%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU616^1 : 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 : n011.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 09:35:38 AM UTC 2026
% Result : Theorem 2.66s 1.75s
% Output : Refutation 2.66s
% Verified :
% SZS Type : Refutation
% Derivation depth : 10
% Number of leaves : 4
% Syntax : Number of formulae : 21 ( 6 unt; 0 typ; 0 def)
% Number of atoms : 2116 ( 559 equ; 0 cnn)
% Maximal formula atoms : 134 ( 100 avg)
% Number of connectives : 1101 ( 19 ~; 8 |; 399 &; 144 @)
% ( 1 <=>; 530 =>; 0 <=; 0 <~>)
% Maximal formula depth : 140 ( 48 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 257 ( 254 usr; 139 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_135,type,
regular: $i > $o ).
thf(func_def_136,type,
setminus: $i > $i > $i ).
thf(func_def_147,type,
symdiff: $i > $i > $i ).
thf(func_def_153,type,
iskpair: $i > $o ).
thf(func_def_167,type,
sP1: $o > $i > $i > $i > $o ).
thf(func_def_168,type,
sP2: $i > $i > $o ).
thf(func_def_169,type,
sP3: $i > $o ).
thf(func_def_170,type,
sP4: $i > $i > $o ).
thf(func_def_171,type,
sP5: $i > $i > $o ).
thf(func_def_175,type,
sK9: $i > $i > $i ).
thf(func_def_181,type,
sK15: $i > $o ).
thf(func_def_191,type,
sK25: ( $i > $o ) > $i > $i ).
thf(func_def_193,type,
sK27: $i > $o ).
thf(func_def_197,type,
sK31: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_198,type,
sK32: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_199,type,
sK33: $i > $o ).
thf(func_def_200,type,
sK34: $i > $o ).
thf(func_def_205,type,
sK39: $i > $o ).
thf(func_def_207,type,
sK41: $i > $i ).
thf(func_def_210,type,
sK44: $i > $o ).
thf(func_def_215,type,
sK49: ( $i > $o ) > $i ).
thf(func_def_216,type,
sK50: $i > $o ).
thf(func_def_217,type,
sK51: $i > $i ).
thf(func_def_230,type,
sK64: $i > $i > $o ).
thf(func_def_231,type,
sK65: $i > $i ).
thf(func_def_232,type,
sK66: $i > $i ).
thf(func_def_233,type,
sK67: ( $i > $i > $o ) > $i > $i ).
thf(func_def_234,type,
sK68: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_235,type,
sK69: ( $i > $i > $o ) > $i > $i ).
thf(func_def_243,type,
sK77: $i > $o ).
thf(func_def_245,type,
sK79: ( $i > $o ) > $i > $i ).
thf(func_def_246,type,
sK80: ( $i > $o ) > $i > $i ).
thf(func_def_248,type,
sK82: $i > $o ).
thf(func_def_255,type,
sK89: $i > $i > $i ).
thf(func_def_258,type,
sK92: $i > $o ).
thf(func_def_262,type,
sK96: $i > $i ).
thf(func_def_265,type,
sK99: $i > $i ).
thf(func_def_278,type,
sK112: $i > $i > $i ).
thf(func_def_281,type,
sK115: $i > $o ).
thf(func_def_283,type,
sK117: $i > ( $i > $o ) > $i ).
thf(func_def_291,type,
sK125: $i > $i ).
thf(func_def_293,type,
sK127: $i > $i > $i ).
thf(func_def_310,type,
sK144: $i > $o ).
thf(func_def_314,type,
sK148: $i > $o ).
thf(func_def_319,type,
sK153: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_320,type,
sK154: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_321,type,
sK155: $i > $o ).
thf(func_def_322,type,
sK156: $i > $o ).
thf(func_def_325,type,
sK159: $i > $i > $i ).
thf(func_def_336,type,
sK170: $i > $o ).
thf(func_def_337,type,
sK171: $i > $o ).
thf(func_def_338,type,
sK172: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_339,type,
sK173: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_347,type,
sK181: $i > $i ).
thf(func_def_365,type,
sK199: $i > $i > $i ).
thf(func_def_369,type,
sK203: $i > $o ).
thf(func_def_384,type,
sK218: $i > $o ).
thf(func_def_392,type,
sK226: $i > $o ).
thf(func_def_393,type,
sK227: ( $i > $o ) > $i ).
thf(func_def_394,type,
sK228: $i > $i ).
thf(func_def_396,type,
sK230: $i > ( $i > $o ) > $i ).
thf(func_def_397,type,
sK231: $i > $o ).
thf(func_def_400,type,
sK234: $i > $o ).
thf(func_def_402,type,
sK236: $i > ( $i > $o ) > $i ).
thf(func_def_418,type,
sK252: $i > $i > $i ).
thf(func_def_419,type,
sK253: $i > $i > $i ).
thf(func_def_420,type,
sK254: $i > $o ).
thf(func_def_423,type,
sK257: $i > ( $i > $o ) > $i ).
thf(func_def_425,type,
sK259: ( $i > $o ) > $i ).
thf(func_def_426,type,
sK260: $i > $o ).
thf(func_def_427,type,
sK261: $i > $i ).
thf(func_def_433,type,
sK267: $i > $o ).
thf(func_def_441,type,
sK275: ( $i > $o ) > $i > $i ).
thf(func_def_443,type,
sK277: $i > $o ).
thf(func_def_447,type,
sK281: $i > $i > $i ).
thf(func_def_451,type,
sK285: $o > $i > $i > $i ).
thf(func_def_452,type,
sK286: $i > $i > $i ).
thf(func_def_453,type,
sK287: $i > $i > $i ).
thf(func_def_454,type,
sK288: $i > $i > $i ).
thf(func_def_455,type,
sK289: $i > $i > $i ).
thf(func_def_456,type,
sK290: $i > $i > $i ).
thf(func_def_457,type,
sK291: $i > $i > $i > $i ).
thf(func_def_458,type,
sK292: $i > $i ).
thf(func_def_459,type,
sK293: $i > $i ).
thf(func_def_460,type,
sK294: $i > $i ).
thf(func_def_461,type,
sK295: $i > $i ).
thf(func_def_462,type,
sK296: $i > $i > $i > $i ).
thf(func_def_463,type,
sK297: $i > $i > $i > $i ).
thf(func_def_464,type,
sK298: $i > $i > $i ).
thf(func_def_465,type,
sK299: $i > $i > $i ).
thf(func_def_466,type,
sK300: $i > $i > $i ).
thf(func_def_468,type,
sK302: $i > $i ).
thf(func_def_469,type,
sK303: $i > $i ).
thf(func_def_470,type,
sK304: $i > $i ).
thf(func_def_473,type,
sK307: $i > $o ).
thf(func_def_490,type,
sK324: $i > $o ).
thf(func_def_495,type,
sK329: ( $i > $o ) > $i ).
thf(func_def_496,type,
sK330: ( $i > $o ) > $i ).
thf(func_def_497,type,
sK331: $i > $o ).
thf(func_def_501,type,
sK335: $i > $o ).
thf(func_def_503,type,
sK337: $i > $i ).
thf(func_def_531,type,
ph365:
!>[X0: $tType] : X0 ).
thf(f2561,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1499,f2496]) ).
thf(f2496,plain,
! [X0: $i,X1: $i] :
( $true
= ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2065]) ).
thf(f2065,plain,
! [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f1562,f1515]) ).
thf(f1515,plain,
upairset2IR = $true,
inference(cnf_transformation,[status(thm)],[f807]) ).
thf(f807,plain,
( ( dsetconstrI = $true )
& ( binintersectI = $true )
& ( upairset2IR = $true )
& ( setminusI = $true )
& ( singletonsswitch = $true )
& ( binunionLsub = $true )
& ( exuE2 = $true )
& ( binintersectSubset2 = $true )
& ( emptysetAx = $true )
& ( emptyE1 = $true )
& ( setadjoinIR = $true )
& ( setunionAx = $true )
& ( emptyinPowerset = $true )
& ( setunionI = $true )
& ( subsetTrans = $true )
& ( nonemptyE1 = $true )
& ( noeltsimpempty = $true )
& ( eqinunit = $true )
& ( ( in @ sK134 @ ( setadjoin @ sK135 @ ( setadjoin @ sK134 @ emptyset ) ) )
!= $true )
& ( emptyInPowerset = $true )
& ( binunionEcases = $true )
& ( symdiffE = $true )
& ( notsubsetI = $true )
& ( omega0Ax = $true )
& ( upairsetIR = $true )
& ( foundationAx = $true )
& ( omega__Cong = $true )
& ( dsetconstr__Cong = $true )
& ( emptyI = $true )
& ( binintersectER = $true )
& ( emptyinunitempty = $true )
& ( binintersectSubset5 = $true )
& ( nonemptyI1 = $true )
& ( exuE3e = $true )
& ( setminusILneg = $true )
& ( prop2setE = $true )
& ( setext = $true )
& ( setunion__Cong = $true )
& ( notequalI2 = $true )
& ( powersetI = $true )
& ( notinsingleton = $true )
& ( powerset__Cong = $true )
& ( setminusELneg = $true )
& ( setadjoinSub = $true )
& ( setoftrueEq = $true )
& ( setminusSubset1 = $true )
& ( dsetconstrER = $true )
& ( exuE3u = $true )
& ( setminusLsub = $true )
& ( powersetAx = $true )
& ( disjointsetsI1 = $true )
& ( setminusER = $true )
& ( upairsetIL = $true )
& ( powersetE1 = $true )
& ( quantDeMorgan2 = $true )
& ( powersetE = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( notinemptyset = $true )
& ( inPowerset = $true )
& ( descrp = $true )
& ( binunionIL = $true )
& ( inCongP = $true )
& ( upairsetE = $true )
& ( exuI1 = $true )
& ( symdiffIneg2 = $true )
& ( emptysetE = $true )
& ( subsetE = $true )
& ( subsetI2 = $true )
& ( binintersectLsub = $true )
& ( symdiffIneg1 = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( exu__Cong = $true )
& ( dsetconstrEL = $true )
& ( setadjoin__Cong = $true )
& ( eqimpsubset2 = $true )
& ( setminusSubset2 = $true )
& ( replAx = $true )
& ( binunionRsub = $true )
& ( omegaIndAx = $true )
& ( subsetE2 = $true )
& ( setadjoinAx = $true )
& ( symdiffI2 = $true )
& ( sepInPowerset = $true )
& ( bs114d = $true )
& ( notequalI1 = $true )
& ( setadjoinOr = $true )
& ( nonemptyI = $true )
& ( setbeta = $true )
& ( exuI3 = $true )
& ( powersetsubset = $true )
& ( uniqinunit = $true )
& ( binintersectEL = $true )
& ( in__Cong = $true )
& ( setunionE = $true )
& ( vacuousDall = $true )
& ( subsetI1 = $true )
& ( exuE1 = $true )
& ( wellorderingAx = $true )
& ( binunionE = $true )
& ( powersetI1 = $true )
& ( setminusIRneg = $true )
& ( binintersectRsub = $true )
& ( subset2powerset = $true )
& ( emptysetimpfalse = $true )
& ( binintersectSubset3 = $true )
& ( setminusERneg = $true )
& ( eqimpsubset1 = $true )
& ( sepSubset = $true )
& ( nonemptyImpWitness = $true )
& ( quantDeMorgan1 = $true )
& ( subPowSU = $true )
& ( binintersectSubset1 = $true )
& ( setadjoinE = $true )
& ( subsetemptysetimpeq = $true )
& ( exuI2 = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinIL = $true )
& ( setextAx = $true )
& ( omegaSAx = $true )
& ( subsetRefl = $true )
& ( prop2setI = $true )
& ( setminusEL = $true )
& ( binintersectSubset4 = $true )
& ( emptyset__Cong = $true )
& ( setextsub = $true )
& ( prop2set2propI = $true )
& ( symdiffI1 = $true )
& ( notdallE = $true )
& ( emptysetsubset = $true )
& ( binunionIR = $true )
& ( notdexE = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK134,sK135])],[f554,f806]) ).
