%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU599^1 : 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 : n027.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:35:31 AM UTC 2026
% Result : Theorem 2.26s 1.54s
% Output : Refutation 2.26s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 4
% Syntax : Number of formulae : 23 ( 8 unt; 0 typ; 0 def)
% Number of atoms : 1858 ( 502 equ; 0 cnn)
% Maximal formula atoms : 117 ( 80 avg)
% Number of connectives : 945 ( 19 ~; 8 |; 353 &; 98 @)
% ( 1 <=>; 466 =>; 0 <=; 0 <~>)
% Maximal formula depth : 122 ( 39 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 234 ( 231 usr; 121 con; 0-4 aty)
% Number of variables : 38 ( 0 ^; 26 !; 12 ?; 38 :)
% 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_145,type,
sP0: $i > $i > $o ).
thf(func_def_146,type,
sP1: $i > $o ).
thf(func_def_147,type,
sP2: $i > $i > $o ).
thf(func_def_148,type,
sP3: $i > $i > $o ).
thf(func_def_155,type,
sK10: $i > ( $i > $o ) > $i ).
thf(func_def_156,type,
sK11: $i > $o ).
thf(func_def_161,type,
sK16: $i > $i > $i ).
thf(func_def_165,type,
sK20: $i > $i ).
thf(func_def_172,type,
sK27: $i > $i > $i ).
thf(func_def_177,type,
sK32: $o > $i > $i > $i ).
thf(func_def_180,type,
sK35: $i > $o ).
thf(func_def_196,type,
sK51: ( $i > $o ) > $i > $i ).
thf(func_def_198,type,
sK53: $i > $o ).
thf(func_def_202,type,
sK57: $i > $i ).
thf(func_def_213,type,
sK68: $i > $o ).
thf(func_def_216,type,
sK71: $i > ( $i > $o ) > $i ).
thf(func_def_217,type,
sK72: $i > $o ).
thf(func_def_220,type,
sK75: $i > $i > $i ).
thf(func_def_221,type,
sK76: $i > $i > $i ).
thf(func_def_222,type,
sK77: $i > $i > $i ).
thf(func_def_223,type,
sK78: $i > $i > $i ).
thf(func_def_224,type,
sK79: $i > $i > $i ).
thf(func_def_225,type,
sK80: $i > $i > $i > $i ).
thf(func_def_226,type,
sK81: $i > $i ).
thf(func_def_227,type,
sK82: $i > $i ).
thf(func_def_228,type,
sK83: $i > $i ).
thf(func_def_229,type,
sK84: $i > $i ).
thf(func_def_230,type,
sK85: $i > $i > $i ).
thf(func_def_231,type,
sK86: $i > $i > $i > $i ).
thf(func_def_232,type,
sK87: $i > $i > $i > $i ).
thf(func_def_233,type,
sK88: $i > $i > $i ).
thf(func_def_234,type,
sK89: $i > $i > $i ).
thf(func_def_236,type,
sK91: $i > $i ).
thf(func_def_237,type,
sK92: $i > $i ).
thf(func_def_238,type,
sK93: $i > $i ).
thf(func_def_239,type,
sK94: $i > $i ).
thf(func_def_244,type,
sK99: $i > $i > $i ).
thf(func_def_250,type,
sK105: $i > $o ).
thf(func_def_251,type,
sK106: $i > $i ).
thf(func_def_252,type,
sK107: ( $i > $o ) > $i ).
thf(func_def_261,type,
sK116: $i > $o ).
thf(func_def_262,type,
sK117: ( $i > $o ) > $i ).
thf(func_def_264,type,
sK119: $i > $o ).
thf(func_def_269,type,
sK124: $i > $o ).
thf(func_def_274,type,
sK129: $i > $o ).
thf(func_def_280,type,
sK135: $i > $i > $i ).
thf(func_def_287,type,
sK142: $i > $o ).
thf(func_def_288,type,
sK143: $i > $o ).
thf(func_def_289,type,
sK144: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_290,type,
sK145: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_292,type,
sK147: $i > $o ).
thf(func_def_297,type,
sK152: $i > $o ).
thf(func_def_302,type,
sK157: ( $i > $o ) > $i ).
thf(func_def_303,type,
sK158: $i > $o ).
thf(func_def_304,type,
sK159: $i > $i ).
thf(func_def_312,type,
sK167: $i > $o ).
thf(func_def_314,type,
sK169: ( $i > $o ) > $i > $i ).
thf(func_def_318,type,
sK173: $i > $i > $i ).
thf(func_def_321,type,
sK176: $i > $i > $o ).
thf(func_def_323,type,
sK178: $i > $i ).
thf(func_def_324,type,
sK179: $i > $i ).
thf(func_def_325,type,
sK180: $i > ( $i > $i > $o ) > $i ).
thf(func_def_326,type,
sK181: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_327,type,
sK182: $i > ( $i > $i > $o ) > $i ).
thf(func_def_328,type,
sK183: $i > $o ).
thf(func_def_333,type,
sK188: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_334,type,
sK189: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_337,type,
sK192: $i > $o ).
thf(func_def_338,type,
sK193: $i > $o ).
thf(func_def_345,type,
sK200: $i > $i ).
thf(func_def_346,type,
sK201: $i > $i ).
thf(func_def_365,type,
sK220: ( $i > $o ) > $i > $i ).
thf(func_def_367,type,
sK222: $i > $o ).
thf(func_def_372,type,
sK227: $i > $i ).
thf(func_def_378,type,
sK233: $i > $o ).
thf(func_def_383,type,
sK238: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_384,type,
sK239: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_385,type,
sK240: $i > $o ).
thf(func_def_386,type,
sK241: $i > $o ).
thf(func_def_391,type,
sK246: $i > $o ).
thf(func_def_394,type,
sK249: $i > ( $i > $o ) > $i ).
thf(func_def_398,type,
sK253: $i > $i > $i ).
thf(func_def_407,type,
sK262: $i > $o ).
thf(func_def_410,type,
sK265: $i > ( $i > $o ) > $i ).
thf(func_def_411,type,
sK266: $i > $o ).
thf(func_def_427,type,
sK282: $i > $o ).
thf(func_def_428,type,
sK283: $i > $o ).
thf(func_def_436,type,
sK291: $i > $o ).
thf(func_def_438,type,
sK293: ( $i > $o ) > $i ).
thf(func_def_439,type,
sK294: ( $i > $o ) > $i ).
thf(func_def_446,type,
sK301: $i > $o ).
thf(func_def_448,type,
sK303: $i > ( $i > $o ) > $i ).
thf(func_def_451,type,
sK306: $i > $i > $i ).
thf(func_def_452,type,
sK307: $i > $i > $i ).
thf(func_def_456,type,
sK311: $i > $i ).
thf(func_def_459,type,
sK314: $i > $i > $i ).
thf(func_def_461,type,
ph316:
!>[X0: $tType] : X0 ).
thf(f2541,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f2537,f1239]) ).
thf(f1239,plain,
( $true
!= ( subset @ sK66 @ sK67 ) ),
inference(cnf_transformation,[status(thm)],[f618]) ).
thf(f618,plain,
( ( dsetconstrEL = $true )
& ( subset2powerset = $true )
& ( emptyE1 = $true )
& ( sK66
= ( binintersect @ sK66 @ sK67 ) )
& ( $true
!= ( subset @ sK66 @ sK67 ) )
& ( eqimpsubset2 = $true )
& ( binintersectRsub = $true )
& ( binunionRsub = $true )
& ( omegaSAx = $true )
& ( quantDeMorgan4 = $true )
& ( inCongP = $true )
& ( nonemptyI = $true )
& ( powersetsubset = $true )
& ( replAx = $true )
& ( emptyinPowerset = $true )
& ( powersetE1 = $true )
& ( setunionAx = $true )
& ( notequalI2 = $true )
& ( notdexE = $true )
& ( binintersectSubset5 = $true )
& ( disjointsetsI1 = $true )
& ( notsubsetI = $true )
& ( setunion__Cong = $true )
& ( binintersectSubset2 = $true )
& ( in__Cong = $true )
& ( subsetRefl = $true )
& ( setadjoinIR = $true )
& ( dsetconstrER = $true )
& ( subsetI1 = $true )
& ( exuI3 = $true )
& ( setadjoinSub = $true )
& ( subsetE = $true )
& ( wellorderingAx = $true )
& ( setunionI = $true )
& ( emptysetimpfalse = $true )
& ( nonemptyE1 = $true )
& ( sepSubset = $true )
& ( foundationAx = $true )
& ( powersetAx = $true )
& ( emptyInPowerset = $true )
& ( singletonsswitch = $true )
& ( binintersectEL = $true )
& ( inPowerset = $true )
& ( subsetE2 = $true )
& ( binunionEcases = $true )
& ( setbeta = $true )
& ( notdallE = $true )
& ( setadjoinAx = $true )
& ( omega__Cong = $true )
& ( prop2set2propI = $true )
& ( sepInPowerset = $true )
& ( exuI1 = $true )
& ( binunionIL = $true )
& ( nonemptyI1 = $true )
& ( subsetI2 = $true )
& ( setadjoinSub2 = $true )
& ( eqinunit = $true )
& ( setunionE = $true )
& ( notequalI1 = $true )
& ( notinsingleton = $true )
& ( exuI2 = $true )
& ( exu__Cong = $true )
& ( exuEu = $true )
& ( descrp = $true )
& ( upairsetIL = $true )
& ( exuE2 = $true )
& ( nonemptyImpWitness = $true )
& ( binunionE = $true )
& ( exuE3u = $true )
& ( notinemptyset = $true )
& ( subsetemptysetimpeq = $true )
& ( subsetTrans = $true )
& ( setadjoin__Cong = $true )
& ( emptyinunitempty = $true )
& ( powersetI1 = $true )
& ( quantDeMorgan3 = $true )
& ( binintersectER = $true )
& ( dsetconstr__Cong = $true )
& ( exuE1 = $true )
& ( binintersectSubset3 = $true )
& ( emptyI = $true )
& ( binunionIR = $true )
& ( descr__Cong = $true )
& ( omegaIndAx = $true )
& ( binintersectSubset4 = $true )
& ( powersetE = $true )
& ( emptyset__Cong = $true )
& ( emptysetAx = $true )
& ( powerset__Cong = $true )
& ( noeltsimpempty = $true )
& ( setadjoinOr = $true )
& ( setoftrueEq = $true )
& ( setadjoinIL = $true )
& ( quantDeMorgan2 = $true )
& ( upairsetIR = $true )
& ( upairsetE = $true )
& ( setextAx = $true )
& ( exuE3e = $true )
& ( omega0Ax = $true )
& ( prop2setI = $true )
& ( vacuousDall = $true )
& ( binintersectLsub = $true )
& ( binintersectI = $true )
& ( prop2setE = $true )
& ( setextsub = $true )
& ( eqimpsubset1 = $true )
& ( setadjoinE = $true )
& ( subPowSU = $true )
& ( setext = $true )
& ( quantDeMorgan1 = $true )
& ( dsetconstrI = $true )
& ( emptysetsubset = $true )
& ( uniqinunit = $true )
& ( binunionLsub = $true )
& ( emptysetE = $true )
& ( powersetI = $true )
& ( upairset2IR = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK66,sK67])],[f490,f617]) ).
thf(f617,plain,
( ? [X0: $i,X1: $i] :
( ( ( binintersect @ X0 @ X1 )
= X0 )
& ( $true
!= ( subset @ X0 @ X1 ) ) )
=> ( ( sK66
= ( binintersect @ sK66 @ sK67 ) )
& ( $true
!= ( subset @ sK66 @ sK67 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f490,plain,
( ( dsetconstrEL = $true )
& ( subset2powerset = $true )
& ( emptyE1 = $true )
& ? [X0: $i,X1: $i] :
( ( ( binintersect @ X0 @ X1 )
= X0 )
& ( $true
!= ( subset @ X0 @ X1 ) ) )
& ( eqimpsubset2 = $true )
& ( binintersectRsub = $true )
& ( binunionRsub = $true )
& ( omegaSAx = $true )
& ( quantDeMorgan4 = $true )
& ( inCongP = $true )
& ( nonemptyI = $true )
& ( powersetsubset = $true )
& ( replAx = $true )
& ( emptyinPowerset = $true )
& ( powersetE1 = $true )
& ( setunionAx = $true )
& ( notequalI2 = $true )
& ( notdexE = $true )
& ( binintersectSubset5 = $true )
& ( disjointsetsI1 = $true )
& ( notsubsetI = $true )
& ( setunion__Cong = $true )
& ( binintersectSubset2 = $true )
& ( in__Cong = $true )
& ( subsetRefl = $true )
& ( setadjoinIR = $true )
& ( dsetconstrER = $true )
& ( subsetI1 = $true )
& ( exuI3 = $true )
& ( setadjoinSub = $true )
& ( subsetE = $true )
& ( wellorderingAx = $true )
& ( setunionI = $true )
& ( emptysetimpfalse = $true )
& ( nonemptyE1 = $true )
& ( sepSubset = $true )
& ( foundationAx = $true )
& ( powersetAx = $true )
& ( emptyInPowerset = $true )
& ( singletonsswitch = $true )
& ( binintersectEL = $true )
& ( inPowerset = $true )
& ( subsetE2 = $true )
& ( binunionEcases = $true )
& ( setbeta = $true )
& ( notdallE = $true )
& ( setadjoinAx = $true )
& ( omega__Cong = $true )
& ( prop2set2propI = $true )
& ( sepInPowerset = $true )
& ( exuI1 = $true )
& ( binunionIL = $true )
& ( nonemptyI1 = $true )
& ( subsetI2 = $true )
& ( setadjoinSub2 = $true )
& ( eqinunit = $true )
& ( setunionE = $true )
& ( notequalI1 = $true )
& ( notinsingleton = $true )
& ( exuI2 = $true )
& ( exu__Cong = $true )
& ( exuEu = $true )
& ( descrp = $true )
& ( upairsetIL = $true )
& ( exuE2 = $true )
& ( nonemptyImpWitness = $true )
& ( binunionE = $true )
& ( exuE3u = $true )
& ( notinemptyset = $true )
& ( subsetemptysetimpeq = $true )
& ( subsetTrans = $true )
& ( setadjoin__Cong = $true )
& ( emptyinunitempty = $true )
& ( powersetI1 = $true )
& ( quantDeMorgan3 = $true )
& ( binintersectER = $true )
& ( dsetconstr__Cong = $true )
& ( exuE1 = $true )
& ( binintersectSubset3 = $true )
& ( emptyI = $true )
& ( binunionIR = $true )
& ( descr__Cong = $true )
& ( omegaIndAx = $true )
& ( binintersectSubset4 = $true )
& ( powersetE = $true )
& ( emptyset__Cong = $true )
& ( emptysetAx = $true )
& ( powerset__Cong = $true )
& ( noeltsimpempty = $true )
& ( setadjoinOr = $true )
& ( setoftrueEq = $true )
& ( setadjoinIL = $true )
& ( quantDeMorgan2 = $true )
& ( upairsetIR = $true )
& ( upairsetE = $true )
& ( setextAx = $true )
& ( exuE3e = $true )
& ( omega0Ax = $true )
& ( prop2setI = $true )
& ( vacuousDall = $true )
& ( binintersectLsub = $true )
& ( binintersectI = $true )
& ( prop2setE = $true )
& ( setextsub = $true )
& ( eqimpsubset1 = $true )
& ( setadjoinE = $true )
& ( subPowSU = $true )
& ( setext = $true )
& ( quantDeMorgan1 = $true )
& ( dsetconstrI = $true )
& ( emptysetsubset = $true )
& ( uniqinunit = $true )
& ( binunionLsub = $true )
& ( emptysetE = $true )
& ( powersetI = $true )
& ( upairset2IR = $true ) ),
inference(flattening,[status(thm)],[f489]) ).
thf(f489,plain,
( ? [X0: $i,X1: $i] :
( ( ( binintersect @ X0 @ X1 )
= X0 )
& ( $true
!= ( subset @ X0 @ X1 ) ) )
& ( 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)],[f126]) ).
thf(f126,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 )
=> ! [X0: $i,X1: $i] :
( ( ( binintersect @ X0 @ X1 )
= X0 )
=> ( $true
= ( subset @ X0 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f125]) ).
thf(f125,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
=> ! [X0: $i,X1: $i] :
( ( ( binintersect @ X0 @ X1 )
= X0 )
=> ( subset @ X0 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f119]) ).
thf(f119,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
=> ! [X3: $i,X4: $i] :
( ( ( binintersect @ X3 @ X4 )
= X3 )
=> ( subset @ X3 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f118]) ).
thf(f118,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
=> ! [X3: $i,X4: $i] :
( ( ( binintersect @ X3 @ X4 )
= X3 )
=> ( subset @ X3 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.5ZrNFR5hgZ/DTF2THF_4965.p',binintersectSubset1) ).
thf(f2537,plain,
( $true
= ( subset @ sK66 @ sK67 ) ),
inference(superposition,[status(thm)],[f2164,f1240]) ).
thf(f1240,plain,
( sK66
= ( binintersect @ sK66 @ sK67 ) ),
inference(cnf_transformation,[status(thm)],[f618]) ).
thf(f2164,plain,
! [X2: $i,X3: $i] :
( $true
= ( subset @ ( binintersect @ X2 @ X3 ) @ X3 ) ),
inference(trivial_inequality_removal,[status(thm)],[f1986]) ).
thf(f1986,plain,
! [X2: $i,X3: $i] :
( ( $true != $true )
| ( $true
= ( subset @ ( binintersect @ X2 @ X3 ) @ X3 ) ) ),
inference(definition_unfolding,[status(thm)],[f1575,f1237]) ).
thf(f1237,plain,
binintersectRsub = $true,
inference(cnf_transformation,[status(thm)],[f618]) ).
thf(f1575,plain,
! [X2: $i,X3: $i] :
( ( $true
= ( subset @ ( binintersect @ X2 @ X3 ) @ X3 ) )
| ( binintersectRsub != $true ) ),
inference(cnf_transformation,[status(thm)],[f1003]) ).
thf(f1003,plain,
( ( ( binintersectRsub = $true )
| ( ( subset @ ( binintersect @ sK295 @ sK296 ) @ sK296 )
!= $true ) )
& ( ! [X2: $i,X3: $i] :
( $true
= ( subset @ ( binintersect @ X2 @ X3 ) @ X3 ) )
| ( binintersectRsub != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK295,sK296])],[f1001,f1002]) ).
thf(f1002,plain,
( ? [X0: $i,X1: $i] :
( ( subset @ ( binintersect @ X0 @ X1 ) @ X1 )
!= $true )
=> ( ( subset @ ( binintersect @ sK295 @ sK296 ) @ sK296 )
!= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1001,plain,
( ( ( binintersectRsub = $true )
| ? [X0: $i,X1: $i] :
( ( subset @ ( binintersect @ X0 @ X1 ) @ X1 )
!= $true ) )
& ( ! [X2: $i,X3: $i] :
( $true
= ( subset @ ( binintersect @ X2 @ X3 ) @ X3 ) )
| ( binintersectRsub != $true ) ) ),
inference(rectify,[status(thm)],[f1000]) ).
thf(f1000,plain,
( ( ( binintersectRsub = $true )
| ? [X0: $i,X1: $i] :
( ( subset @ ( binintersect @ X0 @ X1 ) @ X1 )
!= $true ) )
& ( ! [X0: $i,X1: $i] :
( ( subset @ ( binintersect @ X0 @ X1 ) @ X1 )
= $true )
| ( binintersectRsub != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f160]) ).
thf(f160,plain,
( ( binintersectRsub = $true )
<=> ! [X0: $i,X1: $i] :
( ( subset @ ( binintersect @ X0 @ X1 ) @ X1 )
= $true ) ),
inference(fool_elimination,[status(thm)],[f159]) ).
thf(f159,plain,
( binintersectRsub
= ( ! [X0: $i,X1: $i] : ( subset @ ( binintersect @ X0 @ X1 ) @ X1 ) ) ),
inference(rectify,[status(thm)],[f116]) ).
thf(f116,axiom,
( binintersectRsub
= ( ! [X3: $i,X4: $i] : ( subset @ ( binintersect @ X3 @ X4 ) @ X4 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.5ZrNFR5hgZ/DTF2THF_4965.p',binintersectRsub) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU599^1 : 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
% 0.16/0.33 % Computer : n027.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 12:28:22 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.30/0.45 ---- Original DTF file ---
% 0.30/0.46 thf(spec,logic,$$dhol).
