%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU635^1 : 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 : n021.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 09:35:48 AM UTC 2026
% Result : Theorem 6.41s 2.37s
% Output : Refutation 6.41s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 8
% Syntax : Number of formulae : 49 ( 13 unt; 0 typ; 0 def)
% Number of atoms : 2668 ( 721 equ; 0 cnn)
% Maximal formula atoms : 153 ( 54 avg)
% Number of connectives : 1481 ( 59 ~; 42 |; 472 &; 289 @)
% ( 5 <=>; 614 =>; 0 <=; 0 <~>)
% Maximal formula depth : 158 ( 26 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 322 ( 319 usr; 161 con; 0-4 aty)
% Number of variables : 87 ( 0 ^; 66 !; 21 ?; 87 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_69,type,
set2prop: $i > $o ).
thf(func_def_88,type,
subset: $i > $i > $o ).
thf(func_def_89,type,
disjoint: $i > $i > $o ).
thf(func_def_90,type,
setsmeet: $i > $i > $o ).
thf(func_def_114,type,
binunion: $i > $i > $i ).
thf(func_def_122,type,
binintersect: $i > $i > $i ).
thf(func_def_135,type,
regular: $i > $o ).
thf(func_def_136,type,
setminus: $i > $i > $i ).
thf(func_def_147,type,
symdiff: $i > $i > $i ).
thf(func_def_153,type,
iskpair: $i > $o ).
thf(func_def_158,type,
kpair: $i > $i > $i ).
thf(func_def_160,type,
cartprod: $i > $i > $i ).
thf(func_def_187,type,
sP0: ( $i > $o ) > $o ).
thf(func_def_188,type,
sP1: $i > $i > $o > $o ).
thf(func_def_190,type,
sP3: $i > ( $i > $o ) > $o ).
thf(func_def_192,type,
sP5: ( $i > $o ) > $o ).
thf(func_def_194,type,
sP7: $i > $o > $i > $i > $o ).
thf(func_def_195,type,
sP8: $i > $o > $i > $i > $o ).
thf(func_def_196,type,
sP9: $i > $o ).
thf(func_def_197,type,
sP10: $i > $o ).
thf(func_def_198,type,
sP11: ( $i > $o ) > $i > $o ).
thf(func_def_201,type,
sP14: $i > $o > $i > $o ).
thf(func_def_203,type,
sP16: $i > $i > $i > $o ).
thf(func_def_204,type,
sP17: ( $i > $o ) > $o ).
thf(func_def_205,type,
sP18: $i > $o ).
thf(func_def_207,type,
sP20: $i > $i > $o ).
thf(func_def_209,type,
sP22: $i > $i > $o > $o ).
thf(func_def_210,type,
sP23: $i > $i > $i > $o ).
thf(func_def_211,type,
sP24: $i > ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_212,type,
sP25: ( $i > $o ) > $o ).
thf(func_def_214,type,
sP27: $i > $i > $o ).
thf(func_def_216,type,
sP29: ( $i > $o ) > $i > $o ).
thf(func_def_218,type,
sP31: $i > $i > $i > $o ).
thf(func_def_220,type,
sP33: ( $i > $i > $o ) > $i > $i > $o ).
thf(func_def_221,type,
sP34: ( $i > $i > $o ) > $i > $o ).
thf(func_def_223,type,
sP36: $i > ( $i > $o ) > $o ).
thf(func_def_225,type,
sP38: $i > $i > $i > $o ).
thf(func_def_226,type,
sP39: $i > $i > $o ).
thf(func_def_227,type,
sP40: $i > $o ).
thf(func_def_228,type,
sP41: $i > $i > $o ).
thf(func_def_229,type,
sP42: $i > $i > $o ).
thf(func_def_230,type,
sP43: $i > $i > $o ).
thf(func_def_231,type,
sP44: ( $i > $o ) > $i > $o ).
thf(func_def_235,type,
sP48: $i > ( $i > $o ) > $o ).
thf(func_def_237,type,
sP50: $i > $i > $o ).
thf(func_def_239,type,
sP52: ( $i > $o ) > $o ).
thf(func_def_243,type,
sK56: ( $i > $o ) > $i ).
thf(func_def_244,type,
sK57: ( $i > $o ) > $i ).
thf(func_def_245,type,
sK58: $i > $o ).
thf(func_def_261,type,
sK74: $i > $o ).
thf(func_def_263,type,
sK76: $i > ( $i > $o ) > $i ).
thf(func_def_264,type,
sK77: $i > ( $i > $o ) > $i ).
thf(func_def_265,type,
sK78: ( $i > $o ) > $i ).
thf(func_def_266,type,
sK79: $i > ( $i > $o ) > $i ).
thf(func_def_267,type,
sK80: $i > $o ).
thf(func_def_272,type,
sK85: $i > $i > $i ).
thf(func_def_296,type,
sK109: $i > $o ).
thf(func_def_299,type,
sK112: $i > $i ).
thf(func_def_300,type,
sK113: $i > $i ).
thf(func_def_302,type,
sK115: $i > $o ).
thf(func_def_304,type,
sK117: $i > ( $i > $o ) > $i ).
thf(func_def_305,type,
sK118: ( $i > $o ) > $i > $i ).
thf(func_def_309,type,
sK122: $i > $o ).
thf(func_def_314,type,
sK127: $i > $i > $i ).
thf(func_def_318,type,
sK131: $i > $o > $i > $i ).
thf(func_def_329,type,
sK142: $i > $i > $i ).
thf(func_def_356,type,
sK169: $i > $i > $i ).
thf(func_def_357,type,
sK170: $i > $i > $i > $i ).
thf(func_def_358,type,
sK171: $i > $i > $i > $i ).
thf(func_def_362,type,
sK175: $i > ( $i > $o ) > $i ).
thf(func_def_363,type,
sK176: ( $i > $o ) > $i ).
thf(func_def_364,type,
sK177: $i > $o ).
thf(func_def_365,type,
sK178: $i > $i > $i ).
thf(func_def_366,type,
sK179: $i > $i ).
thf(func_def_371,type,
sK184: $i > $o ).
thf(func_def_379,type,
sK192: $i > $o ).
thf(func_def_403,type,
sK216: $i > $i > $i ).
thf(func_def_406,type,
sK219: $i > $i > $i ).
thf(func_def_407,type,
sK220: $i > $i > $i ).
thf(func_def_408,type,
sK221: $i > $i > $i > $o ).
thf(func_def_419,type,
sK232: $i > $o ).
thf(func_def_426,type,
sK239: $i > ( $i > $o ) > ( $i > $o ) > $i > $i ).
thf(func_def_427,type,
sK240: $i > ( $i > $o ) > ( $i > $o ) > $i > $i ).
thf(func_def_430,type,
sK243: $i > $o ).
thf(func_def_431,type,
sK244: $i > $o ).
thf(func_def_432,type,
sK245: $i > ( $i > $o ) > $i ).
thf(func_def_433,type,
sK246: ( $i > $o ) > $i ).
thf(func_def_434,type,
sK247: $i > $o ).
thf(func_def_440,type,
sK253: $i > $o ).
thf(func_def_441,type,
sK254: $i > $o ).
thf(func_def_442,type,
sK255: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_443,type,
sK256: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_455,type,
sK268: $i > $o ).
thf(func_def_458,type,
sK271: $i > $o ).
thf(func_def_460,type,
sK273: $i > $o ).
thf(func_def_461,type,
sK274: ( $i > $o ) > $i ).
thf(func_def_465,type,
sK278: $i > $i > $i ).
thf(func_def_467,type,
sK280: $i > $o ).
thf(func_def_468,type,
sK281: ( $i > $o ) > $i > $i ).
thf(func_def_469,type,
sK282: ( $i > $o ) > $i > $i ).
thf(func_def_492,type,
sK305: $i > $i ).
thf(func_def_503,type,
sK316: $i > $i > $o ).
thf(func_def_504,type,
sK317: ( $i > $i > $o ) > $i > $i ).
thf(func_def_505,type,
sK318: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_506,type,
sK319: ( $i > $i > $o ) > $i > $i ).
thf(func_def_507,type,
sK320: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_516,type,
sK329: $i > $i ).
thf(func_def_530,type,
sK343: $i > $o ).
thf(func_def_532,type,
sK345: $i > ( $i > $o ) > $i ).
thf(func_def_533,type,
sK346: $i > ( $i > $o ) > $i ).
thf(func_def_540,type,
sK353: $i > $o ).
thf(func_def_543,type,
sK356: $i > $o ).
thf(func_def_544,type,
sK357: $i > $i > $i ).
thf(func_def_545,type,
sK358: $i > $i > $i ).
thf(func_def_546,type,
sK359: $i > $i > $i > $i ).
thf(func_def_547,type,
sK360: $i > $i > $i ).
thf(func_def_548,type,
sK361: $i > $i > $i ).
thf(func_def_549,type,
sK362: $i > $i > $i ).
thf(func_def_550,type,
sK363: $i > $i > $i ).
thf(func_def_551,type,
sK364: $i > $i > $i ).
thf(func_def_552,type,
sK365: $i > $i ).
thf(func_def_553,type,
sK366: $i > $i ).
thf(func_def_554,type,
sK367: $i > $i ).
thf(func_def_555,type,
sK368: $i > $i ).
thf(func_def_556,type,
sK369: $i > $i > $i ).
thf(func_def_557,type,
sK370: $i > $i > $i ).
thf(func_def_558,type,
sK371: $i > $i > $i ).
thf(func_def_559,type,
sK372: $i > $i > $i > $i ).
thf(func_def_560,type,
sK373: $i > $i > $i > $i ).
thf(func_def_562,type,
sK375: $i > $i ).
thf(func_def_564,type,
sK377: $i > $o ).
thf(func_def_566,type,
sK379: $i > $o ).
thf(func_def_567,type,
sK380: ( $i > $o ) > $i > $i ).
thf(func_def_568,type,
sK381: ( $i > $o ) > $i > $i ).
thf(func_def_571,type,
sK384: $i > $i > $i ).
thf(func_def_591,type,
sK404: $i > $i ).
thf(func_def_593,type,
sK406: $i > $o ).
thf(func_def_594,type,
sK407: $i > $o ).
thf(func_def_595,type,
sK408: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_596,type,
sK409: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_611,type,
sK424: $i > $o ).
thf(func_def_613,type,
sK426: $i > ( $i > $o ) > $i ).
thf(func_def_614,type,
sK427: $i > ( $i > $o ) > $i ).
thf(func_def_620,type,
sK433: $i > $o ).
thf(func_def_623,type,
sK436: $i > $i > $i ).
thf(func_def_624,type,
sK437: $i > $o ).
thf(func_def_627,type,
sK440: ( $i > $o ) > $i ).
thf(func_def_628,type,
sK441: $i > ( $i > $o ) > $i ).
thf(func_def_629,type,
sK442: $i > $o ).
thf(func_def_643,type,
sK456: $i > $i ).
thf(func_def_652,type,
ph465:
!>[X0: $tType] : X0 ).
thf(f4108,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f4106,f2883]) ).
thf(f2883,plain,
! [X2: $i,X3: $i] :
( $true
= ( in @ X3 @ ( setadjoin @ X3 @ X2 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2428]) ).
thf(f2428,plain,
! [X2: $i,X3: $i] :
( ( $true != $true )
| ( $true
= ( in @ X3 @ ( setadjoin @ X3 @ X2 ) ) ) ),
inference(definition_unfolding,[status(thm)],[f1665,f2088]) ).
thf(f2088,plain,
setadjoinIL = $true,
inference(cnf_transformation,[status(thm)],[f1266]) ).
thf(f1266,plain,
( ( prop2setI = $true )
& ( $true
!= ( subset @ sK321 @ ( setunion @ ( setadjoin @ sK321 @ emptyset ) ) ) )
& ( emptysetAx = $true )
& ( powersetE1 = $true )
& ( symdiffI1 = $true )
& ( binintersectRsub = $true )
& ( setminusI = $true )
& ( setextAx = $true )
& ( subsetemptysetimpeq = $true )
& ( exuE3u = $true )
& ( noeltsimpempty = $true )
& ( subsetE = $true )
& ( setminusSubset1 = $true )
& ( nonemptyE1 = $true )
& ( eqinunit = $true )
& ( setunionsingleton1 = $true )
& ( cartprodpairin = $true )
& ( emptyInPowerset = $true )
& ( dsetconstr__Cong = $true )
& ( notdallE = $true )
& ( powersetI1 = $true )
& ( setminusIRneg = $true )
& ( descr__Cong = $true )
& ( setadjoinE = $true )
& ( quantDeMorgan2 = $true )
& ( binintersectSubset4 = $true )
& ( notdexE = $true )
& ( powersetI = $true )
& ( emptysetsubset = $true )
& ( secondinupair = $true )
& ( setminusER = $true )
& ( setminusLsub = $true )
& ( quantDeMorgan1 = $true )
& ( binunionLsub = $true )
& ( emptyI = $true )
& ( setadjoin__Cong = $true )
& ( kpairiskpair = $true )
& ( powersetsubset = $true )
& ( setadjoinIL = $true )
& ( exuE2 = $true )
& ( ubforcartprodlem3 = $true )
& ( notequalI2 = $true )
& ( uniqinunit = $true )
& ( upairset2IR = $true )
& ( exu__Cong = $true )
& ( exuI3 = $true )
& ( exuEu = $true )
& ( exuE1 = $true )
& ( subsetTrans = $true )
& ( descrp = $true )
& ( cartprodmempair = $true )
& ( omegaIndAx = $true )
& ( emptyset__Cong = $true )
& ( powersetAx = $true )
& ( nonemptyI = $true )
& ( subsetI2 = $true )
& ( replAx = $true )
& ( sepInPowerset = $true )
& ( upairset2E = $true )
& ( nonemptyI1 = $true )
& ( omega0Ax = $true )
& ( symdiffI2 = $true )
& ( emptyE1 = $true )
& ( setminusSubset2 = $true )
& ( singletoninpowunion = $true )
& ( kpairp = $true )
& ( nonemptyImpWitness = $true )
& ( symdiffIneg1 = $true )
& ( upairsetIR = $true )
& ( setminusERneg = $true )
& ( powersetE = $true )
& ( prop2setE = $true )
& ( notequalI1 = $true )
& ( dsetconstrER = $true )
& ( wellorderingAx = $true )
& ( notinemptyset = $true )
& ( emptyinunitempty = $true )
& ( inCongP = $true )
& ( setext = $true )
& ( foundationAx = $true )
& ( setextsub = $true )
& ( setadjoinAx = $true )
& ( sepSubset = $true )
& ( subsetRefl = $true )
& ( setunion__Cong = $true )
& ( disjointsetsI1 = $true )
& ( setunionAx = $true )
& ( dsetconstrEL = $true )
& ( binintersectSubset3 = $true )
& ( eqimpsubset2 = $true )
& ( binintersectEL = $true )
& ( binintersectER = $true )
& ( setadjoinSub = $true )
& ( subsetE2 = $true )
& ( quantDeMorgan4 = $true )
& ( emptysetE = $true )
& ( upairinpowunion = $true )
& ( quantDeMorgan3 = $true )
& ( binintersectSubset1 = $true )
& ( binunionRsub = $true )
& ( inPowerset = $true )
& ( setunionE = $true )
& ( binintersectLsub = $true )
& ( subsetI1 = $true )
& ( upairsetIL = $true )
& ( setadjoinIR = $true )
& ( setminusILneg = $true )
& ( eqimpsubset1 = $true )
& ( emptyinPowerset = $true )
& ( cartprodmempair1 = $true )
& ( in__Cong = $true )
& ( omega__Cong = $true )
& ( binunionIR = $true )
& ( omegaSAx = $true )
& ( setoftrueEq = $true )
& ( emptysetimpfalse = $true )
& ( setunionI = $true )
& ( setminusELneg = $true )
& ( setadjoinSub2 = $true )
& ( exuI2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairsetE = $true )
& ( prop2set2propI = $true )
& ( powerset__Cong = $true )
& ( singletonsubset = $true )
& ( setukpairIL = $true )
& ( notsubsetI = $true )
& ( binunionE = $true )
& ( symdiffIneg2 = $true )
& ( dsetconstrI = $true )
& ( subset2powerset = $true )
& ( subPowSU = $true )
& ( bs114d = $true )
& ( setadjoinOr = $true )
& ( exuE3e = $true )
& ( binunionIL = $true )
& ( symdiffE = $true )
& ( setbeta = $true )
& ( setunionE2 = $true )
& ( ubforcartprodlem2 = $true )
& ( setukpairIR = $true )
& ( exuI1 = $true )
& ( notinsingleton = $true )
& ( singletoninpowerset = $true )
& ( singletonsswitch = $true )
& ( binunionEcases = $true )
& ( binintersectSubset2 = $true )
& ( binintersectI = $true )
& ( binintersectSubset5 = $true )
& ( vacuousDall = $true )
& ( setminusEL = $true )
& ( upairsubunion = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK321])],[f511,f1265]) ).
thf(f1265,plain,
( ? [X0: $i] :
( ( subset @ X0 @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) )
!= $true )
=> ( $true
!= ( subset @ sK321 @ ( setunion @ ( setadjoin @ sK321 @ emptyset ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f511,plain,
( ( prop2setI = $true )
& ? [X0: $i] :
( ( subset @ X0 @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) )
!= $true )
& ( emptysetAx = $true )
& ( powersetE1 = $true )
& ( symdiffI1 = $true )
& ( binintersectRsub = $true )
& ( setminusI = $true )
& ( setextAx = $true )
& ( subsetemptysetimpeq = $true )
& ( exuE3u = $true )
& ( noeltsimpempty = $true )
& ( subsetE = $true )
& ( setminusSubset1 = $true )
& ( nonemptyE1 = $true )
& ( eqinunit = $true )
& ( setunionsingleton1 = $true )
& ( cartprodpairin = $true )
& ( emptyInPowerset = $true )
& ( dsetconstr__Cong = $true )
& ( notdallE = $true )
& ( powersetI1 = $true )
& ( setminusIRneg = $true )
& ( descr__Cong = $true )
& ( setadjoinE = $true )
& ( quantDeMorgan2 = $true )
& ( binintersectSubset4 = $true )
& ( notdexE = $true )
& ( powersetI = $true )
& ( emptysetsubset = $true )
& ( secondinupair = $true )
& ( setminusER = $true )
& ( setminusLsub = $true )
& ( quantDeMorgan1 = $true )
& ( binunionLsub = $true )
& ( emptyI = $true )
& ( setadjoin__Cong = $true )
& ( kpairiskpair = $true )
& ( powersetsubset = $true )
& ( setadjoinIL = $true )
& ( exuE2 = $true )
& ( ubforcartprodlem3 = $true )
& ( notequalI2 = $true )
& ( uniqinunit = $true )
& ( upairset2IR = $true )
& ( exu__Cong = $true )
& ( exuI3 = $true )
& ( exuEu = $true )
& ( exuE1 = $true )
& ( subsetTrans = $true )
& ( descrp = $true )
& ( cartprodmempair = $true )
& ( omegaIndAx = $true )
& ( emptyset__Cong = $true )
& ( powersetAx = $true )
& ( nonemptyI = $true )
& ( subsetI2 = $true )
& ( replAx = $true )
& ( sepInPowerset = $true )
& ( upairset2E = $true )
& ( nonemptyI1 = $true )
& ( omega0Ax = $true )
& ( symdiffI2 = $true )
& ( emptyE1 = $true )
& ( setminusSubset2 = $true )
& ( singletoninpowunion = $true )
& ( kpairp = $true )
& ( nonemptyImpWitness = $true )
& ( symdiffIneg1 = $true )
& ( upairsetIR = $true )
& ( setminusERneg = $true )
& ( powersetE = $true )
& ( prop2setE = $true )
& ( notequalI1 = $true )
& ( dsetconstrER = $true )
& ( wellorderingAx = $true )
& ( notinemptyset = $true )
& ( emptyinunitempty = $true )
& ( inCongP = $true )
& ( setext = $true )
& ( foundationAx = $true )
& ( setextsub = $true )
& ( setadjoinAx = $true )
& ( sepSubset = $true )
& ( subsetRefl = $true )
& ( setunion__Cong = $true )
& ( disjointsetsI1 = $true )
& ( setunionAx = $true )
& ( dsetconstrEL = $true )
& ( binintersectSubset3 = $true )
& ( eqimpsubset2 = $true )
& ( binintersectEL = $true )
& ( binintersectER = $true )
& ( setadjoinSub = $true )
& ( subsetE2 = $true )
& ( quantDeMorgan4 = $true )
& ( emptysetE = $true )
& ( upairinpowunion = $true )
& ( quantDeMorgan3 = $true )
& ( binintersectSubset1 = $true )
& ( binunionRsub = $true )
& ( inPowerset = $true )
& ( setunionE = $true )
& ( binintersectLsub = $true )
& ( subsetI1 = $true )
& ( upairsetIL = $true )
& ( setadjoinIR = $true )
& ( setminusILneg = $true )
& ( eqimpsubset1 = $true )
& ( emptyinPowerset = $true )
& ( cartprodmempair1 = $true )
& ( in__Cong = $true )
& ( omega__Cong = $true )
& ( binunionIR = $true )
& ( omegaSAx = $true )
& ( setoftrueEq = $true )
& ( emptysetimpfalse = $true )
& ( setunionI = $true )
& ( setminusELneg = $true )
& ( setadjoinSub2 = $true )
& ( exuI2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairsetE = $true )
& ( prop2set2propI = $true )
& ( powerset__Cong = $true )
& ( singletonsubset = $true )
& ( setukpairIL = $true )
& ( notsubsetI = $true )
& ( binunionE = $true )
& ( symdiffIneg2 = $true )
& ( dsetconstrI = $true )
& ( subset2powerset = $true )
& ( subPowSU = $true )
& ( bs114d = $true )
& ( setadjoinOr = $true )
& ( exuE3e = $true )
& ( binunionIL = $true )
& ( symdiffE = $true )
& ( setbeta = $true )
& ( setunionE2 = $true )
& ( ubforcartprodlem2 = $true )
& ( setukpairIR = $true )
& ( exuI1 = $true )
& ( notinsingleton = $true )
& ( singletoninpowerset = $true )
& ( singletonsswitch = $true )
& ( binunionEcases = $true )
& ( binintersectSubset2 = $true )
& ( binintersectI = $true )
& ( binintersectSubset5 = $true )
& ( vacuousDall = $true )
& ( setminusEL = $true )
& ( upairsubunion = $true ) ),
inference(flattening,[status(thm)],[f510]) ).
thf(f510,plain,
( ? [X0: $i] :
( ( subset @ X0 @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) )
!= $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f330]) ).
thf(f330,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ! [X0: $i] :
( ( subset @ X0 @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f329]) ).
thf(f329,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ! [X0: $i] : ( subset @ X0 @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f157]) ).
thf(f157,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ! [X3: $i] : ( subset @ X3 @ ( setunion @ ( setadjoin @ X3 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f156]) ).
thf(f156,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ! [X3: $i] : ( subset @ X3 @ ( setunion @ ( setadjoin @ X3 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.sSg5lqwKQT/DTF2THF_20989.p',setunionsingleton2) ).
thf(f1665,plain,
! [X2: $i,X3: $i] :
( ( $true
= ( in @ X3 @ ( setadjoin @ X3 @ X2 ) ) )
| ( setadjoinIL != $true ) ),
inference(cnf_transformation,[status(thm)],[f893]) ).
thf(f893,plain,
( ( ( setadjoinIL = $true )
| ( $true
!= ( in @ sK120 @ ( setadjoin @ sK120 @ sK119 ) ) ) )
& ( ! [X2: $i,X3: $i] :
( $true
= ( in @ X3 @ ( setadjoin @ X3 @ X2 ) ) )
| ( setadjoinIL != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK119,sK120])],[f891,f892]) ).
thf(f892,plain,
( ? [X0: $i,X1: $i] :
( $true
!= ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) )
=> ( $true
!= ( in @ sK120 @ ( setadjoin @ sK120 @ sK119 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f891,plain,
( ( ( setadjoinIL = $true )
| ? [X0: $i,X1: $i] :
( $true
!= ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) ) )
& ( ! [X2: $i,X3: $i] :
( $true
= ( in @ X3 @ ( setadjoin @ X3 @ X2 ) ) )
| ( setadjoinIL != $true ) ) ),
inference(rectify,[status(thm)],[f890]) ).
thf(f890,plain,
( ( ( setadjoinIL = $true )
| ? [X0: $i,X1: $i] :
( $true
!= ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) ) )
& ( ! [X0: $i,X1: $i] :
( $true
= ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) )
| ( setadjoinIL != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f288]) ).
thf(f288,plain,
( ( setadjoinIL = $true )
<=> ! [X0: $i,X1: $i] :
( $true
= ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) ) ),
inference(fool_elimination,[status(thm)],[f287]) ).
thf(f287,plain,
! [X0: $i,X1: $i] :
( ( in @ X1 @ ( setadjoin @ X1 @ X0 ) )
= setadjoinIL ),
inference(rectify,[status(thm)],[f31]) ).
thf(f31,axiom,
! [X2: $i,X1: $i] :
( ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
= setadjoinIL ),
file('/export/starexec/sandbox/tmp/tmp.sSg5lqwKQT/DTF2THF_20989.p',setadjoinIL) ).
thf(f4106,plain,
( $true
!= ( in @ sK321 @ ( setadjoin @ sK321 @ emptyset ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f4098]) ).
thf(f4098,plain,
( ( $true != $true )
| ( $true
!= ( in @ sK321 @ ( setadjoin @ sK321 @ emptyset ) ) ) ),
inference(superposition,[status(thm)],[f2125,f4009]) ).
thf(f4009,plain,
! [X0: $i,X1: $i] :
( ( $true
= ( subset @ X0 @ ( setunion @ X1 ) ) )
| ( $true
!= ( in @ X0 @ X1 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f4005]) ).
thf(f4005,plain,
! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ X1 ) )
| ( $true != $true )
| ( $true
= ( subset @ X0 @ ( setunion @ X1 ) ) ) ),
inference(superposition,[status(thm)],[f2879,f3012]) ).
thf(f3012,plain,
! [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ ( setunion @ X0 ) ) ) )
| ( $true
!= ( in @ X1 @ X0 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2705]) ).
thf(f2705,plain,
! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X1 @ X0 ) )
| ( $true != $true )
| ( $true
= ( in @ X1 @ ( powerset @ ( setunion @ X0 ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f2239,f1995]) ).
thf(f1995,plain,
subPowSU = $true,
inference(cnf_transformation,[status(thm)],[f1266]) ).
thf(f2239,plain,
! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X1 @ X0 ) )
| ( $true
= ( in @ X1 @ ( powerset @ ( setunion @ X0 ) ) ) )
| ( subPowSU != $true ) ),
inference(cnf_transformation,[status(thm)],[f1397]) ).
thf(f1397,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X1 @ X0 ) )
| ( $true
= ( in @ X1 @ ( powerset @ ( setunion @ X0 ) ) ) ) )
| ( subPowSU != $true ) )
& ( ( subPowSU = $true )
| ( ( $true
= ( in @ sK387 @ sK386 ) )
& ( $true
!= ( in @ sK387 @ ( powerset @ ( setunion @ sK386 ) ) ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK386,sK387])],[f1395,f1396]) ).
thf(f1396,plain,
( ? [X2: $i,X3: $i] :
( ( $true
= ( in @ X3 @ X2 ) )
& ( ( in @ X3 @ ( powerset @ ( setunion @ X2 ) ) )
!= $true ) )
=> ( ( $true
= ( in @ sK387 @ sK386 ) )
& ( $true
!= ( in @ sK387 @ ( powerset @ ( setunion @ sK386 ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1395,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X1 @ X0 ) )
| ( $true
= ( in @ X1 @ ( powerset @ ( setunion @ X0 ) ) ) ) )
| ( subPowSU != $true ) )
& ( ( subPowSU = $true )
| ? [X2: $i,X3: $i] :
( ( $true
= ( in @ X3 @ X2 ) )
& ( ( in @ X3 @ ( powerset @ ( setunion @ X2 ) ) )
!= $true ) ) ) ),
inference(rectify,[status(thm)],[f1394]) ).
thf(f1394,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X1 @ X0 ) )
| ( $true
= ( in @ X1 @ ( powerset @ ( setunion @ X0 ) ) ) ) )
| ( subPowSU != $true ) )
& ( ( subPowSU = $true )
| ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ X0 ) )
& ( $true
!= ( in @ X1 @ ( powerset @ ( setunion @ X0 ) ) ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f561]) ).
thf(f561,plain,
( ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X1 @ X0 ) )
| ( $true
= ( in @ X1 @ ( powerset @ ( setunion @ X0 ) ) ) ) )
<=> ( subPowSU = $true ) ),
inference(ennf_transformation,[status(thm)],[f216]) ).
thf(f216,plain,
( ( subPowSU = $true )
<=> ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ X0 ) )
=> ( $true
= ( in @ X1 @ ( powerset @ ( setunion @ X0 ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f215]) ).
thf(f215,plain,
( subPowSU
= ( ! [X0: $i,X1: $i] :
( ( in @ X1 @ X0 )
=> ( in @ X1 @ ( powerset @ ( setunion @ X0 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f43]) ).
thf(f43,axiom,
( subPowSU
= ( ! [X3: $i,X1: $i] :
( ( in @ X1 @ X3 )
=> ( in @ X1 @ ( powerset @ ( setunion @ X3 ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.sSg5lqwKQT/DTF2THF_20989.p',subPowSU) ).
thf(f2879,plain,
! [X2: $i,X3: $i] :
( ( $true
!= ( in @ X2 @ ( powerset @ X3 ) ) )
| ( $true
= ( subset @ X2 @ X3 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2798]) ).
thf(f2798,plain,
! [X2: $i,X3: $i] :
( ( $true
= ( subset @ X2 @ X3 ) )
| ( $true != $true )
| ( $true
!= ( in @ X2 @ ( powerset @ X3 ) ) ) ),
inference(definition_unfolding,[status(thm)],[f2350,f2123]) ).
thf(f2123,plain,
powersetE1 = $true,
inference(cnf_transformation,[status(thm)],[f1266]) ).
thf(f2350,plain,
! [X2: $i,X3: $i] :
( ( $true
!= ( in @ X2 @ ( powerset @ X3 ) ) )
| ( $true
= ( subset @ X2 @ X3 ) )
| ( powersetE1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f1540]) ).
thf(f1540,plain,
( ( ( powersetE1 = $true )
| ( ( $true
= ( in @ sK462 @ ( powerset @ sK463 ) ) )
& ( $true
!= ( subset @ sK462 @ sK463 ) ) ) )
& ( ! [X2: $i,X3: $i] :
( ( $true
!= ( in @ X2 @ ( powerset @ X3 ) ) )
| ( $true
= ( subset @ X2 @ X3 ) ) )
| ( powersetE1 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK462,sK463])],[f1538,f1539]) ).
thf(f1539,plain,
( ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) )
& ( $true
!= ( subset @ X0 @ X1 ) ) )
=> ( ( $true
= ( in @ sK462 @ ( powerset @ sK463 ) ) )
& ( $true
!= ( subset @ sK462 @ sK463 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1538,plain,
( ( ( powersetE1 = $true )
| ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) )
& ( $true
!= ( subset @ X0 @ X1 ) ) ) )
& ( ! [X2: $i,X3: $i] :
( ( $true
!= ( in @ X2 @ ( powerset @ X3 ) ) )
| ( $true
= ( subset @ X2 @ X3 ) ) )
| ( powersetE1 != $true ) ) ),
inference(rectify,[status(thm)],[f1537]) ).
thf(f1537,plain,
( ( ( powersetE1 = $true )
| ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) )
& ( $true
!= ( subset @ X0 @ X1 ) ) ) )
& ( ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ ( powerset @ X1 ) ) )
| ( $true
= ( subset @ X0 @ X1 ) ) )
| ( powersetE1 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f517]) ).
thf(f517,plain,
( ( powersetE1 = $true )
<=> ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ ( powerset @ X1 ) ) )
| ( $true
= ( subset @ X0 @ X1 ) ) ) ),
inference(ennf_transformation,[status(thm)],[f178]) ).
thf(f178,plain,
( ( powersetE1 = $true )
<=> ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) )
=> ( $true
= ( subset @ X0 @ X1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f177]) ).
thf(f177,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
=> ( subset @ X0 @ X1 ) )
= powersetE1 ),
inference(rectify,[status(thm)],[f96]) ).
thf(f96,axiom,
! [X4: $i,X3: $i] :
( ( ( in @ X4 @ ( powerset @ X3 ) )
=> ( subset @ X4 @ X3 ) )
= powersetE1 ),
file('/export/starexec/sandbox/tmp/tmp.sSg5lqwKQT/DTF2THF_20989.p',powersetE1) ).
thf(f2125,plain,
( $true
!= ( subset @ sK321 @ ( setunion @ ( setadjoin @ sK321 @ emptyset ) ) ) ),
inference(cnf_transformation,[status(thm)],[f1266]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU635^1 : 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
% 0.15/0.34 % Computer : n021.cluster.edu
% 0.15/0.34 % Model : x86_64 x86_64
% 0.15/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.34 % Memory : 8042.1875MB
% 0.15/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34 % CPULimit : 300
% 0.15/0.34 % WCLimit : 300
% 0.15/0.34 % DateTime : Tue Feb 24 12:29:09 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.20/0.46 ---- Original DTF file ---
% 0.20/0.46 thf(spec,logic,$$dhol).