thf(f806,plain,
( ? [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
!= $true )
=> ( ( in @ sK134 @ ( setadjoin @ sK135 @ ( setadjoin @ sK134 @ emptyset ) ) )
!= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f554,plain,
( ( dsetconstrI = $true )
& ( binintersectI = $true )
& ( upairset2IR = $true )
& ( setminusI = $true )
& ( singletonsswitch = $true )
& ( binunionLsub = $true )
& ( exuE2 = $true )
& ( binintersectSubset2 = $true )
& ( emptysetAx = $true )
& ( emptyE1 = $true )
& ( setadjoinIR = $true )
& ( setunionAx = $true )
& ( emptyinPowerset = $true )
& ( setunionI = $true )
& ( subsetTrans = $true )
& ( nonemptyE1 = $true )
& ( noeltsimpempty = $true )
& ( eqinunit = $true )
& ? [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
!= $true )
& ( emptyInPowerset = $true )
& ( binunionEcases = $true )
& ( symdiffE = $true )
& ( notsubsetI = $true )
& ( omega0Ax = $true )
& ( upairsetIR = $true )
& ( foundationAx = $true )
& ( omega__Cong = $true )
& ( dsetconstr__Cong = $true )
& ( emptyI = $true )
& ( binintersectER = $true )
& ( emptyinunitempty = $true )
& ( binintersectSubset5 = $true )
& ( nonemptyI1 = $true )
& ( exuE3e = $true )
& ( setminusILneg = $true )
& ( prop2setE = $true )
& ( setext = $true )
& ( setunion__Cong = $true )
& ( notequalI2 = $true )
& ( powersetI = $true )
& ( notinsingleton = $true )
& ( powerset__Cong = $true )
& ( setminusELneg = $true )
& ( setadjoinSub = $true )
& ( setoftrueEq = $true )
& ( setminusSubset1 = $true )
& ( dsetconstrER = $true )
& ( exuE3u = $true )
& ( setminusLsub = $true )
& ( powersetAx = $true )
& ( disjointsetsI1 = $true )
& ( setminusER = $true )
& ( upairsetIL = $true )
& ( powersetE1 = $true )
& ( quantDeMorgan2 = $true )
& ( powersetE = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( notinemptyset = $true )
& ( inPowerset = $true )
& ( descrp = $true )
& ( binunionIL = $true )
& ( inCongP = $true )
& ( upairsetE = $true )
& ( exuI1 = $true )
& ( symdiffIneg2 = $true )
& ( emptysetE = $true )
& ( subsetE = $true )
& ( subsetI2 = $true )
& ( binintersectLsub = $true )
& ( symdiffIneg1 = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( exu__Cong = $true )
& ( dsetconstrEL = $true )
& ( setadjoin__Cong = $true )
& ( eqimpsubset2 = $true )
& ( setminusSubset2 = $true )
& ( replAx = $true )
& ( binunionRsub = $true )
& ( omegaIndAx = $true )
& ( subsetE2 = $true )
& ( setadjoinAx = $true )
& ( symdiffI2 = $true )
& ( sepInPowerset = $true )
& ( bs114d = $true )
& ( notequalI1 = $true )
& ( setadjoinOr = $true )
& ( nonemptyI = $true )
& ( setbeta = $true )
& ( exuI3 = $true )
& ( powersetsubset = $true )
& ( uniqinunit = $true )
& ( binintersectEL = $true )
& ( in__Cong = $true )
& ( setunionE = $true )
& ( vacuousDall = $true )
& ( subsetI1 = $true )
& ( exuE1 = $true )
& ( wellorderingAx = $true )
& ( binunionE = $true )
& ( powersetI1 = $true )
& ( setminusIRneg = $true )
& ( binintersectRsub = $true )
& ( subset2powerset = $true )
& ( emptysetimpfalse = $true )
& ( binintersectSubset3 = $true )
& ( setminusERneg = $true )
& ( eqimpsubset1 = $true )
& ( sepSubset = $true )
& ( nonemptyImpWitness = $true )
& ( quantDeMorgan1 = $true )
& ( subPowSU = $true )
& ( binintersectSubset1 = $true )
& ( setadjoinE = $true )
& ( subsetemptysetimpeq = $true )
& ( exuI2 = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinIL = $true )
& ( setextAx = $true )
& ( omegaSAx = $true )
& ( subsetRefl = $true )
& ( prop2setI = $true )
& ( setminusEL = $true )
& ( binintersectSubset4 = $true )
& ( emptyset__Cong = $true )
& ( setextsub = $true )
& ( prop2set2propI = $true )
& ( symdiffI1 = $true )
& ( notdallE = $true )
& ( emptysetsubset = $true )
& ( binunionIR = $true )
& ( notdexE = $true ) ),
inference(flattening,[status(thm)],[f553]) ).
thf(f553,plain,
( ? [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
!= $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f256]) ).
thf(f256,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f255]) ).
thf(f255,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ! [X0: $i,X1: $i] : ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f137]) ).
thf(f137,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ! [X2: $i,X1: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f136]) ).
thf(f136,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ! [X2: $i,X1: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.KCJafwTrbB/DTF2THF_13000.p',secondinupair) ).
thf(f1562,plain,
! [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) )
| ( upairset2IR != $true ) ),
inference(cnf_transformation,[status(thm)],[f866]) ).
thf(f866,plain,
( ( ! [X0: $i,X1: $i] :
( $true
= ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) )
| ( upairset2IR != $true ) )
& ( ( upairset2IR = $true )
| ( ( in @ sK167 @ ( setadjoin @ sK166 @ ( setadjoin @ sK167 @ emptyset ) ) )
!= $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK166,sK167])],[f864,f865]) ).
thf(f865,plain,
( ? [X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) )
!= $true )
=> ( ( in @ sK167 @ ( setadjoin @ sK166 @ ( setadjoin @ sK167 @ emptyset ) ) )
!= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f864,plain,
( ( ! [X0: $i,X1: $i] :
( $true
= ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) )
| ( upairset2IR != $true ) )
& ( ( upairset2IR = $true )
| ? [X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) )
!= $true ) ) ),
inference(rectify,[status(thm)],[f863]) ).
thf(f863,plain,
( ( ! [X0: $i,X1: $i] :
( $true
= ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) )
| ( upairset2IR != $true ) )
& ( ( upairset2IR = $true )
| ? [X0: $i,X1: $i] :
( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f168]) ).
thf(f168,plain,
( ! [X0: $i,X1: $i] :
( $true
= ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) )
<=> ( upairset2IR = $true ) ),
inference(fool_elimination,[status(thm)],[f167]) ).
thf(f167,plain,
( upairset2IR
= ( ! [X0: $i,X1: $i] : ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) ) ),
inference(rectify,[status(thm)],[f102]) ).
thf(f102,axiom,
( upairset2IR
= ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.KCJafwTrbB/DTF2THF_13000.p',upairset2IR) ).
thf(f1499,plain,
( ( in @ sK134 @ ( setadjoin @ sK135 @ ( setadjoin @ sK134 @ emptyset ) ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f807]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU616^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.34 % Computer : n011.cluster.edu
% 0.15/0.34 % Model : x86_64 x86_64
% 0.15/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.34 % Memory : 8042.1875MB
% 0.15/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34 % CPULimit : 300
% 0.15/0.34 % WCLimit : 300
% 0.15/0.34 % DateTime : Tue Feb 24 12:28:47 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.44 ---- Original DTF file ---
% 0.19/0.44 thf(spec,logic,$$dhol).
% 0.19/0.44 %------------------------------------------------------------------------------
% 0.19/0.44 % File : SEU616^1 : TPTP v9.2.1. Released v3.7.0.
% 0.19/0.44 % Domain : Set Theory
% 0.19/0.44 % Problem : Ordered Pairs - Kuratowski Pairs
% 0.19/0.44 % Version : Especial.
% 0.19/0.44 % English : (! x:i.! y:i.in y (setadjoin x (setadjoin y emptyset)))
% 0.19/0.44
% 0.19/0.44 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.19/0.44 % Source : [Bro08]
% 0.19/0.44 % Names : ZFC118g [Bro08]
% 0.19/0.44
% 0.19/0.44 % Status : Theorem
% 0.19/0.44 % Rating : 0.11 v9.1.0, 0.25 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 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.33 v4.0.1, 0.67 v4.0.0, 0.33 v3.7.0
% 0.19/0.44 % Syntax : Number of formulae : 289 ( 135 unt; 153 typ; 135 def)
% 0.19/0.44 % Number of atoms : 722 ( 193 equ; 0 cnn)
% 0.19/0.44 % Maximal formula atoms : 133 ( 5 avg)
% 0.19/0.44 % Number of connectives : 1176 ( 47 ~; 6 |; 30 &; 757 @)
% 0.19/0.44 % ( 14 <=>; 322 =>; 0 <=; 0 <~>)
% 0.19/0.44 % Maximal formula depth : 139 ( 2 avg)
% 0.19/0.44 % Number of types : 2 ( 0 usr)
% 0.19/0.44 % Number of type conns : 65 ( 65 >; 0 *; 0 +; 0 <<)
% 0.19/0.44 % Number of symbols : 156 ( 153 usr; 136 con; 0-2 aty)
% 0.19/0.44 % Number of variables : 391 ( 31 ^; 329 !; 31 ?; 391 :)
% 0.19/0.44 % SPC : TH0_THM_EQU_NAR
% 0.19/0.44
% 0.19/0.44 % Comments : http://mathgate.info/detsetitem.php?id=171
% 0.19/0.44 %------------------------------------------------------------------------------
% 0.19/0.44 thf(in_type,type,
% 0.19/0.44 in: $i > $i > $o ).
% 0.19/0.44
% 0.19/0.44 thf(exu_type,type,
% 0.19/0.44 exu: ( $i > $o ) > $o ).
% 0.19/0.44
% 0.19/0.44 thf(exu,definition,
% 0.19/0.44 ( exu
% 0.19/0.44 = ( ^ [Xphi: $i > $o] :
% 0.19/0.44 ? [Xx: $i] :
% 0.19/0.44 ( ( Xphi @ Xx )
% 0.19/0.44 & ! [Xy: $i] :
% 0.19/0.44 ( ( Xphi @ Xy )
% 0.19/0.44 => ( Xx = Xy ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(setextAx_type,type,
% 0.19/0.44 setextAx: $o ).
% 0.19/0.44
% 0.19/0.44 thf(setextAx,definition,
% 0.19/0.44 ( setextAx
% 0.19/0.44 = ( ! [A: $i,B: $i] :
% 0.19/0.44 ( ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 <=> ( in @ Xx @ B ) )
% 0.19/0.44 => ( A = B ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(emptyset_type,type,
% 0.19/0.44 emptyset: $i ).
% 0.19/0.44
% 0.19/0.44 thf(emptysetAx_type,type,
% 0.19/0.44 emptysetAx: $o ).
% 0.19/0.44
% 0.19/0.44 thf(emptysetAx,definition,
% 0.19/0.44 ( emptysetAx
% 0.19/0.44 = ( ! [Xx: $i] :
% 0.19/0.44 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(setadjoin_type,type,
% 0.19/0.44 setadjoin: $i > $i > $i ).
% 0.19/0.44
% 0.19/0.44 thf(setadjoinAx_type,type,
% 0.19/0.44 setadjoinAx: $o ).