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % File : SEU599^1 : TPTP v9.2.1. Released v3.7.0.
% 0.30/0.46 % Domain : Set Theory
% 0.30/0.46 % Problem : Preliminary Notions - Ops on Sets - Unions and Intersections
% 0.30/0.46 % Version : Especial.
% 0.30/0.46 % English : (! A:i.! B:i.binintersect A B = A -> subset A B)
% 0.30/0.46
% 0.30/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.30/0.46 % Source : [Bro08]
% 0.30/0.46 % Names : ZFC101g [Bro08]
% 0.30/0.46
% 0.30/0.46 % Status : Theorem
% 0.30/0.46 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.23 v8.1.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.2.0, 0.60 v5.1.0, 0.80 v5.0.0, 0.60 v4.1.0, 0.67 v3.7.0
% 0.30/0.46 % Syntax : Number of formulae : 250 ( 117 unt; 132 typ; 117 def)
% 0.30/0.46 % Number of atoms : 625 ( 171 equ; 0 cnn)
% 0.30/0.46 % Maximal formula atoms : 117 ( 5 avg)
% 0.30/0.46 % Number of connectives : 977 ( 30 ~; 6 |; 28 &; 621 @)
% 0.30/0.46 % ( 14 <=>; 278 =>; 0 <=; 0 <~>)
% 0.30/0.46 % Maximal formula depth : 121 ( 2 avg)
% 0.30/0.46 % Number of types : 2 ( 0 usr)
% 0.30/0.46 % Number of type conns : 59 ( 59 >; 0 *; 0 +; 0 <<)
% 0.30/0.46 % Number of symbols : 135 ( 132 usr; 119 con; 0-2 aty)
% 0.30/0.46 % Number of variables : 340 ( 30 ^; 281 !; 29 ?; 340 :)
% 0.30/0.46 % SPC : TH0_THM_EQU_NAR
% 0.30/0.46
% 0.30/0.46 % Comments : http://mathgate.info/detsetitem.php?id=362
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 thf(in_type,type,
% 0.30/0.46 in: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(exu_type,type,
% 0.30/0.46 exu: ( $i > $o ) > $o ).
% 0.30/0.46
% 0.30/0.46 thf(exu,definition,
% 0.30/0.46 ( exu
% 0.30/0.46 = ( ^ [Xphi: $i > $o] :
% 0.30/0.46 ? [Xx: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 & ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setextAx_type,type,
% 0.30/0.46 setextAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setextAx,definition,
% 0.30/0.46 ( setextAx
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 <=> ( in @ Xx @ B ) )
% 0.30/0.46 => ( A = B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyset_type,type,
% 0.30/0.46 emptyset: $i ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetAx_type,type,
% 0.30/0.46 emptysetAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetAx,definition,
% 0.30/0.46 ( emptysetAx
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoin_type,type,
% 0.30/0.46 setadjoin: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinAx_type,type,
% 0.30/0.46 setadjoinAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinAx,definition,
% 0.30/0.46 ( setadjoinAx
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 <=> ( ( Xy = Xx )
% 0.30/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powerset_type,type,
% 0.30/0.46 powerset: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(powersetAx_type,type,
% 0.30/0.46 powersetAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetAx,definition,
% 0.30/0.46 ( powersetAx
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.46 <=> ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunion_type,type,
% 0.30/0.46 setunion: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(setunionAx_type,type,
% 0.30/0.46 setunionAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionAx,definition,
% 0.30/0.46 ( setunionAx
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.46 <=> ? [B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omega_type,type,
% 0.30/0.46 omega: $i ).
% 0.30/0.46
% 0.30/0.46 thf(omega0Ax_type,type,
% 0.30/0.46 omega0Ax: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omega0Ax,definition,
% 0.30/0.46 ( omega0Ax
% 0.30/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omegaSAx_type,type,
% 0.30/0.46 omegaSAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omegaSAx,definition,
% 0.30/0.46 ( omegaSAx
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ omega )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omegaIndAx_type,type,
% 0.30/0.46 omegaIndAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omegaIndAx,definition,
% 0.30/0.46 ( omegaIndAx
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( ( in @ emptyset @ A )
% 0.30/0.46 & ! [Xx: $i] :
% 0.30/0.46 ( ( ( in @ Xx @ omega )
% 0.30/0.46 & ( in @ Xx @ A ) )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ omega )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(replAx_type,type,
% 0.30/0.46 replAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(replAx,definition,
% 0.30/0.46 ( replAx
% 0.30/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.30/0.46 => ? [B: $i] :
% 0.30/0.46 ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 <=> ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(foundationAx_type,type,
% 0.30/0.46 foundationAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(foundationAx,definition,
% 0.30/0.46 ( foundationAx
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/0.46 => ? [B: $i] :
% 0.30/0.46 ( ( in @ B @ A )
% 0.30/0.46 & ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(wellorderingAx_type,type,
% 0.30/0.46 wellorderingAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(wellorderingAx,definition,
% 0.30/0.46 ( wellorderingAx
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ? [B: $i] :
% 0.30/0.46 ( ! [C: $i] :
% 0.30/0.46 ( ( in @ C @ B )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ C )
% 0.30/0.46 => ( in @ Xx @ A ) ) )
% 0.30/0.46 & ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( ( in @ Xx @ A )
% 0.30/0.46 & ( in @ Xy @ A ) )
% 0.30/0.46 => ( ! [C: $i] :
% 0.30/0.46 ( ( in @ C @ B )
% 0.30/0.46 => ( ( in @ Xx @ C )
% 0.30/0.46 <=> ( in @ Xy @ C ) ) )
% 0.30/0.46 => ( Xx = Xy ) ) )
% 0.30/0.46 & ! [C: $i,D: $i] :
% 0.30/0.46 ( ( ( in @ C @ B )
% 0.30/0.46 & ( in @ D @ B ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ C )
% 0.30/0.46 => ( in @ Xx @ D ) )
% 0.30/0.46 | ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ D )
% 0.30/0.46 => ( in @ Xx @ C ) ) ) )
% 0.30/0.46 & ! [C: $i] :
% 0.30/0.46 ( ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ C )
% 0.30/0.46 => ( in @ Xx @ A ) )
% 0.30/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.30/0.46 => ? [D: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ D @ B )
% 0.30/0.46 & ( in @ Xx @ C )
% 0.30/0.46 & ~ ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ D )
% 0.30/0.46 & ( in @ Xy @ C ) )
% 0.30/0.46 & ! [E: $i] :
% 0.30/0.46 ( ( in @ E @ B )
% 0.30/0.46 => ( ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ E )
% 0.30/0.46 => ( in @ Xy @ D ) )
% 0.30/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(descr_type,type,
% 0.30/0.46 descr: ( $i > $o ) > $i ).
% 0.30/0.46
% 0.30/0.46 thf(descrp_type,type,
% 0.30/0.46 descrp: $o ).
% 0.30/0.46
% 0.30/0.46 thf(descrp,definition,
% 0.30/0.46 ( descrp
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ( Xphi
% 0.30/0.46 @ ( descr
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstr_type,type,
% 0.30/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrI_type,type,
% 0.30/0.46 dsetconstrI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrI,definition,
% 0.30/0.46 ( dsetconstrI
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrEL_type,type,
% 0.30/0.46 dsetconstrEL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrEL,definition,
% 0.30/0.46 ( dsetconstrEL
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrER_type,type,
% 0.30/0.46 dsetconstrER: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrER,definition,
% 0.30/0.46 ( dsetconstrER
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE1_type,type,
% 0.30/0.46 exuE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE1,definition,
% 0.30/0.46 ( exuE1
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 & ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(prop2set_type,type,
% 0.30/0.46 prop2set: $o > $i ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setE_type,type,
% 0.30/0.46 prop2setE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setE,definition,
% 0.30/0.46 ( prop2setE
% 0.30/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.30/0.46 => Xphi ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetE_type,type,
% 0.30/0.46 emptysetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetE,definition,
% 0.30/0.46 ( emptysetE
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ emptyset )
% 0.30/0.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetimpfalse_type,type,
% 0.30/0.46 emptysetimpfalse: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetimpfalse,definition,
% 0.30/0.46 ( emptysetimpfalse
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ emptyset )
% 0.30/0.46 => $false ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notinemptyset_type,type,
% 0.30/0.46 notinemptyset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notinemptyset,definition,
% 0.30/0.46 ( notinemptyset
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3e_type,type,
% 0.30/0.46 exuE3e: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3e,definition,
% 0.30/0.46 ( exuE3e
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setext_type,type,
% 0.30/0.46 setext: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setext,definition,
% 0.30/0.46 ( setext
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) )
% 0.30/0.46 => ( A = B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyI_type,type,
% 0.30/0.46 emptyI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyI,definition,
% 0.30/0.46 ( emptyI
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ A )
% 0.30/0.46 => ( A = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(noeltsimpempty_type,type,
% 0.30/0.46 noeltsimpempty: $o ).
% 0.30/0.46
% 0.30/0.46 thf(noeltsimpempty,definition,
% 0.30/0.46 ( noeltsimpempty
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ A )
% 0.30/0.46 => ( A = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setbeta_type,type,
% 0.30/0.46 setbeta: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setbeta,definition,
% 0.30/0.46 ( setbeta
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonempty_type,type,
% 0.30/0.46 nonempty: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonempty,definition,
% 0.30/0.46 ( nonempty
% 0.30/0.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyE1_type,type,
% 0.30/0.46 nonemptyE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyE1,definition,
% 0.30/0.46 ( nonemptyE1
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( nonempty @ A )
% 0.30/0.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI_type,type,
% 0.30/0.46 nonemptyI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI,definition,
% 0.30/0.46 ( nonemptyI
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( nonempty
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI1_type,type,
% 0.30/0.46 nonemptyI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI1,definition,
% 0.30/0.46 ( nonemptyI1
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/0.46 => ( nonempty @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIL_type,type,
% 0.30/0.46 setadjoinIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIL,definition,
% 0.30/0.46 ( setadjoinIL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinunitempty_type,type,
% 0.30/0.46 emptyinunitempty: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinunitempty,definition,
% 0.30/0.46 ( emptyinunitempty
% 0.30/0.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIR_type,type,
% 0.30/0.46 setadjoinIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIR,definition,
% 0.30/0.46 ( setadjoinIR
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinE_type,type,
% 0.30/0.46 setadjoinE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinE,definition,
% 0.30/0.46 ( setadjoinE
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ( ( Xy = Xx )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => ( ( ( in @ Xy @ A )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinOr_type,type,
% 0.30/0.46 setadjoinOr: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinOr,definition,
% 0.30/0.46 ( setadjoinOr
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 => ( ( Xy = Xx )
% 0.30/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setoftrueEq_type,type,
% 0.30/0.46 setoftrueEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setoftrueEq,definition,
% 0.30/0.46 ( setoftrueEq
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : $true )
% 0.30/0.46 = A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI_type,type,
% 0.30/0.46 powersetI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI,definition,
% 0.30/0.46 ( powersetI
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) )
% 0.30/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinPowerset_type,type,
% 0.30/0.46 emptyinPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinPowerset,definition,
% 0.30/0.46 ( emptyinPowerset
% 0.30/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyInPowerset_type,type,
% 0.30/0.46 emptyInPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyInPowerset,definition,
% 0.30/0.46 ( emptyInPowerset
% 0.30/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE_type,type,
% 0.30/0.46 powersetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE,definition,
% 0.30/0.46 ( powersetE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionI_type,type,
% 0.30/0.46 setunionI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionI,definition,
% 0.30/0.46 ( setunionI
% 0.30/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( ( in @ B @ A )
% 0.30/0.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE_type,type,
% 0.30/0.46 setunionE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE,definition,
% 0.30/0.46 ( setunionE
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ! [B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( ( in @ B @ A )
% 0.30/0.46 => Xphi ) )
% 0.30/0.46 => Xphi ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subPowSU_type,type,
% 0.30/0.46 subPowSU: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subPowSU,definition,
% 0.30/0.46 ( subPowSU
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE2_type,type,
% 0.30/0.46 exuE2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE2,definition,
% 0.30/0.46 ( exuE2
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyImpWitness_type,type,
% 0.30/0.46 nonemptyImpWitness: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyImpWitness,definition,
% 0.30/0.46 ( nonemptyImpWitness
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( nonempty @ A )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & $true ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(uniqinunit_type,type,
% 0.30/0.46 uniqinunit: $o ).
% 0.30/0.46
% 0.30/0.46 thf(uniqinunit,definition,
% 0.30/0.46 ( uniqinunit
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notinsingleton_type,type,
% 0.30/0.46 notinsingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notinsingleton,definition,
% 0.30/0.46 ( notinsingleton
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx != Xy )
% 0.30/0.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqinunit_type,type,
% 0.30/0.46 eqinunit: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqinunit,definition,
% 0.30/0.46 ( eqinunit
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsswitch_type,type,
% 0.30/0.46 singletonsswitch: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsswitch,definition,
% 0.30/0.46 ( singletonsswitch
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetE_type,type,
% 0.30/0.46 upairsetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetE,definition,
% 0.30/0.46 ( upairsetE
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.30/0.46 => ( ( Xz = Xx )
% 0.30/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIL_type,type,
% 0.30/0.46 upairsetIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIL,definition,
% 0.30/0.46 ( upairsetIL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIR_type,type,
% 0.30/0.46 upairsetIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIR,definition,
% 0.30/0.46 ( upairsetIR
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyE1_type,type,
% 0.30/0.46 emptyE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyE1,definition,
% 0.30/0.46 ( emptyE1
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ( ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 = emptyset )
% 0.30/0.46 => $false ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(vacuousDall_type,type,
% 0.30/0.46 vacuousDall: $o ).