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 % File : SEU635^1 : TPTP v9.2.1. Released v3.7.0.
% 0.20/0.46 % Domain : Set Theory
% 0.20/0.46 % Problem : Ordered Pairs - Singletons
% 0.20/0.46 % Version : Especial.
% 0.20/0.46 % English : (! A:i.subset A (setunion (setadjoin A emptyset)))
% 0.20/0.46
% 0.20/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.20/0.46 % Source : [Bro08]
% 0.20/0.46 % Names : ZFC137g [Bro08]
% 0.20/0.46
% 0.20/0.46 % Status : Theorem
% 0.20/0.46 % Rating : 0.22 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.09 v7.5.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.40 v6.2.0, 0.29 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 0.80 v4.1.0, 1.00 v4.0.1, 0.67 v3.7.0
% 0.20/0.46 % Syntax : Number of formulae : 330 ( 155 unt; 174 typ; 155 def)
% 0.20/0.46 % Number of atoms : 822 ( 216 equ; 0 cnn)
% 0.20/0.46 % Maximal formula atoms : 152 ( 5 avg)
% 0.20/0.46 % Number of connectives : 1414 ( 47 ~; 7 |; 33 &; 953 @)
% 0.20/0.46 % ( 14 <=>; 360 =>; 0 <=; 0 <~>)
% 0.20/0.46 % Maximal formula depth : 157 ( 2 avg)
% 0.20/0.46 % Number of types : 2 ( 0 usr)
% 0.20/0.46 % Number of type conns : 69 ( 69 >; 0 *; 0 +; 0 <<)
% 0.20/0.46 % Number of symbols : 177 ( 174 usr; 155 con; 0-2 aty)
% 0.20/0.46 % Number of variables : 448 ( 33 ^; 381 !; 34 ?; 448 :)
% 0.20/0.46 % SPC : TH0_THM_EQU_NAR
% 0.20/0.46
% 0.20/0.46 % Comments : http://mathgate.info/detsetitem.php?id=192
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 thf(in_type,type,
% 0.20/0.46 in: $i > $i > $o ).
% 0.20/0.46
% 0.20/0.46 thf(exu_type,type,
% 0.20/0.46 exu: ( $i > $o ) > $o ).
% 0.20/0.46
% 0.20/0.46 thf(exu,definition,
% 0.20/0.46 ( exu
% 0.20/0.46 = ( ^ [Xphi: $i > $o] :
% 0.20/0.46 ? [Xx: $i] :
% 0.20/0.46 ( ( Xphi @ Xx )
% 0.20/0.46 & ! [Xy: $i] :
% 0.20/0.46 ( ( Xphi @ Xy )
% 0.20/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setextAx_type,type,
% 0.20/0.46 setextAx: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setextAx,definition,
% 0.20/0.46 ( setextAx
% 0.20/0.46 = ( ! [A: $i,B: $i] :
% 0.20/0.46 ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 <=> ( in @ Xx @ B ) )
% 0.20/0.46 => ( A = B ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(emptyset_type,type,
% 0.20/0.46 emptyset: $i ).
% 0.20/0.46
% 0.20/0.46 thf(emptysetAx_type,type,
% 0.20/0.46 emptysetAx: $o ).
% 0.20/0.46
% 0.20/0.46 thf(emptysetAx,definition,
% 0.20/0.46 ( emptysetAx
% 0.20/0.46 = ( ! [Xx: $i] :
% 0.20/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setadjoin_type,type,
% 0.20/0.46 setadjoin: $i > $i > $i ).
% 0.20/0.46
% 0.20/0.46 thf(setadjoinAx_type,type,
% 0.20/0.46 setadjoinAx: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setadjoinAx,definition,
% 0.20/0.46 ( setadjoinAx
% 0.20/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.20/0.46 <=> ( ( Xy = Xx )
% 0.20/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(powerset_type,type,
% 0.20/0.46 powerset: $i > $i ).
% 0.20/0.46
% 0.20/0.46 thf(powersetAx_type,type,
% 0.20/0.46 powersetAx: $o ).
% 0.20/0.46
% 0.20/0.46 thf(powersetAx,definition,
% 0.20/0.46 ( powersetAx
% 0.20/0.46 = ( ! [A: $i,B: $i] :
% 0.20/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.20/0.46 <=> ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ B )
% 0.20/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setunion_type,type,
% 0.20/0.46 setunion: $i > $i ).
% 0.20/0.46
% 0.20/0.46 thf(setunionAx_type,type,
% 0.20/0.46 setunionAx: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setunionAx,definition,
% 0.20/0.46 ( setunionAx
% 0.20/0.46 = ( ! [A: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.20/0.46 <=> ? [B: $i] :
% 0.20/0.46 ( ( in @ Xx @ B )
% 0.20/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(omega_type,type,
% 0.20/0.46 omega: $i ).
% 0.20/0.46
% 0.20/0.46 thf(omega0Ax_type,type,
% 0.20/0.46 omega0Ax: $o ).
% 0.20/0.46
% 0.20/0.46 thf(omega0Ax,definition,
% 0.20/0.46 ( omega0Ax
% 0.20/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(omegaSAx_type,type,
% 0.20/0.46 omegaSAx: $o ).
% 0.20/0.46
% 0.20/0.46 thf(omegaSAx,definition,
% 0.20/0.46 ( omegaSAx
% 0.20/0.46 = ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ omega )
% 0.20/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(omegaIndAx_type,type,
% 0.20/0.46 omegaIndAx: $o ).
% 0.20/0.46
% 0.20/0.46 thf(omegaIndAx,definition,
% 0.20/0.46 ( omegaIndAx
% 0.20/0.46 = ( ! [A: $i] :
% 0.20/0.46 ( ( ( in @ emptyset @ A )
% 0.20/0.46 & ! [Xx: $i] :
% 0.20/0.46 ( ( ( in @ Xx @ omega )
% 0.20/0.46 & ( in @ Xx @ A ) )
% 0.20/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ omega )
% 0.20/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(replAx_type,type,
% 0.20/0.46 replAx: $o ).
% 0.20/0.46
% 0.20/0.46 thf(replAx,definition,
% 0.20/0.46 ( replAx
% 0.20/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.20/0.46 ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( exu
% 0.20/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.20/0.46 => ? [B: $i] :
% 0.20/0.46 ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ B )
% 0.20/0.46 <=> ? [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ A )
% 0.20/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(foundationAx_type,type,
% 0.20/0.46 foundationAx: $o ).
% 0.20/0.46
% 0.20/0.46 thf(foundationAx,definition,
% 0.20/0.46 ( foundationAx
% 0.20/0.46 = ( ! [A: $i] :
% 0.20/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.20/0.46 => ? [B: $i] :
% 0.20/0.46 ( ( in @ B @ A )
% 0.20/0.46 & ~ ? [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ B )
% 0.20/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(wellorderingAx_type,type,
% 0.20/0.46 wellorderingAx: $o ).
% 0.20/0.46
% 0.20/0.46 thf(wellorderingAx,definition,
% 0.20/0.46 ( wellorderingAx
% 0.20/0.46 = ( ! [A: $i] :
% 0.20/0.46 ? [B: $i] :
% 0.20/0.46 ( ! [C: $i] :
% 0.20/0.46 ( ( in @ C @ B )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ C )
% 0.20/0.46 => ( in @ Xx @ A ) ) )
% 0.20/0.46 & ! [Xx: $i,Xy: $i] :
% 0.20/0.46 ( ( ( in @ Xx @ A )
% 0.20/0.46 & ( in @ Xy @ A ) )
% 0.20/0.46 => ( ! [C: $i] :
% 0.20/0.46 ( ( in @ C @ B )
% 0.20/0.46 => ( ( in @ Xx @ C )
% 0.20/0.46 <=> ( in @ Xy @ C ) ) )
% 0.20/0.46 => ( Xx = Xy ) ) )
% 0.20/0.46 & ! [C: $i,D: $i] :
% 0.20/0.46 ( ( ( in @ C @ B )
% 0.20/0.46 & ( in @ D @ B ) )
% 0.20/0.46 => ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ C )
% 0.20/0.46 => ( in @ Xx @ D ) )
% 0.20/0.46 | ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ D )
% 0.20/0.46 => ( in @ Xx @ C ) ) ) )
% 0.20/0.46 & ! [C: $i] :
% 0.20/0.46 ( ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ C )
% 0.20/0.46 => ( in @ Xx @ A ) )
% 0.20/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.20/0.46 => ? [D: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ D @ B )
% 0.20/0.46 & ( in @ Xx @ C )
% 0.20/0.46 & ~ ? [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ D )
% 0.20/0.46 & ( in @ Xy @ C ) )
% 0.20/0.46 & ! [E: $i] :
% 0.20/0.46 ( ( in @ E @ B )
% 0.20/0.47 => ( ! [Xy: $i] :
% 0.20/0.47 ( ( in @ Xy @ E )
% 0.20/0.47 => ( in @ Xy @ D ) )
% 0.20/0.47 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(descr_type,type,
% 0.20/0.47 descr: ( $i > $o ) > $i ).
% 0.20/0.47
% 0.20/0.47 thf(descrp_type,type,
% 0.20/0.47 descrp: $o ).
% 0.20/0.47
% 0.20/0.47 thf(descrp,definition,
% 0.20/0.47 ( descrp
% 0.20/0.47 = ( ! [Xphi: $i > $o] :
% 0.20/0.47 ( ( exu
% 0.20/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.47 => ( Xphi
% 0.20/0.47 @ ( descr
% 0.20/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(dsetconstr_type,type,
% 0.20/0.47 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.20/0.47
% 0.20/0.47 thf(dsetconstrI_type,type,
% 0.20/0.47 dsetconstrI: $o ).
% 0.20/0.47
% 0.20/0.47 thf(dsetconstrI,definition,
% 0.20/0.47 ( dsetconstrI
% 0.20/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ( ( Xphi @ Xx )
% 0.20/0.47 => ( in @ Xx
% 0.20/0.47 @ ( dsetconstr @ A
% 0.20/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(dsetconstrEL_type,type,
% 0.20/0.47 dsetconstrEL: $o ).
% 0.20/0.47
% 0.20/0.47 thf(dsetconstrEL,definition,
% 0.20/0.47 ( dsetconstrEL
% 0.20/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx
% 0.20/0.47 @ ( dsetconstr @ A
% 0.20/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.20/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(dsetconstrER_type,type,
% 0.20/0.47 dsetconstrER: $o ).
% 0.20/0.47
% 0.20/0.47 thf(dsetconstrER,definition,
% 0.20/0.47 ( dsetconstrER
% 0.20/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx
% 0.20/0.47 @ ( dsetconstr @ A
% 0.20/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.20/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(exuE1_type,type,
% 0.20/0.47 exuE1: $o ).
% 0.20/0.47
% 0.20/0.47 thf(exuE1,definition,
% 0.20/0.47 ( exuE1
% 0.20/0.47 = ( ! [Xphi: $i > $o] :
% 0.20/0.47 ( ( exu
% 0.20/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.47 => ? [Xx: $i] :
% 0.20/0.47 ( ( Xphi @ Xx )
% 0.20/0.47 & ! [Xy: $i] :
% 0.20/0.47 ( ( Xphi @ Xy )
% 0.20/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(prop2set_type,type,
% 0.20/0.47 prop2set: $o > $i ).
% 0.20/0.47
% 0.20/0.47 thf(prop2setE_type,type,
% 0.20/0.47 prop2setE: $o ).
% 0.20/0.47
% 0.20/0.47 thf(prop2setE,definition,
% 0.20/0.47 ( prop2setE
% 0.20/0.47 = ( ! [Xphi: $o,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.20/0.47 => Xphi ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(emptysetE_type,type,
% 0.20/0.47 emptysetE: $o ).
% 0.20/0.47
% 0.20/0.47 thf(emptysetE,definition,
% 0.20/0.47 ( emptysetE
% 0.20/0.47 = ( ! [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ emptyset )
% 0.20/0.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(emptysetimpfalse_type,type,
% 0.20/0.47 emptysetimpfalse: $o ).
% 0.20/0.47
% 0.20/0.47 thf(emptysetimpfalse,definition,
% 0.20/0.47 ( emptysetimpfalse
% 0.20/0.47 = ( ! [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ emptyset )
% 0.20/0.47 => $false ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(notinemptyset_type,type,
% 0.20/0.47 notinemptyset: $o ).
% 0.20/0.47
% 0.20/0.47 thf(notinemptyset,definition,
% 0.20/0.47 ( notinemptyset
% 0.20/0.47 = ( ! [Xx: $i] :
% 0.20/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(exuE3e_type,type,
% 0.20/0.47 exuE3e: $o ).
% 0.20/0.47
% 0.20/0.47 thf(exuE3e,definition,
% 0.20/0.47 ( exuE3e
% 0.20/0.47 = ( ! [Xphi: $i > $o] :
% 0.20/0.47 ( ( exu
% 0.20/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.47 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setext_type,type,
% 0.20/0.47 setext: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setext,definition,
% 0.20/0.47 ( setext
% 0.20/0.47 = ( ! [A: $i,B: $i] :
% 0.20/0.47 ( ! [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ( in @ Xx @ B ) )
% 0.20/0.47 => ( ! [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ B )
% 0.20/0.47 => ( in @ Xx @ A ) )
% 0.20/0.47 => ( A = B ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(emptyI_type,type,
% 0.20/0.47 emptyI: $o ).
% 0.20/0.47
% 0.20/0.47 thf(emptyI,definition,
% 0.20/0.47 ( emptyI
% 0.20/0.47 = ( ! [A: $i] :
% 0.20/0.47 ( ! [Xx: $i] :
% 0.20/0.47 ~ ( in @ Xx @ A )
% 0.20/0.47 => ( A = emptyset ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(noeltsimpempty_type,type,
% 0.20/0.47 noeltsimpempty: $o ).
% 0.20/0.47
% 0.20/0.47 thf(noeltsimpempty,definition,
% 0.20/0.47 ( noeltsimpempty
% 0.20/0.47 = ( ! [A: $i] :
% 0.20/0.47 ( ! [Xx: $i] :
% 0.20/0.47 ~ ( in @ Xx @ A )
% 0.20/0.47 => ( A = emptyset ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setbeta_type,type,
% 0.20/0.47 setbeta: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setbeta,definition,
% 0.20/0.47 ( setbeta
% 0.20/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ( ( in @ Xx
% 0.20/0.47 @ ( dsetconstr @ A
% 0.20/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.20/0.47 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(nonempty_type,type,
% 0.20/0.47 nonempty: $i > $o ).
% 0.20/0.47
% 0.20/0.47 thf(nonempty,definition,
% 0.20/0.47 ( nonempty
% 0.20/0.47 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(nonemptyE1_type,type,
% 0.20/0.47 nonemptyE1: $o ).
% 0.20/0.47
% 0.20/0.47 thf(nonemptyE1,definition,
% 0.20/0.47 ( nonemptyE1
% 0.20/0.47 = ( ! [A: $i] :
% 0.20/0.47 ( ( nonempty @ A )
% 0.20/0.47 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(nonemptyI_type,type,
% 0.20/0.47 nonemptyI: $o ).
% 0.20/0.47
% 0.20/0.47 thf(nonemptyI,definition,
% 0.20/0.47 ( nonemptyI
% 0.20/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ( ( Xphi @ Xx )
% 0.20/0.47 => ( nonempty
% 0.20/0.47 @ ( dsetconstr @ A
% 0.20/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(nonemptyI1_type,type,
% 0.20/0.47 nonemptyI1: $o ).
% 0.20/0.47
% 0.20/0.47 thf(nonemptyI1,definition,
% 0.20/0.47 ( nonemptyI1
% 0.20/0.47 = ( ! [A: $i] :
% 0.20/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.20/0.47 => ( nonempty @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setadjoinIL_type,type,
% 0.20/0.47 setadjoinIL: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setadjoinIL,definition,
% 0.20/0.47 ( setadjoinIL
% 0.20/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(emptyinunitempty_type,type,
% 0.20/0.47 emptyinunitempty: $o ).
% 0.20/0.47
% 0.20/0.47 thf(emptyinunitempty,definition,
% 0.20/0.47 ( emptyinunitempty
% 0.20/0.47 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setadjoinIR_type,type,
% 0.20/0.47 setadjoinIR: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setadjoinIR,definition,
% 0.20/0.47 ( setadjoinIR
% 0.20/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.20/0.47 ( ( in @ Xy @ A )
% 0.20/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setadjoinE_type,type,
% 0.20/0.47 setadjoinE: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setadjoinE,definition,
% 0.20/0.47 ( setadjoinE
% 0.20/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.20/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.20/0.47 => ! [Xphi: $o] :
% 0.20/0.47 ( ( ( Xy = Xx )
% 0.20/0.47 => Xphi )
% 0.20/0.47 => ( ( ( in @ Xy @ A )
% 0.20/0.47 => Xphi )
% 0.20/0.47 => Xphi ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setadjoinOr_type,type,
% 0.20/0.47 setadjoinOr: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setadjoinOr,definition,
% 0.20/0.47 ( setadjoinOr
% 0.20/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.20/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.20/0.47 => ( ( Xy = Xx )
% 0.20/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setoftrueEq_type,type,
% 0.20/0.47 setoftrueEq: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setoftrueEq,definition,
% 0.20/0.47 ( setoftrueEq
% 0.20/0.47 = ( ! [A: $i] :
% 0.20/0.47 ( ( dsetconstr @ A
% 0.20/0.47 @ ^ [Xx: $i] : $true )
% 0.20/0.47 = A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(powersetI_type,type,
% 0.20/0.47 powersetI: $o ).
% 0.20/0.47
% 0.20/0.47 thf(powersetI,definition,
% 0.20/0.47 ( powersetI
% 0.20/0.47 = ( ! [A: $i,B: $i] :
% 0.20/0.47 ( ! [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ B )
% 0.20/0.47 => ( in @ Xx @ A ) )
% 0.20/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(emptyinPowerset_type,type,
% 0.20/0.47 emptyinPowerset: $o ).
% 0.20/0.47
% 0.20/0.47 thf(emptyinPowerset,definition,
% 0.20/0.47 ( emptyinPowerset
% 0.20/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(emptyInPowerset_type,type,
% 0.20/0.47 emptyInPowerset: $o ).
% 0.20/0.47
% 0.20/0.47 thf(emptyInPowerset,definition,
% 0.20/0.47 ( emptyInPowerset
% 0.20/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(powersetE_type,type,
% 0.20/0.47 powersetE: $o ).
% 0.20/0.47
% 0.20/0.47 thf(powersetE,definition,
% 0.20/0.47 ( powersetE
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.20/0.47 => ( ( in @ Xx @ B )
% 0.20/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setunionI_type,type,
% 0.20/0.47 setunionI: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setunionI,definition,
% 0.20/0.47 ( setunionI
% 0.20/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.20/0.47 ( ( in @ Xx @ B )
% 0.20/0.47 => ( ( in @ B @ A )
% 0.20/0.47 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setunionE_type,type,
% 0.20/0.47 setunionE: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setunionE,definition,
% 0.20/0.47 ( setunionE
% 0.20/0.47 = ( ! [A: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.20/0.47 => ! [Xphi: $o] :
% 0.20/0.47 ( ! [B: $i] :
% 0.20/0.47 ( ( in @ Xx @ B )
% 0.20/0.47 => ( ( in @ B @ A )
% 0.20/0.47 => Xphi ) )
% 0.20/0.47 => Xphi ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(subPowSU_type,type,
% 0.20/0.47 subPowSU: $o ).
% 0.20/0.47
% 0.20/0.47 thf(subPowSU,definition,
% 0.20/0.47 ( subPowSU
% 0.20/0.47 = ( ! [A: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(exuE2_type,type,
% 0.20/0.47 exuE2: $o ).
% 0.20/0.47
% 0.20/0.47 thf(exuE2,definition,
% 0.20/0.47 ( exuE2
% 0.20/0.47 = ( ! [Xphi: $i > $o] :
% 0.20/0.47 ( ( exu
% 0.20/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.47 => ? [Xx: $i] :
% 0.20/0.47 ! [Xy: $i] :
% 0.20/0.47 ( ( Xphi @ Xy )
% 0.20/0.47 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(nonemptyImpWitness_type,type,
% 0.20/0.47 nonemptyImpWitness: $o ).
% 0.20/0.47
% 0.20/0.47 thf(nonemptyImpWitness,definition,
% 0.20/0.47 ( nonemptyImpWitness
% 0.20/0.47 = ( ! [A: $i] :
% 0.20/0.47 ( ( nonempty @ A )
% 0.20/0.47 => ? [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 & $true ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(uniqinunit_type,type,
% 0.20/0.47 uniqinunit: $o ).
% 0.20/0.47
% 0.20/0.47 thf(uniqinunit,definition,
% 0.20/0.47 ( uniqinunit
% 0.20/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.20/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.20/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(notinsingleton_type,type,
% 0.20/0.47 notinsingleton: $o ).
% 0.20/0.47
% 0.20/0.47 thf(notinsingleton,definition,
% 0.20/0.47 ( notinsingleton
% 0.20/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.20/0.47 ( ( Xx != Xy )
% 0.20/0.47 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(eqinunit_type,type,
% 0.20/0.47 eqinunit: $o ).
% 0.20/0.47
% 0.20/0.47 thf(eqinunit,definition,
% 0.20/0.47 ( eqinunit
% 0.20/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.20/0.47 ( ( Xx = Xy )
% 0.20/0.47 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(singletonsswitch_type,type,
% 0.20/0.47 singletonsswitch: $o ).
% 0.20/0.47
% 0.20/0.47 thf(singletonsswitch,definition,
% 0.20/0.47 ( singletonsswitch
% 0.20/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.20/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.20/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(upairsetE_type,type,
% 0.20/0.47 upairsetE: $o ).
% 0.20/0.47
% 0.20/0.47 thf(upairsetE,definition,
% 0.20/0.47 ( upairsetE
% 0.20/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.20/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.20/0.47 => ( ( Xz = Xx )
% 0.20/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(upairsetIL_type,type,
% 0.20/0.47 upairsetIL: $o ).
% 0.20/0.47
% 0.20/0.47 thf(upairsetIL,definition,
% 0.20/0.47 ( upairsetIL
% 0.20/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(upairsetIR_type,type,
% 0.20/0.47 upairsetIR: $o ).
% 0.20/0.47
% 0.20/0.47 thf(upairsetIR,definition,
% 0.20/0.47 ( upairsetIR
% 0.20/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(emptyE1_type,type,
% 0.20/0.47 emptyE1: $o ).
% 0.20/0.47
% 0.20/0.47 thf(emptyE1,definition,
% 0.20/0.47 ( emptyE1
% 0.20/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.47 ( ? [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 & ( Xphi @ Xx ) )
% 0.20/0.47 => ( ( ( dsetconstr @ A
% 0.20/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.47 = emptyset )
% 0.20/0.47 => $false ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(vacuousDall_type,type,
% 0.20/0.47 vacuousDall: $o ).
% 0.20/0.47
% 0.20/0.47 thf(vacuousDall,definition,
% 0.20/0.47 ( vacuousDall
% 0.20/0.47 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ emptyset )
% 0.20/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(quantDeMorgan1_type,type,
% 0.20/0.47 quantDeMorgan1: $o ).
% 0.20/0.47
% 0.20/0.47 thf(quantDeMorgan1,definition,
% 0.20/0.47 ( quantDeMorgan1
% 0.20/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.47 ( ~ ! [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ( Xphi @ Xx ) )
% 0.20/0.47 => ? [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(quantDeMorgan2_type,type,
% 0.20/0.47 quantDeMorgan2: $o ).
% 0.20/0.47
% 0.20/0.47 thf(quantDeMorgan2,definition,
% 0.20/0.47 ( quantDeMorgan2
% 0.20/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.47 ( ! [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ~ ( Xphi @ Xx ) )
% 0.20/0.47 => ~ ? [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(quantDeMorgan3_type,type,
% 0.20/0.47 quantDeMorgan3: $o ).
% 0.20/0.47
% 0.20/0.47 thf(quantDeMorgan3,definition,
% 0.20/0.47 ( quantDeMorgan3
% 0.20/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.47 ( ~ ? [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 & ( Xphi @ Xx ) )
% 0.20/0.47 => ! [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(quantDeMorgan4_type,type,
% 0.20/0.47 quantDeMorgan4: $o ).
% 0.20/0.47
% 0.20/0.47 thf(quantDeMorgan4,definition,
% 0.20/0.47 ( quantDeMorgan4
% 0.20/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.47 ( ? [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 & ~ ( Xphi @ Xx ) )
% 0.20/0.47 => ~ ! [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(prop2setI_type,type,
% 0.20/0.47 prop2setI: $o ).
% 0.20/0.47
% 0.20/0.47 thf(prop2setI,definition,
% 0.20/0.47 ( prop2setI
% 0.20/0.47 = ( ! [Xphi: $o] :
% 0.20/0.47 ( Xphi
% 0.20/0.47 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(set2prop_type,type,
% 0.20/0.47 set2prop: $i > $o ).
% 0.20/0.47
% 0.20/0.47 thf(prop2set2propI_type,type,
% 0.20/0.47 prop2set2propI: $o ).
% 0.20/0.47
% 0.20/0.47 thf(prop2set2propI,definition,
% 0.20/0.47 ( prop2set2propI
% 0.20/0.47 = ( ! [Xphi: $o] :
% 0.20/0.47 ( Xphi
% 0.20/0.47 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(notdexE_type,type,
% 0.20/0.47 notdexE: $o ).
% 0.20/0.47
% 0.20/0.47 thf(notdexE,definition,
% 0.20/0.47 ( notdexE
% 0.20/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.47 ( ~ ? [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 & ( Xphi @ Xx ) )
% 0.20/0.47 => ! [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(notdallE_type,type,
% 0.20/0.47 notdallE: $o ).
% 0.20/0.47
% 0.20/0.47 thf(notdallE,definition,
% 0.20/0.47 ( notdallE
% 0.20/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.47 ( ~ ! [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ( Xphi @ Xx ) )
% 0.20/0.47 => ? [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(exuI1_type,type,
% 0.20/0.47 exuI1: $o ).
% 0.20/0.47
% 0.20/0.47 thf(exuI1,definition,
% 0.20/0.47 ( exuI1
% 0.20/0.47 = ( ! [Xphi: $i > $o] :
% 0.20/0.47 ( ? [Xx: $i] :
% 0.20/0.47 ( ( Xphi @ Xx )
% 0.20/0.47 & ! [Xy: $i] :
% 0.20/0.47 ( ( Xphi @ Xy )
% 0.20/0.47 => ( Xx = Xy ) ) )
% 0.20/0.47 => ( exu
% 0.20/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(exuI3_type,type,
% 0.20/0.47 exuI3: $o ).
% 0.20/0.47
% 0.20/0.47 thf(exuI3,definition,
% 0.20/0.47 ( exuI3
% 0.20/0.47 = ( ! [Xphi: $i > $o] :
% 0.20/0.47 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.20/0.47 => ( ! [Xx: $i,Xy: $i] :
% 0.20/0.47 ( ( Xphi @ Xx )
% 0.20/0.47 => ( ( Xphi @ Xy )
% 0.20/0.47 => ( Xx = Xy ) ) )
% 0.20/0.47 => ( exu
% 0.20/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(exuI2_type,type,
% 0.20/0.47 exuI2: $o ).
