitch_type,type,
% 0.14/0.36 singletonsswitch: $o ).
% 0.14/0.36
% 0.14/0.36 thf(singletonsswitch,definition,
% 0.14/0.36 ( singletonsswitch
% 0.14/0.36 = ( ! [Xx: $i,Xy: $i] :
% 0.14/0.36 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.14/0.36 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(upairsetE_type,type,
% 0.14/0.36 upairsetE: $o ).
% 0.14/0.36
% 0.14/0.36 thf(upairsetE,definition,
% 0.14/0.36 ( upairsetE
% 0.14/0.36 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.14/0.36 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.14/0.36 => ( ( Xz = Xx )
% 0.14/0.36 | ( Xz = Xy ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(upairsetIL_type,type,
% 0.14/0.36 upairsetIL: $o ).
% 0.14/0.36
% 0.14/0.36 thf(upairsetIL,definition,
% 0.14/0.36 ( upairsetIL
% 0.14/0.36 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(upairsetIR_type,type,
% 0.14/0.36 upairsetIR: $o ).
% 0.14/0.36
% 0.14/0.36 thf(upairsetIR,definition,
% 0.14/0.36 ( upairsetIR
% 0.14/0.36 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(emptyE1_type,type,
% 0.14/0.36 emptyE1: $o ).
% 0.14/0.36
% 0.14/0.36 thf(emptyE1,definition,
% 0.14/0.36 ( emptyE1
% 0.14/0.36 = ( ! [A: $i,Xphi: $i > $o] :
% 0.14/0.36 ( ? [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 & ( Xphi @ Xx ) )
% 0.14/0.36 => ( ( ( dsetconstr @ A
% 0.14/0.36 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.14/0.36 = emptyset )
% 0.14/0.36 => $false ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(vacuousDall_type,type,
% 0.14/0.36 vacuousDall: $o ).
% 0.14/0.36
% 0.14/0.36 thf(vacuousDall,definition,
% 0.14/0.36 ( vacuousDall
% 0.14/0.36 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ emptyset )
% 0.14/0.36 => ( Xphi @ Xx ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(quantDeMorgan1_type,type,
% 0.14/0.36 quantDeMorgan1: $o ).
% 0.14/0.36
% 0.14/0.36 thf(quantDeMorgan1,definition,
% 0.14/0.36 ( quantDeMorgan1
% 0.14/0.36 = ( ! [A: $i,Xphi: $i > $o] :
% 0.14/0.36 ( ~ ! [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 => ( Xphi @ Xx ) )
% 0.14/0.36 => ? [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(quantDeMorgan2_type,type,
% 0.14/0.36 quantDeMorgan2: $o ).
% 0.14/0.36
% 0.14/0.36 thf(quantDeMorgan2,definition,
% 0.14/0.36 ( quantDeMorgan2
% 0.14/0.36 = ( ! [A: $i,Xphi: $i > $o] :
% 0.14/0.36 ( ! [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 => ~ ( Xphi @ Xx ) )
% 0.14/0.36 => ~ ? [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(quantDeMorgan3_type,type,
% 0.14/0.36 quantDeMorgan3: $o ).
% 0.14/0.36
% 0.14/0.36 thf(quantDeMorgan3,definition,
% 0.14/0.36 ( quantDeMorgan3
% 0.14/0.36 = ( ! [A: $i,Xphi: $i > $o] :
% 0.14/0.36 ( ~ ? [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 & ( Xphi @ Xx ) )
% 0.14/0.36 => ! [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(quantDeMorgan4_type,type,
% 0.14/0.36 quantDeMorgan4: $o ).
% 0.14/0.36
% 0.14/0.36 thf(quantDeMorgan4,definition,
% 0.14/0.36 ( quantDeMorgan4
% 0.14/0.36 = ( ! [A: $i,Xphi: $i > $o] :
% 0.14/0.36 ( ? [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 & ~ ( Xphi @ Xx ) )
% 0.14/0.36 => ~ ! [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(prop2setI_type,type,
% 0.14/0.36 prop2setI: $o ).
% 0.14/0.36
% 0.14/0.36 thf(prop2setI,definition,
% 0.14/0.36 ( prop2setI
% 0.14/0.36 = ( ! [Xphi: $o] :
% 0.14/0.36 ( Xphi
% 0.14/0.36 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(set2prop_type,type,
% 0.14/0.36 set2prop: $i > $o ).
% 0.14/0.36
% 0.14/0.36 thf(prop2set2propI_type,type,
% 0.14/0.36 prop2set2propI: $o ).
% 0.14/0.36
% 0.14/0.36 thf(prop2set2propI,definition,
% 0.14/0.36 ( prop2set2propI
% 0.14/0.36 = ( ! [Xphi: $o] :
% 0.14/0.36 ( Xphi
% 0.14/0.36 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(notdexE_type,type,
% 0.14/0.36 notdexE: $o ).
% 0.14/0.36
% 0.14/0.36 thf(notdexE,definition,
% 0.14/0.36 ( notdexE
% 0.14/0.36 = ( ! [A: $i,Xphi: $i > $o] :
% 0.14/0.36 ( ~ ? [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 & ( Xphi @ Xx ) )
% 0.14/0.36 => ! [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(notdallE_type,type,
% 0.14/0.36 notdallE: $o ).
% 0.14/0.36
% 0.14/0.36 thf(notdallE,definition,
% 0.14/0.36 ( notdallE
% 0.14/0.36 = ( ! [A: $i,Xphi: $i > $o] :
% 0.14/0.36 ( ~ ! [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 => ( Xphi @ Xx ) )
% 0.14/0.36 => ? [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(exuI1_type,type,
% 0.14/0.36 exuI1: $o ).
% 0.14/0.36
% 0.14/0.36 thf(exuI1,definition,
% 0.14/0.36 ( exuI1
% 0.14/0.36 = ( ! [Xphi: $i > $o] :
% 0.14/0.36 ( ? [Xx: $i] :
% 0.14/0.36 ( ( Xphi @ Xx )
% 0.14/0.36 & ! [Xy: $i] :
% 0.14/0.36 ( ( Xphi @ Xy )
% 0.14/0.36 => ( Xx = Xy ) ) )
% 0.14/0.36 => ( exu
% 0.14/0.36 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(exuI3_type,type,
% 0.14/0.36 exuI3: $o ).
% 0.14/0.36
% 0.14/0.36 thf(exuI3,definition,
% 0.14/0.36 ( exuI3
% 0.14/0.36 = ( ! [Xphi: $i > $o] :
% 0.14/0.36 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.14/0.36 => ( ! [Xx: $i,Xy: $i] :
% 0.14/0.36 ( ( Xphi @ Xx )
% 0.14/0.36 => ( ( Xphi @ Xy )
% 0.14/0.36 => ( Xx = Xy ) ) )
% 0.14/0.36 => ( exu
% 0.14/0.36 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(exuI2_type,type,
% 0.14/0.36 exuI2: $o ).
% 0.14/0.36
% 0.14/0.36 thf(exuI2,definition,
% 0.14/0.36 ( exuI2
% 0.14/0.36 = ( ! [Xphi: $i > $o] :
% 0.14/0.36 ( ? [Xx: $i] :
% 0.14/0.36 ! [Xy: $i] :
% 0.14/0.36 ( ( Xphi @ Xy )
% 0.14/0.36 <=> ( Xy = Xx ) )
% 0.14/0.36 => ( exu
% 0.14/0.36 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(inCongP_type,type,
% 0.14/0.36 inCongP: $o ).
% 0.14/0.36
% 0.14/0.36 thf(inCongP,definition,
% 0.14/0.36 ( inCongP
% 0.14/0.36 = ( ! [A: $i,B: $i] :
% 0.14/0.36 ( ( A = B )
% 0.14/0.36 => ! [Xx: $i,Xy: $i] :
% 0.14/0.36 ( ( Xx = Xy )
% 0.14/0.36 => ( ( in @ Xx @ A )
% 0.14/0.36 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(in__Cong_type,type,
% 0.14/0.36 in__Cong: $o ).
% 0.14/0.36
% 0.14/0.36 thf(in__Cong,definition,
% 0.14/0.36 ( in__Cong
% 0.14/0.36 = ( ! [A: $i,B: $i] :
% 0.14/0.36 ( ( A = B )
% 0.14/0.36 => ! [Xx: $i,Xy: $i] :
% 0.14/0.36 ( ( Xx = Xy )
% 0.14/0.36 => ( ( in @ Xx @ A )
% 0.14/0.36 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(exuE3u_type,type,
% 0.14/0.36 exuE3u: $o ).
% 0.14/0.36
% 0.14/0.36 thf(exuE3u,definition,
% 0.14/0.36 ( exuE3u
% 0.14/0.36 = ( ! [Xphi: $i > $o] :
% 0.14/0.36 ( ( exu
% 0.14/0.36 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.14/0.36 => ! [Xx: $i,Xy: $i] :
% 0.14/0.36 ( ( Xphi @ Xx )
% 0.14/0.36 => ( ( Xphi @ Xy )
% 0.14/0.36 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(exu__Cong_type,type,
% 0.14/0.36 exu__Cong: $o ).
% 0.14/0.36
% 0.14/0.36 thf(exu__Cong,definition,
% 0.14/0.36 ( exu__Cong
% 0.14/0.36 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.14/0.36 ( ! [Xx: $i,Xy: $i] :
% 0.14/0.36 ( ( Xx = Xy )
% 0.14/0.36 => ( ( Xphi @ Xx )
% 0.14/0.36 <=> ( Xpsi @ Xy ) ) )
% 0.14/0.36 => ( ( exu
% 0.14/0.36 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.14/0.36 <=> ( exu
% 0.14/0.36 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(emptyset__Cong_type,type,
% 0.14/0.36 emptyset__Cong: $o ).
% 0.14/0.36
% 0.14/0.36 thf(emptyset__Cong,definition,
% 0.14/0.36 ( emptyset__Cong
% 0.14/0.36 = ( emptyset = emptyset ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(setadjoin__Cong_type,type,
% 0.14/0.36 setadjoin__Cong: $o ).
% 0.14/0.36
% 0.14/0.36 thf(setadjoin__Cong,definition,
% 0.14/0.36 ( setadjoin__Cong
% 0.14/0.36 = ( ! [Xx: $i,Xy: $i] :
% 0.14/0.36 ( ( Xx = Xy )
% 0.14/0.36 => ! [Xz: $i,Xu: $i] :
% 0.14/0.36 ( ( Xz = Xu )
% 0.14/0.36 => ( ( setadjoin @ Xx @ Xz )
% 0.14/0.36 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(powerset__Cong_type,type,
% 0.14/0.36 powerset__Cong: $o ).
% 0.14/0.36
% 0.14/0.36 thf(powerset__Cong,definition,
% 0.14/0.36 ( powerset__Cong
% 0.14/0.36 = ( ! [A: $i,B: $i] :
% 0.14/0.36 ( ( A = B )
% 0.14/0.36 => ( ( powerset @ A )
% 0.14/0.36 = ( powerset @ B ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(setunion__Cong_type,type,
% 0.14/0.36 setunion__Cong: $o ).
% 0.14/0.36
% 0.14/0.36 thf(setunion__Cong,definition,
% 0.14/0.36 ( setunion__Cong
% 0.14/0.36 = ( ! [A: $i,B: $i] :
% 0.14/0.36 ( ( A = B )
% 0.14/0.36 => ( ( setunion @ A )
% 0.14/0.36 = ( setunion @ B ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(omega__Cong_type,type,
% 0.14/0.36 omega__Cong: $o ).
% 0.14/0.36
% 0.14/0.36 thf(omega__Cong,definition,
% 0.14/0.36 ( omega__Cong
% 0.14/0.36 = ( omega = omega ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(exuEu_type,type,
% 0.14/0.36 exuEu: $o ).
% 0.14/0.36
% 0.14/0.36 thf(exuEu,definition,
% 0.14/0.36 ( exuEu
% 0.14/0.36 = ( ! [Xphi: $i > $o] :
% 0.14/0.36 ( ( exu
% 0.14/0.36 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.14/0.36 => ! [Xx: $i,Xy: $i] :
% 0.14/0.36 ( ( Xphi @ Xx )
% 0.14/0.36 => ( ( Xphi @ Xy )
% 0.14/0.36 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(descr__Cong_type,type,
% 0.14/0.36 descr__Cong: $o ).
% 0.14/0.36
% 0.14/0.36 thf(descr__Cong,definition,
% 0.14/0.36 ( descr__Cong
% 0.14/0.36 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.14/0.36 ( ! [Xx: $i,Xy: $i] :
% 0.14/0.36 ( ( Xx = Xy )
% 0.14/0.36 => ( ( Xphi @ Xx )
% 0.14/0.36 <=> ( Xpsi @ Xy ) ) )
% 0.14/0.36 => ( ( exu
% 0.14/0.36 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.14/0.36 => ( ( exu
% 0.14/0.36 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.14/0.36 => ( ( descr
% 0.14/0.36 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.14/0.36 = ( descr
% 0.14/0.36 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(dsetconstr__Cong_type,type,
% 0.14/0.36 dsetconstr__Cong: $o ).
% 0.14/0.36
% 0.14/0.36 thf(dsetconstr__Cong,definition,
% 0.14/0.36 ( dsetconstr__Cong
% 0.14/0.36 = ( ! [A: $i,B: $i] :
% 0.14/0.36 ( ( A = B )
% 0.14/0.36 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.14/0.36 ( ! [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 => ! [Xy: $i] :
% 0.14/0.36 ( ( in @ Xy @ B )
% 0.14/0.36 => ( ( Xx = Xy )
% 0.14/0.36 => ( ( Xphi @ Xx )
% 0.14/0.36 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.14/0.36 => ( ( dsetconstr @ A
% 0.14/0.36 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.14/0.36 = ( dsetconstr @ B
% 0.14/0.36 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(subset_type,type,
% 0.14/0.36 subset: $i > $i > $o ).
% 0.14/0.36
% 0.14/0.36 thf(disjoint_type,type,
% 0.14/0.36 disjoint: $i > $i > $o ).
% 0.14/0.36
% 0.14/0.36 thf(setsmeet_type,type,
% 0.14/0.36 setsmeet: $i > $i > $o ).
% 0.14/0.36
% 0.14/0.36 thf(subsetI1_type,type,
% 0.14/0.36 subsetI1: $o ).
% 0.14/0.36
% 0.14/0.36 thf(subsetI1,definition,
% 0.14/0.36 ( subsetI1
% 0.14/0.36 = ( ! [A: $i,B: $i] :
% 0.14/0.36 ( ! [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 => ( in @ Xx @ B ) )
% 0.14/0.36 => ( subset @ A @ B ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(eqimpsubset2_type,type,
% 0.14/0.36 eqimpsubset2: $o ).
% 0.14/0.36
% 0.14/0.36 thf(eqimpsubset2,definition,
% 0.14/0.36 ( eqimpsubset2
% 0.14/0.36 = ( ! [A: $i,B: $i] :
% 0.14/0.36 ( ( A = B )
% 0.14/0.36 => ( subset @ B @ A ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(eqimpsubset1_type,type,
% 0.14/0.36 eqimpsubset1: $o ).
% 0.14/0.36
% 0.14/0.36 thf(eqimpsubset1,definition,
% 0.14/0.36 ( eqimpsubset1
% 0.14/0.36 = ( ! [A: $i,B: $i] :
% 0.14/0.36 ( ( A = B )
% 0.14/0.36 => ( subset @ A @ B ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(subsetI2_type,type,
% 0.14/0.36 subsetI2: $o ).
% 0.14/0.36
% 0.14/0.36 thf(subsetI2,definition,
% 0.14/0.36 ( subsetI2
% 0.14/0.36 = ( ! [A: $i,B: $i] :
% 0.14/0.36 ( ! [Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 => ( in @ Xx @ B ) )
% 0.14/0.36 => ( subset @ A @ B ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(emptysetsubset_type,type,
% 0.14/0.36 emptysetsubset: $o ).