% 0.30/0.46
% 0.30/0.46 thf(vacuousDall,definition,
% 0.30/0.46 ( vacuousDall
% 0.30/0.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ emptyset )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan1_type,type,
% 0.30/0.46 quantDeMorgan1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan1,definition,
% 0.30/0.46 ( quantDeMorgan1
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan2_type,type,
% 0.30/0.46 quantDeMorgan2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan2,definition,
% 0.30/0.46 ( quantDeMorgan2
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ~ ( Xphi @ Xx ) )
% 0.30/0.46 => ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan3_type,type,
% 0.30/0.46 quantDeMorgan3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan3,definition,
% 0.30/0.46 ( quantDeMorgan3
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan4_type,type,
% 0.30/0.46 quantDeMorgan4: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan4,definition,
% 0.30/0.46 ( quantDeMorgan4
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ~ ( Xphi @ Xx ) )
% 0.30/0.46 => ~ ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setI_type,type,
% 0.30/0.46 prop2setI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setI,definition,
% 0.30/0.46 ( prop2setI
% 0.30/0.46 = ( ! [Xphi: $o] :
% 0.30/0.46 ( Xphi
% 0.30/0.46 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(set2prop_type,type,
% 0.30/0.46 set2prop: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2set2propI_type,type,
% 0.30/0.46 prop2set2propI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2set2propI,definition,
% 0.30/0.46 ( prop2set2propI
% 0.30/0.46 = ( ! [Xphi: $o] :
% 0.30/0.46 ( Xphi
% 0.30/0.46 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notdexE_type,type,
% 0.30/0.46 notdexE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notdexE,definition,
% 0.30/0.46 ( notdexE
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notdallE_type,type,
% 0.30/0.46 notdallE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notdallE,definition,
% 0.30/0.46 ( notdallE
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuI1_type,type,
% 0.30/0.46 exuI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuI1,definition,
% 0.30/0.46 ( exuI1
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 & ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuI3_type,type,
% 0.30/0.46 exuI3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuI3,definition,
% 0.30/0.46 ( exuI3
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.30/0.46 => ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuI2_type,type,
% 0.30/0.46 exuI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuI2,definition,
% 0.30/0.46 ( exuI2
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 <=> ( Xy = Xx ) )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inCongP_type,type,
% 0.30/0.46 inCongP: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inCongP,definition,
% 0.30/0.46 ( inCongP
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(in__Cong_type,type,
% 0.30/0.46 in__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(in__Cong,definition,
% 0.30/0.46 ( in__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3u_type,type,
% 0.30/0.46 exuE3u: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3u,definition,
% 0.30/0.46 ( exuE3u
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exu__Cong_type,type,
% 0.30/0.46 exu__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exu__Cong,definition,
% 0.30/0.46 ( exu__Cong
% 0.30/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.30/0.46 => ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 <=> ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyset__Cong_type,type,
% 0.30/0.46 emptyset__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyset__Cong,definition,
% 0.30/0.46 ( emptyset__Cong
% 0.30/0.46 = ( emptyset = emptyset ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoin__Cong_type,type,
% 0.30/0.46 setadjoin__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoin__Cong,definition,
% 0.30/0.46 ( setadjoin__Cong
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ! [Xz: $i,Xu: $i] :
% 0.30/0.46 ( ( Xz = Xu )
% 0.30/0.46 => ( ( setadjoin @ Xx @ Xz )
% 0.30/0.46 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powerset__Cong_type,type,
% 0.30/0.46 powerset__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powerset__Cong,definition,
% 0.30/0.46 ( powerset__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( ( powerset @ A )
% 0.30/0.46 = ( powerset @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunion__Cong_type,type,
% 0.30/0.46 setunion__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunion__Cong,definition,
% 0.30/0.46 ( setunion__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( ( setunion @ A )
% 0.30/0.46 = ( setunion @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omega__Cong_type,type,
% 0.30/0.46 omega__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omega__Cong,definition,
% 0.30/0.46 ( omega__Cong
% 0.30/0.46 = ( omega = omega ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuEu_type,type,
% 0.30/0.46 exuEu: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuEu,definition,
% 0.30/0.46 ( exuEu
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(descr__Cong_type,type,
% 0.30/0.46 descr__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(descr__Cong,definition,
% 0.30/0.46 ( descr__Cong
% 0.30/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.30/0.46 => ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.30/0.46 => ( ( descr
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 = ( descr
% 0.30/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstr__Cong_type,type,
% 0.30/0.46 dsetconstr__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstr__Cong,definition,
% 0.30/0.46 ( dsetconstr__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( Xx = Xy )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.30/0.46 => ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 = ( dsetconstr @ B
% 0.30/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subset_type,type,
% 0.30/0.46 subset: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(disjoint_type,type,
% 0.30/0.46 disjoint: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(setsmeet_type,type,
% 0.30/0.46 setsmeet: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetI1_type,type,
% 0.30/0.46 subsetI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetI1,definition,
% 0.30/0.46 ( subsetI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqimpsubset2_type,type,
% 0.30/0.46 eqimpsubset2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqimpsubset2,definition,
% 0.30/0.46 ( eqimpsubset2
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqimpsubset1_type,type,
% 0.30/0.46 eqimpsubset1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqimpsubset1,definition,
% 0.30/0.46 ( eqimpsubset1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetI2_type,type,
% 0.30/0.46 subsetI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetI2,definition,
% 0.30/0.46 ( subsetI2
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetsubset_type,type,
% 0.30/0.46 emptysetsubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetsubset,definition,
% 0.30/0.46 ( emptysetsubset
% 0.30/0.46 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetE_type,type,
% 0.30/0.46 subsetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetE,definition,
% 0.30/0.46 ( subsetE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetE2_type,type,
% 0.30/0.46 subsetE2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetE2,definition,
% 0.30/0.46 ( subsetE2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notsubsetI_type,type,
% 0.30/0.46 notsubsetI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notsubsetI,definition,
% 0.30/0.46 ( notsubsetI
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI1_type,type,
% 0.30/0.46 notequalI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI1,definition,
% 0.30/0.46 ( notequalI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ~ ( subset @ A @ B )
% 0.30/0.46 => ( A != B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI2_type,type,
% 0.30/0.46 notequalI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI2,definition,
% 0.30/0.46 ( notequalI2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ( A != B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetRefl_type,type,
% 0.30/0.46 subsetRefl: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetRefl,definition,
% 0.30/0.46 ( subsetRefl
% 0.30/0.46 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetTrans_type,type,
% 0.30/0.46 subsetTrans: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetTrans,definition,
% 0.30/0.46 ( subsetTrans
% 0.30/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( subset @ B @ C )
% 0.30/0.46 => ( subset @ A @ C ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub_type,type,
% 0.30/0.46 setadjoinSub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub,definition,
% 0.30/0.46 ( setadjoinSub
% 0.30/0.46 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub2_type,type,
% 0.30/0.46 setadjoinSub2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub2,definition,
% 0.30/0.46 ( setadjoinSub2
% 0.30/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subset2powerset_type,type,
% 0.30/0.46 subset2powerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subset2powerset,definition,
% 0.30/0.46 ( subset2powerset
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setextsub_type,type,
% 0.30/0.46 setextsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setextsub,definition,
% 0.30/0.46 ( setextsub
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( subset @ B @ A )
% 0.30/0.46 => ( A = B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetemptysetimpeq_type,type,
% 0.30/0.46 subsetemptysetimpeq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetemptysetimpeq,definition,
% 0.30/0.46 ( subsetemptysetimpeq
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( subset @ A @ emptyset )
% 0.30/0.46 => ( A = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI1_type,type,
% 0.30/0.46 powersetI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI1,definition,
% 0.30/0.46 ( powersetI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ B @ A )
% 0.30/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE1_type,type,
% 0.30/0.46 powersetE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE1,definition,
% 0.30/0.46 ( powersetE1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inPowerset_type,type,
% 0.30/0.46 inPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inPowerset,definition,
% 0.30/0.46 ( inPowerset
% 0.30/0.46 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetsubset_type,type,
% 0.30/0.46 powersetsubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetsubset,definition,
% 0.30/0.46 ( powersetsubset
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sepInPowerset_type,type,
% 0.30/0.46 sepInPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(sepInPowerset,definition,
% 0.30/0.46 ( sepInPowerset
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( in
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sepSubset_type,type,
% 0.30/0.46 sepSubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(sepSubset,definition,
% 0.30/0.46 ( sepSubset
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( subset
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunion_type,type,
% 0.30/0.46 binunion: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIL_type,type,
% 0.30/0.46 binunionIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIL,definition,
% 0.30/0.46 ( binunionIL
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairset2IR_type,type,
% 0.30/0.46 upairset2IR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairset2IR,definition,
% 0.30/0.46 ( upairset2IR
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIR_type,type,
% 0.30/0.46 binunionIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIR,definition,
% 0.30/0.46 ( binunionIR
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionEcases_type,type,
% 0.30/0.46 binunionEcases: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionEcases,definition,
% 0.30/0.46 ( binunionEcases
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.30/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.30/0.46 => ( ( ( in @ Xx @ A )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => ( ( ( in @ Xx @ B )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionE_type,type,
% 0.30/0.46 binunionE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionE,definition,
% 0.30/0.46 ( binunionE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionLsub_type,type,
% 0.30/0.46 binunionLsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionLsub,definition,
% 0.30/0.46 ( binunionLsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionRsub_type,type,
% 0.30/0.46 binunionRsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionRsub,definition,
% 0.30/0.46 ( binunionRsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersect_type,type,
% 0.30/0.46 binintersect: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectI_type,type,
% 0.30/0.46 binintersectI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectI,definition,
% 0.30/0.46 ( binintersectI
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset5_type,type,
% 0.30/0.46 binintersectSubset5: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset5,definition,
% 0.30/0.46 ( binintersectSubset5
% 0.30/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/0.46 ( ( subset @ C @ A )
% 0.30/0.46 => ( ( subset @ C @ B )
% 0.30/0.46 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectEL_type,type,
% 0.30/0.46 binintersectEL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectEL,definition,
% 0.30/0.46 ( binintersectEL
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectLsub_type,type,
% 0.30/0.46 binintersectLsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectLsub,definition,
% 0.30/0.46 ( binintersectLsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset2_type,type,
% 0.30/0.46 binintersectSubset2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset2,definition,
% 0.30/0.46 ( binintersectSubset2
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( binintersect @ A @ B )
% 0.30/0.46 = A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset3_type,type,
% 0.30/0.46 binintersectSubset3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset3,definition,
% 0.30/0.46 ( binintersectSubset3
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( ( binintersect @ A @ B )
% 0.30/0.46 = B )
% 0.30/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectER_type,type,
% 0.30/0.46 binintersectER: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectER,definition,
% 0.30/0.46 ( binintersectER
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.30/0.46 => ( in @ Xx @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(disjointsetsI1_type,type,
% 0.30/0.46 disjointsetsI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(disjointsetsI1,definition,
% 0.30/0.46 ( disjointsetsI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( in @ Xx @ B ) )
% 0.30/0.46 => ( ( binintersect @ A @ B )
% 0.30/0.46 = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectRsub_type,type,
% 0.30/0.46 binintersectRsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectRsub,definition,
% 0.30/0.46 ( binintersectRsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset4_type,type,
% 0.30/0.46 binintersectSubset4: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset4,definition,
% 0.30/0.46 ( binintersectSubset4
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ B @ A )
% 0.30/0.46 => ( ( binintersect @ A @ B )
% 0.30/0.46 = B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset1,conjecture,
% 0.30/0.46 ( setextAx
% 0.30/0.46 => ( emptysetAx
% 0.30/0.46 => ( setadjoinAx
% 0.30/0.46 => ( powersetAx
% 0.30/0.46 => ( setunionAx
% 0.30/0.46 => ( omega0Ax
% 0.30/0.46 => ( omegaSAx
% 0.30/0.46 => ( omegaIndAx
% 0.30/0.46 => ( replAx
% 0.30/0.46 => ( foundationAx
% 0.30/0.46 => ( wellorderingAx
% 0.30/0.46 => ( descrp
% 0.30/0.46 => ( dsetconstrI
% 0.30/0.46 => ( dsetconstrEL
% 0.30/0.46 => ( dsetconstrER
% 0.30/0.46 => ( exuE1
% 0.30/0.46 => ( prop2setE
% 0.30/0.46 => ( emptysetE
% 0.30/0.46 => ( emptysetimpfalse
% 0.30/0.46 => ( notinemptyset
% 0.30/0.46 => ( exuE3e
% 0.30/0.46 => ( setext
% 0.30/0.46 => ( emptyI
% 0.30/0.46 => ( noeltsimpempty
% 0.30/0.46 => ( setbeta
% 0.30/0.46 => ( nonemptyE1
% 0.30/0.46 => ( nonemptyI
% 0.30/0.46 => ( nonemptyI1
% 0.30/0.46 => ( setadjoinIL
% 0.30/0.46 => ( emptyinunitempty
% 0.30/0.46 => ( setadjoinIR
% 0.30/0.46 => ( setadjoinE
% 0.30/0.46 => ( setadjoinOr
% 0.30/0.46 => ( setoftrueEq
% 0.30/0.46 => ( powersetI
% 0.30/0.46 => ( emptyinPowerset
% 0.30/0.46 => ( emptyInPowerset
% 0.30/0.46 => ( powersetE
% 0.30/0.46 => ( setunionI
% 0.30/0.46 => ( setunionE
% 0.30/0.46 => ( subPowSU
% 0.30/0.46 => ( exuE2
% 0.30/0.46 => ( nonemptyImpWitness
% 0.30/0.46 => ( uniqinunit
% 0.30/0.46 => ( notinsingleton
% 0.30/0.46 => ( eqinunit
% 0.30/0.46 => ( singletonsswitch
% 0.30/0.46 => ( upairsetE
% 0.30/0.46 => ( upairsetIL
% 0.30/0.46 => ( upairsetIR
% 0.30/0.46 => ( emptyE1
% 0.30/0.46 => ( vacuousDall
% 0.30/0.46 => ( quantDeMorgan1
% 0.30/0.46 => ( quantDeMorgan2
% 0.30/0.46 => ( quantDeMorgan3
% 0.30/0.46 => ( quantDeMorgan4
% 0.30/0.46 => ( prop2setI
% 0.30/0.46 => ( prop2set2propI
% 0.30/0.46 => ( notdexE
% 0.30/0.46 => ( notdallE
% 0.30/0.46 => ( exuI1
% 0.30/0.46 => ( exuI3
% 0.30/0.46 => ( exuI2
% 0.30/0.46 => ( inCongP
% 0.30/0.46 => ( in__Cong
% 0.30/0.46 => ( exuE3u
% 0.30/0.46 => ( exu__Cong
% 0.30/0.46 => ( emptyset__Cong
% 0.30/0.46 => ( setadjoin__Cong
% 0.30/0.46 => ( powerset__Cong
% 0.30/0.46 => ( setunion__Cong
% 0.30/0.46 => ( omega__Cong
% 0.30/0.46 => ( exuEu
% 0.30/0.46 => ( descr__Cong
% 0.30/0.46 => ( dsetconstr__Cong
% 0.30/0.46 => ( subsetI1
% 0.30/0.46 => ( eqimpsubset2
% 0.30/0.46 => ( eqimpsubset1
% 0.30/0.46 => ( subsetI2
% 0.30/0.46 => ( emptysetsubset
% 0.30/0.46 => ( subsetE
% 0.30/0.46 => ( subsetE2
% 0.30/0.46 => ( notsubsetI
% 0.30/0.46 => ( notequalI1
% 0.30/0.46 => ( notequalI2
% 0.30/0.46 => ( subsetRefl
% 0.30/0.46 => ( subsetTrans
% 0.30/0.46 => ( setadjoinSub
% 0.30/0.46 => ( setadjoinSub2
% 0.30/0.46 => ( subset2powerset
% 0.30/0.46 => ( setextsub
% 0.30/0.46 => ( subsetemptysetimpeq
% 0.30/0.46 => ( powersetI1
% 0.30/0.46 => ( powersetE1
% 0.30/0.46 => ( inPowerset
% 0.30/0.46 => ( powersetsubset
% 0.30/0.46 => ( sepInPowerset
% 0.30/0.46 => ( sepSubset
% 0.30/0.46 => ( binunionIL
% 0.30/0.46 => ( upairset2IR
% 0.30/0.46 => ( binunionIR
% 0.30/0.46 => ( binunionEcases
% 0.30/0.46 => ( binunionE
% 0.30/0.46 => ( binunionLsub
% 0.30/0.46 => ( binunionRsub
% 0.30/0.46 => ( binintersectI
% 0.30/0.46 => ( binintersectSubset5
% 0.30/0.46 => ( binintersectEL
% 0.30/0.46 => ( binintersectLsub
% 0.30/0.46 => ( binintersectSubset2
% 0.30/0.46 => ( binintersectSubset3
% 0.30/0.46 => ( binintersectER
% 0.30/0.46 => ( disjointsetsI1
% 0.30/0.46 => ( binintersectRsub
% 0.30/0.46 => ( binintersectSubset4
% 0.30/0.46 => ! [A: $i,B: $i] :
% 0.30/0.46 ( ( ( binintersect @ A @ B )
% 0.30/0.46 = A )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 ------------------------
% 2.18/1.42 ---- Embedded in THF ---
% 2.18/1.43 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.18/1.43 %%% Generated on Tue Feb 24 12:28:23 EST 2026
% 2.18/1.43 %%% using '$$dhol' embedding, version 1.3.0.
% 2.18/1.43 %%% Logic specification used:
% 2.18/1.43 %%% thf(spec, logic, $$dhol).
% 2.18/1.43
% 2.18/1.43 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.5ZrNFR5hgZ/DTF2DTF4965.p
% See solution above
% 2.18/1.43 ------------------------
% 2.18/1.45 ---- Cleaned THF ---
% 2.18/1.45 thf(in_type,type,
% 2.18/1.45 in: $i > $i > $o ).
% 2.18/1.45
% 2.18/1.45 thf(exu_type,type,
% 2.18/1.45 exu: ( $i > $o ) > $o ).
% 2.18/1.45
% 2.18/1.45 thf(exu,definition,
% 2.18/1.45 ( exu
% 2.18/1.45 = ( ^ [Xphi: $i > $o] :
% 2.18/1.45 ? [Xx: $i] :
% 2.18/1.45 ( ( Xphi @ Xx )
% 2.18/1.45 & ! [Xy: $i] :
% 2.18/1.45 ( ( Xphi @ Xy )
% 2.18/1.45 => ( Xx = Xy ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(setextAx_type,type,
% 2.18/1.45 setextAx: $o ).
% 2.18/1.45
% 2.18/1.45 thf(setextAx,definition,
% 2.18/1.45 ( setextAx
% 2.18/1.45 = ( ! [A: $i,B: $i] :
% 2.18/1.45 ( ! [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 <=> ( in @ Xx @ B ) )
% 2.18/1.45 => ( A = B ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(emptyset_type,type,
% 2.18/1.45 emptyset: $i ).
% 2.18/1.45
% 2.18/1.45 thf(emptysetAx_type,type,
% 2.18/1.45 emptysetAx: $o ).
% 2.18/1.45
% 2.18/1.45 thf(emptysetAx,definition,
% 2.18/1.45 ( emptysetAx
% 2.18/1.45 = ( ! [Xx: $i] :
% 2.18/1.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(setadjoin_type,type,
% 2.18/1.45 setadjoin: $i > $i > $i ).