% 0.20/0.47
% 0.20/0.47 thf(exuI2,definition,
% 0.20/0.47 ( exuI2
% 0.20/0.47 = ( ! [Xphi: $i > $o] :
% 0.20/0.47 ( ? [Xx: $i] :
% 0.20/0.47 ! [Xy: $i] :
% 0.20/0.47 ( ( Xphi @ Xy )
% 0.20/0.47 <=> ( Xy = Xx ) )
% 0.20/0.47 => ( exu
% 0.20/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(inCongP_type,type,
% 0.20/0.47 inCongP: $o ).
% 0.20/0.47
% 0.20/0.47 thf(inCongP,definition,
% 0.20/0.47 ( inCongP
% 0.20/0.47 = ( ! [A: $i,B: $i] :
% 0.20/0.47 ( ( A = B )
% 0.20/0.47 => ! [Xx: $i,Xy: $i] :
% 0.20/0.47 ( ( Xx = Xy )
% 0.20/0.47 => ( ( in @ Xx @ A )
% 0.20/0.47 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(in__Cong_type,type,
% 0.20/0.47 in__Cong: $o ).
% 0.20/0.47
% 0.20/0.47 thf(in__Cong,definition,
% 0.20/0.47 ( in__Cong
% 0.20/0.47 = ( ! [A: $i,B: $i] :
% 0.20/0.47 ( ( A = B )
% 0.20/0.47 => ! [Xx: $i,Xy: $i] :
% 0.20/0.47 ( ( Xx = Xy )
% 0.20/0.47 => ( ( in @ Xx @ A )
% 0.20/0.47 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(exuE3u_type,type,
% 0.20/0.47 exuE3u: $o ).
% 0.20/0.47
% 0.20/0.47 thf(exuE3u,definition,
% 0.20/0.47 ( exuE3u
% 0.20/0.47 = ( ! [Xphi: $i > $o] :
% 0.20/0.47 ( ( exu
% 0.20/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.47 => ! [Xx: $i,Xy: $i] :
% 0.20/0.47 ( ( Xphi @ Xx )
% 0.20/0.47 => ( ( Xphi @ Xy )
% 0.20/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(exu__Cong_type,type,
% 0.20/0.47 exu__Cong: $o ).
% 0.20/0.47
% 0.20/0.47 thf(exu__Cong,definition,
% 0.20/0.47 ( exu__Cong
% 0.20/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.20/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.20/0.47 ( ( Xx = Xy )
% 0.20/0.47 => ( ( Xphi @ Xx )
% 0.20/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.20/0.47 => ( ( exu
% 0.20/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.47 <=> ( exu
% 0.20/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(emptyset__Cong_type,type,
% 0.20/0.47 emptyset__Cong: $o ).
% 0.20/0.47
% 0.20/0.47 thf(emptyset__Cong,definition,
% 0.20/0.47 ( emptyset__Cong
% 0.20/0.47 = ( emptyset = emptyset ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setadjoin__Cong_type,type,
% 0.20/0.47 setadjoin__Cong: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setadjoin__Cong,definition,
% 0.20/0.47 ( setadjoin__Cong
% 0.20/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.20/0.47 ( ( Xx = Xy )
% 0.20/0.47 => ! [Xz: $i,Xu: $i] :
% 0.20/0.47 ( ( Xz = Xu )
% 0.20/0.47 => ( ( setadjoin @ Xx @ Xz )
% 0.20/0.47 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(powerset__Cong_type,type,
% 0.20/0.47 powerset__Cong: $o ).
% 0.20/0.47
% 0.20/0.47 thf(powerset__Cong,definition,
% 0.20/0.47 ( powerset__Cong
% 0.20/0.47 = ( ! [A: $i,B: $i] :
% 0.20/0.47 ( ( A = B )
% 0.20/0.47 => ( ( powerset @ A )
% 0.20/0.47 = ( powerset @ B ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setunion__Cong_type,type,
% 0.20/0.47 setunion__Cong: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setunion__Cong,definition,
% 0.20/0.47 ( setunion__Cong
% 0.20/0.47 = ( ! [A: $i,B: $i] :
% 0.20/0.47 ( ( A = B )
% 0.20/0.47 => ( ( setunion @ A )
% 0.20/0.47 = ( setunion @ B ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(omega__Cong_type,type,
% 0.20/0.47 omega__Cong: $o ).
% 0.20/0.47
% 0.20/0.47 thf(omega__Cong,definition,
% 0.20/0.47 ( omega__Cong
% 0.20/0.47 = ( omega = omega ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(exuEu_type,type,
% 0.20/0.47 exuEu: $o ).
% 0.20/0.47
% 0.20/0.47 thf(exuEu,definition,
% 0.20/0.47 ( exuEu
% 0.20/0.47 = ( ! [Xphi: $i > $o] :
% 0.20/0.47 ( ( exu
% 0.20/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.47 => ! [Xx: $i,Xy: $i] :
% 0.20/0.47 ( ( Xphi @ Xx )
% 0.20/0.47 => ( ( Xphi @ Xy )
% 0.20/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(descr__Cong_type,type,
% 0.20/0.47 descr__Cong: $o ).
% 0.20/0.47
% 0.20/0.47 thf(descr__Cong,definition,
% 0.20/0.47 ( descr__Cong
% 0.20/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.20/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.20/0.47 ( ( Xx = Xy )
% 0.20/0.47 => ( ( Xphi @ Xx )
% 0.20/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.20/0.47 => ( ( exu
% 0.20/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.47 => ( ( exu
% 0.20/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.20/0.47 => ( ( descr
% 0.20/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.47 = ( descr
% 0.20/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(dsetconstr__Cong_type,type,
% 0.20/0.47 dsetconstr__Cong: $o ).
% 0.20/0.47
% 0.20/0.47 thf(dsetconstr__Cong,definition,
% 0.20/0.47 ( dsetconstr__Cong
% 0.20/0.47 = ( ! [A: $i,B: $i] :
% 0.20/0.47 ( ( A = B )
% 0.20/0.47 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.20/0.47 ( ! [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ! [Xy: $i] :
% 0.20/0.47 ( ( in @ Xy @ B )
% 0.20/0.47 => ( ( Xx = Xy )
% 0.20/0.47 => ( ( Xphi @ Xx )
% 0.20/0.47 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.20/0.47 => ( ( dsetconstr @ A
% 0.20/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.47 = ( dsetconstr @ B
% 0.20/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(subset_type,type,
% 0.20/0.47 subset: $i > $i > $o ).
% 0.20/0.47
% 0.20/0.47 thf(disjoint_type,type,
% 0.20/0.47 disjoint: $i > $i > $o ).
% 0.20/0.47
% 0.20/0.47 thf(setsmeet_type,type,
% 0.20/0.47 setsmeet: $i > $i > $o ).
% 0.20/0.47
% 0.20/0.47 thf(subsetI1_type,type,
% 0.20/0.47 subsetI1: $o ).
% 0.20/0.47
% 0.20/0.47 thf(subsetI1,definition,
% 0.20/0.47 ( subsetI1
% 0.20/0.47 = ( ! [A: $i,B: $i] :
% 0.20/0.47 ( ! [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ( in @ Xx @ B ) )
% 0.20/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(eqimpsubset2_type,type,
% 0.20/0.47 eqimpsubset2: $o ).
% 0.20/0.47
% 0.20/0.47 thf(eqimpsubset2,definition,
% 0.20/0.47 ( eqimpsubset2
% 0.20/0.47 = ( ! [A: $i,B: $i] :
% 0.20/0.47 ( ( A = B )
% 0.20/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(eqimpsubset1_type,type,
% 0.20/0.47 eqimpsubset1: $o ).
% 0.20/0.47
% 0.20/0.47 thf(eqimpsubset1,definition,
% 0.20/0.47 ( eqimpsubset1
% 0.20/0.47 = ( ! [A: $i,B: $i] :
% 0.20/0.47 ( ( A = B )
% 0.20/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(subsetI2_type,type,
% 0.20/0.47 subsetI2: $o ).
% 0.20/0.47
% 0.20/0.47 thf(subsetI2,definition,
% 0.20/0.47 ( subsetI2
% 0.20/0.47 = ( ! [A: $i,B: $i] :
% 0.20/0.47 ( ! [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ( in @ Xx @ B ) )
% 0.20/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(emptysetsubset_type,type,
% 0.20/0.47 emptysetsubset: $o ).
% 0.20/0.47
% 0.20/0.47 thf(emptysetsubset,definition,
% 0.20/0.47 ( emptysetsubset
% 0.20/0.47 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(subsetE_type,type,
% 0.20/0.47 subsetE: $o ).
% 0.20/0.47
% 0.20/0.47 thf(subsetE,definition,
% 0.20/0.47 ( subsetE
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( subset @ A @ B )
% 0.20/0.47 => ( ( in @ Xx @ A )
% 0.20/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(subsetE2_type,type,
% 0.20/0.47 subsetE2: $o ).
% 0.20/0.47
% 0.20/0.47 thf(subsetE2,definition,
% 0.20/0.47 ( subsetE2
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( subset @ A @ B )
% 0.20/0.47 => ( ~ ( in @ Xx @ B )
% 0.20/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(notsubsetI_type,type,
% 0.20/0.47 notsubsetI: $o ).
% 0.20/0.47
% 0.20/0.47 thf(notsubsetI,definition,
% 0.20/0.47 ( notsubsetI
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ( ~ ( in @ Xx @ B )
% 0.20/0.47 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(notequalI1_type,type,
% 0.20/0.47 notequalI1: $o ).
% 0.20/0.47
% 0.20/0.47 thf(notequalI1,definition,
% 0.20/0.47 ( notequalI1
% 0.20/0.47 = ( ! [A: $i,B: $i] :
% 0.20/0.47 ( ~ ( subset @ A @ B )
% 0.20/0.47 => ( A != B ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(notequalI2_type,type,
% 0.20/0.47 notequalI2: $o ).
% 0.20/0.47
% 0.20/0.47 thf(notequalI2,definition,
% 0.20/0.47 ( notequalI2
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ( ~ ( in @ Xx @ B )
% 0.20/0.47 => ( A != B ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(subsetRefl_type,type,
% 0.20/0.47 subsetRefl: $o ).
% 0.20/0.47
% 0.20/0.47 thf(subsetRefl,definition,
% 0.20/0.47 ( subsetRefl
% 0.20/0.47 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(subsetTrans_type,type,
% 0.20/0.47 subsetTrans: $o ).
% 0.20/0.47
% 0.20/0.47 thf(subsetTrans,definition,
% 0.20/0.47 ( subsetTrans
% 0.20/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.20/0.47 ( ( subset @ A @ B )
% 0.20/0.47 => ( ( subset @ B @ C )
% 0.20/0.47 => ( subset @ A @ C ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setadjoinSub_type,type,
% 0.20/0.47 setadjoinSub: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setadjoinSub,definition,
% 0.20/0.47 ( setadjoinSub
% 0.20/0.47 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setadjoinSub2_type,type,
% 0.20/0.47 setadjoinSub2: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setadjoinSub2,definition,
% 0.20/0.47 ( setadjoinSub2
% 0.20/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.20/0.47 ( ( subset @ A @ B )
% 0.20/0.47 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(subset2powerset_type,type,
% 0.20/0.47 subset2powerset: $o ).
% 0.20/0.47
% 0.20/0.47 thf(subset2powerset,definition,
% 0.20/0.47 ( subset2powerset
% 0.20/0.47 = ( ! [A: $i,B: $i] :
% 0.20/0.47 ( ( subset @ A @ B )
% 0.20/0.47 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setextsub_type,type,
% 0.20/0.47 setextsub: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setextsub,definition,
% 0.20/0.47 ( setextsub
% 0.20/0.47 = ( ! [A: $i,B: $i] :
% 0.20/0.47 ( ( subset @ A @ B )
% 0.20/0.47 => ( ( subset @ B @ A )
% 0.20/0.47 => ( A = B ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(subsetemptysetimpeq_type,type,
% 0.20/0.47 subsetemptysetimpeq: $o ).
% 0.20/0.47
% 0.20/0.47 thf(subsetemptysetimpeq,definition,
% 0.20/0.47 ( subsetemptysetimpeq
% 0.20/0.47 = ( ! [A: $i] :
% 0.20/0.47 ( ( subset @ A @ emptyset )
% 0.20/0.47 => ( A = emptyset ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(powersetI1_type,type,
% 0.20/0.47 powersetI1: $o ).
% 0.20/0.47
% 0.20/0.47 thf(powersetI1,definition,
% 0.20/0.47 ( powersetI1
% 0.20/0.47 = ( ! [A: $i,B: $i] :
% 0.20/0.47 ( ( subset @ B @ A )
% 0.20/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(powersetE1_type,type,
% 0.20/0.47 powersetE1: $o ).
% 0.20/0.47
% 0.20/0.47 thf(powersetE1,definition,
% 0.20/0.47 ( powersetE1
% 0.20/0.47 = ( ! [A: $i,B: $i] :
% 0.20/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.20/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(inPowerset_type,type,
% 0.20/0.47 inPowerset: $o ).
% 0.20/0.47
% 0.20/0.47 thf(inPowerset,definition,
% 0.20/0.47 ( inPowerset
% 0.20/0.47 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(powersetsubset_type,type,
% 0.20/0.47 powersetsubset: $o ).
% 0.20/0.47
% 0.20/0.47 thf(powersetsubset,definition,
% 0.20/0.47 ( powersetsubset
% 0.20/0.47 = ( ! [A: $i,B: $i] :
% 0.20/0.47 ( ( subset @ A @ B )
% 0.20/0.47 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sepInPowerset_type,type,
% 0.20/0.47 sepInPowerset: $o ).
% 0.20/0.47
% 0.20/0.47 thf(sepInPowerset,definition,
% 0.20/0.47 ( sepInPowerset
% 0.20/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.47 ( in
% 0.20/0.47 @ ( dsetconstr @ A
% 0.20/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.47 @ ( powerset @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(sepSubset_type,type,
% 0.20/0.47 sepSubset: $o ).
% 0.20/0.47
% 0.20/0.47 thf(sepSubset,definition,
% 0.20/0.47 ( sepSubset
% 0.20/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.47 ( subset
% 0.20/0.47 @ ( dsetconstr @ A
% 0.20/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.47 @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(binunion_type,type,
% 0.20/0.47 binunion: $i > $i > $i ).
% 0.20/0.47
% 0.20/0.47 thf(binunionIL_type,type,
% 0.20/0.47 binunionIL: $o ).
% 0.20/0.47
% 0.20/0.47 thf(binunionIL,definition,
% 0.20/0.47 ( binunionIL
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(upairset2IR_type,type,
% 0.20/0.47 upairset2IR: $o ).
% 0.20/0.47
% 0.20/0.47 thf(upairset2IR,definition,
% 0.20/0.47 ( upairset2IR
% 0.20/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(binunionIR_type,type,
% 0.20/0.47 binunionIR: $o ).
% 0.20/0.47
% 0.20/0.47 thf(binunionIR,definition,
% 0.20/0.47 ( binunionIR
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ B )
% 0.20/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(binunionEcases_type,type,
% 0.20/0.47 binunionEcases: $o ).
% 0.20/0.47
% 0.20/0.47 thf(binunionEcases,definition,
% 0.20/0.47 ( binunionEcases
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.20/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.20/0.47 => ( ( ( in @ Xx @ A )
% 0.20/0.47 => Xphi )
% 0.20/0.47 => ( ( ( in @ Xx @ B )
% 0.20/0.47 => Xphi )
% 0.20/0.47 => Xphi ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(binunionE_type,type,
% 0.20/0.47 binunionE: $o ).
% 0.20/0.47
% 0.20/0.47 thf(binunionE,definition,
% 0.20/0.47 ( binunionE
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.20/0.47 => ( ( in @ Xx @ A )
% 0.20/0.47 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(binunionLsub_type,type,
% 0.20/0.47 binunionLsub: $o ).
% 0.20/0.47
% 0.20/0.47 thf(binunionLsub,definition,
% 0.20/0.47 ( binunionLsub
% 0.20/0.47 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(binunionRsub_type,type,
% 0.20/0.47 binunionRsub: $o ).
% 0.20/0.47
% 0.20/0.47 thf(binunionRsub,definition,
% 0.20/0.47 ( binunionRsub
% 0.20/0.47 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(binintersect_type,type,
% 0.20/0.47 binintersect: $i > $i > $i ).
% 0.20/0.47
% 0.20/0.47 thf(binintersectI_type,type,
% 0.20/0.47 binintersectI: $o ).
% 0.20/0.47
% 0.20/0.47 thf(binintersectI,definition,
% 0.20/0.47 ( binintersectI
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ( ( in @ Xx @ B )
% 0.20/0.47 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(binintersectSubset5_type,type,
% 0.20/0.47 binintersectSubset5: $o ).
% 0.20/0.47
% 0.20/0.47 thf(binintersectSubset5,definition,
% 0.20/0.47 ( binintersectSubset5
% 0.20/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.20/0.47 ( ( subset @ C @ A )
% 0.20/0.47 => ( ( subset @ C @ B )
% 0.20/0.47 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(binintersectEL_type,type,
% 0.20/0.47 binintersectEL: $o ).
% 0.20/0.47
% 0.20/0.47 thf(binintersectEL,definition,
% 0.20/0.47 ( binintersectEL
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.20/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(binintersectLsub_type,type,
% 0.20/0.47 binintersectLsub: $o ).
% 0.20/0.47
% 0.20/0.47 thf(binintersectLsub,definition,
% 0.20/0.47 ( binintersectLsub
% 0.20/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(binintersectSubset2_type,type,
% 0.20/0.47 binintersectSubset2: $o ).
% 0.20/0.47
% 0.20/0.47 thf(binintersectSubset2,definition,
% 0.20/0.47 ( binintersectSubset2
% 0.20/0.47 = ( ! [A: $i,B: $i] :
% 0.20/0.47 ( ( subset @ A @ B )
% 0.20/0.47 => ( ( binintersect @ A @ B )
% 0.20/0.47 = A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(binintersectSubset3_type,type,
% 0.20/0.47 binintersectSubset3: $o ).
% 0.20/0.47
% 0.20/0.47 thf(binintersectSubset3,definition,
% 0.20/0.47 ( binintersectSubset3
% 0.20/0.47 = ( ! [A: $i,B: $i] :
% 0.20/0.47 ( ( ( binintersect @ A @ B )
% 0.20/0.47 = B )
% 0.20/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(binintersectER_type,type,
% 0.20/0.47 binintersectER: $o ).
% 0.20/0.47
% 0.20/0.47 thf(binintersectER,definition,
% 0.20/0.47 ( binintersectER
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.20/0.47 => ( in @ Xx @ B ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(disjointsetsI1_type,type,
% 0.20/0.47 disjointsetsI1: $o ).
% 0.20/0.47
% 0.20/0.47 thf(disjointsetsI1,definition,
% 0.20/0.47 ( disjointsetsI1
% 0.20/0.47 = ( ! [A: $i,B: $i] :
% 0.20/0.47 ( ~ ? [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 & ( in @ Xx @ B ) )
% 0.20/0.47 => ( ( binintersect @ A @ B )
% 0.20/0.47 = emptyset ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(binintersectRsub_type,type,
% 0.20/0.47 binintersectRsub: $o ).
% 0.20/0.47
% 0.20/0.47 thf(binintersectRsub,definition,
% 0.20/0.47 ( binintersectRsub
% 0.20/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(binintersectSubset4_type,type,
% 0.20/0.47 binintersectSubset4: $o ).
% 0.20/0.47
% 0.20/0.47 thf(binintersectSubset4,definition,
% 0.20/0.47 ( binintersectSubset4
% 0.20/0.47 = ( ! [A: $i,B: $i] :
% 0.20/0.47 ( ( subset @ B @ A )
% 0.20/0.47 => ( ( binintersect @ A @ B )
% 0.20/0.47 = B ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(binintersectSubset1_type,type,
% 0.20/0.47 binintersectSubset1: $o ).
% 0.20/0.47
% 0.20/0.47 thf(binintersectSubset1,definition,
% 0.20/0.47 ( binintersectSubset1
% 0.20/0.47 = ( ! [A: $i,B: $i] :
% 0.20/0.47 ( ( ( binintersect @ A @ B )
% 0.20/0.47 = A )
% 0.20/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(bs114d_type,type,
% 0.20/0.47 bs114d: $o ).
% 0.20/0.47
% 0.20/0.47 thf(bs114d,definition,
% 0.20/0.47 ( bs114d
% 0.20/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.20/0.47 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.20/0.47 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(regular_type,type,
% 0.20/0.47 regular: $i > $o ).
% 0.20/0.47
% 0.20/0.47 thf(setminus_type,type,
% 0.20/0.47 setminus: $i > $i > $i ).
% 0.20/0.47
% 0.20/0.47 thf(setminusI_type,type,
% 0.20/0.47 setminusI: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setminusI,definition,
% 0.20/0.47 ( setminusI
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ( ~ ( in @ Xx @ B )
% 0.20/0.47 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setminusEL_type,type,
% 0.20/0.47 setminusEL: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setminusEL,definition,
% 0.20/0.47 ( setminusEL
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.20/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setminusER_type,type,
% 0.20/0.47 setminusER: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setminusER,definition,
% 0.20/0.47 ( setminusER
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.20/0.47 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setminusSubset2_type,type,
% 0.20/0.47 setminusSubset2: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setminusSubset2,definition,
% 0.20/0.47 ( setminusSubset2
% 0.20/0.47 = ( ! [A: $i,B: $i] :
% 0.20/0.47 ( ( subset @ A @ B )
% 0.20/0.47 => ( ( setminus @ A @ B )
% 0.20/0.47 = emptyset ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setminusERneg_type,type,
% 0.20/0.47 setminusERneg: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setminusERneg,definition,
% 0.20/0.47 ( setminusERneg
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.20/0.47 => ( ( in @ Xx @ A )
% 0.20/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setminusELneg_type,type,
% 0.20/0.47 setminusELneg: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setminusELneg,definition,
% 0.20/0.47 ( setminusELneg
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.20/0.47 => ( ~ ( in @ Xx @ B )
% 0.20/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setminusILneg_type,type,
% 0.20/0.47 setminusILneg: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setminusILneg,definition,
% 0.20/0.47 ( setminusILneg
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ~ ( in @ Xx @ A )
% 0.20/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setminusIRneg_type,type,
% 0.20/0.47 setminusIRneg: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setminusIRneg,definition,
% 0.20/0.47 ( setminusIRneg
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ B )
% 0.20/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setminusLsub_type,type,
% 0.20/0.47 setminusLsub: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setminusLsub,definition,
% 0.20/0.47 ( setminusLsub
% 0.20/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setminusSubset1_type,type,
% 0.20/0.47 setminusSubset1: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setminusSubset1,definition,
% 0.20/0.47 ( setminusSubset1
% 0.20/0.47 = ( ! [A: $i,B: $i] :
% 0.20/0.47 ( ( ( setminus @ A @ B )
% 0.20/0.47 = emptyset )
% 0.20/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(symdiff_type,type,
% 0.20/0.47 symdiff: $i > $i > $i ).
% 0.20/0.47
% 0.20/0.47 thf(symdiffE_type,type,
% 0.20/0.47 symdiffE: $o ).
% 0.20/0.47
% 0.20/0.47 thf(symdiffE,definition,
% 0.20/0.47 ( symdiffE
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.20/0.47 => ! [Xphi: $o] :
% 0.20/0.47 ( ( ( in @ Xx @ A )
% 0.20/0.47 => ( ~ ( in @ Xx @ B )
% 0.20/0.47 => Xphi ) )
% 0.20/0.47 => ( ( ~ ( in @ Xx @ A )
% 0.20/0.47 => ( ( in @ Xx @ B )
% 0.20/0.47 => Xphi ) )
% 0.20/0.47 => Xphi ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(symdiffI1_type,type,
% 0.20/0.47 symdiffI1: $o ).
% 0.20/0.47
% 0.20/0.47 thf(symdiffI1,definition,
% 0.20/0.47 ( symdiffI1
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ( ~ ( in @ Xx @ B )
% 0.20/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(symdiffI2_type,type,
% 0.20/0.47 symdiffI2: $o ).
% 0.20/0.47
% 0.20/0.47 thf(symdiffI2,definition,
% 0.20/0.47 ( symdiffI2
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ~ ( in @ Xx @ A )
% 0.20/0.47 => ( ( in @ Xx @ B )
% 0.20/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(symdiffIneg1_type,type,
% 0.20/0.47 symdiffIneg1: $o ).
% 0.20/0.47
% 0.20/0.47 thf(symdiffIneg1,definition,
% 0.20/0.47 ( symdiffIneg1
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ( ( in @ Xx @ B )
% 0.20/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(symdiffIneg2_type,type,
% 0.20/0.47 symdiffIneg2: $o ).
% 0.20/0.47
% 0.20/0.47 thf(symdiffIneg2,definition,
% 0.20/0.47 ( symdiffIneg2
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ~ ( in @ Xx @ A )
% 0.20/0.47 => ( ~ ( in @ Xx @ B )
% 0.20/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(iskpair_type,type,
% 0.20/0.47 iskpair: $i > $o ).
% 0.20/0.47
% 0.20/0.47 thf(iskpair,definition,
% 0.20/0.47 ( iskpair
% 0.20/0.47 = ( ^ [A: $i] :
% 0.20/0.47 ? [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.20/0.47 & ? [Xy: $i] :
% 0.20/0.47 ( ( in @ Xy @ ( setunion @ A ) )
% 0.20/0.47 & ( A
% 0.20/0.47 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(secondinupair_type,type,
% 0.20/0.47 secondinupair: $o ).
% 0.20/0.47
% 0.20/0.47 thf(secondinupair,definition,
% 0.20/0.47 ( secondinupair
% 0.20/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setukpairIL_type,type,
% 0.20/0.47 setukpairIL: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setukpairIL,definition,
% 0.20/0.47 ( setukpairIL
% 0.20/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setukpairIR_type,type,
% 0.20/0.47 setukpairIR: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setukpairIR,definition,
% 0.20/0.47 ( setukpairIR
% 0.20/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(kpairiskpair_type,type,
% 0.20/0.47 kpairiskpair: $o ).
% 0.20/0.47
% 0.20/0.47 thf(kpairiskpair,definition,
% 0.20/0.47 ( kpairiskpair
% 0.20/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(kpair_type,type,
% 0.20/0.47 kpair: $i > $i > $i ).
% 0.20/0.47
% 0.20/0.47 thf(kpair,definition,
% 0.20/0.47 ( kpair
% 0.20/0.47 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(kpairp_type,type,
% 0.20/0.47 kpairp: $o ).
% 0.20/0.47
% 0.20/0.47 thf(kpairp,definition,
% 0.20/0.47 ( kpairp
% 0.20/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(cartprod_type,type,
% 0.20/0.47 cartprod: $i > $i > $i ).
% 0.20/0.47
% 0.20/0.47 thf(singletonsubset_type,type,
% 0.20/0.47 singletonsubset: $o ).
% 0.20/0.47
% 0.20/0.47 thf(singletonsubset,definition,
% 0.20/0.47 ( singletonsubset
% 0.20/0.47 = ( ! [A: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(singletoninpowerset_type,type,
% 0.20/0.47 singletoninpowerset: $o ).
% 0.20/0.47
% 0.20/0.47 thf(singletoninpowerset,definition,
% 0.20/0.47 ( singletoninpowerset
% 0.20/0.47 = ( ! [A: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(singletoninpowunion_type,type,
% 0.20/0.47 singletoninpowunion: $o ).
% 0.20/0.47
% 0.20/0.47 thf(singletoninpowunion,definition,
% 0.20/0.47 ( singletoninpowunion
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(upairset2E_type,type,
% 0.20/0.47 upairset2E: $o ).
% 0.20/0.47
% 0.20/0.47 thf(upairset2E,definition,
% 0.20/0.47 ( upairset2E
% 0.20/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.20/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.20/0.47 => ( ( Xz = Xx )
% 0.20/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(upairsubunion_type,type,
% 0.20/0.47 upairsubunion: $o ).
% 0.20/0.47
% 0.20/0.47 thf(upairsubunion,definition,
% 0.20/0.47 ( upairsubunion
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ! [Xy: $i] :
% 0.20/0.47 ( ( in @ Xy @ B )
% 0.20/0.47 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(upairinpowunion_type,type,
% 0.20/0.47 upairinpowunion: $o ).
% 0.20/0.47
% 0.20/0.47 thf(upairinpowunion,definition,
% 0.20/0.47 ( upairinpowunion
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ! [Xy: $i] :
% 0.20/0.47 ( ( in @ Xy @ B )
% 0.20/0.47 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(ubforcartprodlem1_type,type,
% 0.20/0.47 ubforcartprodlem1: $o ).
% 0.20/0.47
% 0.20/0.47 thf(ubforcartprodlem1,definition,
% 0.20/0.47 ( ubforcartprodlem1
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ! [Xy: $i] :
% 0.20/0.47 ( ( in @ Xy @ B )
% 0.20/0.47 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(ubforcartprodlem2_type,type,
% 0.20/0.47 ubforcartprodlem2: $o ).
% 0.20/0.47
% 0.20/0.47 thf(ubforcartprodlem2,definition,
% 0.20/0.47 ( ubforcartprodlem2
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ! [Xy: $i] :
% 0.20/0.47 ( ( in @ Xy @ B )
% 0.20/0.47 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(ubforcartprodlem3_type,type,
% 0.20/0.47 ubforcartprodlem3: $o ).
% 0.20/0.47
% 0.20/0.47 thf(ubforcartprodlem3,definition,
% 0.20/0.47 ( ubforcartprodlem3
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ! [Xy: $i] :
% 0.20/0.47 ( ( in @ Xy @ B )
% 0.20/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(cartprodpairin_type,type,
% 0.20/0.47 cartprodpairin: $o ).
% 0.20/0.47
% 0.20/0.47 thf(cartprodpairin,definition,
% 0.20/0.47 ( cartprodpairin
% 0.20/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 => ! [Xy: $i] :
% 0.20/0.47 ( ( in @ Xy @ B )
% 0.20/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(cartprodmempair1_type,type,
% 0.20/0.47 cartprodmempair1: $o ).