% 0.14/0.36
% 0.14/0.36 thf(emptysetsubset,definition,
% 0.14/0.36 ( emptysetsubset
% 0.14/0.36 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(subsetE_type,type,
% 0.14/0.36 subsetE: $o ).
% 0.14/0.36
% 0.14/0.36 thf(subsetE,definition,
% 0.14/0.36 ( subsetE
% 0.14/0.36 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.14/0.36 ( ( subset @ A @ B )
% 0.14/0.36 => ( ( in @ Xx @ A )
% 0.14/0.36 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(subsetE2_type,type,
% 0.14/0.36 subsetE2: $o ).
% 0.14/0.36
% 0.14/0.36 thf(subsetE2,definition,
% 0.14/0.36 ( subsetE2
% 0.14/0.36 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.14/0.36 ( ( subset @ A @ B )
% 0.14/0.36 => ( ~ ( in @ Xx @ B )
% 0.14/0.36 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(notsubsetI_type,type,
% 0.14/0.36 notsubsetI: $o ).
% 0.14/0.36
% 0.14/0.36 thf(notsubsetI,definition,
% 0.14/0.36 ( notsubsetI
% 0.14/0.36 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 => ( ~ ( in @ Xx @ B )
% 0.14/0.36 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(notequalI1_type,type,
% 0.14/0.36 notequalI1: $o ).
% 0.14/0.36
% 0.14/0.36 thf(notequalI1,definition,
% 0.14/0.36 ( notequalI1
% 0.14/0.36 = ( ! [A: $i,B: $i] :
% 0.14/0.36 ( ~ ( subset @ A @ B )
% 0.14/0.36 => ( A != B ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(notequalI2_type,type,
% 0.14/0.36 notequalI2: $o ).
% 0.14/0.36
% 0.14/0.36 thf(notequalI2,definition,
% 0.14/0.36 ( notequalI2
% 0.14/0.36 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 => ( ~ ( in @ Xx @ B )
% 0.14/0.36 => ( A != B ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(subsetRefl_type,type,
% 0.14/0.36 subsetRefl: $o ).
% 0.14/0.36
% 0.14/0.36 thf(subsetRefl,definition,
% 0.14/0.36 ( subsetRefl
% 0.14/0.36 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(subsetTrans_type,type,
% 0.14/0.36 subsetTrans: $o ).
% 0.14/0.36
% 0.14/0.36 thf(subsetTrans,definition,
% 0.14/0.36 ( subsetTrans
% 0.14/0.36 = ( ! [A: $i,B: $i,C: $i] :
% 0.14/0.36 ( ( subset @ A @ B )
% 0.14/0.36 => ( ( subset @ B @ C )
% 0.14/0.36 => ( subset @ A @ C ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(setadjoinSub_type,type,
% 0.14/0.36 setadjoinSub: $o ).
% 0.14/0.36
% 0.14/0.36 thf(setadjoinSub,definition,
% 0.14/0.36 ( setadjoinSub
% 0.14/0.36 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(setadjoinSub2_type,type,
% 0.14/0.36 setadjoinSub2: $o ).
% 0.14/0.36
% 0.14/0.36 thf(setadjoinSub2,definition,
% 0.14/0.36 ( setadjoinSub2
% 0.14/0.36 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.14/0.36 ( ( subset @ A @ B )
% 0.14/0.36 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(subset2powerset_type,type,
% 0.14/0.36 subset2powerset: $o ).
% 0.14/0.36
% 0.14/0.36 thf(subset2powerset,definition,
% 0.14/0.36 ( subset2powerset
% 0.14/0.36 = ( ! [A: $i,B: $i] :
% 0.14/0.36 ( ( subset @ A @ B )
% 0.14/0.36 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(setextsub_type,type,
% 0.14/0.36 setextsub: $o ).
% 0.14/0.36
% 0.14/0.36 thf(setextsub,definition,
% 0.14/0.36 ( setextsub
% 0.14/0.36 = ( ! [A: $i,B: $i] :
% 0.14/0.36 ( ( subset @ A @ B )
% 0.14/0.36 => ( ( subset @ B @ A )
% 0.14/0.36 => ( A = B ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(subsetemptysetimpeq_type,type,
% 0.14/0.36 subsetemptysetimpeq: $o ).
% 0.14/0.36
% 0.14/0.36 thf(subsetemptysetimpeq,definition,
% 0.14/0.36 ( subsetemptysetimpeq
% 0.14/0.36 = ( ! [A: $i] :
% 0.14/0.36 ( ( subset @ A @ emptyset )
% 0.14/0.36 => ( A = emptyset ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(powersetI1_type,type,
% 0.14/0.36 powersetI1: $o ).
% 0.14/0.36
% 0.14/0.36 thf(powersetI1,definition,
% 0.14/0.36 ( powersetI1
% 0.14/0.36 = ( ! [A: $i,B: $i] :
% 0.14/0.36 ( ( subset @ B @ A )
% 0.14/0.36 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(powersetE1_type,type,
% 0.14/0.36 powersetE1: $o ).
% 0.14/0.36
% 0.14/0.36 thf(powersetE1,definition,
% 0.14/0.36 ( powersetE1
% 0.14/0.36 = ( ! [A: $i,B: $i] :
% 0.14/0.36 ( ( in @ B @ ( powerset @ A ) )
% 0.14/0.36 => ( subset @ B @ A ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(inPowerset_type,type,
% 0.14/0.36 inPowerset: $o ).
% 0.14/0.36
% 0.14/0.36 thf(inPowerset,definition,
% 0.14/0.36 ( inPowerset
% 0.14/0.36 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(powersetsubset_type,type,
% 0.14/0.36 powersetsubset: $o ).
% 0.14/0.36
% 0.14/0.36 thf(powersetsubset,definition,
% 0.14/0.36 ( powersetsubset
% 0.14/0.36 = ( ! [A: $i,B: $i] :
% 0.14/0.36 ( ( subset @ A @ B )
% 0.14/0.36 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(sepInPowerset_type,type,
% 0.14/0.36 sepInPowerset: $o ).
% 0.14/0.36
% 0.14/0.36 thf(sepInPowerset,definition,
% 0.14/0.36 ( sepInPowerset
% 0.14/0.36 = ( ! [A: $i,Xphi: $i > $o] :
% 0.14/0.36 ( in
% 0.14/0.36 @ ( dsetconstr @ A
% 0.14/0.36 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.14/0.36 @ ( powerset @ A ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(sepSubset_type,type,
% 0.14/0.36 sepSubset: $o ).
% 0.14/0.36
% 0.14/0.36 thf(sepSubset,definition,
% 0.14/0.36 ( sepSubset
% 0.14/0.36 = ( ! [A: $i,Xphi: $i > $o] :
% 0.14/0.36 ( subset
% 0.14/0.36 @ ( dsetconstr @ A
% 0.14/0.36 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.14/0.36 @ A ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(binunion_type,type,
% 0.14/0.36 binunion: $i > $i > $i ).
% 0.14/0.36
% 0.14/0.36 thf(binunionIL_type,type,
% 0.14/0.36 binunionIL: $o ).
% 0.14/0.36
% 0.14/0.36 thf(binunionIL,definition,
% 0.14/0.36 ( binunionIL
% 0.14/0.36 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ A )
% 0.14/0.36 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(upairset2IR_type,type,
% 0.14/0.36 upairset2IR: $o ).
% 0.14/0.36
% 0.14/0.36 thf(upairset2IR,definition,
% 0.14/0.36 ( upairset2IR
% 0.14/0.36 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(binunionIR_type,type,
% 0.14/0.36 binunionIR: $o ).
% 0.14/0.36
% 0.14/0.36 thf(binunionIR,definition,
% 0.14/0.36 ( binunionIR
% 0.14/0.36 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.14/0.36 ( ( in @ Xx @ B )
% 0.14/0.36 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(binunionEcases_type,type,
% 0.14/0.36 binunionEcases: $o ).
% 0.14/0.36
% 0.14/0.36 thf(binunionEcases,definition,
% 0.14/0.36 ( binunionEcases
% 0.14/0.36 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.14/0.36 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.14/0.36 => ( ( ( in @ Xx @ A )
% 0.14/0.36 => Xphi )
% 0.14/0.36 => ( ( ( in @ Xx @ B )
% 0.14/0.36 => Xphi )
% 0.14/0.36 => Xphi ) ) ) ) ) ).
% 0.14/0.36
% 0.14/0.36 thf(binunionE,conjecture,
% 0.14/0.36 ( setextAx
% 0.14/0.36 => ( emptysetAx
% 0.14/0.36 => ( setadjoinAx
% 0.14/0.36 => ( powersetAx
% 0.14/0.36 => ( setunionAx
% 0.14/0.36 => ( omega0Ax
% 0.14/0.36 => ( omegaSAx
% 0.14/0.36 => ( omegaIndAx
% 0.14/0.36 => ( replAx
% 0.14/0.36 => ( foundationAx
% 0.14/0.36 => ( wellorderingAx
% 0.14/0.36 => ( descrp
% 0.14/0.36 => ( dsetconstrI
% 0.14/0.36 => ( dsetconstrEL
% 0.14/0.36 => ( dsetconstrER
% 0.14/0.36 => ( exuE1
% 0.14/0.36 => ( prop2setE
% 0.14/0.36 => ( emptysetE
% 0.14/0.36 => ( emptysetimpfalse
% 0.14/0.36 => ( notinemptyset
% 0.14/0.36 => ( exuE3e
% 0.14/0.36 => ( setext
% 0.14/0.36 => ( emptyI
% 0.14/0.36 => ( noeltsimpempty
% 0.14/0.36 => ( setbeta
% 0.14/0.36 => ( nonemptyE1
% 0.14/0.36 => ( nonemptyI
% 0.14/0.36 => ( nonemptyI1
% 0.14/0.36 => ( setadjoinIL
% 0.14/0.36 => ( emptyinunitempty
% 0.14/0.36 => ( setadjoinIR
% 0.14/0.36 => ( setadjoinE
% 0.14/0.36 => ( setadjoinOr
% 0.14/0.36 => ( setoftrueEq
% 0.14/0.36 => ( powersetI
% 0.14/0.36 => ( emptyinPowerset
% 0.14/0.36 => ( emptyInPowerset
% 0.14/0.36 => ( powersetE
% 0.14/0.36 => ( setunionI
% 0.14/0.36 => ( setunionE
% 0.14/0.36 => ( subPowSU
% 0.14/0.36 => ( exuE2
% 0.14/0.36 => ( nonemptyImpWitness
% 0.14/0.36 => ( uniqinunit
% 0.14/0.36 => ( notinsingleton
% 0.14/0.36 => ( eqinunit
% 0.14/0.36 => ( singletonsswitch
% 0.14/0.36 => ( upairsetE
% 0.14/0.36 => ( upairsetIL
% 0.14/0.36 => ( upairsetIR
% 0.14/0.36 => ( emptyE1
% 0.14/0.36 => ( vacuousDall
% 0.14/0.36 => ( quantDeMorgan1
% 0.14/0.36 => ( quantDeMorgan2
% 0.14/0.36 => ( quantDeMorgan3
% 0.14/0.36 => ( quantDeMorgan4
% 0.14/0.36 => ( prop2setI
% 0.14/0.36 => ( prop2set2propI
% 0.14/0.36 => ( notdexE
% 0.14/0.36 => ( notdallE
% 0.14/0.36 => ( exuI1
% 0.14/0.36 => ( exuI3
% 0.14/0.36 => ( exuI2
% 0.14/0.36 => ( inCongP
% 0.14/0.36 => ( in__Cong
% 0.14/0.36 => ( exuE3u
% 0.14/0.36 => ( exu__Cong
% 0.14/0.36 => ( emptyset__Cong
% 0.14/0.36 => ( setadjoin__Cong
% 0.14/0.36 => ( powerset__Cong
% 0.14/0.36 => ( setunion__Cong
% 0.14/0.36 => ( omega__Cong
% 0.14/0.36 => ( exuEu
% 0.14/0.36 => ( descr__Cong
% 0.14/0.36 => ( dsetconstr__Cong
% 0.14/0.36 => ( subsetI1
% 0.14/0.36 => ( eqimpsubset2
% 0.14/0.36 => ( eqimpsubset1
% 0.14/0.36 => ( subsetI2
% 0.14/0.36 => ( emptysetsubset
% 0.14/0.36 => ( subsetE
% 0.14/0.36 => ( subsetE2
% 0.14/0.36 => ( notsubsetI
% 0.14/0.37 => ( notequalI1
% 0.14/0.37 => ( notequalI2
% 0.14/0.37 => ( subsetRefl
% 0.14/0.37 => ( subsetTrans
% 0.14/0.37 => ( setadjoinSub
% 0.14/0.37 => ( setadjoinSub2
% 0.14/0.37 => ( subset2powerset
% 0.14/0.37 => ( setextsub
% 0.14/0.37 => ( subsetemptysetimpeq
% 0.14/0.37 => ( powersetI1
% 0.14/0.37 => ( powersetE1
% 0.14/0.37 => ( inPowerset
% 0.14/0.37 => ( powersetsubset
% 0.14/0.37 => ( sepInPowerset
% 0.14/0.37 => ( sepSubset
% 0.14/0.37 => ( binunionIL
% 0.14/0.37 => ( upairset2IR
% 0.14/0.37 => ( binunionIR
% 0.14/0.37 => ( binunionEcases
% 0.14/0.37 => ! [A: $i,B: $i,Xx: $i] :
% 0.14/0.37 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.14/0.37 => ( ( in @ Xx @ A )
% 0.14/0.37 | ( in @ Xx @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.14/0.37
% 0.14/0.37 %------------------------------------------------------------------------------
% 0.14/0.37 ------------------------
% 2.44/1.57 ---- Embedded in THF ---
% 2.44/1.58 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.44/1.58 %%% Generated on Tue Feb 24 12:33:46 EST 2026
% 2.44/1.58 %%% using '$$dhol' embedding, version 1.3.0.
% 2.44/1.58 %%% Logic specification used:
% 2.44/1.58 %%% thf(spec, logic, $$dhol).
% 2.44/1.58
% 2.44/1.58 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.9nobNTBT08/DTF2DTF21348.p
% See solution above
% 2.44/1.58 ------------------------
% 2.61/1.60 ---- Cleaned THF ---
% 2.61/1.60 thf(in_type,type,
% 2.61/1.60 in: $i > $i > $o ).
% 2.61/1.60
% 2.61/1.60 thf(exu_type,type,
% 2.61/1.60 exu: ( $i > $o ) > $o ).
% 2.61/1.60
% 2.61/1.60 thf(exu,definition,
% 2.61/1.60 ( exu
% 2.61/1.60 = ( ^ [Xphi: $i > $o] :
% 2.61/1.60 ? [Xx: $i] :
% 2.61/1.60 ( ( Xphi @ Xx )
% 2.61/1.60 & ! [Xy: $i] :
% 2.61/1.60 ( ( Xphi @ Xy )
% 2.61/1.60 => ( Xx = Xy ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(setextAx_type,type,
% 2.61/1.60 setextAx: $o ).
% 2.61/1.60
% 2.61/1.60 thf(setextAx,definition,
% 2.61/1.60 ( setextAx
% 2.61/1.60 = ( ! [A: $i,B: $i] :
% 2.61/1.60 ( ! [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 <=> ( in @ Xx @ B ) )
% 2.61/1.60 => ( A = B ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(emptyset_type,type,
% 2.61/1.60 emptyset: $i ).
% 2.61/1.60
% 2.61/1.60 thf(emptysetAx_type,type,
% 2.61/1.60 emptysetAx: $o ).
% 2.61/1.60
% 2.61/1.60 thf(emptysetAx,definition,
% 2.61/1.60 ( emptysetAx
% 2.61/1.60 = ( ! [Xx: $i] :
% 2.61/1.60 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(setadjoin_type,type,
% 2.61/1.60 setadjoin: $i > $i > $i ).
% 2.61/1.60
% 2.61/1.60 thf(setadjoinAx_type,type,
% 2.61/1.60 setadjoinAx: $o ).