% 2.18/1.45
% 2.18/1.45 thf(setadjoinAx_type,type,
% 2.18/1.45 setadjoinAx: $o ).
% 2.18/1.45
% 2.18/1.45 thf(setadjoinAx,definition,
% 2.18/1.45 ( setadjoinAx
% 2.18/1.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.18/1.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.18/1.45 <=> ( ( Xy = Xx )
% 2.18/1.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(powerset_type,type,
% 2.18/1.45 powerset: $i > $i ).
% 2.18/1.45
% 2.18/1.45 thf(powersetAx_type,type,
% 2.18/1.45 powersetAx: $o ).
% 2.18/1.45
% 2.18/1.45 thf(powersetAx,definition,
% 2.18/1.45 ( powersetAx
% 2.18/1.45 = ( ! [A: $i,B: $i] :
% 2.18/1.45 ( ( in @ B @ ( powerset @ A ) )
% 2.18/1.45 <=> ! [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ B )
% 2.18/1.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(setunion_type,type,
% 2.18/1.45 setunion: $i > $i ).
% 2.18/1.45
% 2.18/1.45 thf(setunionAx_type,type,
% 2.18/1.45 setunionAx: $o ).
% 2.18/1.45
% 2.18/1.45 thf(setunionAx,definition,
% 2.18/1.45 ( setunionAx
% 2.18/1.45 = ( ! [A: $i,Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ ( setunion @ A ) )
% 2.18/1.45 <=> ? [B: $i] :
% 2.18/1.45 ( ( in @ Xx @ B )
% 2.18/1.45 & ( in @ B @ A ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(omega_type,type,
% 2.18/1.45 omega: $i ).
% 2.18/1.45
% 2.18/1.45 thf(omega0Ax_type,type,
% 2.18/1.45 omega0Ax: $o ).
% 2.18/1.45
% 2.18/1.45 thf(omega0Ax,definition,
% 2.18/1.45 ( omega0Ax
% 2.18/1.45 = ( in @ emptyset @ omega ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(omegaSAx_type,type,
% 2.18/1.45 omegaSAx: $o ).
% 2.18/1.45
% 2.18/1.45 thf(omegaSAx,definition,
% 2.18/1.45 ( omegaSAx
% 2.18/1.45 = ( ! [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ omega )
% 2.18/1.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(omegaIndAx_type,type,
% 2.18/1.45 omegaIndAx: $o ).
% 2.18/1.45
% 2.18/1.45 thf(omegaIndAx,definition,
% 2.18/1.45 ( omegaIndAx
% 2.18/1.45 = ( ! [A: $i] :
% 2.18/1.45 ( ( ( in @ emptyset @ A )
% 2.18/1.45 & ! [Xx: $i] :
% 2.18/1.45 ( ( ( in @ Xx @ omega )
% 2.18/1.45 & ( in @ Xx @ A ) )
% 2.18/1.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.18/1.45 => ! [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ omega )
% 2.18/1.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(replAx_type,type,
% 2.18/1.45 replAx: $o ).
% 2.18/1.45
% 2.18/1.45 thf(replAx,definition,
% 2.18/1.45 ( replAx
% 2.18/1.45 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.18/1.45 ( ! [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 => ( exu
% 2.18/1.45 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.18/1.45 => ? [B: $i] :
% 2.18/1.45 ! [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ B )
% 2.18/1.45 <=> ? [Xy: $i] :
% 2.18/1.45 ( ( in @ Xy @ A )
% 2.18/1.45 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(foundationAx_type,type,
% 2.18/1.45 foundationAx: $o ).
% 2.18/1.45
% 2.18/1.45 thf(foundationAx,definition,
% 2.18/1.45 ( foundationAx
% 2.18/1.45 = ( ! [A: $i] :
% 2.18/1.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.18/1.45 => ? [B: $i] :
% 2.18/1.45 ( ( in @ B @ A )
% 2.18/1.45 & ~ ? [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ B )
% 2.18/1.45 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(wellorderingAx_type,type,
% 2.18/1.45 wellorderingAx: $o ).
% 2.18/1.45
% 2.18/1.45 thf(wellorderingAx,definition,
% 2.18/1.45 ( wellorderingAx
% 2.18/1.45 = ( ! [A: $i] :
% 2.18/1.45 ? [B: $i] :
% 2.18/1.45 ( ! [C: $i] :
% 2.18/1.45 ( ( in @ C @ B )
% 2.18/1.45 => ! [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ C )
% 2.18/1.45 => ( in @ Xx @ A ) ) )
% 2.18/1.45 & ! [Xx: $i,Xy: $i] :
% 2.18/1.45 ( ( ( in @ Xx @ A )
% 2.18/1.45 & ( in @ Xy @ A ) )
% 2.18/1.45 => ( ! [C: $i] :
% 2.18/1.45 ( ( in @ C @ B )
% 2.18/1.45 => ( ( in @ Xx @ C )
% 2.18/1.45 <=> ( in @ Xy @ C ) ) )
% 2.18/1.45 => ( Xx = Xy ) ) )
% 2.18/1.45 & ! [C: $i,D: $i] :
% 2.18/1.45 ( ( ( in @ C @ B )
% 2.18/1.45 & ( in @ D @ B ) )
% 2.18/1.45 => ( ! [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ C )
% 2.18/1.45 => ( in @ Xx @ D ) )
% 2.18/1.45 | ! [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ D )
% 2.18/1.45 => ( in @ Xx @ C ) ) ) )
% 2.18/1.45 & ! [C: $i] :
% 2.18/1.45 ( ( ! [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ C )
% 2.18/1.45 => ( in @ Xx @ A ) )
% 2.18/1.45 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.18/1.45 => ? [D: $i,Xx: $i] :
% 2.18/1.45 ( ( in @ D @ B )
% 2.18/1.45 & ( in @ Xx @ C )
% 2.18/1.45 & ~ ? [Xy: $i] :
% 2.18/1.45 ( ( in @ Xy @ D )
% 2.18/1.45 & ( in @ Xy @ C ) )
% 2.18/1.45 & ! [E: $i] :
% 2.18/1.45 ( ( in @ E @ B )
% 2.18/1.45 => ( ! [Xy: $i] :
% 2.18/1.45 ( ( in @ Xy @ E )
% 2.18/1.45 => ( in @ Xy @ D ) )
% 2.18/1.45 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(descr_type,type,
% 2.18/1.45 descr: ( $i > $o ) > $i ).
% 2.18/1.45
% 2.18/1.45 thf(descrp_type,type,
% 2.18/1.45 descrp: $o ).
% 2.18/1.45
% 2.18/1.45 thf(descrp,definition,
% 2.18/1.45 ( descrp
% 2.18/1.45 = ( ! [Xphi: $i > $o] :
% 2.18/1.45 ( ( exu
% 2.18/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.45 => ( Xphi
% 2.18/1.45 @ ( descr
% 2.18/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(dsetconstr_type,type,
% 2.18/1.45 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.18/1.45
% 2.18/1.45 thf(dsetconstrI_type,type,
% 2.18/1.45 dsetconstrI: $o ).
% 2.18/1.45
% 2.18/1.45 thf(dsetconstrI,definition,
% 2.18/1.45 ( dsetconstrI
% 2.18/1.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 => ( ( Xphi @ Xx )
% 2.18/1.45 => ( in @ Xx
% 2.18/1.45 @ ( dsetconstr @ A
% 2.18/1.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(dsetconstrEL_type,type,
% 2.18/1.45 dsetconstrEL: $o ).
% 2.18/1.45
% 2.18/1.45 thf(dsetconstrEL,definition,
% 2.18/1.45 ( dsetconstrEL
% 2.18/1.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.18/1.45 ( ( in @ Xx
% 2.18/1.45 @ ( dsetconstr @ A
% 2.18/1.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.18/1.45 => ( in @ Xx @ A ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(dsetconstrER_type,type,
% 2.18/1.45 dsetconstrER: $o ).
% 2.18/1.45
% 2.18/1.45 thf(dsetconstrER,definition,
% 2.18/1.45 ( dsetconstrER
% 2.18/1.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.18/1.45 ( ( in @ Xx
% 2.18/1.45 @ ( dsetconstr @ A
% 2.18/1.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.18/1.45 => ( Xphi @ Xx ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(exuE1_type,type,
% 2.18/1.45 exuE1: $o ).
% 2.18/1.45
% 2.18/1.45 thf(exuE1,definition,
% 2.18/1.45 ( exuE1
% 2.18/1.45 = ( ! [Xphi: $i > $o] :
% 2.18/1.45 ( ( exu
% 2.18/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.45 => ? [Xx: $i] :
% 2.18/1.45 ( ( Xphi @ Xx )
% 2.18/1.45 & ! [Xy: $i] :
% 2.18/1.45 ( ( Xphi @ Xy )
% 2.18/1.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(prop2set_type,type,
% 2.18/1.45 prop2set: $o > $i ).
% 2.18/1.45
% 2.18/1.45 thf(prop2setE_type,type,
% 2.18/1.45 prop2setE: $o ).
% 2.18/1.45
% 2.18/1.45 thf(prop2setE,definition,
% 2.18/1.45 ( prop2setE
% 2.18/1.45 = ( ! [Xphi: $o,Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.18/1.45 => Xphi ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(emptysetE_type,type,
% 2.18/1.45 emptysetE: $o ).
% 2.18/1.45
% 2.18/1.45 thf(emptysetE,definition,
% 2.18/1.45 ( emptysetE
% 2.18/1.45 = ( ! [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ emptyset )
% 2.18/1.45 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(emptysetimpfalse_type,type,
% 2.18/1.45 emptysetimpfalse: $o ).
% 2.18/1.45
% 2.18/1.45 thf(emptysetimpfalse,definition,
% 2.18/1.45 ( emptysetimpfalse
% 2.18/1.45 = ( ! [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ emptyset )
% 2.18/1.45 => $false ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(notinemptyset_type,type,
% 2.18/1.45 notinemptyset: $o ).
% 2.18/1.45
% 2.18/1.45 thf(notinemptyset,definition,
% 2.18/1.45 ( notinemptyset
% 2.18/1.45 = ( ! [Xx: $i] :
% 2.18/1.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(exuE3e_type,type,
% 2.18/1.45 exuE3e: $o ).
% 2.18/1.45
% 2.18/1.45 thf(exuE3e,definition,
% 2.18/1.45 ( exuE3e
% 2.18/1.45 = ( ! [Xphi: $i > $o] :
% 2.18/1.45 ( ( exu
% 2.18/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.45 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(setext_type,type,
% 2.18/1.45 setext: $o ).
% 2.18/1.45
% 2.18/1.45 thf(setext,definition,
% 2.18/1.45 ( setext
% 2.18/1.45 = ( ! [A: $i,B: $i] :
% 2.18/1.45 ( ! [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 => ( in @ Xx @ B ) )
% 2.18/1.45 => ( ! [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ B )
% 2.18/1.45 => ( in @ Xx @ A ) )
% 2.18/1.45 => ( A = B ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(emptyI_type,type,
% 2.18/1.45 emptyI: $o ).
% 2.18/1.45
% 2.18/1.45 thf(emptyI,definition,
% 2.18/1.45 ( emptyI
% 2.18/1.45 = ( ! [A: $i] :
% 2.18/1.45 ( ! [Xx: $i] :
% 2.18/1.45 ~ ( in @ Xx @ A )
% 2.18/1.45 => ( A = emptyset ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(noeltsimpempty_type,type,
% 2.18/1.45 noeltsimpempty: $o ).
% 2.18/1.45
% 2.18/1.45 thf(noeltsimpempty,definition,
% 2.18/1.45 ( noeltsimpempty
% 2.18/1.45 = ( ! [A: $i] :
% 2.18/1.45 ( ! [Xx: $i] :
% 2.18/1.45 ~ ( in @ Xx @ A )
% 2.18/1.45 => ( A = emptyset ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(setbeta_type,type,
% 2.18/1.45 setbeta: $o ).
% 2.18/1.45
% 2.18/1.45 thf(setbeta,definition,
% 2.18/1.45 ( setbeta
% 2.18/1.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 => ( ( in @ Xx
% 2.18/1.45 @ ( dsetconstr @ A
% 2.18/1.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.18/1.45 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(nonempty_type,type,
% 2.18/1.45 nonempty: $i > $o ).
% 2.18/1.45
% 2.18/1.45 thf(nonempty,definition,
% 2.18/1.45 ( nonempty
% 2.18/1.45 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(nonemptyE1_type,type,
% 2.18/1.45 nonemptyE1: $o ).
% 2.18/1.45
% 2.18/1.45 thf(nonemptyE1,definition,
% 2.18/1.45 ( nonemptyE1
% 2.18/1.45 = ( ! [A: $i] :
% 2.18/1.45 ( ( nonempty @ A )
% 2.18/1.45 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(nonemptyI_type,type,
% 2.18/1.45 nonemptyI: $o ).
% 2.18/1.45
% 2.18/1.45 thf(nonemptyI,definition,
% 2.18/1.45 ( nonemptyI
% 2.18/1.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 => ( ( Xphi @ Xx )
% 2.18/1.45 => ( nonempty
% 2.18/1.45 @ ( dsetconstr @ A
% 2.18/1.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(nonemptyI1_type,type,
% 2.18/1.45 nonemptyI1: $o ).
% 2.18/1.45
% 2.18/1.45 thf(nonemptyI1,definition,
% 2.18/1.45 ( nonemptyI1
% 2.18/1.45 = ( ! [A: $i] :
% 2.18/1.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.18/1.45 => ( nonempty @ A ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(setadjoinIL_type,type,
% 2.18/1.45 setadjoinIL: $o ).
% 2.18/1.45
% 2.18/1.45 thf(setadjoinIL,definition,
% 2.18/1.45 ( setadjoinIL
% 2.18/1.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(emptyinunitempty_type,type,
% 2.18/1.45 emptyinunitempty: $o ).
% 2.18/1.45
% 2.18/1.45 thf(emptyinunitempty,definition,
% 2.18/1.45 ( emptyinunitempty
% 2.18/1.45 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(setadjoinIR_type,type,
% 2.18/1.45 setadjoinIR: $o ).
% 2.18/1.45
% 2.18/1.45 thf(setadjoinIR,definition,
% 2.18/1.45 ( setadjoinIR
% 2.18/1.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.18/1.45 ( ( in @ Xy @ A )
% 2.18/1.45 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(setadjoinE_type,type,
% 2.18/1.45 setadjoinE: $o ).
% 2.18/1.45
% 2.18/1.45 thf(setadjoinE,definition,
% 2.18/1.45 ( setadjoinE
% 2.18/1.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.18/1.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.18/1.45 => ! [Xphi: $o] :
% 2.18/1.45 ( ( ( Xy = Xx )
% 2.18/1.45 => Xphi )
% 2.18/1.45 => ( ( ( in @ Xy @ A )
% 2.18/1.45 => Xphi )
% 2.18/1.45 => Xphi ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(setadjoinOr_type,type,
% 2.18/1.45 setadjoinOr: $o ).
% 2.18/1.45
% 2.18/1.45 thf(setadjoinOr,definition,
% 2.18/1.45 ( setadjoinOr
% 2.18/1.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.18/1.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.18/1.45 => ( ( Xy = Xx )
% 2.18/1.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(setoftrueEq_type,type,
% 2.18/1.45 setoftrueEq: $o ).
% 2.18/1.45
% 2.18/1.45 thf(setoftrueEq,definition,
% 2.18/1.45 ( setoftrueEq
% 2.18/1.45 = ( ! [A: $i] :
% 2.18/1.45 ( ( dsetconstr @ A
% 2.18/1.45 @ ^ [Xx: $i] : $true )
% 2.18/1.45 = A ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(powersetI_type,type,
% 2.18/1.45 powersetI: $o ).
% 2.18/1.45
% 2.18/1.45 thf(powersetI,definition,
% 2.18/1.45 ( powersetI
% 2.18/1.45 = ( ! [A: $i,B: $i] :
% 2.18/1.45 ( ! [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ B )
% 2.18/1.45 => ( in @ Xx @ A ) )
% 2.18/1.45 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(emptyinPowerset_type,type,
% 2.18/1.45 emptyinPowerset: $o ).
% 2.18/1.45
% 2.18/1.45 thf(emptyinPowerset,definition,
% 2.18/1.45 ( emptyinPowerset
% 2.18/1.45 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(emptyInPowerset_type,type,
% 2.18/1.45 emptyInPowerset: $o ).
% 2.18/1.45
% 2.18/1.45 thf(emptyInPowerset,definition,
% 2.18/1.45 ( emptyInPowerset
% 2.18/1.45 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(powersetE_type,type,
% 2.18/1.45 powersetE: $o ).
% 2.18/1.45
% 2.18/1.45 thf(powersetE,definition,
% 2.18/1.45 ( powersetE
% 2.18/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.45 ( ( in @ B @ ( powerset @ A ) )
% 2.18/1.45 => ( ( in @ Xx @ B )
% 2.18/1.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(setunionI_type,type,
% 2.18/1.45 setunionI: $o ).
% 2.18/1.45
% 2.18/1.45 thf(setunionI,definition,
% 2.18/1.45 ( setunionI
% 2.18/1.45 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.18/1.45 ( ( in @ Xx @ B )
% 2.18/1.45 => ( ( in @ B @ A )
% 2.18/1.45 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(setunionE_type,type,
% 2.18/1.45 setunionE: $o ).
% 2.18/1.45
% 2.18/1.45 thf(setunionE,definition,
% 2.18/1.45 ( setunionE
% 2.18/1.45 = ( ! [A: $i,Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ ( setunion @ A ) )
% 2.18/1.45 => ! [Xphi: $o] :
% 2.18/1.45 ( ! [B: $i] :
% 2.18/1.45 ( ( in @ Xx @ B )
% 2.18/1.45 => ( ( in @ B @ A )
% 2.18/1.45 => Xphi ) )
% 2.18/1.45 => Xphi ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(subPowSU_type,type,
% 2.18/1.45 subPowSU: $o ).