% 0.20/0.47
% 0.20/0.47 thf(cartprodmempair1,definition,
% 0.20/0.47 ( cartprodmempair1
% 0.20/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.20/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.20/0.47 => ? [Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ A )
% 0.20/0.47 & ? [Xy: $i] :
% 0.20/0.47 ( ( in @ Xy @ B )
% 0.20/0.47 & ( Xu
% 0.20/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(cartprodmempair_type,type,
% 0.20/0.47 cartprodmempair: $o ).
% 0.20/0.47
% 0.20/0.47 thf(cartprodmempair,definition,
% 0.20/0.47 ( cartprodmempair
% 0.20/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.20/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.20/0.47 => ( iskpair @ Xu ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setunionE2_type,type,
% 0.20/0.47 setunionE2: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setunionE2,definition,
% 0.20/0.47 ( setunionE2
% 0.20/0.47 = ( ! [A: $i,Xx: $i] :
% 0.20/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.20/0.47 => ? [X: $i] :
% 0.20/0.47 ( ( in @ X @ A )
% 0.20/0.47 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setunionsingleton1_type,type,
% 0.20/0.47 setunionsingleton1: $o ).
% 0.20/0.47
% 0.20/0.47 thf(setunionsingleton1,definition,
% 0.20/0.47 ( setunionsingleton1
% 0.20/0.47 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 thf(setunionsingleton2,conjecture,
% 0.20/0.47 ( setextAx
% 0.20/0.47 => ( emptysetAx
% 0.20/0.47 => ( setadjoinAx
% 0.20/0.47 => ( powersetAx
% 0.20/0.47 => ( setunionAx
% 0.20/0.47 => ( omega0Ax
% 0.20/0.47 => ( omegaSAx
% 0.20/0.47 => ( omegaIndAx
% 0.20/0.47 => ( replAx
% 0.20/0.47 => ( foundationAx
% 0.20/0.47 => ( wellorderingAx
% 0.20/0.47 => ( descrp
% 0.20/0.47 => ( dsetconstrI
% 0.20/0.47 => ( dsetconstrEL
% 0.20/0.47 => ( dsetconstrER
% 0.20/0.47 => ( exuE1
% 0.20/0.47 => ( prop2setE
% 0.20/0.47 => ( emptysetE
% 0.20/0.47 => ( emptysetimpfalse
% 0.20/0.47 => ( notinemptyset
% 0.20/0.47 => ( exuE3e
% 0.20/0.47 => ( setext
% 0.20/0.47 => ( emptyI
% 0.20/0.47 => ( noeltsimpempty
% 0.20/0.47 => ( setbeta
% 0.20/0.47 => ( nonemptyE1
% 0.20/0.47 => ( nonemptyI
% 0.20/0.47 => ( nonemptyI1
% 0.20/0.47 => ( setadjoinIL
% 0.20/0.47 => ( emptyinunitempty
% 0.20/0.47 => ( setadjoinIR
% 0.20/0.47 => ( setadjoinE
% 0.20/0.47 => ( setadjoinOr
% 0.20/0.47 => ( setoftrueEq
% 0.20/0.47 => ( powersetI
% 0.20/0.47 => ( emptyinPowerset
% 0.20/0.47 => ( emptyInPowerset
% 0.20/0.47 => ( powersetE
% 0.20/0.47 => ( setunionI
% 0.20/0.47 => ( setunionE
% 0.20/0.47 => ( subPowSU
% 0.20/0.47 => ( exuE2
% 0.20/0.47 => ( nonemptyImpWitness
% 0.20/0.47 => ( uniqinunit
% 0.20/0.47 => ( notinsingleton
% 0.20/0.47 => ( eqinunit
% 0.20/0.47 => ( singletonsswitch
% 0.20/0.47 => ( upairsetE
% 0.20/0.47 => ( upairsetIL
% 0.20/0.47 => ( upairsetIR
% 0.20/0.47 => ( emptyE1
% 0.20/0.47 => ( vacuousDall
% 0.20/0.47 => ( quantDeMorgan1
% 0.20/0.47 => ( quantDeMorgan2
% 0.20/0.47 => ( quantDeMorgan3
% 0.20/0.47 => ( quantDeMorgan4
% 0.20/0.47 => ( prop2setI
% 0.20/0.47 => ( prop2set2propI
% 0.20/0.47 => ( notdexE
% 0.20/0.47 => ( notdallE
% 0.20/0.47 => ( exuI1
% 0.20/0.47 => ( exuI3
% 0.20/0.47 => ( exuI2
% 0.20/0.47 => ( inCongP
% 0.20/0.47 => ( in__Cong
% 0.20/0.47 => ( exuE3u
% 0.20/0.47 => ( exu__Cong
% 0.20/0.47 => ( emptyset__Cong
% 0.20/0.47 => ( setadjoin__Cong
% 0.20/0.47 => ( powerset__Cong
% 0.20/0.47 => ( setunion__Cong
% 0.20/0.47 => ( omega__Cong
% 0.20/0.47 => ( exuEu
% 0.20/0.47 => ( descr__Cong
% 0.20/0.47 => ( dsetconstr__Cong
% 0.20/0.47 => ( subsetI1
% 0.20/0.47 => ( eqimpsubset2
% 0.20/0.47 => ( eqimpsubset1
% 0.20/0.47 => ( subsetI2
% 0.20/0.47 => ( emptysetsubset
% 0.20/0.47 => ( subsetE
% 0.20/0.47 => ( subsetE2
% 0.20/0.47 => ( notsubsetI
% 0.20/0.47 => ( notequalI1
% 0.20/0.47 => ( notequalI2
% 0.20/0.47 => ( subsetRefl
% 0.20/0.47 => ( subsetTrans
% 0.20/0.47 => ( setadjoinSub
% 0.20/0.47 => ( setadjoinSub2
% 0.20/0.47 => ( subset2powerset
% 0.20/0.47 => ( setextsub
% 0.20/0.47 => ( subsetemptysetimpeq
% 0.20/0.47 => ( powersetI1
% 0.20/0.47 => ( powersetE1
% 0.20/0.47 => ( inPowerset
% 0.20/0.47 => ( powersetsubset
% 0.20/0.47 => ( sepInPowerset
% 0.20/0.47 => ( sepSubset
% 0.20/0.47 => ( binunionIL
% 0.20/0.47 => ( upairset2IR
% 0.20/0.47 => ( binunionIR
% 0.20/0.47 => ( binunionEcases
% 0.20/0.47 => ( binunionE
% 0.20/0.47 => ( binunionLsub
% 0.20/0.47 => ( binunionRsub
% 0.20/0.47 => ( binintersectI
% 0.20/0.47 => ( binintersectSubset5
% 0.20/0.47 => ( binintersectEL
% 0.20/0.47 => ( binintersectLsub
% 0.20/0.47 => ( binintersectSubset2
% 0.20/0.47 => ( binintersectSubset3
% 0.20/0.47 => ( binintersectER
% 0.20/0.47 => ( disjointsetsI1
% 0.20/0.47 => ( binintersectRsub
% 0.20/0.47 => ( binintersectSubset4
% 0.20/0.47 => ( binintersectSubset1
% 0.20/0.47 => ( bs114d
% 0.20/0.47 => ( setminusI
% 0.20/0.47 => ( setminusEL
% 0.20/0.47 => ( setminusER
% 0.20/0.47 => ( setminusSubset2
% 0.20/0.47 => ( setminusERneg
% 0.20/0.47 => ( setminusELneg
% 0.20/0.47 => ( setminusILneg
% 0.20/0.47 => ( setminusIRneg
% 0.20/0.47 => ( setminusLsub
% 0.20/0.47 => ( setminusSubset1
% 0.20/0.47 => ( symdiffE
% 0.20/0.47 => ( symdiffI1
% 0.20/0.47 => ( symdiffI2
% 0.20/0.47 => ( symdiffIneg1
% 0.20/0.47 => ( symdiffIneg2
% 0.20/0.47 => ( secondinupair
% 0.20/0.47 => ( setukpairIL
% 0.20/0.47 => ( setukpairIR
% 0.20/0.47 => ( kpairiskpair
% 0.20/0.47 => ( kpairp
% 0.20/0.47 => ( singletonsubset
% 0.20/0.47 => ( singletoninpowerset
% 0.20/0.47 => ( singletoninpowunion
% 0.20/0.47 => ( upairset2E
% 0.20/0.47 => ( upairsubunion
% 0.20/0.47 => ( upairinpowunion
% 0.20/0.47 => ( ubforcartprodlem1
% 0.20/0.47 => ( ubforcartprodlem2
% 0.20/0.47 => ( ubforcartprodlem3
% 0.20/0.47 => ( cartprodpairin
% 0.20/0.47 => ( cartprodmempair1
% 0.20/0.47 => ( cartprodmempair
% 0.20/0.47 => ( setunionE2
% 0.20/0.47 => ( setunionsingleton1
% 0.20/0.47 => ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 %------------------------------------------------------------------------------
% 0.20/0.47 ------------------------
% 2.61/1.66 ---- Embedded in THF ---
% 2.61/1.67 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.61/1.67 %%% Generated on Tue Feb 24 12:29:10 EST 2026
% 2.61/1.67 %%% using '$$dhol' embedding, version 1.3.0.
% 2.61/1.67 %%% Logic specification used:
% 2.61/1.67 %%% thf(spec, logic, $$dhol).
% 2.61/1.67
% 2.61/1.67 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.sSg5lqwKQT/DTF2DTF20989.p
% See solution above
% 2.61/1.67 ------------------------
% 2.78/1.70 ---- Cleaned THF ---
% 2.78/1.70 thf(in_type,type,
% 2.78/1.70 in: $i > $i > $o ).
% 2.78/1.70
% 2.78/1.70 thf(exu_type,type,
% 2.78/1.70 exu: ( $i > $o ) > $o ).
% 2.78/1.70
% 2.78/1.70 thf(exu,definition,
% 2.78/1.70 ( exu
% 2.78/1.70 = ( ^ [Xphi: $i > $o] :
% 2.78/1.70 ? [Xx: $i] :
% 2.78/1.70 ( ( Xphi @ Xx )
% 2.78/1.70 & ! [Xy: $i] :
% 2.78/1.70 ( ( Xphi @ Xy )
% 2.78/1.70 => ( Xx = Xy ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setextAx_type,type,
% 2.78/1.70 setextAx: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setextAx,definition,
% 2.78/1.70 ( setextAx
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ! [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 <=> ( in @ Xx @ B ) )
% 2.78/1.70 => ( A = B ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(emptyset_type,type,
% 2.78/1.70 emptyset: $i ).
% 2.78/1.70
% 2.78/1.70 thf(emptysetAx_type,type,
% 2.78/1.70 emptysetAx: $o ).
% 2.78/1.70
% 2.78/1.70 thf(emptysetAx,definition,
% 2.78/1.70 ( emptysetAx
% 2.78/1.70 = ( ! [Xx: $i] :
% 2.78/1.70 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setadjoin_type,type,
% 2.78/1.70 setadjoin: $i > $i > $i ).
% 2.78/1.70
% 2.78/1.70 thf(setadjoinAx_type,type,
% 2.78/1.70 setadjoinAx: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setadjoinAx,definition,
% 2.78/1.70 ( setadjoinAx
% 2.78/1.70 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.78/1.70 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.78/1.70 <=> ( ( Xy = Xx )
% 2.78/1.70 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(powerset_type,type,
% 2.78/1.70 powerset: $i > $i ).
% 2.78/1.70
% 2.78/1.70 thf(powersetAx_type,type,
% 2.78/1.70 powersetAx: $o ).
% 2.78/1.70
% 2.78/1.70 thf(powersetAx,definition,
% 2.78/1.70 ( powersetAx
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ( in @ B @ ( powerset @ A ) )
% 2.78/1.70 <=> ! [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ B )
% 2.78/1.70 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setunion_type,type,
% 2.78/1.70 setunion: $i > $i ).
% 2.78/1.70
% 2.78/1.70 thf(setunionAx_type,type,
% 2.78/1.70 setunionAx: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setunionAx,definition,
% 2.78/1.70 ( setunionAx
% 2.78/1.70 = ( ! [A: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ ( setunion @ A ) )
% 2.78/1.70 <=> ? [B: $i] :
% 2.78/1.70 ( ( in @ Xx @ B )
% 2.78/1.70 & ( in @ B @ A ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(omega_type,type,
% 2.78/1.70 omega: $i ).
% 2.78/1.70
% 2.78/1.70 thf(omega0Ax_type,type,
% 2.78/1.70 omega0Ax: $o ).
% 2.78/1.70
% 2.78/1.70 thf(omega0Ax,definition,
% 2.78/1.70 ( omega0Ax
% 2.78/1.70 = ( in @ emptyset @ omega ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(omegaSAx_type,type,
% 2.78/1.70 omegaSAx: $o ).
% 2.78/1.70
% 2.78/1.70 thf(omegaSAx,definition,
% 2.78/1.70 ( omegaSAx
% 2.78/1.70 = ( ! [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ omega )
% 2.78/1.70 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(omegaIndAx_type,type,
% 2.78/1.70 omegaIndAx: $o ).
% 2.78/1.70
% 2.78/1.70 thf(omegaIndAx,definition,
% 2.78/1.70 ( omegaIndAx
% 2.78/1.70 = ( ! [A: $i] :
% 2.78/1.70 ( ( ( in @ emptyset @ A )
% 2.78/1.70 & ! [Xx: $i] :
% 2.78/1.70 ( ( ( in @ Xx @ omega )
% 2.78/1.70 & ( in @ Xx @ A ) )
% 2.78/1.70 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.78/1.70 => ! [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ omega )
% 2.78/1.70 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(replAx_type,type,
% 2.78/1.70 replAx: $o ).
% 2.78/1.70
% 2.78/1.70 thf(replAx,definition,
% 2.78/1.70 ( replAx
% 2.78/1.70 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.78/1.70 ( ! [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ( exu
% 2.78/1.70 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.78/1.70 => ? [B: $i] :
% 2.78/1.70 ! [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ B )
% 2.78/1.70 <=> ? [Xy: $i] :
% 2.78/1.70 ( ( in @ Xy @ A )
% 2.78/1.70 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(foundationAx_type,type,
% 2.78/1.70 foundationAx: $o ).
% 2.78/1.70
% 2.78/1.70 thf(foundationAx,definition,
% 2.78/1.70 ( foundationAx
% 2.78/1.70 = ( ! [A: $i] :
% 2.78/1.70 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.78/1.70 => ? [B: $i] :
% 2.78/1.70 ( ( in @ B @ A )
% 2.78/1.70 & ~ ? [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ B )
% 2.78/1.70 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(wellorderingAx_type,type,
% 2.78/1.70 wellorderingAx: $o ).
% 2.78/1.70
% 2.78/1.70 thf(wellorderingAx,definition,
% 2.78/1.70 ( wellorderingAx
% 2.78/1.70 = ( ! [A: $i] :
% 2.78/1.70 ? [B: $i] :
% 2.78/1.70 ( ! [C: $i] :
% 2.78/1.70 ( ( in @ C @ B )
% 2.78/1.70 => ! [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ C )
% 2.78/1.70 => ( in @ Xx @ A ) ) )
% 2.78/1.70 & ! [Xx: $i,Xy: $i] :
% 2.78/1.70 ( ( ( in @ Xx @ A )
% 2.78/1.70 & ( in @ Xy @ A ) )
% 2.78/1.70 => ( ! [C: $i] :
% 2.78/1.70 ( ( in @ C @ B )
% 2.78/1.70 => ( ( in @ Xx @ C )
% 2.78/1.70 <=> ( in @ Xy @ C ) ) )
% 2.78/1.70 => ( Xx = Xy ) ) )
% 2.78/1.70 & ! [C: $i,D: $i] :
% 2.78/1.70 ( ( ( in @ C @ B )
% 2.78/1.70 & ( in @ D @ B ) )
% 2.78/1.70 => ( ! [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ C )
% 2.78/1.70 => ( in @ Xx @ D ) )
% 2.78/1.70 | ! [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ D )
% 2.78/1.70 => ( in @ Xx @ C ) ) ) )
% 2.78/1.70 & ! [C: $i] :
% 2.78/1.70 ( ( ! [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ C )
% 2.78/1.70 => ( in @ Xx @ A ) )
% 2.78/1.70 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.78/1.70 => ? [D: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ D @ B )
% 2.78/1.70 & ( in @ Xx @ C )
% 2.78/1.70 & ~ ? [Xy: $i] :
% 2.78/1.70 ( ( in @ Xy @ D )
% 2.78/1.70 & ( in @ Xy @ C ) )
% 2.78/1.70 & ! [E: $i] :
% 2.78/1.70 ( ( in @ E @ B )
% 2.78/1.70 => ( ! [Xy: $i] :
% 2.78/1.70 ( ( in @ Xy @ E )
% 2.78/1.70 => ( in @ Xy @ D ) )
% 2.78/1.70 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(descr_type,type,
% 2.78/1.70 descr: ( $i > $o ) > $i ).
% 2.78/1.70
% 2.78/1.70 thf(descrp_type,type,
% 2.78/1.70 descrp: $o ).
% 2.78/1.70
% 2.78/1.70 thf(descrp,definition,
% 2.78/1.70 ( descrp
% 2.78/1.70 = ( ! [Xphi: $i > $o] :
% 2.78/1.70 ( ( exu
% 2.78/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.70 => ( Xphi
% 2.78/1.70 @ ( descr
% 2.78/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(dsetconstr_type,type,
% 2.78/1.70 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.78/1.70
% 2.78/1.70 thf(dsetconstrI_type,type,
% 2.78/1.70 dsetconstrI: $o ).
% 2.78/1.70
% 2.78/1.70 thf(dsetconstrI,definition,
% 2.78/1.70 ( dsetconstrI
% 2.78/1.70 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ( ( Xphi @ Xx )
% 2.78/1.70 => ( in @ Xx
% 2.78/1.70 @ ( dsetconstr @ A
% 2.78/1.70 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(dsetconstrEL_type,type,
% 2.78/1.70 dsetconstrEL: $o ).
% 2.78/1.70
% 2.78/1.70 thf(dsetconstrEL,definition,
% 2.78/1.70 ( dsetconstrEL
% 2.78/1.70 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx
% 2.78/1.70 @ ( dsetconstr @ A
% 2.78/1.70 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.78/1.70 => ( in @ Xx @ A ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(dsetconstrER_type,type,
% 2.78/1.70 dsetconstrER: $o ).
% 2.78/1.70
% 2.78/1.70 thf(dsetconstrER,definition,
% 2.78/1.70 ( dsetconstrER
% 2.78/1.70 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx
% 2.78/1.70 @ ( dsetconstr @ A
% 2.78/1.70 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.78/1.70 => ( Xphi @ Xx ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(exuE1_type,type,
% 2.78/1.70 exuE1: $o ).
% 2.78/1.70
% 2.78/1.70 thf(exuE1,definition,
% 2.78/1.70 ( exuE1
% 2.78/1.70 = ( ! [Xphi: $i > $o] :
% 2.78/1.70 ( ( exu
% 2.78/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.70 => ? [Xx: $i] :
% 2.78/1.70 ( ( Xphi @ Xx )
% 2.78/1.70 & ! [Xy: $i] :
% 2.78/1.70 ( ( Xphi @ Xy )
% 2.78/1.70 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(prop2set_type,type,
% 2.78/1.70 prop2set: $o > $i ).
% 2.78/1.70
% 2.78/1.70 thf(prop2setE_type,type,
% 2.78/1.70 prop2setE: $o ).
% 2.78/1.70
% 2.78/1.70 thf(prop2setE,definition,
% 2.78/1.70 ( prop2setE
% 2.78/1.70 = ( ! [Xphi: $o,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.78/1.70 => Xphi ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(emptysetE_type,type,
% 2.78/1.70 emptysetE: $o ).
% 2.78/1.70
% 2.78/1.70 thf(emptysetE,definition,
% 2.78/1.70 ( emptysetE
% 2.78/1.70 = ( ! [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ emptyset )
% 2.78/1.70 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(emptysetimpfalse_type,type,
% 2.78/1.70 emptysetimpfalse: $o ).
% 2.78/1.70
% 2.78/1.70 thf(emptysetimpfalse,definition,
% 2.78/1.70 ( emptysetimpfalse
% 2.78/1.70 = ( ! [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ emptyset )
% 2.78/1.70 => $false ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(notinemptyset_type,type,
% 2.78/1.70 notinemptyset: $o ).
% 2.78/1.70
% 2.78/1.70 thf(notinemptyset,definition,
% 2.78/1.70 ( notinemptyset
% 2.78/1.70 = ( ! [Xx: $i] :
% 2.78/1.70 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(exuE3e_type,type,
% 2.78/1.70 exuE3e: $o ).
% 2.78/1.70
% 2.78/1.70 thf(exuE3e,definition,
% 2.78/1.70 ( exuE3e
% 2.78/1.70 = ( ! [Xphi: $i > $o] :
% 2.78/1.70 ( ( exu
% 2.78/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.70 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setext_type,type,
% 2.78/1.70 setext: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setext,definition,
% 2.78/1.70 ( setext
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ! [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ( in @ Xx @ B ) )
% 2.78/1.70 => ( ! [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ B )
% 2.78/1.70 => ( in @ Xx @ A ) )
% 2.78/1.70 => ( A = B ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(emptyI_type,type,
% 2.78/1.70 emptyI: $o ).
% 2.78/1.70
% 2.78/1.70 thf(emptyI,definition,
% 2.78/1.70 ( emptyI
% 2.78/1.70 = ( ! [A: $i] :
% 2.78/1.70 ( ! [Xx: $i] :
% 2.78/1.70 ~ ( in @ Xx @ A )
% 2.78/1.70 => ( A = emptyset ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(noeltsimpempty_type,type,
% 2.78/1.70 noeltsimpempty: $o ).
% 2.78/1.70
% 2.78/1.70 thf(noeltsimpempty,definition,
% 2.78/1.70 ( noeltsimpempty
% 2.78/1.70 = ( ! [A: $i] :
% 2.78/1.70 ( ! [Xx: $i] :
% 2.78/1.70 ~ ( in @ Xx @ A )
% 2.78/1.70 => ( A = emptyset ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setbeta_type,type,
% 2.78/1.70 setbeta: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setbeta,definition,
% 2.78/1.70 ( setbeta
% 2.78/1.70 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ( ( in @ Xx
% 2.78/1.70 @ ( dsetconstr @ A
% 2.78/1.70 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.78/1.70 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(nonempty_type,type,
% 2.78/1.70 nonempty: $i > $o ).
% 2.78/1.70
% 2.78/1.70 thf(nonempty,definition,
% 2.78/1.70 ( nonempty
% 2.78/1.70 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(nonemptyE1_type,type,
% 2.78/1.70 nonemptyE1: $o ).
% 2.78/1.70
% 2.78/1.70 thf(nonemptyE1,definition,
% 2.78/1.70 ( nonemptyE1
% 2.78/1.70 = ( ! [A: $i] :
% 2.78/1.70 ( ( nonempty @ A )
% 2.78/1.70 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(nonemptyI_type,type,
% 2.78/1.70 nonemptyI: $o ).
% 2.78/1.70
% 2.78/1.70 thf(nonemptyI,definition,
% 2.78/1.70 ( nonemptyI
% 2.78/1.70 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ( ( Xphi @ Xx )
% 2.78/1.70 => ( nonempty
% 2.78/1.70 @ ( dsetconstr @ A
% 2.78/1.70 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(nonemptyI1_type,type,
% 2.78/1.70 nonemptyI1: $o ).
% 2.78/1.70
% 2.78/1.70 thf(nonemptyI1,definition,
% 2.78/1.70 ( nonemptyI1
% 2.78/1.70 = ( ! [A: $i] :
% 2.78/1.70 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.78/1.70 => ( nonempty @ A ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setadjoinIL_type,type,
% 2.78/1.70 setadjoinIL: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setadjoinIL,definition,
% 2.78/1.70 ( setadjoinIL
% 2.78/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(emptyinunitempty_type,type,
% 2.78/1.70 emptyinunitempty: $o ).
% 2.78/1.70
% 2.78/1.70 thf(emptyinunitempty,definition,
% 2.78/1.70 ( emptyinunitempty
% 2.78/1.70 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setadjoinIR_type,type,
% 2.78/1.70 setadjoinIR: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setadjoinIR,definition,
% 2.78/1.70 ( setadjoinIR
% 2.78/1.70 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.78/1.70 ( ( in @ Xy @ A )
% 2.78/1.70 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setadjoinE_type,type,
% 2.78/1.70 setadjoinE: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setadjoinE,definition,
% 2.78/1.70 ( setadjoinE
% 2.78/1.70 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.78/1.70 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.78/1.70 => ! [Xphi: $o] :
% 2.78/1.70 ( ( ( Xy = Xx )
% 2.78/1.70 => Xphi )
% 2.78/1.70 => ( ( ( in @ Xy @ A )
% 2.78/1.70 => Xphi )
% 2.78/1.70 => Xphi ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setadjoinOr_type,type,
% 2.78/1.70 setadjoinOr: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setadjoinOr,definition,
% 2.78/1.70 ( setadjoinOr
% 2.78/1.70 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.78/1.70 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.78/1.70 => ( ( Xy = Xx )
% 2.78/1.70 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setoftrueEq_type,type,
% 2.78/1.70 setoftrueEq: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setoftrueEq,definition,
% 2.78/1.70 ( setoftrueEq
% 2.78/1.70 = ( ! [A: $i] :
% 2.78/1.70 ( ( dsetconstr @ A
% 2.78/1.70 @ ^ [Xx: $i] : $true )
% 2.78/1.70 = A ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(powersetI_type,type,
% 2.78/1.70 powersetI: $o ).
% 2.78/1.70
% 2.78/1.70 thf(powersetI,definition,
% 2.78/1.70 ( powersetI
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ! [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ B )
% 2.78/1.70 => ( in @ Xx @ A ) )
% 2.78/1.70 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(emptyinPowerset_type,type,
% 2.78/1.70 emptyinPowerset: $o ).
% 2.78/1.70
% 2.78/1.70 thf(emptyinPowerset,definition,
% 2.78/1.70 ( emptyinPowerset
% 2.78/1.70 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(emptyInPowerset_type,type,
% 2.78/1.70 emptyInPowerset: $o ).
% 2.78/1.70
% 2.78/1.70 thf(emptyInPowerset,definition,
% 2.78/1.70 ( emptyInPowerset
% 2.78/1.70 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(powersetE_type,type,
% 2.78/1.70 powersetE: $o ).
% 2.78/1.70
% 2.78/1.70 thf(powersetE,definition,
% 2.78/1.70 ( powersetE
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ B @ ( powerset @ A ) )
% 2.78/1.70 => ( ( in @ Xx @ B )
% 2.78/1.70 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setunionI_type,type,
% 2.78/1.70 setunionI: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setunionI,definition,
% 2.78/1.70 ( setunionI
% 2.78/1.70 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.78/1.70 ( ( in @ Xx @ B )
% 2.78/1.70 => ( ( in @ B @ A )
% 2.78/1.70 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setunionE_type,type,
% 2.78/1.70 setunionE: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setunionE,definition,
% 2.78/1.70 ( setunionE
% 2.78/1.70 = ( ! [A: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ ( setunion @ A ) )
% 2.78/1.70 => ! [Xphi: $o] :
% 2.78/1.70 ( ! [B: $i] :
% 2.78/1.70 ( ( in @ Xx @ B )
% 2.78/1.70 => ( ( in @ B @ A )
% 2.78/1.70 => Xphi ) )
% 2.78/1.70 => Xphi ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(subPowSU_type,type,
% 2.78/1.70 subPowSU: $o ).
% 2.78/1.70
% 2.78/1.70 thf(subPowSU,definition,
% 2.78/1.70 ( subPowSU
% 2.78/1.70 = ( ! [A: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(exuE2_type,type,
% 2.78/1.70 exuE2: $o ).
% 2.78/1.70
% 2.78/1.70 thf(exuE2,definition,
% 2.78/1.70 ( exuE2
% 2.78/1.70 = ( ! [Xphi: $i > $o] :
% 2.78/1.70 ( ( exu
% 2.78/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.70 => ? [Xx: $i] :
% 2.78/1.70 ! [Xy: $i] :
% 2.78/1.70 ( ( Xphi @ Xy )
% 2.78/1.70 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(nonemptyImpWitness_type,type,
% 2.78/1.70 nonemptyImpWitness: $o ).
% 2.78/1.70
% 2.78/1.70 thf(nonemptyImpWitness,definition,
% 2.78/1.70 ( nonemptyImpWitness
% 2.78/1.70 = ( ! [A: $i] :
% 2.78/1.70 ( ( nonempty @ A )
% 2.78/1.70 => ? [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 & $true ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(uniqinunit_type,type,
% 2.78/1.70 uniqinunit: $o ).
% 2.78/1.70
% 2.78/1.70 thf(uniqinunit,definition,
% 2.78/1.70 ( uniqinunit
% 2.78/1.70 = ( ! [Xx: $i,Xy: $i] :
% 2.78/1.70 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.78/1.70 => ( Xx = Xy ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(notinsingleton_type,type,
% 2.78/1.70 notinsingleton: $o ).
% 2.78/1.70
% 2.78/1.70 thf(notinsingleton,definition,
% 2.78/1.70 ( notinsingleton
% 2.78/1.70 = ( ! [Xx: $i,Xy: $i] :
% 2.78/1.70 ( ( Xx != Xy )
% 2.78/1.70 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(eqinunit_type,type,
% 2.78/1.70 eqinunit: $o ).
% 2.78/1.70
% 2.78/1.70 thf(eqinunit,definition,
% 2.78/1.70 ( eqinunit
% 2.78/1.70 = ( ! [Xx: $i,Xy: $i] :
% 2.78/1.70 ( ( Xx = Xy )
% 2.78/1.70 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(singletonsswitch_type,type,
% 2.78/1.70 singletonsswitch: $o ).
% 2.78/1.70
% 2.78/1.70 thf(singletonsswitch,definition,
% 2.78/1.70 ( singletonsswitch
% 2.78/1.70 = ( ! [Xx: $i,Xy: $i] :
% 2.78/1.70 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.78/1.70 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(upairsetE_type,type,
% 2.78/1.70 upairsetE: $o ).
% 2.78/1.70
% 2.78/1.70 thf(upairsetE,definition,
% 2.78/1.70 ( upairsetE
% 2.78/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.78/1.70 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.78/1.70 => ( ( Xz = Xx )
% 2.78/1.70 | ( Xz = Xy ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(upairsetIL_type,type,
% 2.78/1.70 upairsetIL: $o ).
