%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU572^1 : 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 : n013.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 09:35:20 AM UTC 2026
% Result : Theorem 2.24s 1.60s
% Output : Refutation 2.24s
% Verified :
% SZS Type : Refutation
% Derivation depth : 18
% Number of leaves : 6
% Syntax : Number of formulae : 40 ( 11 unt; 0 typ; 0 def)
% Number of atoms : 1699 ( 457 equ; 0 cnn)
% Maximal formula atoms : 90 ( 42 avg)
% Number of connectives : 976 ( 60 ~; 38 |; 285 &; 224 @)
% ( 4 <=>; 365 =>; 0 <=; 0 <~>)
% Maximal formula depth : 97 ( 20 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 206 ( 203 usr; 98 con; 0-4 aty)
% Number of variables : 81 ( 0 ^; 57 !; 24 ?; 81 :)
% 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_117,type,
sP1: $i > $i > $o ).
thf(func_def_118,type,
sP2: $i > $o ).
thf(func_def_119,type,
sP3: $i > $i > $o ).
thf(func_def_120,type,
sP4: $i > $i > $o ).
thf(func_def_121,type,
sK5: $i > $i > $i ).
thf(func_def_124,type,
sK8: $i > $o ).
thf(func_def_127,type,
sK11: $i > ( $i > $o ) > $i ).
thf(func_def_130,type,
sK14: $i > $o ).
thf(func_def_133,type,
sK17: ( $i > $o ) > $i ).
thf(func_def_134,type,
sK18: ( $i > $o ) > $i ).
thf(func_def_135,type,
sK19: $i > $o ).
thf(func_def_137,type,
sK21: ( $i > $o ) > $i ).
thf(func_def_138,type,
sK22: $i > $o ).
thf(func_def_139,type,
sK23: $i > $o ).
thf(func_def_144,type,
sK28: $i > $i ).
thf(func_def_150,type,
sK34: $i > $i > $i ).
thf(func_def_151,type,
sK35: $i > $i > $i ).
thf(func_def_154,type,
sK38: $i > ( $i > $o ) > $i ).
thf(func_def_155,type,
sK39: $i > $o ).
thf(func_def_171,type,
sK55: $i > $o ).
thf(func_def_182,type,
sK66: ( $i > $o ) > $i > $i ).
thf(func_def_184,type,
sK68: $i > $o ).
thf(func_def_187,type,
sK71: $i > $o ).
thf(func_def_194,type,
sK78: $i > $i ).
thf(func_def_198,type,
sK82: $i > $i ).
thf(func_def_200,type,
sK84: $i > $i ).
thf(func_def_203,type,
sK87: $i > $o ).
thf(func_def_206,type,
sK90: $i > $o ).
thf(func_def_207,type,
sK91: $i > $i ).
thf(func_def_208,type,
sK92: ( $i > $o ) > $i ).
thf(func_def_209,type,
sK93: $i > $o ).
thf(func_def_210,type,
sK94: $i > $o ).
thf(func_def_211,type,
sK95: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_212,type,
sK96: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_215,type,
sK99: $i > $o ).
thf(func_def_217,type,
sK101: ( $i > $o ) > $i > $i ).
thf(func_def_219,type,
sK103: $i > $o ).
thf(func_def_221,type,
sK105: $i > $o ).
thf(func_def_222,type,
sK106: $i > $i ).
thf(func_def_223,type,
sK107: ( $i > $o ) > $i ).
thf(func_def_228,type,
sK112: $i > $o ).
thf(func_def_229,type,
sK113: $i > $o ).
thf(func_def_230,type,
sK114: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_231,type,
sK115: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_232,type,
sK116: $i > ( $i > $o ) > $i ).
thf(func_def_233,type,
sK117: $i > $o ).
thf(func_def_236,type,
sK120: $i > $o ).
thf(func_def_238,type,
sK122: $i > ( $i > $o ) > $i ).
thf(func_def_239,type,
sK123: $i > $o ).
thf(func_def_252,type,
sK136: $i > $i > $i ).
thf(func_def_253,type,
sK137: $i > $o ).
thf(func_def_258,type,
sK142: $i > $i ).
thf(func_def_261,type,
sK145: $i > $i > $o ).
thf(func_def_263,type,
sK147: $i > $i ).
thf(func_def_264,type,
sK148: $i > $i ).
thf(func_def_265,type,
sK149: $i > ( $i > $i > $o ) > $i ).
thf(func_def_266,type,
sK150: $i > ( $i > $i > $o ) > $i ).
thf(func_def_267,type,
sK151: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_278,type,
sK162: $i > $i ).
thf(func_def_280,type,
sK164: $i > ( $i > $o ) > $i ).
thf(func_def_281,type,
sK165: $i > $o ).
thf(func_def_285,type,
sK169: $i > $i ).
thf(func_def_289,type,
sK173: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_290,type,
sK174: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_291,type,
sK175: $i > $o ).
thf(func_def_292,type,
sK176: $i > $o ).
thf(func_def_296,type,
sK180: $o > $i > $i > $i ).
thf(func_def_304,type,
sK188: $i > $i > $i ).
thf(func_def_309,type,
sK193: $i > $i > $i ).
thf(func_def_312,type,
sK196: $i > $i > $i ).
thf(func_def_313,type,
sK197: $i > $i > $i ).
thf(func_def_314,type,
sK198: $i > $i > $i ).
thf(func_def_315,type,
sK199: $i > $i > $i ).
thf(func_def_316,type,
sK200: $i > $i > $i > $i ).
thf(func_def_317,type,
sK201: $i > $i ).
thf(func_def_318,type,
sK202: $i > $i ).
thf(func_def_319,type,
sK203: $i > $i ).
thf(func_def_320,type,
sK204: $i > $i ).
thf(func_def_321,type,
sK205: $i > $i > $i > $i ).
thf(func_def_322,type,
sK206: $i > $i > $i > $i ).
thf(func_def_323,type,
sK207: $i > $i > $i ).
thf(func_def_324,type,
sK208: $i > $i > $i ).
thf(func_def_325,type,
sK209: $i > $i > $i ).
thf(func_def_327,type,
sK211: $i > $i ).
thf(func_def_328,type,
sK212: $i > $i ).
thf(func_def_329,type,
sK213: $i > $i ).
thf(func_def_330,type,
sK214: $i > $i > $i ).
thf(func_def_333,type,
sK217: $i > $o ).
thf(func_def_349,type,
sK233: $i > $o ).
thf(func_def_352,type,
sK236: $i > $i > $i ).
thf(func_def_361,type,
sK245: $i > $o ).
thf(func_def_363,type,
sK247: ( $i > $o ) > $i > $i ).
thf(func_def_367,type,
sK251: $i > $i > $i ).
thf(func_def_371,type,
ph255:
!>[X0: $tType] : X0 ).
thf(f1429,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f1426]) ).
thf(f1426,plain,
$true != $true,
inference(superposition,[status(thm)],[f1419,f1094]) ).
thf(f1094,plain,
( ( subset @ sK132 @ sK131 )
= $true ),
inference(cnf_transformation,[status(thm)],[f617]) ).
thf(f617,plain,
( ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( upairsetIL = $true )
& ( exuE2 = $true )
& ( subsetI2 = $true )
& ( setoftrueEq = $true )
& ( notdallE = $true )
& ( descrp = $true )
& ( nonemptyI1 = $true )
& ( eqimpsubset1 = $true )
& ( $true
!= ( subset @ sK132 @ ( setadjoin @ sK133 @ sK131 ) ) )
& ( ( subset @ sK132 @ sK131 )
= $true )
& ( emptysetsubset = $true )
& ( subsetI1 = $true )
& ( prop2setI = $true )
& ( prop2set2propI = $true )
& ( notsubsetI = $true )
& ( emptysetE = $true )
& ( emptyI = $true )
& ( exuE3e = $true )
& ( setadjoinAx = $true )
& ( exu__Cong = $true )
& ( setadjoinOr = $true )
& ( exuE1 = $true )
& ( notdexE = $true )
& ( emptyInPowerset = $true )
& ( exuI2 = $true )
& ( replAx = $true )
& ( notequalI1 = $true )
& ( subsetE = $true )
& ( upairsetE = $true )
& ( emptysetimpfalse = $true )
& ( uniqinunit = $true )
& ( notinemptyset = $true )
& ( notinsingleton = $true )
& ( exuI3 = $true )
& ( inCongP = $true )
& ( in__Cong = $true )
& ( noeltsimpempty = $true )
& ( emptyinunitempty = $true )
& ( setunionI = $true )
& ( emptyinPowerset = $true )
& ( setextAx = $true )
& ( exuI1 = $true )
& ( upairsetIR = $true )
& ( quantDeMorgan4 = $true )
& ( nonemptyImpWitness = $true )
& ( setunionE = $true )
& ( subPowSU = $true )
& ( quantDeMorgan3 = $true )
& ( descr__Cong = $true )
& ( setadjoinIL = $true )
& ( wellorderingAx = $true )
& ( emptysetAx = $true )
& ( dsetconstr__Cong = $true )
& ( notequalI2 = $true )
& ( nonemptyE1 = $true )
& ( powersetI = $true )
& ( eqimpsubset2 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrI = $true )
& ( subsetRefl = $true )
& ( omega__Cong = $true )
& ( emptyset__Cong = $true )
& ( omegaSAx = $true )
& ( setadjoinIR = $true )
& ( exuE3u = $true )
& ( powersetAx = $true )
& ( emptyE1 = $true )
& ( omega0Ax = $true )
& ( singletonsswitch = $true )
& ( omegaIndAx = $true )
& ( subsetTrans = $true )
& ( setbeta = $true )
& ( powerset__Cong = $true )
& ( setext = $true )
& ( nonemptyI = $true )
& ( setunionAx = $true )
& ( dsetconstrEL = $true )
& ( powersetE = $true )
& ( eqinunit = $true )
& ( setadjoin__Cong = $true )
& ( vacuousDall = $true )
& ( foundationAx = $true )
& ( setadjoinSub = $true )
& ( prop2setE = $true )
& ( setunion__Cong = $true )
& ( subsetE2 = $true )
& ( exuEu = $true )
& ( setadjoinE = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK131,sK132,sK133])],[f375,f616]) ).
thf(f616,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( subset @ X1 @ ( setadjoin @ X2 @ X0 ) ) )
& ( $true
= ( subset @ X1 @ X0 ) ) )
=> ( ( $true
!= ( subset @ sK132 @ ( setadjoin @ sK133 @ sK131 ) ) )
& ( ( subset @ sK132 @ sK131 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f375,plain,
( ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( upairsetIL = $true )
& ( exuE2 = $true )
& ( subsetI2 = $true )
& ( setoftrueEq = $true )
& ( notdallE = $true )
& ( descrp = $true )
& ( nonemptyI1 = $true )
& ( eqimpsubset1 = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( subset @ X1 @ ( setadjoin @ X2 @ X0 ) ) )
& ( $true
= ( subset @ X1 @ X0 ) ) )
& ( emptysetsubset = $true )
& ( subsetI1 = $true )
& ( prop2setI = $true )
& ( prop2set2propI = $true )
& ( notsubsetI = $true )
& ( emptysetE = $true )
& ( emptyI = $true )
& ( exuE3e = $true )
& ( setadjoinAx = $true )
& ( exu__Cong = $true )
& ( setadjoinOr = $true )
& ( exuE1 = $true )
& ( notdexE = $true )
& ( emptyInPowerset = $true )
& ( exuI2 = $true )
& ( replAx = $true )
& ( notequalI1 = $true )
& ( subsetE = $true )
& ( upairsetE = $true )
& ( emptysetimpfalse = $true )
& ( uniqinunit = $true )
& ( notinemptyset = $true )
& ( notinsingleton = $true )
& ( exuI3 = $true )
& ( inCongP = $true )
& ( in__Cong = $true )
& ( noeltsimpempty = $true )
& ( emptyinunitempty = $true )
& ( setunionI = $true )
& ( emptyinPowerset = $true )
& ( setextAx = $true )
& ( exuI1 = $true )
& ( upairsetIR = $true )
& ( quantDeMorgan4 = $true )
& ( nonemptyImpWitness = $true )
& ( setunionE = $true )
& ( subPowSU = $true )
& ( quantDeMorgan3 = $true )
& ( descr__Cong = $true )
& ( setadjoinIL = $true )
& ( wellorderingAx = $true )
& ( emptysetAx = $true )
& ( dsetconstr__Cong = $true )
& ( notequalI2 = $true )
& ( nonemptyE1 = $true )
& ( powersetI = $true )
& ( eqimpsubset2 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrI = $true )
& ( subsetRefl = $true )
& ( omega__Cong = $true )
& ( emptyset__Cong = $true )
& ( omegaSAx = $true )
& ( setadjoinIR = $true )
& ( exuE3u = $true )
& ( powersetAx = $true )
& ( emptyE1 = $true )
& ( omega0Ax = $true )
& ( singletonsswitch = $true )
& ( omegaIndAx = $true )
& ( subsetTrans = $true )
& ( setbeta = $true )
& ( powerset__Cong = $true )
& ( setext = $true )
& ( nonemptyI = $true )
& ( setunionAx = $true )
& ( dsetconstrEL = $true )
& ( powersetE = $true )
& ( eqinunit = $true )
& ( setadjoin__Cong = $true )
& ( vacuousDall = $true )
& ( foundationAx = $true )
& ( setadjoinSub = $true )
& ( prop2setE = $true )
& ( setunion__Cong = $true )
& ( subsetE2 = $true )
& ( exuEu = $true )
& ( setadjoinE = $true ) ),
inference(flattening,[status(thm)],[f374]) ).
thf(f374,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( subset @ X1 @ ( setadjoin @ X2 @ X0 ) ) )
& ( $true
= ( subset @ X1 @ X0 ) ) )
& ( 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)],[f188]) ).
thf(f188,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 )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( subset @ X1 @ X0 ) )
=> ( $true
= ( subset @ X1 @ ( setadjoin @ X2 @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f187]) ).
thf(f187,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
=> ! [X0: $i,X1: $i,X2: $i] :
( ( subset @ X1 @ X0 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f92]) ).
thf(f92,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
=> ! [X4: $i,X3: $i,X1: $i] :
( ( subset @ X3 @ X4 )
=> ( subset @ X3 @ ( setadjoin @ X1 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f91]) ).
thf(f91,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
=> ! [X4: $i,X3: $i,X1: $i] :
( ( subset @ X3 @ X4 )
=> ( subset @ X3 @ ( setadjoin @ X1 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.pvugr8aAKv/DTF2THF_16254.p',setadjoinSub2) ).
thf(f1419,plain,
( ( subset @ sK132 @ sK131 )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f1418]) ).
thf(f1418,plain,
( ( $true != $true )
| ( ( subset @ sK132 @ sK131 )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f1391,f1021]) ).
thf(f1021,plain,
setadjoinSub = $true,
inference(cnf_transformation,[status(thm)],[f617]) ).
thf(f1391,plain,
( ( ( subset @ sK132 @ sK131 )
!= $true )
| ( setadjoinSub != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f1380]) ).
thf(f1380,plain,
( ( $true != $true )
| ( setadjoinSub != $true )
| ( ( subset @ sK132 @ sK131 )
!= $true ) ),
inference(superposition,[status(thm)],[f1379,f940]) ).
thf(f940,plain,
! [X2: $i,X3: $i] :
( ( ( subset @ X3 @ ( setadjoin @ X2 @ X3 ) )
= $true )
| ( setadjoinSub != $true ) ),
inference(cnf_transformation,[status(thm)],[f528]) ).
thf(f528,plain,
( ( ( setadjoinSub = $true )
| ( $true
!= ( subset @ sK81 @ ( setadjoin @ sK80 @ sK81 ) ) ) )
& ( ! [X2: $i,X3: $i] :
( ( subset @ X3 @ ( setadjoin @ X2 @ X3 ) )
= $true )
| ( setadjoinSub != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK80,sK81])],[f526,f527]) ).
thf(f527,plain,
( ? [X0: $i,X1: $i] :
( ( subset @ X1 @ ( setadjoin @ X0 @ X1 ) )
!= $true )
=> ( $true
!= ( subset @ sK81 @ ( setadjoin @ sK80 @ sK81 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f526,plain,
( ( ( setadjoinSub = $true )
| ? [X0: $i,X1: $i] :
( ( subset @ X1 @ ( setadjoin @ X0 @ X1 ) )
!= $true ) )
& ( ! [X2: $i,X3: $i] :
( ( subset @ X3 @ ( setadjoin @ X2 @ X3 ) )
= $true )
| ( setadjoinSub != $true ) ) ),
inference(rectify,[status(thm)],[f525]) ).
thf(f525,plain,
( ( ( setadjoinSub = $true )
| ? [X0: $i,X1: $i] :
( ( subset @ X1 @ ( setadjoin @ X0 @ X1 ) )
!= $true ) )
& ( ! [X0: $i,X1: $i] :
( ( subset @ X1 @ ( setadjoin @ X0 @ X1 ) )
= $true )
| ( setadjoinSub != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f150]) ).
thf(f150,plain,
( ( setadjoinSub = $true )
<=> ! [X0: $i,X1: $i] :
( ( subset @ X1 @ ( setadjoin @ X0 @ X1 ) )
= $true ) ),
inference(fool_elimination,[status(thm)],[f149]) ).
thf(f149,plain,
( setadjoinSub
= ( ! [X0: $i,X1: $i] : ( subset @ X1 @ ( setadjoin @ X0 @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f90]) ).
thf(f90,axiom,
( setadjoinSub
= ( ! [X1: $i,X3: $i] : ( subset @ X3 @ ( setadjoin @ X1 @ X3 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.pvugr8aAKv/DTF2THF_16254.p',setadjoinSub) ).
thf(f1379,plain,
! [X0: $i] :
( ( $true
!= ( subset @ X0 @ ( setadjoin @ sK133 @ sK131 ) ) )
| ( $true
!= ( subset @ sK132 @ X0 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f1378]) ).
thf(f1378,plain,
! [X0: $i] :
( ( $true != $true )
| ( $true
!= ( subset @ X0 @ ( setadjoin @ sK133 @ sK131 ) ) )
| ( $true
!= ( subset @ sK132 @ X0 ) ) ),
inference(forward_demodulation,[status(thm)],[f1377,f1033]) ).