% 2.18/1.45
% 2.18/1.45 thf(subPowSU,definition,
% 2.18/1.45 ( subPowSU
% 2.18/1.45 = ( ! [A: $i,Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(exuE2_type,type,
% 2.18/1.45 exuE2: $o ).
% 2.18/1.45
% 2.18/1.45 thf(exuE2,definition,
% 2.18/1.45 ( exuE2
% 2.18/1.45 = ( ! [Xphi: $i > $o] :
% 2.18/1.45 ( ( exu
% 2.18/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.45 => ? [Xx: $i] :
% 2.18/1.45 ! [Xy: $i] :
% 2.18/1.45 ( ( Xphi @ Xy )
% 2.18/1.45 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(nonemptyImpWitness_type,type,
% 2.18/1.45 nonemptyImpWitness: $o ).
% 2.18/1.45
% 2.18/1.45 thf(nonemptyImpWitness,definition,
% 2.18/1.45 ( nonemptyImpWitness
% 2.18/1.45 = ( ! [A: $i] :
% 2.18/1.45 ( ( nonempty @ A )
% 2.18/1.45 => ? [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 & $true ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(uniqinunit_type,type,
% 2.18/1.45 uniqinunit: $o ).
% 2.18/1.45
% 2.18/1.45 thf(uniqinunit,definition,
% 2.18/1.45 ( uniqinunit
% 2.18/1.45 = ( ! [Xx: $i,Xy: $i] :
% 2.18/1.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.18/1.45 => ( Xx = Xy ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(notinsingleton_type,type,
% 2.18/1.45 notinsingleton: $o ).
% 2.18/1.45
% 2.18/1.45 thf(notinsingleton,definition,
% 2.18/1.45 ( notinsingleton
% 2.18/1.45 = ( ! [Xx: $i,Xy: $i] :
% 2.18/1.45 ( ( Xx != Xy )
% 2.18/1.45 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(eqinunit_type,type,
% 2.18/1.45 eqinunit: $o ).
% 2.18/1.45
% 2.18/1.45 thf(eqinunit,definition,
% 2.18/1.45 ( eqinunit
% 2.18/1.45 = ( ! [Xx: $i,Xy: $i] :
% 2.18/1.45 ( ( Xx = Xy )
% 2.18/1.45 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(singletonsswitch_type,type,
% 2.18/1.45 singletonsswitch: $o ).
% 2.18/1.45
% 2.18/1.45 thf(singletonsswitch,definition,
% 2.18/1.45 ( singletonsswitch
% 2.18/1.45 = ( ! [Xx: $i,Xy: $i] :
% 2.18/1.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.18/1.45 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(upairsetE_type,type,
% 2.18/1.45 upairsetE: $o ).
% 2.18/1.45
% 2.18/1.45 thf(upairsetE,definition,
% 2.18/1.45 ( upairsetE
% 2.18/1.45 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.18/1.45 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.18/1.45 => ( ( Xz = Xx )
% 2.18/1.45 | ( Xz = Xy ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(upairsetIL_type,type,
% 2.18/1.45 upairsetIL: $o ).
% 2.18/1.45
% 2.18/1.45 thf(upairsetIL,definition,
% 2.18/1.45 ( upairsetIL
% 2.18/1.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(upairsetIR_type,type,
% 2.18/1.45 upairsetIR: $o ).
% 2.18/1.45
% 2.18/1.45 thf(upairsetIR,definition,
% 2.18/1.45 ( upairsetIR
% 2.18/1.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(emptyE1_type,type,
% 2.18/1.45 emptyE1: $o ).
% 2.18/1.45
% 2.18/1.45 thf(emptyE1,definition,
% 2.18/1.45 ( emptyE1
% 2.18/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.45 ( ? [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 & ( Xphi @ Xx ) )
% 2.18/1.45 => ( ( ( dsetconstr @ A
% 2.18/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.45 = emptyset )
% 2.18/1.45 => $false ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(vacuousDall_type,type,
% 2.18/1.45 vacuousDall: $o ).
% 2.18/1.45
% 2.18/1.45 thf(vacuousDall,definition,
% 2.18/1.45 ( vacuousDall
% 2.18/1.45 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ emptyset )
% 2.18/1.45 => ( Xphi @ Xx ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(quantDeMorgan1_type,type,
% 2.18/1.45 quantDeMorgan1: $o ).
% 2.18/1.45
% 2.18/1.45 thf(quantDeMorgan1,definition,
% 2.18/1.45 ( quantDeMorgan1
% 2.18/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.45 ( ~ ! [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 => ( Xphi @ Xx ) )
% 2.18/1.45 => ? [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(quantDeMorgan2_type,type,
% 2.18/1.45 quantDeMorgan2: $o ).
% 2.18/1.45
% 2.18/1.45 thf(quantDeMorgan2,definition,
% 2.18/1.45 ( quantDeMorgan2
% 2.18/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.45 ( ! [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 => ~ ( Xphi @ Xx ) )
% 2.18/1.45 => ~ ? [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(quantDeMorgan3_type,type,
% 2.18/1.45 quantDeMorgan3: $o ).
% 2.18/1.45
% 2.18/1.45 thf(quantDeMorgan3,definition,
% 2.18/1.45 ( quantDeMorgan3
% 2.18/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.45 ( ~ ? [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 & ( Xphi @ Xx ) )
% 2.18/1.45 => ! [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(quantDeMorgan4_type,type,
% 2.18/1.45 quantDeMorgan4: $o ).
% 2.18/1.45
% 2.18/1.45 thf(quantDeMorgan4,definition,
% 2.18/1.45 ( quantDeMorgan4
% 2.18/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.45 ( ? [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 & ~ ( Xphi @ Xx ) )
% 2.18/1.45 => ~ ! [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(prop2setI_type,type,
% 2.18/1.45 prop2setI: $o ).
% 2.18/1.45
% 2.18/1.45 thf(prop2setI,definition,
% 2.18/1.45 ( prop2setI
% 2.18/1.45 = ( ! [Xphi: $o] :
% 2.18/1.45 ( Xphi
% 2.18/1.45 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(set2prop_type,type,
% 2.18/1.45 set2prop: $i > $o ).
% 2.18/1.45
% 2.18/1.45 thf(prop2set2propI_type,type,
% 2.18/1.45 prop2set2propI: $o ).
% 2.18/1.45
% 2.18/1.45 thf(prop2set2propI,definition,
% 2.18/1.45 ( prop2set2propI
% 2.18/1.45 = ( ! [Xphi: $o] :
% 2.18/1.45 ( Xphi
% 2.18/1.45 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(notdexE_type,type,
% 2.18/1.45 notdexE: $o ).
% 2.18/1.45
% 2.18/1.45 thf(notdexE,definition,
% 2.18/1.45 ( notdexE
% 2.18/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.45 ( ~ ? [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 & ( Xphi @ Xx ) )
% 2.18/1.45 => ! [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(notdallE_type,type,
% 2.18/1.45 notdallE: $o ).
% 2.18/1.45
% 2.18/1.45 thf(notdallE,definition,
% 2.18/1.45 ( notdallE
% 2.18/1.45 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.45 ( ~ ! [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 => ( Xphi @ Xx ) )
% 2.18/1.45 => ? [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(exuI1_type,type,
% 2.18/1.45 exuI1: $o ).
% 2.18/1.45
% 2.18/1.45 thf(exuI1,definition,
% 2.18/1.45 ( exuI1
% 2.18/1.45 = ( ! [Xphi: $i > $o] :
% 2.18/1.45 ( ? [Xx: $i] :
% 2.18/1.45 ( ( Xphi @ Xx )
% 2.18/1.45 & ! [Xy: $i] :
% 2.18/1.45 ( ( Xphi @ Xy )
% 2.18/1.45 => ( Xx = Xy ) ) )
% 2.18/1.45 => ( exu
% 2.18/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(exuI3_type,type,
% 2.18/1.45 exuI3: $o ).
% 2.18/1.45
% 2.18/1.45 thf(exuI3,definition,
% 2.18/1.45 ( exuI3
% 2.18/1.45 = ( ! [Xphi: $i > $o] :
% 2.18/1.45 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.18/1.45 => ( ! [Xx: $i,Xy: $i] :
% 2.18/1.45 ( ( Xphi @ Xx )
% 2.18/1.45 => ( ( Xphi @ Xy )
% 2.18/1.45 => ( Xx = Xy ) ) )
% 2.18/1.45 => ( exu
% 2.18/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(exuI2_type,type,
% 2.18/1.45 exuI2: $o ).
% 2.18/1.45
% 2.18/1.45 thf(exuI2,definition,
% 2.18/1.45 ( exuI2
% 2.18/1.45 = ( ! [Xphi: $i > $o] :
% 2.18/1.45 ( ? [Xx: $i] :
% 2.18/1.45 ! [Xy: $i] :
% 2.18/1.45 ( ( Xphi @ Xy )
% 2.18/1.45 <=> ( Xy = Xx ) )
% 2.18/1.45 => ( exu
% 2.18/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(inCongP_type,type,
% 2.18/1.45 inCongP: $o ).
% 2.18/1.45
% 2.18/1.45 thf(inCongP,definition,
% 2.18/1.45 ( inCongP
% 2.18/1.45 = ( ! [A: $i,B: $i] :
% 2.18/1.45 ( ( A = B )
% 2.18/1.45 => ! [Xx: $i,Xy: $i] :
% 2.18/1.45 ( ( Xx = Xy )
% 2.18/1.45 => ( ( in @ Xx @ A )
% 2.18/1.45 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(in__Cong_type,type,
% 2.18/1.45 in__Cong: $o ).
% 2.18/1.45
% 2.18/1.45 thf(in__Cong,definition,
% 2.18/1.45 ( in__Cong
% 2.18/1.45 = ( ! [A: $i,B: $i] :
% 2.18/1.45 ( ( A = B )
% 2.18/1.45 => ! [Xx: $i,Xy: $i] :
% 2.18/1.45 ( ( Xx = Xy )
% 2.18/1.45 => ( ( in @ Xx @ A )
% 2.18/1.45 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(exuE3u_type,type,
% 2.18/1.45 exuE3u: $o ).
% 2.18/1.45
% 2.18/1.45 thf(exuE3u,definition,
% 2.18/1.45 ( exuE3u
% 2.18/1.45 = ( ! [Xphi: $i > $o] :
% 2.18/1.45 ( ( exu
% 2.18/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.45 => ! [Xx: $i,Xy: $i] :
% 2.18/1.45 ( ( Xphi @ Xx )
% 2.18/1.45 => ( ( Xphi @ Xy )
% 2.18/1.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(exu__Cong_type,type,
% 2.18/1.45 exu__Cong: $o ).
% 2.18/1.45
% 2.18/1.45 thf(exu__Cong,definition,
% 2.18/1.45 ( exu__Cong
% 2.18/1.45 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.18/1.45 ( ! [Xx: $i,Xy: $i] :
% 2.18/1.45 ( ( Xx = Xy )
% 2.18/1.45 => ( ( Xphi @ Xx )
% 2.18/1.45 <=> ( Xpsi @ Xy ) ) )
% 2.18/1.45 => ( ( exu
% 2.18/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.45 <=> ( exu
% 2.18/1.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(emptyset__Cong_type,type,
% 2.18/1.45 emptyset__Cong: $o ).
% 2.18/1.45
% 2.18/1.45 thf(emptyset__Cong,definition,
% 2.18/1.45 ( emptyset__Cong
% 2.18/1.45 = ( emptyset = emptyset ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(setadjoin__Cong_type,type,
% 2.18/1.45 setadjoin__Cong: $o ).
% 2.18/1.45
% 2.18/1.45 thf(setadjoin__Cong,definition,
% 2.18/1.45 ( setadjoin__Cong
% 2.18/1.45 = ( ! [Xx: $i,Xy: $i] :
% 2.18/1.45 ( ( Xx = Xy )
% 2.18/1.45 => ! [Xz: $i,Xu: $i] :
% 2.18/1.45 ( ( Xz = Xu )
% 2.18/1.45 => ( ( setadjoin @ Xx @ Xz )
% 2.18/1.45 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(powerset__Cong_type,type,
% 2.18/1.45 powerset__Cong: $o ).
% 2.18/1.45
% 2.18/1.45 thf(powerset__Cong,definition,
% 2.18/1.45 ( powerset__Cong
% 2.18/1.45 = ( ! [A: $i,B: $i] :
% 2.18/1.45 ( ( A = B )
% 2.18/1.45 => ( ( powerset @ A )
% 2.18/1.45 = ( powerset @ B ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(setunion__Cong_type,type,
% 2.18/1.45 setunion__Cong: $o ).
% 2.18/1.45
% 2.18/1.45 thf(setunion__Cong,definition,
% 2.18/1.45 ( setunion__Cong
% 2.18/1.45 = ( ! [A: $i,B: $i] :
% 2.18/1.45 ( ( A = B )
% 2.18/1.45 => ( ( setunion @ A )
% 2.18/1.45 = ( setunion @ B ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(omega__Cong_type,type,
% 2.18/1.45 omega__Cong: $o ).
% 2.18/1.45
% 2.18/1.45 thf(omega__Cong,definition,
% 2.18/1.45 ( omega__Cong
% 2.18/1.45 = ( omega = omega ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(exuEu_type,type,
% 2.18/1.45 exuEu: $o ).
% 2.18/1.45
% 2.18/1.45 thf(exuEu,definition,
% 2.18/1.45 ( exuEu
% 2.18/1.45 = ( ! [Xphi: $i > $o] :
% 2.18/1.45 ( ( exu
% 2.18/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.45 => ! [Xx: $i,Xy: $i] :
% 2.18/1.45 ( ( Xphi @ Xx )
% 2.18/1.45 => ( ( Xphi @ Xy )
% 2.18/1.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(descr__Cong_type,type,
% 2.18/1.45 descr__Cong: $o ).
% 2.18/1.45
% 2.18/1.45 thf(descr__Cong,definition,
% 2.18/1.45 ( descr__Cong
% 2.18/1.45 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.18/1.45 ( ! [Xx: $i,Xy: $i] :
% 2.18/1.45 ( ( Xx = Xy )
% 2.18/1.45 => ( ( Xphi @ Xx )
% 2.18/1.45 <=> ( Xpsi @ Xy ) ) )
% 2.18/1.45 => ( ( exu
% 2.18/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.45 => ( ( exu
% 2.18/1.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.18/1.45 => ( ( descr
% 2.18/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.45 = ( descr
% 2.18/1.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(dsetconstr__Cong_type,type,
% 2.18/1.45 dsetconstr__Cong: $o ).
% 2.18/1.45
% 2.18/1.45 thf(dsetconstr__Cong,definition,
% 2.18/1.45 ( dsetconstr__Cong
% 2.18/1.45 = ( ! [A: $i,B: $i] :
% 2.18/1.45 ( ( A = B )
% 2.18/1.45 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.18/1.45 ( ! [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 => ! [Xy: $i] :
% 2.18/1.45 ( ( in @ Xy @ B )
% 2.18/1.45 => ( ( Xx = Xy )
% 2.18/1.45 => ( ( Xphi @ Xx )
% 2.18/1.45 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.18/1.45 => ( ( dsetconstr @ A
% 2.18/1.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.45 = ( dsetconstr @ B
% 2.18/1.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(subset_type,type,
% 2.18/1.45 subset: $i > $i > $o ).
% 2.18/1.45
% 2.18/1.45 thf(disjoint_type,type,
% 2.18/1.45 disjoint: $i > $i > $o ).
% 2.18/1.45
% 2.18/1.45 thf(setsmeet_type,type,
% 2.18/1.45 setsmeet: $i > $i > $o ).
% 2.18/1.45
% 2.18/1.45 thf(subsetI1_type,type,
% 2.18/1.45 subsetI1: $o ).
% 2.18/1.45
% 2.18/1.45 thf(subsetI1,definition,
% 2.18/1.45 ( subsetI1
% 2.18/1.45 = ( ! [A: $i,B: $i] :
% 2.18/1.45 ( ! [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 => ( in @ Xx @ B ) )
% 2.18/1.45 => ( subset @ A @ B ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(eqimpsubset2_type,type,
% 2.18/1.45 eqimpsubset2: $o ).
% 2.18/1.45
% 2.18/1.45 thf(eqimpsubset2,definition,
% 2.18/1.45 ( eqimpsubset2
% 2.18/1.45 = ( ! [A: $i,B: $i] :
% 2.18/1.45 ( ( A = B )
% 2.18/1.45 => ( subset @ B @ A ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(eqimpsubset1_type,type,
% 2.18/1.45 eqimpsubset1: $o ).
% 2.18/1.45
% 2.18/1.45 thf(eqimpsubset1,definition,
% 2.18/1.45 ( eqimpsubset1
% 2.18/1.45 = ( ! [A: $i,B: $i] :
% 2.18/1.45 ( ( A = B )
% 2.18/1.45 => ( subset @ A @ B ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(subsetI2_type,type,
% 2.18/1.45 subsetI2: $o ).
% 2.18/1.45
% 2.18/1.45 thf(subsetI2,definition,
% 2.18/1.45 ( subsetI2
% 2.18/1.45 = ( ! [A: $i,B: $i] :
% 2.18/1.45 ( ! [Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 => ( in @ Xx @ B ) )
% 2.18/1.45 => ( subset @ A @ B ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(emptysetsubset_type,type,
% 2.18/1.45 emptysetsubset: $o ).
% 2.18/1.45
% 2.18/1.45 thf(emptysetsubset,definition,
% 2.18/1.45 ( emptysetsubset
% 2.18/1.45 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(subsetE_type,type,
% 2.18/1.45 subsetE: $o ).