% 2.78/1.70
% 2.78/1.70 thf(upairsetIL,definition,
% 2.78/1.70 ( upairsetIL
% 2.78/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(upairsetIR_type,type,
% 2.78/1.70 upairsetIR: $o ).
% 2.78/1.70
% 2.78/1.70 thf(upairsetIR,definition,
% 2.78/1.70 ( upairsetIR
% 2.78/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(emptyE1_type,type,
% 2.78/1.70 emptyE1: $o ).
% 2.78/1.70
% 2.78/1.70 thf(emptyE1,definition,
% 2.78/1.70 ( emptyE1
% 2.78/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 2.78/1.70 ( ? [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 & ( Xphi @ Xx ) )
% 2.78/1.70 => ( ( ( dsetconstr @ A
% 2.78/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.70 = emptyset )
% 2.78/1.70 => $false ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(vacuousDall_type,type,
% 2.78/1.70 vacuousDall: $o ).
% 2.78/1.70
% 2.78/1.70 thf(vacuousDall,definition,
% 2.78/1.70 ( vacuousDall
% 2.78/1.70 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ emptyset )
% 2.78/1.70 => ( Xphi @ Xx ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(quantDeMorgan1_type,type,
% 2.78/1.70 quantDeMorgan1: $o ).
% 2.78/1.70
% 2.78/1.70 thf(quantDeMorgan1,definition,
% 2.78/1.70 ( quantDeMorgan1
% 2.78/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 2.78/1.70 ( ~ ! [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ( Xphi @ Xx ) )
% 2.78/1.70 => ? [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(quantDeMorgan2_type,type,
% 2.78/1.70 quantDeMorgan2: $o ).
% 2.78/1.70
% 2.78/1.70 thf(quantDeMorgan2,definition,
% 2.78/1.70 ( quantDeMorgan2
% 2.78/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 2.78/1.70 ( ! [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ~ ( Xphi @ Xx ) )
% 2.78/1.70 => ~ ? [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(quantDeMorgan3_type,type,
% 2.78/1.70 quantDeMorgan3: $o ).
% 2.78/1.70
% 2.78/1.70 thf(quantDeMorgan3,definition,
% 2.78/1.70 ( quantDeMorgan3
% 2.78/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 2.78/1.70 ( ~ ? [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 & ( Xphi @ Xx ) )
% 2.78/1.70 => ! [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(quantDeMorgan4_type,type,
% 2.78/1.70 quantDeMorgan4: $o ).
% 2.78/1.70
% 2.78/1.70 thf(quantDeMorgan4,definition,
% 2.78/1.70 ( quantDeMorgan4
% 2.78/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 2.78/1.70 ( ? [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 & ~ ( Xphi @ Xx ) )
% 2.78/1.70 => ~ ! [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(prop2setI_type,type,
% 2.78/1.70 prop2setI: $o ).
% 2.78/1.70
% 2.78/1.70 thf(prop2setI,definition,
% 2.78/1.70 ( prop2setI
% 2.78/1.70 = ( ! [Xphi: $o] :
% 2.78/1.70 ( Xphi
% 2.78/1.70 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(set2prop_type,type,
% 2.78/1.70 set2prop: $i > $o ).
% 2.78/1.70
% 2.78/1.70 thf(prop2set2propI_type,type,
% 2.78/1.70 prop2set2propI: $o ).
% 2.78/1.70
% 2.78/1.70 thf(prop2set2propI,definition,
% 2.78/1.70 ( prop2set2propI
% 2.78/1.70 = ( ! [Xphi: $o] :
% 2.78/1.70 ( Xphi
% 2.78/1.70 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(notdexE_type,type,
% 2.78/1.70 notdexE: $o ).
% 2.78/1.70
% 2.78/1.70 thf(notdexE,definition,
% 2.78/1.70 ( notdexE
% 2.78/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 2.78/1.70 ( ~ ? [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 & ( Xphi @ Xx ) )
% 2.78/1.70 => ! [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(notdallE_type,type,
% 2.78/1.70 notdallE: $o ).
% 2.78/1.70
% 2.78/1.70 thf(notdallE,definition,
% 2.78/1.70 ( notdallE
% 2.78/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 2.78/1.70 ( ~ ! [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ( Xphi @ Xx ) )
% 2.78/1.70 => ? [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(exuI1_type,type,
% 2.78/1.70 exuI1: $o ).
% 2.78/1.70
% 2.78/1.70 thf(exuI1,definition,
% 2.78/1.70 ( exuI1
% 2.78/1.70 = ( ! [Xphi: $i > $o] :
% 2.78/1.70 ( ? [Xx: $i] :
% 2.78/1.70 ( ( Xphi @ Xx )
% 2.78/1.70 & ! [Xy: $i] :
% 2.78/1.70 ( ( Xphi @ Xy )
% 2.78/1.70 => ( Xx = Xy ) ) )
% 2.78/1.70 => ( exu
% 2.78/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(exuI3_type,type,
% 2.78/1.70 exuI3: $o ).
% 2.78/1.70
% 2.78/1.70 thf(exuI3,definition,
% 2.78/1.70 ( exuI3
% 2.78/1.70 = ( ! [Xphi: $i > $o] :
% 2.78/1.70 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.78/1.70 => ( ! [Xx: $i,Xy: $i] :
% 2.78/1.70 ( ( Xphi @ Xx )
% 2.78/1.70 => ( ( Xphi @ Xy )
% 2.78/1.70 => ( Xx = Xy ) ) )
% 2.78/1.70 => ( exu
% 2.78/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(exuI2_type,type,
% 2.78/1.70 exuI2: $o ).
% 2.78/1.70
% 2.78/1.70 thf(exuI2,definition,
% 2.78/1.70 ( exuI2
% 2.78/1.70 = ( ! [Xphi: $i > $o] :
% 2.78/1.70 ( ? [Xx: $i] :
% 2.78/1.70 ! [Xy: $i] :
% 2.78/1.70 ( ( Xphi @ Xy )
% 2.78/1.70 <=> ( Xy = Xx ) )
% 2.78/1.70 => ( exu
% 2.78/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(inCongP_type,type,
% 2.78/1.70 inCongP: $o ).
% 2.78/1.70
% 2.78/1.70 thf(inCongP,definition,
% 2.78/1.70 ( inCongP
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ( A = B )
% 2.78/1.70 => ! [Xx: $i,Xy: $i] :
% 2.78/1.70 ( ( Xx = Xy )
% 2.78/1.70 => ( ( in @ Xx @ A )
% 2.78/1.70 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(in__Cong_type,type,
% 2.78/1.70 in__Cong: $o ).
% 2.78/1.70
% 2.78/1.70 thf(in__Cong,definition,
% 2.78/1.70 ( in__Cong
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ( A = B )
% 2.78/1.70 => ! [Xx: $i,Xy: $i] :
% 2.78/1.70 ( ( Xx = Xy )
% 2.78/1.70 => ( ( in @ Xx @ A )
% 2.78/1.70 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(exuE3u_type,type,
% 2.78/1.70 exuE3u: $o ).
% 2.78/1.70
% 2.78/1.70 thf(exuE3u,definition,
% 2.78/1.70 ( exuE3u
% 2.78/1.70 = ( ! [Xphi: $i > $o] :
% 2.78/1.70 ( ( exu
% 2.78/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.70 => ! [Xx: $i,Xy: $i] :
% 2.78/1.70 ( ( Xphi @ Xx )
% 2.78/1.70 => ( ( Xphi @ Xy )
% 2.78/1.70 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(exu__Cong_type,type,
% 2.78/1.70 exu__Cong: $o ).
% 2.78/1.70
% 2.78/1.70 thf(exu__Cong,definition,
% 2.78/1.70 ( exu__Cong
% 2.78/1.70 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.78/1.70 ( ! [Xx: $i,Xy: $i] :
% 2.78/1.70 ( ( Xx = Xy )
% 2.78/1.70 => ( ( Xphi @ Xx )
% 2.78/1.70 <=> ( Xpsi @ Xy ) ) )
% 2.78/1.70 => ( ( exu
% 2.78/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.70 <=> ( exu
% 2.78/1.70 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(emptyset__Cong_type,type,
% 2.78/1.70 emptyset__Cong: $o ).
% 2.78/1.70
% 2.78/1.70 thf(emptyset__Cong,definition,
% 2.78/1.70 ( emptyset__Cong
% 2.78/1.70 = ( emptyset = emptyset ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setadjoin__Cong_type,type,
% 2.78/1.70 setadjoin__Cong: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setadjoin__Cong,definition,
% 2.78/1.70 ( setadjoin__Cong
% 2.78/1.70 = ( ! [Xx: $i,Xy: $i] :
% 2.78/1.70 ( ( Xx = Xy )
% 2.78/1.70 => ! [Xz: $i,Xu: $i] :
% 2.78/1.70 ( ( Xz = Xu )
% 2.78/1.70 => ( ( setadjoin @ Xx @ Xz )
% 2.78/1.70 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(powerset__Cong_type,type,
% 2.78/1.70 powerset__Cong: $o ).
% 2.78/1.70
% 2.78/1.70 thf(powerset__Cong,definition,
% 2.78/1.70 ( powerset__Cong
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ( A = B )
% 2.78/1.70 => ( ( powerset @ A )
% 2.78/1.70 = ( powerset @ B ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setunion__Cong_type,type,
% 2.78/1.70 setunion__Cong: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setunion__Cong,definition,
% 2.78/1.70 ( setunion__Cong
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ( A = B )
% 2.78/1.70 => ( ( setunion @ A )
% 2.78/1.70 = ( setunion @ B ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(omega__Cong_type,type,
% 2.78/1.70 omega__Cong: $o ).
% 2.78/1.70
% 2.78/1.70 thf(omega__Cong,definition,
% 2.78/1.70 ( omega__Cong
% 2.78/1.70 = ( omega = omega ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(exuEu_type,type,
% 2.78/1.70 exuEu: $o ).
% 2.78/1.70
% 2.78/1.70 thf(exuEu,definition,
% 2.78/1.70 ( exuEu
% 2.78/1.70 = ( ! [Xphi: $i > $o] :
% 2.78/1.70 ( ( exu
% 2.78/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.70 => ! [Xx: $i,Xy: $i] :
% 2.78/1.70 ( ( Xphi @ Xx )
% 2.78/1.70 => ( ( Xphi @ Xy )
% 2.78/1.70 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(descr__Cong_type,type,
% 2.78/1.70 descr__Cong: $o ).
% 2.78/1.70
% 2.78/1.70 thf(descr__Cong,definition,
% 2.78/1.70 ( descr__Cong
% 2.78/1.70 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.78/1.70 ( ! [Xx: $i,Xy: $i] :
% 2.78/1.70 ( ( Xx = Xy )
% 2.78/1.70 => ( ( Xphi @ Xx )
% 2.78/1.70 <=> ( Xpsi @ Xy ) ) )
% 2.78/1.70 => ( ( exu
% 2.78/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.70 => ( ( exu
% 2.78/1.70 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.78/1.70 => ( ( descr
% 2.78/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.70 = ( descr
% 2.78/1.70 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(dsetconstr__Cong_type,type,
% 2.78/1.70 dsetconstr__Cong: $o ).
% 2.78/1.70
% 2.78/1.70 thf(dsetconstr__Cong,definition,
% 2.78/1.70 ( dsetconstr__Cong
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ( A = B )
% 2.78/1.70 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.78/1.70 ( ! [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ! [Xy: $i] :
% 2.78/1.70 ( ( in @ Xy @ B )
% 2.78/1.70 => ( ( Xx = Xy )
% 2.78/1.70 => ( ( Xphi @ Xx )
% 2.78/1.70 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.78/1.70 => ( ( dsetconstr @ A
% 2.78/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.70 = ( dsetconstr @ B
% 2.78/1.70 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(subset_type,type,
% 2.78/1.70 subset: $i > $i > $o ).
% 2.78/1.70
% 2.78/1.70 thf(disjoint_type,type,
% 2.78/1.70 disjoint: $i > $i > $o ).
% 2.78/1.70
% 2.78/1.70 thf(setsmeet_type,type,
% 2.78/1.70 setsmeet: $i > $i > $o ).
% 2.78/1.70
% 2.78/1.70 thf(subsetI1_type,type,
% 2.78/1.70 subsetI1: $o ).
% 2.78/1.70
% 2.78/1.70 thf(subsetI1,definition,
% 2.78/1.70 ( subsetI1
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ! [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ( in @ Xx @ B ) )
% 2.78/1.70 => ( subset @ A @ B ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(eqimpsubset2_type,type,
% 2.78/1.70 eqimpsubset2: $o ).
% 2.78/1.70
% 2.78/1.70 thf(eqimpsubset2,definition,
% 2.78/1.70 ( eqimpsubset2
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ( A = B )
% 2.78/1.70 => ( subset @ B @ A ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(eqimpsubset1_type,type,
% 2.78/1.70 eqimpsubset1: $o ).
% 2.78/1.70
% 2.78/1.70 thf(eqimpsubset1,definition,
% 2.78/1.70 ( eqimpsubset1
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ( A = B )
% 2.78/1.70 => ( subset @ A @ B ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(subsetI2_type,type,
% 2.78/1.70 subsetI2: $o ).
% 2.78/1.70
% 2.78/1.70 thf(subsetI2,definition,
% 2.78/1.70 ( subsetI2
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ! [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ( in @ Xx @ B ) )
% 2.78/1.70 => ( subset @ A @ B ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(emptysetsubset_type,type,
% 2.78/1.70 emptysetsubset: $o ).
% 2.78/1.70
% 2.78/1.70 thf(emptysetsubset,definition,
% 2.78/1.70 ( emptysetsubset
% 2.78/1.70 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(subsetE_type,type,
% 2.78/1.70 subsetE: $o ).
% 2.78/1.70
% 2.78/1.70 thf(subsetE,definition,
% 2.78/1.70 ( subsetE
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ( subset @ A @ B )
% 2.78/1.70 => ( ( in @ Xx @ A )
% 2.78/1.70 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(subsetE2_type,type,
% 2.78/1.70 subsetE2: $o ).
% 2.78/1.70
% 2.78/1.70 thf(subsetE2,definition,
% 2.78/1.70 ( subsetE2
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ( subset @ A @ B )
% 2.78/1.70 => ( ~ ( in @ Xx @ B )
% 2.78/1.70 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(notsubsetI_type,type,
% 2.78/1.70 notsubsetI: $o ).
% 2.78/1.70
% 2.78/1.70 thf(notsubsetI,definition,
% 2.78/1.70 ( notsubsetI
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ( ~ ( in @ Xx @ B )
% 2.78/1.70 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(notequalI1_type,type,
% 2.78/1.70 notequalI1: $o ).
% 2.78/1.70
% 2.78/1.70 thf(notequalI1,definition,
% 2.78/1.70 ( notequalI1
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ~ ( subset @ A @ B )
% 2.78/1.70 => ( A != B ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(notequalI2_type,type,
% 2.78/1.70 notequalI2: $o ).
% 2.78/1.70
% 2.78/1.70 thf(notequalI2,definition,
% 2.78/1.70 ( notequalI2
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ( ~ ( in @ Xx @ B )
% 2.78/1.70 => ( A != B ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(subsetRefl_type,type,
% 2.78/1.70 subsetRefl: $o ).
% 2.78/1.70
% 2.78/1.70 thf(subsetRefl,definition,
% 2.78/1.70 ( subsetRefl
% 2.78/1.70 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(subsetTrans_type,type,
% 2.78/1.70 subsetTrans: $o ).
% 2.78/1.70
% 2.78/1.70 thf(subsetTrans,definition,
% 2.78/1.70 ( subsetTrans
% 2.78/1.70 = ( ! [A: $i,B: $i,C: $i] :
% 2.78/1.70 ( ( subset @ A @ B )
% 2.78/1.70 => ( ( subset @ B @ C )
% 2.78/1.70 => ( subset @ A @ C ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setadjoinSub_type,type,
% 2.78/1.70 setadjoinSub: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setadjoinSub,definition,
% 2.78/1.70 ( setadjoinSub
% 2.78/1.70 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setadjoinSub2_type,type,
% 2.78/1.70 setadjoinSub2: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setadjoinSub2,definition,
% 2.78/1.70 ( setadjoinSub2
% 2.78/1.70 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.78/1.70 ( ( subset @ A @ B )
% 2.78/1.70 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(subset2powerset_type,type,
% 2.78/1.70 subset2powerset: $o ).
% 2.78/1.70
% 2.78/1.70 thf(subset2powerset,definition,
% 2.78/1.70 ( subset2powerset
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ( subset @ A @ B )
% 2.78/1.70 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setextsub_type,type,
% 2.78/1.70 setextsub: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setextsub,definition,
% 2.78/1.70 ( setextsub
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ( subset @ A @ B )
% 2.78/1.70 => ( ( subset @ B @ A )
% 2.78/1.70 => ( A = B ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(subsetemptysetimpeq_type,type,
% 2.78/1.70 subsetemptysetimpeq: $o ).
% 2.78/1.70
% 2.78/1.70 thf(subsetemptysetimpeq,definition,
% 2.78/1.70 ( subsetemptysetimpeq
% 2.78/1.70 = ( ! [A: $i] :
% 2.78/1.70 ( ( subset @ A @ emptyset )
% 2.78/1.70 => ( A = emptyset ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(powersetI1_type,type,
% 2.78/1.70 powersetI1: $o ).
% 2.78/1.70
% 2.78/1.70 thf(powersetI1,definition,
% 2.78/1.70 ( powersetI1
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ( subset @ B @ A )
% 2.78/1.70 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(powersetE1_type,type,
% 2.78/1.70 powersetE1: $o ).
% 2.78/1.70
% 2.78/1.70 thf(powersetE1,definition,
% 2.78/1.70 ( powersetE1
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ( in @ B @ ( powerset @ A ) )
% 2.78/1.70 => ( subset @ B @ A ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(inPowerset_type,type,
% 2.78/1.70 inPowerset: $o ).
% 2.78/1.70
% 2.78/1.70 thf(inPowerset,definition,
% 2.78/1.70 ( inPowerset
% 2.78/1.70 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(powersetsubset_type,type,
% 2.78/1.70 powersetsubset: $o ).
% 2.78/1.70
% 2.78/1.70 thf(powersetsubset,definition,
% 2.78/1.70 ( powersetsubset
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ( subset @ A @ B )
% 2.78/1.70 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(sepInPowerset_type,type,
% 2.78/1.70 sepInPowerset: $o ).
% 2.78/1.70
% 2.78/1.70 thf(sepInPowerset,definition,
% 2.78/1.70 ( sepInPowerset
% 2.78/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 2.78/1.70 ( in
% 2.78/1.70 @ ( dsetconstr @ A
% 2.78/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.70 @ ( powerset @ A ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(sepSubset_type,type,
% 2.78/1.70 sepSubset: $o ).
% 2.78/1.70
% 2.78/1.70 thf(sepSubset,definition,
% 2.78/1.70 ( sepSubset
% 2.78/1.70 = ( ! [A: $i,Xphi: $i > $o] :
% 2.78/1.70 ( subset
% 2.78/1.70 @ ( dsetconstr @ A
% 2.78/1.70 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.78/1.70 @ A ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(binunion_type,type,
% 2.78/1.70 binunion: $i > $i > $i ).
% 2.78/1.70
% 2.78/1.70 thf(binunionIL_type,type,
% 2.78/1.70 binunionIL: $o ).
% 2.78/1.70
% 2.78/1.70 thf(binunionIL,definition,
% 2.78/1.70 ( binunionIL
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(upairset2IR_type,type,
% 2.78/1.70 upairset2IR: $o ).
% 2.78/1.70
% 2.78/1.70 thf(upairset2IR,definition,
% 2.78/1.70 ( upairset2IR
% 2.78/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(binunionIR_type,type,
% 2.78/1.70 binunionIR: $o ).
% 2.78/1.70
% 2.78/1.70 thf(binunionIR,definition,
% 2.78/1.70 ( binunionIR
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ B )
% 2.78/1.70 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(binunionEcases_type,type,
% 2.78/1.70 binunionEcases: $o ).
% 2.78/1.70
% 2.78/1.70 thf(binunionEcases,definition,
% 2.78/1.70 ( binunionEcases
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.78/1.70 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.78/1.70 => ( ( ( in @ Xx @ A )
% 2.78/1.70 => Xphi )
% 2.78/1.70 => ( ( ( in @ Xx @ B )
% 2.78/1.70 => Xphi )
% 2.78/1.70 => Xphi ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(binunionE_type,type,
% 2.78/1.70 binunionE: $o ).
% 2.78/1.70
% 2.78/1.70 thf(binunionE,definition,
% 2.78/1.70 ( binunionE
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.78/1.70 => ( ( in @ Xx @ A )
% 2.78/1.70 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(binunionLsub_type,type,
% 2.78/1.70 binunionLsub: $o ).
% 2.78/1.70
% 2.78/1.70 thf(binunionLsub,definition,
% 2.78/1.70 ( binunionLsub
% 2.78/1.70 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(binunionRsub_type,type,
% 2.78/1.70 binunionRsub: $o ).
% 2.78/1.70
% 2.78/1.70 thf(binunionRsub,definition,
% 2.78/1.70 ( binunionRsub
% 2.78/1.70 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(binintersect_type,type,
% 2.78/1.70 binintersect: $i > $i > $i ).
% 2.78/1.70
% 2.78/1.70 thf(binintersectI_type,type,
% 2.78/1.70 binintersectI: $o ).
% 2.78/1.70
% 2.78/1.70 thf(binintersectI,definition,
% 2.78/1.70 ( binintersectI
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ( ( in @ Xx @ B )
% 2.78/1.70 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(binintersectSubset5_type,type,
% 2.78/1.70 binintersectSubset5: $o ).
% 2.78/1.70
% 2.78/1.70 thf(binintersectSubset5,definition,
% 2.78/1.70 ( binintersectSubset5
% 2.78/1.70 = ( ! [A: $i,B: $i,C: $i] :
% 2.78/1.70 ( ( subset @ C @ A )
% 2.78/1.70 => ( ( subset @ C @ B )
% 2.78/1.70 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(binintersectEL_type,type,
% 2.78/1.70 binintersectEL: $o ).
% 2.78/1.70
% 2.78/1.70 thf(binintersectEL,definition,
% 2.78/1.70 ( binintersectEL
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.78/1.70 => ( in @ Xx @ A ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(binintersectLsub_type,type,
% 2.78/1.70 binintersectLsub: $o ).
% 2.78/1.70
% 2.78/1.70 thf(binintersectLsub,definition,
% 2.78/1.70 ( binintersectLsub
% 2.78/1.70 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(binintersectSubset2_type,type,
% 2.78/1.70 binintersectSubset2: $o ).
% 2.78/1.70
% 2.78/1.70 thf(binintersectSubset2,definition,
% 2.78/1.70 ( binintersectSubset2
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ( subset @ A @ B )
% 2.78/1.70 => ( ( binintersect @ A @ B )
% 2.78/1.70 = A ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(binintersectSubset3_type,type,
% 2.78/1.70 binintersectSubset3: $o ).
% 2.78/1.70
% 2.78/1.70 thf(binintersectSubset3,definition,
% 2.78/1.70 ( binintersectSubset3
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ( ( binintersect @ A @ B )
% 2.78/1.70 = B )
% 2.78/1.70 => ( subset @ B @ A ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(binintersectER_type,type,
% 2.78/1.70 binintersectER: $o ).
% 2.78/1.70
% 2.78/1.70 thf(binintersectER,definition,
% 2.78/1.70 ( binintersectER
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.78/1.70 => ( in @ Xx @ B ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(disjointsetsI1_type,type,
% 2.78/1.70 disjointsetsI1: $o ).
% 2.78/1.70
% 2.78/1.70 thf(disjointsetsI1,definition,
% 2.78/1.70 ( disjointsetsI1
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ~ ? [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 & ( in @ Xx @ B ) )
% 2.78/1.70 => ( ( binintersect @ A @ B )
% 2.78/1.70 = emptyset ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(binintersectRsub_type,type,
% 2.78/1.70 binintersectRsub: $o ).
% 2.78/1.70
% 2.78/1.70 thf(binintersectRsub,definition,
% 2.78/1.70 ( binintersectRsub
% 2.78/1.70 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(binintersectSubset4_type,type,
% 2.78/1.70 binintersectSubset4: $o ).
% 2.78/1.70
% 2.78/1.70 thf(binintersectSubset4,definition,
% 2.78/1.70 ( binintersectSubset4
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ( subset @ B @ A )
% 2.78/1.70 => ( ( binintersect @ A @ B )
% 2.78/1.70 = B ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(binintersectSubset1_type,type,
% 2.78/1.70 binintersectSubset1: $o ).
% 2.78/1.70
% 2.78/1.70 thf(binintersectSubset1,definition,
% 2.78/1.70 ( binintersectSubset1
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ( ( binintersect @ A @ B )
% 2.78/1.70 = A )
% 2.78/1.70 => ( subset @ A @ B ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(bs114d_type,type,
% 2.78/1.70 bs114d: $o ).
% 2.78/1.70
% 2.78/1.70 thf(bs114d,definition,
% 2.78/1.70 ( bs114d
% 2.78/1.70 = ( ! [A: $i,B: $i,C: $i] :
% 2.78/1.70 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.78/1.70 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(regular_type,type,
% 2.78/1.70 regular: $i > $o ).
% 2.78/1.70
% 2.78/1.70 thf(setminus_type,type,
% 2.78/1.70 setminus: $i > $i > $i ).
% 2.78/1.70
% 2.78/1.70 thf(setminusI_type,type,
% 2.78/1.70 setminusI: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setminusI,definition,
% 2.78/1.70 ( setminusI
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ( ~ ( in @ Xx @ B )
% 2.78/1.70 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setminusEL_type,type,
% 2.78/1.70 setminusEL: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setminusEL,definition,
% 2.78/1.70 ( setminusEL
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.78/1.70 => ( in @ Xx @ A ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setminusER_type,type,
% 2.78/1.70 setminusER: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setminusER,definition,
% 2.78/1.70 ( setminusER
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.78/1.70 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setminusSubset2_type,type,
% 2.78/1.70 setminusSubset2: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setminusSubset2,definition,
% 2.78/1.70 ( setminusSubset2
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ( subset @ A @ B )
% 2.78/1.70 => ( ( setminus @ A @ B )
% 2.78/1.70 = emptyset ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setminusERneg_type,type,
% 2.78/1.70 setminusERneg: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setminusERneg,definition,
% 2.78/1.70 ( setminusERneg
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.78/1.70 => ( ( in @ Xx @ A )
% 2.78/1.70 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setminusELneg_type,type,
% 2.78/1.70 setminusELneg: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setminusELneg,definition,
% 2.78/1.70 ( setminusELneg
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.78/1.70 => ( ~ ( in @ Xx @ B )
% 2.78/1.70 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setminusILneg_type,type,
% 2.78/1.70 setminusILneg: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setminusILneg,definition,
% 2.78/1.70 ( setminusILneg
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ~ ( in @ Xx @ A )
% 2.78/1.70 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setminusIRneg_type,type,
% 2.78/1.70 setminusIRneg: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setminusIRneg,definition,
% 2.78/1.70 ( setminusIRneg
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ B )
% 2.78/1.70 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setminusLsub_type,type,
% 2.78/1.70 setminusLsub: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setminusLsub,definition,
% 2.78/1.70 ( setminusLsub
% 2.78/1.70 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setminusSubset1_type,type,
% 2.78/1.70 setminusSubset1: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setminusSubset1,definition,
% 2.78/1.70 ( setminusSubset1
% 2.78/1.70 = ( ! [A: $i,B: $i] :
% 2.78/1.70 ( ( ( setminus @ A @ B )
% 2.78/1.70 = emptyset )
% 2.78/1.70 => ( subset @ A @ B ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(symdiff_type,type,
% 2.78/1.70 symdiff: $i > $i > $i ).
% 2.78/1.70
% 2.78/1.70 thf(symdiffE_type,type,
% 2.78/1.70 symdiffE: $o ).
% 2.78/1.70
% 2.78/1.70 thf(symdiffE,definition,
% 2.78/1.70 ( symdiffE
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.78/1.70 => ! [Xphi: $o] :
% 2.78/1.70 ( ( ( in @ Xx @ A )
% 2.78/1.70 => ( ~ ( in @ Xx @ B )
% 2.78/1.70 => Xphi ) )
% 2.78/1.70 => ( ( ~ ( in @ Xx @ A )
% 2.78/1.70 => ( ( in @ Xx @ B )
% 2.78/1.70 => Xphi ) )
% 2.78/1.70 => Xphi ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(symdiffI1_type,type,
% 2.78/1.70 symdiffI1: $o ).
% 2.78/1.70
% 2.78/1.70 thf(symdiffI1,definition,
% 2.78/1.70 ( symdiffI1
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ( ~ ( in @ Xx @ B )
% 2.78/1.70 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(symdiffI2_type,type,
% 2.78/1.70 symdiffI2: $o ).
% 2.78/1.70
% 2.78/1.70 thf(symdiffI2,definition,
% 2.78/1.70 ( symdiffI2
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ~ ( in @ Xx @ A )
% 2.78/1.70 => ( ( in @ Xx @ B )
% 2.78/1.70 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(symdiffIneg1_type,type,
% 2.78/1.70 symdiffIneg1: $o ).
% 2.78/1.70
% 2.78/1.70 thf(symdiffIneg1,definition,
% 2.78/1.70 ( symdiffIneg1
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ( ( in @ Xx @ B )
% 2.78/1.70 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(symdiffIneg2_type,type,
% 2.78/1.70 symdiffIneg2: $o ).
% 2.78/1.70
% 2.78/1.70 thf(symdiffIneg2,definition,
% 2.78/1.70 ( symdiffIneg2
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ~ ( in @ Xx @ A )
% 2.78/1.70 => ( ~ ( in @ Xx @ B )
% 2.78/1.70 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(iskpair_type,type,
% 2.78/1.70 iskpair: $i > $o ).
% 2.78/1.70
% 2.78/1.70 thf(iskpair,definition,
% 2.78/1.70 ( iskpair
% 2.78/1.70 = ( ^ [A: $i] :
% 2.78/1.70 ? [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ ( setunion @ A ) )
% 2.78/1.70 & ? [Xy: $i] :
% 2.78/1.70 ( ( in @ Xy @ ( setunion @ A ) )
% 2.78/1.70 & ( A
% 2.78/1.70 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(secondinupair_type,type,
% 2.78/1.70 secondinupair: $o ).