% 2.61/1.60
% 2.61/1.60 thf(setadjoinAx,definition,
% 2.61/1.60 ( setadjoinAx
% 2.61/1.60 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.61/1.60 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.61/1.60 <=> ( ( Xy = Xx )
% 2.61/1.60 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(powerset_type,type,
% 2.61/1.60 powerset: $i > $i ).
% 2.61/1.60
% 2.61/1.60 thf(powersetAx_type,type,
% 2.61/1.60 powersetAx: $o ).
% 2.61/1.60
% 2.61/1.60 thf(powersetAx,definition,
% 2.61/1.60 ( powersetAx
% 2.61/1.60 = ( ! [A: $i,B: $i] :
% 2.61/1.60 ( ( in @ B @ ( powerset @ A ) )
% 2.61/1.60 <=> ! [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ B )
% 2.61/1.60 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(setunion_type,type,
% 2.61/1.60 setunion: $i > $i ).
% 2.61/1.60
% 2.61/1.60 thf(setunionAx_type,type,
% 2.61/1.60 setunionAx: $o ).
% 2.61/1.60
% 2.61/1.60 thf(setunionAx,definition,
% 2.61/1.60 ( setunionAx
% 2.61/1.60 = ( ! [A: $i,Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ ( setunion @ A ) )
% 2.61/1.60 <=> ? [B: $i] :
% 2.61/1.60 ( ( in @ Xx @ B )
% 2.61/1.60 & ( in @ B @ A ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(omega_type,type,
% 2.61/1.60 omega: $i ).
% 2.61/1.60
% 2.61/1.60 thf(omega0Ax_type,type,
% 2.61/1.60 omega0Ax: $o ).
% 2.61/1.60
% 2.61/1.60 thf(omega0Ax,definition,
% 2.61/1.60 ( omega0Ax
% 2.61/1.60 = ( in @ emptyset @ omega ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(omegaSAx_type,type,
% 2.61/1.60 omegaSAx: $o ).
% 2.61/1.60
% 2.61/1.60 thf(omegaSAx,definition,
% 2.61/1.60 ( omegaSAx
% 2.61/1.60 = ( ! [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ omega )
% 2.61/1.60 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(omegaIndAx_type,type,
% 2.61/1.60 omegaIndAx: $o ).
% 2.61/1.60
% 2.61/1.60 thf(omegaIndAx,definition,
% 2.61/1.60 ( omegaIndAx
% 2.61/1.60 = ( ! [A: $i] :
% 2.61/1.60 ( ( ( in @ emptyset @ A )
% 2.61/1.60 & ! [Xx: $i] :
% 2.61/1.60 ( ( ( in @ Xx @ omega )
% 2.61/1.60 & ( in @ Xx @ A ) )
% 2.61/1.60 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.61/1.60 => ! [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ omega )
% 2.61/1.60 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(replAx_type,type,
% 2.61/1.60 replAx: $o ).
% 2.61/1.60
% 2.61/1.60 thf(replAx,definition,
% 2.61/1.60 ( replAx
% 2.61/1.60 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.61/1.60 ( ! [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 => ( exu
% 2.61/1.60 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.61/1.60 => ? [B: $i] :
% 2.61/1.60 ! [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ B )
% 2.61/1.60 <=> ? [Xy: $i] :
% 2.61/1.60 ( ( in @ Xy @ A )
% 2.61/1.60 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(foundationAx_type,type,
% 2.61/1.60 foundationAx: $o ).
% 2.61/1.60
% 2.61/1.60 thf(foundationAx,definition,
% 2.61/1.60 ( foundationAx
% 2.61/1.60 = ( ! [A: $i] :
% 2.61/1.60 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.61/1.60 => ? [B: $i] :
% 2.61/1.60 ( ( in @ B @ A )
% 2.61/1.60 & ~ ? [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ B )
% 2.61/1.60 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(wellorderingAx_type,type,
% 2.61/1.60 wellorderingAx: $o ).
% 2.61/1.60
% 2.61/1.60 thf(wellorderingAx,definition,
% 2.61/1.60 ( wellorderingAx
% 2.61/1.60 = ( ! [A: $i] :
% 2.61/1.60 ? [B: $i] :
% 2.61/1.60 ( ! [C: $i] :
% 2.61/1.60 ( ( in @ C @ B )
% 2.61/1.60 => ! [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ C )
% 2.61/1.60 => ( in @ Xx @ A ) ) )
% 2.61/1.60 & ! [Xx: $i,Xy: $i] :
% 2.61/1.60 ( ( ( in @ Xx @ A )
% 2.61/1.60 & ( in @ Xy @ A ) )
% 2.61/1.60 => ( ! [C: $i] :
% 2.61/1.60 ( ( in @ C @ B )
% 2.61/1.60 => ( ( in @ Xx @ C )
% 2.61/1.60 <=> ( in @ Xy @ C ) ) )
% 2.61/1.60 => ( Xx = Xy ) ) )
% 2.61/1.60 & ! [C: $i,D: $i] :
% 2.61/1.60 ( ( ( in @ C @ B )
% 2.61/1.60 & ( in @ D @ B ) )
% 2.61/1.60 => ( ! [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ C )
% 2.61/1.60 => ( in @ Xx @ D ) )
% 2.61/1.60 | ! [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ D )
% 2.61/1.60 => ( in @ Xx @ C ) ) ) )
% 2.61/1.60 & ! [C: $i] :
% 2.61/1.60 ( ( ! [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ C )
% 2.61/1.60 => ( in @ Xx @ A ) )
% 2.61/1.60 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.61/1.60 => ? [D: $i,Xx: $i] :
% 2.61/1.60 ( ( in @ D @ B )
% 2.61/1.60 & ( in @ Xx @ C )
% 2.61/1.60 & ~ ? [Xy: $i] :
% 2.61/1.60 ( ( in @ Xy @ D )
% 2.61/1.60 & ( in @ Xy @ C ) )
% 2.61/1.60 & ! [E: $i] :
% 2.61/1.60 ( ( in @ E @ B )
% 2.61/1.60 => ( ! [Xy: $i] :
% 2.61/1.60 ( ( in @ Xy @ E )
% 2.61/1.60 => ( in @ Xy @ D ) )
% 2.61/1.60 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(descr_type,type,
% 2.61/1.60 descr: ( $i > $o ) > $i ).
% 2.61/1.60
% 2.61/1.60 thf(descrp_type,type,
% 2.61/1.60 descrp: $o ).
% 2.61/1.60
% 2.61/1.60 thf(descrp,definition,
% 2.61/1.60 ( descrp
% 2.61/1.60 = ( ! [Xphi: $i > $o] :
% 2.61/1.60 ( ( exu
% 2.61/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.60 => ( Xphi
% 2.61/1.60 @ ( descr
% 2.61/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(dsetconstr_type,type,
% 2.61/1.60 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.61/1.60
% 2.61/1.60 thf(dsetconstrI_type,type,
% 2.61/1.60 dsetconstrI: $o ).
% 2.61/1.60
% 2.61/1.60 thf(dsetconstrI,definition,
% 2.61/1.60 ( dsetconstrI
% 2.61/1.60 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 => ( ( Xphi @ Xx )
% 2.61/1.60 => ( in @ Xx
% 2.61/1.60 @ ( dsetconstr @ A
% 2.61/1.60 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(dsetconstrEL_type,type,
% 2.61/1.60 dsetconstrEL: $o ).
% 2.61/1.60
% 2.61/1.60 thf(dsetconstrEL,definition,
% 2.61/1.60 ( dsetconstrEL
% 2.61/1.60 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.61/1.60 ( ( in @ Xx
% 2.61/1.60 @ ( dsetconstr @ A
% 2.61/1.60 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.61/1.60 => ( in @ Xx @ A ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(dsetconstrER_type,type,
% 2.61/1.60 dsetconstrER: $o ).
% 2.61/1.60
% 2.61/1.60 thf(dsetconstrER,definition,
% 2.61/1.60 ( dsetconstrER
% 2.61/1.60 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.61/1.60 ( ( in @ Xx
% 2.61/1.60 @ ( dsetconstr @ A
% 2.61/1.60 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.61/1.60 => ( Xphi @ Xx ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(exuE1_type,type,
% 2.61/1.60 exuE1: $o ).
% 2.61/1.60
% 2.61/1.60 thf(exuE1,definition,
% 2.61/1.60 ( exuE1
% 2.61/1.60 = ( ! [Xphi: $i > $o] :
% 2.61/1.60 ( ( exu
% 2.61/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.60 => ? [Xx: $i] :
% 2.61/1.60 ( ( Xphi @ Xx )
% 2.61/1.60 & ! [Xy: $i] :
% 2.61/1.60 ( ( Xphi @ Xy )
% 2.61/1.60 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(prop2set_type,type,
% 2.61/1.60 prop2set: $o > $i ).
% 2.61/1.60
% 2.61/1.60 thf(prop2setE_type,type,
% 2.61/1.60 prop2setE: $o ).
% 2.61/1.60
% 2.61/1.60 thf(prop2setE,definition,
% 2.61/1.60 ( prop2setE
% 2.61/1.60 = ( ! [Xphi: $o,Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.61/1.60 => Xphi ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(emptysetE_type,type,
% 2.61/1.60 emptysetE: $o ).
% 2.61/1.60
% 2.61/1.60 thf(emptysetE,definition,
% 2.61/1.60 ( emptysetE
% 2.61/1.60 = ( ! [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ emptyset )
% 2.61/1.60 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(emptysetimpfalse_type,type,
% 2.61/1.60 emptysetimpfalse: $o ).
% 2.61/1.60
% 2.61/1.60 thf(emptysetimpfalse,definition,
% 2.61/1.60 ( emptysetimpfalse
% 2.61/1.60 = ( ! [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ emptyset )
% 2.61/1.60 => $false ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(notinemptyset_type,type,
% 2.61/1.60 notinemptyset: $o ).
% 2.61/1.60
% 2.61/1.60 thf(notinemptyset,definition,
% 2.61/1.60 ( notinemptyset
% 2.61/1.60 = ( ! [Xx: $i] :
% 2.61/1.60 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(exuE3e_type,type,
% 2.61/1.60 exuE3e: $o ).
% 2.61/1.60
% 2.61/1.60 thf(exuE3e,definition,
% 2.61/1.60 ( exuE3e
% 2.61/1.60 = ( ! [Xphi: $i > $o] :
% 2.61/1.60 ( ( exu
% 2.61/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.60 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(setext_type,type,
% 2.61/1.60 setext: $o ).
% 2.61/1.60
% 2.61/1.60 thf(setext,definition,
% 2.61/1.60 ( setext
% 2.61/1.60 = ( ! [A: $i,B: $i] :
% 2.61/1.60 ( ! [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 => ( in @ Xx @ B ) )
% 2.61/1.60 => ( ! [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ B )
% 2.61/1.60 => ( in @ Xx @ A ) )
% 2.61/1.60 => ( A = B ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(emptyI_type,type,
% 2.61/1.60 emptyI: $o ).
% 2.61/1.60
% 2.61/1.60 thf(emptyI,definition,
% 2.61/1.60 ( emptyI
% 2.61/1.60 = ( ! [A: $i] :
% 2.61/1.60 ( ! [Xx: $i] :
% 2.61/1.60 ~ ( in @ Xx @ A )
% 2.61/1.60 => ( A = emptyset ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(noeltsimpempty_type,type,
% 2.61/1.60 noeltsimpempty: $o ).
% 2.61/1.60
% 2.61/1.60 thf(noeltsimpempty,definition,
% 2.61/1.60 ( noeltsimpempty
% 2.61/1.60 = ( ! [A: $i] :
% 2.61/1.60 ( ! [Xx: $i] :
% 2.61/1.60 ~ ( in @ Xx @ A )
% 2.61/1.60 => ( A = emptyset ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(setbeta_type,type,
% 2.61/1.60 setbeta: $o ).
% 2.61/1.60
% 2.61/1.60 thf(setbeta,definition,
% 2.61/1.60 ( setbeta
% 2.61/1.60 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 => ( ( in @ Xx
% 2.61/1.60 @ ( dsetconstr @ A
% 2.61/1.60 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.61/1.60 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(nonempty_type,type,
% 2.61/1.60 nonempty: $i > $o ).
% 2.61/1.60
% 2.61/1.60 thf(nonempty,definition,
% 2.61/1.60 ( nonempty
% 2.61/1.60 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(nonemptyE1_type,type,
% 2.61/1.60 nonemptyE1: $o ).
% 2.61/1.60
% 2.61/1.60 thf(nonemptyE1,definition,
% 2.61/1.60 ( nonemptyE1
% 2.61/1.60 = ( ! [A: $i] :
% 2.61/1.60 ( ( nonempty @ A )
% 2.61/1.60 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(nonemptyI_type,type,
% 2.61/1.60 nonemptyI: $o ).
% 2.61/1.60
% 2.61/1.60 thf(nonemptyI,definition,
% 2.61/1.60 ( nonemptyI
% 2.61/1.60 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 => ( ( Xphi @ Xx )
% 2.61/1.60 => ( nonempty
% 2.61/1.60 @ ( dsetconstr @ A
% 2.61/1.60 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(nonemptyI1_type,type,
% 2.61/1.60 nonemptyI1: $o ).
% 2.61/1.60
% 2.61/1.60 thf(nonemptyI1,definition,
% 2.61/1.60 ( nonemptyI1
% 2.61/1.60 = ( ! [A: $i] :
% 2.61/1.60 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.61/1.60 => ( nonempty @ A ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(setadjoinIL_type,type,
% 2.61/1.60 setadjoinIL: $o ).
% 2.61/1.60
% 2.61/1.60 thf(setadjoinIL,definition,
% 2.61/1.60 ( setadjoinIL
% 2.61/1.60 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(emptyinunitempty_type,type,
% 2.61/1.60 emptyinunitempty: $o ).
% 2.61/1.60
% 2.61/1.60 thf(emptyinunitempty,definition,
% 2.61/1.60 ( emptyinunitempty
% 2.61/1.60 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(setadjoinIR_type,type,
% 2.61/1.60 setadjoinIR: $o ).
% 2.61/1.60
% 2.61/1.60 thf(setadjoinIR,definition,
% 2.61/1.60 ( setadjoinIR
% 2.61/1.60 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.61/1.60 ( ( in @ Xy @ A )
% 2.61/1.60 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(setadjoinE_type,type,
% 2.61/1.60 setadjoinE: $o ).
% 2.61/1.60
% 2.61/1.60 thf(setadjoinE,definition,
% 2.61/1.60 ( setadjoinE
% 2.61/1.60 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.61/1.60 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.61/1.60 => ! [Xphi: $o] :
% 2.61/1.60 ( ( ( Xy = Xx )
% 2.61/1.60 => Xphi )
% 2.61/1.60 => ( ( ( in @ Xy @ A )
% 2.61/1.60 => Xphi )
% 2.61/1.60 => Xphi ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(setadjoinOr_type,type,
% 2.61/1.60 setadjoinOr: $o ).
% 2.61/1.60
% 2.61/1.60 thf(setadjoinOr,definition,
% 2.61/1.60 ( setadjoinOr
% 2.61/1.60 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.61/1.60 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.61/1.60 => ( ( Xy = Xx )
% 2.61/1.60 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(setoftrueEq_type,type,
% 2.61/1.60 setoftrueEq: $o ).
% 2.61/1.60
% 2.61/1.60 thf(setoftrueEq,definition,
% 2.61/1.60 ( setoftrueEq
% 2.61/1.60 = ( ! [A: $i] :
% 2.61/1.60 ( ( dsetconstr @ A
% 2.61/1.60 @ ^ [Xx: $i] : $true )
% 2.61/1.60 = A ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(powersetI_type,type,
% 2.61/1.60 powersetI: $o ).
% 2.61/1.60
% 2.61/1.60 thf(powersetI,definition,
% 2.61/1.60 ( powersetI
% 2.61/1.60 = ( ! [A: $i,B: $i] :
% 2.61/1.60 ( ! [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ B )
% 2.61/1.60 => ( in @ Xx @ A ) )
% 2.61/1.60 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(emptyinPowerset_type,type,
% 2.61/1.60 emptyinPowerset: $o ).