% 0.19/0.44
% 0.19/0.44 thf(setadjoinAx,definition,
% 0.19/0.44 ( setadjoinAx
% 0.19/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.44 <=> ( ( Xy = Xx )
% 0.19/0.44 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(powerset_type,type,
% 0.19/0.44 powerset: $i > $i ).
% 0.19/0.44
% 0.19/0.44 thf(powersetAx_type,type,
% 0.19/0.44 powersetAx: $o ).
% 0.19/0.44
% 0.19/0.44 thf(powersetAx,definition,
% 0.19/0.44 ( powersetAx
% 0.19/0.44 = ( ! [A: $i,B: $i] :
% 0.19/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.44 <=> ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ B )
% 0.19/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(setunion_type,type,
% 0.19/0.44 setunion: $i > $i ).
% 0.19/0.44
% 0.19/0.44 thf(setunionAx_type,type,
% 0.19/0.44 setunionAx: $o ).
% 0.19/0.44
% 0.19/0.44 thf(setunionAx,definition,
% 0.19/0.44 ( setunionAx
% 0.19/0.44 = ( ! [A: $i,Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.44 <=> ? [B: $i] :
% 0.19/0.44 ( ( in @ Xx @ B )
% 0.19/0.44 & ( in @ B @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(omega_type,type,
% 0.19/0.44 omega: $i ).
% 0.19/0.44
% 0.19/0.44 thf(omega0Ax_type,type,
% 0.19/0.44 omega0Ax: $o ).
% 0.19/0.44
% 0.19/0.44 thf(omega0Ax,definition,
% 0.19/0.44 ( omega0Ax
% 0.19/0.44 = ( in @ emptyset @ omega ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(omegaSAx_type,type,
% 0.19/0.44 omegaSAx: $o ).
% 0.19/0.44
% 0.19/0.44 thf(omegaSAx,definition,
% 0.19/0.44 ( omegaSAx
% 0.19/0.44 = ( ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ omega )
% 0.19/0.44 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(omegaIndAx_type,type,
% 0.19/0.44 omegaIndAx: $o ).
% 0.19/0.44
% 0.19/0.44 thf(omegaIndAx,definition,
% 0.19/0.44 ( omegaIndAx
% 0.19/0.44 = ( ! [A: $i] :
% 0.19/0.44 ( ( ( in @ emptyset @ A )
% 0.19/0.44 & ! [Xx: $i] :
% 0.19/0.44 ( ( ( in @ Xx @ omega )
% 0.19/0.44 & ( in @ Xx @ A ) )
% 0.19/0.44 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.19/0.44 => ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ omega )
% 0.19/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(replAx_type,type,
% 0.19/0.44 replAx: $o ).
% 0.19/0.44
% 0.19/0.44 thf(replAx,definition,
% 0.19/0.44 ( replAx
% 0.19/0.44 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.19/0.44 ( ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 => ( exu
% 0.19/0.44 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.19/0.44 => ? [B: $i] :
% 0.19/0.44 ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ B )
% 0.19/0.44 <=> ? [Xy: $i] :
% 0.19/0.44 ( ( in @ Xy @ A )
% 0.19/0.44 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(foundationAx_type,type,
% 0.19/0.44 foundationAx: $o ).
% 0.19/0.44
% 0.19/0.44 thf(foundationAx,definition,
% 0.19/0.44 ( foundationAx
% 0.19/0.44 = ( ! [A: $i] :
% 0.19/0.44 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.44 => ? [B: $i] :
% 0.19/0.44 ( ( in @ B @ A )
% 0.19/0.44 & ~ ? [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ B )
% 0.19/0.44 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(wellorderingAx_type,type,
% 0.19/0.44 wellorderingAx: $o ).
% 0.19/0.44
% 0.19/0.44 thf(wellorderingAx,definition,
% 0.19/0.44 ( wellorderingAx
% 0.19/0.44 = ( ! [A: $i] :
% 0.19/0.44 ? [B: $i] :
% 0.19/0.44 ( ! [C: $i] :
% 0.19/0.44 ( ( in @ C @ B )
% 0.19/0.44 => ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ C )
% 0.19/0.44 => ( in @ Xx @ A ) ) )
% 0.19/0.44 & ! [Xx: $i,Xy: $i] :
% 0.19/0.44 ( ( ( in @ Xx @ A )
% 0.19/0.44 & ( in @ Xy @ A ) )
% 0.19/0.44 => ( ! [C: $i] :
% 0.19/0.44 ( ( in @ C @ B )
% 0.19/0.44 => ( ( in @ Xx @ C )
% 0.19/0.44 <=> ( in @ Xy @ C ) ) )
% 0.19/0.44 => ( Xx = Xy ) ) )
% 0.19/0.44 & ! [C: $i,D: $i] :
% 0.19/0.44 ( ( ( in @ C @ B )
% 0.19/0.44 & ( in @ D @ B ) )
% 0.19/0.44 => ( ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ C )
% 0.19/0.44 => ( in @ Xx @ D ) )
% 0.19/0.44 | ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ D )
% 0.19/0.44 => ( in @ Xx @ C ) ) ) )
% 0.19/0.44 & ! [C: $i] :
% 0.19/0.44 ( ( ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ C )
% 0.19/0.44 => ( in @ Xx @ A ) )
% 0.19/0.44 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.19/0.44 => ? [D: $i,Xx: $i] :
% 0.19/0.44 ( ( in @ D @ B )
% 0.19/0.44 & ( in @ Xx @ C )
% 0.19/0.44 & ~ ? [Xy: $i] :
% 0.19/0.44 ( ( in @ Xy @ D )
% 0.19/0.44 & ( in @ Xy @ C ) )
% 0.19/0.44 & ! [E: $i] :
% 0.19/0.44 ( ( in @ E @ B )
% 0.19/0.44 => ( ! [Xy: $i] :
% 0.19/0.44 ( ( in @ Xy @ E )
% 0.19/0.44 => ( in @ Xy @ D ) )
% 0.19/0.44 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(descr_type,type,
% 0.19/0.44 descr: ( $i > $o ) > $i ).
% 0.19/0.44
% 0.19/0.44 thf(descrp_type,type,
% 0.19/0.44 descrp: $o ).
% 0.19/0.44
% 0.19/0.44 thf(descrp,definition,
% 0.19/0.44 ( descrp
% 0.19/0.44 = ( ! [Xphi: $i > $o] :
% 0.19/0.44 ( ( exu
% 0.19/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.44 => ( Xphi
% 0.19/0.44 @ ( descr
% 0.19/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(dsetconstr_type,type,
% 0.19/0.44 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.19/0.44
% 0.19/0.44 thf(dsetconstrI_type,type,
% 0.19/0.44 dsetconstrI: $o ).
% 0.19/0.44
% 0.19/0.44 thf(dsetconstrI,definition,
% 0.19/0.44 ( dsetconstrI
% 0.19/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 => ( ( Xphi @ Xx )
% 0.19/0.44 => ( in @ Xx
% 0.19/0.44 @ ( dsetconstr @ A
% 0.19/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(dsetconstrEL_type,type,
% 0.19/0.44 dsetconstrEL: $o ).
% 0.19/0.44
% 0.19/0.44 thf(dsetconstrEL,definition,
% 0.19/0.44 ( dsetconstrEL
% 0.19/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.44 ( ( in @ Xx
% 0.19/0.44 @ ( dsetconstr @ A
% 0.19/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(dsetconstrER_type,type,
% 0.19/0.44 dsetconstrER: $o ).
% 0.19/0.44
% 0.19/0.44 thf(dsetconstrER,definition,
% 0.19/0.44 ( dsetconstrER
% 0.19/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.44 ( ( in @ Xx
% 0.19/0.44 @ ( dsetconstr @ A
% 0.19/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.44 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(exuE1_type,type,
% 0.19/0.44 exuE1: $o ).
% 0.19/0.44
% 0.19/0.44 thf(exuE1,definition,
% 0.19/0.44 ( exuE1
% 0.19/0.44 = ( ! [Xphi: $i > $o] :
% 0.19/0.44 ( ( exu
% 0.19/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.44 => ? [Xx: $i] :
% 0.19/0.44 ( ( Xphi @ Xx )
% 0.19/0.44 & ! [Xy: $i] :
% 0.19/0.44 ( ( Xphi @ Xy )
% 0.19/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(prop2set_type,type,
% 0.19/0.44 prop2set: $o > $i ).
% 0.19/0.44
% 0.19/0.44 thf(prop2setE_type,type,
% 0.19/0.44 prop2setE: $o ).
% 0.19/0.44
% 0.19/0.44 thf(prop2setE,definition,
% 0.19/0.44 ( prop2setE
% 0.19/0.44 = ( ! [Xphi: $o,Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.19/0.44 => Xphi ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(emptysetE_type,type,
% 0.19/0.44 emptysetE: $o ).
% 0.19/0.44
% 0.19/0.44 thf(emptysetE,definition,
% 0.19/0.44 ( emptysetE
% 0.19/0.44 = ( ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ emptyset )
% 0.19/0.44 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(emptysetimpfalse_type,type,
% 0.19/0.44 emptysetimpfalse: $o ).
% 0.19/0.44
% 0.19/0.44 thf(emptysetimpfalse,definition,
% 0.19/0.44 ( emptysetimpfalse
% 0.19/0.44 = ( ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ emptyset )
% 0.19/0.44 => $false ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(notinemptyset_type,type,
% 0.19/0.44 notinemptyset: $o ).
% 0.19/0.44
% 0.19/0.44 thf(notinemptyset,definition,
% 0.19/0.44 ( notinemptyset
% 0.19/0.44 = ( ! [Xx: $i] :
% 0.19/0.44 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(exuE3e_type,type,
% 0.19/0.44 exuE3e: $o ).
% 0.19/0.44
% 0.19/0.44 thf(exuE3e,definition,
% 0.19/0.44 ( exuE3e
% 0.19/0.44 = ( ! [Xphi: $i > $o] :
% 0.19/0.44 ( ( exu
% 0.19/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.44 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(setext_type,type,
% 0.19/0.44 setext: $o ).
% 0.19/0.44
% 0.19/0.44 thf(setext,definition,
% 0.19/0.44 ( setext
% 0.19/0.44 = ( ! [A: $i,B: $i] :
% 0.19/0.44 ( ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 => ( in @ Xx @ B ) )
% 0.19/0.44 => ( ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ B )
% 0.19/0.44 => ( in @ Xx @ A ) )
% 0.19/0.44 => ( A = B ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(emptyI_type,type,
% 0.19/0.44 emptyI: $o ).
% 0.19/0.44
% 0.19/0.44 thf(emptyI,definition,
% 0.19/0.44 ( emptyI
% 0.19/0.44 = ( ! [A: $i] :
% 0.19/0.44 ( ! [Xx: $i] :
% 0.19/0.44 ~ ( in @ Xx @ A )
% 0.19/0.44 => ( A = emptyset ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(noeltsimpempty_type,type,
% 0.19/0.44 noeltsimpempty: $o ).
% 0.19/0.44
% 0.19/0.44 thf(noeltsimpempty,definition,
% 0.19/0.44 ( noeltsimpempty
% 0.19/0.44 = ( ! [A: $i] :
% 0.19/0.44 ( ! [Xx: $i] :
% 0.19/0.44 ~ ( in @ Xx @ A )
% 0.19/0.44 => ( A = emptyset ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(setbeta_type,type,
% 0.19/0.44 setbeta: $o ).
% 0.19/0.44
% 0.19/0.44 thf(setbeta,definition,
% 0.19/0.44 ( setbeta
% 0.19/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 => ( ( in @ Xx
% 0.19/0.44 @ ( dsetconstr @ A
% 0.19/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.44 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(nonempty_type,type,
% 0.19/0.44 nonempty: $i > $o ).