% 2.78/1.70
% 2.78/1.70 thf(secondinupair,definition,
% 2.78/1.70 ( secondinupair
% 2.78/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setukpairIL_type,type,
% 2.78/1.70 setukpairIL: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setukpairIL,definition,
% 2.78/1.70 ( setukpairIL
% 2.78/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setukpairIR_type,type,
% 2.78/1.70 setukpairIR: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setukpairIR,definition,
% 2.78/1.70 ( setukpairIR
% 2.78/1.70 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(kpairiskpair_type,type,
% 2.78/1.70 kpairiskpair: $o ).
% 2.78/1.70
% 2.78/1.70 thf(kpairiskpair,definition,
% 2.78/1.70 ( kpairiskpair
% 2.78/1.70 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(kpair_type,type,
% 2.78/1.70 kpair: $i > $i > $i ).
% 2.78/1.70
% 2.78/1.70 thf(kpair,definition,
% 2.78/1.70 ( kpair
% 2.78/1.70 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(kpairp_type,type,
% 2.78/1.70 kpairp: $o ).
% 2.78/1.70
% 2.78/1.70 thf(kpairp,definition,
% 2.78/1.70 ( kpairp
% 2.78/1.70 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(cartprod_type,type,
% 2.78/1.70 cartprod: $i > $i > $i ).
% 2.78/1.70
% 2.78/1.70 thf(singletonsubset_type,type,
% 2.78/1.70 singletonsubset: $o ).
% 2.78/1.70
% 2.78/1.70 thf(singletonsubset,definition,
% 2.78/1.70 ( singletonsubset
% 2.78/1.70 = ( ! [A: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(singletoninpowerset_type,type,
% 2.78/1.70 singletoninpowerset: $o ).
% 2.78/1.70
% 2.78/1.70 thf(singletoninpowerset,definition,
% 2.78/1.70 ( singletoninpowerset
% 2.78/1.70 = ( ! [A: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(singletoninpowunion_type,type,
% 2.78/1.70 singletoninpowunion: $o ).
% 2.78/1.70
% 2.78/1.70 thf(singletoninpowunion,definition,
% 2.78/1.70 ( singletoninpowunion
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(upairset2E_type,type,
% 2.78/1.70 upairset2E: $o ).
% 2.78/1.70
% 2.78/1.70 thf(upairset2E,definition,
% 2.78/1.70 ( upairset2E
% 2.78/1.70 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.78/1.70 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.78/1.70 => ( ( Xz = Xx )
% 2.78/1.70 | ( Xz = Xy ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(upairsubunion_type,type,
% 2.78/1.70 upairsubunion: $o ).
% 2.78/1.70
% 2.78/1.70 thf(upairsubunion,definition,
% 2.78/1.70 ( upairsubunion
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ! [Xy: $i] :
% 2.78/1.70 ( ( in @ Xy @ B )
% 2.78/1.70 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(upairinpowunion_type,type,
% 2.78/1.70 upairinpowunion: $o ).
% 2.78/1.70
% 2.78/1.70 thf(upairinpowunion,definition,
% 2.78/1.70 ( upairinpowunion
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ! [Xy: $i] :
% 2.78/1.70 ( ( in @ Xy @ B )
% 2.78/1.70 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(ubforcartprodlem1_type,type,
% 2.78/1.70 ubforcartprodlem1: $o ).
% 2.78/1.70
% 2.78/1.70 thf(ubforcartprodlem1,definition,
% 2.78/1.70 ( ubforcartprodlem1
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ! [Xy: $i] :
% 2.78/1.70 ( ( in @ Xy @ B )
% 2.78/1.70 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(ubforcartprodlem2_type,type,
% 2.78/1.70 ubforcartprodlem2: $o ).
% 2.78/1.70
% 2.78/1.70 thf(ubforcartprodlem2,definition,
% 2.78/1.70 ( ubforcartprodlem2
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ! [Xy: $i] :
% 2.78/1.70 ( ( in @ Xy @ B )
% 2.78/1.70 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(ubforcartprodlem3_type,type,
% 2.78/1.70 ubforcartprodlem3: $o ).
% 2.78/1.70
% 2.78/1.70 thf(ubforcartprodlem3,definition,
% 2.78/1.70 ( ubforcartprodlem3
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ! [Xy: $i] :
% 2.78/1.70 ( ( in @ Xy @ B )
% 2.78/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(cartprodpairin_type,type,
% 2.78/1.70 cartprodpairin: $o ).
% 2.78/1.70
% 2.78/1.70 thf(cartprodpairin,definition,
% 2.78/1.70 ( cartprodpairin
% 2.78/1.70 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 => ! [Xy: $i] :
% 2.78/1.70 ( ( in @ Xy @ B )
% 2.78/1.70 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(cartprodmempair1_type,type,
% 2.78/1.70 cartprodmempair1: $o ).
% 2.78/1.70
% 2.78/1.70 thf(cartprodmempair1,definition,
% 2.78/1.70 ( cartprodmempair1
% 2.78/1.70 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.78/1.70 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.78/1.70 => ? [Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ A )
% 2.78/1.70 & ? [Xy: $i] :
% 2.78/1.70 ( ( in @ Xy @ B )
% 2.78/1.70 & ( Xu
% 2.78/1.70 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(cartprodmempair_type,type,
% 2.78/1.70 cartprodmempair: $o ).
% 2.78/1.70
% 2.78/1.70 thf(cartprodmempair,definition,
% 2.78/1.70 ( cartprodmempair
% 2.78/1.70 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.78/1.70 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.78/1.70 => ( iskpair @ Xu ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setunionE2_type,type,
% 2.78/1.70 setunionE2: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setunionE2,definition,
% 2.78/1.70 ( setunionE2
% 2.78/1.70 = ( ! [A: $i,Xx: $i] :
% 2.78/1.70 ( ( in @ Xx @ ( setunion @ A ) )
% 2.78/1.70 => ? [X: $i] :
% 2.78/1.70 ( ( in @ X @ A )
% 2.78/1.70 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setunionsingleton1_type,type,
% 2.78/1.70 setunionsingleton1: $o ).
% 2.78/1.70
% 2.78/1.70 thf(setunionsingleton1,definition,
% 2.78/1.70 ( setunionsingleton1
% 2.78/1.70 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.78/1.70
% 2.78/1.70 thf(setunionsingleton2,conjecture,
% 2.78/1.70 ( setextAx
% 2.78/1.70 => ( emptysetAx
% 2.78/1.70 => ( setadjoinAx
% 2.78/1.70 => ( powersetAx
% 2.78/1.70 => ( setunionAx
% 2.78/1.70 => ( omega0Ax
% 2.78/1.70 => ( omegaSAx
% 2.78/1.70 => ( omegaIndAx
% 2.78/1.70 => ( replAx
% 2.78/1.70 => ( foundationAx
% 2.78/1.70 => ( wellorderingAx
% 2.78/1.70 => ( descrp
% 2.78/1.70 => ( dsetconstrI
% 2.78/1.70 => ( dsetconstrEL
% 2.78/1.70 => ( dsetconstrER
% 2.78/1.70 => ( exuE1
% 2.78/1.70 => ( prop2setE
% 2.78/1.70 => ( emptysetE
% 2.78/1.70 => ( emptysetimpfalse
% 2.78/1.70 => ( notinemptyset
% 2.78/1.70 => ( exuE3e
% 2.78/1.70 => ( setext
% 2.78/1.70 => ( emptyI
% 2.78/1.70 => ( noeltsimpempty
% 2.78/1.70 => ( setbeta
% 2.78/1.70 => ( nonemptyE1
% 2.78/1.70 => ( nonemptyI
% 2.78/1.70 => ( nonemptyI1
% 2.78/1.70 => ( setadjoinIL
% 2.78/1.70 => ( emptyinunitempty
% 2.78/1.70 => ( setadjoinIR
% 2.78/1.70 => ( setadjoinE
% 2.78/1.70 => ( setadjoinOr
% 2.78/1.70 => ( setoftrueEq
% 2.78/1.70 => ( powersetI
% 2.78/1.70 => ( emptyinPowerset
% 2.78/1.70 => ( emptyInPowerset
% 2.78/1.70 => ( powersetE
% 2.78/1.70 => ( setunionI
% 2.78/1.70 => ( setunionE
% 2.78/1.70 => ( subPowSU
% 2.78/1.70 => ( exuE2
% 2.78/1.70 => ( nonemptyImpWitness
% 2.78/1.70 => ( uniqinunit
% 2.78/1.70 => ( notinsingleton
% 2.78/1.70 => ( eqinunit
% 2.78/1.70 => ( singletonsswitch
% 2.78/1.70 => ( upairsetE
% 2.78/1.70 => ( upairsetIL
% 2.78/1.70 => ( upairsetIR
% 2.78/1.70 => ( emptyE1
% 2.78/1.70 => ( vacuousDall
% 2.78/1.70 => ( quantDeMorgan1
% 2.78/1.70 => ( quantDeMorgan2
% 2.78/1.70 => ( quantDeMorgan3
% 2.78/1.70 => ( quantDeMorgan4
% 2.78/1.70 => ( prop2setI
% 2.78/1.70 => ( prop2set2propI
% 2.78/1.70 => ( notdexE
% 2.78/1.70 => ( notdallE
% 2.78/1.70 => ( exuI1
% 2.78/1.70 => ( exuI3
% 2.78/1.70 => ( exuI2
% 2.78/1.70 => ( inCongP
% 2.78/1.70 => ( in__Cong
% 2.78/1.70 => ( exuE3u
% 2.78/1.70 => ( exu__Cong
% 2.78/1.70 => ( emptyset__Cong
% 2.78/1.70 => ( setadjoin__Cong
% 2.78/1.70 => ( powerset__Cong
% 2.78/1.70 => ( setunion__Cong
% 2.78/1.70 => ( omega__Cong
% 2.78/1.70 => ( exuEu
% 2.78/1.70 => ( descr__Cong
% 2.78/1.70 => ( dsetconstr__Cong
% 2.78/1.70 => ( subsetI1
% 2.78/1.70 => ( eqimpsubset2
% 2.78/1.70 => ( eqimpsubset1
% 2.78/1.70 => ( subsetI2
% 2.78/1.70 => ( emptysetsubset
% 2.78/1.70 => ( subsetE
% 2.78/1.70 => ( subsetE2
% 2.78/1.70 => ( notsubsetI
% 2.78/1.70 => ( notequalI1
% 2.78/1.70 => ( notequalI2
% 2.78/1.70 => ( subsetRefl
% 2.78/1.70 => ( subsetTrans
% 2.78/1.70 => ( setadjoinSub
% 2.78/1.70 => ( setadjoinSub2
% 2.78/1.70 => ( subset2powerset
% 2.78/1.70 => ( setextsub
% 2.78/1.70 => ( subsetemptysetimpeq
% 2.78/1.70 => ( powersetI1
% 2.78/1.70 => ( powersetE1
% 2.78/1.70 => ( inPowerset
% 2.78/1.70 => ( powersetsubset
% 2.78/1.70 => ( sepInPowerset
% 2.78/1.70 => ( sepSubset
% 2.78/1.70 => ( binunionIL
% 2.78/1.70 => ( upairset2IR
% 2.78/1.70 => ( binunionIR
% 2.78/1.70 => ( binunionEcases
% 2.78/1.70 => ( binunionE
% 2.78/1.70 => ( binunionLsub
% 2.78/1.70 => ( binunionRsub
% 2.78/1.70 => ( binintersectI
% 2.78/1.70 => ( binintersectSubset5
% 2.78/1.70 => ( binintersectEL
% 2.78/1.70 => ( binintersectLsub
% 2.78/1.70 => ( binintersectSubset2
% 2.78/1.70 => ( binintersectSubset3
% 2.78/1.70 => ( binintersectER
% 2.78/1.70 => ( disjointsetsI1
% 2.78/1.70 => ( binintersectRsub
% 2.78/1.70 => ( binintersectSubset4
% 2.78/1.70 => ( binintersectSubset1
% 2.78/1.70 => ( bs114d
% 2.78/1.70 => ( setminusI
% 2.78/1.70 => ( setminusEL
% 2.78/1.70 => ( setminusER
% 2.78/1.70 => ( setminusSubset2
% 2.78/1.70 => ( setminusERneg
% 2.78/1.70 => ( setminusELneg
% 2.78/1.70 => ( setminusILneg
% 2.78/1.70 => ( setminusIRneg
% 2.78/1.70 => ( setminusLsub
% 2.78/1.70 => ( setminusSubset1
% 2.78/1.70 => ( symdiffE
% 2.78/1.70 => ( symdiffI1
% 2.78/1.70 => ( symdiffI2
% 2.78/1.70 => ( symdiffIneg1
% 2.78/1.70 => ( symdiffIneg2
% 2.78/1.70 => ( secondinupair
% 2.78/1.70 => ( setukpairIL
% 2.78/1.70 => ( setukpairIR
% 2.78/1.70 => ( kpairiskpair
% 2.78/1.70 => ( kpairp
% 2.78/1.70 => ( singletonsubset
% 2.78/1.70 => ( singletoninpowerset
% 2.78/1.70 => ( singletoninpowunion
% 2.78/1.70 => ( upairset2E
% 2.78/1.70 => ( upairsubunion
% 2.78/1.70 => ( upairinpowunion
% 2.78/1.70 => ( ubforcartprodlem1
% 2.78/1.70 => ( ubforcartprodlem2
% 2.78/1.70 => ( ubforcartprodlem3
% 2.78/1.70 => ( cartprodpairin
% 2.78/1.70 => ( cartprodmempair1
% 2.78/1.70 => ( cartprodmempair
% 2.78/1.70 => ( setunionE2
% 2.78/1.70 => ( setunionsingleton1
% 2.78/1.70 => ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.78/1.70 ------------------------
% 2.78/1.73 % (21093)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_20989 for (2999ds/275Mi)
% 2.78/1.73 % (21091)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_20989 for (2999ds/2Mi)
% 2.78/1.73 % (21088)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_20989 for (2999ds/183Mi)
% 2.78/1.73 % (21089)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_20989 for (2999ds/4Mi)
% 2.78/1.73 % (21090)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_20989 for (2999ds/27Mi)
% 2.78/1.73 % (21092)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_20989 for (2999ds/2Mi)
% 2.78/1.73 % (21094)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_20989 for (2999ds/18Mi)
% 2.78/1.73 % (21091)Instruction limit reached!
% 2.78/1.73 % (21091)------------------------------
% 2.78/1.73 % (21091)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.73 % (21091)Termination reason: Unknown
% 2.78/1.73 % (21091)Termination phase: shuffling
% 2.78/1.73 % (21092)Instruction limit reached!
% 2.78/1.73 % (21092)------------------------------
% 2.78/1.73 % (21092)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.73 % (21092)Termination reason: Unknown
% 2.78/1.73 % (21092)Termination phase: shuffling
% 2.78/1.73
% 2.78/1.73 % (21092)Memory used [KB]: 1279
% 2.78/1.73 % (21092)Time elapsed: 0.003 s
% 2.78/1.73 % (21092)Instructions burned: 3 (million)
% 2.78/1.73 % (21092)------------------------------
% 2.78/1.73 % (21092)------------------------------
% 2.78/1.73
% 2.78/1.73 % (21091)Memory used [KB]: 1279
% 2.78/1.73 % (21091)Time elapsed: 0.003 s
% 2.78/1.73 % (21091)Instructions burned: 3 (million)
% 2.78/1.73 % (21091)------------------------------
% 2.78/1.73 % (21091)------------------------------
% 2.78/1.73 % (21089)Instruction limit reached!
% 2.78/1.73 % (21089)------------------------------
% 2.78/1.73 % (21089)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.73 % (21089)Termination reason: Unknown
% 2.78/1.73 % (21089)Termination phase: shuffling
% 2.78/1.73
% 2.78/1.73 % (21089)Memory used [KB]: 1279
% 2.78/1.73 % (21089)Time elapsed: 0.004 s
% 2.78/1.73 % (21089)Instructions burned: 5 (million)
% 2.78/1.73 % (21089)------------------------------
% 2.78/1.73 % (21089)------------------------------
% 2.78/1.74 % (21094)Instruction limit reached!
% 2.78/1.74 % (21094)------------------------------
% 2.78/1.74 % (21094)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.74 % (21094)Termination reason: Unknown
% 2.78/1.74 % (21094)Termination phase: Property scanning
% 2.78/1.74
% 2.78/1.74 % (21094)Memory used [KB]: 1535
% 2.78/1.74 % (21094)Time elapsed: 0.011 s
% 2.78/1.74 % (21094)Instructions burned: 19 (million)
% 2.78/1.74 % (21094)------------------------------
% 2.78/1.74 % (21094)------------------------------
% 2.78/1.74 % (21090)Instruction limit reached!
% 2.78/1.74 % (21090)------------------------------
% 2.78/1.74 % (21090)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.74 % (21090)Termination reason: Unknown
% 2.78/1.74 % (21090)Termination phase: Property scanning
% 2.78/1.74
% 2.78/1.74 % (21090)Memory used [KB]: 1791
% 2.78/1.74 % (21090)Time elapsed: 0.015 s
% 2.78/1.74 % (21090)Instructions burned: 29 (million)
% 2.78/1.74 % (21090)------------------------------
% 2.78/1.74 % (21090)------------------------------
% 2.78/1.75 % (21096)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_20989 for (2999ds/37Mi)
% 2.78/1.75 % (21095)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_20989 for (2999ds/3Mi)
% 2.78/1.75 % (21097)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_20989 for (2999ds/15Mi)
% 2.78/1.75 % (21095)Instruction limit reached!
% 2.78/1.75 % (21095)------------------------------
% 2.78/1.75 % (21095)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.75 % (21095)Termination reason: Unknown
% 2.78/1.75 % (21095)Termination phase: shuffling
% 2.78/1.75
% 2.78/1.75 % (21095)Memory used [KB]: 1279
% 2.78/1.75 % (21095)Time elapsed: 0.003 s
% 2.78/1.75 % (21095)Instructions burned: 3 (million)
% 2.78/1.75 % (21095)------------------------------
% 2.78/1.75 % (21095)------------------------------
% 2.78/1.75 % (21097)Instruction limit reached!
% 2.78/1.75 % (21097)------------------------------
% 2.78/1.75 % (21097)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.75 % (21097)Termination reason: Unknown
% 2.78/1.75 % (21097)Termination phase: Property scanning
% 2.78/1.75
% 2.78/1.75 % (21097)Memory used [KB]: 1535
% 2.78/1.75 % (21097)Time elapsed: 0.009 s
% 2.78/1.75 % (21097)Instructions burned: 15 (million)
% 2.78/1.75 % (21097)------------------------------
% 2.78/1.75 % (21097)------------------------------
% 2.78/1.75 % (21098)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_20989 for (2999ds/3Mi)
% 2.78/1.76 % (21098)Instruction limit reached!
% 2.78/1.76 % (21098)------------------------------
% 2.78/1.76 % (21098)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.76 % (21098)Termination reason: Unknown
% 2.78/1.76 % (21098)Termination phase: shuffling
% 2.78/1.76
% 2.78/1.76 % (21098)Memory used [KB]: 1279
% 2.78/1.76 % (21098)Time elapsed: 0.004 s
% 2.78/1.76 % (21098)Instructions burned: 5 (million)
% 2.78/1.76 % (21098)------------------------------
% 2.78/1.76 % (21098)------------------------------
% 2.78/1.76 % (21099)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_20989 for (2999ds/1041Mi)
% 2.78/1.76 % (21096)Instruction limit reached!
% 2.78/1.76 % (21096)------------------------------
% 2.78/1.76 % (21096)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.76 % (21096)Termination reason: Unknown
% 2.78/1.76 % (21096)Termination phase: Property scanning
% 2.78/1.76
% 2.78/1.76 % (21096)Memory used [KB]: 1663
% 2.78/1.76 % (21096)Time elapsed: 0.018 s
% 2.78/1.76 % (21096)Instructions burned: 37 (million)
% 2.78/1.76 % (21096)------------------------------
% 2.78/1.76 % (21096)------------------------------
% 2.78/1.76 % (21100)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_20989 for (2999ds/7Mi)
% 2.78/1.77 % (21100)Instruction limit reached!
% 2.78/1.77 % (21100)------------------------------
% 2.78/1.77 % (21100)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.77 % (21100)Termination reason: Unknown
% 2.78/1.77 % (21100)Termination phase: shuffling
% 2.78/1.77
% 2.78/1.77 % (21100)Memory used [KB]: 1407
% 2.78/1.77 % (21100)Time elapsed: 0.005 s
% 2.78/1.77 % (21100)Instructions burned: 7 (million)
% 2.78/1.77 % (21100)------------------------------
% 2.78/1.77 % (21100)------------------------------
% 2.78/1.77 % (21101)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_20989 for (2999ds/16Mi)
% 2.78/1.77 % (21102)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_20989 for (2999ds/3Mi)
% 2.78/1.77 % (21102)Instruction limit reached!
% 2.78/1.77 % (21102)------------------------------
% 2.78/1.77 % (21102)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.77 % (21102)Termination reason: Unknown
% 2.78/1.77 % (21102)Termination phase: shuffling
% 2.78/1.77
% 2.78/1.77 % (21102)Memory used [KB]: 1279
% 2.78/1.77 % (21102)Time elapsed: 0.004 s
% 2.78/1.77 % (21102)Instructions burned: 5 (million)
% 2.78/1.77 % (21102)------------------------------
% 2.78/1.77 % (21102)------------------------------
% 2.78/1.78 % (21101)Instruction limit reached!
% 2.78/1.78 % (21101)------------------------------
% 2.78/1.78 % (21101)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.78 % (21101)Termination reason: Unknown
% 2.78/1.78 % (21101)Termination phase: shuffling
% 2.78/1.78
% 2.78/1.78 % (21101)Memory used [KB]: 1535
% 2.78/1.78 % (21101)Time elapsed: 0.009 s
% 2.78/1.78 % (21101)Instructions burned: 16 (million)
% 2.78/1.78 % (21101)------------------------------
% 2.78/1.78 % (21101)------------------------------
% 2.78/1.78 % (21103)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_20989 for (2999ds/3Mi)
% 2.78/1.78 % (21103)Instruction limit reached!
% 2.78/1.78 % (21103)------------------------------
% 2.78/1.78 % (21103)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.78 % (21103)Termination reason: Unknown
% 2.78/1.78 % (21103)Termination phase: shuffling
% 2.78/1.78
% 2.78/1.78 % (21103)Memory used [KB]: 1279
% 2.78/1.78 % (21103)Time elapsed: 0.003 s
% 2.78/1.78 % (21103)Instructions burned: 3 (million)
% 2.78/1.78 % (21103)------------------------------
% 2.78/1.78 % (21103)------------------------------
% 2.78/1.78 % (21104)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_20989 for (2999ds/7Mi)
% 2.78/1.78 % (21104)Instruction limit reached!
% 2.78/1.78 % (21104)------------------------------
% 2.78/1.78 % (21104)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.78 % (21104)Termination reason: Unknown
% 2.78/1.78 % (21104)Termination phase: shuffling
% 2.78/1.78
% 2.78/1.78 % (21104)Memory used [KB]: 1407
% 2.78/1.78 % (21104)Time elapsed: 0.005 s
% 2.78/1.78 % (21104)Instructions burned: 8 (million)
% 2.78/1.78 % (21104)------------------------------
% 2.78/1.78 % (21104)------------------------------
% 2.78/1.79 % (21105)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_20989 for (2999ds/3Mi)
% 2.78/1.79 % (21105)Instruction limit reached!
% 2.78/1.79 % (21105)------------------------------
% 2.78/1.79 % (21105)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.79 % (21105)Termination reason: Unknown
% 2.78/1.79 % (21105)Termination phase: shuffling
% 2.78/1.79
% 2.78/1.79 % (21105)Memory used [KB]: 1279
% 2.78/1.79 % (21105)Time elapsed: 0.004 s
% 2.78/1.79 % (21105)Instructions burned: 5 (million)
% 2.78/1.79 % (21105)------------------------------
% 2.78/1.79 % (21105)------------------------------
% 2.78/1.79 % (21106)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_20989 for (2999ds/4Mi)
% 2.78/1.80 % (21106)Instruction limit reached!
% 2.78/1.80 % (21106)------------------------------
% 2.78/1.80 % (21106)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.80 % (21106)Termination reason: Unknown
% 2.78/1.80 % (21106)Termination phase: shuffling
% 2.78/1.80
% 2.78/1.80 % (21106)Memory used [KB]: 1279
% 2.78/1.80 % (21106)Time elapsed: 0.004 s
% 2.78/1.80 % (21106)Instructions burned: 5 (million)
% 2.78/1.80 % (21106)------------------------------
% 2.78/1.80 % (21106)------------------------------
% 2.78/1.80 % (21107)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_20989 for (2999ds/18Mi)
% 2.81/1.80 % (21108)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_20989 for (2999ds/710Mi)
% 2.81/1.80 % (21107)Instruction limit reached!
% 2.81/1.80 % (21107)------------------------------
% 2.81/1.80 % (21107)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.80 % (21107)Termination reason: Unknown
% 2.81/1.80 % (21107)Termination phase: shuffling
% 2.81/1.80
% 2.81/1.80 % (21107)Memory used [KB]: 1535
% 2.81/1.80 % (21107)Time elapsed: 0.010 s
% 2.81/1.80 % (21107)Instructions burned: 18 (million)
% 2.81/1.80 % (21107)------------------------------
% 2.81/1.80 % (21107)------------------------------
% 2.81/1.81 % (21109)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_20989 for (2999ds/6Mi)
% 2.81/1.81 % (21088)Instruction limit reached!
% 2.81/1.81 % (21088)------------------------------
% 2.81/1.81 % (21088)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.81 % (21088)Termination reason: Unknown
% 2.81/1.81 % (21088)Termination phase: Saturation
% 2.81/1.81
% 2.81/1.81 % (21088)Memory used [KB]: 7419
% 2.81/1.81 % (21088)Time elapsed: 0.084 s
% 2.81/1.81 % (21088)Instructions burned: 184 (million)
% 2.81/1.81 % (21088)------------------------------
% 2.81/1.81 % (21088)------------------------------
% 2.81/1.81 % (21109)Instruction limit reached!
% 2.81/1.81 % (21109)------------------------------
% 2.81/1.81 % (21109)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.81 % (21109)Termination reason: Unknown
% 2.81/1.81 % (21109)Termination phase: shuffling
% 2.81/1.81
% 2.81/1.81 % (21109)Memory used [KB]: 1407
% 2.81/1.81 % (21109)Time elapsed: 0.005 s
% 2.81/1.81 % (21109)Instructions burned: 7 (million)
% 2.81/1.81 % (21109)------------------------------
% 2.81/1.81 % (21109)------------------------------
% 2.81/1.81 % (21110)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_20989 for (2999ds/902Mi)
% 2.81/1.82 % (21111)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_20989 for (2999ds/21Mi)
% 2.81/1.83 % (21112)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_20989 for (2999ds/5Mi)
% 2.81/1.83 % (21113)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_20989 for (2999ds/6Mi)
% 2.81/1.83 % (21112)Instruction limit reached!
% 2.81/1.83 % (21112)------------------------------
% 2.81/1.83 % (21112)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.83 % (21112)Termination reason: Unknown
% 2.81/1.83 % (21112)Termination phase: shuffling
% 2.81/1.83
% 2.81/1.83 % (21112)Memory used [KB]: 1279
% 2.81/1.83 % (21112)Time elapsed: 0.004 s
% 2.81/1.83 % (21112)Instructions burned: 6 (million)
% 2.81/1.83 % (21112)------------------------------
% 2.81/1.83 % (21112)------------------------------
% 2.81/1.83 % (21113)Instruction limit reached!
% 2.81/1.83 % (21113)------------------------------
% 2.81/1.83 % (21113)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.83 % (21113)Termination reason: Unknown
% 2.81/1.83 % (21113)Termination phase: shuffling
% 2.81/1.83
% 2.81/1.83 % (21113)Memory used [KB]: 1407
% 2.81/1.83 % (21113)Time elapsed: 0.005 s
% 2.81/1.83 % (21113)Instructions burned: 8 (million)
% 2.81/1.83 % (21113)------------------------------
% 2.81/1.83 % (21113)------------------------------
% 2.81/1.83 % (21111)Instruction limit reached!
% 2.81/1.83 % (21111)------------------------------
% 2.81/1.83 % (21111)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.83 % (21111)Termination reason: Unknown
% 2.81/1.83 % (21111)Termination phase: shuffling
% 2.81/1.83
% 2.81/1.83 % (21111)Memory used [KB]: 1663
% 2.81/1.83 % (21111)Time elapsed: 0.012 s
% 2.81/1.83 % (21111)Instructions burned: 21 (million)
% 2.81/1.83 % (21111)------------------------------
% 2.81/1.83 % (21111)------------------------------
% 2.81/1.84 % (21115)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_20989 for (2998ds/779Mi)
% 2.81/1.85 % (21114)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_20989 for (2998ds/377Mi)
% 2.81/1.85 % (21116)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_20989 for (2998ds/19Mi)
% 2.81/1.86 % (21116)Instruction limit reached!
% 2.81/1.86 % (21116)------------------------------
% 2.81/1.86 % (21116)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.86 % (21116)Termination reason: Unknown
% 2.81/1.86 % (21116)Termination phase: shuffling
% 2.81/1.86
% 2.81/1.86 % (21116)Memory used [KB]: 1663
% 2.81/1.86 % (21116)Time elapsed: 0.010 s
% 2.81/1.86 % (21116)Instructions burned: 19 (million)
% 2.81/1.86 % (21116)------------------------------
% 2.81/1.86 % (21116)------------------------------
% 2.81/1.86 % (21093)Instruction limit reached!
% 2.81/1.86 % (21093)------------------------------
% 2.81/1.86 % (21093)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.86 % (21093)Termination reason: Unknown
% 2.81/1.86 % (21093)Termination phase: Saturation
% 2.81/1.86
% 2.81/1.86 % (21093)Memory used [KB]: 8059
% 2.81/1.86 % (21093)Time elapsed: 0.137 s
% 2.81/1.86 % (21093)Instructions burned: 276 (million)
% 2.81/1.86 % (21093)------------------------------
% 2.81/1.86 % (21093)------------------------------
% 2.81/1.87 % (21117)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_20989 for (2998ds/879Mi)
% 2.81/1.88 % (21118)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_20989 for (2998ds/17Mi)
% 2.81/1.89 % (21118)Instruction limit reached!