thf(f1033,plain,
subsetTrans = $true,
inference(cnf_transformation,[status(thm)],[f617]) ).
thf(f1377,plain,
! [X0: $i] :
( ( $true
!= ( subset @ sK132 @ X0 ) )
| ( subsetTrans != $true )
| ( $true
!= ( subset @ X0 @ ( setadjoin @ sK133 @ sK131 ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f1376]) ).
thf(f1376,plain,
! [X0: $i] :
( ( $true != $true )
| ( $true
!= ( subset @ sK132 @ X0 ) )
| ( $true
!= ( subset @ X0 @ ( setadjoin @ sK133 @ sK131 ) ) )
| ( subsetTrans != $true ) ),
inference(superposition,[status(thm)],[f1095,f1140]) ).
thf(f1140,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true
= ( subset @ X1 @ X0 ) )
| ( ( subset @ X1 @ X2 )
!= $true )
| ( ( subset @ X2 @ X0 )
!= $true )
| ( subsetTrans != $true ) ),
inference(cnf_transformation,[status(thm)],[f658]) ).
thf(f658,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X2 @ X0 )
!= $true )
| ( $true
= ( subset @ X1 @ X0 ) )
| ( ( subset @ X1 @ X2 )
!= $true ) )
| ( subsetTrans != $true ) )
& ( ( subsetTrans = $true )
| ( ( ( subset @ sK154 @ sK152 )
= $true )
& ( $true
!= ( subset @ sK153 @ sK152 ) )
& ( ( subset @ sK153 @ sK154 )
= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK152,sK153,sK154])],[f656,f657]) ).
thf(f657,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( ( subset @ X5 @ X3 )
= $true )
& ( ( subset @ X4 @ X3 )
!= $true )
& ( ( subset @ X4 @ X5 )
= $true ) )
=> ( ( ( subset @ sK154 @ sK152 )
= $true )
& ( $true
!= ( subset @ sK153 @ sK152 ) )
& ( ( subset @ sK153 @ sK154 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f656,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X2 @ X0 )
!= $true )
| ( $true
= ( subset @ X1 @ X0 ) )
| ( ( subset @ X1 @ X2 )
!= $true ) )
| ( subsetTrans != $true ) )
& ( ( subsetTrans = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( ( subset @ X5 @ X3 )
= $true )
& ( ( subset @ X4 @ X3 )
!= $true )
& ( ( subset @ X4 @ X5 )
= $true ) ) ) ),
inference(rectify,[status(thm)],[f655]) ).
thf(f655,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X2 @ X0 )
!= $true )
| ( $true
= ( subset @ X1 @ X0 ) )
| ( ( subset @ X1 @ X2 )
!= $true ) )
| ( subsetTrans != $true ) )
& ( ( subsetTrans = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X2 @ X0 )
= $true )
& ( $true
!= ( subset @ X1 @ X0 ) )
& ( ( subset @ X1 @ X2 )
= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f323]) ).
thf(f323,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X2 @ X0 )
!= $true )
| ( $true
= ( subset @ X1 @ X0 ) )
| ( ( subset @ X1 @ X2 )
!= $true ) )
<=> ( subsetTrans = $true ) ),
inference(flattening,[status(thm)],[f322]) ).
thf(f322,plain,
( ( subsetTrans = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( subset @ X1 @ X0 ) )
| ( ( subset @ X2 @ X0 )
!= $true )
| ( ( subset @ X1 @ X2 )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f152]) ).
thf(f152,plain,
( ( subsetTrans = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X1 @ X2 )
= $true )
=> ( ( ( subset @ X2 @ X0 )
= $true )
=> ( $true
= ( subset @ X1 @ X0 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f151]) ).
thf(f151,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X0 )
=> ( subset @ X1 @ X0 ) ) )
= subsetTrans ),
inference(rectify,[status(thm)],[f89]) ).
thf(f89,axiom,
! [X5: $i,X3: $i,X4: $i] :
( ( ( subset @ X3 @ X4 )
=> ( ( subset @ X4 @ X5 )
=> ( subset @ X3 @ X5 ) ) )
= subsetTrans ),
file('/export/starexec/sandbox/tmp/tmp.pvugr8aAKv/DTF2THF_16254.p',subsetTrans) ).
thf(f1095,plain,
( $true
!= ( subset @ sK132 @ ( setadjoin @ sK133 @ sK131 ) ) ),
inference(cnf_transformation,[status(thm)],[f617]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU572^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.18/0.34 % Computer : n013.cluster.edu
% 0.18/0.34 % Model : x86_64 x86_64
% 0.18/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.34 % Memory : 8042.1875MB
% 0.18/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.18/0.34 % CPULimit : 300
% 0.18/0.34 % WCLimit : 300
% 0.18/0.34 % DateTime : Tue Feb 24 12:27:08 EST 2026
% 0.18/0.34 % CPUTime :
% 0.18/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.20/0.46 ---- Original DTF file ---
% 0.32/0.47 thf(spec,logic,$$dhol).
% 0.32/0.47 %------------------------------------------------------------------------------
% 0.32/0.47 % File : SEU572^1 : TPTP v9.2.1. Released v3.7.0.
% 0.32/0.47 % Domain : Set Theory
% 0.32/0.47 % Problem : Preliminary Notions - Relations on Sets - Subsets
% 0.32/0.47 % Version : Especial.
% 0.32/0.47 % English : (! A:i.! x:i.! B:i.subset A B -> subset A (setadjoin x B))
% 0.32/0.47
% 0.32/0.47 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.32/0.47 % Source : [Bro08]
% 0.32/0.47 % Names : ZFC074g [Bro08]
% 0.32/0.47
% 0.32/0.47 % Status : Theorem
% 0.32/0.47 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.31 v8.1.0, 0.09 v7.5.0, 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 v6.0.0, 0.43 v5.5.0, 0.33 v5.4.0, 0.40 v5.3.0, 0.60 v5.2.0, 0.40 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.67 v4.0.1, 1.00 v3.7.0
% 0.32/0.47 % Syntax : Number of formulae : 194 ( 90 unt; 103 typ; 90 def)
% 0.32/0.47 % Number of atoms : 492 ( 137 equ; 0 cnn)
% 0.32/0.47 % Maximal formula atoms : 90 ( 5 avg)
% 0.32/0.47 % Number of connectives : 780 ( 29 ~; 5 |; 27 &; 480 @)
% 0.32/0.47 % ( 14 <=>; 225 =>; 0 <=; 0 <~>)
% 0.32/0.47 % Maximal formula depth : 96 ( 2 avg)
% 0.32/0.47 % Number of types : 2 ( 0 usr)
% 0.32/0.47 % Number of type conns : 53 ( 53 >; 0 *; 0 +; 0 <<)
% 0.32/0.47 % Number of symbols : 106 ( 103 usr; 92 con; 0-2 aty)
% 0.32/0.47 % Number of variables : 276 ( 28 ^; 220 !; 28 ?; 276 :)
% 0.32/0.47 % SPC : TH0_THM_EQU_NAR
% 0.32/0.47
% 0.32/0.47 % Comments : http://mathgate.info/detsetitem.php?id=478
% 0.32/0.47 %------------------------------------------------------------------------------
% 0.32/0.47 thf(in_type,type,
% 0.32/0.47 in: $i > $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(exu_type,type,
% 0.32/0.47 exu: ( $i > $o ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(exu,definition,
% 0.32/0.47 ( exu
% 0.32/0.47 = ( ^ [Xphi: $i > $o] :
% 0.32/0.47 ? [Xx: $i] :
% 0.32/0.47 ( ( Xphi @ Xx )
% 0.32/0.47 & ! [Xy: $i] :
% 0.32/0.47 ( ( Xphi @ Xy )
% 0.32/0.47 => ( Xx = Xy ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setextAx_type,type,
% 0.32/0.47 setextAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setextAx,definition,
% 0.32/0.47 ( setextAx
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 <=> ( in @ Xx @ B ) )
% 0.32/0.47 => ( A = B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptyset_type,type,
% 0.32/0.47 emptyset: $i ).
% 0.32/0.47
% 0.32/0.47 thf(emptysetAx_type,type,
% 0.32/0.47 emptysetAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptysetAx,definition,
% 0.32/0.47 ( emptysetAx
% 0.32/0.47 = ( ! [Xx: $i] :
% 0.32/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoin_type,type,
% 0.32/0.47 setadjoin: $i > $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinAx_type,type,
% 0.32/0.47 setadjoinAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinAx,definition,
% 0.32/0.47 ( setadjoinAx
% 0.32/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.32/0.47 <=> ( ( Xy = Xx )
% 0.32/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(powerset_type,type,
% 0.32/0.47 powerset: $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(powersetAx_type,type,
% 0.32/0.47 powersetAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(powersetAx,definition,
% 0.32/0.47 ( powersetAx
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.32/0.47 <=> ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setunion_type,type,
% 0.32/0.47 setunion: $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(setunionAx_type,type,
% 0.32/0.47 setunionAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setunionAx,definition,
% 0.32/0.47 ( setunionAx
% 0.32/0.47 = ( ! [A: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.32/0.47 <=> ? [B: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 & ( in @ B @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(omega_type,type,
% 0.32/0.47 omega: $i ).
% 0.32/0.47
% 0.32/0.47 thf(omega0Ax_type,type,
% 0.32/0.47 omega0Ax: $o ).
% 0.32/0.47
% 0.32/0.47 thf(omega0Ax,definition,
% 0.32/0.47 ( omega0Ax
% 0.32/0.47 = ( in @ emptyset @ omega ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(omegaSAx_type,type,
% 0.32/0.47 omegaSAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(omegaSAx,definition,
% 0.32/0.47 ( omegaSAx
% 0.32/0.47 = ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ omega )
% 0.32/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(omegaIndAx_type,type,
% 0.32/0.47 omegaIndAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(omegaIndAx,definition,
% 0.32/0.47 ( omegaIndAx
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ( ( in @ emptyset @ A )
% 0.32/0.47 & ! [Xx: $i] :
% 0.32/0.47 ( ( ( in @ Xx @ omega )
% 0.32/0.47 & ( in @ Xx @ A ) )
% 0.32/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ omega )
% 0.32/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(replAx_type,type,
% 0.32/0.47 replAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(replAx,definition,
% 0.32/0.47 ( replAx
% 0.32/0.47 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( exu
% 0.32/0.47 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.32/0.47 => ? [B: $i] :
% 0.32/0.47 ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 <=> ? [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ A )
% 0.32/0.47 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(foundationAx_type,type,
% 0.32/0.47 foundationAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(foundationAx,definition,
% 0.32/0.47 ( foundationAx
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.32/0.47 => ? [B: $i] :
% 0.32/0.47 ( ( in @ B @ A )
% 0.32/0.47 & ~ ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(wellorderingAx_type,type,
% 0.32/0.47 wellorderingAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(wellorderingAx,definition,
% 0.32/0.47 ( wellorderingAx
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ? [B: $i] :
% 0.32/0.47 ( ! [C: $i] :
% 0.32/0.47 ( ( in @ C @ B )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ C )
% 0.32/0.47 => ( in @ Xx @ A ) ) )
% 0.32/0.47 & ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( ( in @ Xx @ A )
% 0.32/0.47 & ( in @ Xy @ A ) )
% 0.32/0.47 => ( ! [C: $i] :
% 0.32/0.47 ( ( in @ C @ B )
% 0.32/0.47 => ( ( in @ Xx @ C )
% 0.32/0.47 <=> ( in @ Xy @ C ) ) )
% 0.32/0.47 => ( Xx = Xy ) ) )
% 0.32/0.47 & ! [C: $i,D: $i] :
% 0.32/0.47 ( ( ( in @ C @ B )
% 0.32/0.47 & ( in @ D @ B ) )
% 0.32/0.47 => ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ C )
% 0.32/0.47 => ( in @ Xx @ D ) )
% 0.32/0.47 | ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ D )
% 0.32/0.47 => ( in @ Xx @ C ) ) ) )
% 0.32/0.47 & ! [C: $i] :
% 0.32/0.47 ( ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ C )
% 0.32/0.47 => ( in @ Xx @ A ) )
% 0.32/0.47 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.32/0.47 => ? [D: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ D @ B )
% 0.32/0.47 & ( in @ Xx @ C )
% 0.32/0.47 & ~ ? [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ D )
% 0.32/0.47 & ( in @ Xy @ C ) )
% 0.32/0.47 & ! [E: $i] :
% 0.32/0.47 ( ( in @ E @ B )
% 0.32/0.47 => ( ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ E )
% 0.32/0.47 => ( in @ Xy @ D ) )
% 0.32/0.47 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(descr_type,type,
% 0.32/0.47 descr: ( $i > $o ) > $i ).
% 0.32/0.47
% 0.32/0.47 thf(descrp_type,type,
% 0.32/0.47 descrp: $o ).
% 0.32/0.47
% 0.32/0.47 thf(descrp,definition,
% 0.32/0.47 ( descrp
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 => ( Xphi
% 0.32/0.47 @ ( descr
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstr_type,type,
% 0.32/0.47 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstrI_type,type,
% 0.32/0.47 dsetconstrI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstrI,definition,
% 0.32/0.47 ( dsetconstrI
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ( Xphi @ Xx )
% 0.32/0.47 => ( in @ Xx
% 0.32/0.47 @ ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstrEL_type,type,
% 0.32/0.47 dsetconstrEL: $o ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstrEL,definition,
% 0.32/0.47 ( dsetconstrEL
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx
% 0.32/0.47 @ ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.32/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstrER_type,type,
% 0.32/0.47 dsetconstrER: $o ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstrER,definition,
% 0.32/0.47 ( dsetconstrER
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx
% 0.32/0.47 @ ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.32/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exuE1_type,type,
% 0.32/0.47 exuE1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exuE1,definition,
% 0.32/0.47 ( exuE1
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 => ? [Xx: $i] :
% 0.32/0.47 ( ( Xphi @ Xx )
% 0.32/0.47 & ! [Xy: $i] :
% 0.32/0.47 ( ( Xphi @ Xy )
% 0.32/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(prop2set_type,type,
% 0.32/0.47 prop2set: $o > $i ).
% 0.32/0.47
% 0.32/0.47 thf(prop2setE_type,type,
% 0.32/0.47 prop2setE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(prop2setE,definition,
% 0.32/0.47 ( prop2setE
% 0.32/0.47 = ( ! [Xphi: $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.32/0.47 => Xphi ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptysetE_type,type,
% 0.32/0.47 emptysetE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptysetE,definition,
% 0.32/0.47 ( emptysetE
% 0.32/0.47 = ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ emptyset )
% 0.32/0.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptysetimpfalse_type,type,
% 0.32/0.47 emptysetimpfalse: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptysetimpfalse,definition,
% 0.32/0.47 ( emptysetimpfalse
% 0.32/0.47 = ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ emptyset )
% 0.32/0.47 => $false ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(notinemptyset_type,type,
% 0.32/0.47 notinemptyset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(notinemptyset,definition,
% 0.32/0.47 ( notinemptyset
% 0.32/0.47 = ( ! [Xx: $i] :
% 0.32/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exuE3e_type,type,
% 0.32/0.47 exuE3e: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exuE3e,definition,
% 0.32/0.47 ( exuE3e
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setext_type,type,
% 0.32/0.47 setext: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setext,definition,
% 0.32/0.47 ( setext
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ Xx @ B ) )
% 0.32/0.47 => ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 => ( in @ Xx @ A ) )
% 0.32/0.47 => ( A = B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptyI_type,type,
% 0.32/0.47 emptyI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptyI,definition,
% 0.32/0.47 ( emptyI
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ~ ( in @ Xx @ A )
% 0.32/0.47 => ( A = emptyset ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(noeltsimpempty_type,type,
% 0.32/0.47 noeltsimpempty: $o ).