% 2.61/1.60
% 2.61/1.60 thf(emptyinPowerset,definition,
% 2.61/1.60 ( emptyinPowerset
% 2.61/1.60 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(emptyInPowerset_type,type,
% 2.61/1.60 emptyInPowerset: $o ).
% 2.61/1.60
% 2.61/1.60 thf(emptyInPowerset,definition,
% 2.61/1.60 ( emptyInPowerset
% 2.61/1.60 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(powersetE_type,type,
% 2.61/1.60 powersetE: $o ).
% 2.61/1.60
% 2.61/1.60 thf(powersetE,definition,
% 2.61/1.60 ( powersetE
% 2.61/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.60 ( ( in @ B @ ( powerset @ A ) )
% 2.61/1.60 => ( ( in @ Xx @ B )
% 2.61/1.60 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(setunionI_type,type,
% 2.61/1.60 setunionI: $o ).
% 2.61/1.60
% 2.61/1.60 thf(setunionI,definition,
% 2.61/1.60 ( setunionI
% 2.61/1.60 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.61/1.60 ( ( in @ Xx @ B )
% 2.61/1.60 => ( ( in @ B @ A )
% 2.61/1.60 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(setunionE_type,type,
% 2.61/1.60 setunionE: $o ).
% 2.61/1.60
% 2.61/1.60 thf(setunionE,definition,
% 2.61/1.60 ( setunionE
% 2.61/1.60 = ( ! [A: $i,Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ ( setunion @ A ) )
% 2.61/1.60 => ! [Xphi: $o] :
% 2.61/1.60 ( ! [B: $i] :
% 2.61/1.60 ( ( in @ Xx @ B )
% 2.61/1.60 => ( ( in @ B @ A )
% 2.61/1.60 => Xphi ) )
% 2.61/1.60 => Xphi ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(subPowSU_type,type,
% 2.61/1.60 subPowSU: $o ).
% 2.61/1.60
% 2.61/1.60 thf(subPowSU,definition,
% 2.61/1.60 ( subPowSU
% 2.61/1.60 = ( ! [A: $i,Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(exuE2_type,type,
% 2.61/1.60 exuE2: $o ).
% 2.61/1.60
% 2.61/1.60 thf(exuE2,definition,
% 2.61/1.60 ( exuE2
% 2.61/1.60 = ( ! [Xphi: $i > $o] :
% 2.61/1.60 ( ( exu
% 2.61/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.60 => ? [Xx: $i] :
% 2.61/1.60 ! [Xy: $i] :
% 2.61/1.60 ( ( Xphi @ Xy )
% 2.61/1.60 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(nonemptyImpWitness_type,type,
% 2.61/1.60 nonemptyImpWitness: $o ).
% 2.61/1.60
% 2.61/1.60 thf(nonemptyImpWitness,definition,
% 2.61/1.60 ( nonemptyImpWitness
% 2.61/1.60 = ( ! [A: $i] :
% 2.61/1.60 ( ( nonempty @ A )
% 2.61/1.60 => ? [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 & $true ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(uniqinunit_type,type,
% 2.61/1.60 uniqinunit: $o ).
% 2.61/1.60
% 2.61/1.60 thf(uniqinunit,definition,
% 2.61/1.60 ( uniqinunit
% 2.61/1.60 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.60 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.61/1.60 => ( Xx = Xy ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(notinsingleton_type,type,
% 2.61/1.60 notinsingleton: $o ).
% 2.61/1.60
% 2.61/1.60 thf(notinsingleton,definition,
% 2.61/1.60 ( notinsingleton
% 2.61/1.60 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.60 ( ( Xx != Xy )
% 2.61/1.60 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(eqinunit_type,type,
% 2.61/1.60 eqinunit: $o ).
% 2.61/1.60
% 2.61/1.60 thf(eqinunit,definition,
% 2.61/1.60 ( eqinunit
% 2.61/1.60 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.60 ( ( Xx = Xy )
% 2.61/1.60 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(singletonsswitch_type,type,
% 2.61/1.60 singletonsswitch: $o ).
% 2.61/1.60
% 2.61/1.60 thf(singletonsswitch,definition,
% 2.61/1.60 ( singletonsswitch
% 2.61/1.60 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.60 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.61/1.60 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(upairsetE_type,type,
% 2.61/1.60 upairsetE: $o ).
% 2.61/1.60
% 2.61/1.60 thf(upairsetE,definition,
% 2.61/1.60 ( upairsetE
% 2.61/1.60 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.61/1.60 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.61/1.60 => ( ( Xz = Xx )
% 2.61/1.60 | ( Xz = Xy ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(upairsetIL_type,type,
% 2.61/1.60 upairsetIL: $o ).
% 2.61/1.60
% 2.61/1.60 thf(upairsetIL,definition,
% 2.61/1.60 ( upairsetIL
% 2.61/1.60 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(upairsetIR_type,type,
% 2.61/1.60 upairsetIR: $o ).
% 2.61/1.60
% 2.61/1.60 thf(upairsetIR,definition,
% 2.61/1.60 ( upairsetIR
% 2.61/1.60 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(emptyE1_type,type,
% 2.61/1.60 emptyE1: $o ).
% 2.61/1.60
% 2.61/1.60 thf(emptyE1,definition,
% 2.61/1.60 ( emptyE1
% 2.61/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.60 ( ? [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 & ( Xphi @ Xx ) )
% 2.61/1.60 => ( ( ( dsetconstr @ A
% 2.61/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.60 = emptyset )
% 2.61/1.60 => $false ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(vacuousDall_type,type,
% 2.61/1.60 vacuousDall: $o ).
% 2.61/1.60
% 2.61/1.60 thf(vacuousDall,definition,
% 2.61/1.60 ( vacuousDall
% 2.61/1.60 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ emptyset )
% 2.61/1.60 => ( Xphi @ Xx ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(quantDeMorgan1_type,type,
% 2.61/1.60 quantDeMorgan1: $o ).
% 2.61/1.60
% 2.61/1.60 thf(quantDeMorgan1,definition,
% 2.61/1.60 ( quantDeMorgan1
% 2.61/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.60 ( ~ ! [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 => ( Xphi @ Xx ) )
% 2.61/1.60 => ? [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(quantDeMorgan2_type,type,
% 2.61/1.60 quantDeMorgan2: $o ).
% 2.61/1.60
% 2.61/1.60 thf(quantDeMorgan2,definition,
% 2.61/1.60 ( quantDeMorgan2
% 2.61/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.60 ( ! [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 => ~ ( Xphi @ Xx ) )
% 2.61/1.60 => ~ ? [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(quantDeMorgan3_type,type,
% 2.61/1.60 quantDeMorgan3: $o ).
% 2.61/1.60
% 2.61/1.60 thf(quantDeMorgan3,definition,
% 2.61/1.60 ( quantDeMorgan3
% 2.61/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.60 ( ~ ? [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 & ( Xphi @ Xx ) )
% 2.61/1.60 => ! [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(quantDeMorgan4_type,type,
% 2.61/1.60 quantDeMorgan4: $o ).
% 2.61/1.60
% 2.61/1.60 thf(quantDeMorgan4,definition,
% 2.61/1.60 ( quantDeMorgan4
% 2.61/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.60 ( ? [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 & ~ ( Xphi @ Xx ) )
% 2.61/1.60 => ~ ! [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(prop2setI_type,type,
% 2.61/1.60 prop2setI: $o ).
% 2.61/1.60
% 2.61/1.60 thf(prop2setI,definition,
% 2.61/1.60 ( prop2setI
% 2.61/1.60 = ( ! [Xphi: $o] :
% 2.61/1.60 ( Xphi
% 2.61/1.60 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(set2prop_type,type,
% 2.61/1.60 set2prop: $i > $o ).
% 2.61/1.60
% 2.61/1.60 thf(prop2set2propI_type,type,
% 2.61/1.60 prop2set2propI: $o ).
% 2.61/1.60
% 2.61/1.60 thf(prop2set2propI,definition,
% 2.61/1.60 ( prop2set2propI
% 2.61/1.60 = ( ! [Xphi: $o] :
% 2.61/1.60 ( Xphi
% 2.61/1.60 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(notdexE_type,type,
% 2.61/1.60 notdexE: $o ).
% 2.61/1.60
% 2.61/1.60 thf(notdexE,definition,
% 2.61/1.60 ( notdexE
% 2.61/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.60 ( ~ ? [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 & ( Xphi @ Xx ) )
% 2.61/1.60 => ! [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(notdallE_type,type,
% 2.61/1.60 notdallE: $o ).
% 2.61/1.60
% 2.61/1.60 thf(notdallE,definition,
% 2.61/1.60 ( notdallE
% 2.61/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.60 ( ~ ! [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 => ( Xphi @ Xx ) )
% 2.61/1.60 => ? [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(exuI1_type,type,
% 2.61/1.60 exuI1: $o ).
% 2.61/1.60
% 2.61/1.60 thf(exuI1,definition,
% 2.61/1.60 ( exuI1
% 2.61/1.60 = ( ! [Xphi: $i > $o] :
% 2.61/1.60 ( ? [Xx: $i] :
% 2.61/1.60 ( ( Xphi @ Xx )
% 2.61/1.60 & ! [Xy: $i] :
% 2.61/1.60 ( ( Xphi @ Xy )
% 2.61/1.60 => ( Xx = Xy ) ) )
% 2.61/1.60 => ( exu
% 2.61/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(exuI3_type,type,
% 2.61/1.60 exuI3: $o ).
% 2.61/1.60
% 2.61/1.60 thf(exuI3,definition,
% 2.61/1.60 ( exuI3
% 2.61/1.60 = ( ! [Xphi: $i > $o] :
% 2.61/1.60 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.61/1.60 => ( ! [Xx: $i,Xy: $i] :
% 2.61/1.60 ( ( Xphi @ Xx )
% 2.61/1.60 => ( ( Xphi @ Xy )
% 2.61/1.60 => ( Xx = Xy ) ) )
% 2.61/1.60 => ( exu
% 2.61/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(exuI2_type,type,
% 2.61/1.60 exuI2: $o ).
% 2.61/1.60
% 2.61/1.60 thf(exuI2,definition,
% 2.61/1.60 ( exuI2
% 2.61/1.60 = ( ! [Xphi: $i > $o] :
% 2.61/1.60 ( ? [Xx: $i] :
% 2.61/1.60 ! [Xy: $i] :
% 2.61/1.60 ( ( Xphi @ Xy )
% 2.61/1.60 <=> ( Xy = Xx ) )
% 2.61/1.60 => ( exu
% 2.61/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(inCongP_type,type,
% 2.61/1.60 inCongP: $o ).
% 2.61/1.60
% 2.61/1.60 thf(inCongP,definition,
% 2.61/1.60 ( inCongP
% 2.61/1.60 = ( ! [A: $i,B: $i] :
% 2.61/1.60 ( ( A = B )
% 2.61/1.60 => ! [Xx: $i,Xy: $i] :
% 2.61/1.60 ( ( Xx = Xy )
% 2.61/1.60 => ( ( in @ Xx @ A )
% 2.61/1.60 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(in__Cong_type,type,
% 2.61/1.60 in__Cong: $o ).
% 2.61/1.60
% 2.61/1.60 thf(in__Cong,definition,
% 2.61/1.60 ( in__Cong
% 2.61/1.60 = ( ! [A: $i,B: $i] :
% 2.61/1.60 ( ( A = B )
% 2.61/1.60 => ! [Xx: $i,Xy: $i] :
% 2.61/1.60 ( ( Xx = Xy )
% 2.61/1.60 => ( ( in @ Xx @ A )
% 2.61/1.60 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(exuE3u_type,type,
% 2.61/1.60 exuE3u: $o ).
% 2.61/1.60
% 2.61/1.60 thf(exuE3u,definition,
% 2.61/1.60 ( exuE3u
% 2.61/1.60 = ( ! [Xphi: $i > $o] :
% 2.61/1.60 ( ( exu
% 2.61/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.60 => ! [Xx: $i,Xy: $i] :
% 2.61/1.60 ( ( Xphi @ Xx )
% 2.61/1.60 => ( ( Xphi @ Xy )
% 2.61/1.60 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(exu__Cong_type,type,
% 2.61/1.60 exu__Cong: $o ).
% 2.61/1.60
% 2.61/1.60 thf(exu__Cong,definition,
% 2.61/1.60 ( exu__Cong
% 2.61/1.60 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.61/1.60 ( ! [Xx: $i,Xy: $i] :
% 2.61/1.60 ( ( Xx = Xy )
% 2.61/1.60 => ( ( Xphi @ Xx )
% 2.61/1.60 <=> ( Xpsi @ Xy ) ) )
% 2.61/1.60 => ( ( exu
% 2.61/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.60 <=> ( exu
% 2.61/1.60 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(emptyset__Cong_type,type,
% 2.61/1.60 emptyset__Cong: $o ).
% 2.61/1.60
% 2.61/1.60 thf(emptyset__Cong,definition,
% 2.61/1.60 ( emptyset__Cong
% 2.61/1.60 = ( emptyset = emptyset ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(setadjoin__Cong_type,type,
% 2.61/1.60 setadjoin__Cong: $o ).
% 2.61/1.60
% 2.61/1.60 thf(setadjoin__Cong,definition,
% 2.61/1.60 ( setadjoin__Cong
% 2.61/1.60 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.60 ( ( Xx = Xy )
% 2.61/1.60 => ! [Xz: $i,Xu: $i] :
% 2.61/1.60 ( ( Xz = Xu )
% 2.61/1.60 => ( ( setadjoin @ Xx @ Xz )
% 2.61/1.60 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(powerset__Cong_type,type,
% 2.61/1.60 powerset__Cong: $o ).
% 2.61/1.60
% 2.61/1.60 thf(powerset__Cong,definition,
% 2.61/1.60 ( powerset__Cong
% 2.61/1.60 = ( ! [A: $i,B: $i] :
% 2.61/1.60 ( ( A = B )
% 2.61/1.60 => ( ( powerset @ A )
% 2.61/1.60 = ( powerset @ B ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(setunion__Cong_type,type,
% 2.61/1.60 setunion__Cong: $o ).
% 2.61/1.60
% 2.61/1.60 thf(setunion__Cong,definition,
% 2.61/1.60 ( setunion__Cong
% 2.61/1.60 = ( ! [A: $i,B: $i] :
% 2.61/1.60 ( ( A = B )
% 2.61/1.60 => ( ( setunion @ A )
% 2.61/1.60 = ( setunion @ B ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(omega__Cong_type,type,
% 2.61/1.60 omega__Cong: $o ).
% 2.61/1.60
% 2.61/1.60 thf(omega__Cong,definition,
% 2.61/1.60 ( omega__Cong
% 2.61/1.60 = ( omega = omega ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(exuEu_type,type,
% 2.61/1.60 exuEu: $o ).
% 2.61/1.60
% 2.61/1.60 thf(exuEu,definition,
% 2.61/1.60 ( exuEu
% 2.61/1.60 = ( ! [Xphi: $i > $o] :
% 2.61/1.60 ( ( exu
% 2.61/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.60 => ! [Xx: $i,Xy: $i] :
% 2.61/1.60 ( ( Xphi @ Xx )
% 2.61/1.60 => ( ( Xphi @ Xy )
% 2.61/1.60 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(descr__Cong_type,type,
% 2.61/1.60 descr__Cong: $o ).
% 2.61/1.60
% 2.61/1.60 thf(descr__Cong,definition,
% 2.61/1.60 ( descr__Cong
% 2.61/1.60 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.61/1.60 ( ! [Xx: $i,Xy: $i] :
% 2.61/1.60 ( ( Xx = Xy )
% 2.61/1.60 => ( ( Xphi @ Xx )
% 2.61/1.60 <=> ( Xpsi @ Xy ) ) )
% 2.61/1.60 => ( ( exu
% 2.61/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.60 => ( ( exu
% 2.61/1.60 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.61/1.60 => ( ( descr
% 2.61/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.60 = ( descr
% 2.61/1.60 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(dsetconstr__Cong_type,type,
% 2.61/1.60 dsetconstr__Cong: $o ).