% 0.19/0.44
% 0.19/0.44 thf(nonempty,definition,
% 0.19/0.44 ( nonempty
% 0.19/0.44 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(nonemptyE1_type,type,
% 0.19/0.44 nonemptyE1: $o ).
% 0.19/0.44
% 0.19/0.44 thf(nonemptyE1,definition,
% 0.19/0.44 ( nonemptyE1
% 0.19/0.44 = ( ! [A: $i] :
% 0.19/0.44 ( ( nonempty @ A )
% 0.19/0.44 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(nonemptyI_type,type,
% 0.19/0.44 nonemptyI: $o ).
% 0.19/0.44
% 0.19/0.44 thf(nonemptyI,definition,
% 0.19/0.44 ( nonemptyI
% 0.19/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 => ( ( Xphi @ Xx )
% 0.19/0.44 => ( nonempty
% 0.19/0.44 @ ( dsetconstr @ A
% 0.19/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(nonemptyI1_type,type,
% 0.19/0.44 nonemptyI1: $o ).
% 0.19/0.44
% 0.19/0.44 thf(nonemptyI1,definition,
% 0.19/0.44 ( nonemptyI1
% 0.19/0.44 = ( ! [A: $i] :
% 0.19/0.44 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.44 => ( nonempty @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(setadjoinIL_type,type,
% 0.19/0.44 setadjoinIL: $o ).
% 0.19/0.44
% 0.19/0.44 thf(setadjoinIL,definition,
% 0.19/0.44 ( setadjoinIL
% 0.19/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(emptyinunitempty_type,type,
% 0.19/0.44 emptyinunitempty: $o ).
% 0.19/0.44
% 0.19/0.44 thf(emptyinunitempty,definition,
% 0.19/0.44 ( emptyinunitempty
% 0.19/0.44 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(setadjoinIR_type,type,
% 0.19/0.44 setadjoinIR: $o ).
% 0.19/0.44
% 0.19/0.44 thf(setadjoinIR,definition,
% 0.19/0.44 ( setadjoinIR
% 0.19/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.44 ( ( in @ Xy @ A )
% 0.19/0.44 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(setadjoinE_type,type,
% 0.19/0.44 setadjoinE: $o ).
% 0.19/0.44
% 0.19/0.44 thf(setadjoinE,definition,
% 0.19/0.44 ( setadjoinE
% 0.19/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.44 => ! [Xphi: $o] :
% 0.19/0.44 ( ( ( Xy = Xx )
% 0.19/0.44 => Xphi )
% 0.19/0.44 => ( ( ( in @ Xy @ A )
% 0.19/0.44 => Xphi )
% 0.19/0.44 => Xphi ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(setadjoinOr_type,type,
% 0.19/0.44 setadjoinOr: $o ).
% 0.19/0.44
% 0.19/0.44 thf(setadjoinOr,definition,
% 0.19/0.44 ( setadjoinOr
% 0.19/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.44 => ( ( Xy = Xx )
% 0.19/0.44 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(setoftrueEq_type,type,
% 0.19/0.44 setoftrueEq: $o ).
% 0.19/0.44
% 0.19/0.44 thf(setoftrueEq,definition,
% 0.19/0.44 ( setoftrueEq
% 0.19/0.44 = ( ! [A: $i] :
% 0.19/0.44 ( ( dsetconstr @ A
% 0.19/0.44 @ ^ [Xx: $i] : $true )
% 0.19/0.44 = A ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(powersetI_type,type,
% 0.19/0.44 powersetI: $o ).
% 0.19/0.44
% 0.19/0.44 thf(powersetI,definition,
% 0.19/0.44 ( powersetI
% 0.19/0.44 = ( ! [A: $i,B: $i] :
% 0.19/0.44 ( ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ B )
% 0.19/0.44 => ( in @ Xx @ A ) )
% 0.19/0.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(emptyinPowerset_type,type,
% 0.19/0.44 emptyinPowerset: $o ).
% 0.19/0.44
% 0.19/0.44 thf(emptyinPowerset,definition,
% 0.19/0.44 ( emptyinPowerset
% 0.19/0.44 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(emptyInPowerset_type,type,
% 0.19/0.44 emptyInPowerset: $o ).
% 0.19/0.44
% 0.19/0.44 thf(emptyInPowerset,definition,
% 0.19/0.44 ( emptyInPowerset
% 0.19/0.44 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(powersetE_type,type,
% 0.19/0.44 powersetE: $o ).
% 0.19/0.44
% 0.19/0.44 thf(powersetE,definition,
% 0.19/0.44 ( powersetE
% 0.19/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.44 => ( ( in @ Xx @ B )
% 0.19/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(setunionI_type,type,
% 0.19/0.44 setunionI: $o ).
% 0.19/0.44
% 0.19/0.44 thf(setunionI,definition,
% 0.19/0.44 ( setunionI
% 0.19/0.44 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.19/0.44 ( ( in @ Xx @ B )
% 0.19/0.44 => ( ( in @ B @ A )
% 0.19/0.44 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(setunionE_type,type,
% 0.19/0.44 setunionE: $o ).
% 0.19/0.44
% 0.19/0.44 thf(setunionE,definition,
% 0.19/0.44 ( setunionE
% 0.19/0.44 = ( ! [A: $i,Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.44 => ! [Xphi: $o] :
% 0.19/0.44 ( ! [B: $i] :
% 0.19/0.44 ( ( in @ Xx @ B )
% 0.19/0.44 => ( ( in @ B @ A )
% 0.19/0.44 => Xphi ) )
% 0.19/0.44 => Xphi ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(subPowSU_type,type,
% 0.19/0.44 subPowSU: $o ).
% 0.19/0.44
% 0.19/0.44 thf(subPowSU,definition,
% 0.19/0.44 ( subPowSU
% 0.19/0.44 = ( ! [A: $i,Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(exuE2_type,type,
% 0.19/0.44 exuE2: $o ).
% 0.19/0.44
% 0.19/0.44 thf(exuE2,definition,
% 0.19/0.44 ( exuE2
% 0.19/0.44 = ( ! [Xphi: $i > $o] :
% 0.19/0.44 ( ( exu
% 0.19/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.44 => ? [Xx: $i] :
% 0.19/0.44 ! [Xy: $i] :
% 0.19/0.44 ( ( Xphi @ Xy )
% 0.19/0.44 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(nonemptyImpWitness_type,type,
% 0.19/0.44 nonemptyImpWitness: $o ).
% 0.19/0.44
% 0.19/0.44 thf(nonemptyImpWitness,definition,
% 0.19/0.44 ( nonemptyImpWitness
% 0.19/0.44 = ( ! [A: $i] :
% 0.19/0.44 ( ( nonempty @ A )
% 0.19/0.44 => ? [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 & $true ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(uniqinunit_type,type,
% 0.19/0.44 uniqinunit: $o ).
% 0.19/0.44
% 0.19/0.44 thf(uniqinunit,definition,
% 0.19/0.44 ( uniqinunit
% 0.19/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.44 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(notinsingleton_type,type,
% 0.19/0.44 notinsingleton: $o ).
% 0.19/0.44
% 0.19/0.44 thf(notinsingleton,definition,
% 0.19/0.44 ( notinsingleton
% 0.19/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.44 ( ( Xx != Xy )
% 0.19/0.44 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(eqinunit_type,type,
% 0.19/0.44 eqinunit: $o ).
% 0.19/0.44
% 0.19/0.44 thf(eqinunit,definition,
% 0.19/0.44 ( eqinunit
% 0.19/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.44 ( ( Xx = Xy )
% 0.19/0.44 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(singletonsswitch_type,type,
% 0.19/0.44 singletonsswitch: $o ).
% 0.19/0.44
% 0.19/0.44 thf(singletonsswitch,definition,
% 0.19/0.44 ( singletonsswitch
% 0.19/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.44 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(upairsetE_type,type,
% 0.19/0.44 upairsetE: $o ).
% 0.19/0.44
% 0.19/0.44 thf(upairsetE,definition,
% 0.19/0.44 ( upairsetE
% 0.19/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.44 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.19/0.44 => ( ( Xz = Xx )
% 0.19/0.44 | ( Xz = Xy ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(upairsetIL_type,type,
% 0.19/0.44 upairsetIL: $o ).
% 0.19/0.44
% 0.19/0.44 thf(upairsetIL,definition,
% 0.19/0.44 ( upairsetIL
% 0.19/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(upairsetIR_type,type,
% 0.19/0.44 upairsetIR: $o ).
% 0.19/0.44
% 0.19/0.44 thf(upairsetIR,definition,
% 0.19/0.44 ( upairsetIR
% 0.19/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(emptyE1_type,type,
% 0.19/0.44 emptyE1: $o ).
% 0.19/0.44
% 0.19/0.44 thf(emptyE1,definition,
% 0.19/0.44 ( emptyE1
% 0.19/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.44 ( ? [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 & ( Xphi @ Xx ) )
% 0.19/0.44 => ( ( ( dsetconstr @ A
% 0.19/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.44 = emptyset )
% 0.19/0.44 => $false ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(vacuousDall_type,type,
% 0.19/0.44 vacuousDall: $o ).
% 0.19/0.44
% 0.19/0.44 thf(vacuousDall,definition,
% 0.19/0.44 ( vacuousDall
% 0.19/0.44 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ emptyset )
% 0.19/0.44 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(quantDeMorgan1_type,type,
% 0.19/0.44 quantDeMorgan1: $o ).
% 0.19/0.44
% 0.19/0.44 thf(quantDeMorgan1,definition,
% 0.19/0.44 ( quantDeMorgan1
% 0.19/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.44 ( ~ ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 => ( Xphi @ Xx ) )
% 0.19/0.44 => ? [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(quantDeMorgan2_type,type,
% 0.19/0.44 quantDeMorgan2: $o ).
% 0.19/0.44
% 0.19/0.44 thf(quantDeMorgan2,definition,
% 0.19/0.44 ( quantDeMorgan2
% 0.19/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.44 ( ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 => ~ ( Xphi @ Xx ) )
% 0.19/0.44 => ~ ? [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(quantDeMorgan3_type,type,
% 0.19/0.44 quantDeMorgan3: $o ).
% 0.19/0.44
% 0.19/0.44 thf(quantDeMorgan3,definition,
% 0.19/0.44 ( quantDeMorgan3
% 0.19/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.44 ( ~ ? [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 & ( Xphi @ Xx ) )
% 0.19/0.44 => ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(quantDeMorgan4_type,type,
% 0.19/0.44 quantDeMorgan4: $o ).
% 0.19/0.44
% 0.19/0.44 thf(quantDeMorgan4,definition,
% 0.19/0.44 ( quantDeMorgan4
% 0.19/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.44 ( ? [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 & ~ ( Xphi @ Xx ) )
% 0.19/0.44 => ~ ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(prop2setI_type,type,
% 0.19/0.44 prop2setI: $o ).
% 0.19/0.44
% 0.19/0.44 thf(prop2setI,definition,
% 0.19/0.44 ( prop2setI
% 0.19/0.44 = ( ! [Xphi: $o] :
% 0.19/0.44 ( Xphi
% 0.19/0.44 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(set2prop_type,type,
% 0.19/0.44 set2prop: $i > $o ).
% 0.19/0.44
% 0.19/0.44 thf(prop2set2propI_type,type,
% 0.19/0.44 prop2set2propI: $o ).
% 0.19/0.44
% 0.19/0.44 thf(prop2set2propI,definition,
% 0.19/0.44 ( prop2set2propI
% 0.19/0.44 = ( ! [Xphi: $o] :
% 0.19/0.44 ( Xphi
% 0.19/0.44 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(notdexE_type,type,
% 0.19/0.44 notdexE: $o ).