% 2.18/1.45
% 2.18/1.45 thf(subsetE,definition,
% 2.18/1.45 ( subsetE
% 2.18/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.45 ( ( subset @ A @ B )
% 2.18/1.45 => ( ( in @ Xx @ A )
% 2.18/1.45 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(subsetE2_type,type,
% 2.18/1.45 subsetE2: $o ).
% 2.18/1.45
% 2.18/1.45 thf(subsetE2,definition,
% 2.18/1.45 ( subsetE2
% 2.18/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.45 ( ( subset @ A @ B )
% 2.18/1.45 => ( ~ ( in @ Xx @ B )
% 2.18/1.45 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(notsubsetI_type,type,
% 2.18/1.45 notsubsetI: $o ).
% 2.18/1.45
% 2.18/1.45 thf(notsubsetI,definition,
% 2.18/1.45 ( notsubsetI
% 2.18/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 => ( ~ ( in @ Xx @ B )
% 2.18/1.45 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(notequalI1_type,type,
% 2.18/1.45 notequalI1: $o ).
% 2.18/1.45
% 2.18/1.45 thf(notequalI1,definition,
% 2.18/1.45 ( notequalI1
% 2.18/1.45 = ( ! [A: $i,B: $i] :
% 2.18/1.45 ( ~ ( subset @ A @ B )
% 2.18/1.45 => ( A != B ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(notequalI2_type,type,
% 2.18/1.45 notequalI2: $o ).
% 2.18/1.45
% 2.18/1.45 thf(notequalI2,definition,
% 2.18/1.45 ( notequalI2
% 2.18/1.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.45 ( ( in @ Xx @ A )
% 2.18/1.45 => ( ~ ( in @ Xx @ B )
% 2.18/1.45 => ( A != B ) ) ) ) ) ).
% 2.18/1.45
% 2.18/1.45 thf(subsetRefl_type,type,
% 2.18/1.45 subsetRefl: $o ).
% 2.18/1.45
% 2.18/1.45 thf(subsetRefl,definition,
% 2.18/1.45 ( subsetRefl
% 2.18/1.45 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(subsetTrans_type,type,
% 2.18/1.46 subsetTrans: $o ).
% 2.18/1.46
% 2.18/1.46 thf(subsetTrans,definition,
% 2.18/1.46 ( subsetTrans
% 2.18/1.46 = ( ! [A: $i,B: $i,C: $i] :
% 2.18/1.46 ( ( subset @ A @ B )
% 2.18/1.46 => ( ( subset @ B @ C )
% 2.18/1.46 => ( subset @ A @ C ) ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(setadjoinSub_type,type,
% 2.18/1.46 setadjoinSub: $o ).
% 2.18/1.46
% 2.18/1.46 thf(setadjoinSub,definition,
% 2.18/1.46 ( setadjoinSub
% 2.18/1.46 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(setadjoinSub2_type,type,
% 2.18/1.46 setadjoinSub2: $o ).
% 2.18/1.46
% 2.18/1.46 thf(setadjoinSub2,definition,
% 2.18/1.46 ( setadjoinSub2
% 2.18/1.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.18/1.46 ( ( subset @ A @ B )
% 2.18/1.46 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(subset2powerset_type,type,
% 2.18/1.46 subset2powerset: $o ).
% 2.18/1.46
% 2.18/1.46 thf(subset2powerset,definition,
% 2.18/1.46 ( subset2powerset
% 2.18/1.46 = ( ! [A: $i,B: $i] :
% 2.18/1.46 ( ( subset @ A @ B )
% 2.18/1.46 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(setextsub_type,type,
% 2.18/1.46 setextsub: $o ).
% 2.18/1.46
% 2.18/1.46 thf(setextsub,definition,
% 2.18/1.46 ( setextsub
% 2.18/1.46 = ( ! [A: $i,B: $i] :
% 2.18/1.46 ( ( subset @ A @ B )
% 2.18/1.46 => ( ( subset @ B @ A )
% 2.18/1.46 => ( A = B ) ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(subsetemptysetimpeq_type,type,
% 2.18/1.46 subsetemptysetimpeq: $o ).
% 2.18/1.46
% 2.18/1.46 thf(subsetemptysetimpeq,definition,
% 2.18/1.46 ( subsetemptysetimpeq
% 2.18/1.46 = ( ! [A: $i] :
% 2.18/1.46 ( ( subset @ A @ emptyset )
% 2.18/1.46 => ( A = emptyset ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(powersetI1_type,type,
% 2.18/1.46 powersetI1: $o ).
% 2.18/1.46
% 2.18/1.46 thf(powersetI1,definition,
% 2.18/1.46 ( powersetI1
% 2.18/1.46 = ( ! [A: $i,B: $i] :
% 2.18/1.46 ( ( subset @ B @ A )
% 2.18/1.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(powersetE1_type,type,
% 2.18/1.46 powersetE1: $o ).
% 2.18/1.46
% 2.18/1.46 thf(powersetE1,definition,
% 2.18/1.46 ( powersetE1
% 2.18/1.46 = ( ! [A: $i,B: $i] :
% 2.18/1.46 ( ( in @ B @ ( powerset @ A ) )
% 2.18/1.46 => ( subset @ B @ A ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(inPowerset_type,type,
% 2.18/1.46 inPowerset: $o ).
% 2.18/1.46
% 2.18/1.46 thf(inPowerset,definition,
% 2.18/1.46 ( inPowerset
% 2.18/1.46 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(powersetsubset_type,type,
% 2.18/1.46 powersetsubset: $o ).
% 2.18/1.46
% 2.18/1.46 thf(powersetsubset,definition,
% 2.18/1.46 ( powersetsubset
% 2.18/1.46 = ( ! [A: $i,B: $i] :
% 2.18/1.46 ( ( subset @ A @ B )
% 2.18/1.46 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(sepInPowerset_type,type,
% 2.18/1.46 sepInPowerset: $o ).
% 2.18/1.46
% 2.18/1.46 thf(sepInPowerset,definition,
% 2.18/1.46 ( sepInPowerset
% 2.18/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.46 ( in
% 2.18/1.46 @ ( dsetconstr @ A
% 2.18/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.46 @ ( powerset @ A ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(sepSubset_type,type,
% 2.18/1.46 sepSubset: $o ).
% 2.18/1.46
% 2.18/1.46 thf(sepSubset,definition,
% 2.18/1.46 ( sepSubset
% 2.18/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.46 ( subset
% 2.18/1.46 @ ( dsetconstr @ A
% 2.18/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.46 @ A ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(binunion_type,type,
% 2.18/1.46 binunion: $i > $i > $i ).
% 2.18/1.46
% 2.18/1.46 thf(binunionIL_type,type,
% 2.18/1.46 binunionIL: $o ).
% 2.18/1.46
% 2.18/1.46 thf(binunionIL,definition,
% 2.18/1.46 ( binunionIL
% 2.18/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.46 ( ( in @ Xx @ A )
% 2.18/1.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(upairset2IR_type,type,
% 2.18/1.46 upairset2IR: $o ).
% 2.18/1.46
% 2.18/1.46 thf(upairset2IR,definition,
% 2.18/1.46 ( upairset2IR
% 2.18/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(binunionIR_type,type,
% 2.18/1.46 binunionIR: $o ).
% 2.18/1.46
% 2.18/1.46 thf(binunionIR,definition,
% 2.18/1.46 ( binunionIR
% 2.18/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.46 ( ( in @ Xx @ B )
% 2.18/1.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(binunionEcases_type,type,
% 2.18/1.46 binunionEcases: $o ).
% 2.18/1.46
% 2.18/1.46 thf(binunionEcases,definition,
% 2.18/1.46 ( binunionEcases
% 2.18/1.46 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.18/1.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.18/1.46 => ( ( ( in @ Xx @ A )
% 2.18/1.46 => Xphi )
% 2.18/1.46 => ( ( ( in @ Xx @ B )
% 2.18/1.46 => Xphi )
% 2.18/1.46 => Xphi ) ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(binunionE_type,type,
% 2.18/1.46 binunionE: $o ).
% 2.18/1.46
% 2.18/1.46 thf(binunionE,definition,
% 2.18/1.46 ( binunionE
% 2.18/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.18/1.46 => ( ( in @ Xx @ A )
% 2.18/1.46 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(binunionLsub_type,type,
% 2.18/1.46 binunionLsub: $o ).
% 2.18/1.46
% 2.18/1.46 thf(binunionLsub,definition,
% 2.18/1.46 ( binunionLsub
% 2.18/1.46 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(binunionRsub_type,type,
% 2.18/1.46 binunionRsub: $o ).
% 2.18/1.46
% 2.18/1.46 thf(binunionRsub,definition,
% 2.18/1.46 ( binunionRsub
% 2.18/1.46 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(binintersect_type,type,
% 2.18/1.46 binintersect: $i > $i > $i ).
% 2.18/1.46
% 2.18/1.46 thf(binintersectI_type,type,
% 2.18/1.46 binintersectI: $o ).
% 2.18/1.46
% 2.18/1.46 thf(binintersectI,definition,
% 2.18/1.46 ( binintersectI
% 2.18/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.46 ( ( in @ Xx @ A )
% 2.18/1.46 => ( ( in @ Xx @ B )
% 2.18/1.46 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(binintersectSubset5_type,type,
% 2.18/1.46 binintersectSubset5: $o ).
% 2.18/1.46
% 2.18/1.46 thf(binintersectSubset5,definition,
% 2.18/1.46 ( binintersectSubset5
% 2.18/1.46 = ( ! [A: $i,B: $i,C: $i] :
% 2.18/1.46 ( ( subset @ C @ A )
% 2.18/1.46 => ( ( subset @ C @ B )
% 2.18/1.46 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(binintersectEL_type,type,
% 2.18/1.46 binintersectEL: $o ).
% 2.18/1.46
% 2.18/1.46 thf(binintersectEL,definition,
% 2.18/1.46 ( binintersectEL
% 2.18/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.18/1.46 => ( in @ Xx @ A ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(binintersectLsub_type,type,
% 2.18/1.46 binintersectLsub: $o ).
% 2.18/1.46
% 2.18/1.46 thf(binintersectLsub,definition,
% 2.18/1.46 ( binintersectLsub
% 2.18/1.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(binintersectSubset2_type,type,
% 2.18/1.46 binintersectSubset2: $o ).
% 2.18/1.46
% 2.18/1.46 thf(binintersectSubset2,definition,
% 2.18/1.46 ( binintersectSubset2
% 2.18/1.46 = ( ! [A: $i,B: $i] :
% 2.18/1.46 ( ( subset @ A @ B )
% 2.18/1.46 => ( ( binintersect @ A @ B )
% 2.18/1.46 = A ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(binintersectSubset3_type,type,
% 2.18/1.46 binintersectSubset3: $o ).
% 2.18/1.46
% 2.18/1.46 thf(binintersectSubset3,definition,
% 2.18/1.46 ( binintersectSubset3
% 2.18/1.46 = ( ! [A: $i,B: $i] :
% 2.18/1.46 ( ( ( binintersect @ A @ B )
% 2.18/1.46 = B )
% 2.18/1.46 => ( subset @ B @ A ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(binintersectER_type,type,
% 2.18/1.46 binintersectER: $o ).
% 2.18/1.46
% 2.18/1.46 thf(binintersectER,definition,
% 2.18/1.46 ( binintersectER
% 2.18/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.18/1.46 => ( in @ Xx @ B ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(disjointsetsI1_type,type,
% 2.18/1.46 disjointsetsI1: $o ).
% 2.18/1.46
% 2.18/1.46 thf(disjointsetsI1,definition,
% 2.18/1.46 ( disjointsetsI1
% 2.18/1.46 = ( ! [A: $i,B: $i] :
% 2.18/1.46 ( ~ ? [Xx: $i] :
% 2.18/1.46 ( ( in @ Xx @ A )
% 2.18/1.46 & ( in @ Xx @ B ) )
% 2.18/1.46 => ( ( binintersect @ A @ B )
% 2.18/1.46 = emptyset ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(binintersectRsub_type,type,
% 2.18/1.46 binintersectRsub: $o ).
% 2.18/1.46
% 2.18/1.46 thf(binintersectRsub,definition,
% 2.18/1.46 ( binintersectRsub
% 2.18/1.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(binintersectSubset4_type,type,
% 2.18/1.46 binintersectSubset4: $o ).
% 2.18/1.46
% 2.18/1.46 thf(binintersectSubset4,definition,
% 2.18/1.46 ( binintersectSubset4
% 2.18/1.46 = ( ! [A: $i,B: $i] :
% 2.18/1.46 ( ( subset @ B @ A )
% 2.18/1.46 => ( ( binintersect @ A @ B )
% 2.18/1.46 = B ) ) ) ) ).
% 2.18/1.46
% 2.18/1.46 thf(binintersectSubset1,conjecture,
% 2.18/1.46 ( setextAx
% 2.18/1.46 => ( emptysetAx
% 2.18/1.46 => ( setadjoinAx
% 2.18/1.46 => ( powersetAx
% 2.18/1.46 => ( setunionAx
% 2.18/1.46 => ( omega0Ax
% 2.18/1.46 => ( omegaSAx
% 2.18/1.46 => ( omegaIndAx
% 2.18/1.46 => ( replAx
% 2.18/1.46 => ( foundationAx
% 2.18/1.46 => ( wellorderingAx
% 2.18/1.46 => ( descrp
% 2.18/1.46 => ( dsetconstrI
% 2.18/1.46 => ( dsetconstrEL
% 2.18/1.46 => ( dsetconstrER
% 2.18/1.46 => ( exuE1
% 2.18/1.46 => ( prop2setE
% 2.18/1.46 => ( emptysetE
% 2.18/1.46 => ( emptysetimpfalse
% 2.18/1.46 => ( notinemptyset
% 2.18/1.46 => ( exuE3e
% 2.18/1.46 => ( setext
% 2.18/1.46 => ( emptyI
% 2.18/1.46 => ( noeltsimpempty
% 2.18/1.46 => ( setbeta
% 2.18/1.46 => ( nonemptyE1
% 2.18/1.46 => ( nonemptyI
% 2.18/1.46 => ( nonemptyI1
% 2.18/1.46 => ( setadjoinIL
% 2.18/1.46 => ( emptyinunitempty
% 2.18/1.46 => ( setadjoinIR
% 2.18/1.46 => ( setadjoinE
% 2.18/1.46 => ( setadjoinOr
% 2.18/1.46 => ( setoftrueEq
% 2.18/1.46 => ( powersetI
% 2.18/1.46 => ( emptyinPowerset
% 2.18/1.46 => ( emptyInPowerset
% 2.18/1.46 => ( powersetE
% 2.18/1.46 => ( setunionI
% 2.18/1.46 => ( setunionE
% 2.18/1.46 => ( subPowSU
% 2.18/1.46 => ( exuE2
% 2.18/1.46 => ( nonemptyImpWitness
% 2.18/1.46 => ( uniqinunit
% 2.18/1.46 => ( notinsingleton
% 2.18/1.46 => ( eqinunit
% 2.18/1.46 => ( singletonsswitch
% 2.18/1.46 => ( upairsetE
% 2.18/1.46 => ( upairsetIL
% 2.18/1.46 => ( upairsetIR
% 2.18/1.46 => ( emptyE1
% 2.18/1.46 => ( vacuousDall
% 2.18/1.46 => ( quantDeMorgan1
% 2.18/1.46 => ( quantDeMorgan2
% 2.18/1.46 => ( quantDeMorgan3
% 2.18/1.46 => ( quantDeMorgan4
% 2.18/1.46 => ( prop2setI
% 2.18/1.46 => ( prop2set2propI
% 2.18/1.46 => ( notdexE
% 2.18/1.46 => ( notdallE
% 2.18/1.46 => ( exuI1
% 2.18/1.46 => ( exuI3
% 2.18/1.46 => ( exuI2
% 2.18/1.46 => ( inCongP
% 2.18/1.46 => ( in__Cong
% 2.18/1.46 => ( exuE3u
% 2.18/1.46 => ( exu__Cong
% 2.18/1.46 => ( emptyset__Cong
% 2.18/1.46 => ( setadjoin__Cong
% 2.18/1.46 => ( powerset__Cong
% 2.18/1.46 => ( setunion__Cong
% 2.18/1.46 => ( omega__Cong
% 2.18/1.46 => ( exuEu
% 2.18/1.46 => ( descr__Cong
% 2.18/1.46 => ( dsetconstr__Cong
% 2.18/1.46 => ( subsetI1
% 2.18/1.46 => ( eqimpsubset2
% 2.18/1.46 => ( eqimpsubset1
% 2.18/1.46 => ( subsetI2
% 2.18/1.46 => ( emptysetsubset
% 2.18/1.46 => ( subsetE
% 2.18/1.46 => ( subsetE2
% 2.18/1.46 => ( notsubsetI
% 2.18/1.46 => ( notequalI1
% 2.18/1.46 => ( notequalI2
% 2.18/1.46 => ( subsetRefl
% 2.18/1.46 => ( subsetTrans
% 2.18/1.46 => ( setadjoinSub
% 2.18/1.46 => ( setadjoinSub2
% 2.18/1.46 => ( subset2powerset
% 2.18/1.46 => ( setextsub
% 2.18/1.46 => ( subsetemptysetimpeq
% 2.18/1.46 => ( powersetI1
% 2.18/1.46 => ( powersetE1
% 2.18/1.46 => ( inPowerset
% 2.18/1.46 => ( powersetsubset
% 2.18/1.46 => ( sepInPowerset
% 2.18/1.46 => ( sepSubset
% 2.18/1.46 => ( binunionIL
% 2.18/1.46 => ( upairset2IR
% 2.18/1.46 => ( binunionIR
% 2.18/1.46 => ( binunionEcases
% 2.18/1.46 => ( binunionE
% 2.18/1.46 => ( binunionLsub
% 2.18/1.46 => ( binunionRsub
% 2.18/1.46 => ( binintersectI
% 2.18/1.46 => ( binintersectSubset5
% 2.18/1.46 => ( binintersectEL
% 2.18/1.46 => ( binintersectLsub
% 2.18/1.46 => ( binintersectSubset2
% 2.18/1.46 => ( binintersectSubset3
% 2.18/1.46 => ( binintersectER
% 2.18/1.46 => ( disjointsetsI1
% 2.18/1.46 => ( binintersectRsub
% 2.18/1.46 => ( binintersectSubset4
% 2.18/1.46 => ! [A: $i,B: $i] :
% 2.18/1.46 ( ( ( binintersect @ A @ B )
% 2.18/1.46 = A )
% 2.18/1.46 => ( subset @ A @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.18/1.46 ------------------------
% 2.18/1.47 % (5243)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_4965 for (2999ds/275Mi)
% 2.18/1.47 % (5242)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_4965 for (2999ds/2Mi)
% 2.18/1.47 % (5242)Instruction limit reached!