% 2.81/1.89 % (21118)------------------------------
% 2.81/1.89 % (21118)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.89 % (21118)Termination reason: Unknown
% 2.81/1.89 % (21118)Termination phase: shuffling
% 2.81/1.89
% 2.81/1.89 % (21118)Memory used [KB]: 1535
% 2.81/1.89 % (21118)Time elapsed: 0.010 s
% 2.81/1.89 % (21118)Instructions burned: 17 (million)
% 2.81/1.89 % (21118)------------------------------
% 2.81/1.89 % (21118)------------------------------
% 2.81/1.91 % (21119)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_20989 for (2998ds/3Mi)
% 2.81/1.91 % (21119)Instruction limit reached!
% 2.81/1.91 % (21119)------------------------------
% 2.81/1.91 % (21119)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.91 % (21119)Termination reason: Unknown
% 2.81/1.91 % (21119)Termination phase: shuffling
% 2.81/1.91
% 2.81/1.91 % (21119)Memory used [KB]: 1279
% 2.81/1.91 % (21119)Time elapsed: 0.004 s
% 2.81/1.91 % (21119)Instructions burned: 4 (million)
% 2.81/1.91 % (21119)------------------------------
% 2.81/1.91 % (21119)------------------------------
% 2.81/1.92 % (21120)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_20989 for (2997ds/30Mi)
% 2.81/1.94 % (21120)Instruction limit reached!
% 2.81/1.94 % (21120)------------------------------
% 2.81/1.94 % (21120)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/1.94 % (21120)Termination reason: Unknown
% 2.81/1.94 % (21120)Termination phase: Property scanning
% 2.81/1.94
% 2.81/1.94 % (21120)Memory used [KB]: 1663
% 2.81/1.94 % (21120)Time elapsed: 0.015 s
% 2.81/1.94 % (21120)Instructions burned: 30 (million)
% 2.81/1.94 % (21120)------------------------------
% 2.81/1.94 % (21120)------------------------------
% 2.81/1.95 % (21121)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_20989 for (2997ds/127Mi)
% 2.81/2.02 % (21121)Instruction limit reached!
% 2.81/2.02 % (21121)------------------------------
% 2.81/2.02 % (21121)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.81/2.02 % (21121)Termination reason: Unknown
% 2.81/2.02 % (21121)Termination phase: Saturation
% 2.81/2.02
% 2.81/2.02 % (21121)Memory used [KB]: 6780
% 2.81/2.02 % (21121)Time elapsed: 0.058 s
% 2.81/2.02 % (21121)Instructions burned: 127 (million)
% 2.81/2.02 % (21121)------------------------------
% 2.81/2.02 % (21121)------------------------------
% 4.91/2.03 % (21114)Instruction limit reached!
% 4.91/2.03 % (21114)------------------------------
% 4.91/2.03 % (21114)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.91/2.03 % (21114)Termination reason: Unknown
% 4.91/2.03 % (21114)Termination phase: Saturation
% 4.91/2.03
% 4.91/2.03 % (21114)Memory used [KB]: 7675
% 4.91/2.03 % (21114)Time elapsed: 0.209 s
% 4.91/2.03 % (21114)Instructions burned: 377 (million)
% 4.91/2.03 % (21114)------------------------------
% 4.91/2.03 % (21114)------------------------------
% 4.91/2.04 % (21122)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_20989 for (2996ds/100Mi)
% 4.91/2.05 % (21123)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_20989 for (2996ds/3Mi)
% 4.91/2.05 % (21123)Instruction limit reached!
% 4.91/2.05 % (21123)------------------------------
% 4.91/2.05 % (21123)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.91/2.05 % (21123)Termination reason: Unknown
% 4.91/2.05 % (21123)Termination phase: shuffling
% 4.91/2.05
% 4.91/2.05 % (21123)Memory used [KB]: 1279
% 4.91/2.05 % (21123)Time elapsed: 0.004 s
% 4.91/2.05 % (21123)Instructions burned: 5 (million)
% 4.91/2.05 % (21123)------------------------------
% 4.91/2.05 % (21123)------------------------------
% 4.91/2.07 % (21124)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_20989 for (2996ds/20Mi)
% 4.91/2.08 % (21124)Instruction limit reached!
% 4.91/2.08 % (21124)------------------------------
% 4.91/2.08 % (21124)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.91/2.08 % (21124)Termination reason: Unknown
% 4.91/2.08 % (21124)Termination phase: SInE selection
% 4.91/2.08
% 4.91/2.08 % (21124)Memory used [KB]: 1535
% 4.91/2.08 % (21124)Time elapsed: 0.012 s
% 4.91/2.08 % (21124)Instructions burned: 21 (million)
% 4.91/2.08 % (21124)------------------------------
% 4.91/2.08 % (21124)------------------------------
% 4.91/2.08 % (21122)Instruction limit reached!
% 4.91/2.08 % (21122)------------------------------
% 4.91/2.08 % (21122)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.91/2.08 % (21122)Termination reason: Unknown
% 4.91/2.08 % (21122)Termination phase: Saturation
% 4.91/2.08
% 4.91/2.08 % (21122)Memory used [KB]: 6396
% 4.91/2.08 % (21122)Time elapsed: 0.047 s
% 4.91/2.08 % (21122)Instructions burned: 101 (million)
% 4.91/2.08 % (21122)------------------------------
% 4.91/2.08 % (21122)------------------------------
% 4.91/2.09 % (21125)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_20989 for (2996ds/86Mi)
% 4.91/2.10 % (21126)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_20989 for (2996ds/107Mi)
% 5.58/2.13 % (21115)Instruction limit reached!
% 5.58/2.13 % (21115)------------------------------
% 5.58/2.13 % (21115)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.58/2.13 % (21115)Termination reason: Unknown
% 5.58/2.13 % (21115)Termination phase: Saturation
% 5.58/2.13
% 5.58/2.13 % (21115)Memory used [KB]: 6652
% 5.58/2.13 % (21115)Time elapsed: 0.285 s
% 5.58/2.13 % (21115)Instructions burned: 780 (million)
% 5.58/2.13 % (21115)------------------------------
% 5.58/2.13 % (21115)------------------------------
% 5.58/2.13 % (21125)Instruction limit reached!
% 5.58/2.13 % (21125)------------------------------
% 5.58/2.13 % (21125)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.58/2.13 % (21125)Termination reason: Unknown
% 5.58/2.13 % (21125)Termination phase: Saturation
% 5.58/2.13
% 5.58/2.13 % (21125)Memory used [KB]: 6524
% 5.58/2.13 % (21125)Time elapsed: 0.041 s
% 5.58/2.13 % (21125)Instructions burned: 88 (million)
% 5.58/2.13 % (21125)------------------------------
% 5.58/2.13 % (21125)------------------------------
% 5.58/2.14 % (21099)Instruction limit reached!
% 5.58/2.14 % (21099)------------------------------
% 5.58/2.14 % (21099)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.58/2.14 % (21099)Termination reason: Unknown
% 5.58/2.14 % (21099)Termination phase: Saturation
% 5.58/2.14
% 5.58/2.14 % (21099)Memory used [KB]: 9466
% 5.58/2.14 % (21099)Time elapsed: 0.384 s
% 5.58/2.14 % (21099)Instructions burned: 1041 (million)
% 5.58/2.14 % (21099)------------------------------
% 5.58/2.14 % (21099)------------------------------
% 5.58/2.14 % (21127)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_20989 for (2995ds/39Mi)
% 5.58/2.15 % (21126)Instruction limit reached!
% 5.58/2.15 % (21126)------------------------------
% 5.58/2.15 % (21126)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.58/2.15 % (21126)Termination reason: Unknown
% 5.58/2.15 % (21126)Termination phase: Saturation
% 5.58/2.15
% 5.58/2.15 % (21126)Memory used [KB]: 6780
% 5.58/2.15 % (21126)Time elapsed: 0.049 s
% 5.58/2.15 % (21126)Instructions burned: 107 (million)
% 5.58/2.15 % (21126)------------------------------
% 5.58/2.15 % (21126)------------------------------
% 5.58/2.15 % (21128)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_20989 for (2995ds/448Mi)
% 5.58/2.16 % (21129)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_20989 for (2995ds/46Mi)
% 5.58/2.16 % (21108)Instruction limit reached!
% 5.58/2.16 % (21108)------------------------------
% 5.58/2.16 % (21108)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.58/2.16 % (21108)Termination reason: Unknown
% 5.58/2.16 % (21108)Termination phase: Saturation
% 5.58/2.16
% 5.58/2.16 % (21108)Memory used [KB]: 9466
% 5.58/2.16 % (21108)Time elapsed: 0.363 s
% 5.58/2.16 % (21108)Instructions burned: 711 (million)
% 5.58/2.16 % (21108)------------------------------
% 5.58/2.16 % (21108)------------------------------
% 5.58/2.16 % (21127)Instruction limit reached!
% 5.58/2.16 % (21127)------------------------------
% 5.58/2.16 % (21127)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.58/2.16 % (21127)Termination reason: Unknown
% 5.58/2.16 % (21127)Termination phase: Property scanning
% 5.58/2.16
% 5.58/2.16 % (21127)Memory used [KB]: 1791
% 5.58/2.16 % (21127)Time elapsed: 0.019 s
% 5.58/2.16 % (21127)Instructions burned: 40 (million)
% 5.58/2.16 % (21127)------------------------------
% 5.58/2.16 % (21127)------------------------------
% 5.58/2.16 % (21130)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_20989 for (2995ds/98Mi)
% 5.58/2.18 % (21129)Instruction limit reached!
% 5.58/2.18 % (21129)------------------------------
% 5.58/2.18 % (21129)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.58/2.18 % (21129)Termination reason: Unknown
% 5.58/2.18 % (21129)Termination phase: Preprocessing 3
% 5.58/2.18
% 5.58/2.18 % (21129)Memory used [KB]: 1791
% 5.58/2.18 % (21129)Time elapsed: 0.022 s
% 5.58/2.18 % (21129)Instructions burned: 47 (million)
% 5.58/2.18 % (21129)------------------------------
% 5.58/2.18 % (21129)------------------------------
% 5.58/2.18 % (21132)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_20989 for (2995ds/149Mi)
% 5.58/2.18 % (21131)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_20989 for (2995ds/507Mi)
% 5.58/2.19 % (21133)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_20989 for (2995ds/39Mi)
% 5.96/2.20 % (21130)Instruction limit reached!
% 5.96/2.20 % (21130)------------------------------
% 5.96/2.20 % (21130)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.96/2.20 % (21130)Termination reason: Unknown
% 5.96/2.20 % (21130)Termination phase: Saturation
% 5.96/2.20
% 5.96/2.20 % (21130)Memory used [KB]: 2046
% 5.96/2.20 % (21130)Time elapsed: 0.044 s
% 5.96/2.20 % (21130)Instructions burned: 98 (million)
% 5.96/2.20 % (21130)------------------------------
% 5.96/2.20 % (21130)------------------------------
% 5.96/2.21 % (21133)Instruction limit reached!
% 5.96/2.21 % (21133)------------------------------
% 5.96/2.21 % (21133)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.96/2.21 % (21133)Termination reason: Unknown
% 5.96/2.21 % (21133)Termination phase: Property scanning
% 5.96/2.21
% 5.96/2.21 % (21133)Memory used [KB]: 1918
% 5.96/2.21 % (21133)Time elapsed: 0.019 s
% 5.96/2.21 % (21133)Instructions burned: 39 (million)
% 5.96/2.21 % (21133)------------------------------
% 5.96/2.21 % (21133)------------------------------
% 5.96/2.22 % (21134)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_20989 for (2994ds/6Mi)
% 5.96/2.22 % (21134)Instruction limit reached!
% 5.96/2.22 % (21134)------------------------------
% 5.96/2.22 % (21134)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.96/2.22 % (21134)Termination reason: Unknown
% 5.96/2.22 % (21134)Termination phase: shuffling
% 5.96/2.22
% 5.96/2.22 % (21134)Memory used [KB]: 1407
% 5.96/2.22 % (21134)Time elapsed: 0.005 s
% 5.96/2.22 % (21134)Instructions burned: 6 (million)
% 5.96/2.22 % (21134)------------------------------
% 5.96/2.22 % (21134)------------------------------
% 5.96/2.23 % (21135)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_20989 for (2994ds/1087Mi)
% 5.96/2.24 % (21136)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_20989 for (2994ds/47Mi)
% 5.96/2.25 % (21132)Instruction limit reached!
% 5.96/2.25 % (21132)------------------------------
% 5.96/2.25 % (21132)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.96/2.25 % (21132)Termination reason: Unknown
% 5.96/2.25 % (21132)Termination phase: Saturation
% 5.96/2.25
% 5.96/2.25 % (21132)Memory used [KB]: 7164
% 5.96/2.25 % (21132)Time elapsed: 0.072 s
% 5.96/2.25 % (21132)Instructions burned: 149 (million)
% 5.96/2.25 % (21132)------------------------------
% 5.96/2.25 % (21132)------------------------------
% 5.96/2.26 % (21110)Instruction limit reached!
% 5.96/2.26 % (21110)------------------------------
% 5.96/2.26 % (21110)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.96/2.26 % (21110)Termination reason: Unknown
% 5.96/2.26 % (21110)Termination phase: Saturation
% 5.96/2.26
% 5.96/2.26 % (21110)Memory used [KB]: 10362
% 5.96/2.26 % (21110)Time elapsed: 0.451 s
% 5.96/2.26 % (21110)Instructions burned: 902 (million)
% 5.96/2.26 % (21110)------------------------------
% 5.96/2.26 % (21110)------------------------------
% 5.96/2.26 % (21136)Instruction limit reached!
% 5.96/2.26 % (21136)------------------------------
% 5.96/2.26 % (21136)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.96/2.26 % (21136)Termination reason: Unknown
% 5.96/2.26 % (21136)Termination phase: Property scanning
% 5.96/2.26
% 5.96/2.26 % (21136)Memory used [KB]: 1791
% 5.96/2.26 % (21136)Time elapsed: 0.024 s
% 5.96/2.26 % (21136)Instructions burned: 49 (million)
% 5.96/2.26 % (21136)------------------------------
% 5.96/2.26 % (21136)------------------------------
% 5.96/2.26 % (21137)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_20989 for (2994ds/44Mi)
% 5.96/2.28 % (21138)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_20989 for (2994ds/432Mi)
% 5.96/2.28 % (21139)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_20989 for (2994ds/62Mi)
% 5.96/2.28 % (21137)Instruction limit reached!
% 5.96/2.28 % (21137)------------------------------
% 5.96/2.28 % (21137)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.96/2.28 % (21137)Termination reason: Unknown
% 5.96/2.28 % (21137)Termination phase: SInE selection
% 5.96/2.28
% 5.96/2.28 % (21137)Memory used [KB]: 1791
% 5.96/2.28 % (21137)Time elapsed: 0.022 s
% 5.96/2.28 % (21137)Instructions burned: 46 (million)
% 5.96/2.28 % (21137)------------------------------
% 5.96/2.28 % (21137)------------------------------
% 6.41/2.30 % (21140)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_20989 for (2994ds/31Mi)
% 6.41/2.31 % (21139)Instruction limit reached!
% 6.41/2.31 % (21139)------------------------------
% 6.41/2.31 % (21139)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.41/2.31 % (21139)Termination reason: Unknown
% 6.41/2.31 % (21139)Termination phase: Property scanning
% 6.41/2.31
% 6.41/2.31 % (21139)Memory used [KB]: 1791
% 6.41/2.31 % (21139)Time elapsed: 0.029 s
% 6.41/2.31 % (21139)Instructions burned: 64 (million)
% 6.41/2.31 % (21139)------------------------------
% 6.41/2.31 % (21139)------------------------------
% 6.41/2.31 % (21117)Instruction limit reached!
% 6.41/2.31 % (21117)------------------------------
% 6.41/2.31 % (21117)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.41/2.31 % (21117)Termination reason: Unknown
% 6.41/2.31 % (21117)Termination phase: Saturation
% 6.41/2.31
% 6.41/2.31 % (21117)Memory used [KB]: 8187
% 6.41/2.31 % (21117)Time elapsed: 0.460 s
% 6.41/2.31 % (21117)Instructions burned: 880 (million)
% 6.41/2.31 % (21117)------------------------------
% 6.41/2.31 % (21117)------------------------------
% 6.41/2.32 % (21140)Instruction limit reached!
% 6.41/2.32 % (21140)------------------------------
% 6.41/2.32 % (21140)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.41/2.32 % (21140)Termination reason: Unknown
% 6.41/2.32 % (21140)Termination phase: Property scanning
% 6.41/2.32
% 6.41/2.32 % (21140)Memory used [KB]: 1791
% 6.41/2.32 % (21140)Time elapsed: 0.016 s
% 6.41/2.32 % (21140)Instructions burned: 32 (million)
% 6.41/2.32 % (21140)------------------------------
% 6.41/2.32 % (21140)------------------------------
% 6.41/2.32 % (21128)Instruction limit reached!
% 6.41/2.32 % (21128)------------------------------
% 6.41/2.32 % (21128)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.41/2.32 % (21128)Termination reason: Unknown
% 6.41/2.32 % (21128)Termination phase: Saturation
% 6.41/2.32
% 6.41/2.32 % (21128)Memory used [KB]: 6524
% 6.41/2.32 % (21128)Time elapsed: 0.170 s
% 6.41/2.32 % (21128)Instructions burned: 448 (million)
% 6.41/2.32 % (21128)------------------------------
% 6.41/2.32 % (21128)------------------------------
% 6.41/2.32 % (21141)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_20989 for (2994ds/96Mi)
% 6.41/2.32 % (21142)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_20989 for (2993ds/453Mi)
% 6.41/2.33 % (21143)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_20989 for (2993ds/71Mi)
% 6.41/2.33 % (21144)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_20989 for (2993ds/83Mi)
% 6.41/2.36 % (21143)Instruction limit reached!
% 6.41/2.36 % (21143)------------------------------
% 6.41/2.36 % (21143)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.41/2.36 % (21143)Termination reason: Unknown
% 6.41/2.36 % (21143)Termination phase: Saturation
% 6.41/2.36
% 6.41/2.36 % (21143)Memory used [KB]: 6396
% 6.41/2.36 % (21143)Time elapsed: 0.033 s
% 6.41/2.36 % (21143)Instructions burned: 72 (million)
% 6.41/2.36 % (21143)------------------------------
% 6.41/2.36 % (21143)------------------------------
% 6.41/2.36 % (21135)First to succeed.
% 6.41/2.37 % (21141)Instruction limit reached!
% 6.41/2.37 % (21141)------------------------------
% 6.41/2.37 % (21141)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.41/2.37 % (21141)Termination reason: Unknown
% 6.41/2.37 % (21141)Termination phase: Saturation
% 6.41/2.37
% 6.41/2.37 % (21141)Memory used [KB]: 6780
% 6.41/2.37 % (21141)Time elapsed: 0.045 s
% 6.41/2.37 % (21141)Instructions burned: 96 (million)
% 6.41/2.37 % (21141)------------------------------
% 6.41/2.37 % (21141)------------------------------
% 6.41/2.37 % (21144)Instruction limit reached!
% 6.41/2.37 % (21144)------------------------------
% 6.41/2.37 % (21144)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.41/2.37 % (21144)Termination reason: Unknown
% 6.41/2.37 % (21144)Termination phase: Saturation
% 6.41/2.37
% 6.41/2.37 % (21144)Memory used [KB]: 6396
% 6.41/2.37 % (21144)Time elapsed: 0.037 s
% 6.41/2.37 % (21144)Instructions burned: 84 (million)
% 6.41/2.37 % (21144)------------------------------
% 6.41/2.37 % (21144)------------------------------
% 6.41/2.37 % (21135)Refutation found. Thanks to Tanya!
% 6.41/2.37 % SZS status Theorem for DTF2THF_20989
% 6.41/2.37 % SZS output start Proof for DTF2THF_20989
% 6.41/2.37 thf(func_def_0, type, in: $i > $i > $o).
% 6.41/2.37 thf(func_def_1, type, exu: ($i > $o) > $o).
% 6.41/2.37 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 6.41/2.37 thf(func_def_8, type, powerset: $i > $i).
% 6.41/2.37 thf(func_def_10, type, setunion: $i > $i).
% 6.41/2.37 thf(func_def_19, type, descr: ($i > $o) > $i).
% 6.41/2.37 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 6.41/2.37 thf(func_def_26, type, prop2set: $o > $i).
% 6.41/2.37 thf(func_def_36, type, nonempty: $i > $o).
% 6.41/2.37 thf(func_def_69, type, set2prop: $i > $o).
% 6.41/2.37 thf(func_def_88, type, subset: $i > $i > $o).
% 6.41/2.37 thf(func_def_89, type, disjoint: $i > $i > $o).
% 6.41/2.37 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 6.41/2.37 thf(func_def_114, type, binunion: $i > $i > $i).
% 6.41/2.37 thf(func_def_122, type, binintersect: $i > $i > $i).
% 6.41/2.37 thf(func_def_135, type, regular: $i > $o).
% 6.41/2.37 thf(func_def_136, type, setminus: $i > $i > $i).
% 6.41/2.37 thf(func_def_147, type, symdiff: $i > $i > $i).
% 6.41/2.37 thf(func_def_153, type, iskpair: $i > $o).
% 6.41/2.37 thf(func_def_158, type, kpair: $i > $i > $i).
% 6.41/2.37 thf(func_def_160, type, cartprod: $i > $i > $i).
% 6.41/2.37 thf(func_def_187, type, sP0: ($i > $o) > $o).
% 6.41/2.37 thf(func_def_188, type, sP1: $i > $i > $o > $o).
% 6.41/2.37 thf(func_def_190, type, sP3: $i > ($i > $o) > $o).
% 6.41/2.37 thf(func_def_192, type, sP5: ($i > $o) > $o).
% 6.41/2.37 thf(func_def_194, type, sP7: $i > $o > $i > $i > $o).
% 6.41/2.37 thf(func_def_195, type, sP8: $i > $o > $i > $i > $o).
% 6.41/2.37 thf(func_def_196, type, sP9: $i > $o).
% 6.41/2.37 thf(func_def_197, type, sP10: $i > $o).
% 6.41/2.37 thf(func_def_198, type, sP11: ($i > $o) > $i > $o).
% 6.41/2.37 thf(func_def_201, type, sP14: $i > $o > $i > $o).
% 6.41/2.37 thf(func_def_203, type, sP16: $i > $i > $i > $o).
% 6.41/2.37 thf(func_def_204, type, sP17: ($i > $o) > $o).
% 6.41/2.37 thf(func_def_205, type, sP18: $i > $o).
% 6.41/2.37 thf(func_def_207, type, sP20: $i > $i > $o).
% 6.41/2.37 thf(func_def_209, type, sP22: $i > $i > $o > $o).
% 6.41/2.37 thf(func_def_210, type, sP23: $i > $i > $i > $o).
% 6.41/2.37 thf(func_def_211, type, sP24: $i > ($i > $o) > ($i > $o) > $i > $o).
% 6.41/2.37 thf(func_def_212, type, sP25: ($i > $o) > $o).
% 6.41/2.37 thf(func_def_214, type, sP27: $i > $i > $o).
% 6.41/2.37 thf(func_def_216, type, sP29: ($i > $o) > $i > $o).
% 6.41/2.37 thf(func_def_218, type, sP31: $i > $i > $i > $o).
% 6.41/2.37 thf(func_def_220, type, sP33: ($i > $i > $o) > $i > $i > $o).
% 6.41/2.37 thf(func_def_221, type, sP34: ($i > $i > $o) > $i > $o).
% 6.41/2.37 thf(func_def_223, type, sP36: $i > ($i > $o) > $o).
% 6.41/2.37 thf(func_def_225, type, sP38: $i > $i > $i > $o).
% 6.41/2.37 thf(func_def_226, type, sP39: $i > $i > $o).
% 6.41/2.37 thf(func_def_227, type, sP40: $i > $o).
% 6.41/2.37 thf(func_def_228, type, sP41: $i > $i > $o).
% 6.41/2.37 thf(func_def_229, type, sP42: $i > $i > $o).
% 6.41/2.37 thf(func_def_230, type, sP43: $i > $i > $o).
% 6.41/2.37 thf(func_def_231, type, sP44: ($i > $o) > $i > $o).
% 6.41/2.37 thf(func_def_235, type, sP48: $i > ($i > $o) > $o).
% 6.41/2.37 thf(func_def_237, type, sP50: $i > $i > $o).
% 6.41/2.37 thf(func_def_239, type, sP52: ($i > $o) > $o).
% 6.41/2.37 thf(func_def_243, type, sK56: ($i > $o) > $i).
% 6.41/2.37 thf(func_def_244, type, sK57: ($i > $o) > $i).
% 6.41/2.37 thf(func_def_245, type, sK58: $i > $o).
% 6.41/2.37 thf(func_def_261, type, sK74: $i > $o).
% 6.41/2.37 thf(func_def_263, type, sK76: $i > ($i > $o) > $i).
% 6.41/2.37 thf(func_def_264, type, sK77: $i > ($i > $o) > $i).
% 6.41/2.37 thf(func_def_265, type, sK78: ($i > $o) > $i).
% 6.41/2.37 thf(func_def_266, type, sK79: $i > ($i > $o) > $i).
% 6.41/2.37 thf(func_def_267, type, sK80: $i > $o).
% 6.41/2.37 thf(func_def_272, type, sK85: $i > $i > $i).
% 6.41/2.37 thf(func_def_296, type, sK109: $i > $o).
% 6.41/2.37 thf(func_def_299, type, sK112: $i > $i).
% 6.41/2.37 thf(func_def_300, type, sK113: $i > $i).
% 6.41/2.37 thf(func_def_302, type, sK115: $i > $o).
% 6.41/2.37 thf(func_def_304, type, sK117: $i > ($i > $o) > $i).
% 6.41/2.37 thf(func_def_305, type, sK118: ($i > $o) > $i > $i).
% 6.41/2.37 thf(func_def_309, type, sK122: $i > $o).
% 6.41/2.37 thf(func_def_314, type, sK127: $i > $i > $i).
% 6.41/2.37 thf(func_def_318, type, sK131: $i > $o > $i > $i).
% 6.41/2.37 thf(func_def_329, type, sK142: $i > $i > $i).
% 6.41/2.37 thf(func_def_356, type, sK169: $i > $i > $i).
% 6.41/2.37 thf(func_def_357, type, sK170: $i > $i > $i > $i).
% 6.41/2.37 thf(func_def_358, type, sK171: $i > $i > $i > $i).
% 6.41/2.37 thf(func_def_362, type, sK175: $i > ($i > $o) > $i).
% 6.41/2.37 thf(func_def_363, type, sK176: ($i > $o) > $i).
% 6.41/2.37 thf(func_def_364, type, sK177: $i > $o).
% 6.41/2.37 thf(func_def_365, type, sK178: $i > $i > $i).
% 6.41/2.37 thf(func_def_366, type, sK179: $i > $i).
% 6.41/2.37 thf(func_def_371, type, sK184: $i > $o).
% 6.41/2.37 thf(func_def_379, type, sK192: $i > $o).
% 6.41/2.37 thf(func_def_403, type, sK216: $i > $i > $i).
% 6.41/2.37 thf(func_def_406, type, sK219: $i > $i > $i).
% 6.41/2.37 thf(func_def_407, type, sK220: $i > $i > $i).
% 6.41/2.37 thf(func_def_408, type, sK221: $i > $i > $i > $o).
% 6.41/2.37 thf(func_def_419, type, sK232: $i > $o).
% 6.41/2.37 thf(func_def_426, type, sK239: $i > ($i > $o) > ($i > $o) > $i > $i).
% 6.41/2.37 thf(func_def_427, type, sK240: $i > ($i > $o) > ($i > $o) > $i > $i).
% 6.41/2.37 thf(func_def_430, type, sK243: $i > $o).
% 6.41/2.37 thf(func_def_431, type, sK244: $i > $o).
% 6.41/2.37 thf(func_def_432, type, sK245: $i > ($i > $o) > $i).
% 6.41/2.37 thf(func_def_433, type, sK246: ($i > $o) > $i).
% 6.41/2.37 thf(func_def_434, type, sK247: $i > $o).
% 6.41/2.37 thf(func_def_440, type, sK253: $i > $o).
% 6.41/2.37 thf(func_def_441, type, sK254: $i > $o).
% 6.41/2.37 thf(func_def_442, type, sK255: ($i > $o) > ($i > $o) > $i).
% 6.41/2.37 thf(func_def_443, type, sK256: ($i > $o) > ($i > $o) > $i).
% 6.41/2.37 thf(func_def_455, type, sK268: $i > $o).
% 6.41/2.37 thf(func_def_458, type, sK271: $i > $o).
% 6.41/2.37 thf(func_def_460, type, sK273: $i > $o).
% 6.41/2.37 thf(func_def_461, type, sK274: ($i > $o) > $i).
% 6.41/2.37 thf(func_def_465, type, sK278: $i > $i > $i).
% 6.41/2.37 thf(func_def_467, type, sK280: $i > $o).
% 6.41/2.37 thf(func_def_468, type, sK281: ($i > $o) > $i > $i).
% 6.41/2.37 thf(func_def_469, type, sK282: ($i > $o) > $i > $i).
% 6.41/2.37 thf(func_def_492, type, sK305: $i > $i).
% 6.41/2.37 thf(func_def_503, type, sK316: $i > $i > $o).
% 6.41/2.37 thf(func_def_504, type, sK317: ($i > $i > $o) > $i > $i).
% 6.41/2.37 thf(func_def_505, type, sK318: $i > ($i > $i > $o) > $i > $i).
% 6.41/2.37 thf(func_def_506, type, sK319: ($i > $i > $o) > $i > $i).
% 6.41/2.37 thf(func_def_507, type, sK320: ($i > $i > $o) > $i > $i > $i).
% 6.41/2.37 thf(func_def_516, type, sK329: $i > $i).
% 6.41/2.37 thf(func_def_530, type, sK343: $i > $o).