% 0.19/0.44
% 0.19/0.44 thf(notdexE,definition,
% 0.19/0.44 ( notdexE
% 0.19/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.44 ( ~ ? [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 & ( Xphi @ Xx ) )
% 0.19/0.44 => ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(notdallE_type,type,
% 0.19/0.44 notdallE: $o ).
% 0.19/0.44
% 0.19/0.44 thf(notdallE,definition,
% 0.19/0.44 ( notdallE
% 0.19/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.44 ( ~ ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 => ( Xphi @ Xx ) )
% 0.19/0.44 => ? [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(exuI1_type,type,
% 0.19/0.44 exuI1: $o ).
% 0.19/0.44
% 0.19/0.44 thf(exuI1,definition,
% 0.19/0.44 ( exuI1
% 0.19/0.44 = ( ! [Xphi: $i > $o] :
% 0.19/0.44 ( ? [Xx: $i] :
% 0.19/0.44 ( ( Xphi @ Xx )
% 0.19/0.44 & ! [Xy: $i] :
% 0.19/0.44 ( ( Xphi @ Xy )
% 0.19/0.44 => ( Xx = Xy ) ) )
% 0.19/0.44 => ( exu
% 0.19/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(exuI3_type,type,
% 0.19/0.44 exuI3: $o ).
% 0.19/0.44
% 0.19/0.44 thf(exuI3,definition,
% 0.19/0.44 ( exuI3
% 0.19/0.44 = ( ! [Xphi: $i > $o] :
% 0.19/0.44 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.19/0.44 => ( ! [Xx: $i,Xy: $i] :
% 0.19/0.44 ( ( Xphi @ Xx )
% 0.19/0.44 => ( ( Xphi @ Xy )
% 0.19/0.44 => ( Xx = Xy ) ) )
% 0.19/0.44 => ( exu
% 0.19/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(exuI2_type,type,
% 0.19/0.44 exuI2: $o ).
% 0.19/0.44
% 0.19/0.44 thf(exuI2,definition,
% 0.19/0.44 ( exuI2
% 0.19/0.44 = ( ! [Xphi: $i > $o] :
% 0.19/0.44 ( ? [Xx: $i] :
% 0.19/0.44 ! [Xy: $i] :
% 0.19/0.44 ( ( Xphi @ Xy )
% 0.19/0.44 <=> ( Xy = Xx ) )
% 0.19/0.44 => ( exu
% 0.19/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(inCongP_type,type,
% 0.19/0.44 inCongP: $o ).
% 0.19/0.44
% 0.19/0.44 thf(inCongP,definition,
% 0.19/0.44 ( inCongP
% 0.19/0.44 = ( ! [A: $i,B: $i] :
% 0.19/0.44 ( ( A = B )
% 0.19/0.44 => ! [Xx: $i,Xy: $i] :
% 0.19/0.44 ( ( Xx = Xy )
% 0.19/0.44 => ( ( in @ Xx @ A )
% 0.19/0.44 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(in__Cong_type,type,
% 0.19/0.44 in__Cong: $o ).
% 0.19/0.44
% 0.19/0.44 thf(in__Cong,definition,
% 0.19/0.44 ( in__Cong
% 0.19/0.44 = ( ! [A: $i,B: $i] :
% 0.19/0.44 ( ( A = B )
% 0.19/0.44 => ! [Xx: $i,Xy: $i] :
% 0.19/0.44 ( ( Xx = Xy )
% 0.19/0.44 => ( ( in @ Xx @ A )
% 0.19/0.44 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(exuE3u_type,type,
% 0.19/0.44 exuE3u: $o ).
% 0.19/0.44
% 0.19/0.44 thf(exuE3u,definition,
% 0.19/0.44 ( exuE3u
% 0.19/0.44 = ( ! [Xphi: $i > $o] :
% 0.19/0.44 ( ( exu
% 0.19/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.44 => ! [Xx: $i,Xy: $i] :
% 0.19/0.44 ( ( Xphi @ Xx )
% 0.19/0.44 => ( ( Xphi @ Xy )
% 0.19/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(exu__Cong_type,type,
% 0.19/0.44 exu__Cong: $o ).
% 0.19/0.44
% 0.19/0.44 thf(exu__Cong,definition,
% 0.19/0.44 ( exu__Cong
% 0.19/0.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.44 ( ! [Xx: $i,Xy: $i] :
% 0.19/0.44 ( ( Xx = Xy )
% 0.19/0.44 => ( ( Xphi @ Xx )
% 0.19/0.44 <=> ( Xpsi @ Xy ) ) )
% 0.19/0.44 => ( ( exu
% 0.19/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.44 <=> ( exu
% 0.19/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(emptyset__Cong_type,type,
% 0.19/0.44 emptyset__Cong: $o ).
% 0.19/0.44
% 0.19/0.44 thf(emptyset__Cong,definition,
% 0.19/0.44 ( emptyset__Cong
% 0.19/0.44 = ( emptyset = emptyset ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(setadjoin__Cong_type,type,
% 0.19/0.44 setadjoin__Cong: $o ).
% 0.19/0.44
% 0.19/0.44 thf(setadjoin__Cong,definition,
% 0.19/0.44 ( setadjoin__Cong
% 0.19/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.44 ( ( Xx = Xy )
% 0.19/0.44 => ! [Xz: $i,Xu: $i] :
% 0.19/0.44 ( ( Xz = Xu )
% 0.19/0.44 => ( ( setadjoin @ Xx @ Xz )
% 0.19/0.44 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(powerset__Cong_type,type,
% 0.19/0.44 powerset__Cong: $o ).
% 0.19/0.44
% 0.19/0.44 thf(powerset__Cong,definition,
% 0.19/0.44 ( powerset__Cong
% 0.19/0.44 = ( ! [A: $i,B: $i] :
% 0.19/0.44 ( ( A = B )
% 0.19/0.44 => ( ( powerset @ A )
% 0.19/0.44 = ( powerset @ B ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(setunion__Cong_type,type,
% 0.19/0.44 setunion__Cong: $o ).
% 0.19/0.44
% 0.19/0.44 thf(setunion__Cong,definition,
% 0.19/0.44 ( setunion__Cong
% 0.19/0.44 = ( ! [A: $i,B: $i] :
% 0.19/0.44 ( ( A = B )
% 0.19/0.44 => ( ( setunion @ A )
% 0.19/0.44 = ( setunion @ B ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(omega__Cong_type,type,
% 0.19/0.44 omega__Cong: $o ).
% 0.19/0.44
% 0.19/0.44 thf(omega__Cong,definition,
% 0.19/0.44 ( omega__Cong
% 0.19/0.44 = ( omega = omega ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(exuEu_type,type,
% 0.19/0.44 exuEu: $o ).
% 0.19/0.44
% 0.19/0.44 thf(exuEu,definition,
% 0.19/0.44 ( exuEu
% 0.19/0.44 = ( ! [Xphi: $i > $o] :
% 0.19/0.44 ( ( exu
% 0.19/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.44 => ! [Xx: $i,Xy: $i] :
% 0.19/0.44 ( ( Xphi @ Xx )
% 0.19/0.44 => ( ( Xphi @ Xy )
% 0.19/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(descr__Cong_type,type,
% 0.19/0.44 descr__Cong: $o ).
% 0.19/0.44
% 0.19/0.44 thf(descr__Cong,definition,
% 0.19/0.44 ( descr__Cong
% 0.19/0.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.44 ( ! [Xx: $i,Xy: $i] :
% 0.19/0.44 ( ( Xx = Xy )
% 0.19/0.44 => ( ( Xphi @ Xx )
% 0.19/0.44 <=> ( Xpsi @ Xy ) ) )
% 0.19/0.44 => ( ( exu
% 0.19/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.44 => ( ( exu
% 0.19/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.19/0.44 => ( ( descr
% 0.19/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.44 = ( descr
% 0.19/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(dsetconstr__Cong_type,type,
% 0.19/0.44 dsetconstr__Cong: $o ).
% 0.19/0.44
% 0.19/0.44 thf(dsetconstr__Cong,definition,
% 0.19/0.44 ( dsetconstr__Cong
% 0.19/0.44 = ( ! [A: $i,B: $i] :
% 0.19/0.44 ( ( A = B )
% 0.19/0.44 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.44 ( ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 => ! [Xy: $i] :
% 0.19/0.44 ( ( in @ Xy @ B )
% 0.19/0.44 => ( ( Xx = Xy )
% 0.19/0.44 => ( ( Xphi @ Xx )
% 0.19/0.44 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.19/0.44 => ( ( dsetconstr @ A
% 0.19/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.44 = ( dsetconstr @ B
% 0.19/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(subset_type,type,
% 0.19/0.44 subset: $i > $i > $o ).
% 0.19/0.44
% 0.19/0.44 thf(disjoint_type,type,
% 0.19/0.44 disjoint: $i > $i > $o ).
% 0.19/0.44
% 0.19/0.44 thf(setsmeet_type,type,
% 0.19/0.44 setsmeet: $i > $i > $o ).
% 0.19/0.44
% 0.19/0.44 thf(subsetI1_type,type,
% 0.19/0.44 subsetI1: $o ).
% 0.19/0.44
% 0.19/0.44 thf(subsetI1,definition,
% 0.19/0.44 ( subsetI1
% 0.19/0.44 = ( ! [A: $i,B: $i] :
% 0.19/0.44 ( ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 => ( in @ Xx @ B ) )
% 0.19/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(eqimpsubset2_type,type,
% 0.19/0.44 eqimpsubset2: $o ).
% 0.19/0.44
% 0.19/0.44 thf(eqimpsubset2,definition,
% 0.19/0.44 ( eqimpsubset2
% 0.19/0.44 = ( ! [A: $i,B: $i] :
% 0.19/0.44 ( ( A = B )
% 0.19/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(eqimpsubset1_type,type,
% 0.19/0.44 eqimpsubset1: $o ).
% 0.19/0.44
% 0.19/0.44 thf(eqimpsubset1,definition,
% 0.19/0.44 ( eqimpsubset1
% 0.19/0.44 = ( ! [A: $i,B: $i] :
% 0.19/0.44 ( ( A = B )
% 0.19/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(subsetI2_type,type,
% 0.19/0.44 subsetI2: $o ).
% 0.19/0.44
% 0.19/0.44 thf(subsetI2,definition,
% 0.19/0.44 ( subsetI2
% 0.19/0.44 = ( ! [A: $i,B: $i] :
% 0.19/0.44 ( ! [Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 => ( in @ Xx @ B ) )
% 0.19/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(emptysetsubset_type,type,
% 0.19/0.44 emptysetsubset: $o ).
% 0.19/0.44
% 0.19/0.44 thf(emptysetsubset,definition,
% 0.19/0.44 ( emptysetsubset
% 0.19/0.44 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(subsetE_type,type,
% 0.19/0.44 subsetE: $o ).
% 0.19/0.44
% 0.19/0.44 thf(subsetE,definition,
% 0.19/0.44 ( subsetE
% 0.19/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.44 ( ( subset @ A @ B )
% 0.19/0.44 => ( ( in @ Xx @ A )
% 0.19/0.44 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(subsetE2_type,type,
% 0.19/0.44 subsetE2: $o ).
% 0.19/0.44
% 0.19/0.44 thf(subsetE2,definition,
% 0.19/0.44 ( subsetE2
% 0.19/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.44 ( ( subset @ A @ B )
% 0.19/0.44 => ( ~ ( in @ Xx @ B )
% 0.19/0.44 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 thf(notsubsetI_type,type,
% 0.19/0.44 notsubsetI: $o ).