% 0.32/0.47
% 0.32/0.47 thf(noeltsimpempty,definition,
% 0.32/0.47 ( noeltsimpempty
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ~ ( in @ Xx @ A )
% 0.32/0.47 => ( A = emptyset ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setbeta_type,type,
% 0.32/0.47 setbeta: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setbeta,definition,
% 0.32/0.47 ( setbeta
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ( in @ Xx
% 0.32/0.47 @ ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.32/0.47 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(nonempty_type,type,
% 0.32/0.47 nonempty: $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(nonempty,definition,
% 0.32/0.47 ( nonempty
% 0.32/0.47 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(nonemptyE1_type,type,
% 0.32/0.47 nonemptyE1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(nonemptyE1,definition,
% 0.32/0.47 ( nonemptyE1
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ( nonempty @ A )
% 0.32/0.47 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(nonemptyI_type,type,
% 0.32/0.47 nonemptyI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(nonemptyI,definition,
% 0.32/0.47 ( nonemptyI
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ( Xphi @ Xx )
% 0.32/0.47 => ( nonempty
% 0.32/0.47 @ ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(nonemptyI1_type,type,
% 0.32/0.47 nonemptyI1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(nonemptyI1,definition,
% 0.32/0.47 ( nonemptyI1
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.32/0.47 => ( nonempty @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinIL_type,type,
% 0.32/0.47 setadjoinIL: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinIL,definition,
% 0.32/0.47 ( setadjoinIL
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptyinunitempty_type,type,
% 0.32/0.47 emptyinunitempty: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptyinunitempty,definition,
% 0.32/0.47 ( emptyinunitempty
% 0.32/0.47 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinIR_type,type,
% 0.32/0.47 setadjoinIR: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinIR,definition,
% 0.32/0.47 ( setadjoinIR
% 0.32/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ A )
% 0.32/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinE_type,type,
% 0.32/0.47 setadjoinE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinE,definition,
% 0.32/0.47 ( setadjoinE
% 0.32/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.32/0.47 => ! [Xphi: $o] :
% 0.32/0.47 ( ( ( Xy = Xx )
% 0.32/0.47 => Xphi )
% 0.32/0.47 => ( ( ( in @ Xy @ A )
% 0.32/0.47 => Xphi )
% 0.32/0.47 => Xphi ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinOr_type,type,
% 0.32/0.47 setadjoinOr: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinOr,definition,
% 0.32/0.47 ( setadjoinOr
% 0.32/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.32/0.47 => ( ( Xy = Xx )
% 0.32/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setoftrueEq_type,type,
% 0.32/0.47 setoftrueEq: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setoftrueEq,definition,
% 0.32/0.47 ( setoftrueEq
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xx: $i] : $true )
% 0.32/0.47 = A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(powersetI_type,type,
% 0.32/0.47 powersetI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(powersetI,definition,
% 0.32/0.47 ( powersetI
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 => ( in @ Xx @ A ) )
% 0.32/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptyinPowerset_type,type,
% 0.32/0.47 emptyinPowerset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptyinPowerset,definition,
% 0.32/0.47 ( emptyinPowerset
% 0.32/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptyInPowerset_type,type,
% 0.32/0.47 emptyInPowerset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptyInPowerset,definition,
% 0.32/0.47 ( emptyInPowerset
% 0.32/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(powersetE_type,type,
% 0.32/0.47 powersetE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(powersetE,definition,
% 0.32/0.47 ( powersetE
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.32/0.47 => ( ( in @ Xx @ B )
% 0.32/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setunionI_type,type,
% 0.32/0.47 setunionI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setunionI,definition,
% 0.32/0.47 ( setunionI
% 0.32/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 => ( ( in @ B @ A )
% 0.32/0.47 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setunionE_type,type,
% 0.32/0.47 setunionE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setunionE,definition,
% 0.32/0.47 ( setunionE
% 0.32/0.47 = ( ! [A: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.32/0.47 => ! [Xphi: $o] :
% 0.32/0.47 ( ! [B: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 => ( ( in @ B @ A )
% 0.32/0.47 => Xphi ) )
% 0.32/0.47 => Xphi ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subPowSU_type,type,
% 0.32/0.47 subPowSU: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subPowSU,definition,
% 0.32/0.47 ( subPowSU
% 0.32/0.47 = ( ! [A: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exuE2_type,type,
% 0.32/0.47 exuE2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exuE2,definition,
% 0.32/0.47 ( exuE2
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 => ? [Xx: $i] :
% 0.32/0.47 ! [Xy: $i] :
% 0.32/0.47 ( ( Xphi @ Xy )
% 0.32/0.47 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(nonemptyImpWitness_type,type,
% 0.32/0.47 nonemptyImpWitness: $o ).
% 0.32/0.47
% 0.32/0.47 thf(nonemptyImpWitness,definition,
% 0.32/0.47 ( nonemptyImpWitness
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ( nonempty @ A )
% 0.32/0.47 => ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & $true ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(uniqinunit_type,type,
% 0.32/0.47 uniqinunit: $o ).
% 0.32/0.47
% 0.32/0.47 thf(uniqinunit,definition,
% 0.32/0.47 ( uniqinunit
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.32/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(notinsingleton_type,type,
% 0.32/0.47 notinsingleton: $o ).
% 0.32/0.47
% 0.32/0.47 thf(notinsingleton,definition,
% 0.32/0.47 ( notinsingleton
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xx != Xy )
% 0.32/0.47 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(eqinunit_type,type,
% 0.32/0.47 eqinunit: $o ).
% 0.32/0.47
% 0.32/0.47 thf(eqinunit,definition,
% 0.32/0.47 ( eqinunit
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xx = Xy )
% 0.32/0.47 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(singletonsswitch_type,type,
% 0.32/0.47 singletonsswitch: $o ).
% 0.32/0.47
% 0.32/0.47 thf(singletonsswitch,definition,
% 0.32/0.47 ( singletonsswitch
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.32/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(upairsetE_type,type,
% 0.32/0.47 upairsetE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(upairsetE,definition,
% 0.32/0.47 ( upairsetE
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.32/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.32/0.47 => ( ( Xz = Xx )
% 0.32/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(upairsetIL_type,type,
% 0.32/0.47 upairsetIL: $o ).
% 0.32/0.47
% 0.32/0.47 thf(upairsetIL,definition,
% 0.32/0.47 ( upairsetIL
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(upairsetIR_type,type,
% 0.32/0.47 upairsetIR: $o ).
% 0.32/0.47
% 0.32/0.47 thf(upairsetIR,definition,
% 0.32/0.47 ( upairsetIR
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptyE1_type,type,
% 0.32/0.47 emptyE1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptyE1,definition,
% 0.32/0.47 ( emptyE1
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ( Xphi @ Xx ) )
% 0.32/0.47 => ( ( ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 = emptyset )
% 0.32/0.47 => $false ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(vacuousDall_type,type,
% 0.32/0.47 vacuousDall: $o ).
% 0.32/0.47
% 0.32/0.47 thf(vacuousDall,definition,
% 0.32/0.47 ( vacuousDall
% 0.32/0.47 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ emptyset )
% 0.32/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(quantDeMorgan1_type,type,
% 0.32/0.47 quantDeMorgan1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(quantDeMorgan1,definition,
% 0.32/0.47 ( quantDeMorgan1
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ~ ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( Xphi @ Xx ) )
% 0.32/0.47 => ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(quantDeMorgan2_type,type,
% 0.32/0.47 quantDeMorgan2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(quantDeMorgan2,definition,
% 0.32/0.47 ( quantDeMorgan2
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ~ ( Xphi @ Xx ) )
% 0.32/0.47 => ~ ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(quantDeMorgan3_type,type,
% 0.32/0.47 quantDeMorgan3: $o ).
% 0.32/0.47
% 0.32/0.47 thf(quantDeMorgan3,definition,
% 0.32/0.47 ( quantDeMorgan3
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ~ ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ( Xphi @ Xx ) )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(quantDeMorgan4_type,type,
% 0.32/0.47 quantDeMorgan4: $o ).
% 0.32/0.47
% 0.32/0.47 thf(quantDeMorgan4,definition,
% 0.32/0.47 ( quantDeMorgan4
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ~ ( Xphi @ Xx ) )
% 0.32/0.47 => ~ ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(prop2setI_type,type,
% 0.32/0.47 prop2setI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(prop2setI,definition,
% 0.32/0.47 ( prop2setI
% 0.32/0.47 = ( ! [Xphi: $o] :
% 0.32/0.47 ( Xphi
% 0.32/0.47 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(set2prop_type,type,
% 0.32/0.47 set2prop: $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(prop2set2propI_type,type,
% 0.32/0.47 prop2set2propI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(prop2set2propI,definition,
% 0.32/0.47 ( prop2set2propI
% 0.32/0.47 = ( ! [Xphi: $o] :
% 0.32/0.47 ( Xphi
% 0.32/0.47 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(notdexE_type,type,
% 0.32/0.47 notdexE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(notdexE,definition,
% 0.32/0.47 ( notdexE
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ~ ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ( Xphi @ Xx ) )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(notdallE_type,type,
% 0.32/0.47 notdallE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(notdallE,definition,
% 0.32/0.47 ( notdallE
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ~ ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( Xphi @ Xx ) )
% 0.32/0.47 => ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exuI1_type,type,
% 0.32/0.47 exuI1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exuI1,definition,
% 0.32/0.47 ( exuI1
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ? [Xx: $i] :
% 0.32/0.47 ( ( Xphi @ Xx )
% 0.32/0.47 & ! [Xy: $i] :
% 0.32/0.47 ( ( Xphi @ Xy )
% 0.32/0.47 => ( Xx = Xy ) ) )
% 0.32/0.47 => ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exuI3_type,type,
% 0.32/0.47 exuI3: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exuI3,definition,
% 0.32/0.47 ( exuI3
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.32/0.47 => ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xphi @ Xx )
% 0.32/0.47 => ( ( Xphi @ Xy )
% 0.32/0.47 => ( Xx = Xy ) ) )
% 0.32/0.47 => ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exuI2_type,type,
% 0.32/0.47 exuI2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exuI2,definition,
% 0.32/0.47 ( exuI2
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ? [Xx: $i] :
% 0.32/0.47 ! [Xy: $i] :
% 0.32/0.47 ( ( Xphi @ Xy )
% 0.32/0.47 <=> ( Xy = Xx ) )
% 0.32/0.47 => ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(inCongP_type,type,
% 0.32/0.47 inCongP: $o ).
% 0.32/0.47
% 0.32/0.47 thf(inCongP,definition,
% 0.32/0.47 ( inCongP
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( A = B )
% 0.32/0.47 => ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xx = Xy )
% 0.32/0.47 => ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(in__Cong_type,type,
% 0.32/0.47 in__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(in__Cong,definition,
% 0.32/0.47 ( in__Cong
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( A = B )
% 0.32/0.47 => ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xx = Xy )
% 0.32/0.47 => ( ( in @ Xx @ A )
% 0.32/0.47 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exuE3u_type,type,
% 0.32/0.47 exuE3u: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exuE3u,definition,
% 0.32/0.47 ( exuE3u
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 => ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xphi @ Xx )
% 0.32/0.47 => ( ( Xphi @ Xy )
% 0.32/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exu__Cong_type,type,
% 0.32/0.47 exu__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exu__Cong,definition,
% 0.32/0.47 ( exu__Cong
% 0.32/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.32/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xx = Xy )
% 0.32/0.47 => ( ( Xphi @ Xx )
% 0.32/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.32/0.47 => ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 <=> ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptyset__Cong_type,type,
% 0.32/0.47 emptyset__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptyset__Cong,definition,
% 0.32/0.47 ( emptyset__Cong
% 0.32/0.47 = ( emptyset = emptyset ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoin__Cong_type,type,
% 0.32/0.47 setadjoin__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoin__Cong,definition,
% 0.32/0.47 ( setadjoin__Cong
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xx = Xy )
% 0.32/0.47 => ! [Xz: $i,Xu: $i] :
% 0.32/0.47 ( ( Xz = Xu )
% 0.32/0.47 => ( ( setadjoin @ Xx @ Xz )
% 0.32/0.47 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(powerset__Cong_type,type,
% 0.32/0.47 powerset__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(powerset__Cong,definition,
% 0.32/0.47 ( powerset__Cong
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( A = B )
% 0.32/0.47 => ( ( powerset @ A )
% 0.32/0.47 = ( powerset @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setunion__Cong_type,type,
% 0.32/0.47 setunion__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setunion__Cong,definition,
% 0.32/0.47 ( setunion__Cong
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( A = B )
% 0.32/0.47 => ( ( setunion @ A )
% 0.32/0.47 = ( setunion @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(omega__Cong_type,type,
% 0.32/0.47 omega__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(omega__Cong,definition,
% 0.32/0.47 ( omega__Cong
% 0.32/0.47 = ( omega = omega ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exuEu_type,type,
% 0.32/0.47 exuEu: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exuEu,definition,
% 0.32/0.47 ( exuEu
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 => ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xphi @ Xx )
% 0.32/0.47 => ( ( Xphi @ Xy )
% 0.32/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(descr__Cong_type,type,
% 0.32/0.47 descr__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(descr__Cong,definition,
% 0.32/0.47 ( descr__Cong
% 0.32/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.32/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xx = Xy )
% 0.32/0.47 => ( ( Xphi @ Xx )
% 0.32/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.32/0.47 => ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 => ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.32/0.47 => ( ( descr
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 = ( descr
% 0.32/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstr__Cong_type,type,
% 0.32/0.47 dsetconstr__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstr__Cong,definition,
% 0.32/0.47 ( dsetconstr__Cong
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( A = B )
% 0.32/0.47 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( ( Xx = Xy )
% 0.32/0.47 => ( ( Xphi @ Xx )
% 0.32/0.47 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.32/0.47 => ( ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 = ( dsetconstr @ B
% 0.32/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subset_type,type,
% 0.32/0.47 subset: $i > $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(disjoint_type,type,
% 0.32/0.47 disjoint: $i > $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(setsmeet_type,type,
% 0.32/0.47 setsmeet: $i > $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(subsetI1_type,type,
% 0.32/0.47 subsetI1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subsetI1,definition,
% 0.32/0.47 ( subsetI1
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ Xx @ B ) )
% 0.32/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(eqimpsubset2_type,type,
% 0.32/0.47 eqimpsubset2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(eqimpsubset2,definition,
% 0.32/0.47 ( eqimpsubset2
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( A = B )
% 0.32/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(eqimpsubset1_type,type,
% 0.32/0.47 eqimpsubset1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(eqimpsubset1,definition,
% 0.32/0.47 ( eqimpsubset1
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( A = B )
% 0.32/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subsetI2_type,type,
% 0.32/0.47 subsetI2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subsetI2,definition,
% 0.32/0.47 ( subsetI2
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ Xx @ B ) )
% 0.32/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptysetsubset_type,type,
% 0.32/0.47 emptysetsubset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptysetsubset,definition,
% 0.32/0.47 ( emptysetsubset
% 0.32/0.47 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subsetE_type,type,
% 0.32/0.47 subsetE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subsetE,definition,
% 0.32/0.47 ( subsetE
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( subset @ A @ B )
% 0.32/0.47 => ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subsetE2_type,type,
% 0.32/0.47 subsetE2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subsetE2,definition,
% 0.32/0.47 ( subsetE2
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( subset @ A @ B )
% 0.32/0.47 => ( ~ ( in @ Xx @ B )
% 0.32/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(notsubsetI_type,type,
% 0.32/0.47 notsubsetI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(notsubsetI,definition,
% 0.32/0.47 ( notsubsetI
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ~ ( in @ Xx @ B )
% 0.32/0.47 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(notequalI1_type,type,
% 0.32/0.47 notequalI1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(notequalI1,definition,
% 0.32/0.47 ( notequalI1
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ~ ( subset @ A @ B )
% 0.32/0.47 => ( A != B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(notequalI2_type,type,
% 0.32/0.47 notequalI2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(notequalI2,definition,
% 0.32/0.47 ( notequalI2
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ~ ( in @ Xx @ B )
% 0.32/0.47 => ( A != B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subsetRefl_type,type,
% 0.32/0.47 subsetRefl: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subsetRefl,definition,
% 0.32/0.47 ( subsetRefl
% 0.32/0.47 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subsetTrans_type,type,
% 0.32/0.47 subsetTrans: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subsetTrans,definition,
% 0.32/0.47 ( subsetTrans
% 0.32/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.32/0.47 ( ( subset @ A @ B )
% 0.32/0.47 => ( ( subset @ B @ C )
% 0.32/0.47 => ( subset @ A @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinSub_type,type,
% 0.32/0.47 setadjoinSub: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinSub,definition,
% 0.32/0.47 ( setadjoinSub
% 0.32/0.47 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinSub2,conjecture,
% 0.32/0.47 ( setextAx
% 0.32/0.47 => ( emptysetAx
% 0.32/0.47 => ( setadjoinAx
% 0.32/0.47 => ( powersetAx
% 0.32/0.47 => ( setunionAx
% 0.32/0.47 => ( omega0Ax
% 0.32/0.47 => ( omegaSAx
% 0.32/0.47 => ( omegaIndAx
% 0.32/0.47 => ( replAx
% 0.32/0.47 => ( foundationAx
% 0.32/0.47 => ( wellorderingAx
% 0.32/0.47 => ( descrp
% 0.32/0.47 => ( dsetconstrI
% 0.32/0.47 => ( dsetconstrEL
% 0.32/0.47 => ( dsetconstrER
% 0.32/0.47 => ( exuE1
% 0.32/0.47 => ( prop2setE
% 0.32/0.47 => ( emptysetE
% 0.32/0.47 => ( emptysetimpfalse
% 0.32/0.47 => ( notinemptyset
% 0.32/0.47 => ( exuE3e
% 0.32/0.47 => ( setext
% 0.32/0.47 => ( emptyI
% 0.32/0.47 => ( noeltsimpempty
% 0.32/0.47 => ( setbeta
% 0.32/0.47 => ( nonemptyE1
% 0.32/0.47 => ( nonemptyI
% 0.32/0.47 => ( nonemptyI1
% 0.32/0.47 => ( setadjoinIL
% 0.32/0.47 => ( emptyinunitempty
% 0.32/0.47 => ( setadjoinIR
% 0.32/0.47 => ( setadjoinE
% 0.32/0.47 => ( setadjoinOr
% 0.32/0.47 => ( setoftrueEq
% 0.32/0.47 => ( powersetI
% 0.32/0.47 => ( emptyinPowerset
% 0.32/0.47 => ( emptyInPowerset
% 0.32/0.47 => ( powersetE
% 0.32/0.47 => ( setunionI
% 0.32/0.47 => ( setunionE
% 0.32/0.47 => ( subPowSU
% 0.32/0.47 => ( exuE2
% 0.32/0.47 => ( nonemptyImpWitness
% 0.32/0.47 => ( uniqinunit
% 0.32/0.47 => ( notinsingleton
% 0.32/0.47 => ( eqinunit
% 0.32/0.47 => ( singletonsswitch
% 0.32/0.47 => ( upairsetE
% 0.32/0.47 => ( upairsetIL
% 0.32/0.47 => ( upairsetIR
% 0.32/0.47 => ( emptyE1
% 0.32/0.47 => ( vacuousDall
% 0.32/0.47 => ( quantDeMorgan1
% 0.32/0.47 => ( quantDeMorgan2
% 0.32/0.47 => ( quantDeMorgan3
% 0.32/0.47 => ( quantDeMorgan4
% 0.32/0.47 => ( prop2setI
% 0.32/0.47 => ( prop2set2propI
% 0.32/0.47 => ( notdexE
% 0.32/0.47 => ( notdallE
% 0.32/0.47 => ( exuI1
% 0.32/0.47 => ( exuI3
% 0.32/0.47 => ( exuI2
% 0.32/0.47 => ( inCongP
% 0.32/0.47 => ( in__Cong
% 0.32/0.47 => ( exuE3u
% 0.32/0.47 => ( exu__Cong
% 0.32/0.47 => ( emptyset__Cong
% 0.32/0.47 => ( setadjoin__Cong
% 0.32/0.47 => ( powerset__Cong
% 0.32/0.47 => ( setunion__Cong
% 0.32/0.47 => ( omega__Cong
% 0.32/0.47 => ( exuEu
% 0.32/0.47 => ( descr__Cong
% 0.32/0.47 => ( dsetconstr__Cong
% 0.32/0.47 => ( subsetI1
% 0.32/0.47 => ( eqimpsubset2
% 0.32/0.47 => ( eqimpsubset1
% 0.32/0.47 => ( subsetI2
% 0.32/0.47 => ( emptysetsubset
% 0.32/0.47 => ( subsetE
% 0.32/0.47 => ( subsetE2
% 0.32/0.47 => ( notsubsetI
% 0.32/0.47 => ( notequalI1
% 0.32/0.47 => ( notequalI2
% 0.32/0.47 => ( subsetRefl
% 0.32/0.47 => ( subsetTrans
% 0.32/0.47 => ( setadjoinSub
% 0.32/0.47 => ! [A: $i,Xx: $i,B: $i] :
% 0.32/0.47 ( ( subset @ A @ B )
% 0.32/0.47 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 %------------------------------------------------------------------------------
% 0.32/0.47 ------------------------
% 2.11/1.43 ---- Embedded in THF ---
% 2.11/1.44 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.11/1.44 %%% Generated on Tue Feb 24 12:27:09 EST 2026
% 2.11/1.44 %%% using '$$dhol' embedding, version 1.3.0.