% 2.18/1.47 % (5242)------------------------------
% 2.18/1.47 % (5242)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.47 % (5242)Termination reason: Unknown
% 2.18/1.47 % (5242)Termination phase: shuffling
% 2.18/1.47
% 2.18/1.47 % (5242)Memory used [KB]: 1151
% 2.18/1.47 % (5242)Time elapsed: 0.002 s
% 2.18/1.47 % (5242)Instructions burned: 3 (million)
% 2.18/1.47 % (5242)------------------------------
% 2.18/1.47 % (5242)------------------------------
% 2.18/1.48 % (5238)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_4965 for (2999ds/183Mi)
% 2.18/1.48 % (5239)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_4965 for (2999ds/4Mi)
% 2.18/1.48 % (5240)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_4965 for (2999ds/27Mi)
% 2.18/1.48 % (5241)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_4965 for (2999ds/2Mi)
% 2.18/1.48 % (5244)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_4965 for (2999ds/18Mi)
% 2.18/1.48 % (5239)Instruction limit reached!
% 2.18/1.48 % (5239)------------------------------
% 2.18/1.48 % (5239)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.48 % (5239)Termination reason: Unknown
% 2.18/1.48 % (5239)Termination phase: shuffling
% 2.18/1.48
% 2.18/1.48 % (5239)Memory used [KB]: 1151
% 2.18/1.48 % (5239)Time elapsed: 0.004 s
% 2.18/1.48 % (5239)Instructions burned: 4 (million)
% 2.18/1.48 % (5239)------------------------------
% 2.18/1.48 % (5239)------------------------------
% 2.18/1.48 % (5241)Instruction limit reached!
% 2.18/1.48 % (5241)------------------------------
% 2.18/1.48 % (5241)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.48 % (5241)Termination reason: Unknown
% 2.18/1.48 % (5241)Termination phase: shuffling
% 2.18/1.48
% 2.18/1.48 % (5241)Memory used [KB]: 1151
% 2.18/1.48 % (5241)Time elapsed: 0.004 s
% 2.18/1.48 % (5241)Instructions burned: 3 (million)
% 2.18/1.48 % (5241)------------------------------
% 2.18/1.48 % (5241)------------------------------
% 2.18/1.48 % (5245)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_4965 for (2999ds/3Mi)
% 2.18/1.48 % (5245)Instruction limit reached!
% 2.18/1.48 % (5245)------------------------------
% 2.18/1.48 % (5245)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.48 % (5245)Termination reason: Unknown
% 2.18/1.48 % (5245)Termination phase: shuffling
% 2.18/1.48
% 2.18/1.48 % (5245)Memory used [KB]: 1151
% 2.18/1.48 % (5245)Time elapsed: 0.003 s
% 2.18/1.48 % (5245)Instructions burned: 4 (million)
% 2.18/1.48 % (5245)------------------------------
% 2.18/1.48 % (5245)------------------------------
% 2.18/1.49 % (5244)Instruction limit reached!
% 2.18/1.49 % (5244)------------------------------
% 2.18/1.49 % (5244)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.49 % (5244)Termination reason: Unknown
% 2.18/1.49 % (5244)Termination phase: Preprocessing 3
% 2.18/1.49
% 2.18/1.49 % (5244)Memory used [KB]: 1535
% 2.18/1.49 % (5244)Time elapsed: 0.013 s
% 2.18/1.49 % (5244)Instructions burned: 19 (million)
% 2.18/1.49 % (5244)------------------------------
% 2.18/1.49 % (5244)------------------------------
% 2.18/1.49 % (5240)Instruction limit reached!
% 2.18/1.49 % (5240)------------------------------
% 2.18/1.49 % (5240)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.49 % (5240)Termination reason: Unknown
% 2.18/1.49 % (5240)Termination phase: Preprocessing 3
% 2.18/1.49
% 2.18/1.49 % (5240)Memory used [KB]: 1535
% 2.18/1.49 % (5240)Time elapsed: 0.016 s
% 2.18/1.49 % (5240)Instructions burned: 28 (million)
% 2.18/1.49 % (5240)------------------------------
% 2.18/1.49 % (5240)------------------------------
% 2.18/1.49 % (5248)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_4965 for (2999ds/3Mi)
% 2.18/1.50 % (5248)Instruction limit reached!
% 2.18/1.50 % (5248)------------------------------
% 2.18/1.50 % (5248)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.50 % (5248)Termination reason: Unknown
% 2.18/1.50 % (5248)Termination phase: shuffling
% 2.18/1.50
% 2.18/1.50 % (5248)Memory used [KB]: 1279
% 2.18/1.50 % (5248)Time elapsed: 0.003 s
% 2.18/1.50 % (5248)Instructions burned: 4 (million)
% 2.18/1.50 % (5248)------------------------------
% 2.18/1.50 % (5248)------------------------------
% 2.26/1.50 % (5246)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_4965 for (2999ds/37Mi)
% 2.26/1.50 % (5247)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_4965 for (2999ds/15Mi)
% 2.26/1.51 % (5251)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_4965 for (2999ds/16Mi)
% 2.26/1.51 % (5247)Instruction limit reached!
% 2.26/1.51 % (5247)------------------------------
% 2.26/1.51 % (5247)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.51 % (5247)Termination reason: Unknown
% 2.26/1.51 % (5247)Termination phase: shuffling
% 2.26/1.51
% 2.26/1.51 % (5247)Memory used [KB]: 1407
% 2.26/1.51 % (5247)Time elapsed: 0.011 s
% 2.26/1.51 % (5247)Instructions burned: 16 (million)
% 2.26/1.51 % (5247)------------------------------
% 2.26/1.51 % (5247)------------------------------
% 2.26/1.51 % (5249)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_4965 for (2999ds/1041Mi)
% 2.26/1.51 % (5250)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_4965 for (2999ds/7Mi)
% 2.26/1.51 % (5251)Instruction limit reached!
% 2.26/1.51 % (5251)------------------------------
% 2.26/1.51 % (5251)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.51 % (5251)Termination reason: Unknown
% 2.26/1.51 % (5251)Termination phase: Property scanning
% 2.26/1.51
% 2.26/1.51 % (5251)Memory used [KB]: 1535
% 2.26/1.51 % (5251)Time elapsed: 0.007 s
% 2.26/1.51 % (5251)Instructions burned: 18 (million)
% 2.26/1.51 % (5251)------------------------------
% 2.26/1.51 % (5251)------------------------------
% 2.26/1.51 % (5250)Instruction limit reached!
% 2.26/1.51 % (5250)------------------------------
% 2.26/1.51 % (5250)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.51 % (5250)Termination reason: Unknown
% 2.26/1.51 % (5250)Termination phase: shuffling
% 2.26/1.51
% 2.26/1.51 % (5250)Memory used [KB]: 1279
% 2.26/1.51 % (5250)Time elapsed: 0.006 s
% 2.26/1.51 % (5250)Instructions burned: 8 (million)
% 2.26/1.51 % (5250)------------------------------
% 2.26/1.51 % (5250)------------------------------
% 2.26/1.52 % (5246)Instruction limit reached!
% 2.26/1.52 % (5246)------------------------------
% 2.26/1.52 % (5246)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.52 % (5246)Termination reason: Unknown
% 2.26/1.52 % (5246)Termination phase: Property scanning
% 2.26/1.52
% 2.26/1.52 % (5246)Memory used [KB]: 1663
% 2.26/1.52 % (5246)Time elapsed: 0.020 s
% 2.26/1.52 % (5246)Instructions burned: 37 (million)
% 2.26/1.52 % (5246)------------------------------
% 2.26/1.52 % (5246)------------------------------
% 2.26/1.52 % (5253)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_4965 for (2999ds/3Mi)
% 2.26/1.52 % (5253)Instruction limit reached!
% 2.26/1.52 % (5253)------------------------------
% 2.26/1.52 % (5253)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.52 % (5253)Termination reason: Unknown
% 2.26/1.52 % (5253)Termination phase: shuffling
% 2.26/1.52
% 2.26/1.52 % (5253)Memory used [KB]: 1279
% 2.26/1.52 % (5253)Time elapsed: 0.002 s
% 2.26/1.52 % (5253)Instructions burned: 3 (million)
% 2.26/1.52 % (5253)------------------------------
% 2.26/1.52 % (5253)------------------------------
% 2.26/1.53 % (5252)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_4965 for (2999ds/3Mi)
% 2.26/1.53 % (5252)Instruction limit reached!
% 2.26/1.53 % (5252)------------------------------
% 2.26/1.53 % (5252)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.53 % (5252)Termination reason: Unknown
% 2.26/1.53 % (5252)Termination phase: shuffling
% 2.26/1.53
% 2.26/1.53 % (5252)Memory used [KB]: 1151
% 2.26/1.53 % (5252)Time elapsed: 0.004 s
% 2.26/1.53 % (5252)Instructions burned: 4 (million)
% 2.26/1.53 % (5252)------------------------------
% 2.26/1.53 % (5252)------------------------------
% 2.26/1.53 % (5254)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_4965 for (2999ds/7Mi)
% 2.26/1.53 % (5256)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_4965 for (2999ds/4Mi)
% 2.26/1.53 % (5256)Instruction limit reached!
% 2.26/1.53 % (5256)------------------------------
% 2.26/1.53 % (5256)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.53 % (5256)Termination reason: Unknown
% 2.26/1.53 % (5256)Termination phase: shuffling
% 2.26/1.53
% 2.26/1.53 % (5256)Memory used [KB]: 1279
% 2.26/1.53 % (5256)Time elapsed: 0.003 s
% 2.26/1.53 % (5256)Instructions burned: 7 (million)
% 2.26/1.53 % (5256)------------------------------
% 2.26/1.53 % (5256)------------------------------
% 2.26/1.53 % (5254)Instruction limit reached!
% 2.26/1.53 % (5254)------------------------------
% 2.26/1.53 % (5254)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.53 % (5254)Termination reason: Unknown
% 2.26/1.53 % (5254)Termination phase: shuffling
% 2.26/1.53
% 2.26/1.53 % (5254)Memory used [KB]: 1279
% 2.26/1.53 % (5254)Time elapsed: 0.004 s
% 2.26/1.53 % (5254)Instructions burned: 7 (million)
% 2.26/1.53 % (5254)------------------------------
% 2.26/1.53 % (5254)------------------------------
% 2.26/1.53 % (5255)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_4965 for (2999ds/3Mi)
% 2.26/1.53 % (5243)First to succeed.
% 2.26/1.54 % (5255)Instruction limit reached!
% 2.26/1.54 % (5255)------------------------------
% 2.26/1.54 % (5255)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.54 % (5255)Termination reason: Unknown
% 2.26/1.54 % (5255)Termination phase: shuffling
% 2.26/1.54
% 2.26/1.54 % (5255)Memory used [KB]: 1279
% 2.26/1.54 % (5255)Time elapsed: 0.004 s
% 2.26/1.54 % (5255)Instructions burned: 4 (million)
% 2.26/1.54 % (5255)------------------------------
% 2.26/1.54 % (5255)------------------------------
% 2.26/1.54 % (5243)Refutation found. Thanks to Tanya!
% 2.26/1.54 % SZS status Theorem for DTF2THF_4965
% 2.26/1.54 % SZS output start Proof for DTF2THF_4965
% 2.26/1.54 thf(func_def_0, type, in: $i > $i > $o).
% 2.26/1.54 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.26/1.54 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.26/1.54 thf(func_def_8, type, powerset: $i > $i).
% 2.26/1.54 thf(func_def_10, type, setunion: $i > $i).
% 2.26/1.54 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.26/1.54 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.26/1.54 thf(func_def_26, type, prop2set: $o > $i).
% 2.26/1.54 thf(func_def_36, type, nonempty: $i > $o).
% 2.26/1.54 thf(func_def_69, type, set2prop: $i > $o).
% 2.26/1.54 thf(func_def_88, type, subset: $i > $i > $o).
% 2.26/1.54 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.26/1.54 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.26/1.54 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.26/1.54 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.26/1.54 thf(func_def_145, type, sP0: $i > $i > $o).
% 2.26/1.54 thf(func_def_146, type, sP1: $i > $o).
% 2.26/1.54 thf(func_def_147, type, sP2: $i > $i > $o).
% 2.26/1.54 thf(func_def_148, type, sP3: $i > $i > $o).
% 2.26/1.54 thf(func_def_155, type, sK10: $i > ($i > $o) > $i).
% 2.26/1.54 thf(func_def_156, type, sK11: $i > $o).
% 2.26/1.54 thf(func_def_161, type, sK16: $i > $i > $i).
% 2.26/1.54 thf(func_def_165, type, sK20: $i > $i).
% 2.26/1.54 thf(func_def_172, type, sK27: $i > $i > $i).
% 2.26/1.54 thf(func_def_177, type, sK32: $o > $i > $i > $i).
% 2.26/1.54 thf(func_def_180, type, sK35: $i > $o).
% 2.26/1.54 thf(func_def_196, type, sK51: ($i > $o) > $i > $i).
% 2.26/1.54 thf(func_def_198, type, sK53: $i > $o).
% 2.26/1.54 thf(func_def_202, type, sK57: $i > $i).
% 2.26/1.54 thf(func_def_213, type, sK68: $i > $o).
% 2.26/1.54 thf(func_def_216, type, sK71: $i > ($i > $o) > $i).
% 2.26/1.54 thf(func_def_217, type, sK72: $i > $o).
% 2.26/1.54 thf(func_def_220, type, sK75: $i > $i > $i).
% 2.26/1.54 thf(func_def_221, type, sK76: $i > $i > $i).
% 2.26/1.54 thf(func_def_222, type, sK77: $i > $i > $i).
% 2.26/1.54 thf(func_def_223, type, sK78: $i > $i > $i).
% 2.26/1.54 thf(func_def_224, type, sK79: $i > $i > $i).
% 2.26/1.54 thf(func_def_225, type, sK80: $i > $i > $i > $i).
% 2.26/1.54 thf(func_def_226, type, sK81: $i > $i).
% 2.26/1.54 thf(func_def_227, type, sK82: $i > $i).
% 2.26/1.54 thf(func_def_228, type, sK83: $i > $i).
% 2.26/1.54 thf(func_def_229, type, sK84: $i > $i).
% 2.26/1.54 thf(func_def_230, type, sK85: $i > $i > $i).
% 2.26/1.54 thf(func_def_231, type, sK86: $i > $i > $i > $i).
% 2.26/1.54 thf(func_def_232, type, sK87: $i > $i > $i > $i).
% 2.26/1.54 thf(func_def_233, type, sK88: $i > $i > $i).
% 2.26/1.54 thf(func_def_234, type, sK89: $i > $i > $i).
% 2.26/1.54 thf(func_def_236, type, sK91: $i > $i).
% 2.26/1.54 thf(func_def_237, type, sK92: $i > $i).
% 2.26/1.54 thf(func_def_238, type, sK93: $i > $i).
% 2.26/1.54 thf(func_def_239, type, sK94: $i > $i).
% 2.26/1.54 thf(func_def_244, type, sK99: $i > $i > $i).
% 2.26/1.54 thf(func_def_250, type, sK105: $i > $o).
% 2.26/1.54 thf(func_def_251, type, sK106: $i > $i).
% 2.26/1.54 thf(func_def_252, type, sK107: ($i > $o) > $i).
% 2.26/1.54 thf(func_def_261, type, sK116: $i > $o).
% 2.26/1.54 thf(func_def_262, type, sK117: ($i > $o) > $i).
% 2.26/1.54 thf(func_def_264, type, sK119: $i > $o).
% 2.26/1.54 thf(func_def_269, type, sK124: $i > $o).
% 2.26/1.54 thf(func_def_274, type, sK129: $i > $o).
% 2.26/1.54 thf(func_def_280, type, sK135: $i > $i > $i).
% 2.26/1.54 thf(func_def_287, type, sK142: $i > $o).
% 2.26/1.54 thf(func_def_288, type, sK143: $i > $o).
% 2.26/1.54 thf(func_def_289, type, sK144: ($i > $o) > ($i > $o) > $i).
% 2.26/1.54 thf(func_def_290, type, sK145: ($i > $o) > ($i > $o) > $i).
% 2.26/1.54 thf(func_def_292, type, sK147: $i > $o).
% 2.26/1.54 thf(func_def_297, type, sK152: $i > $o).
% 2.26/1.54 thf(func_def_302, type, sK157: ($i > $o) > $i).
% 2.26/1.54 thf(func_def_303, type, sK158: $i > $o).
% 2.26/1.54 thf(func_def_304, type, sK159: $i > $i).
% 2.26/1.54 thf(func_def_312, type, sK167: $i > $o).
% 2.26/1.54 thf(func_def_314, type, sK169: ($i > $o) > $i > $i).
% 2.26/1.54 thf(func_def_318, type, sK173: $i > $i > $i).
% 2.26/1.54 thf(func_def_321, type, sK176: $i > $i > $o).
% 2.26/1.54 thf(func_def_323, type, sK178: $i > $i).
% 2.26/1.54 thf(func_def_324, type, sK179: $i > $i).