% 6.41/2.37 thf(func_def_532, type, sK345: $i > ($i > $o) > $i).
% 6.41/2.37 thf(func_def_533, type, sK346: $i > ($i > $o) > $i).
% 6.41/2.37 thf(func_def_540, type, sK353: $i > $o).
% 6.41/2.37 thf(func_def_543, type, sK356: $i > $o).
% 6.41/2.37 thf(func_def_544, type, sK357: $i > $i > $i).
% 6.41/2.37 thf(func_def_545, type, sK358: $i > $i > $i).
% 6.41/2.37 thf(func_def_546, type, sK359: $i > $i > $i > $i).
% 6.41/2.37 thf(func_def_547, type, sK360: $i > $i > $i).
% 6.41/2.37 thf(func_def_548, type, sK361: $i > $i > $i).
% 6.41/2.37 thf(func_def_549, type, sK362: $i > $i > $i).
% 6.41/2.37 thf(func_def_550, type, sK363: $i > $i > $i).
% 6.41/2.37 thf(func_def_551, type, sK364: $i > $i > $i).
% 6.41/2.37 thf(func_def_552, type, sK365: $i > $i).
% 6.41/2.37 thf(func_def_553, type, sK366: $i > $i).
% 6.41/2.37 thf(func_def_554, type, sK367: $i > $i).
% 6.41/2.37 thf(func_def_555, type, sK368: $i > $i).
% 6.41/2.37 thf(func_def_556, type, sK369: $i > $i > $i).
% 6.41/2.37 thf(func_def_557, type, sK370: $i > $i > $i).
% 6.41/2.37 thf(func_def_558, type, sK371: $i > $i > $i).
% 6.41/2.37 thf(func_def_559, type, sK372: $i > $i > $i > $i).
% 6.41/2.37 thf(func_def_560, type, sK373: $i > $i > $i > $i).
% 6.41/2.37 thf(func_def_562, type, sK375: $i > $i).
% 6.41/2.37 thf(func_def_564, type, sK377: $i > $o).
% 6.41/2.37 thf(func_def_566, type, sK379: $i > $o).
% 6.41/2.37 thf(func_def_567, type, sK380: ($i > $o) > $i > $i).
% 6.41/2.37 thf(func_def_568, type, sK381: ($i > $o) > $i > $i).
% 6.41/2.37 thf(func_def_571, type, sK384: $i > $i > $i).
% 6.41/2.37 thf(func_def_591, type, sK404: $i > $i).
% 6.41/2.37 thf(func_def_593, type, sK406: $i > $o).
% 6.41/2.37 thf(func_def_594, type, sK407: $i > $o).
% 6.41/2.37 thf(func_def_595, type, sK408: ($i > $o) > ($i > $o) > $i).
% 6.41/2.37 thf(func_def_596, type, sK409: ($i > $o) > ($i > $o) > $i).
% 6.41/2.37 thf(func_def_611, type, sK424: $i > $o).
% 6.41/2.37 thf(func_def_613, type, sK426: $i > ($i > $o) > $i).
% 6.41/2.37 thf(func_def_614, type, sK427: $i > ($i > $o) > $i).
% 6.41/2.37 thf(func_def_620, type, sK433: $i > $o).
% 6.41/2.37 thf(func_def_623, type, sK436: $i > $i > $i).
% 6.41/2.37 thf(func_def_624, type, sK437: $i > $o).
% 6.41/2.37 thf(func_def_627, type, sK440: ($i > $o) > $i).
% 6.41/2.37 thf(func_def_628, type, sK441: $i > ($i > $o) > $i).
% 6.41/2.37 thf(func_def_629, type, sK442: $i > $o).
% 6.41/2.37 thf(func_def_643, type, sK456: $i > $i).
% 6.41/2.37 thf(func_def_652, type, ph465: !>[X0: $tType]:(X0)).
% 6.41/2.37 thf(f4108,plain,(
% 6.41/2.37 $false),
% 6.41/2.37 inference(subsumption_resolution,[status(thm)],[f4106,f2883])).
% 6.41/2.37 thf(f2883,plain,(
% 6.41/2.37 ( ! [X2 : $i,X3 : $i] : (($true = (in @ X3 @ (setadjoin @ X3 @ X2)))) )),
% 6.41/2.37 inference(trivial_inequality_removal,[status(thm)],[f2428])).
% 6.41/2.37 thf(f2428,plain,(
% 6.41/2.37 ( ! [X2 : $i,X3 : $i] : (($true != $true) | ($true = (in @ X3 @ (setadjoin @ X3 @ X2)))) )),
% 6.41/2.37 inference(definition_unfolding,[status(thm)],[f1665,f2088])).
% 6.41/2.37 thf(f2088,plain,(
% 6.41/2.37 (setadjoinIL = $true)),
% 6.41/2.37 inference(cnf_transformation,[status(thm)],[f1266])).
% 6.41/2.37 thf(f1266,plain,(
% 6.41/2.37 (prop2setI = $true) & ($true != (subset @ sK321 @ (setunion @ (setadjoin @ sK321 @ emptyset)))) & (emptysetAx = $true) & (powersetE1 = $true) & (symdiffI1 = $true) & (binintersectRsub = $true) & (setminusI = $true) & (setextAx = $true) & (subsetemptysetimpeq = $true) & (exuE3u = $true) & (noeltsimpempty = $true) & (subsetE = $true) & (setminusSubset1 = $true) & (nonemptyE1 = $true) & (eqinunit = $true) & (setunionsingleton1 = $true) & (cartprodpairin = $true) & (emptyInPowerset = $true) & (dsetconstr__Cong = $true) & (notdallE = $true) & (powersetI1 = $true) & (setminusIRneg = $true) & (descr__Cong = $true) & (setadjoinE = $true) & (quantDeMorgan2 = $true) & (binintersectSubset4 = $true) & (notdexE = $true) & (powersetI = $true) & (emptysetsubset = $true) & (secondinupair = $true) & (setminusER = $true) & (setminusLsub = $true) & (quantDeMorgan1 = $true) & (binunionLsub = $true) & (emptyI = $true) & (setadjoin__Cong = $true) & (kpairiskpair = $true) & (powersetsubset = $true) & (setadjoinIL = $true) & (exuE2 = $true) & (ubforcartprodlem3 = $true) & (notequalI2 = $true) & (uniqinunit = $true) & (upairset2IR = $true) & (exu__Cong = $true) & (exuI3 = $true) & (exuEu = $true) & (exuE1 = $true) & (subsetTrans = $true) & (descrp = $true) & (cartprodmempair = $true) & (omegaIndAx = $true) & (emptyset__Cong = $true) & (powersetAx = $true) & (nonemptyI = $true) & (subsetI2 = $true) & (replAx = $true) & (sepInPowerset = $true) & (upairset2E = $true) & (nonemptyI1 = $true) & (omega0Ax = $true) & (symdiffI2 = $true) & (emptyE1 = $true) & (setminusSubset2 = $true) & (singletoninpowunion = $true) & (kpairp = $true) & (nonemptyImpWitness = $true) & (symdiffIneg1 = $true) & (upairsetIR = $true) & (setminusERneg = $true) & (powersetE = $true) & (prop2setE = $true) & (notequalI1 = $true) & (dsetconstrER = $true) & (wellorderingAx = $true) & (notinemptyset = $true) & (emptyinunitempty = $true) & (inCongP = $true) & (setext = $true) & (foundationAx = $true) & (setextsub = $true) & (setadjoinAx = $true) & (sepSubset = $true) & (subsetRefl = $true) & (setunion__Cong = $true) & (disjointsetsI1 = $true) & (setunionAx = $true) & (dsetconstrEL = $true) & (binintersectSubset3 = $true) & (eqimpsubset2 = $true) & (binintersectEL = $true) & (binintersectER = $true) & (setadjoinSub = $true) & (subsetE2 = $true) & (quantDeMorgan4 = $true) & (emptysetE = $true) & (upairinpowunion = $true) & (quantDeMorgan3 = $true) & (binintersectSubset1 = $true) & (binunionRsub = $true) & (inPowerset = $true) & (setunionE = $true) & (binintersectLsub = $true) & (subsetI1 = $true) & (upairsetIL = $true) & (setadjoinIR = $true) & (setminusILneg = $true) & (eqimpsubset1 = $true) & (emptyinPowerset = $true) & (cartprodmempair1 = $true) & (in__Cong = $true) & (omega__Cong = $true) & (binunionIR = $true) & (omegaSAx = $true) & (setoftrueEq = $true) & (emptysetimpfalse = $true) & (setunionI = $true) & (setminusELneg = $true) & (setadjoinSub2 = $true) & (exuI2 = $true) & (ubforcartprodlem1 = $true) & (upairsetE = $true) & (prop2set2propI = $true) & (powerset__Cong = $true) & (singletonsubset = $true) & (setukpairIL = $true) & (notsubsetI = $true) & (binunionE = $true) & (symdiffIneg2 = $true) & (dsetconstrI = $true) & (subset2powerset = $true) & (subPowSU = $true) & (bs114d = $true) & (setadjoinOr = $true) & (exuE3e = $true) & (binunionIL = $true) & (symdiffE = $true) & (setbeta = $true) & (setunionE2 = $true) & (ubforcartprodlem2 = $true) & (setukpairIR = $true) & (exuI1 = $true) & (notinsingleton = $true) & (singletoninpowerset = $true) & (singletonsswitch = $true) & (binunionEcases = $true) & (binintersectSubset2 = $true) & (binintersectI = $true) & (binintersectSubset5 = $true) & (vacuousDall = $true) & (setminusEL = $true) & (upairsubunion = $true)),
% 6.41/2.37 inference(skolemisation,[status(esa),new_symbols(skolem,[sK321])],[f511,f1265])).
% 6.41/2.37 thf(f1265,plain,(
% 6.41/2.37 ? [X0 : $i] : ((subset @ X0 @ (setunion @ (setadjoin @ X0 @ emptyset))) != $true) => ($true != (subset @ sK321 @ (setunion @ (setadjoin @ sK321 @ emptyset))))),
% 6.41/2.37 introduced(definition,[],[choice_axiom])).
% 6.41/2.37 thf(f511,plain,(
% 6.41/2.37 (prop2setI = $true) & ? [X0 : $i] : ((subset @ X0 @ (setunion @ (setadjoin @ X0 @ emptyset))) != $true) & (emptysetAx = $true) & (powersetE1 = $true) & (symdiffI1 = $true) & (binintersectRsub = $true) & (setminusI = $true) & (setextAx = $true) & (subsetemptysetimpeq = $true) & (exuE3u = $true) & (noeltsimpempty = $true) & (subsetE = $true) & (setminusSubset1 = $true) & (nonemptyE1 = $true) & (eqinunit = $true) & (setunionsingleton1 = $true) & (cartprodpairin = $true) & (emptyInPowerset = $true) & (dsetconstr__Cong = $true) & (notdallE = $true) & (powersetI1 = $true) & (setminusIRneg = $true) & (descr__Cong = $true) & (setadjoinE = $true) & (quantDeMorgan2 = $true) & (binintersectSubset4 = $true) & (notdexE = $true) & (powersetI = $true) & (emptysetsubset = $true) & (secondinupair = $true) & (setminusER = $true) & (setminusLsub = $true) & (quantDeMorgan1 = $true) & (binunionLsub = $true) & (emptyI = $true) & (setadjoin__Cong = $true) & (kpairiskpair = $true) & (powersetsubset = $true) & (setadjoinIL = $true) & (exuE2 = $true) & (ubforcartprodlem3 = $true) & (notequalI2 = $true) & (uniqinunit = $true) & (upairset2IR = $true) & (exu__Cong = $true) & (exuI3 = $true) & (exuEu = $true) & (exuE1 = $true) & (subsetTrans = $true) & (descrp = $true) & (cartprodmempair = $true) & (omegaIndAx = $true) & (emptyset__Cong = $true) & (powersetAx = $true) & (nonemptyI = $true) & (subsetI2 = $true) & (replAx = $true) & (sepInPowerset = $true) & (upairset2E = $true) & (nonemptyI1 = $true) & (omega0Ax = $true) & (symdiffI2 = $true) & (emptyE1 = $true) & (setminusSubset2 = $true) & (singletoninpowunion = $true) & (kpairp = $true) & (nonemptyImpWitness = $true) & (symdiffIneg1 = $true) & (upairsetIR = $true) & (setminusERneg = $true) & (powersetE = $true) & (prop2setE = $true) & (notequalI1 = $true) & (dsetconstrER = $true) & (wellorderingAx = $true) & (notinemptyset = $true) & (emptyinunitempty = $true) & (inCongP = $true) & (setext = $true) & (foundationAx = $true) & (setextsub = $true) & (setadjoinAx = $true) & (sepSubset = $true) & (subsetRefl = $true) & (setunion__Cong = $true) & (disjointsetsI1 = $true) & (setunionAx = $true) & (dsetconstrEL = $true) & (binintersectSubset3 = $true) & (eqimpsubset2 = $true) & (binintersectEL = $true) & (binintersectER = $true) & (setadjoinSub = $true) & (subsetE2 = $true) & (quantDeMorgan4 = $true) & (emptysetE = $true) & (upairinpowunion = $true) & (quantDeMorgan3 = $true) & (binintersectSubset1 = $true) & (binunionRsub = $true) & (inPowerset = $true) & (setunionE = $true) & (binintersectLsub = $true) & (subsetI1 = $true) & (upairsetIL = $true) & (setadjoinIR = $true) & (setminusILneg = $true) & (eqimpsubset1 = $true) & (emptyinPowerset = $true) & (cartprodmempair1 = $true) & (in__Cong = $true) & (omega__Cong = $true) & (binunionIR = $true) & (omegaSAx = $true) & (setoftrueEq = $true) & (emptysetimpfalse = $true) & (setunionI = $true) & (setminusELneg = $true) & (setadjoinSub2 = $true) & (exuI2 = $true) & (ubforcartprodlem1 = $true) & (upairsetE = $true) & (prop2set2propI = $true) & (powerset__Cong = $true) & (singletonsubset = $true) & (setukpairIL = $true) & (notsubsetI = $true) & (binunionE = $true) & (symdiffIneg2 = $true) & (dsetconstrI = $true) & (subset2powerset = $true) & (subPowSU = $true) & (bs114d = $true) & (setadjoinOr = $true) & (exuE3e = $true) & (binunionIL = $true) & (symdiffE = $true) & (setbeta = $true) & (setunionE2 = $true) & (ubforcartprodlem2 = $true) & (setukpairIR = $true) & (exuI1 = $true) & (notinsingleton = $true) & (singletoninpowerset = $true) & (singletonsswitch = $true) & (binunionEcases = $true) & (binintersectSubset2 = $true) & (binintersectI = $true) & (binintersectSubset5 = $true) & (vacuousDall = $true) & (setminusEL = $true) & (upairsubunion = $true)),
% 6.41/2.37 inference(flattening,[status(thm)],[f510])).
% 6.41/2.37 thf(f510,plain,(
% 6.41/2.37 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i] : ((subset @ X0 @ (setunion @ (setadjoin @ X0 @ emptyset))) != $true) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 6.41/2.37 inference(ennf_transformation,[status(thm)],[f330])).
% 6.41/2.37 thf(f330,plain,(
% 6.41/2.37 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ! [X0 : $i] : ((subset @ X0 @ (setunion @ (setadjoin @ X0 @ emptyset))) = $true))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 6.41/2.37 inference(fool_elimination,[status(thm)],[f329])).
% 6.41/2.37 thf(f329,plain,(
% 6.41/2.37 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => ! [X0 : $i] : (subset @ X0 @ (setunion @ (setadjoin @ X0 @ emptyset))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 6.41/2.37 inference(rectify,[status(thm)],[f157])).
% 6.41/2.37 thf(f157,negated_conjecture,(
% 6.41/2.37 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => ! [X3 : $i] : (subset @ X3 @ (setunion @ (setadjoin @ X3 @ emptyset))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 6.41/2.37 inference(negated_conjecture,[status(cth)],[f156])).
% 6.41/2.37 thf(f156,conjecture,(
% 6.41/2.37 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => ! [X3 : $i] : (subset @ X3 @ (setunion @ (setadjoin @ X3 @ emptyset)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 6.41/2.37 file('/export/starexec/sandbox/tmp/tmp.sSg5lqwKQT/DTF2THF_20989.p',setunionsingleton2)).
% 6.41/2.37 thf(f1665,plain,(
% 6.41/2.37 ( ! [X2 : $i,X3 : $i] : (($true = (in @ X3 @ (setadjoin @ X3 @ X2))) | (setadjoinIL != $true)) )),
% 6.41/2.37 inference(cnf_transformation,[status(thm)],[f893])).
% 6.41/2.37 thf(f893,plain,(
% 6.41/2.37 ((setadjoinIL = $true) | ($true != (in @ sK120 @ (setadjoin @ sK120 @ sK119)))) & (! [X2 : $i,X3 : $i] : ($true = (in @ X3 @ (setadjoin @ X3 @ X2))) | (setadjoinIL != $true))),
% 6.41/2.37 inference(skolemisation,[status(esa),new_symbols(skolem,[sK119,sK120])],[f891,f892])).
% 6.41/2.37 thf(f892,plain,(
% 6.41/2.37 ? [X0 : $i,X1 : $i] : ($true != (in @ X1 @ (setadjoin @ X1 @ X0))) => ($true != (in @ sK120 @ (setadjoin @ sK120 @ sK119)))),
% 6.41/2.37 introduced(definition,[],[choice_axiom])).
% 6.41/2.37 thf(f891,plain,(
% 6.41/2.37 ((setadjoinIL = $true) | ? [X0 : $i,X1 : $i] : ($true != (in @ X1 @ (setadjoin @ X1 @ X0)))) & (! [X2 : $i,X3 : $i] : ($true = (in @ X3 @ (setadjoin @ X3 @ X2))) | (setadjoinIL != $true))),
% 6.41/2.37 inference(rectify,[status(thm)],[f890])).
% 6.41/2.37 thf(f890,plain,(
% 6.41/2.37 ((setadjoinIL = $true) | ? [X0 : $i,X1 : $i] : ($true != (in @ X1 @ (setadjoin @ X1 @ X0)))) & (! [X0 : $i,X1 : $i] : ($true = (in @ X1 @ (setadjoin @ X1 @ X0))) | (setadjoinIL != $true))),
% 6.41/2.37 inference(nnf_transformation,[status(thm)],[f288])).
% 6.41/2.37 thf(f288,plain,(
% 6.41/2.37 (setadjoinIL = $true) <=> ! [X0 : $i,X1 : $i] : ($true = (in @ X1 @ (setadjoin @ X1 @ X0)))),
% 6.41/2.37 inference(fool_elimination,[status(thm)],[f287])).
% 6.41/2.37 thf(f287,plain,(
% 6.41/2.37 (! [X0 : $i,X1 : $i] : (in @ X1 @ (setadjoin @ X1 @ X0)) = setadjoinIL)),
% 6.41/2.37 inference(rectify,[status(thm)],[f31])).
% 6.41/2.37 thf(f31,axiom,(
% 6.41/2.37 (! [X2 : $i,X1 : $i] : (in @ X1 @ (setadjoin @ X1 @ X2)) = setadjoinIL)),
% 6.41/2.37 file('/export/starexec/sandbox/tmp/tmp.sSg5lqwKQT/DTF2THF_20989.p',setadjoinIL)).
% 6.41/2.37 thf(f4106,plain,(
% 6.41/2.37 ($true != (in @ sK321 @ (setadjoin @ sK321 @ emptyset)))),
% 6.41/2.37 inference(trivial_inequality_removal,[status(thm)],[f4098])).
% 6.41/2.37 thf(f4098,plain,(
% 6.41/2.37 ($true != $true) | ($true != (in @ sK321 @ (setadjoin @ sK321 @ emptyset)))),
% 6.41/2.37 inference(superposition,[status(thm)],[f2125,f4009])).
% 6.41/2.37 thf(f4009,plain,(
% 6.41/2.37 ( ! [X0 : $i,X1 : $i] : (($true = (subset @ X0 @ (setunion @ X1))) | ($true != (in @ X0 @ X1))) )),
% 6.41/2.37 inference(trivial_inequality_removal,[status(thm)],[f4005])).
% 6.41/2.37 thf(f4005,plain,(
% 6.41/2.37 ( ! [X0 : $i,X1 : $i] : (($true != (in @ X0 @ X1)) | ($true != $true) | ($true = (subset @ X0 @ (setunion @ X1)))) )),
% 6.41/2.37 inference(superposition,[status(thm)],[f2879,f3012])).
% 6.41/2.37 thf(f3012,plain,(
% 6.41/2.37 ( ! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ (setunion @ X0)))) | ($true != (in @ X1 @ X0))) )),
% 6.41/2.37 inference(trivial_inequality_removal,[status(thm)],[f2705])).
% 6.41/2.37 thf(f2705,plain,(
% 6.41/2.37 ( ! [X0 : $i,X1 : $i] : (($true != (in @ X1 @ X0)) | ($true != $true) | ($true = (in @ X1 @ (powerset @ (setunion @ X0))))) )),
% 6.41/2.37 inference(definition_unfolding,[status(thm)],[f2239,f1995])).
% 6.41/2.37 thf(f1995,plain,(
% 6.41/2.37 (subPowSU = $true)),
% 6.41/2.37 inference(cnf_transformation,[status(thm)],[f1266])).
% 6.41/2.37 thf(f2239,plain,(
% 6.41/2.37 ( ! [X0 : $i,X1 : $i] : (($true != (in @ X1 @ X0)) | ($true = (in @ X1 @ (powerset @ (setunion @ X0)))) | (subPowSU != $true)) )),
% 6.41/2.37 inference(cnf_transformation,[status(thm)],[f1397])).
% 6.41/2.37 thf(f1397,plain,(
% 6.41/2.37 (! [X0 : $i,X1 : $i] : (($true != (in @ X1 @ X0)) | ($true = (in @ X1 @ (powerset @ (setunion @ X0))))) | (subPowSU != $true)) & ((subPowSU = $true) | (($true = (in @ sK387 @ sK386)) & ($true != (in @ sK387 @ (powerset @ (setunion @ sK386))))))),
% 6.41/2.37 inference(skolemisation,[status(esa),new_symbols(skolem,[sK386,sK387])],[f1395,f1396])).
% 6.41/2.37 thf(f1396,plain,(
% 6.41/2.37 ? [X2 : $i,X3 : $i] : (($true = (in @ X3 @ X2)) & ((in @ X3 @ (powerset @ (setunion @ X2))) != $true)) => (($true = (in @ sK387 @ sK386)) & ($true != (in @ sK387 @ (powerset @ (setunion @ sK386)))))),
% 6.41/2.37 introduced(definition,[],[choice_axiom])).
% 6.41/2.37 thf(f1395,plain,(
% 6.41/2.37 (! [X0 : $i,X1 : $i] : (($true != (in @ X1 @ X0)) | ($true = (in @ X1 @ (powerset @ (setunion @ X0))))) | (subPowSU != $true)) & ((subPowSU = $true) | ? [X2 : $i,X3 : $i] : (($true = (in @ X3 @ X2)) & ((in @ X3 @ (powerset @ (setunion @ X2))) != $true)))),
% 6.41/2.37 inference(rectify,[status(thm)],[f1394])).
% 6.41/2.37 thf(f1394,plain,(
% 6.41/2.37 (! [X0 : $i,X1 : $i] : (($true != (in @ X1 @ X0)) | ($true = (in @ X1 @ (powerset @ (setunion @ X0))))) | (subPowSU != $true)) & ((subPowSU = $true) | ? [X0 : $i,X1 : $i] : (($true = (in @ X1 @ X0)) & ($true != (in @ X1 @ (powerset @ (setunion @ X0))))))),
% 6.41/2.37 inference(nnf_transformation,[status(thm)],[f561])).
% 6.41/2.37 thf(f561,plain,(
% 6.41/2.37 ! [X0 : $i,X1 : $i] : (($true != (in @ X1 @ X0)) | ($true = (in @ X1 @ (powerset @ (setunion @ X0))))) <=> (subPowSU = $true)),
% 6.41/2.37 inference(ennf_transformation,[status(thm)],[f216])).
% 6.41/2.37 thf(f216,plain,(
% 6.41/2.37 (subPowSU = $true) <=> ! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ X0)) => ($true = (in @ X1 @ (powerset @ (setunion @ X0)))))),
% 6.41/2.37 inference(fool_elimination,[status(thm)],[f215])).
% 6.41/2.37 thf(f215,plain,(
% 6.41/2.37 (subPowSU = ! [X0 : $i,X1 : $i] : ((in @ X1 @ X0) => (in @ X1 @ (powerset @ (setunion @ X0)))))),
% 6.41/2.37 inference(rectify,[status(thm)],[f43])).
% 6.41/2.37 thf(f43,axiom,(
% 6.41/2.37 (subPowSU = ! [X3 : $i,X1 : $i] : ((in @ X1 @ X3) => (in @ X1 @ (powerset @ (setunion @ X3)))))),
% 6.41/2.37 file('/export/starexec/sandbox/tmp/tmp.sSg5lqwKQT/DTF2THF_20989.p',subPowSU)).
% 6.41/2.37 thf(f2879,plain,(
% 6.41/2.37 ( ! [X2 : $i,X3 : $i] : (($true != (in @ X2 @ (powerset @ X3))) | ($true = (subset @ X2 @ X3))) )),
% 6.41/2.37 inference(trivial_inequality_removal,[status(thm)],[f2798])).
% 6.41/2.37 thf(f2798,plain,(
% 6.41/2.37 ( ! [X2 : $i,X3 : $i] : (($true = (subset @ X2 @ X3)) | ($true != $true) | ($true != (in @ X2 @ (powerset @ X3)))) )),
% 6.41/2.37 inference(definition_unfolding,[status(thm)],[f2350,f2123])).
% 6.41/2.37 thf(f2123,plain,(
% 6.41/2.37 (powersetE1 = $true)),
% 6.41/2.37 inference(cnf_transformation,[status(thm)],[f1266])).
% 6.41/2.37 thf(f2350,plain,(
% 6.41/2.37 ( ! [X2 : $i,X3 : $i] : (($true != (in @ X2 @ (powerset @ X3))) | ($true = (subset @ X2 @ X3)) | (powersetE1 != $true)) )),
% 6.41/2.37 inference(cnf_transformation,[status(thm)],[f1540])).
% 6.41/2.37 thf(f1540,plain,(
% 6.41/2.37 ((powersetE1 = $true) | (($true = (in @ sK462 @ (powerset @ sK463))) & ($true != (subset @ sK462 @ sK463)))) & (! [X2 : $i,X3 : $i] : (($true != (in @ X2 @ (powerset @ X3))) | ($true = (subset @ X2 @ X3))) | (powersetE1 != $true))),
% 6.41/2.37 inference(skolemisation,[status(esa),new_symbols(skolem,[sK462,sK463])],[f1538,f1539])).
% 6.41/2.37 thf(f1539,plain,(
% 6.41/2.37 ? [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (powerset @ X1))) & ($true != (subset @ X0 @ X1))) => (($true = (in @ sK462 @ (powerset @ sK463))) & ($true != (subset @ sK462 @ sK463)))),
% 6.41/2.37 introduced(definition,[],[choice_axiom])).
% 6.41/2.37 thf(f1538,plain,(
% 6.41/2.37 ((powersetE1 = $true) | ? [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (powerset @ X1))) & ($true != (subset @ X0 @ X1)))) & (! [X2 : $i,X3 : $i] : (($true != (in @ X2 @ (powerset @ X3))) | ($true = (subset @ X2 @ X3))) | (powersetE1 != $true))),
% 6.41/2.37 inference(rectify,[status(thm)],[f1537])).
% 6.41/2.37 thf(f1537,plain,(
% 6.41/2.37 ((powersetE1 = $true) | ? [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (powerset @ X1))) & ($true != (subset @ X0 @ X1)))) & (! [X0 : $i,X1 : $i] : (($true != (in @ X0 @ (powerset @ X1))) | ($true = (subset @ X0 @ X1))) | (powersetE1 != $true))),
% 6.41/2.37 inference(nnf_transformation,[status(thm)],[f517])).
% 6.41/2.37 thf(f517,plain,(
% 6.41/2.37 (powersetE1 = $true) <=> ! [X0 : $i,X1 : $i] : (($true != (in @ X0 @ (powerset @ X1))) | ($true = (subset @ X0 @ X1)))),
% 6.41/2.37 inference(ennf_transformation,[status(thm)],[f178])).
% 6.41/2.37 thf(f178,plain,(
% 6.41/2.37 (powersetE1 = $true) <=> ! [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (powerset @ X1))) => ($true = (subset @ X0 @ X1)))),
% 6.41/2.37 inference(fool_elimination,[status(thm)],[f177])).
% 6.41/2.37 thf(f177,plain,(
% 6.41/2.37 (! [X0 : $i,X1 : $i] : ((in @ X0 @ (powerset @ X1)) => (subset @ X0 @ X1)) = powersetE1)),
% 6.41/2.37 inference(rectify,[status(thm)],[f96])).
% 6.41/2.37 thf(f96,axiom,(
% 6.41/2.37 (! [X4 : $i,X3 : $i] : ((in @ X4 @ (powerset @ X3)) => (subset @ X4 @ X3)) = powersetE1)),
% 6.41/2.37 file('/export/starexec/sandbox/tmp/tmp.sSg5lqwKQT/DTF2THF_20989.p',powersetE1)).
% 6.41/2.37 thf(f2125,plain,(
% 6.41/2.37 ($true != (subset @ sK321 @ (setunion @ (setadjoin @ sK321 @ emptyset))))),
% 6.41/2.37 inference(cnf_transformation,[status(thm)],[f1266])).
% 6.41/2.37 % SZS output end Proof for DTF2THF_20989
% 6.41/2.37 % (21135)------------------------------
% 6.41/2.37 % (21135)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.41/2.37 % (21135)Termination reason: Refutation
% 6.41/2.37
% 6.41/2.37 % (21135)Memory used [KB]: 8443
% 6.41/2.37 % (21135)Time elapsed: 0.143 s
% 6.41/2.37 % (21135)Instructions burned: 275 (million)
% 6.41/2.37 % (21135)------------------------------
% 6.41/2.37 % (21135)------------------------------
% 6.41/2.37 % (21087)Success in time 0.658 s
%------------------------------------------------------------------------------