% 0.19/0.44
% 0.19/0.44 thf(notsubsetI,definition,
% 0.19/0.44 ( notsubsetI
% 0.19/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.44 ( ( in @ Xx @ A )
% 0.19/0.44 => ( ~ ( in @ Xx @ B )
% 0.19/0.44 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(notequalI1_type,type,
% 0.19/0.45 notequalI1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(notequalI1,definition,
% 0.19/0.45 ( notequalI1
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ~ ( subset @ A @ B )
% 0.19/0.45 => ( A != B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(notequalI2_type,type,
% 0.19/0.45 notequalI2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(notequalI2,definition,
% 0.19/0.45 ( notequalI2
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ~ ( in @ Xx @ B )
% 0.19/0.45 => ( A != B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(subsetRefl_type,type,
% 0.19/0.45 subsetRefl: $o ).
% 0.19/0.45
% 0.19/0.45 thf(subsetRefl,definition,
% 0.19/0.45 ( subsetRefl
% 0.19/0.45 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(subsetTrans_type,type,
% 0.19/0.45 subsetTrans: $o ).
% 0.19/0.45
% 0.19/0.45 thf(subsetTrans,definition,
% 0.19/0.45 ( subsetTrans
% 0.19/0.45 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.45 ( ( subset @ A @ B )
% 0.19/0.45 => ( ( subset @ B @ C )
% 0.19/0.45 => ( subset @ A @ C ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinSub_type,type,
% 0.19/0.45 setadjoinSub: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinSub,definition,
% 0.19/0.45 ( setadjoinSub
% 0.19/0.45 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinSub2_type,type,
% 0.19/0.45 setadjoinSub2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinSub2,definition,
% 0.19/0.45 ( setadjoinSub2
% 0.19/0.45 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.19/0.45 ( ( subset @ A @ B )
% 0.19/0.45 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(subset2powerset_type,type,
% 0.19/0.45 subset2powerset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(subset2powerset,definition,
% 0.19/0.45 ( subset2powerset
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( subset @ A @ B )
% 0.19/0.45 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setextsub_type,type,
% 0.19/0.45 setextsub: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setextsub,definition,
% 0.19/0.45 ( setextsub
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( subset @ A @ B )
% 0.19/0.45 => ( ( subset @ B @ A )
% 0.19/0.45 => ( A = B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(subsetemptysetimpeq_type,type,
% 0.19/0.45 subsetemptysetimpeq: $o ).
% 0.19/0.45
% 0.19/0.45 thf(subsetemptysetimpeq,definition,
% 0.19/0.45 ( subsetemptysetimpeq
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ( subset @ A @ emptyset )
% 0.19/0.45 => ( A = emptyset ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(powersetI1_type,type,
% 0.19/0.45 powersetI1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(powersetI1,definition,
% 0.19/0.45 ( powersetI1
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( subset @ B @ A )
% 0.19/0.45 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(powersetE1_type,type,
% 0.19/0.45 powersetE1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(powersetE1,definition,
% 0.19/0.45 ( powersetE1
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.45 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(inPowerset_type,type,
% 0.19/0.45 inPowerset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(inPowerset,definition,
% 0.19/0.45 ( inPowerset
% 0.19/0.45 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(powersetsubset_type,type,
% 0.19/0.45 powersetsubset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(powersetsubset,definition,
% 0.19/0.45 ( powersetsubset
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( subset @ A @ B )
% 0.19/0.45 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sepInPowerset_type,type,
% 0.19/0.45 sepInPowerset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(sepInPowerset,definition,
% 0.19/0.45 ( sepInPowerset
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( in
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 @ ( powerset @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sepSubset_type,type,
% 0.19/0.45 sepSubset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(sepSubset,definition,
% 0.19/0.45 ( sepSubset
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( subset
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binunion_type,type,
% 0.19/0.45 binunion: $i > $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(binunionIL_type,type,
% 0.19/0.45 binunionIL: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binunionIL,definition,
% 0.19/0.45 ( binunionIL
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(upairset2IR_type,type,
% 0.19/0.45 upairset2IR: $o ).
% 0.19/0.45
% 0.19/0.45 thf(upairset2IR,definition,
% 0.19/0.45 ( upairset2IR
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binunionIR_type,type,
% 0.19/0.45 binunionIR: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binunionIR,definition,
% 0.19/0.45 ( binunionIR
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binunionEcases_type,type,
% 0.19/0.45 binunionEcases: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binunionEcases,definition,
% 0.19/0.45 ( binunionEcases
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.19/0.45 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.19/0.45 => ( ( ( in @ Xx @ A )
% 0.19/0.45 => Xphi )
% 0.19/0.45 => ( ( ( in @ Xx @ B )
% 0.19/0.45 => Xphi )
% 0.19/0.45 => Xphi ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binunionE_type,type,
% 0.19/0.45 binunionE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binunionE,definition,
% 0.19/0.45 ( binunionE
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.19/0.45 => ( ( in @ Xx @ A )
% 0.19/0.45 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binunionLsub_type,type,
% 0.19/0.45 binunionLsub: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binunionLsub,definition,
% 0.19/0.45 ( binunionLsub
% 0.19/0.45 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binunionRsub_type,type,
% 0.19/0.45 binunionRsub: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binunionRsub,definition,
% 0.19/0.45 ( binunionRsub
% 0.19/0.45 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersect_type,type,
% 0.19/0.45 binintersect: $i > $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectI_type,type,
% 0.19/0.45 binintersectI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectI,definition,
% 0.19/0.45 ( binintersectI
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset5_type,type,
% 0.19/0.45 binintersectSubset5: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset5,definition,
% 0.19/0.45 ( binintersectSubset5
% 0.19/0.45 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.45 ( ( subset @ C @ A )
% 0.19/0.45 => ( ( subset @ C @ B )
% 0.19/0.45 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectEL_type,type,
% 0.19/0.45 binintersectEL: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectEL,definition,
% 0.19/0.45 ( binintersectEL
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.19/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectLsub_type,type,
% 0.19/0.45 binintersectLsub: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectLsub,definition,
% 0.19/0.45 ( binintersectLsub
% 0.19/0.45 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset2_type,type,
% 0.19/0.45 binintersectSubset2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset2,definition,
% 0.19/0.45 ( binintersectSubset2
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( subset @ A @ B )
% 0.19/0.45 => ( ( binintersect @ A @ B )
% 0.19/0.45 = A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset3_type,type,
% 0.19/0.45 binintersectSubset3: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset3,definition,
% 0.19/0.45 ( binintersectSubset3
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( ( binintersect @ A @ B )
% 0.19/0.45 = B )
% 0.19/0.45 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectER_type,type,
% 0.19/0.45 binintersectER: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectER,definition,
% 0.19/0.45 ( binintersectER
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.19/0.45 => ( in @ Xx @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(disjointsetsI1_type,type,
% 0.19/0.45 disjointsetsI1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(disjointsetsI1,definition,
% 0.19/0.45 ( disjointsetsI1
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ~ ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ( in @ Xx @ B ) )
% 0.19/0.45 => ( ( binintersect @ A @ B )
% 0.19/0.45 = emptyset ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectRsub_type,type,
% 0.19/0.45 binintersectRsub: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectRsub,definition,
% 0.19/0.45 ( binintersectRsub
% 0.19/0.45 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset4_type,type,
% 0.19/0.45 binintersectSubset4: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset4,definition,
% 0.19/0.45 ( binintersectSubset4
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( subset @ B @ A )
% 0.19/0.45 => ( ( binintersect @ A @ B )
% 0.19/0.45 = B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset1_type,type,
% 0.19/0.45 binintersectSubset1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset1,definition,
% 0.19/0.45 ( binintersectSubset1
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( ( binintersect @ A @ B )
% 0.19/0.45 = A )
% 0.19/0.45 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(bs114d_type,type,
% 0.19/0.45 bs114d: $o ).
% 0.19/0.45
% 0.19/0.45 thf(bs114d,definition,
% 0.19/0.45 ( bs114d
% 0.19/0.45 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.45 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.19/0.45 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(regular_type,type,
% 0.19/0.45 regular: $i > $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminus_type,type,
% 0.19/0.45 setminus: $i > $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(setminusI_type,type,
% 0.19/0.45 setminusI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusI,definition,
% 0.19/0.45 ( setminusI
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ~ ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusEL_type,type,
% 0.19/0.45 setminusEL: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusEL,definition,
% 0.19/0.45 ( setminusEL
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusER_type,type,
% 0.19/0.45 setminusER: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusER,definition,
% 0.19/0.45 ( setminusER
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.45 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusSubset2_type,type,
% 0.19/0.45 setminusSubset2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusSubset2,definition,
% 0.19/0.45 ( setminusSubset2
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( subset @ A @ B )
% 0.19/0.45 => ( ( setminus @ A @ B )
% 0.19/0.45 = emptyset ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusERneg_type,type,
% 0.19/0.45 setminusERneg: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusERneg,definition,
% 0.19/0.45 ( setminusERneg
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.45 => ( ( in @ Xx @ A )
% 0.19/0.45 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusELneg_type,type,
% 0.19/0.45 setminusELneg: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusELneg,definition,
% 0.19/0.45 ( setminusELneg
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.45 => ( ~ ( in @ Xx @ B )
% 0.19/0.45 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusILneg_type,type,
% 0.19/0.45 setminusILneg: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusILneg,definition,
% 0.19/0.45 ( setminusILneg
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ~ ( in @ Xx @ A )
% 0.19/0.45 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusIRneg_type,type,
% 0.19/0.45 setminusIRneg: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusIRneg,definition,
% 0.19/0.45 ( setminusIRneg
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusLsub_type,type,
% 0.19/0.45 setminusLsub: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusLsub,definition,
% 0.19/0.45 ( setminusLsub
% 0.19/0.45 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusSubset1_type,type,
% 0.19/0.45 setminusSubset1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusSubset1,definition,
% 0.19/0.45 ( setminusSubset1
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( ( setminus @ A @ B )
% 0.19/0.45 = emptyset )
% 0.19/0.45 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(symdiff_type,type,
% 0.19/0.45 symdiff: $i > $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffE_type,type,
% 0.19/0.45 symdiffE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffE,definition,
% 0.19/0.45 ( symdiffE
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.19/0.45 => ! [Xphi: $o] :
% 0.19/0.45 ( ( ( in @ Xx @ A )
% 0.19/0.45 => ( ~ ( in @ Xx @ B )
% 0.19/0.45 => Xphi ) )
% 0.19/0.45 => ( ( ~ ( in @ Xx @ A )
% 0.19/0.45 => ( ( in @ Xx @ B )
% 0.19/0.45 => Xphi ) )
% 0.19/0.45 => Xphi ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffI1_type,type,
% 0.19/0.45 symdiffI1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffI1,definition,
% 0.19/0.45 ( symdiffI1
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ~ ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffI2_type,type,
% 0.19/0.45 symdiffI2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffI2,definition,
% 0.19/0.45 ( symdiffI2
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ~ ( in @ Xx @ A )
% 0.19/0.45 => ( ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffIneg1_type,type,
% 0.19/0.45 symdiffIneg1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffIneg1,definition,
% 0.19/0.45 ( symdiffIneg1
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ( in @ Xx @ B )
% 0.19/0.45 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffIneg2_type,type,
% 0.19/0.45 symdiffIneg2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffIneg2,definition,
% 0.19/0.45 ( symdiffIneg2
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ~ ( in @ Xx @ A )
% 0.19/0.45 => ( ~ ( in @ Xx @ B )
% 0.19/0.45 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(iskpair_type,type,
% 0.19/0.45 iskpair: $i > $o ).