% 2.26/1.54 thf(func_def_325, type, sK180: $i > ($i > $i > $o) > $i).
% 2.26/1.54 thf(func_def_326, type, sK181: $i > $i > ($i > $i > $o) > $i).
% 2.26/1.54 thf(func_def_327, type, sK182: $i > ($i > $i > $o) > $i).
% 2.26/1.54 thf(func_def_328, type, sK183: $i > $o).
% 2.26/1.54 thf(func_def_333, type, sK188: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.26/1.54 thf(func_def_334, type, sK189: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.26/1.54 thf(func_def_337, type, sK192: $i > $o).
% 2.26/1.54 thf(func_def_338, type, sK193: $i > $o).
% 2.26/1.54 thf(func_def_345, type, sK200: $i > $i).
% 2.26/1.54 thf(func_def_346, type, sK201: $i > $i).
% 2.26/1.54 thf(func_def_365, type, sK220: ($i > $o) > $i > $i).
% 2.26/1.54 thf(func_def_367, type, sK222: $i > $o).
% 2.26/1.54 thf(func_def_372, type, sK227: $i > $i).
% 2.26/1.54 thf(func_def_378, type, sK233: $i > $o).
% 2.26/1.54 thf(func_def_383, type, sK238: ($i > $o) > ($i > $o) > $i).
% 2.26/1.54 thf(func_def_384, type, sK239: ($i > $o) > ($i > $o) > $i).
% 2.26/1.54 thf(func_def_385, type, sK240: $i > $o).
% 2.26/1.54 thf(func_def_386, type, sK241: $i > $o).
% 2.26/1.54 thf(func_def_391, type, sK246: $i > $o).
% 2.26/1.54 thf(func_def_394, type, sK249: $i > ($i > $o) > $i).
% 2.26/1.54 thf(func_def_398, type, sK253: $i > $i > $i).
% 2.26/1.54 thf(func_def_407, type, sK262: $i > $o).
% 2.26/1.54 thf(func_def_410, type, sK265: $i > ($i > $o) > $i).
% 2.26/1.54 thf(func_def_411, type, sK266: $i > $o).
% 2.26/1.54 thf(func_def_427, type, sK282: $i > $o).
% 2.26/1.54 thf(func_def_428, type, sK283: $i > $o).
% 2.26/1.54 thf(func_def_436, type, sK291: $i > $o).
% 2.26/1.54 thf(func_def_438, type, sK293: ($i > $o) > $i).
% 2.26/1.54 thf(func_def_439, type, sK294: ($i > $o) > $i).
% 2.26/1.54 thf(func_def_446, type, sK301: $i > $o).
% 2.26/1.54 thf(func_def_448, type, sK303: $i > ($i > $o) > $i).
% 2.26/1.54 thf(func_def_451, type, sK306: $i > $i > $i).
% 2.26/1.54 thf(func_def_452, type, sK307: $i > $i > $i).
% 2.26/1.54 thf(func_def_456, type, sK311: $i > $i).
% 2.26/1.54 thf(func_def_459, type, sK314: $i > $i > $i).
% 2.26/1.54 thf(func_def_461, type, ph316: !>[X0: $tType]:(X0)).
% 2.26/1.54 thf(f2541,plain,(
% 2.26/1.54 $false),
% 2.26/1.54 inference(subsumption_resolution,[status(thm)],[f2537,f1239])).
% 2.26/1.54 thf(f1239,plain,(
% 2.26/1.54 ($true != (subset @ sK66 @ sK67))),
% 2.26/1.54 inference(cnf_transformation,[status(thm)],[f618])).
% 2.26/1.54 thf(f618,plain,(
% 2.26/1.54 (dsetconstrEL = $true) & (subset2powerset = $true) & (emptyE1 = $true) & ((sK66 = (binintersect @ sK66 @ sK67)) & ($true != (subset @ sK66 @ sK67))) & (eqimpsubset2 = $true) & (binintersectRsub = $true) & (binunionRsub = $true) & (omegaSAx = $true) & (quantDeMorgan4 = $true) & (inCongP = $true) & (nonemptyI = $true) & (powersetsubset = $true) & (replAx = $true) & (emptyinPowerset = $true) & (powersetE1 = $true) & (setunionAx = $true) & (notequalI2 = $true) & (notdexE = $true) & (binintersectSubset5 = $true) & (disjointsetsI1 = $true) & (notsubsetI = $true) & (setunion__Cong = $true) & (binintersectSubset2 = $true) & (in__Cong = $true) & (subsetRefl = $true) & (setadjoinIR = $true) & (dsetconstrER = $true) & (subsetI1 = $true) & (exuI3 = $true) & (setadjoinSub = $true) & (subsetE = $true) & (wellorderingAx = $true) & (setunionI = $true) & (emptysetimpfalse = $true) & (nonemptyE1 = $true) & (sepSubset = $true) & (foundationAx = $true) & (powersetAx = $true) & (emptyInPowerset = $true) & (singletonsswitch = $true) & (binintersectEL = $true) & (inPowerset = $true) & (subsetE2 = $true) & (binunionEcases = $true) & (setbeta = $true) & (notdallE = $true) & (setadjoinAx = $true) & (omega__Cong = $true) & (prop2set2propI = $true) & (sepInPowerset = $true) & (exuI1 = $true) & (binunionIL = $true) & (nonemptyI1 = $true) & (subsetI2 = $true) & (setadjoinSub2 = $true) & (eqinunit = $true) & (setunionE = $true) & (notequalI1 = $true) & (notinsingleton = $true) & (exuI2 = $true) & (exu__Cong = $true) & (exuEu = $true) & (descrp = $true) & (upairsetIL = $true) & (exuE2 = $true) & (nonemptyImpWitness = $true) & (binunionE = $true) & (exuE3u = $true) & (notinemptyset = $true) & (subsetemptysetimpeq = $true) & (subsetTrans = $true) & (setadjoin__Cong = $true) & (emptyinunitempty = $true) & (powersetI1 = $true) & (quantDeMorgan3 = $true) & (binintersectER = $true) & (dsetconstr__Cong = $true) & (exuE1 = $true) & (binintersectSubset3 = $true) & (emptyI = $true) & (binunionIR = $true) & (descr__Cong = $true) & (omegaIndAx = $true) & (binintersectSubset4 = $true) & (powersetE = $true) & (emptyset__Cong = $true) & (emptysetAx = $true) & (powerset__Cong = $true) & (noeltsimpempty = $true) & (setadjoinOr = $true) & (setoftrueEq = $true) & (setadjoinIL = $true) & (quantDeMorgan2 = $true) & (upairsetIR = $true) & (upairsetE = $true) & (setextAx = $true) & (exuE3e = $true) & (omega0Ax = $true) & (prop2setI = $true) & (vacuousDall = $true) & (binintersectLsub = $true) & (binintersectI = $true) & (prop2setE = $true) & (setextsub = $true) & (eqimpsubset1 = $true) & (setadjoinE = $true) & (subPowSU = $true) & (setext = $true) & (quantDeMorgan1 = $true) & (dsetconstrI = $true) & (emptysetsubset = $true) & (uniqinunit = $true) & (binunionLsub = $true) & (emptysetE = $true) & (powersetI = $true) & (upairset2IR = $true)),
% 2.26/1.54 inference(skolemisation,[status(esa),new_symbols(skolem,[sK66,sK67])],[f490,f617])).
% 2.26/1.54 thf(f617,plain,(
% 2.26/1.54 ? [X0 : $i,X1 : $i] : (((binintersect @ X0 @ X1) = X0) & ($true != (subset @ X0 @ X1))) => ((sK66 = (binintersect @ sK66 @ sK67)) & ($true != (subset @ sK66 @ sK67)))),
% 2.26/1.54 introduced(definition,[],[choice_axiom])).
% 2.26/1.54 thf(f490,plain,(
% 2.26/1.54 (dsetconstrEL = $true) & (subset2powerset = $true) & (emptyE1 = $true) & ? [X0 : $i,X1 : $i] : (((binintersect @ X0 @ X1) = X0) & ($true != (subset @ X0 @ X1))) & (eqimpsubset2 = $true) & (binintersectRsub = $true) & (binunionRsub = $true) & (omegaSAx = $true) & (quantDeMorgan4 = $true) & (inCongP = $true) & (nonemptyI = $true) & (powersetsubset = $true) & (replAx = $true) & (emptyinPowerset = $true) & (powersetE1 = $true) & (setunionAx = $true) & (notequalI2 = $true) & (notdexE = $true) & (binintersectSubset5 = $true) & (disjointsetsI1 = $true) & (notsubsetI = $true) & (setunion__Cong = $true) & (binintersectSubset2 = $true) & (in__Cong = $true) & (subsetRefl = $true) & (setadjoinIR = $true) & (dsetconstrER = $true) & (subsetI1 = $true) & (exuI3 = $true) & (setadjoinSub = $true) & (subsetE = $true) & (wellorderingAx = $true) & (setunionI = $true) & (emptysetimpfalse = $true) & (nonemptyE1 = $true) & (sepSubset = $true) & (foundationAx = $true) & (powersetAx = $true) & (emptyInPowerset = $true) & (singletonsswitch = $true) & (binintersectEL = $true) & (inPowerset = $true) & (subsetE2 = $true) & (binunionEcases = $true) & (setbeta = $true) & (notdallE = $true) & (setadjoinAx = $true) & (omega__Cong = $true) & (prop2set2propI = $true) & (sepInPowerset = $true) & (exuI1 = $true) & (binunionIL = $true) & (nonemptyI1 = $true) & (subsetI2 = $true) & (setadjoinSub2 = $true) & (eqinunit = $true) & (setunionE = $true) & (notequalI1 = $true) & (notinsingleton = $true) & (exuI2 = $true) & (exu__Cong = $true) & (exuEu = $true) & (descrp = $true) & (upairsetIL = $true) & (exuE2 = $true) & (nonemptyImpWitness = $true) & (binunionE = $true) & (exuE3u = $true) & (notinemptyset = $true) & (subsetemptysetimpeq = $true) & (subsetTrans = $true) & (setadjoin__Cong = $true) & (emptyinunitempty = $true) & (powersetI1 = $true) & (quantDeMorgan3 = $true) & (binintersectER = $true) & (dsetconstr__Cong = $true) & (exuE1 = $true) & (binintersectSubset3 = $true) & (emptyI = $true) & (binunionIR = $true) & (descr__Cong = $true) & (omegaIndAx = $true) & (binintersectSubset4 = $true) & (powersetE = $true) & (emptyset__Cong = $true) & (emptysetAx = $true) & (powerset__Cong = $true) & (noeltsimpempty = $true) & (setadjoinOr = $true) & (setoftrueEq = $true) & (setadjoinIL = $true) & (quantDeMorgan2 = $true) & (upairsetIR = $true) & (upairsetE = $true) & (setextAx = $true) & (exuE3e = $true) & (omega0Ax = $true) & (prop2setI = $true) & (vacuousDall = $true) & (binintersectLsub = $true) & (binintersectI = $true) & (prop2setE = $true) & (setextsub = $true) & (eqimpsubset1 = $true) & (setadjoinE = $true) & (subPowSU = $true) & (setext = $true) & (quantDeMorgan1 = $true) & (dsetconstrI = $true) & (emptysetsubset = $true) & (uniqinunit = $true) & (binunionLsub = $true) & (emptysetE = $true) & (powersetI = $true) & (upairset2IR = $true)),
% 2.26/1.54 inference(flattening,[status(thm)],[f489])).
% 2.26/1.54 thf(f489,plain,(
% 2.26/1.54 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (((binintersect @ X0 @ X1) = X0) & ($true != (subset @ X0 @ X1))) & (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)),
% 2.26/1.54 inference(ennf_transformation,[status(thm)],[f126])).
% 2.26/1.54 thf(f126,plain,(
% 2.26/1.54 ~((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) => ! [X0 : $i,X1 : $i] : (((binintersect @ X0 @ X1) = X0) => ($true = (subset @ X0 @ X1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.26/1.54 inference(fool_elimination,[status(thm)],[f125])).
% 2.26/1.54 thf(f125,plain,(
% 2.26/1.54 ~(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 => ! [X0 : $i,X1 : $i] : (((binintersect @ X0 @ X1) = X0) => (subset @ X0 @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.26/1.54 inference(rectify,[status(thm)],[f119])).
% 2.26/1.54 thf(f119,negated_conjecture,(
% 2.26/1.54 ~(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 => ! [X3 : $i,X4 : $i] : (((binintersect @ X3 @ X4) = X3) => (subset @ X3 @ X4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.26/1.54 inference(negated_conjecture,[status(cth)],[f118])).
% 2.26/1.54 thf(f118,conjecture,(
% 2.26/1.54 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 => ! [X3 : $i,X4 : $i] : (((binintersect @ X3 @ X4) = X3) => (subset @ X3 @ X4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.26/1.54 file('/export/starexec/sandbox/tmp/tmp.5ZrNFR5hgZ/DTF2THF_4965.p',binintersectSubset1)).
% 2.26/1.54 thf(f2537,plain,(
% 2.26/1.54 ($true = (subset @ sK66 @ sK67))),
% 2.26/1.54 inference(superposition,[status(thm)],[f2164,f1240])).
% 2.26/1.54 thf(f1240,plain,(
% 2.26/1.54 (sK66 = (binintersect @ sK66 @ sK67))),
% 2.26/1.54 inference(cnf_transformation,[status(thm)],[f618])).
% 2.26/1.54 thf(f2164,plain,(
% 2.26/1.54 ( ! [X2 : $i,X3 : $i] : (($true = (subset @ (binintersect @ X2 @ X3) @ X3))) )),
% 2.26/1.54 inference(trivial_inequality_removal,[status(thm)],[f1986])).
% 2.26/1.54 thf(f1986,plain,(
% 2.26/1.54 ( ! [X2 : $i,X3 : $i] : (($true != $true) | ($true = (subset @ (binintersect @ X2 @ X3) @ X3))) )),
% 2.26/1.54 inference(definition_unfolding,[status(thm)],[f1575,f1237])).
% 2.26/1.54 thf(f1237,plain,(
% 2.26/1.54 (binintersectRsub = $true)),
% 2.26/1.54 inference(cnf_transformation,[status(thm)],[f618])).
% 2.26/1.54 thf(f1575,plain,(
% 2.26/1.54 ( ! [X2 : $i,X3 : $i] : (($true = (subset @ (binintersect @ X2 @ X3) @ X3)) | (binintersectRsub != $true)) )),
% 2.26/1.54 inference(cnf_transformation,[status(thm)],[f1003])).
% 2.26/1.54 thf(f1003,plain,(
% 2.26/1.54 ((binintersectRsub = $true) | ((subset @ (binintersect @ sK295 @ sK296) @ sK296) != $true)) & (! [X2 : $i,X3 : $i] : ($true = (subset @ (binintersect @ X2 @ X3) @ X3)) | (binintersectRsub != $true))),
% 2.26/1.54 inference(skolemisation,[status(esa),new_symbols(skolem,[sK295,sK296])],[f1001,f1002])).
% 2.26/1.54 thf(f1002,plain,(
% 2.26/1.54 ? [X0 : $i,X1 : $i] : ((subset @ (binintersect @ X0 @ X1) @ X1) != $true) => ((subset @ (binintersect @ sK295 @ sK296) @ sK296) != $true)),
% 2.26/1.54 introduced(definition,[],[choice_axiom])).
% 2.26/1.54 thf(f1001,plain,(
% 2.26/1.54 ((binintersectRsub = $true) | ? [X0 : $i,X1 : $i] : ((subset @ (binintersect @ X0 @ X1) @ X1) != $true)) & (! [X2 : $i,X3 : $i] : ($true = (subset @ (binintersect @ X2 @ X3) @ X3)) | (binintersectRsub != $true))),
% 2.26/1.54 inference(rectify,[status(thm)],[f1000])).
% 2.26/1.54 thf(f1000,plain,(
% 2.26/1.54 ((binintersectRsub = $true) | ? [X0 : $i,X1 : $i] : ((subset @ (binintersect @ X0 @ X1) @ X1) != $true)) & (! [X0 : $i,X1 : $i] : ((subset @ (binintersect @ X0 @ X1) @ X1) = $true) | (binintersectRsub != $true))),
% 2.26/1.54 inference(nnf_transformation,[status(thm)],[f160])).
% 2.26/1.54 thf(f160,plain,(
% 2.26/1.54 (binintersectRsub = $true) <=> ! [X0 : $i,X1 : $i] : ((subset @ (binintersect @ X0 @ X1) @ X1) = $true)),
% 2.26/1.54 inference(fool_elimination,[status(thm)],[f159])).
% 2.26/1.54 thf(f159,plain,(
% 2.26/1.54 (binintersectRsub = ! [X0 : $i,X1 : $i] : (subset @ (binintersect @ X0 @ X1) @ X1))),
% 2.26/1.54 inference(rectify,[status(thm)],[f116])).
% 2.26/1.54 thf(f116,axiom,(
% 2.26/1.54 (binintersectRsub = ! [X3 : $i,X4 : $i] : (subset @ (binintersect @ X3 @ X4) @ X4))),
% 2.26/1.54 file('/export/starexec/sandbox/tmp/tmp.5ZrNFR5hgZ/DTF2THF_4965.p',binintersectRsub)).
% 2.26/1.54 % SZS output end Proof for DTF2THF_4965
% 2.26/1.54 % (5243)------------------------------
% 2.26/1.54 % (5243)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.54 % (5243)Termination reason: Refutation
% 2.26/1.54
% 2.26/1.54 % (5243)Memory used [KB]: 7675
% 2.26/1.54 % (5243)Time elapsed: 0.066 s
% 2.26/1.54 % (5243)Instructions burned: 201 (million)
% 2.26/1.54 % (5243)------------------------------
% 2.26/1.54 % (5243)------------------------------
% 2.26/1.54 % (5237)Success in time 0.07 s
%------------------------------------------------------------------------------