% 2.61/1.60
% 2.61/1.60 thf(dsetconstr__Cong,definition,
% 2.61/1.60 ( dsetconstr__Cong
% 2.61/1.60 = ( ! [A: $i,B: $i] :
% 2.61/1.60 ( ( A = B )
% 2.61/1.60 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.61/1.60 ( ! [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 => ! [Xy: $i] :
% 2.61/1.60 ( ( in @ Xy @ B )
% 2.61/1.60 => ( ( Xx = Xy )
% 2.61/1.60 => ( ( Xphi @ Xx )
% 2.61/1.60 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.61/1.60 => ( ( dsetconstr @ A
% 2.61/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.60 = ( dsetconstr @ B
% 2.61/1.60 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(subset_type,type,
% 2.61/1.60 subset: $i > $i > $o ).
% 2.61/1.60
% 2.61/1.60 thf(disjoint_type,type,
% 2.61/1.60 disjoint: $i > $i > $o ).
% 2.61/1.60
% 2.61/1.60 thf(setsmeet_type,type,
% 2.61/1.60 setsmeet: $i > $i > $o ).
% 2.61/1.60
% 2.61/1.60 thf(subsetI1_type,type,
% 2.61/1.60 subsetI1: $o ).
% 2.61/1.60
% 2.61/1.60 thf(subsetI1,definition,
% 2.61/1.60 ( subsetI1
% 2.61/1.60 = ( ! [A: $i,B: $i] :
% 2.61/1.60 ( ! [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 => ( in @ Xx @ B ) )
% 2.61/1.60 => ( subset @ A @ B ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(eqimpsubset2_type,type,
% 2.61/1.60 eqimpsubset2: $o ).
% 2.61/1.60
% 2.61/1.60 thf(eqimpsubset2,definition,
% 2.61/1.60 ( eqimpsubset2
% 2.61/1.60 = ( ! [A: $i,B: $i] :
% 2.61/1.60 ( ( A = B )
% 2.61/1.60 => ( subset @ B @ A ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(eqimpsubset1_type,type,
% 2.61/1.60 eqimpsubset1: $o ).
% 2.61/1.60
% 2.61/1.60 thf(eqimpsubset1,definition,
% 2.61/1.60 ( eqimpsubset1
% 2.61/1.60 = ( ! [A: $i,B: $i] :
% 2.61/1.60 ( ( A = B )
% 2.61/1.60 => ( subset @ A @ B ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(subsetI2_type,type,
% 2.61/1.60 subsetI2: $o ).
% 2.61/1.60
% 2.61/1.60 thf(subsetI2,definition,
% 2.61/1.60 ( subsetI2
% 2.61/1.60 = ( ! [A: $i,B: $i] :
% 2.61/1.60 ( ! [Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 => ( in @ Xx @ B ) )
% 2.61/1.60 => ( subset @ A @ B ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(emptysetsubset_type,type,
% 2.61/1.60 emptysetsubset: $o ).
% 2.61/1.60
% 2.61/1.60 thf(emptysetsubset,definition,
% 2.61/1.60 ( emptysetsubset
% 2.61/1.60 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(subsetE_type,type,
% 2.61/1.60 subsetE: $o ).
% 2.61/1.60
% 2.61/1.60 thf(subsetE,definition,
% 2.61/1.60 ( subsetE
% 2.61/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.60 ( ( subset @ A @ B )
% 2.61/1.60 => ( ( in @ Xx @ A )
% 2.61/1.60 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(subsetE2_type,type,
% 2.61/1.60 subsetE2: $o ).
% 2.61/1.60
% 2.61/1.60 thf(subsetE2,definition,
% 2.61/1.60 ( subsetE2
% 2.61/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.60 ( ( subset @ A @ B )
% 2.61/1.60 => ( ~ ( in @ Xx @ B )
% 2.61/1.60 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(notsubsetI_type,type,
% 2.61/1.60 notsubsetI: $o ).
% 2.61/1.60
% 2.61/1.60 thf(notsubsetI,definition,
% 2.61/1.60 ( notsubsetI
% 2.61/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 => ( ~ ( in @ Xx @ B )
% 2.61/1.60 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(notequalI1_type,type,
% 2.61/1.60 notequalI1: $o ).
% 2.61/1.60
% 2.61/1.60 thf(notequalI1,definition,
% 2.61/1.60 ( notequalI1
% 2.61/1.60 = ( ! [A: $i,B: $i] :
% 2.61/1.60 ( ~ ( subset @ A @ B )
% 2.61/1.60 => ( A != B ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(notequalI2_type,type,
% 2.61/1.60 notequalI2: $o ).
% 2.61/1.60
% 2.61/1.60 thf(notequalI2,definition,
% 2.61/1.60 ( notequalI2
% 2.61/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 => ( ~ ( in @ Xx @ B )
% 2.61/1.60 => ( A != B ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(subsetRefl_type,type,
% 2.61/1.60 subsetRefl: $o ).
% 2.61/1.60
% 2.61/1.60 thf(subsetRefl,definition,
% 2.61/1.60 ( subsetRefl
% 2.61/1.60 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(subsetTrans_type,type,
% 2.61/1.60 subsetTrans: $o ).
% 2.61/1.60
% 2.61/1.60 thf(subsetTrans,definition,
% 2.61/1.60 ( subsetTrans
% 2.61/1.60 = ( ! [A: $i,B: $i,C: $i] :
% 2.61/1.60 ( ( subset @ A @ B )
% 2.61/1.60 => ( ( subset @ B @ C )
% 2.61/1.60 => ( subset @ A @ C ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(setadjoinSub_type,type,
% 2.61/1.60 setadjoinSub: $o ).
% 2.61/1.60
% 2.61/1.60 thf(setadjoinSub,definition,
% 2.61/1.60 ( setadjoinSub
% 2.61/1.60 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(setadjoinSub2_type,type,
% 2.61/1.60 setadjoinSub2: $o ).
% 2.61/1.60
% 2.61/1.60 thf(setadjoinSub2,definition,
% 2.61/1.60 ( setadjoinSub2
% 2.61/1.60 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.61/1.60 ( ( subset @ A @ B )
% 2.61/1.60 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(subset2powerset_type,type,
% 2.61/1.60 subset2powerset: $o ).
% 2.61/1.60
% 2.61/1.60 thf(subset2powerset,definition,
% 2.61/1.60 ( subset2powerset
% 2.61/1.60 = ( ! [A: $i,B: $i] :
% 2.61/1.60 ( ( subset @ A @ B )
% 2.61/1.60 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(setextsub_type,type,
% 2.61/1.60 setextsub: $o ).
% 2.61/1.60
% 2.61/1.60 thf(setextsub,definition,
% 2.61/1.60 ( setextsub
% 2.61/1.60 = ( ! [A: $i,B: $i] :
% 2.61/1.60 ( ( subset @ A @ B )
% 2.61/1.60 => ( ( subset @ B @ A )
% 2.61/1.60 => ( A = B ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(subsetemptysetimpeq_type,type,
% 2.61/1.60 subsetemptysetimpeq: $o ).
% 2.61/1.60
% 2.61/1.60 thf(subsetemptysetimpeq,definition,
% 2.61/1.60 ( subsetemptysetimpeq
% 2.61/1.60 = ( ! [A: $i] :
% 2.61/1.60 ( ( subset @ A @ emptyset )
% 2.61/1.60 => ( A = emptyset ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(powersetI1_type,type,
% 2.61/1.60 powersetI1: $o ).
% 2.61/1.60
% 2.61/1.60 thf(powersetI1,definition,
% 2.61/1.60 ( powersetI1
% 2.61/1.60 = ( ! [A: $i,B: $i] :
% 2.61/1.60 ( ( subset @ B @ A )
% 2.61/1.60 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(powersetE1_type,type,
% 2.61/1.60 powersetE1: $o ).
% 2.61/1.60
% 2.61/1.60 thf(powersetE1,definition,
% 2.61/1.60 ( powersetE1
% 2.61/1.60 = ( ! [A: $i,B: $i] :
% 2.61/1.60 ( ( in @ B @ ( powerset @ A ) )
% 2.61/1.60 => ( subset @ B @ A ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(inPowerset_type,type,
% 2.61/1.60 inPowerset: $o ).
% 2.61/1.60
% 2.61/1.60 thf(inPowerset,definition,
% 2.61/1.60 ( inPowerset
% 2.61/1.60 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(powersetsubset_type,type,
% 2.61/1.60 powersetsubset: $o ).
% 2.61/1.60
% 2.61/1.60 thf(powersetsubset,definition,
% 2.61/1.60 ( powersetsubset
% 2.61/1.60 = ( ! [A: $i,B: $i] :
% 2.61/1.60 ( ( subset @ A @ B )
% 2.61/1.60 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(sepInPowerset_type,type,
% 2.61/1.60 sepInPowerset: $o ).
% 2.61/1.60
% 2.61/1.60 thf(sepInPowerset,definition,
% 2.61/1.60 ( sepInPowerset
% 2.61/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.60 ( in
% 2.61/1.60 @ ( dsetconstr @ A
% 2.61/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.60 @ ( powerset @ A ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(sepSubset_type,type,
% 2.61/1.60 sepSubset: $o ).
% 2.61/1.60
% 2.61/1.60 thf(sepSubset,definition,
% 2.61/1.60 ( sepSubset
% 2.61/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.60 ( subset
% 2.61/1.60 @ ( dsetconstr @ A
% 2.61/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.60 @ A ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(binunion_type,type,
% 2.61/1.60 binunion: $i > $i > $i ).
% 2.61/1.60
% 2.61/1.60 thf(binunionIL_type,type,
% 2.61/1.60 binunionIL: $o ).
% 2.61/1.60
% 2.61/1.60 thf(binunionIL,definition,
% 2.61/1.60 ( binunionIL
% 2.61/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ A )
% 2.61/1.60 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(upairset2IR_type,type,
% 2.61/1.60 upairset2IR: $o ).
% 2.61/1.60
% 2.61/1.60 thf(upairset2IR,definition,
% 2.61/1.60 ( upairset2IR
% 2.61/1.60 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(binunionIR_type,type,
% 2.61/1.60 binunionIR: $o ).
% 2.61/1.60
% 2.61/1.60 thf(binunionIR,definition,
% 2.61/1.60 ( binunionIR
% 2.61/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ B )
% 2.61/1.60 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(binunionEcases_type,type,
% 2.61/1.60 binunionEcases: $o ).
% 2.61/1.60
% 2.61/1.60 thf(binunionEcases,definition,
% 2.61/1.60 ( binunionEcases
% 2.61/1.60 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.61/1.60 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.61/1.60 => ( ( ( in @ Xx @ A )
% 2.61/1.60 => Xphi )
% 2.61/1.60 => ( ( ( in @ Xx @ B )
% 2.61/1.60 => Xphi )
% 2.61/1.60 => Xphi ) ) ) ) ) ).
% 2.61/1.60
% 2.61/1.60 thf(binunionE,conjecture,
% 2.61/1.60 ( setextAx
% 2.61/1.60 => ( emptysetAx
% 2.61/1.60 => ( setadjoinAx
% 2.61/1.60 => ( powersetAx
% 2.61/1.60 => ( setunionAx
% 2.61/1.60 => ( omega0Ax
% 2.61/1.60 => ( omegaSAx
% 2.61/1.60 => ( omegaIndAx
% 2.61/1.60 => ( replAx
% 2.61/1.60 => ( foundationAx
% 2.61/1.60 => ( wellorderingAx
% 2.61/1.60 => ( descrp
% 2.61/1.60 => ( dsetconstrI
% 2.61/1.60 => ( dsetconstrEL
% 2.61/1.60 => ( dsetconstrER
% 2.61/1.60 => ( exuE1
% 2.61/1.60 => ( prop2setE
% 2.61/1.60 => ( emptysetE
% 2.61/1.60 => ( emptysetimpfalse
% 2.61/1.60 => ( notinemptyset
% 2.61/1.60 => ( exuE3e
% 2.61/1.60 => ( setext
% 2.61/1.60 => ( emptyI
% 2.61/1.60 => ( noeltsimpempty
% 2.61/1.60 => ( setbeta
% 2.61/1.60 => ( nonemptyE1
% 2.61/1.60 => ( nonemptyI
% 2.61/1.60 => ( nonemptyI1
% 2.61/1.60 => ( setadjoinIL
% 2.61/1.60 => ( emptyinunitempty
% 2.61/1.60 => ( setadjoinIR
% 2.61/1.60 => ( setadjoinE
% 2.61/1.60 => ( setadjoinOr
% 2.61/1.60 => ( setoftrueEq
% 2.61/1.60 => ( powersetI
% 2.61/1.60 => ( emptyinPowerset
% 2.61/1.60 => ( emptyInPowerset
% 2.61/1.60 => ( powersetE
% 2.61/1.60 => ( setunionI
% 2.61/1.60 => ( setunionE
% 2.61/1.60 => ( subPowSU
% 2.61/1.60 => ( exuE2
% 2.61/1.60 => ( nonemptyImpWitness
% 2.61/1.60 => ( uniqinunit
% 2.61/1.60 => ( notinsingleton
% 2.61/1.60 => ( eqinunit
% 2.61/1.60 => ( singletonsswitch
% 2.61/1.60 => ( upairsetE
% 2.61/1.60 => ( upairsetIL
% 2.61/1.60 => ( upairsetIR
% 2.61/1.60 => ( emptyE1
% 2.61/1.60 => ( vacuousDall
% 2.61/1.60 => ( quantDeMorgan1
% 2.61/1.60 => ( quantDeMorgan2
% 2.61/1.60 => ( quantDeMorgan3
% 2.61/1.60 => ( quantDeMorgan4
% 2.61/1.60 => ( prop2setI
% 2.61/1.60 => ( prop2set2propI
% 2.61/1.60 => ( notdexE
% 2.61/1.60 => ( notdallE
% 2.61/1.60 => ( exuI1
% 2.61/1.60 => ( exuI3
% 2.61/1.60 => ( exuI2
% 2.61/1.60 => ( inCongP
% 2.61/1.60 => ( in__Cong
% 2.61/1.60 => ( exuE3u
% 2.61/1.60 => ( exu__Cong
% 2.61/1.60 => ( emptyset__Cong
% 2.61/1.60 => ( setadjoin__Cong
% 2.61/1.60 => ( powerset__Cong
% 2.61/1.60 => ( setunion__Cong
% 2.61/1.60 => ( omega__Cong
% 2.61/1.60 => ( exuEu
% 2.61/1.60 => ( descr__Cong
% 2.61/1.60 => ( dsetconstr__Cong
% 2.61/1.60 => ( subsetI1
% 2.61/1.60 => ( eqimpsubset2
% 2.61/1.60 => ( eqimpsubset1
% 2.61/1.60 => ( subsetI2
% 2.61/1.60 => ( emptysetsubset
% 2.61/1.60 => ( subsetE
% 2.61/1.60 => ( subsetE2
% 2.61/1.60 => ( notsubsetI
% 2.61/1.60 => ( notequalI1
% 2.61/1.60 => ( notequalI2
% 2.61/1.60 => ( subsetRefl
% 2.61/1.60 => ( subsetTrans
% 2.61/1.60 => ( setadjoinSub
% 2.61/1.60 => ( setadjoinSub2
% 2.61/1.60 => ( subset2powerset
% 2.61/1.60 => ( setextsub
% 2.61/1.60 => ( subsetemptysetimpeq
% 2.61/1.60 => ( powersetI1
% 2.61/1.60 => ( powersetE1
% 2.61/1.60 => ( inPowerset
% 2.61/1.60 => ( powersetsubset
% 2.61/1.60 => ( sepInPowerset
% 2.61/1.60 => ( sepSubset
% 2.61/1.60 => ( binunionIL
% 2.61/1.60 => ( upairset2IR
% 2.61/1.60 => ( binunionIR
% 2.61/1.60 => ( binunionEcases
% 2.61/1.60 => ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.60 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.61/1.60 => ( ( in @ Xx @ A )
% 2.61/1.60 | ( in @ Xx @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.61/1.60 ------------------------
% 2.61/1.63 % (21449)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_21348 for (2999ds/27Mi)
% 2.61/1.63 % (21452)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_21348 for (2999ds/275Mi)
% 2.61/1.63 % (21450)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_21348 for (2999ds/2Mi)
% 2.61/1.63 % (21447)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_21348 for (2999ds/183Mi)
% 2.61/1.63 % (21451)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_21348 for (2999ds/2Mi)
% 2.61/1.63 % (21450)Instruction limit reached!