% 2.11/1.44 %%% Logic specification used:
% 2.11/1.44 %%% thf(spec, logic, $$dhol).
% 2.11/1.44
% 2.11/1.44 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.pvugr8aAKv/DTF2DTF16254.p
% See solution above
% 2.11/1.44 ------------------------
% 2.11/1.46 ---- Cleaned THF ---
% 2.11/1.46 thf(in_type,type,
% 2.11/1.46 in: $i > $i > $o ).
% 2.11/1.46
% 2.11/1.46 thf(exu_type,type,
% 2.11/1.46 exu: ( $i > $o ) > $o ).
% 2.11/1.46
% 2.11/1.46 thf(exu,definition,
% 2.11/1.46 ( exu
% 2.11/1.46 = ( ^ [Xphi: $i > $o] :
% 2.11/1.46 ? [Xx: $i] :
% 2.11/1.46 ( ( Xphi @ Xx )
% 2.11/1.46 & ! [Xy: $i] :
% 2.11/1.46 ( ( Xphi @ Xy )
% 2.11/1.46 => ( Xx = Xy ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(setextAx_type,type,
% 2.11/1.46 setextAx: $o ).
% 2.11/1.46
% 2.11/1.46 thf(setextAx,definition,
% 2.11/1.46 ( setextAx
% 2.11/1.46 = ( ! [A: $i,B: $i] :
% 2.11/1.46 ( ! [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 <=> ( in @ Xx @ B ) )
% 2.11/1.46 => ( A = B ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(emptyset_type,type,
% 2.11/1.46 emptyset: $i ).
% 2.11/1.46
% 2.11/1.46 thf(emptysetAx_type,type,
% 2.11/1.46 emptysetAx: $o ).
% 2.11/1.46
% 2.11/1.46 thf(emptysetAx,definition,
% 2.11/1.46 ( emptysetAx
% 2.11/1.46 = ( ! [Xx: $i] :
% 2.11/1.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(setadjoin_type,type,
% 2.11/1.46 setadjoin: $i > $i > $i ).
% 2.11/1.46
% 2.11/1.46 thf(setadjoinAx_type,type,
% 2.11/1.46 setadjoinAx: $o ).
% 2.11/1.46
% 2.11/1.46 thf(setadjoinAx,definition,
% 2.11/1.46 ( setadjoinAx
% 2.11/1.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.11/1.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.11/1.46 <=> ( ( Xy = Xx )
% 2.11/1.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(powerset_type,type,
% 2.11/1.46 powerset: $i > $i ).
% 2.11/1.46
% 2.11/1.46 thf(powersetAx_type,type,
% 2.11/1.46 powersetAx: $o ).
% 2.11/1.46
% 2.11/1.46 thf(powersetAx,definition,
% 2.11/1.46 ( powersetAx
% 2.11/1.46 = ( ! [A: $i,B: $i] :
% 2.11/1.46 ( ( in @ B @ ( powerset @ A ) )
% 2.11/1.46 <=> ! [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ B )
% 2.11/1.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(setunion_type,type,
% 2.11/1.46 setunion: $i > $i ).
% 2.11/1.46
% 2.11/1.46 thf(setunionAx_type,type,
% 2.11/1.46 setunionAx: $o ).
% 2.11/1.46
% 2.11/1.46 thf(setunionAx,definition,
% 2.11/1.46 ( setunionAx
% 2.11/1.46 = ( ! [A: $i,Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ ( setunion @ A ) )
% 2.11/1.46 <=> ? [B: $i] :
% 2.11/1.46 ( ( in @ Xx @ B )
% 2.11/1.46 & ( in @ B @ A ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(omega_type,type,
% 2.11/1.46 omega: $i ).
% 2.11/1.46
% 2.11/1.46 thf(omega0Ax_type,type,
% 2.11/1.46 omega0Ax: $o ).
% 2.11/1.46
% 2.11/1.46 thf(omega0Ax,definition,
% 2.11/1.46 ( omega0Ax
% 2.11/1.46 = ( in @ emptyset @ omega ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(omegaSAx_type,type,
% 2.11/1.46 omegaSAx: $o ).
% 2.11/1.46
% 2.11/1.46 thf(omegaSAx,definition,
% 2.11/1.46 ( omegaSAx
% 2.11/1.46 = ( ! [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ omega )
% 2.11/1.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(omegaIndAx_type,type,
% 2.11/1.46 omegaIndAx: $o ).
% 2.11/1.46
% 2.11/1.46 thf(omegaIndAx,definition,
% 2.11/1.46 ( omegaIndAx
% 2.11/1.46 = ( ! [A: $i] :
% 2.11/1.46 ( ( ( in @ emptyset @ A )
% 2.11/1.46 & ! [Xx: $i] :
% 2.11/1.46 ( ( ( in @ Xx @ omega )
% 2.11/1.46 & ( in @ Xx @ A ) )
% 2.11/1.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.11/1.46 => ! [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ omega )
% 2.11/1.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(replAx_type,type,
% 2.11/1.46 replAx: $o ).
% 2.11/1.46
% 2.11/1.46 thf(replAx,definition,
% 2.11/1.46 ( replAx
% 2.11/1.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.11/1.46 ( ! [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 => ( exu
% 2.11/1.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.11/1.46 => ? [B: $i] :
% 2.11/1.46 ! [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ B )
% 2.11/1.46 <=> ? [Xy: $i] :
% 2.11/1.46 ( ( in @ Xy @ A )
% 2.11/1.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(foundationAx_type,type,
% 2.11/1.46 foundationAx: $o ).
% 2.11/1.46
% 2.11/1.46 thf(foundationAx,definition,
% 2.11/1.46 ( foundationAx
% 2.11/1.46 = ( ! [A: $i] :
% 2.11/1.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.11/1.46 => ? [B: $i] :
% 2.11/1.46 ( ( in @ B @ A )
% 2.11/1.46 & ~ ? [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ B )
% 2.11/1.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(wellorderingAx_type,type,
% 2.11/1.46 wellorderingAx: $o ).
% 2.11/1.46
% 2.11/1.46 thf(wellorderingAx,definition,
% 2.11/1.46 ( wellorderingAx
% 2.11/1.46 = ( ! [A: $i] :
% 2.11/1.46 ? [B: $i] :
% 2.11/1.46 ( ! [C: $i] :
% 2.11/1.46 ( ( in @ C @ B )
% 2.11/1.46 => ! [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ C )
% 2.11/1.46 => ( in @ Xx @ A ) ) )
% 2.11/1.46 & ! [Xx: $i,Xy: $i] :
% 2.11/1.46 ( ( ( in @ Xx @ A )
% 2.11/1.46 & ( in @ Xy @ A ) )
% 2.11/1.46 => ( ! [C: $i] :
% 2.11/1.46 ( ( in @ C @ B )
% 2.11/1.46 => ( ( in @ Xx @ C )
% 2.11/1.46 <=> ( in @ Xy @ C ) ) )
% 2.11/1.46 => ( Xx = Xy ) ) )
% 2.11/1.46 & ! [C: $i,D: $i] :
% 2.11/1.46 ( ( ( in @ C @ B )
% 2.11/1.46 & ( in @ D @ B ) )
% 2.11/1.46 => ( ! [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ C )
% 2.11/1.46 => ( in @ Xx @ D ) )
% 2.11/1.46 | ! [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ D )
% 2.11/1.46 => ( in @ Xx @ C ) ) ) )
% 2.11/1.46 & ! [C: $i] :
% 2.11/1.46 ( ( ! [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ C )
% 2.11/1.46 => ( in @ Xx @ A ) )
% 2.11/1.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.11/1.46 => ? [D: $i,Xx: $i] :
% 2.11/1.46 ( ( in @ D @ B )
% 2.11/1.46 & ( in @ Xx @ C )
% 2.11/1.46 & ~ ? [Xy: $i] :
% 2.11/1.46 ( ( in @ Xy @ D )
% 2.11/1.46 & ( in @ Xy @ C ) )
% 2.11/1.46 & ! [E: $i] :
% 2.11/1.46 ( ( in @ E @ B )
% 2.11/1.46 => ( ! [Xy: $i] :
% 2.11/1.46 ( ( in @ Xy @ E )
% 2.11/1.46 => ( in @ Xy @ D ) )
% 2.11/1.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(descr_type,type,
% 2.11/1.46 descr: ( $i > $o ) > $i ).
% 2.11/1.46
% 2.11/1.46 thf(descrp_type,type,
% 2.11/1.46 descrp: $o ).
% 2.11/1.46
% 2.11/1.46 thf(descrp,definition,
% 2.11/1.46 ( descrp
% 2.11/1.46 = ( ! [Xphi: $i > $o] :
% 2.11/1.46 ( ( exu
% 2.11/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.11/1.46 => ( Xphi
% 2.11/1.46 @ ( descr
% 2.11/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(dsetconstr_type,type,
% 2.11/1.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.11/1.46
% 2.11/1.46 thf(dsetconstrI_type,type,
% 2.11/1.46 dsetconstrI: $o ).
% 2.11/1.46
% 2.11/1.46 thf(dsetconstrI,definition,
% 2.11/1.46 ( dsetconstrI
% 2.11/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 => ( ( Xphi @ Xx )
% 2.11/1.46 => ( in @ Xx
% 2.11/1.46 @ ( dsetconstr @ A
% 2.11/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(dsetconstrEL_type,type,
% 2.11/1.46 dsetconstrEL: $o ).
% 2.11/1.46
% 2.11/1.46 thf(dsetconstrEL,definition,
% 2.11/1.46 ( dsetconstrEL
% 2.11/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.11/1.46 ( ( in @ Xx
% 2.11/1.46 @ ( dsetconstr @ A
% 2.11/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.11/1.46 => ( in @ Xx @ A ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(dsetconstrER_type,type,
% 2.11/1.46 dsetconstrER: $o ).
% 2.11/1.46
% 2.11/1.46 thf(dsetconstrER,definition,
% 2.11/1.46 ( dsetconstrER
% 2.11/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.11/1.46 ( ( in @ Xx
% 2.11/1.46 @ ( dsetconstr @ A
% 2.11/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.11/1.46 => ( Xphi @ Xx ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(exuE1_type,type,
% 2.11/1.46 exuE1: $o ).
% 2.11/1.46
% 2.11/1.46 thf(exuE1,definition,
% 2.11/1.46 ( exuE1
% 2.11/1.46 = ( ! [Xphi: $i > $o] :
% 2.11/1.46 ( ( exu
% 2.11/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.11/1.46 => ? [Xx: $i] :
% 2.11/1.46 ( ( Xphi @ Xx )
% 2.11/1.46 & ! [Xy: $i] :
% 2.11/1.46 ( ( Xphi @ Xy )
% 2.11/1.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(prop2set_type,type,
% 2.11/1.46 prop2set: $o > $i ).
% 2.11/1.46
% 2.11/1.46 thf(prop2setE_type,type,
% 2.11/1.46 prop2setE: $o ).
% 2.11/1.46
% 2.11/1.46 thf(prop2setE,definition,
% 2.11/1.46 ( prop2setE
% 2.11/1.46 = ( ! [Xphi: $o,Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.11/1.46 => Xphi ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(emptysetE_type,type,
% 2.11/1.46 emptysetE: $o ).
% 2.11/1.46
% 2.11/1.46 thf(emptysetE,definition,
% 2.11/1.46 ( emptysetE
% 2.11/1.46 = ( ! [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ emptyset )
% 2.11/1.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(emptysetimpfalse_type,type,
% 2.11/1.46 emptysetimpfalse: $o ).
% 2.11/1.46
% 2.11/1.46 thf(emptysetimpfalse,definition,
% 2.11/1.46 ( emptysetimpfalse
% 2.11/1.46 = ( ! [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ emptyset )
% 2.11/1.46 => $false ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(notinemptyset_type,type,
% 2.11/1.46 notinemptyset: $o ).
% 2.11/1.46
% 2.11/1.46 thf(notinemptyset,definition,
% 2.11/1.46 ( notinemptyset
% 2.11/1.46 = ( ! [Xx: $i] :
% 2.11/1.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(exuE3e_type,type,
% 2.11/1.46 exuE3e: $o ).
% 2.11/1.46
% 2.11/1.46 thf(exuE3e,definition,
% 2.11/1.46 ( exuE3e
% 2.11/1.46 = ( ! [Xphi: $i > $o] :
% 2.11/1.46 ( ( exu
% 2.11/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.11/1.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(setext_type,type,
% 2.11/1.46 setext: $o ).
% 2.11/1.46
% 2.11/1.46 thf(setext,definition,
% 2.11/1.46 ( setext
% 2.11/1.46 = ( ! [A: $i,B: $i] :
% 2.11/1.46 ( ! [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 => ( in @ Xx @ B ) )
% 2.11/1.46 => ( ! [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ B )
% 2.11/1.46 => ( in @ Xx @ A ) )
% 2.11/1.46 => ( A = B ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(emptyI_type,type,
% 2.11/1.46 emptyI: $o ).
% 2.11/1.46
% 2.11/1.46 thf(emptyI,definition,
% 2.11/1.46 ( emptyI
% 2.11/1.46 = ( ! [A: $i] :
% 2.11/1.46 ( ! [Xx: $i] :
% 2.11/1.46 ~ ( in @ Xx @ A )
% 2.11/1.46 => ( A = emptyset ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(noeltsimpempty_type,type,
% 2.11/1.46 noeltsimpempty: $o ).
% 2.11/1.46
% 2.11/1.46 thf(noeltsimpempty,definition,
% 2.11/1.46 ( noeltsimpempty
% 2.11/1.46 = ( ! [A: $i] :
% 2.11/1.46 ( ! [Xx: $i] :
% 2.11/1.46 ~ ( in @ Xx @ A )
% 2.11/1.46 => ( A = emptyset ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(setbeta_type,type,
% 2.11/1.46 setbeta: $o ).
% 2.11/1.46
% 2.11/1.46 thf(setbeta,definition,
% 2.11/1.46 ( setbeta
% 2.11/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 => ( ( in @ Xx
% 2.11/1.46 @ ( dsetconstr @ A
% 2.11/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.11/1.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(nonempty_type,type,
% 2.11/1.46 nonempty: $i > $o ).
% 2.11/1.46
% 2.11/1.46 thf(nonempty,definition,
% 2.11/1.46 ( nonempty
% 2.11/1.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(nonemptyE1_type,type,
% 2.11/1.46 nonemptyE1: $o ).
% 2.11/1.46
% 2.11/1.46 thf(nonemptyE1,definition,
% 2.11/1.46 ( nonemptyE1
% 2.11/1.46 = ( ! [A: $i] :
% 2.11/1.46 ( ( nonempty @ A )
% 2.11/1.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(nonemptyI_type,type,
% 2.11/1.46 nonemptyI: $o ).
% 2.11/1.46
% 2.11/1.46 thf(nonemptyI,definition,
% 2.11/1.46 ( nonemptyI
% 2.11/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 => ( ( Xphi @ Xx )
% 2.11/1.46 => ( nonempty
% 2.11/1.46 @ ( dsetconstr @ A
% 2.11/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(nonemptyI1_type,type,
% 2.11/1.46 nonemptyI1: $o ).
% 2.11/1.46
% 2.11/1.46 thf(nonemptyI1,definition,
% 2.11/1.46 ( nonemptyI1
% 2.11/1.46 = ( ! [A: $i] :
% 2.11/1.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.11/1.46 => ( nonempty @ A ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(setadjoinIL_type,type,
% 2.11/1.46 setadjoinIL: $o ).
% 2.11/1.46
% 2.11/1.46 thf(setadjoinIL,definition,
% 2.11/1.46 ( setadjoinIL
% 2.11/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(emptyinunitempty_type,type,
% 2.11/1.46 emptyinunitempty: $o ).
% 2.11/1.46
% 2.11/1.46 thf(emptyinunitempty,definition,
% 2.11/1.46 ( emptyinunitempty
% 2.11/1.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(setadjoinIR_type,type,
% 2.11/1.46 setadjoinIR: $o ).
% 2.11/1.46
% 2.11/1.46 thf(setadjoinIR,definition,
% 2.11/1.46 ( setadjoinIR
% 2.11/1.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.11/1.46 ( ( in @ Xy @ A )
% 2.11/1.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(setadjoinE_type,type,
% 2.11/1.46 setadjoinE: $o ).
% 2.11/1.46
% 2.11/1.46 thf(setadjoinE,definition,
% 2.11/1.46 ( setadjoinE
% 2.11/1.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.11/1.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.11/1.46 => ! [Xphi: $o] :
% 2.11/1.46 ( ( ( Xy = Xx )
% 2.11/1.46 => Xphi )
% 2.11/1.46 => ( ( ( in @ Xy @ A )
% 2.11/1.46 => Xphi )
% 2.11/1.46 => Xphi ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(setadjoinOr_type,type,
% 2.11/1.46 setadjoinOr: $o ).