% 0.19/0.45
% 0.19/0.45 thf(iskpair,definition,
% 0.19/0.45 ( iskpair
% 0.19/0.45 = ( ^ [A: $i] :
% 0.19/0.45 ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.45 & ? [Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ ( setunion @ A ) )
% 0.19/0.45 & ( A
% 0.19/0.45 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(secondinupair,conjecture,
% 0.19/0.45 ( setextAx
% 0.19/0.45 => ( emptysetAx
% 0.19/0.45 => ( setadjoinAx
% 0.19/0.45 => ( powersetAx
% 0.19/0.45 => ( setunionAx
% 0.19/0.45 => ( omega0Ax
% 0.19/0.45 => ( omegaSAx
% 0.19/0.45 => ( omegaIndAx
% 0.19/0.45 => ( replAx
% 0.19/0.45 => ( foundationAx
% 0.19/0.45 => ( wellorderingAx
% 0.19/0.45 => ( descrp
% 0.19/0.45 => ( dsetconstrI
% 0.19/0.45 => ( dsetconstrEL
% 0.19/0.45 => ( dsetconstrER
% 0.19/0.45 => ( exuE1
% 0.19/0.45 => ( prop2setE
% 0.19/0.45 => ( emptysetE
% 0.19/0.45 => ( emptysetimpfalse
% 0.19/0.45 => ( notinemptyset
% 0.19/0.45 => ( exuE3e
% 0.19/0.45 => ( setext
% 0.19/0.45 => ( emptyI
% 0.19/0.45 => ( noeltsimpempty
% 0.19/0.45 => ( setbeta
% 0.19/0.45 => ( nonemptyE1
% 0.19/0.45 => ( nonemptyI
% 0.19/0.45 => ( nonemptyI1
% 0.19/0.45 => ( setadjoinIL
% 0.19/0.45 => ( emptyinunitempty
% 0.19/0.45 => ( setadjoinIR
% 0.19/0.45 => ( setadjoinE
% 0.19/0.45 => ( setadjoinOr
% 0.19/0.45 => ( setoftrueEq
% 0.19/0.45 => ( powersetI
% 0.19/0.45 => ( emptyinPowerset
% 0.19/0.45 => ( emptyInPowerset
% 0.19/0.45 => ( powersetE
% 0.19/0.45 => ( setunionI
% 0.19/0.45 => ( setunionE
% 0.19/0.45 => ( subPowSU
% 0.19/0.45 => ( exuE2
% 0.19/0.45 => ( nonemptyImpWitness
% 0.19/0.45 => ( uniqinunit
% 0.19/0.45 => ( notinsingleton
% 0.19/0.45 => ( eqinunit
% 0.19/0.45 => ( singletonsswitch
% 0.19/0.45 => ( upairsetE
% 0.19/0.45 => ( upairsetIL
% 0.19/0.45 => ( upairsetIR
% 0.19/0.45 => ( emptyE1
% 0.19/0.45 => ( vacuousDall
% 0.19/0.45 => ( quantDeMorgan1
% 0.19/0.45 => ( quantDeMorgan2
% 0.19/0.45 => ( quantDeMorgan3
% 0.19/0.45 => ( quantDeMorgan4
% 0.19/0.45 => ( prop2setI
% 0.19/0.45 => ( prop2set2propI
% 0.19/0.45 => ( notdexE
% 0.19/0.45 => ( notdallE
% 0.19/0.45 => ( exuI1
% 0.19/0.45 => ( exuI3
% 0.19/0.45 => ( exuI2
% 0.19/0.45 => ( inCongP
% 0.19/0.45 => ( in__Cong
% 0.19/0.45 => ( exuE3u
% 0.19/0.45 => ( exu__Cong
% 0.19/0.45 => ( emptyset__Cong
% 0.19/0.45 => ( setadjoin__Cong
% 0.19/0.45 => ( powerset__Cong
% 0.19/0.45 => ( setunion__Cong
% 0.19/0.45 => ( omega__Cong
% 0.19/0.45 => ( exuEu
% 0.19/0.45 => ( descr__Cong
% 0.19/0.45 => ( dsetconstr__Cong
% 0.19/0.45 => ( subsetI1
% 0.19/0.45 => ( eqimpsubset2
% 0.19/0.45 => ( eqimpsubset1
% 0.19/0.45 => ( subsetI2
% 0.19/0.45 => ( emptysetsubset
% 0.19/0.45 => ( subsetE
% 0.19/0.45 => ( subsetE2
% 0.19/0.45 => ( notsubsetI
% 0.19/0.45 => ( notequalI1
% 0.19/0.45 => ( notequalI2
% 0.19/0.45 => ( subsetRefl
% 0.19/0.45 => ( subsetTrans
% 0.19/0.45 => ( setadjoinSub
% 0.19/0.45 => ( setadjoinSub2
% 0.19/0.45 => ( subset2powerset
% 0.19/0.45 => ( setextsub
% 0.19/0.45 => ( subsetemptysetimpeq
% 0.19/0.45 => ( powersetI1
% 0.19/0.45 => ( powersetE1
% 0.19/0.45 => ( inPowerset
% 0.19/0.45 => ( powersetsubset
% 0.19/0.45 => ( sepInPowerset
% 0.19/0.45 => ( sepSubset
% 0.19/0.45 => ( binunionIL
% 0.19/0.45 => ( upairset2IR
% 0.19/0.45 => ( binunionIR
% 0.19/0.45 => ( binunionEcases
% 0.19/0.45 => ( binunionE
% 0.19/0.45 => ( binunionLsub
% 0.19/0.45 => ( binunionRsub
% 0.19/0.45 => ( binintersectI
% 0.19/0.45 => ( binintersectSubset5
% 0.19/0.45 => ( binintersectEL
% 0.19/0.45 => ( binintersectLsub
% 0.19/0.45 => ( binintersectSubset2
% 0.19/0.45 => ( binintersectSubset3
% 0.19/0.45 => ( binintersectER
% 0.19/0.45 => ( disjointsetsI1
% 0.19/0.45 => ( binintersectRsub
% 0.19/0.45 => ( binintersectSubset4
% 0.19/0.45 => ( binintersectSubset1
% 0.19/0.45 => ( bs114d
% 0.19/0.45 => ( setminusI
% 0.19/0.45 => ( setminusEL
% 0.19/0.45 => ( setminusER
% 0.19/0.45 => ( setminusSubset2
% 0.19/0.45 => ( setminusERneg
% 0.19/0.45 => ( setminusELneg
% 0.19/0.45 => ( setminusILneg
% 0.19/0.45 => ( setminusIRneg
% 0.19/0.45 => ( setminusLsub
% 0.19/0.45 => ( setminusSubset1
% 0.19/0.45 => ( symdiffE
% 0.19/0.45 => ( symdiffI1
% 0.19/0.45 => ( symdiffI2
% 0.19/0.45 => ( symdiffIneg1
% 0.19/0.45 => ( symdiffIneg2
% 0.19/0.45 => ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 ------------------------
% 2.61/1.63 ---- Embedded in THF ---
% 2.61/1.63 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.61/1.63 %%% Generated on Tue Feb 24 12:28:48 EST 2026
% 2.61/1.63 %%% using '$$dhol' embedding, version 1.3.0.
% 2.61/1.63 %%% Logic specification used:
% 2.61/1.63 %%% thf(spec, logic, $$dhol).
% 2.61/1.63
% 2.61/1.63 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.KCJafwTrbB/DTF2DTF13000.p
% See solution above
% 2.61/1.63 ------------------------
% 2.61/1.66 ---- Cleaned THF ---
% 2.61/1.66 thf(in_type,type,
% 2.61/1.66 in: $i > $i > $o ).
% 2.61/1.66
% 2.61/1.66 thf(exu_type,type,
% 2.61/1.66 exu: ( $i > $o ) > $o ).
% 2.61/1.66
% 2.61/1.66 thf(exu,definition,
% 2.61/1.66 ( exu
% 2.61/1.66 = ( ^ [Xphi: $i > $o] :
% 2.61/1.66 ? [Xx: $i] :
% 2.61/1.66 ( ( Xphi @ Xx )
% 2.61/1.66 & ! [Xy: $i] :
% 2.61/1.66 ( ( Xphi @ Xy )
% 2.61/1.66 => ( Xx = Xy ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setextAx_type,type,
% 2.61/1.66 setextAx: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setextAx,definition,
% 2.61/1.66 ( setextAx
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ! [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 <=> ( in @ Xx @ B ) )
% 2.61/1.66 => ( A = B ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(emptyset_type,type,
% 2.61/1.66 emptyset: $i ).
% 2.61/1.66
% 2.61/1.66 thf(emptysetAx_type,type,
% 2.61/1.66 emptysetAx: $o ).
% 2.61/1.66
% 2.61/1.66 thf(emptysetAx,definition,
% 2.61/1.66 ( emptysetAx
% 2.61/1.66 = ( ! [Xx: $i] :
% 2.61/1.66 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setadjoin_type,type,
% 2.61/1.66 setadjoin: $i > $i > $i ).
% 2.61/1.66
% 2.61/1.66 thf(setadjoinAx_type,type,
% 2.61/1.66 setadjoinAx: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setadjoinAx,definition,
% 2.61/1.66 ( setadjoinAx
% 2.61/1.66 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.61/1.66 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.61/1.66 <=> ( ( Xy = Xx )
% 2.61/1.66 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(powerset_type,type,
% 2.61/1.66 powerset: $i > $i ).
% 2.61/1.66
% 2.61/1.66 thf(powersetAx_type,type,
% 2.61/1.66 powersetAx: $o ).
% 2.61/1.66
% 2.61/1.66 thf(powersetAx,definition,
% 2.61/1.66 ( powersetAx
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ( in @ B @ ( powerset @ A ) )
% 2.61/1.66 <=> ! [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ B )
% 2.61/1.66 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setunion_type,type,
% 2.61/1.66 setunion: $i > $i ).
% 2.61/1.66
% 2.61/1.66 thf(setunionAx_type,type,
% 2.61/1.66 setunionAx: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setunionAx,definition,
% 2.61/1.66 ( setunionAx
% 2.61/1.66 = ( ! [A: $i,Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ ( setunion @ A ) )
% 2.61/1.66 <=> ? [B: $i] :
% 2.61/1.66 ( ( in @ Xx @ B )
% 2.61/1.66 & ( in @ B @ A ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(omega_type,type,
% 2.61/1.66 omega: $i ).
% 2.61/1.66
% 2.61/1.66 thf(omega0Ax_type,type,
% 2.61/1.66 omega0Ax: $o ).
% 2.61/1.66
% 2.61/1.66 thf(omega0Ax,definition,
% 2.61/1.66 ( omega0Ax
% 2.61/1.66 = ( in @ emptyset @ omega ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(omegaSAx_type,type,
% 2.61/1.66 omegaSAx: $o ).
% 2.61/1.66
% 2.61/1.66 thf(omegaSAx,definition,
% 2.61/1.66 ( omegaSAx
% 2.61/1.66 = ( ! [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ omega )
% 2.61/1.66 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(omegaIndAx_type,type,
% 2.61/1.66 omegaIndAx: $o ).
% 2.61/1.66
% 2.61/1.66 thf(omegaIndAx,definition,
% 2.61/1.66 ( omegaIndAx
% 2.61/1.66 = ( ! [A: $i] :
% 2.61/1.66 ( ( ( in @ emptyset @ A )
% 2.61/1.66 & ! [Xx: $i] :
% 2.61/1.66 ( ( ( in @ Xx @ omega )
% 2.61/1.66 & ( in @ Xx @ A ) )
% 2.61/1.66 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.61/1.66 => ! [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ omega )
% 2.61/1.66 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(replAx_type,type,
% 2.61/1.66 replAx: $o ).
% 2.61/1.66
% 2.61/1.66 thf(replAx,definition,
% 2.61/1.66 ( replAx
% 2.61/1.66 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.61/1.66 ( ! [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 => ( exu
% 2.61/1.66 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.61/1.66 => ? [B: $i] :
% 2.61/1.66 ! [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ B )
% 2.61/1.66 <=> ? [Xy: $i] :
% 2.61/1.66 ( ( in @ Xy @ A )
% 2.61/1.66 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(foundationAx_type,type,
% 2.61/1.66 foundationAx: $o ).