% 2.61/1.63 % (21450)------------------------------
% 2.61/1.63 % (21450)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.63 % (21450)Termination reason: Unknown
% 2.61/1.63 % (21450)Termination phase: shuffling
% 2.61/1.63
% 2.61/1.63 % (21450)Memory used [KB]: 1151
% 2.61/1.63 % (21450)Time elapsed: 0.003 s
% 2.61/1.63 % (21450)Instructions burned: 2 (million)
% 2.61/1.63 % (21450)------------------------------
% 2.61/1.63 % (21450)------------------------------
% 2.61/1.63 % (21453)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_21348 for (2999ds/18Mi)
% 2.61/1.63 % (21448)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_21348 for (2999ds/4Mi)
% 2.61/1.63 % (21451)Instruction limit reached!
% 2.61/1.63 % (21451)------------------------------
% 2.61/1.63 % (21451)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.63 % (21451)Termination reason: Unknown
% 2.61/1.63 % (21451)Termination phase: shuffling
% 2.61/1.63
% 2.61/1.63 % (21451)Memory used [KB]: 1151
% 2.61/1.63 % (21451)Time elapsed: 0.003 s
% 2.61/1.63 % (21451)Instructions burned: 3 (million)
% 2.61/1.63 % (21451)------------------------------
% 2.61/1.63 % (21451)------------------------------
% 2.61/1.63 % (21448)Instruction limit reached!
% 2.61/1.63 % (21448)------------------------------
% 2.61/1.63 % (21448)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.63 % (21448)Termination reason: Unknown
% 2.61/1.63 % (21448)Termination phase: shuffling
% 2.61/1.63
% 2.61/1.63 % (21448)Memory used [KB]: 1279
% 2.61/1.63 % (21448)Time elapsed: 0.005 s
% 2.61/1.63 % (21448)Instructions burned: 6 (million)
% 2.61/1.63 % (21448)------------------------------
% 2.61/1.63 % (21448)------------------------------
% 2.61/1.64 % (21453)Instruction limit reached!
% 2.61/1.64 % (21453)------------------------------
% 2.61/1.64 % (21453)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.64 % (21453)Termination reason: Unknown
% 2.61/1.64 % (21453)Termination phase: Preprocessing 3
% 2.61/1.64
% 2.61/1.64 % (21453)Memory used [KB]: 1535
% 2.61/1.64 % (21453)Time elapsed: 0.012 s
% 2.61/1.64 % (21453)Instructions burned: 18 (million)
% 2.61/1.64 % (21453)------------------------------
% 2.61/1.64 % (21453)------------------------------
% 2.61/1.64 % (21449)Instruction limit reached!
% 2.61/1.64 % (21449)------------------------------
% 2.61/1.64 % (21449)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.64 % (21449)Termination reason: Unknown
% 2.61/1.64 % (21449)Termination phase: Property scanning
% 2.61/1.64
% 2.61/1.64 % (21449)Memory used [KB]: 1535
% 2.61/1.64 % (21449)Time elapsed: 0.015 s
% 2.61/1.64 % (21449)Instructions burned: 27 (million)
% 2.61/1.64 % (21449)------------------------------
% 2.61/1.64 % (21449)------------------------------
% 2.61/1.64 % (21454)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_21348 for (2999ds/3Mi)
% 2.61/1.65 % (21454)Instruction limit reached!
% 2.61/1.65 % (21454)------------------------------
% 2.61/1.65 % (21454)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.65 % (21454)Termination reason: Unknown
% 2.61/1.65 % (21454)Termination phase: shuffling
% 2.61/1.65
% 2.61/1.65 % (21454)Memory used [KB]: 1151
% 2.61/1.65 % (21454)Time elapsed: 0.003 s
% 2.61/1.65 % (21454)Instructions burned: 4 (million)
% 2.61/1.65 % (21454)------------------------------
% 2.61/1.65 % (21454)------------------------------
% 2.61/1.65 % (21455)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_21348 for (2999ds/37Mi)
% 2.61/1.65 % (21456)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_21348 for (2999ds/15Mi)
% 2.61/1.65 % (21456)Instruction limit reached!
% 2.61/1.65 % (21456)------------------------------
% 2.61/1.65 % (21456)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.65 % (21456)Termination reason: Unknown
% 2.61/1.65 % (21456)Termination phase: Naming
% 2.61/1.65
% 2.61/1.65 % (21456)Memory used [KB]: 1407
% 2.61/1.65 % (21456)Time elapsed: 0.009 s
% 2.61/1.65 % (21456)Instructions burned: 15 (million)
% 2.61/1.65 % (21456)------------------------------
% 2.61/1.65 % (21456)------------------------------
% 2.61/1.65 % (21457)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_21348 for (2999ds/3Mi)
% 2.61/1.66 % (21457)Instruction limit reached!
% 2.61/1.66 % (21457)------------------------------
% 2.61/1.66 % (21457)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.66 % (21457)Termination reason: Unknown
% 2.61/1.66 % (21457)Termination phase: shuffling
% 2.61/1.66
% 2.61/1.66 % (21457)Memory used [KB]: 1151
% 2.61/1.66 % (21457)Time elapsed: 0.004 s
% 2.61/1.66 % (21457)Instructions burned: 4 (million)
% 2.61/1.66 % (21457)------------------------------
% 2.61/1.66 % (21457)------------------------------
% 2.61/1.66 % (21458)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_21348 for (2999ds/1041Mi)
% 2.61/1.66 % (21459)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_21348 for (2999ds/7Mi)
% 2.61/1.66 % (21455)Instruction limit reached!
% 2.61/1.66 % (21455)------------------------------
% 2.61/1.66 % (21455)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.66 % (21455)Termination reason: Unknown
% 2.61/1.66 % (21455)Termination phase: Function definition elimination
% 2.61/1.66
% 2.61/1.66 % (21455)Memory used [KB]: 1535
% 2.61/1.66 % (21455)Time elapsed: 0.019 s
% 2.61/1.66 % (21455)Instructions burned: 38 (million)
% 2.61/1.66 % (21455)------------------------------
% 2.61/1.66 % (21455)------------------------------
% 2.61/1.67 % (21459)Instruction limit reached!
% 2.61/1.67 % (21459)------------------------------
% 2.61/1.67 % (21459)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.67 % (21459)Termination reason: Unknown
% 2.61/1.67 % (21459)Termination phase: shuffling
% 2.61/1.67
% 2.61/1.67 % (21459)Memory used [KB]: 1279
% 2.61/1.67 % (21459)Time elapsed: 0.006 s
% 2.61/1.67 % (21459)Instructions burned: 8 (million)
% 2.61/1.67 % (21459)------------------------------
% 2.61/1.67 % (21459)------------------------------
% 2.61/1.67 % (21460)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_21348 for (2999ds/16Mi)
% 2.61/1.67 % (21461)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_21348 for (2999ds/3Mi)
% 2.61/1.67 % (21461)Instruction limit reached!
% 2.61/1.67 % (21461)------------------------------
% 2.61/1.67 % (21461)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.67 % (21461)Termination reason: Unknown
% 2.61/1.67 % (21461)Termination phase: shuffling
% 2.61/1.67
% 2.61/1.67 % (21461)Memory used [KB]: 1151
% 2.61/1.67 % (21461)Time elapsed: 0.003 s
% 2.61/1.67 % (21461)Instructions burned: 4 (million)
% 2.61/1.67 % (21461)------------------------------
% 2.61/1.67 % (21461)------------------------------
% 2.61/1.68 % (21460)Instruction limit reached!
% 2.61/1.68 % (21460)------------------------------
% 2.61/1.68 % (21460)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.68 % (21460)Termination reason: Unknown
% 2.61/1.68 % (21460)Termination phase: Property scanning
% 2.61/1.68
% 2.61/1.68 % (21460)Memory used [KB]: 1407
% 2.61/1.68 % (21460)Time elapsed: 0.009 s
% 2.61/1.68 % (21460)Instructions burned: 16 (million)
% 2.61/1.68 % (21460)------------------------------
% 2.61/1.68 % (21460)------------------------------
% 2.61/1.68 % (21462)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_21348 for (2999ds/3Mi)
% 2.61/1.68 % (21462)Instruction limit reached!
% 2.61/1.68 % (21462)------------------------------
% 2.61/1.68 % (21462)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.68 % (21462)Termination reason: Unknown
% 2.61/1.68 % (21462)Termination phase: shuffling
% 2.61/1.68
% 2.61/1.68 % (21462)Memory used [KB]: 1151
% 2.61/1.68 % (21462)Time elapsed: 0.003 s
% 2.61/1.68 % (21462)Instructions burned: 3 (million)
% 2.61/1.68 % (21462)------------------------------
% 2.61/1.68 % (21462)------------------------------
% 2.61/1.68 % (21463)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_21348 for (2999ds/7Mi)
% 2.61/1.69 % (21463)Instruction limit reached!
% 2.61/1.69 % (21463)------------------------------
% 2.61/1.69 % (21463)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.69 % (21463)Termination reason: Unknown
% 2.61/1.69 % (21463)Termination phase: shuffling
% 2.61/1.69
% 2.61/1.69 % (21463)Memory used [KB]: 1279
% 2.61/1.69 % (21463)Time elapsed: 0.007 s
% 2.61/1.69 % (21463)Instructions burned: 8 (million)
% 2.61/1.69 % (21463)------------------------------
% 2.61/1.69 % (21463)------------------------------
% 2.61/1.69 % (21464)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_21348 for (2999ds/3Mi)
% 2.61/1.69 % (21464)Instruction limit reached!
% 2.61/1.69 % (21464)------------------------------
% 2.61/1.69 % (21464)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.69 % (21464)Termination reason: Unknown
% 2.61/1.69 % (21464)Termination phase: shuffling
% 2.61/1.69
% 2.61/1.69 % (21464)Memory used [KB]: 1151
% 2.61/1.69 % (21464)Time elapsed: 0.003 s
% 2.61/1.69 % (21464)Instructions burned: 4 (million)
% 2.61/1.69 % (21464)------------------------------
% 2.61/1.69 % (21464)------------------------------
% 2.62/1.70 % (21465)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_21348 for (2999ds/4Mi)
% 2.62/1.70 % (21466)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_21348 for (2999ds/18Mi)
% 2.62/1.70 % (21465)Instruction limit reached!
% 2.62/1.70 % (21465)------------------------------
% 2.62/1.70 % (21465)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.70 % (21465)Termination reason: Unknown
% 2.62/1.70 % (21465)Termination phase: shuffling
% 2.62/1.70
% 2.62/1.70 % (21465)Memory used [KB]: 1151
% 2.62/1.70 % (21465)Time elapsed: 0.003 s
% 2.62/1.70 % (21465)Instructions burned: 4 (million)
% 2.62/1.70 % (21465)------------------------------
% 2.62/1.70 % (21465)------------------------------
% 2.62/1.71 % (21466)Instruction limit reached!
% 2.62/1.71 % (21466)------------------------------
% 2.62/1.71 % (21466)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.71 % (21466)Termination reason: Unknown
% 2.62/1.71 % (21466)Termination phase: Property scanning
% 2.62/1.71
% 2.62/1.71 % (21466)Memory used [KB]: 1407
% 2.62/1.71 % (21466)Time elapsed: 0.011 s
% 2.62/1.71 % (21466)Instructions burned: 19 (million)
% 2.62/1.71 % (21466)------------------------------
% 2.62/1.71 % (21466)------------------------------
% 2.62/1.71 % (21467)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_21348 for (2999ds/710Mi)
% 2.62/1.71 % (21468)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_21348 for (2999ds/6Mi)
% 2.62/1.71 % (21468)Instruction limit reached!
% 2.62/1.71 % (21468)------------------------------
% 2.62/1.71 % (21468)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.71 % (21468)Termination reason: Unknown
% 2.62/1.71 % (21468)Termination phase: shuffling
% 2.62/1.71
% 2.62/1.71 % (21468)Memory used [KB]: 1279
% 2.62/1.71 % (21468)Time elapsed: 0.005 s
% 2.62/1.71 % (21468)Instructions burned: 8 (million)
% 2.62/1.71 % (21468)------------------------------
% 2.62/1.71 % (21468)------------------------------
% 2.62/1.72 % (21469)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_21348 for (2999ds/902Mi)
% 2.62/1.72 % (21447)Instruction limit reached!
% 2.62/1.72 % (21447)------------------------------
% 2.62/1.72 % (21447)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.72 % (21447)Termination reason: Unknown
% 2.62/1.72 % (21447)Termination phase: Saturation
% 2.62/1.72
% 2.62/1.72 % (21447)Memory used [KB]: 6908
% 2.62/1.72 % (21447)Time elapsed: 0.094 s
% 2.62/1.72 % (21447)Instructions burned: 184 (million)
% 2.62/1.72 % (21447)------------------------------
% 2.62/1.72 % (21447)------------------------------
% 2.62/1.72 % (21470)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_21348 for (2998ds/21Mi)
% 2.62/1.73 % (21471)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_21348 for (2998ds/5Mi)
% 2.62/1.73 % (21471)Instruction limit reached!
% 2.62/1.73 % (21471)------------------------------
% 2.62/1.73 % (21471)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.73 % (21471)Termination reason: Unknown
% 2.62/1.73 % (21471)Termination phase: shuffling
% 2.62/1.73
% 2.62/1.73 % (21471)Memory used [KB]: 1279
% 2.62/1.73 % (21471)Time elapsed: 0.004 s
% 2.62/1.73 % (21471)Instructions burned: 5 (million)
% 2.62/1.73 % (21471)------------------------------
% 2.62/1.73 % (21471)------------------------------
% 2.62/1.73 % (21470)Instruction limit reached!
% 2.62/1.73 % (21470)------------------------------
% 2.62/1.73 % (21470)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.73 % (21470)Termination reason: Unknown
% 2.62/1.73 % (21470)Termination phase: Property scanning
% 2.62/1.73
% 2.62/1.73 % (21470)Memory used [KB]: 1407
% 2.62/1.73 % (21470)Time elapsed: 0.011 s
% 2.62/1.73 % (21470)Instructions burned: 21 (million)
% 2.62/1.73 % (21470)------------------------------
% 2.62/1.73 % (21470)------------------------------
% 2.62/1.74 % (21472)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_21348 for (2998ds/6Mi)
% 2.62/1.74 % (21472)Instruction limit reached!
% 2.62/1.74 % (21472)------------------------------
% 2.62/1.74 % (21472)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.74 % (21472)Termination reason: Unknown
% 2.62/1.74 % (21472)Termination phase: shuffling
% 2.62/1.74
% 2.62/1.74 % (21472)Memory used [KB]: 1279
% 2.62/1.74 % (21472)Time elapsed: 0.005 s
% 2.62/1.74 % (21472)Instructions burned: 6 (million)
% 2.62/1.74 % (21472)------------------------------
% 2.62/1.74 % (21472)------------------------------
% 2.62/1.75 % (21473)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_21348 for (2998ds/377Mi)
% 2.62/1.75 % (21474)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_21348 for (2998ds/779Mi)
% 2.62/1.76 % (21475)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_21348 for (2998ds/19Mi)
% 2.62/1.77 % (21475)Instruction limit reached!