% 2.11/1.46
% 2.11/1.46 thf(setadjoinOr,definition,
% 2.11/1.46 ( setadjoinOr
% 2.11/1.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.11/1.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.11/1.46 => ( ( Xy = Xx )
% 2.11/1.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(setoftrueEq_type,type,
% 2.11/1.46 setoftrueEq: $o ).
% 2.11/1.46
% 2.11/1.46 thf(setoftrueEq,definition,
% 2.11/1.46 ( setoftrueEq
% 2.11/1.46 = ( ! [A: $i] :
% 2.11/1.46 ( ( dsetconstr @ A
% 2.11/1.46 @ ^ [Xx: $i] : $true )
% 2.11/1.46 = A ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(powersetI_type,type,
% 2.11/1.46 powersetI: $o ).
% 2.11/1.46
% 2.11/1.46 thf(powersetI,definition,
% 2.11/1.46 ( powersetI
% 2.11/1.46 = ( ! [A: $i,B: $i] :
% 2.11/1.46 ( ! [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ B )
% 2.11/1.46 => ( in @ Xx @ A ) )
% 2.11/1.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(emptyinPowerset_type,type,
% 2.11/1.46 emptyinPowerset: $o ).
% 2.11/1.46
% 2.11/1.46 thf(emptyinPowerset,definition,
% 2.11/1.46 ( emptyinPowerset
% 2.11/1.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(emptyInPowerset_type,type,
% 2.11/1.46 emptyInPowerset: $o ).
% 2.11/1.46
% 2.11/1.46 thf(emptyInPowerset,definition,
% 2.11/1.46 ( emptyInPowerset
% 2.11/1.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(powersetE_type,type,
% 2.11/1.46 powersetE: $o ).
% 2.11/1.46
% 2.11/1.46 thf(powersetE,definition,
% 2.11/1.46 ( powersetE
% 2.11/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.11/1.46 ( ( in @ B @ ( powerset @ A ) )
% 2.11/1.46 => ( ( in @ Xx @ B )
% 2.11/1.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(setunionI_type,type,
% 2.11/1.46 setunionI: $o ).
% 2.11/1.46
% 2.11/1.46 thf(setunionI,definition,
% 2.11/1.46 ( setunionI
% 2.11/1.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.11/1.46 ( ( in @ Xx @ B )
% 2.11/1.46 => ( ( in @ B @ A )
% 2.11/1.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(setunionE_type,type,
% 2.11/1.46 setunionE: $o ).
% 2.11/1.46
% 2.11/1.46 thf(setunionE,definition,
% 2.11/1.46 ( setunionE
% 2.11/1.46 = ( ! [A: $i,Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ ( setunion @ A ) )
% 2.11/1.46 => ! [Xphi: $o] :
% 2.11/1.46 ( ! [B: $i] :
% 2.11/1.46 ( ( in @ Xx @ B )
% 2.11/1.46 => ( ( in @ B @ A )
% 2.11/1.46 => Xphi ) )
% 2.11/1.46 => Xphi ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(subPowSU_type,type,
% 2.11/1.46 subPowSU: $o ).
% 2.11/1.46
% 2.11/1.46 thf(subPowSU,definition,
% 2.11/1.46 ( subPowSU
% 2.11/1.46 = ( ! [A: $i,Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(exuE2_type,type,
% 2.11/1.46 exuE2: $o ).
% 2.11/1.46
% 2.11/1.46 thf(exuE2,definition,
% 2.11/1.46 ( exuE2
% 2.11/1.46 = ( ! [Xphi: $i > $o] :
% 2.11/1.46 ( ( exu
% 2.11/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.11/1.46 => ? [Xx: $i] :
% 2.11/1.46 ! [Xy: $i] :
% 2.11/1.46 ( ( Xphi @ Xy )
% 2.11/1.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(nonemptyImpWitness_type,type,
% 2.11/1.46 nonemptyImpWitness: $o ).
% 2.11/1.46
% 2.11/1.46 thf(nonemptyImpWitness,definition,
% 2.11/1.46 ( nonemptyImpWitness
% 2.11/1.46 = ( ! [A: $i] :
% 2.11/1.46 ( ( nonempty @ A )
% 2.11/1.46 => ? [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 & $true ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(uniqinunit_type,type,
% 2.11/1.46 uniqinunit: $o ).
% 2.11/1.46
% 2.11/1.46 thf(uniqinunit,definition,
% 2.11/1.46 ( uniqinunit
% 2.11/1.46 = ( ! [Xx: $i,Xy: $i] :
% 2.11/1.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.11/1.46 => ( Xx = Xy ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(notinsingleton_type,type,
% 2.11/1.46 notinsingleton: $o ).
% 2.11/1.46
% 2.11/1.46 thf(notinsingleton,definition,
% 2.11/1.46 ( notinsingleton
% 2.11/1.46 = ( ! [Xx: $i,Xy: $i] :
% 2.11/1.46 ( ( Xx != Xy )
% 2.11/1.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(eqinunit_type,type,
% 2.11/1.46 eqinunit: $o ).
% 2.11/1.46
% 2.11/1.46 thf(eqinunit,definition,
% 2.11/1.46 ( eqinunit
% 2.11/1.46 = ( ! [Xx: $i,Xy: $i] :
% 2.11/1.46 ( ( Xx = Xy )
% 2.11/1.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(singletonsswitch_type,type,
% 2.11/1.46 singletonsswitch: $o ).
% 2.11/1.46
% 2.11/1.46 thf(singletonsswitch,definition,
% 2.11/1.46 ( singletonsswitch
% 2.11/1.46 = ( ! [Xx: $i,Xy: $i] :
% 2.11/1.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.11/1.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(upairsetE_type,type,
% 2.11/1.46 upairsetE: $o ).
% 2.11/1.46
% 2.11/1.46 thf(upairsetE,definition,
% 2.11/1.46 ( upairsetE
% 2.11/1.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.11/1.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.11/1.46 => ( ( Xz = Xx )
% 2.11/1.46 | ( Xz = Xy ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(upairsetIL_type,type,
% 2.11/1.46 upairsetIL: $o ).
% 2.11/1.46
% 2.11/1.46 thf(upairsetIL,definition,
% 2.11/1.46 ( upairsetIL
% 2.11/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(upairsetIR_type,type,
% 2.11/1.46 upairsetIR: $o ).
% 2.11/1.46
% 2.11/1.46 thf(upairsetIR,definition,
% 2.11/1.46 ( upairsetIR
% 2.11/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(emptyE1_type,type,
% 2.11/1.46 emptyE1: $o ).
% 2.11/1.46
% 2.11/1.46 thf(emptyE1,definition,
% 2.11/1.46 ( emptyE1
% 2.11/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.11/1.46 ( ? [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 & ( Xphi @ Xx ) )
% 2.11/1.46 => ( ( ( dsetconstr @ A
% 2.11/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.11/1.46 = emptyset )
% 2.11/1.46 => $false ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(vacuousDall_type,type,
% 2.11/1.46 vacuousDall: $o ).
% 2.11/1.46
% 2.11/1.46 thf(vacuousDall,definition,
% 2.11/1.46 ( vacuousDall
% 2.11/1.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ emptyset )
% 2.11/1.46 => ( Xphi @ Xx ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(quantDeMorgan1_type,type,
% 2.11/1.46 quantDeMorgan1: $o ).
% 2.11/1.46
% 2.11/1.46 thf(quantDeMorgan1,definition,
% 2.11/1.46 ( quantDeMorgan1
% 2.11/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.11/1.46 ( ~ ! [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 => ( Xphi @ Xx ) )
% 2.11/1.46 => ? [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(quantDeMorgan2_type,type,
% 2.11/1.46 quantDeMorgan2: $o ).
% 2.11/1.46
% 2.11/1.46 thf(quantDeMorgan2,definition,
% 2.11/1.46 ( quantDeMorgan2
% 2.11/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.11/1.46 ( ! [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 => ~ ( Xphi @ Xx ) )
% 2.11/1.46 => ~ ? [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(quantDeMorgan3_type,type,
% 2.11/1.46 quantDeMorgan3: $o ).
% 2.11/1.46
% 2.11/1.46 thf(quantDeMorgan3,definition,
% 2.11/1.46 ( quantDeMorgan3
% 2.11/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.11/1.46 ( ~ ? [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 & ( Xphi @ Xx ) )
% 2.11/1.46 => ! [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(quantDeMorgan4_type,type,
% 2.11/1.46 quantDeMorgan4: $o ).
% 2.11/1.46
% 2.11/1.46 thf(quantDeMorgan4,definition,
% 2.11/1.46 ( quantDeMorgan4
% 2.11/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.11/1.46 ( ? [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 & ~ ( Xphi @ Xx ) )
% 2.11/1.46 => ~ ! [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(prop2setI_type,type,
% 2.11/1.46 prop2setI: $o ).
% 2.11/1.46
% 2.11/1.46 thf(prop2setI,definition,
% 2.11/1.46 ( prop2setI
% 2.11/1.46 = ( ! [Xphi: $o] :
% 2.11/1.46 ( Xphi
% 2.11/1.46 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(set2prop_type,type,
% 2.11/1.46 set2prop: $i > $o ).
% 2.11/1.46
% 2.11/1.46 thf(prop2set2propI_type,type,
% 2.11/1.46 prop2set2propI: $o ).
% 2.11/1.46
% 2.11/1.46 thf(prop2set2propI,definition,
% 2.11/1.46 ( prop2set2propI
% 2.11/1.46 = ( ! [Xphi: $o] :
% 2.11/1.46 ( Xphi
% 2.11/1.46 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(notdexE_type,type,
% 2.11/1.46 notdexE: $o ).
% 2.11/1.46
% 2.11/1.46 thf(notdexE,definition,
% 2.11/1.46 ( notdexE
% 2.11/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.11/1.46 ( ~ ? [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 & ( Xphi @ Xx ) )
% 2.11/1.46 => ! [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(notdallE_type,type,
% 2.11/1.46 notdallE: $o ).
% 2.11/1.46
% 2.11/1.46 thf(notdallE,definition,
% 2.11/1.46 ( notdallE
% 2.11/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.11/1.46 ( ~ ! [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 => ( Xphi @ Xx ) )
% 2.11/1.46 => ? [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(exuI1_type,type,
% 2.11/1.46 exuI1: $o ).
% 2.11/1.46
% 2.11/1.46 thf(exuI1,definition,
% 2.11/1.46 ( exuI1
% 2.11/1.46 = ( ! [Xphi: $i > $o] :
% 2.11/1.46 ( ? [Xx: $i] :
% 2.11/1.46 ( ( Xphi @ Xx )
% 2.11/1.46 & ! [Xy: $i] :
% 2.11/1.46 ( ( Xphi @ Xy )
% 2.11/1.46 => ( Xx = Xy ) ) )
% 2.11/1.46 => ( exu
% 2.11/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(exuI3_type,type,
% 2.11/1.46 exuI3: $o ).
% 2.11/1.46
% 2.11/1.46 thf(exuI3,definition,
% 2.11/1.46 ( exuI3
% 2.11/1.46 = ( ! [Xphi: $i > $o] :
% 2.11/1.46 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.11/1.46 => ( ! [Xx: $i,Xy: $i] :
% 2.11/1.46 ( ( Xphi @ Xx )
% 2.11/1.46 => ( ( Xphi @ Xy )
% 2.11/1.46 => ( Xx = Xy ) ) )
% 2.11/1.46 => ( exu
% 2.11/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(exuI2_type,type,
% 2.11/1.46 exuI2: $o ).
% 2.11/1.46
% 2.11/1.46 thf(exuI2,definition,
% 2.11/1.46 ( exuI2
% 2.11/1.46 = ( ! [Xphi: $i > $o] :
% 2.11/1.46 ( ? [Xx: $i] :
% 2.11/1.46 ! [Xy: $i] :
% 2.11/1.46 ( ( Xphi @ Xy )
% 2.11/1.46 <=> ( Xy = Xx ) )
% 2.11/1.46 => ( exu
% 2.11/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(inCongP_type,type,
% 2.11/1.46 inCongP: $o ).
% 2.11/1.46
% 2.11/1.46 thf(inCongP,definition,
% 2.11/1.46 ( inCongP
% 2.11/1.46 = ( ! [A: $i,B: $i] :
% 2.11/1.46 ( ( A = B )
% 2.11/1.46 => ! [Xx: $i,Xy: $i] :
% 2.11/1.46 ( ( Xx = Xy )
% 2.11/1.46 => ( ( in @ Xx @ A )
% 2.11/1.46 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(in__Cong_type,type,
% 2.11/1.46 in__Cong: $o ).
% 2.11/1.46
% 2.11/1.46 thf(in__Cong,definition,
% 2.11/1.46 ( in__Cong
% 2.11/1.46 = ( ! [A: $i,B: $i] :
% 2.11/1.46 ( ( A = B )
% 2.11/1.46 => ! [Xx: $i,Xy: $i] :
% 2.11/1.46 ( ( Xx = Xy )
% 2.11/1.46 => ( ( in @ Xx @ A )
% 2.11/1.46 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(exuE3u_type,type,
% 2.11/1.46 exuE3u: $o ).
% 2.11/1.46
% 2.11/1.46 thf(exuE3u,definition,
% 2.11/1.46 ( exuE3u
% 2.11/1.46 = ( ! [Xphi: $i > $o] :
% 2.11/1.46 ( ( exu
% 2.11/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.11/1.46 => ! [Xx: $i,Xy: $i] :
% 2.11/1.46 ( ( Xphi @ Xx )
% 2.11/1.46 => ( ( Xphi @ Xy )
% 2.11/1.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(exu__Cong_type,type,
% 2.11/1.46 exu__Cong: $o ).
% 2.11/1.46
% 2.11/1.46 thf(exu__Cong,definition,
% 2.11/1.46 ( exu__Cong
% 2.11/1.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.11/1.46 ( ! [Xx: $i,Xy: $i] :
% 2.11/1.46 ( ( Xx = Xy )
% 2.11/1.46 => ( ( Xphi @ Xx )
% 2.11/1.46 <=> ( Xpsi @ Xy ) ) )
% 2.11/1.46 => ( ( exu
% 2.11/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.11/1.46 <=> ( exu
% 2.11/1.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(emptyset__Cong_type,type,
% 2.11/1.46 emptyset__Cong: $o ).
% 2.11/1.46
% 2.11/1.46 thf(emptyset__Cong,definition,
% 2.11/1.46 ( emptyset__Cong
% 2.11/1.46 = ( emptyset = emptyset ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(setadjoin__Cong_type,type,
% 2.11/1.46 setadjoin__Cong: $o ).
% 2.11/1.46
% 2.11/1.46 thf(setadjoin__Cong,definition,
% 2.11/1.46 ( setadjoin__Cong
% 2.11/1.46 = ( ! [Xx: $i,Xy: $i] :
% 2.11/1.46 ( ( Xx = Xy )
% 2.11/1.46 => ! [Xz: $i,Xu: $i] :
% 2.11/1.46 ( ( Xz = Xu )
% 2.11/1.46 => ( ( setadjoin @ Xx @ Xz )
% 2.11/1.46 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(powerset__Cong_type,type,
% 2.11/1.46 powerset__Cong: $o ).
% 2.11/1.46
% 2.11/1.46 thf(powerset__Cong,definition,
% 2.11/1.46 ( powerset__Cong
% 2.11/1.46 = ( ! [A: $i,B: $i] :
% 2.11/1.46 ( ( A = B )
% 2.11/1.46 => ( ( powerset @ A )
% 2.11/1.46 = ( powerset @ B ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(setunion__Cong_type,type,
% 2.11/1.46 setunion__Cong: $o ).
% 2.11/1.46
% 2.11/1.46 thf(setunion__Cong,definition,
% 2.11/1.46 ( setunion__Cong
% 2.11/1.46 = ( ! [A: $i,B: $i] :
% 2.11/1.46 ( ( A = B )
% 2.11/1.46 => ( ( setunion @ A )
% 2.11/1.46 = ( setunion @ B ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(omega__Cong_type,type,
% 2.11/1.46 omega__Cong: $o ).
% 2.11/1.46
% 2.11/1.46 thf(omega__Cong,definition,
% 2.11/1.46 ( omega__Cong
% 2.11/1.46 = ( omega = omega ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(exuEu_type,type,
% 2.11/1.46 exuEu: $o ).
% 2.11/1.46
% 2.11/1.46 thf(exuEu,definition,
% 2.11/1.46 ( exuEu
% 2.11/1.46 = ( ! [Xphi: $i > $o] :
% 2.11/1.46 ( ( exu
% 2.11/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.11/1.46 => ! [Xx: $i,Xy: $i] :
% 2.11/1.46 ( ( Xphi @ Xx )
% 2.11/1.46 => ( ( Xphi @ Xy )
% 2.11/1.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(descr__Cong_type,type,
% 2.11/1.46 descr__Cong: $o ).
% 2.11/1.46
% 2.11/1.46 thf(descr__Cong,definition,
% 2.11/1.46 ( descr__Cong
% 2.11/1.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.11/1.46 ( ! [Xx: $i,Xy: $i] :
% 2.11/1.46 ( ( Xx = Xy )
% 2.11/1.46 => ( ( Xphi @ Xx )
% 2.11/1.46 <=> ( Xpsi @ Xy ) ) )
% 2.11/1.46 => ( ( exu
% 2.11/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.11/1.46 => ( ( exu
% 2.11/1.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.11/1.46 => ( ( descr
% 2.11/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.11/1.46 = ( descr
% 2.11/1.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(dsetconstr__Cong_type,type,
% 2.11/1.46 dsetconstr__Cong: $o ).
% 2.11/1.46
% 2.11/1.46 thf(dsetconstr__Cong,definition,
% 2.11/1.46 ( dsetconstr__Cong
% 2.11/1.46 = ( ! [A: $i,B: $i] :
% 2.11/1.46 ( ( A = B )
% 2.11/1.46 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.11/1.46 ( ! [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 => ! [Xy: $i] :
% 2.11/1.46 ( ( in @ Xy @ B )
% 2.11/1.46 => ( ( Xx = Xy )
% 2.11/1.46 => ( ( Xphi @ Xx )
% 2.11/1.46 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.11/1.46 => ( ( dsetconstr @ A
% 2.11/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.11/1.46 = ( dsetconstr @ B
% 2.11/1.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(subset_type,type,
% 2.11/1.46 subset: $i > $i > $o ).