% 2.62/1.77 % (21475)------------------------------
% 2.62/1.77 % (21475)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.77 % (21475)Termination reason: Unknown
% 2.62/1.77 % (21475)Termination phase: Property scanning
% 2.62/1.77
% 2.62/1.77 % (21475)Memory used [KB]: 1407
% 2.62/1.77 % (21475)Time elapsed: 0.011 s
% 2.62/1.77 % (21475)Instructions burned: 20 (million)
% 2.62/1.77 % (21475)------------------------------
% 2.62/1.77 % (21475)------------------------------
% 2.62/1.77 % (21452)Instruction limit reached!
% 2.62/1.77 % (21452)------------------------------
% 2.62/1.77 % (21452)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.77 % (21452)Termination reason: Unknown
% 2.62/1.77 % (21452)Termination phase: Saturation
% 2.62/1.77
% 2.62/1.77 % (21452)Memory used [KB]: 7675
% 2.62/1.77 % (21452)Time elapsed: 0.147 s
% 2.62/1.77 % (21452)Instructions burned: 275 (million)
% 2.62/1.77 % (21452)------------------------------
% 2.62/1.77 % (21452)------------------------------
% 2.62/1.79 % (21476)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_21348 for (2998ds/879Mi)
% 2.62/1.79 % (21469)First to succeed.
% 2.62/1.79 % (21477)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_21348 for (2998ds/17Mi)
% 2.62/1.79 % (21469)Refutation found. Thanks to Tanya!
% 2.62/1.79 % SZS status Theorem for DTF2THF_21348
% 2.62/1.79 % SZS output start Proof for DTF2THF_21348
% 2.62/1.79 thf(func_def_0, type, in: $i > $i > $o).
% 2.62/1.79 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.62/1.79 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.62/1.79 thf(func_def_8, type, powerset: $i > $i).
% 2.62/1.79 thf(func_def_10, type, setunion: $i > $i).
% 2.62/1.79 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.62/1.79 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.62/1.79 thf(func_def_26, type, prop2set: $o > $i).
% 2.62/1.79 thf(func_def_36, type, nonempty: $i > $o).
% 2.62/1.79 thf(func_def_69, type, set2prop: $i > $o).
% 2.62/1.79 thf(func_def_88, type, subset: $i > $i > $o).
% 2.62/1.79 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.62/1.79 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.62/1.79 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.62/1.79 thf(func_def_132, type, sP1: $i > $i > $o).
% 2.62/1.79 thf(func_def_133, type, sP2: $i > $o).
% 2.62/1.79 thf(func_def_134, type, sP3: $i > $i > $o).
% 2.62/1.79 thf(func_def_135, type, sP4: $i > $i > $o).
% 2.62/1.79 thf(func_def_137, type, sK6: $i > $i).
% 2.62/1.79 thf(func_def_138, type, sK7: ($i > $o) > ($i > $o) > $i).
% 2.62/1.79 thf(func_def_139, type, sK8: ($i > $o) > ($i > $o) > $i).
% 2.62/1.79 thf(func_def_140, type, sK9: $i > $o).
% 2.62/1.79 thf(func_def_141, type, sK10: $i > $o).
% 2.62/1.79 thf(func_def_149, type, sK18: $i > $o).
% 2.62/1.79 thf(func_def_153, type, sK22: $i > $o).
% 2.62/1.79 thf(func_def_157, type, sK26: $i > $o).
% 2.62/1.79 thf(func_def_159, type, sK28: $i > $o).
% 2.62/1.79 thf(func_def_161, type, sK30: ($i > $o) > $i > $i).
% 2.62/1.79 thf(func_def_162, type, sK31: $i > $o).
% 2.62/1.79 thf(func_def_163, type, sK32: $i > $i > $i).
% 2.62/1.79 thf(func_def_175, type, sK44: $i > $i > $o).
% 2.62/1.79 thf(func_def_177, type, sK46: $i > $i).
% 2.62/1.79 thf(func_def_178, type, sK47: $i > $i).
% 2.62/1.79 thf(func_def_179, type, sK48: $i > ($i > $i > $o) > $i).
% 2.62/1.79 thf(func_def_180, type, sK49: $i > ($i > $i > $o) > $i).
% 2.62/1.79 thf(func_def_181, type, sK50: $i > $i > ($i > $i > $o) > $i).
% 2.62/1.79 thf(func_def_187, type, sK56: $i > $i).
% 2.62/1.79 thf(func_def_189, type, sK58: $i > $i).
% 2.62/1.79 thf(func_def_191, type, sK60: $i > $o).
% 2.62/1.79 thf(func_def_192, type, sK61: $i > $i).
% 2.62/1.79 thf(func_def_193, type, sK62: ($i > $o) > $i).
% 2.62/1.79 thf(func_def_198, type, sK67: $i > ($i > $o) > $i).
% 2.62/1.79 thf(func_def_199, type, sK68: $i > $o).
% 2.62/1.79 thf(func_def_205, type, sK74: $i > $o).
% 2.62/1.79 thf(func_def_208, type, sK77: $i > $i).
% 2.62/1.79 thf(func_def_210, type, sK79: $i > $i).
% 2.62/1.79 thf(func_def_212, type, sK81: $i > $o).
% 2.62/1.79 thf(func_def_225, type, sK94: $i > $i).
% 2.62/1.79 thf(func_def_227, type, sK96: $i > $i > $i).
% 2.62/1.79 thf(func_def_231, type, sK100: $i > $i > $i).
% 2.62/1.79 thf(func_def_232, type, sK101: $i > $i > $i).
% 2.62/1.79 thf(func_def_238, type, sK107: $i > $o).
% 2.62/1.79 thf(func_def_244, type, sK113: $i > $o).
% 2.62/1.79 thf(func_def_247, type, sK116: $i > ($i > $o) > $i).
% 2.62/1.79 thf(func_def_248, type, sK117: $i > $o).
% 2.62/1.79 thf(func_def_253, type, sK122: $i > $o).
% 2.62/1.79 thf(func_def_255, type, sK124: ($i > $o) > $i).
% 2.62/1.79 thf(func_def_256, type, sK125: ($i > $o) > $i).
% 2.62/1.79 thf(func_def_267, type, sK136: ($i > $o) > $i > $i).
% 2.62/1.79 thf(func_def_269, type, sK138: $i > $o).
% 2.62/1.79 thf(func_def_273, type, sK142: $i > $o).
% 2.62/1.79 thf(func_def_292, type, sK161: $i > $o).
% 2.62/1.79 thf(func_def_295, type, sK164: ($i > $o) > ($i > $o) > $i).
% 2.62/1.79 thf(func_def_296, type, sK165: ($i > $o) > ($i > $o) > $i).
% 2.62/1.79 thf(func_def_297, type, sK166: $i > $o).
% 2.62/1.79 thf(func_def_298, type, sK167: $i > $o).
% 2.62/1.79 thf(func_def_305, type, sK174: $i > ($i > $o) > $i).
% 2.62/1.79 thf(func_def_306, type, sK175: $i > $o).
% 2.62/1.79 thf(func_def_322, type, sK191: $o > $i > $i > $i).
% 2.62/1.79 thf(func_def_329, type, sK198: $i > $o).
% 2.62/1.79 thf(func_def_330, type, sK199: $i > $o).
% 2.62/1.79 thf(func_def_331, type, sK200: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.62/1.79 thf(func_def_332, type, sK201: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.62/1.79 thf(func_def_336, type, sK205: $i > $i > $i).
% 2.62/1.79 thf(func_def_337, type, sK206: $i > $i > $i).
% 2.62/1.79 thf(func_def_338, type, sK207: $i > $i > $i).
% 2.62/1.79 thf(func_def_339, type, sK208: $i > $i > $i).
% 2.62/1.79 thf(func_def_340, type, sK209: $i > $i > $i).
% 2.62/1.79 thf(func_def_341, type, sK210: $i > $i > $i > $i).
% 2.62/1.79 thf(func_def_342, type, sK211: $i > $i).
% 2.62/1.79 thf(func_def_343, type, sK212: $i > $i).
% 2.62/1.79 thf(func_def_344, type, sK213: $i > $i).
% 2.62/1.79 thf(func_def_345, type, sK214: $i > $i).
% 2.62/1.79 thf(func_def_346, type, sK215: $i > $i > $i).
% 2.62/1.79 thf(func_def_347, type, sK216: $i > $i > $i > $i).
% 2.62/1.79 thf(func_def_348, type, sK217: $i > $i > $i > $i).
% 2.62/1.79 thf(func_def_349, type, sK218: $i > $i > $i).
% 2.62/1.79 thf(func_def_350, type, sK219: $i > $i > $i).
% 2.62/1.79 thf(func_def_352, type, sK221: $i > $i).
% 2.62/1.79 thf(func_def_353, type, sK222: $i > $i).
% 2.62/1.79 thf(func_def_354, type, sK223: $i > $i).
% 2.62/1.79 thf(func_def_360, type, sK229: ($i > $o) > $i).
% 2.62/1.79 thf(func_def_361, type, sK230: $i > $o).
% 2.62/1.79 thf(func_def_362, type, sK231: $i > $i).
% 2.62/1.79 thf(func_def_364, type, sK233: $i > $i > $i).
% 2.62/1.79 thf(func_def_370, type, sK239: $i > $i > $i).
% 2.62/1.79 thf(func_def_371, type, sK240: $i > $i > $i).
% 2.62/1.79 thf(func_def_374, type, sK243: $i > $o).
% 2.62/1.79 thf(func_def_375, type, sK244: ($i > $o) > $i).
% 2.62/1.79 thf(func_def_378, type, sK247: $i > $o).
% 2.62/1.79 thf(func_def_380, type, sK249: ($i > $o) > $i > $i).
% 2.62/1.79 thf(func_def_381, type, sK250: $i > $o).
% 2.62/1.79 thf(func_def_383, type, sK252: $i > ($i > $o) > $i).
% 2.62/1.79 thf(func_def_386, type, sK255: $i > $o).
% 2.62/1.79 thf(func_def_388, type, sK257: $i > $o).
% 2.62/1.79 thf(func_def_390, type, sK259: ($i > $o) > $i > $i).
% 2.62/1.79 thf(func_def_397, type, sK266: $i > $i > $i).
% 2.62/1.79 thf(func_def_401, type, sK270: $i > $o).
% 2.62/1.79 thf(func_def_404, type, sK273: $i > $i).
% 2.62/1.79 thf(func_def_417, type, ph286: !>[X0: $tType]:(X0)).
% 2.62/1.79 thf(f2054,plain,(
% 2.62/1.79 $false),
% 2.62/1.79 inference(avatar_sat_refutation,[status(thm)],[f1599,f1604,f1915,f2051])).
% 2.62/1.79 thf(f2051,plain,(
% 2.62/1.79 ~spl285_1),
% 2.62/1.79 inference(avatar_contradiction_clause,[status(thm)],[f2050])).
% 2.62/1.79 thf(f2050,plain,(
% 2.62/1.79 $false | ~spl285_1),
% 2.62/1.79 inference(trivial_inequality_removal,[status(thm)],[f2047])).
% 2.62/1.79 thf(f2047,plain,(
% 2.62/1.79 ($true != $true) | ~spl285_1),
% 2.62/1.79 inference(superposition,[status(thm)],[f1011,f1591])).
% 2.62/1.79 thf(f1591,plain,(
% 2.62/1.79 ($true = (in @ sK37 @ sK36)) | ~spl285_1),
% 2.62/1.79 inference(avatar_component_clause,[status(thm)],[f1589])).
% 2.62/1.79 thf(f1589,definition,(
% 2.62/1.79 spl285_1 <=> ($true = (in @ sK37 @ sK36))),
% 2.62/1.79 introduced(definition,[new_symbols(naming,[spl285_1])],[avatar_definition])).
% 2.62/1.79 thf(f1011,plain,(
% 2.62/1.79 ($true != (in @ sK37 @ sK36))),
% 2.62/1.79 inference(cnf_transformation,[status(thm)],[f502])).
% 2.62/1.79 thf(f502,plain,(
% 2.62/1.79 (exuI1 = $true) & (descrp = $true) & (binunionIL = $true) & (setadjoinE = $true) & (subsetE = $true) & (in__Cong = $true) & (setadjoinSub2 = $true) & (emptysetAx = $true) & (setadjoinOr = $true) & (eqimpsubset1 = $true) & (exuEu = $true) & (nonemptyI = $true) & (exuE1 = $true) & (emptyinunitempty = $true) & (notequalI1 = $true) & (nonemptyE1 = $true) & (emptyinPowerset = $true) & (setadjoinSub = $true) & (exuE3e = $true) & (replAx = $true) & (sepInPowerset = $true) & (emptyset__Cong = $true) & (setbeta = $true) & (emptyInPowerset = $true) & (subPowSU = $true) & (inPowerset = $true) & (subset2powerset = $true) & (exu__Cong = $true) & (subsetI1 = $true) & (eqimpsubset2 = $true) & (upairset2IR = $true) & (emptysetsubset = $true) & (powersetsubset = $true) & (setadjoin__Cong = $true) & (setextAx = $true) & (wellorderingAx = $true) & (exuI3 = $true) & (powersetAx = $true) & (eqinunit = $true) & (setoftrueEq = $true) & (singletonsswitch = $true) & (setext = $true) & (subsetI2 = $true) & (notdexE = $true) & (descr__Cong = $true) & (emptysetE = $true) & (subsetemptysetimpeq = $true) & (setunionE = $true) & (powersetI1 = $true) & (dsetconstrER = $true) & (sepSubset = $true) & (notequalI2 = $true) & (powersetE = $true) & (upairsetE = $true) & (subsetE2 = $true) & (quantDeMorgan2 = $true) & (binunionIR = $true) & (dsetconstr__Cong = $true) & (omega0Ax = $true) & (noeltsimpempty = $true) & (setadjoinAx = $true) & (quantDeMorgan3 = $true) & (upairsetIR = $true) & (prop2setI = $true) & (upairsetIL = $true) & (notinsingleton = $true) & (uniqinunit = $true) & (subsetRefl = $true) & (exuE2 = $true) & (quantDeMorgan1 = $true) & (emptysetimpfalse = $true) & (nonemptyImpWitness = $true) & (($true != (in @ sK37 @ sK35)) & ($true = (in @ sK37 @ (binunion @ sK36 @ sK35))) & ($true != (in @ sK37 @ sK36))) & (prop2set2propI = $true) & (inCongP = $true) & (powerset__Cong = $true) & (omegaIndAx = $true) & (notsubsetI = $true) & (foundationAx = $true) & (exuE3u = $true) & (setunionI = $true) & (binunionEcases = $true) & (notdallE = $true) & (vacuousDall = $true) & (setunion__Cong = $true) & (subsetTrans = $true) & (emptyI = $true) & (nonemptyI1 = $true) & (dsetconstrEL = $true) & (exuI2 = $true) & (omegaSAx = $true) & (dsetconstrI = $true) & (setadjoinIL = $true) & (omega__Cong = $true) & (setextsub = $true) & (setadjoinIR = $true) & (powersetI = $true) & (quantDeMorgan4 = $true) & (powersetE1 = $true) & (notinemptyset = $true) & (prop2setE = $true) & (emptyE1 = $true) & (setunionAx = $true)),
% 2.62/1.79 inference(skolemisation,[status(esa),new_symbols(skolem,[sK35,sK36,sK37])],[f355,f501])).
% 2.62/1.79 thf(f501,plain,(
% 2.62/1.79 ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X0) != $true) & ((in @ X2 @ (binunion @ X1 @ X0)) = $true) & ($true != (in @ X2 @ X1))) => (($true != (in @ sK37 @ sK35)) & ($true = (in @ sK37 @ (binunion @ sK36 @ sK35))) & ($true != (in @ sK37 @ sK36)))),
% 2.62/1.79 introduced(definition,[],[choice_axiom])).