% 2.11/1.46
% 2.11/1.46 thf(disjoint_type,type,
% 2.11/1.46 disjoint: $i > $i > $o ).
% 2.11/1.46
% 2.11/1.46 thf(setsmeet_type,type,
% 2.11/1.46 setsmeet: $i > $i > $o ).
% 2.11/1.46
% 2.11/1.46 thf(subsetI1_type,type,
% 2.11/1.46 subsetI1: $o ).
% 2.11/1.46
% 2.11/1.46 thf(subsetI1,definition,
% 2.11/1.46 ( subsetI1
% 2.11/1.46 = ( ! [A: $i,B: $i] :
% 2.11/1.46 ( ! [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 => ( in @ Xx @ B ) )
% 2.11/1.46 => ( subset @ A @ B ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(eqimpsubset2_type,type,
% 2.11/1.46 eqimpsubset2: $o ).
% 2.11/1.46
% 2.11/1.46 thf(eqimpsubset2,definition,
% 2.11/1.46 ( eqimpsubset2
% 2.11/1.46 = ( ! [A: $i,B: $i] :
% 2.11/1.46 ( ( A = B )
% 2.11/1.46 => ( subset @ B @ A ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(eqimpsubset1_type,type,
% 2.11/1.46 eqimpsubset1: $o ).
% 2.11/1.46
% 2.11/1.46 thf(eqimpsubset1,definition,
% 2.11/1.46 ( eqimpsubset1
% 2.11/1.46 = ( ! [A: $i,B: $i] :
% 2.11/1.46 ( ( A = B )
% 2.11/1.46 => ( subset @ A @ B ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(subsetI2_type,type,
% 2.11/1.46 subsetI2: $o ).
% 2.11/1.46
% 2.11/1.46 thf(subsetI2,definition,
% 2.11/1.46 ( subsetI2
% 2.11/1.46 = ( ! [A: $i,B: $i] :
% 2.11/1.46 ( ! [Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 => ( in @ Xx @ B ) )
% 2.11/1.46 => ( subset @ A @ B ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(emptysetsubset_type,type,
% 2.11/1.46 emptysetsubset: $o ).
% 2.11/1.46
% 2.11/1.46 thf(emptysetsubset,definition,
% 2.11/1.46 ( emptysetsubset
% 2.11/1.46 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(subsetE_type,type,
% 2.11/1.46 subsetE: $o ).
% 2.11/1.46
% 2.11/1.46 thf(subsetE,definition,
% 2.11/1.46 ( subsetE
% 2.11/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.11/1.46 ( ( subset @ A @ B )
% 2.11/1.46 => ( ( in @ Xx @ A )
% 2.11/1.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(subsetE2_type,type,
% 2.11/1.46 subsetE2: $o ).
% 2.11/1.46
% 2.11/1.46 thf(subsetE2,definition,
% 2.11/1.46 ( subsetE2
% 2.11/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.11/1.46 ( ( subset @ A @ B )
% 2.11/1.46 => ( ~ ( in @ Xx @ B )
% 2.11/1.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(notsubsetI_type,type,
% 2.11/1.46 notsubsetI: $o ).
% 2.11/1.46
% 2.11/1.46 thf(notsubsetI,definition,
% 2.11/1.46 ( notsubsetI
% 2.11/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 => ( ~ ( in @ Xx @ B )
% 2.11/1.46 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(notequalI1_type,type,
% 2.11/1.46 notequalI1: $o ).
% 2.11/1.46
% 2.11/1.46 thf(notequalI1,definition,
% 2.11/1.46 ( notequalI1
% 2.11/1.46 = ( ! [A: $i,B: $i] :
% 2.11/1.46 ( ~ ( subset @ A @ B )
% 2.11/1.46 => ( A != B ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(notequalI2_type,type,
% 2.11/1.46 notequalI2: $o ).
% 2.11/1.46
% 2.11/1.46 thf(notequalI2,definition,
% 2.11/1.46 ( notequalI2
% 2.11/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.11/1.46 ( ( in @ Xx @ A )
% 2.11/1.46 => ( ~ ( in @ Xx @ B )
% 2.11/1.46 => ( A != B ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(subsetRefl_type,type,
% 2.11/1.46 subsetRefl: $o ).
% 2.11/1.46
% 2.11/1.46 thf(subsetRefl,definition,
% 2.11/1.46 ( subsetRefl
% 2.11/1.46 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(subsetTrans_type,type,
% 2.11/1.46 subsetTrans: $o ).
% 2.11/1.46
% 2.11/1.46 thf(subsetTrans,definition,
% 2.11/1.46 ( subsetTrans
% 2.11/1.46 = ( ! [A: $i,B: $i,C: $i] :
% 2.11/1.46 ( ( subset @ A @ B )
% 2.11/1.46 => ( ( subset @ B @ C )
% 2.11/1.46 => ( subset @ A @ C ) ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(setadjoinSub_type,type,
% 2.11/1.46 setadjoinSub: $o ).
% 2.11/1.46
% 2.11/1.46 thf(setadjoinSub,definition,
% 2.11/1.46 ( setadjoinSub
% 2.11/1.46 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.11/1.46
% 2.11/1.46 thf(setadjoinSub2,conjecture,
% 2.11/1.46 ( setextAx
% 2.11/1.46 => ( emptysetAx
% 2.11/1.46 => ( setadjoinAx
% 2.11/1.46 => ( powersetAx
% 2.11/1.46 => ( setunionAx
% 2.11/1.46 => ( omega0Ax
% 2.11/1.46 => ( omegaSAx
% 2.11/1.46 => ( omegaIndAx
% 2.11/1.46 => ( replAx
% 2.11/1.46 => ( foundationAx
% 2.11/1.46 => ( wellorderingAx
% 2.11/1.46 => ( descrp
% 2.11/1.46 => ( dsetconstrI
% 2.11/1.46 => ( dsetconstrEL
% 2.11/1.46 => ( dsetconstrER
% 2.11/1.46 => ( exuE1
% 2.11/1.46 => ( prop2setE
% 2.11/1.46 => ( emptysetE
% 2.11/1.46 => ( emptysetimpfalse
% 2.11/1.46 => ( notinemptyset
% 2.11/1.46 => ( exuE3e
% 2.11/1.46 => ( setext
% 2.11/1.46 => ( emptyI
% 2.11/1.46 => ( noeltsimpempty
% 2.11/1.46 => ( setbeta
% 2.11/1.46 => ( nonemptyE1
% 2.11/1.46 => ( nonemptyI
% 2.11/1.46 => ( nonemptyI1
% 2.11/1.46 => ( setadjoinIL
% 2.11/1.46 => ( emptyinunitempty
% 2.11/1.46 => ( setadjoinIR
% 2.11/1.46 => ( setadjoinE
% 2.11/1.46 => ( setadjoinOr
% 2.11/1.46 => ( setoftrueEq
% 2.11/1.46 => ( powersetI
% 2.11/1.46 => ( emptyinPowerset
% 2.11/1.46 => ( emptyInPowerset
% 2.11/1.46 => ( powersetE
% 2.11/1.46 => ( setunionI
% 2.11/1.46 => ( setunionE
% 2.11/1.46 => ( subPowSU
% 2.11/1.46 => ( exuE2
% 2.11/1.46 => ( nonemptyImpWitness
% 2.11/1.46 => ( uniqinunit
% 2.11/1.46 => ( notinsingleton
% 2.11/1.46 => ( eqinunit
% 2.11/1.46 => ( singletonsswitch
% 2.11/1.46 => ( upairsetE
% 2.11/1.46 => ( upairsetIL
% 2.11/1.46 => ( upairsetIR
% 2.11/1.46 => ( emptyE1
% 2.11/1.46 => ( vacuousDall
% 2.11/1.46 => ( quantDeMorgan1
% 2.11/1.46 => ( quantDeMorgan2
% 2.11/1.46 => ( quantDeMorgan3
% 2.11/1.46 => ( quantDeMorgan4
% 2.11/1.46 => ( prop2setI
% 2.11/1.46 => ( prop2set2propI
% 2.11/1.46 => ( notdexE
% 2.11/1.46 => ( notdallE
% 2.11/1.46 => ( exuI1
% 2.11/1.46 => ( exuI3
% 2.11/1.46 => ( exuI2
% 2.11/1.46 => ( inCongP
% 2.11/1.46 => ( in__Cong
% 2.11/1.46 => ( exuE3u
% 2.11/1.46 => ( exu__Cong
% 2.11/1.46 => ( emptyset__Cong
% 2.11/1.46 => ( setadjoin__Cong
% 2.11/1.46 => ( powerset__Cong
% 2.11/1.46 => ( setunion__Cong
% 2.11/1.46 => ( omega__Cong
% 2.11/1.46 => ( exuEu
% 2.11/1.46 => ( descr__Cong
% 2.11/1.46 => ( dsetconstr__Cong
% 2.11/1.46 => ( subsetI1
% 2.11/1.46 => ( eqimpsubset2
% 2.11/1.46 => ( eqimpsubset1
% 2.11/1.46 => ( subsetI2
% 2.11/1.46 => ( emptysetsubset
% 2.11/1.46 => ( subsetE
% 2.11/1.46 => ( subsetE2
% 2.11/1.46 => ( notsubsetI
% 2.11/1.46 => ( notequalI1
% 2.11/1.46 => ( notequalI2
% 2.11/1.46 => ( subsetRefl
% 2.11/1.46 => ( subsetTrans
% 2.11/1.46 => ( setadjoinSub
% 2.11/1.46 => ! [A: $i,Xx: $i,B: $i] :
% 2.11/1.46 ( ( subset @ A @ B )
% 2.11/1.46 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.11/1.46 ------------------------
% 2.11/1.47 % (16590)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_16254 for (2999ds/18Mi)
% 2.11/1.47 % (16585)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_16254 for (2999ds/4Mi)
% 2.11/1.48 % (16585)Instruction limit reached!
% 2.11/1.48 % (16585)------------------------------
% 2.11/1.48 % (16585)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.11/1.48 % (16585)Termination reason: Unknown
% 2.11/1.48 % (16585)Termination phase: shuffling
% 2.11/1.48
% 2.11/1.48 % (16585)Memory used [KB]: 1279
% 2.11/1.48 % (16585)Time elapsed: 0.003 s
% 2.11/1.48 % (16585)Instructions burned: 6 (million)
% 2.11/1.48 % (16585)------------------------------
% 2.11/1.48 % (16585)------------------------------
% 2.11/1.48 % (16590)Instruction limit reached!
% 2.11/1.48 % (16590)------------------------------
% 2.11/1.48 % (16590)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.11/1.48 % (16590)Termination reason: Unknown
% 2.11/1.48 % (16590)Termination phase: Preprocessing 3
% 2.11/1.48
% 2.11/1.48 % (16590)Memory used [KB]: 1663
% 2.11/1.48 % (16590)Time elapsed: 0.007 s
% 2.11/1.48 % (16590)Instructions burned: 19 (million)
% 2.11/1.48 % (16590)------------------------------
% 2.11/1.48 % (16590)------------------------------
% 2.11/1.48 % (16584)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_16254 for (2999ds/183Mi)
% 2.11/1.48 % (16586)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_16254 for (2999ds/27Mi)
% 2.11/1.48 % (16587)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_16254 for (2999ds/2Mi)
% 2.11/1.48 % (16588)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_16254 for (2999ds/2Mi)
% 2.11/1.48 % (16589)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_16254 for (2999ds/275Mi)
% 2.11/1.48 % (16587)Instruction limit reached!
% 2.11/1.48 % (16587)------------------------------
% 2.11/1.48 % (16587)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.11/1.48 % (16587)Termination reason: Unknown
% 2.11/1.48 % (16587)Termination phase: shuffling
% 2.11/1.48
% 2.11/1.48 % (16587)Memory used [KB]: 1151
% 2.11/1.48 % (16587)Time elapsed: 0.003 s
% 2.11/1.48 % (16587)Instructions burned: 2 (million)
% 2.11/1.48 % (16587)------------------------------
% 2.11/1.48 % (16587)------------------------------
% 2.11/1.48 % (16588)Instruction limit reached!
% 2.11/1.48 % (16588)------------------------------
% 2.11/1.48 % (16588)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.11/1.48 % (16588)Termination reason: Unknown
% 2.11/1.48 % (16588)Termination phase: shuffling
% 2.11/1.48
% 2.11/1.48 % (16588)Memory used [KB]: 1151
% 2.11/1.48 % (16588)Time elapsed: 0.004 s
% 2.11/1.48 % (16588)Instructions burned: 3 (million)
% 2.11/1.48 % (16588)------------------------------
% 2.11/1.48 % (16588)------------------------------
% 2.11/1.49 % (16591)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_16254 for (2999ds/3Mi)
% 2.11/1.49 % (16591)Instruction limit reached!
% 2.11/1.49 % (16591)------------------------------
% 2.11/1.49 % (16591)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.11/1.49 % (16591)Termination reason: Unknown
% 2.11/1.49 % (16591)Termination phase: shuffling
% 2.11/1.49
% 2.11/1.49 % (16591)Memory used [KB]: 1151
% 2.11/1.49 % (16591)Time elapsed: 0.003 s
% 2.11/1.49 % (16591)Instructions burned: 4 (million)
% 2.11/1.49 % (16591)------------------------------
% 2.11/1.49 % (16591)------------------------------
% 2.11/1.49 % (16592)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_16254 for (2999ds/37Mi)
% 2.11/1.50 % (16586)Instruction limit reached!
% 2.11/1.50 % (16586)------------------------------
% 2.11/1.50 % (16586)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.11/1.50 % (16586)Termination reason: Unknown
% 2.11/1.50 % (16586)Termination phase: Property scanning
% 2.11/1.50
% 2.11/1.50 % (16586)Memory used [KB]: 1407
% 2.11/1.50 % (16586)Time elapsed: 0.016 s
% 2.11/1.50 % (16586)Instructions burned: 28 (million)
% 2.11/1.50 % (16586)------------------------------
% 2.11/1.50 % (16586)------------------------------
% 2.24/1.50 % (16595)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_16254 for (2999ds/1041Mi)
% 2.24/1.50 % (16593)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_16254 for (2999ds/15Mi)
% 2.24/1.50 % (16592)Instruction limit reached!
% 2.24/1.50 % (16592)------------------------------
% 2.24/1.50 % (16592)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.24/1.50 % (16592)Termination reason: Unknown
% 2.24/1.50 % (16592)Termination phase: Property scanning
% 2.24/1.50
% 2.24/1.50 % (16592)Memory used [KB]: 1535
% 2.24/1.50 % (16592)Time elapsed: 0.011 s
% 2.24/1.50 % (16592)Instructions burned: 37 (million)
% 2.24/1.50 % (16592)------------------------------
% 2.24/1.50 % (16592)------------------------------
% 2.24/1.50 % (16594)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_16254 for (2999ds/3Mi)
% 2.24/1.50 % (16594)Instruction limit reached!
% 2.24/1.50 % (16594)------------------------------
% 2.24/1.50 % (16594)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.24/1.50 % (16594)Termination reason: Unknown
% 2.24/1.50 % (16594)Termination phase: shuffling
% 2.24/1.50
% 2.24/1.50 % (16594)Memory used [KB]: 1151
% 2.24/1.50 % (16594)Time elapsed: 0.004 s
% 2.24/1.50 % (16594)Instructions burned: 4 (million)
% 2.24/1.50 % (16594)------------------------------
% 2.24/1.50 % (16594)------------------------------
% 2.24/1.50 % (16593)Instruction limit reached!
% 2.24/1.50 % (16593)------------------------------
% 2.24/1.50 % (16593)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.24/1.50 % (16593)Termination reason: Unknown
% 2.24/1.50 % (16593)Termination phase: Preprocessing 3
% 2.24/1.50
% 2.24/1.50 % (16593)Memory used [KB]: 1407
% 2.24/1.50 % (16593)Time elapsed: 0.007 s
% 2.24/1.50 % (16593)Instructions burned: 15 (million)
% 2.24/1.50 % (16593)------------------------------
% 2.24/1.50 % (16593)------------------------------
% 2.24/1.51 % (16597)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_16254 for (2999ds/16Mi)
% 2.24/1.51 % (16596)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_16254 for (2999ds/7Mi)
% 2.24/1.51 % (16597)Instruction limit reached!
% 2.24/1.51 % (16597)------------------------------
% 2.24/1.51 % (16597)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.24/1.51 % (16597)Termination reason: Unknown
% 2.24/1.51 % (16597)Termination phase: Property scanning
% 2.24/1.51
% 2.24/1.51 % (16597)Memory used [KB]: 1407
% 2.24/1.51 % (16597)Time elapsed: 0.006 s
% 2.24/1.51 % (16597)Instructions burned: 19 (million)
% 2.24/1.51 % (16597)------------------------------
% 2.24/1.51 % (16597)------------------------------
% 2.24/1.51 % (16599)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_16254 for (2999ds/3Mi)
% 2.24/1.51 % (16599)Instruction limit reached!
% 2.24/1.51 % (16599)------------------------------
% 2.24/1.51 % (16599)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.24/1.51 % (16599)Termination reason: Unknown
% 2.24/1.51 % (16599)Termination phase: shuffling
% 2.24/1.51
% 2.24/1.51 % (16599)Memory used [KB]: 1151
% 2.24/1.51 % (16599)Time elapsed: 0.003 s
% 2.24/1.51 % (16599)Instructions burned: 4 (million)
% 2.24/1.51 % (16599)------------------------------
% 2.24/1.51 % (16599)------------------------------
% 2.24/1.52 % (16596)Instruction limit reached!
% 2.24/1.52 % (16596)------------------------------
% 2.24/1.52 % (16596)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.24/1.52 % (16596)Termination reason: Unknown
% 2.24/1.52 % (16596)Termination phase: shuffling
% 2.24/1.52
% 2.24/1.52 % (16596)Memory used [KB]: 1279
% 2.24/1.52 % (16596)Time elapsed: 0.006 s
% 2.24/1.52 % (16596)Instructions burned: 8 (million)
% 2.24/1.52 % (16596)------------------------------
% 2.24/1.52 % (16596)------------------------------
% 2.24/1.52 % (16598)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_16254 for (2999ds/3Mi)
% 2.24/1.52 % (16598)Instruction limit reached!
% 2.24/1.52 % (16598)------------------------------
% 2.24/1.52 % (16598)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.24/1.52 % (16598)Termination reason: Unknown
% 2.24/1.52 % (16598)Termination phase: shuffling
% 2.24/1.52
% 2.24/1.52 % (16598)Memory used [KB]: 1151
% 2.24/1.52 % (16598)Time elapsed: 0.003 s
% 2.24/1.52 % (16598)Instructions burned: 4 (million)
% 2.24/1.52 % (16598)------------------------------
% 2.24/1.52 % (16598)------------------------------
% 2.24/1.52 % (16600)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_16254 for (2999ds/7Mi)
% 2.24/1.52 % (16600)Instruction limit reached!
% 2.24/1.52 % (16600)------------------------------
% 2.24/1.52 % (16600)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.24/1.52 % (16600)Termination reason: Unknown
% 2.24/1.52 % (16600)Termination phase: shuffling
% 2.24/1.52
% 2.24/1.52 % (16600)Memory used [KB]: 1279
% 2.24/1.52 % (16600)Time elapsed: 0.003 s
% 2.24/1.52 % (16600)Instructions burned: 7 (million)
% 2.24/1.52 % (16600)------------------------------
% 2.24/1.52 % (16600)------------------------------
% 2.24/1.52 % (16601)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_16254 for (2999ds/3Mi)
% 2.24/1.53 % (16601)Instruction limit reached!
% 2.24/1.53 % (16601)------------------------------
% 2.24/1.53 % (16601)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.24/1.53 % (16601)Termination reason: Unknown
% 2.24/1.53 % (16601)Termination phase: shuffling
% 2.24/1.53
% 2.24/1.53 % (16601)Memory used [KB]: 1151
% 2.24/1.53 % (16601)Time elapsed: 0.003 s
% 2.24/1.53 % (16601)Instructions burned: 5 (million)
% 2.24/1.53 % (16601)------------------------------
% 2.24/1.53 % (16601)------------------------------
% 2.24/1.53 % (16604)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_16254 for (2999ds/710Mi)
% 2.24/1.53 % (16602)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_16254 for (2999ds/4Mi)
% 2.24/1.53 % (16603)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_16254 for (2999ds/18Mi)
% 2.24/1.54 % (16602)Instruction limit reached!
% 2.24/1.54 % (16602)------------------------------
% 2.24/1.54 % (16602)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.24/1.54 % (16602)Termination reason: Unknown
% 2.24/1.54 % (16602)Termination phase: shuffling
% 2.24/1.54
% 2.24/1.54 % (16602)Memory used [KB]: 1151
% 2.24/1.54 % (16602)Time elapsed: 0.004 s
% 2.24/1.54 % (16602)Instructions burned: 4 (million)
% 2.24/1.54 % (16602)------------------------------
% 2.24/1.54 % (16602)------------------------------
% 2.24/1.54 % (16605)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_16254 for (2999ds/6Mi)
% 2.24/1.54 % (16605)Instruction limit reached!
% 2.24/1.54 % (16605)------------------------------
% 2.24/1.54 % (16605)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.24/1.54 % (16605)Termination reason: Unknown
% 2.24/1.54 % (16605)Termination phase: shuffling
% 2.24/1.54
% 2.24/1.54 % (16605)Memory used [KB]: 1279
% 2.24/1.54 % (16605)Time elapsed: 0.004 s
% 2.24/1.54 % (16605)Instructions burned: 8 (million)
% 2.24/1.54 % (16605)------------------------------
% 2.24/1.54 % (16605)------------------------------
% 2.24/1.54 % (16603)Instruction limit reached!
% 2.24/1.54 % (16603)------------------------------
% 2.24/1.54 % (16603)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.24/1.54 % (16603)Termination reason: Unknown
% 2.24/1.54 % (16603)Termination phase: Property scanning
% 2.24/1.54
% 2.24/1.54 % (16603)Memory used [KB]: 1407
% 2.24/1.54 % (16603)Time elapsed: 0.007 s
% 2.24/1.54 % (16603)Instructions burned: 21 (million)
% 2.24/1.54 % (16603)------------------------------
% 2.24/1.54 % (16603)------------------------------
% 2.24/1.55 % (16607)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_16254 for (2999ds/21Mi)
% 2.24/1.55 % (16608)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_16254 for (2999ds/5Mi)
% 2.24/1.55 % (16608)Instruction limit reached!
% 2.24/1.55 % (16608)------------------------------
% 2.24/1.55 % (16608)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.24/1.55 % (16608)Termination reason: Unknown
% 2.24/1.55 % (16608)Termination phase: shuffling
% 2.24/1.55
% 2.24/1.55 % (16608)Memory used [KB]: 1151
% 2.24/1.55 % (16608)Time elapsed: 0.003 s
% 2.24/1.55 % (16608)Instructions burned: 5 (million)
% 2.24/1.55 % (16608)------------------------------
% 2.24/1.55 % (16608)------------------------------
% 2.24/1.55 % (16606)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_16254 for (2999ds/902Mi)
% 2.24/1.55 % (16607)Instruction limit reached!
% 2.24/1.55 % (16607)------------------------------
% 2.24/1.55 % (16607)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.24/1.55 % (16607)Termination reason: Unknown
% 2.24/1.55 % (16607)Termination phase: shuffling
% 2.24/1.55
% 2.24/1.55 % (16607)Memory used [KB]: 1407
% 2.24/1.55 % (16607)Time elapsed: 0.008 s
% 2.24/1.55 % (16607)Instructions burned: 22 (million)
% 2.24/1.55 % (16607)------------------------------
% 2.24/1.55 % (16607)------------------------------
% 2.24/1.56 % (16609)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_16254 for (2999ds/6Mi)
% 2.24/1.56 % (16609)Instruction limit reached!
% 2.24/1.56 % (16609)------------------------------
% 2.24/1.56 % (16609)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.24/1.56 % (16609)Termination reason: Unknown
% 2.24/1.56 % (16609)Termination phase: shuffling
% 2.24/1.56
% 2.24/1.56 % (16609)Memory used [KB]: 1279
% 2.24/1.56 % (16609)Time elapsed: 0.004 s
% 2.24/1.56 % (16609)Instructions burned: 8 (million)
% 2.24/1.56 % (16609)------------------------------
% 2.24/1.56 % (16609)------------------------------
% 2.24/1.56 % (16610)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_16254 for (2998ds/377Mi)
% 2.24/1.57 % (16611)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_16254 for (2998ds/779Mi)
% 2.24/1.58 % (16584)Instruction limit reached!
% 2.24/1.58 % (16584)------------------------------
% 2.24/1.58 % (16584)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.24/1.58 % (16584)Termination reason: Unknown
% 2.24/1.58 % (16584)Termination phase: Saturation
% 2.24/1.58
% 2.24/1.58 % (16584)Memory used [KB]: 6908
% 2.24/1.58 % (16584)Time elapsed: 0.098 s
% 2.24/1.58 % (16584)Instructions burned: 183 (million)
% 2.24/1.58 % (16584)------------------------------
% 2.24/1.58 % (16584)------------------------------
% 2.24/1.59 % (16612)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_16254 for (2998ds/19Mi)
% 2.24/1.59 % (16612)Instruction limit reached!
% 2.24/1.59 % (16612)------------------------------
% 2.24/1.59 % (16612)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.24/1.59 % (16612)Termination reason: Unknown
% 2.24/1.59 % (16612)Termination phase: Property scanning
% 2.24/1.59
% 2.24/1.59 % (16612)Memory used [KB]: 1407
% 2.24/1.59 % (16612)Time elapsed: 0.007 s
% 2.24/1.59 % (16612)Instructions burned: 22 (million)
% 2.24/1.59 % (16612)------------------------------
% 2.24/1.59 % (16612)------------------------------
% 2.24/1.60 % (16606)First to succeed.
% 2.24/1.60 % (16606)Refutation found. Thanks to Tanya!
% 2.24/1.60 % SZS status Theorem for DTF2THF_16254
% 2.24/1.60 % SZS output start Proof for DTF2THF_16254
% 2.24/1.60 thf(func_def_0, type, in: $i > $i > $o).
% 2.24/1.60 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.24/1.60 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.24/1.60 thf(func_def_8, type, powerset: $i > $i).
% 2.24/1.60 thf(func_def_10, type, setunion: $i > $i).
% 2.24/1.60 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.24/1.60 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.24/1.60 thf(func_def_26, type, prop2set: $o > $i).
% 2.24/1.60 thf(func_def_36, type, nonempty: $i > $o).
% 2.24/1.60 thf(func_def_69, type, set2prop: $i > $o).
% 2.24/1.60 thf(func_def_88, type, subset: $i > $i > $o).
% 2.24/1.60 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.24/1.60 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.24/1.60 thf(func_def_117, type, sP1: $i > $i > $o).
% 2.24/1.60 thf(func_def_118, type, sP2: $i > $o).
% 2.24/1.60 thf(func_def_119, type, sP3: $i > $i > $o).
% 2.24/1.60 thf(func_def_120, type, sP4: $i > $i > $o).
% 2.24/1.60 thf(func_def_121, type, sK5: $i > $i > $i).
% 2.24/1.60 thf(func_def_124, type, sK8: $i > $o).
% 2.24/1.60 thf(func_def_127, type, sK11: $i > ($i > $o) > $i).
% 2.24/1.60 thf(func_def_130, type, sK14: $i > $o).
% 2.24/1.60 thf(func_def_133, type, sK17: ($i > $o) > $i).
% 2.24/1.60 thf(func_def_134, type, sK18: ($i > $o) > $i).
% 2.24/1.60 thf(func_def_135, type, sK19: $i > $o).
% 2.24/1.60 thf(func_def_137, type, sK21: ($i > $o) > $i).
% 2.24/1.60 thf(func_def_138, type, sK22: $i > $o).
% 2.24/1.60 thf(func_def_139, type, sK23: $i > $o).
% 2.24/1.60 thf(func_def_144, type, sK28: $i > $i).
% 2.24/1.60 thf(func_def_150, type, sK34: $i > $i > $i).
% 2.24/1.60 thf(func_def_151, type, sK35: $i > $i > $i).
% 2.24/1.60 thf(func_def_154, type, sK38: $i > ($i > $o) > $i).
% 2.24/1.60 thf(func_def_155, type, sK39: $i > $o).
% 2.24/1.60 thf(func_def_171, type, sK55: $i > $o).
% 2.24/1.60 thf(func_def_182, type, sK66: ($i > $o) > $i > $i).
% 2.24/1.60 thf(func_def_184, type, sK68: $i > $o).
% 2.24/1.60 thf(func_def_187, type, sK71: $i > $o).
% 2.24/1.60 thf(func_def_194, type, sK78: $i > $i).
% 2.24/1.60 thf(func_def_198, type, sK82: $i > $i).
% 2.24/1.60 thf(func_def_200, type, sK84: $i > $i).
% 2.24/1.60 thf(func_def_203, type, sK87: $i > $o).
% 2.24/1.60 thf(func_def_206, type, sK90: $i > $o).
% 2.24/1.60 thf(func_def_207, type, sK91: $i > $i).
% 2.24/1.60 thf(func_def_208, type, sK92: ($i > $o) > $i).
% 2.24/1.60 thf(func_def_209, type, sK93: $i > $o).
% 2.24/1.60 thf(func_def_210, type, sK94: $i > $o).
% 2.24/1.60 thf(func_def_211, type, sK95: ($i > $o) > ($i > $o) > $i).
% 2.24/1.60 thf(func_def_212, type, sK96: ($i > $o) > ($i > $o) > $i).
% 2.24/1.60 thf(func_def_215, type, sK99: $i > $o).
% 2.24/1.60 thf(func_def_217, type, sK101: ($i > $o) > $i > $i).
% 2.24/1.60 thf(func_def_219, type, sK103: $i > $o).
% 2.24/1.60 thf(func_def_221, type, sK105: $i > $o).
% 2.24/1.60 thf(func_def_222, type, sK106: $i > $i).
% 2.24/1.60 thf(func_def_223, type, sK107: ($i > $o) > $i).
% 2.24/1.60 thf(func_def_228, type, sK112: $i > $o).
% 2.24/1.60 thf(func_def_229, type, sK113: $i > $o).
% 2.24/1.60 thf(func_def_230, type, sK114: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.24/1.60 thf(func_def_231, type, sK115: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.24/1.60 thf(func_def_232, type, sK116: $i > ($i > $o) > $i).
% 2.24/1.60 thf(func_def_233, type, sK117: $i > $o).
% 2.24/1.60 thf(func_def_236, type, sK120: $i > $o).
% 2.24/1.60 thf(func_def_238, type, sK122: $i > ($i > $o) > $i).
% 2.24/1.60 thf(func_def_239, type, sK123: $i > $o).
% 2.24/1.60 thf(func_def_252, type, sK136: $i > $i > $i).
% 2.24/1.60 thf(func_def_253, type, sK137: $i > $o).
% 2.24/1.60 thf(func_def_258, type, sK142: $i > $i).
% 2.24/1.60 thf(func_def_261, type, sK145: $i > $i > $o).
% 2.24/1.60 thf(func_def_263, type, sK147: $i > $i).
% 2.24/1.60 thf(func_def_264, type, sK148: $i > $i).
% 2.24/1.60 thf(func_def_265, type, sK149: $i > ($i > $i > $o) > $i).
% 2.24/1.60 thf(func_def_266, type, sK150: $i > ($i > $i > $o) > $i).
% 2.24/1.60 thf(func_def_267, type, sK151: $i > $i > ($i > $i > $o) > $i).
% 2.24/1.60 thf(func_def_278, type, sK162: $i > $i).
% 2.24/1.60 thf(func_def_280, type, sK164: $i > ($i > $o) > $i).
% 2.24/1.60 thf(func_def_281, type, sK165: $i > $o).
% 2.24/1.60 thf(func_def_285, type, sK169: $i > $i).
% 2.24/1.60 thf(func_def_289, type, sK173: ($i > $o) > ($i > $o) > $i).
% 2.24/1.60 thf(func_def_290, type, sK174: ($i > $o) > ($i > $o) > $i).
% 2.24/1.60 thf(func_def_291, type, sK175: $i > $o).
% 2.24/1.60 thf(func_def_292, type, sK176: $i > $o).
% 2.24/1.60 thf(func_def_296, type, sK180: $o > $i > $i > $i).
% 2.24/1.60 thf(func_def_304, type, sK188: $i > $i > $i).
% 2.24/1.60 thf(func_def_309, type, sK193: $i > $i > $i).
% 2.24/1.60 thf(func_def_312, type, sK196: $i > $i > $i).
% 2.24/1.60 thf(func_def_313, type, sK197: $i > $i > $i).
% 2.24/1.60 thf(func_def_314, type, sK198: $i > $i > $i).
% 2.24/1.60 thf(func_def_315, type, sK199: $i > $i > $i).
% 2.24/1.60 thf(func_def_316, type, sK200: $i > $i > $i > $i).
% 2.24/1.60 thf(func_def_317, type, sK201: $i > $i).
% 2.24/1.60 thf(func_def_318, type, sK202: $i > $i).
% 2.24/1.60 thf(func_def_319, type, sK203: $i > $i).
% 2.24/1.60 thf(func_def_320, type, sK204: $i > $i).
% 2.24/1.60 thf(func_def_321, type, sK205: $i > $i > $i > $i).
% 2.24/1.60 thf(func_def_322, type, sK206: $i > $i > $i > $i).
% 2.24/1.60 thf(func_def_323, type, sK207: $i > $i > $i).
% 2.24/1.60 thf(func_def_324, type, sK208: $i > $i > $i).
% 2.24/1.60 thf(func_def_325, type, sK209: $i > $i > $i).
% 2.24/1.60 thf(func_def_327, type, sK211: $i > $i).
% 2.24/1.60 thf(func_def_328, type, sK212: $i > $i).
% 2.24/1.60 thf(func_def_329, type, sK213: $i > $i).
% 2.24/1.60 thf(func_def_330, type, sK214: $i > $i > $i).
% 2.24/1.60 thf(func_def_333, type, sK217: $i > $o).
% 2.24/1.60 thf(func_def_349, type, sK233: $i > $o).
% 2.24/1.60 thf(func_def_352, type, sK236: $i > $i > $i).
% 2.24/1.60 thf(func_def_361, type, sK245: $i > $o).
% 2.24/1.60 thf(func_def_363, type, sK247: ($i > $o) > $i > $i).
% 2.24/1.60 thf(func_def_367, type, sK251: $i > $i > $i).
% 2.24/1.60 thf(func_def_371, type, ph255: !>[X0: $tType]:(X0)).
% 2.24/1.60 thf(f1429,plain,(
% 2.24/1.60 $false),
% 2.24/1.60 inference(trivial_inequality_removal,[status(thm)],[f1426])).
% 2.24/1.60 thf(f1426,plain,(
% 2.24/1.60 ($true != $true)),
% 2.24/1.60 inference(superposition,[status(thm)],[f1419,f1094])).
% 2.24/1.60 thf(f1094,plain,(
% 2.24/1.60 ((subset @ sK132 @ sK131) = $true)),
% 2.24/1.60 inference(cnf_transformation,[status(thm)],[f617])).