% 2.62/1.79 thf(f355,plain,(
% 2.62/1.79 (exuI1 = $true) & (descrp = $true) & (binunionIL = $true) & (setadjoinE = $true) & (subsetE = $true) & (in__Cong = $true) & (setadjoinSub2 = $true) & (emptysetAx = $true) & (setadjoinOr = $true) & (eqimpsubset1 = $true) & (exuEu = $true) & (nonemptyI = $true) & (exuE1 = $true) & (emptyinunitempty = $true) & (notequalI1 = $true) & (nonemptyE1 = $true) & (emptyinPowerset = $true) & (setadjoinSub = $true) & (exuE3e = $true) & (replAx = $true) & (sepInPowerset = $true) & (emptyset__Cong = $true) & (setbeta = $true) & (emptyInPowerset = $true) & (subPowSU = $true) & (inPowerset = $true) & (subset2powerset = $true) & (exu__Cong = $true) & (subsetI1 = $true) & (eqimpsubset2 = $true) & (upairset2IR = $true) & (emptysetsubset = $true) & (powersetsubset = $true) & (setadjoin__Cong = $true) & (setextAx = $true) & (wellorderingAx = $true) & (exuI3 = $true) & (powersetAx = $true) & (eqinunit = $true) & (setoftrueEq = $true) & (singletonsswitch = $true) & (setext = $true) & (subsetI2 = $true) & (notdexE = $true) & (descr__Cong = $true) & (emptysetE = $true) & (subsetemptysetimpeq = $true) & (setunionE = $true) & (powersetI1 = $true) & (dsetconstrER = $true) & (sepSubset = $true) & (notequalI2 = $true) & (powersetE = $true) & (upairsetE = $true) & (subsetE2 = $true) & (quantDeMorgan2 = $true) & (binunionIR = $true) & (dsetconstr__Cong = $true) & (omega0Ax = $true) & (noeltsimpempty = $true) & (setadjoinAx = $true) & (quantDeMorgan3 = $true) & (upairsetIR = $true) & (prop2setI = $true) & (upairsetIL = $true) & (notinsingleton = $true) & (uniqinunit = $true) & (subsetRefl = $true) & (exuE2 = $true) & (quantDeMorgan1 = $true) & (emptysetimpfalse = $true) & (nonemptyImpWitness = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ X0) != $true) & ((in @ X2 @ (binunion @ X1 @ X0)) = $true) & ($true != (in @ X2 @ X1))) & (prop2set2propI = $true) & (inCongP = $true) & (powerset__Cong = $true) & (omegaIndAx = $true) & (notsubsetI = $true) & (foundationAx = $true) & (exuE3u = $true) & (setunionI = $true) & (binunionEcases = $true) & (notdallE = $true) & (vacuousDall = $true) & (setunion__Cong = $true) & (subsetTrans = $true) & (emptyI = $true) & (nonemptyI1 = $true) & (dsetconstrEL = $true) & (exuI2 = $true) & (omegaSAx = $true) & (dsetconstrI = $true) & (setadjoinIL = $true) & (omega__Cong = $true) & (setextsub = $true) & (setadjoinIR = $true) & (powersetI = $true) & (quantDeMorgan4 = $true) & (powersetE1 = $true) & (notinemptyset = $true) & (prop2setE = $true) & (emptyE1 = $true) & (setunionAx = $true)),
% 2.62/1.79 inference(flattening,[status(thm)],[f354])).
% 2.62/1.79 thf(f354,plain,(
% 2.62/1.79 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i,X2 : $i] : ((($true != (in @ X2 @ X1)) & ((in @ X2 @ X0) != $true)) & ((in @ X2 @ (binunion @ X1 @ X0)) = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (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)),
% 2.62/1.79 inference(ennf_transformation,[status(thm)],[f261])).
% 2.62/1.79 thf(f261,plain,(
% 2.62/1.79 ~((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) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ (binunion @ X1 @ X0)) = $true) => (($true = (in @ X2 @ X1)) | ((in @ X2 @ X0) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.62/1.79 inference(fool_elimination,[status(thm)],[f260])).
% 2.62/1.79 thf(f260,plain,(
% 2.62/1.79 ~(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 => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ (binunion @ X1 @ X0)) => ((in @ X2 @ X1) | (in @ X2 @ X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.62/1.79 inference(rectify,[status(thm)],[f106])).
% 2.62/1.79 thf(f106,negated_conjecture,(
% 2.62/1.79 ~(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 => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => ! [X4 : $i,X3 : $i,X1 : $i] : ((in @ X1 @ (binunion @ X3 @ X4)) => ((in @ X1 @ X3) | (in @ X1 @ X4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.62/1.79 inference(negated_conjecture,[status(cth)],[f105])).
% 2.62/1.79 thf(f105,conjecture,(
% 2.62/1.79 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 => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => ! [X4 : $i,X3 : $i,X1 : $i] : ((in @ X1 @ (binunion @ X3 @ X4)) => ((in @ X1 @ X3) | (in @ X1 @ X4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.62/1.79 file('/export/starexec/sandbox/tmp/tmp.9nobNTBT08/DTF2THF_21348.p',binunionE)).
% 2.62/1.79 thf(f1915,plain,(
% 2.62/1.79 ~spl285_3),
% 2.62/1.79 inference(avatar_contradiction_clause,[status(thm)],[f1914])).
% 2.62/1.79 thf(f1914,plain,(
% 2.62/1.79 $false | ~spl285_3),
% 2.62/1.79 inference(trivial_inequality_removal,[status(thm)],[f1691])).
% 2.62/1.79 thf(f1691,plain,(
% 2.62/1.79 ($true != $true) | ~spl285_3),
% 2.62/1.79 inference(superposition,[status(thm)],[f1011,f1598])).
% 2.62/1.79 thf(f1598,plain,(
% 2.62/1.79 ( ! [X0 : $o] : (($true = X0)) ) | ~spl285_3),
% 2.62/1.79 inference(avatar_component_clause,[status(thm)],[f1597])).
% 2.62/1.79 thf(f1597,definition,(
% 2.62/1.79 spl285_3 <=> ! [X0 : $o] : ($true = X0)),
% 2.62/1.79 introduced(definition,[new_symbols(naming,[spl285_3])],[avatar_definition])).
% 2.62/1.79 thf(f1604,plain,(
% 2.62/1.79 ~spl285_2),
% 2.62/1.79 inference(avatar_contradiction_clause,[status(thm)],[f1603])).
% 2.62/1.79 thf(f1603,plain,(
% 2.62/1.79 $false | ~spl285_2),
% 2.62/1.79 inference(trivial_inequality_removal,[status(thm)],[f1600])).
% 2.62/1.79 thf(f1600,plain,(
% 2.62/1.79 ($true != $true) | ~spl285_2),
% 2.62/1.79 inference(superposition,[status(thm)],[f1013,f1595])).
% 2.62/1.79 thf(f1595,plain,(
% 2.62/1.79 ($true = (in @ sK37 @ sK35)) | ~spl285_2),
% 2.62/1.79 inference(avatar_component_clause,[status(thm)],[f1593])).
% 2.62/1.79 thf(f1593,definition,(
% 2.62/1.79 spl285_2 <=> ($true = (in @ sK37 @ sK35))),
% 2.62/1.79 introduced(definition,[new_symbols(naming,[spl285_2])],[avatar_definition])).
% 2.62/1.79 thf(f1013,plain,(
% 2.62/1.79 ($true != (in @ sK37 @ sK35))),
% 2.62/1.79 inference(cnf_transformation,[status(thm)],[f502])).
% 2.62/1.79 thf(f1599,plain,(
% 2.62/1.79 spl285_1 | spl285_2 | spl285_3),
% 2.62/1.79 inference(avatar_split_clause,[status(thm)],[f1587,f1597,f1593,f1589])).
% 2.62/1.79 thf(f1587,plain,(
% 2.62/1.79 ( ! [X0 : $o] : (($true = X0) | ($true = (in @ sK37 @ sK36)) | ($true = (in @ sK37 @ sK35))) )),
% 2.62/1.79 inference(trivial_inequality_removal,[status(thm)],[f1586])).
% 2.62/1.79 thf(f1586,plain,(
% 2.62/1.79 ( ! [X0 : $o] : (($true != $true) | ($true = X0) | ($true = (in @ sK37 @ sK35)) | ($true = (in @ sK37 @ sK36))) )),
% 2.62/1.79 inference(forward_demodulation,[status(thm)],[f1582,f1002])).
% 2.62/1.79 thf(f1002,plain,(
% 2.62/1.79 (binunionEcases = $true)),
% 2.62/1.79 inference(cnf_transformation,[status(thm)],[f502])).
% 2.62/1.79 thf(f1582,plain,(
% 2.62/1.79 ( ! [X0 : $o] : (($true = (in @ sK37 @ sK36)) | ($true = X0) | (binunionEcases != $true) | ($true = (in @ sK37 @ sK35))) )),
% 2.62/1.79 inference(trivial_inequality_removal,[status(thm)],[f1580])).
% 2.62/1.79 thf(f1580,plain,(
% 2.62/1.79 ( ! [X0 : $o] : (($true != $true) | ($true = (in @ sK37 @ sK35)) | ($true = X0) | ($true = (in @ sK37 @ sK36)) | (binunionEcases != $true)) )),
% 2.62/1.79 inference(superposition,[status(thm)],[f1260,f1012])).
% 2.62/1.79 thf(f1012,plain,(
% 2.62/1.79 ($true = (in @ sK37 @ (binunion @ sK36 @ sK35)))),
% 2.62/1.79 inference(cnf_transformation,[status(thm)],[f502])).
% 2.62/1.79 thf(f1260,plain,(
% 2.62/1.79 ( ! [X6 : $i,X7 : $i,X4 : $i,X5 : $o] : (($true != (in @ X7 @ (binunion @ X4 @ X6))) | ($true = X5) | ($true = (in @ X7 @ X6)) | ((in @ X7 @ X4) = $true) | (binunionEcases != $true)) )),
% 2.62/1.79 inference(cnf_transformation,[status(thm)],[f711])).
% 2.62/1.79 thf(f711,plain,(
% 2.62/1.79 ((binunionEcases = $true) | ((($true = sK154) | ($true != (in @ sK156 @ sK153))) & (($true = sK154) | ($true != (in @ sK156 @ sK155))) & ((in @ sK156 @ (binunion @ sK153 @ sK155)) = $true) & ($true != sK154))) & (! [X4 : $i,X5 : $o,X6 : $i,X7 : $i] : ((($true != X5) & ((in @ X7 @ X4) = $true)) | (($true != X5) & ($true = (in @ X7 @ X6))) | ($true != (in @ X7 @ (binunion @ X4 @ X6))) | ($true = X5)) | (binunionEcases != $true))),
% 2.62/1.79 inference(skolemisation,[status(esa),new_symbols(skolem,[sK153,sK154,sK155,sK156])],[f709,f710])).
% 2.62/1.79 thf(f710,plain,(
% 2.62/1.79 ? [X0 : $i,X1 : $o,X2 : $i,X3 : $i] : ((($true = X1) | ($true != (in @ X3 @ X0))) & (($true = X1) | ($true != (in @ X3 @ X2))) & ((in @ X3 @ (binunion @ X0 @ X2)) = $true) & ($true != X1)) => ((($true = sK154) | ($true != (in @ sK156 @ sK153))) & (($true = sK154) | ($true != (in @ sK156 @ sK155))) & ((in @ sK156 @ (binunion @ sK153 @ sK155)) = $true) & ($true != sK154))),
% 2.62/1.79 introduced(definition,[],[choice_axiom])).
% 2.62/1.79 thf(f709,plain,(
% 2.62/1.79 ((binunionEcases = $true) | ? [X0 : $i,X1 : $o,X2 : $i,X3 : $i] : ((($true = X1) | ($true != (in @ X3 @ X0))) & (($true = X1) | ($true != (in @ X3 @ X2))) & ((in @ X3 @ (binunion @ X0 @ X2)) = $true) & ($true != X1))) & (! [X4 : $i,X5 : $o,X6 : $i,X7 : $i] : ((($true != X5) & ((in @ X7 @ X4) = $true)) | (($true != X5) & ($true = (in @ X7 @ X6))) | ($true != (in @ X7 @ (binunion @ X4 @ X6))) | ($true = X5)) | (binunionEcases != $true))),
% 2.62/1.79 inference(rectify,[status(thm)],[f708])).
% 2.62/1.79 thf(f708,plain,(
% 2.62/1.79 ((binunionEcases = $true) | ? [X0 : $i,X1 : $o,X2 : $i,X3 : $i] : ((($true = X1) | ($true != (in @ X3 @ X0))) & (($true = X1) | ($true != (in @ X3 @ X2))) & ((in @ X3 @ (binunion @ X0 @ X2)) = $true) & ($true != X1))) & (! [X0 : $i,X1 : $o,X2 : $i,X3 : $i] : ((($true != X1) & ($true = (in @ X3 @ X0))) | (($true != X1) & ($true = (in @ X3 @ X2))) | ((in @ X3 @ (binunion @ X0 @ X2)) != $true) | ($true = X1)) | (binunionEcases != $true))),
% 2.62/1.79 inference(nnf_transformation,[status(thm)],[f429])).
% 2.62/1.79 thf(f429,plain,(
% 2.62/1.79 (binunionEcases = $true) <=> ! [X0 : $i,X1 : $o,X2 : $i,X3 : $i] : ((($true != X1) & ($true = (in @ X3 @ X0))) | (($true != X1) & ($true = (in @ X3 @ X2))) | ((in @ X3 @ (binunion @ X0 @ X2)) != $true) | ($true = X1))),
% 2.62/1.79 inference(flattening,[status(thm)],[f428])).
% 2.62/1.79 thf(f428,plain,(
% 2.62/1.79 ! [X0 : $i,X1 : $o,X2 : $i,X3 : $i] : (((($true = X1) | (($true != X1) & ($true = (in @ X3 @ X2)))) | (($true != X1) & ($true = (in @ X3 @ X0)))) | ((in @ X3 @ (binunion @ X0 @ X2)) != $true)) <=> (binunionEcases = $true)),
% 2.62/1.79 inference(ennf_transformation,[status(thm)],[f133])).
% 2.62/1.79 thf(f133,plain,(
% 2.62/1.79 ! [X0 : $i,X1 : $o,X2 : $i,X3 : $i] : (((in @ X3 @ (binunion @ X0 @ X2)) = $true) => ((($true = (in @ X3 @ X0)) => ($true = X1)) => ((($true = (in @ X3 @ X2)) => ($true = X1)) => ($true = X1)))) <=> (binunionEcases = $true)),
% 2.62/1.79 inference(fool_elimination,[status(thm)],[f132])).
% 2.62/1.79 thf(f132,plain,(
% 2.62/1.79 (binunionEcases = ! [X0 : $i,X1 : $o,X2 : $i,X3 : $i] : ((in @ X3 @ (binunion @ X0 @ X2)) => (((in @ X3 @ X0) => X1) => (((in @ X3 @ X2) => X1) => X1))))),
% 2.62/1.79 inference(rectify,[status(thm)],[f104])).
% 2.62/1.79 thf(f104,axiom,(
% 2.62/1.79 (binunionEcases = ! [X3 : $i,X0 : $o,X4 : $i,X1 : $i] : ((in @ X1 @ (binunion @ X3 @ X4)) => (((in @ X1 @ X3) => X0) => (((in @ X1 @ X4) => X0) => X0))))),
% 2.62/1.79 file('/export/starexec/sandbox/tmp/tmp.9nobNTBT08/DTF2THF_21348.p',binunionEcases)).
% 2.62/1.79 % SZS output end Proof for DTF2THF_21348
% 2.62/1.79 % (21469)------------------------------
% 2.62/1.79 % (21469)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.79 % (21469)Termination reason: Refutation
% 2.62/1.79
% 2.62/1.79 % (21469)Memory used [KB]: 7419
% 2.62/1.79 % (21469)Time elapsed: 0.075 s
% 2.62/1.79 % (21469)Instructions burned: 136 (million)
% 2.62/1.79 % (21469)------------------------------
% 2.62/1.79 % (21469)------------------------------
% 2.62/1.79 % (21446)Success in time 0.182 s
%------------------------------------------------------------------------------