% 2.24/1.60 thf(f617,plain,(
% 2.24/1.60 (quantDeMorgan2 = $true) & (quantDeMorgan1 = $true) & (upairsetIL = $true) & (exuE2 = $true) & (subsetI2 = $true) & (setoftrueEq = $true) & (notdallE = $true) & (descrp = $true) & (nonemptyI1 = $true) & (eqimpsubset1 = $true) & (($true != (subset @ sK132 @ (setadjoin @ sK133 @ sK131))) & ((subset @ sK132 @ sK131) = $true)) & (emptysetsubset = $true) & (subsetI1 = $true) & (prop2setI = $true) & (prop2set2propI = $true) & (notsubsetI = $true) & (emptysetE = $true) & (emptyI = $true) & (exuE3e = $true) & (setadjoinAx = $true) & (exu__Cong = $true) & (setadjoinOr = $true) & (exuE1 = $true) & (notdexE = $true) & (emptyInPowerset = $true) & (exuI2 = $true) & (replAx = $true) & (notequalI1 = $true) & (subsetE = $true) & (upairsetE = $true) & (emptysetimpfalse = $true) & (uniqinunit = $true) & (notinemptyset = $true) & (notinsingleton = $true) & (exuI3 = $true) & (inCongP = $true) & (in__Cong = $true) & (noeltsimpempty = $true) & (emptyinunitempty = $true) & (setunionI = $true) & (emptyinPowerset = $true) & (setextAx = $true) & (exuI1 = $true) & (upairsetIR = $true) & (quantDeMorgan4 = $true) & (nonemptyImpWitness = $true) & (setunionE = $true) & (subPowSU = $true) & (quantDeMorgan3 = $true) & (descr__Cong = $true) & (setadjoinIL = $true) & (wellorderingAx = $true) & (emptysetAx = $true) & (dsetconstr__Cong = $true) & (notequalI2 = $true) & (nonemptyE1 = $true) & (powersetI = $true) & (eqimpsubset2 = $true) & (dsetconstrER = $true) & (dsetconstrI = $true) & (subsetRefl = $true) & (omega__Cong = $true) & (emptyset__Cong = $true) & (omegaSAx = $true) & (setadjoinIR = $true) & (exuE3u = $true) & (powersetAx = $true) & (emptyE1 = $true) & (omega0Ax = $true) & (singletonsswitch = $true) & (omegaIndAx = $true) & (subsetTrans = $true) & (setbeta = $true) & (powerset__Cong = $true) & (setext = $true) & (nonemptyI = $true) & (setunionAx = $true) & (dsetconstrEL = $true) & (powersetE = $true) & (eqinunit = $true) & (setadjoin__Cong = $true) & (vacuousDall = $true) & (foundationAx = $true) & (setadjoinSub = $true) & (prop2setE = $true) & (setunion__Cong = $true) & (subsetE2 = $true) & (exuEu = $true) & (setadjoinE = $true)),
% 2.24/1.60 inference(skolemisation,[status(esa),new_symbols(skolem,[sK131,sK132,sK133])],[f375,f616])).
% 2.24/1.60 thf(f616,plain,(
% 2.24/1.60 ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (subset @ X1 @ (setadjoin @ X2 @ X0))) & ($true = (subset @ X1 @ X0))) => (($true != (subset @ sK132 @ (setadjoin @ sK133 @ sK131))) & ((subset @ sK132 @ sK131) = $true))),
% 2.24/1.60 introduced(definition,[],[choice_axiom])).
% 2.24/1.60 thf(f375,plain,(
% 2.24/1.60 (quantDeMorgan2 = $true) & (quantDeMorgan1 = $true) & (upairsetIL = $true) & (exuE2 = $true) & (subsetI2 = $true) & (setoftrueEq = $true) & (notdallE = $true) & (descrp = $true) & (nonemptyI1 = $true) & (eqimpsubset1 = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (subset @ X1 @ (setadjoin @ X2 @ X0))) & ($true = (subset @ X1 @ X0))) & (emptysetsubset = $true) & (subsetI1 = $true) & (prop2setI = $true) & (prop2set2propI = $true) & (notsubsetI = $true) & (emptysetE = $true) & (emptyI = $true) & (exuE3e = $true) & (setadjoinAx = $true) & (exu__Cong = $true) & (setadjoinOr = $true) & (exuE1 = $true) & (notdexE = $true) & (emptyInPowerset = $true) & (exuI2 = $true) & (replAx = $true) & (notequalI1 = $true) & (subsetE = $true) & (upairsetE = $true) & (emptysetimpfalse = $true) & (uniqinunit = $true) & (notinemptyset = $true) & (notinsingleton = $true) & (exuI3 = $true) & (inCongP = $true) & (in__Cong = $true) & (noeltsimpempty = $true) & (emptyinunitempty = $true) & (setunionI = $true) & (emptyinPowerset = $true) & (setextAx = $true) & (exuI1 = $true) & (upairsetIR = $true) & (quantDeMorgan4 = $true) & (nonemptyImpWitness = $true) & (setunionE = $true) & (subPowSU = $true) & (quantDeMorgan3 = $true) & (descr__Cong = $true) & (setadjoinIL = $true) & (wellorderingAx = $true) & (emptysetAx = $true) & (dsetconstr__Cong = $true) & (notequalI2 = $true) & (nonemptyE1 = $true) & (powersetI = $true) & (eqimpsubset2 = $true) & (dsetconstrER = $true) & (dsetconstrI = $true) & (subsetRefl = $true) & (omega__Cong = $true) & (emptyset__Cong = $true) & (omegaSAx = $true) & (setadjoinIR = $true) & (exuE3u = $true) & (powersetAx = $true) & (emptyE1 = $true) & (omega0Ax = $true) & (singletonsswitch = $true) & (omegaIndAx = $true) & (subsetTrans = $true) & (setbeta = $true) & (powerset__Cong = $true) & (setext = $true) & (nonemptyI = $true) & (setunionAx = $true) & (dsetconstrEL = $true) & (powersetE = $true) & (eqinunit = $true) & (setadjoin__Cong = $true) & (vacuousDall = $true) & (foundationAx = $true) & (setadjoinSub = $true) & (prop2setE = $true) & (setunion__Cong = $true) & (subsetE2 = $true) & (exuEu = $true) & (setadjoinE = $true)),
% 2.24/1.60 inference(flattening,[status(thm)],[f374])).
% 2.24/1.60 thf(f374,plain,(
% 2.24/1.60 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i,X2 : $i] : (($true != (subset @ X1 @ (setadjoin @ X2 @ X0))) & ($true = (subset @ X1 @ X0))) & (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)),
% 2.24/1.60 inference(ennf_transformation,[status(thm)],[f188])).
% 2.24/1.60 thf(f188,plain,(
% 2.24/1.60 ~((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) => ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (subset @ X1 @ X0)) => ($true = (subset @ X1 @ (setadjoin @ X2 @ X0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.24/1.60 inference(fool_elimination,[status(thm)],[f187])).
% 2.24/1.60 thf(f187,plain,(
% 2.24/1.60 ~(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 => ! [X0 : $i,X1 : $i,X2 : $i] : ((subset @ X1 @ X0) => (subset @ X1 @ (setadjoin @ X2 @ X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.24/1.60 inference(rectify,[status(thm)],[f92])).
% 2.24/1.60 thf(f92,negated_conjecture,(
% 2.24/1.60 ~(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 => ! [X4 : $i,X3 : $i,X1 : $i] : ((subset @ X3 @ X4) => (subset @ X3 @ (setadjoin @ X1 @ X4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.24/1.60 inference(negated_conjecture,[status(cth)],[f91])).
% 2.24/1.60 thf(f91,conjecture,(
% 2.24/1.60 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 => ! [X4 : $i,X3 : $i,X1 : $i] : ((subset @ X3 @ X4) => (subset @ X3 @ (setadjoin @ X1 @ X4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.24/1.60 file('/export/starexec/sandbox/tmp/tmp.pvugr8aAKv/DTF2THF_16254.p',setadjoinSub2)).
% 2.24/1.60 thf(f1419,plain,(
% 2.24/1.60 ((subset @ sK132 @ sK131) != $true)),
% 2.24/1.60 inference(trivial_inequality_removal,[status(thm)],[f1418])).
% 2.24/1.60 thf(f1418,plain,(
% 2.24/1.60 ($true != $true) | ((subset @ sK132 @ sK131) != $true)),
% 2.24/1.60 inference(forward_demodulation,[status(thm)],[f1391,f1021])).
% 2.24/1.60 thf(f1021,plain,(
% 2.24/1.60 (setadjoinSub = $true)),
% 2.24/1.60 inference(cnf_transformation,[status(thm)],[f617])).
% 2.24/1.60 thf(f1391,plain,(
% 2.24/1.60 ((subset @ sK132 @ sK131) != $true) | (setadjoinSub != $true)),
% 2.24/1.60 inference(trivial_inequality_removal,[status(thm)],[f1380])).
% 2.24/1.60 thf(f1380,plain,(
% 2.24/1.60 ($true != $true) | (setadjoinSub != $true) | ((subset @ sK132 @ sK131) != $true)),
% 2.24/1.60 inference(superposition,[status(thm)],[f1379,f940])).
% 2.24/1.60 thf(f940,plain,(
% 2.24/1.60 ( ! [X2 : $i,X3 : $i] : (((subset @ X3 @ (setadjoin @ X2 @ X3)) = $true) | (setadjoinSub != $true)) )),
% 2.24/1.60 inference(cnf_transformation,[status(thm)],[f528])).
% 2.24/1.60 thf(f528,plain,(
% 2.24/1.60 ((setadjoinSub = $true) | ($true != (subset @ sK81 @ (setadjoin @ sK80 @ sK81)))) & (! [X2 : $i,X3 : $i] : ((subset @ X3 @ (setadjoin @ X2 @ X3)) = $true) | (setadjoinSub != $true))),
% 2.24/1.60 inference(skolemisation,[status(esa),new_symbols(skolem,[sK80,sK81])],[f526,f527])).
% 2.24/1.60 thf(f527,plain,(
% 2.24/1.60 ? [X0 : $i,X1 : $i] : ((subset @ X1 @ (setadjoin @ X0 @ X1)) != $true) => ($true != (subset @ sK81 @ (setadjoin @ sK80 @ sK81)))),
% 2.24/1.60 introduced(definition,[],[choice_axiom])).
% 2.24/1.60 thf(f526,plain,(
% 2.24/1.60 ((setadjoinSub = $true) | ? [X0 : $i,X1 : $i] : ((subset @ X1 @ (setadjoin @ X0 @ X1)) != $true)) & (! [X2 : $i,X3 : $i] : ((subset @ X3 @ (setadjoin @ X2 @ X3)) = $true) | (setadjoinSub != $true))),
% 2.24/1.60 inference(rectify,[status(thm)],[f525])).
% 2.24/1.60 thf(f525,plain,(
% 2.24/1.60 ((setadjoinSub = $true) | ? [X0 : $i,X1 : $i] : ((subset @ X1 @ (setadjoin @ X0 @ X1)) != $true)) & (! [X0 : $i,X1 : $i] : ((subset @ X1 @ (setadjoin @ X0 @ X1)) = $true) | (setadjoinSub != $true))),
% 2.24/1.60 inference(nnf_transformation,[status(thm)],[f150])).
% 2.24/1.60 thf(f150,plain,(
% 2.24/1.60 (setadjoinSub = $true) <=> ! [X0 : $i,X1 : $i] : ((subset @ X1 @ (setadjoin @ X0 @ X1)) = $true)),
% 2.24/1.60 inference(fool_elimination,[status(thm)],[f149])).
% 2.24/1.60 thf(f149,plain,(
% 2.24/1.60 (setadjoinSub = ! [X0 : $i,X1 : $i] : (subset @ X1 @ (setadjoin @ X0 @ X1)))),
% 2.24/1.60 inference(rectify,[status(thm)],[f90])).
% 2.24/1.60 thf(f90,axiom,(
% 2.24/1.60 (setadjoinSub = ! [X1 : $i,X3 : $i] : (subset @ X3 @ (setadjoin @ X1 @ X3)))),
% 2.24/1.60 file('/export/starexec/sandbox/tmp/tmp.pvugr8aAKv/DTF2THF_16254.p',setadjoinSub)).
% 2.24/1.60 thf(f1379,plain,(
% 2.24/1.60 ( ! [X0 : $i] : (($true != (subset @ X0 @ (setadjoin @ sK133 @ sK131))) | ($true != (subset @ sK132 @ X0))) )),
% 2.24/1.60 inference(trivial_inequality_removal,[status(thm)],[f1378])).
% 2.24/1.60 thf(f1378,plain,(
% 2.24/1.60 ( ! [X0 : $i] : (($true != $true) | ($true != (subset @ X0 @ (setadjoin @ sK133 @ sK131))) | ($true != (subset @ sK132 @ X0))) )),
% 2.24/1.60 inference(forward_demodulation,[status(thm)],[f1377,f1033])).
% 2.24/1.60 thf(f1033,plain,(
% 2.24/1.60 (subsetTrans = $true)),
% 2.24/1.60 inference(cnf_transformation,[status(thm)],[f617])).
% 2.24/1.60 thf(f1377,plain,(
% 2.24/1.60 ( ! [X0 : $i] : (($true != (subset @ sK132 @ X0)) | (subsetTrans != $true) | ($true != (subset @ X0 @ (setadjoin @ sK133 @ sK131)))) )),
% 2.24/1.60 inference(trivial_inequality_removal,[status(thm)],[f1376])).
% 2.24/1.60 thf(f1376,plain,(
% 2.24/1.60 ( ! [X0 : $i] : (($true != $true) | ($true != (subset @ sK132 @ X0)) | ($true != (subset @ X0 @ (setadjoin @ sK133 @ sK131))) | (subsetTrans != $true)) )),
% 2.24/1.60 inference(superposition,[status(thm)],[f1095,f1140])).
% 2.24/1.60 thf(f1140,plain,(
% 2.24/1.60 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true = (subset @ X1 @ X0)) | ((subset @ X1 @ X2) != $true) | ((subset @ X2 @ X0) != $true) | (subsetTrans != $true)) )),
% 2.24/1.60 inference(cnf_transformation,[status(thm)],[f658])).
% 2.24/1.60 thf(f658,plain,(
% 2.24/1.60 (! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X2 @ X0) != $true) | ($true = (subset @ X1 @ X0)) | ((subset @ X1 @ X2) != $true)) | (subsetTrans != $true)) & ((subsetTrans = $true) | (((subset @ sK154 @ sK152) = $true) & ($true != (subset @ sK153 @ sK152)) & ((subset @ sK153 @ sK154) = $true)))),
% 2.24/1.60 inference(skolemisation,[status(esa),new_symbols(skolem,[sK152,sK153,sK154])],[f656,f657])).
% 2.24/1.60 thf(f657,plain,(
% 2.24/1.60 ? [X3 : $i,X4 : $i,X5 : $i] : (((subset @ X5 @ X3) = $true) & ((subset @ X4 @ X3) != $true) & ((subset @ X4 @ X5) = $true)) => (((subset @ sK154 @ sK152) = $true) & ($true != (subset @ sK153 @ sK152)) & ((subset @ sK153 @ sK154) = $true))),
% 2.24/1.60 introduced(definition,[],[choice_axiom])).
% 2.24/1.60 thf(f656,plain,(
% 2.24/1.60 (! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X2 @ X0) != $true) | ($true = (subset @ X1 @ X0)) | ((subset @ X1 @ X2) != $true)) | (subsetTrans != $true)) & ((subsetTrans = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (((subset @ X5 @ X3) = $true) & ((subset @ X4 @ X3) != $true) & ((subset @ X4 @ X5) = $true)))),
% 2.24/1.60 inference(rectify,[status(thm)],[f655])).
% 2.24/1.60 thf(f655,plain,(
% 2.24/1.60 (! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X2 @ X0) != $true) | ($true = (subset @ X1 @ X0)) | ((subset @ X1 @ X2) != $true)) | (subsetTrans != $true)) & ((subsetTrans = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X2 @ X0) = $true) & ($true != (subset @ X1 @ X0)) & ((subset @ X1 @ X2) = $true)))),
% 2.24/1.60 inference(nnf_transformation,[status(thm)],[f323])).
% 2.24/1.60 thf(f323,plain,(
% 2.24/1.60 ! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X2 @ X0) != $true) | ($true = (subset @ X1 @ X0)) | ((subset @ X1 @ X2) != $true)) <=> (subsetTrans = $true)),
% 2.24/1.60 inference(flattening,[status(thm)],[f322])).
% 2.24/1.60 thf(f322,plain,(
% 2.24/1.60 (subsetTrans = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : ((($true = (subset @ X1 @ X0)) | ((subset @ X2 @ X0) != $true)) | ((subset @ X1 @ X2) != $true))),
% 2.24/1.60 inference(ennf_transformation,[status(thm)],[f152])).
% 2.24/1.60 thf(f152,plain,(
% 2.24/1.60 (subsetTrans = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X1 @ X2) = $true) => (((subset @ X2 @ X0) = $true) => ($true = (subset @ X1 @ X0))))),
% 2.24/1.60 inference(fool_elimination,[status(thm)],[f151])).
% 2.24/1.60 thf(f151,plain,(
% 2.24/1.60 (! [X0 : $i,X1 : $i,X2 : $i] : ((subset @ X1 @ X2) => ((subset @ X2 @ X0) => (subset @ X1 @ X0))) = subsetTrans)),
% 2.24/1.60 inference(rectify,[status(thm)],[f89])).
% 2.24/1.60 thf(f89,axiom,(
% 2.24/1.60 (! [X5 : $i,X3 : $i,X4 : $i] : ((subset @ X3 @ X4) => ((subset @ X4 @ X5) => (subset @ X3 @ X5))) = subsetTrans)),
% 2.24/1.60 file('/export/starexec/sandbox/tmp/tmp.pvugr8aAKv/DTF2THF_16254.p',subsetTrans)).
% 2.24/1.60 thf(f1095,plain,(
% 2.24/1.60 ($true != (subset @ sK132 @ (setadjoin @ sK133 @ sK131)))),
% 2.24/1.60 inference(cnf_transformation,[status(thm)],[f617])).
% 2.24/1.60 % SZS output end Proof for DTF2THF_16254
% 2.24/1.60 % (16606)------------------------------
% 2.24/1.60 % (16606)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.24/1.60 % (16606)Termination reason: Refutation
% 2.24/1.60
% 2.24/1.60 % (16606)Memory used [KB]: 6908
% 2.24/1.60 % (16606)Time elapsed: 0.070 s
% 2.24/1.60 % (16606)Instructions burned: 94 (million)
% 2.24/1.60 % (16606)------------------------------
% 2.24/1.60 % (16606)------------------------------
% 2.24/1.60 % (16583)Success in time 0.148 s
%------------------------------------------------------------------------------