% 2.61/1.66
% 2.61/1.66 thf(foundationAx,definition,
% 2.61/1.66 ( foundationAx
% 2.61/1.66 = ( ! [A: $i] :
% 2.61/1.66 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.61/1.66 => ? [B: $i] :
% 2.61/1.66 ( ( in @ B @ A )
% 2.61/1.66 & ~ ? [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ B )
% 2.61/1.66 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(wellorderingAx_type,type,
% 2.61/1.66 wellorderingAx: $o ).
% 2.61/1.66
% 2.61/1.66 thf(wellorderingAx,definition,
% 2.61/1.66 ( wellorderingAx
% 2.61/1.66 = ( ! [A: $i] :
% 2.61/1.66 ? [B: $i] :
% 2.61/1.66 ( ! [C: $i] :
% 2.61/1.66 ( ( in @ C @ B )
% 2.61/1.66 => ! [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ C )
% 2.61/1.66 => ( in @ Xx @ A ) ) )
% 2.61/1.66 & ! [Xx: $i,Xy: $i] :
% 2.61/1.66 ( ( ( in @ Xx @ A )
% 2.61/1.66 & ( in @ Xy @ A ) )
% 2.61/1.66 => ( ! [C: $i] :
% 2.61/1.66 ( ( in @ C @ B )
% 2.61/1.66 => ( ( in @ Xx @ C )
% 2.61/1.66 <=> ( in @ Xy @ C ) ) )
% 2.61/1.66 => ( Xx = Xy ) ) )
% 2.61/1.66 & ! [C: $i,D: $i] :
% 2.61/1.66 ( ( ( in @ C @ B )
% 2.61/1.66 & ( in @ D @ B ) )
% 2.61/1.66 => ( ! [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ C )
% 2.61/1.66 => ( in @ Xx @ D ) )
% 2.61/1.66 | ! [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ D )
% 2.61/1.66 => ( in @ Xx @ C ) ) ) )
% 2.61/1.66 & ! [C: $i] :
% 2.61/1.66 ( ( ! [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ C )
% 2.61/1.66 => ( in @ Xx @ A ) )
% 2.61/1.66 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.61/1.66 => ? [D: $i,Xx: $i] :
% 2.61/1.66 ( ( in @ D @ B )
% 2.61/1.66 & ( in @ Xx @ C )
% 2.61/1.66 & ~ ? [Xy: $i] :
% 2.61/1.66 ( ( in @ Xy @ D )
% 2.61/1.66 & ( in @ Xy @ C ) )
% 2.61/1.66 & ! [E: $i] :
% 2.61/1.66 ( ( in @ E @ B )
% 2.61/1.66 => ( ! [Xy: $i] :
% 2.61/1.66 ( ( in @ Xy @ E )
% 2.61/1.66 => ( in @ Xy @ D ) )
% 2.61/1.66 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(descr_type,type,
% 2.61/1.66 descr: ( $i > $o ) > $i ).
% 2.61/1.66
% 2.61/1.66 thf(descrp_type,type,
% 2.61/1.66 descrp: $o ).
% 2.61/1.66
% 2.61/1.66 thf(descrp,definition,
% 2.61/1.66 ( descrp
% 2.61/1.66 = ( ! [Xphi: $i > $o] :
% 2.61/1.66 ( ( exu
% 2.61/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.66 => ( Xphi
% 2.61/1.66 @ ( descr
% 2.61/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(dsetconstr_type,type,
% 2.61/1.66 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.61/1.66
% 2.61/1.66 thf(dsetconstrI_type,type,
% 2.61/1.66 dsetconstrI: $o ).
% 2.61/1.66
% 2.61/1.66 thf(dsetconstrI,definition,
% 2.61/1.66 ( dsetconstrI
% 2.61/1.66 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 => ( ( Xphi @ Xx )
% 2.61/1.66 => ( in @ Xx
% 2.61/1.66 @ ( dsetconstr @ A
% 2.61/1.66 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(dsetconstrEL_type,type,
% 2.61/1.66 dsetconstrEL: $o ).
% 2.61/1.66
% 2.61/1.66 thf(dsetconstrEL,definition,
% 2.61/1.66 ( dsetconstrEL
% 2.61/1.66 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.61/1.66 ( ( in @ Xx
% 2.61/1.66 @ ( dsetconstr @ A
% 2.61/1.66 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.61/1.66 => ( in @ Xx @ A ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(dsetconstrER_type,type,
% 2.61/1.66 dsetconstrER: $o ).
% 2.61/1.66
% 2.61/1.66 thf(dsetconstrER,definition,
% 2.61/1.66 ( dsetconstrER
% 2.61/1.66 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.61/1.66 ( ( in @ Xx
% 2.61/1.66 @ ( dsetconstr @ A
% 2.61/1.66 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.61/1.66 => ( Xphi @ Xx ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(exuE1_type,type,
% 2.61/1.66 exuE1: $o ).
% 2.61/1.66
% 2.61/1.66 thf(exuE1,definition,
% 2.61/1.66 ( exuE1
% 2.61/1.66 = ( ! [Xphi: $i > $o] :
% 2.61/1.66 ( ( exu
% 2.61/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.66 => ? [Xx: $i] :
% 2.61/1.66 ( ( Xphi @ Xx )
% 2.61/1.66 & ! [Xy: $i] :
% 2.61/1.66 ( ( Xphi @ Xy )
% 2.61/1.66 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(prop2set_type,type,
% 2.61/1.66 prop2set: $o > $i ).
% 2.61/1.66
% 2.61/1.66 thf(prop2setE_type,type,
% 2.61/1.66 prop2setE: $o ).
% 2.61/1.66
% 2.61/1.66 thf(prop2setE,definition,
% 2.61/1.66 ( prop2setE
% 2.61/1.66 = ( ! [Xphi: $o,Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.61/1.66 => Xphi ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(emptysetE_type,type,
% 2.61/1.66 emptysetE: $o ).
% 2.61/1.66
% 2.61/1.66 thf(emptysetE,definition,
% 2.61/1.66 ( emptysetE
% 2.61/1.66 = ( ! [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ emptyset )
% 2.61/1.66 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(emptysetimpfalse_type,type,
% 2.61/1.66 emptysetimpfalse: $o ).
% 2.61/1.66
% 2.61/1.66 thf(emptysetimpfalse,definition,
% 2.61/1.66 ( emptysetimpfalse
% 2.61/1.66 = ( ! [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ emptyset )
% 2.61/1.66 => $false ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(notinemptyset_type,type,
% 2.61/1.66 notinemptyset: $o ).
% 2.61/1.66
% 2.61/1.66 thf(notinemptyset,definition,
% 2.61/1.66 ( notinemptyset
% 2.61/1.66 = ( ! [Xx: $i] :
% 2.61/1.66 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(exuE3e_type,type,
% 2.61/1.66 exuE3e: $o ).
% 2.61/1.66
% 2.61/1.66 thf(exuE3e,definition,
% 2.61/1.66 ( exuE3e
% 2.61/1.66 = ( ! [Xphi: $i > $o] :
% 2.61/1.66 ( ( exu
% 2.61/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.66 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setext_type,type,
% 2.61/1.66 setext: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setext,definition,
% 2.61/1.66 ( setext
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ! [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 => ( in @ Xx @ B ) )
% 2.61/1.66 => ( ! [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ B )
% 2.61/1.66 => ( in @ Xx @ A ) )
% 2.61/1.66 => ( A = B ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(emptyI_type,type,
% 2.61/1.66 emptyI: $o ).
% 2.61/1.66
% 2.61/1.66 thf(emptyI,definition,
% 2.61/1.66 ( emptyI
% 2.61/1.66 = ( ! [A: $i] :
% 2.61/1.66 ( ! [Xx: $i] :
% 2.61/1.66 ~ ( in @ Xx @ A )
% 2.61/1.66 => ( A = emptyset ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(noeltsimpempty_type,type,
% 2.61/1.66 noeltsimpempty: $o ).
% 2.61/1.66
% 2.61/1.66 thf(noeltsimpempty,definition,
% 2.61/1.66 ( noeltsimpempty
% 2.61/1.66 = ( ! [A: $i] :
% 2.61/1.66 ( ! [Xx: $i] :
% 2.61/1.66 ~ ( in @ Xx @ A )
% 2.61/1.66 => ( A = emptyset ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setbeta_type,type,
% 2.61/1.66 setbeta: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setbeta,definition,
% 2.61/1.66 ( setbeta
% 2.61/1.66 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 => ( ( in @ Xx
% 2.61/1.66 @ ( dsetconstr @ A
% 2.61/1.66 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.61/1.66 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(nonempty_type,type,
% 2.61/1.66 nonempty: $i > $o ).
% 2.61/1.66
% 2.61/1.66 thf(nonempty,definition,
% 2.61/1.66 ( nonempty
% 2.61/1.66 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(nonemptyE1_type,type,
% 2.61/1.66 nonemptyE1: $o ).
% 2.61/1.66
% 2.61/1.66 thf(nonemptyE1,definition,
% 2.61/1.66 ( nonemptyE1
% 2.61/1.66 = ( ! [A: $i] :
% 2.61/1.66 ( ( nonempty @ A )
% 2.61/1.66 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(nonemptyI_type,type,
% 2.61/1.66 nonemptyI: $o ).
% 2.61/1.66
% 2.61/1.66 thf(nonemptyI,definition,
% 2.61/1.66 ( nonemptyI
% 2.61/1.66 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 => ( ( Xphi @ Xx )
% 2.61/1.66 => ( nonempty
% 2.61/1.66 @ ( dsetconstr @ A
% 2.61/1.66 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(nonemptyI1_type,type,
% 2.61/1.66 nonemptyI1: $o ).
% 2.61/1.66
% 2.61/1.66 thf(nonemptyI1,definition,
% 2.61/1.66 ( nonemptyI1
% 2.61/1.66 = ( ! [A: $i] :
% 2.61/1.66 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.61/1.66 => ( nonempty @ A ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setadjoinIL_type,type,
% 2.61/1.66 setadjoinIL: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setadjoinIL,definition,
% 2.61/1.66 ( setadjoinIL
% 2.61/1.66 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(emptyinunitempty_type,type,
% 2.61/1.66 emptyinunitempty: $o ).
% 2.61/1.66
% 2.61/1.66 thf(emptyinunitempty,definition,
% 2.61/1.66 ( emptyinunitempty
% 2.61/1.66 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setadjoinIR_type,type,
% 2.61/1.66 setadjoinIR: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setadjoinIR,definition,
% 2.61/1.66 ( setadjoinIR
% 2.61/1.66 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.61/1.66 ( ( in @ Xy @ A )
% 2.61/1.66 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setadjoinE_type,type,
% 2.61/1.66 setadjoinE: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setadjoinE,definition,
% 2.61/1.66 ( setadjoinE
% 2.61/1.66 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.61/1.66 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.61/1.66 => ! [Xphi: $o] :
% 2.61/1.66 ( ( ( Xy = Xx )
% 2.61/1.66 => Xphi )
% 2.61/1.66 => ( ( ( in @ Xy @ A )
% 2.61/1.66 => Xphi )
% 2.61/1.66 => Xphi ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setadjoinOr_type,type,
% 2.61/1.66 setadjoinOr: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setadjoinOr,definition,
% 2.61/1.66 ( setadjoinOr
% 2.61/1.66 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.61/1.66 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.61/1.66 => ( ( Xy = Xx )
% 2.61/1.66 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setoftrueEq_type,type,
% 2.61/1.66 setoftrueEq: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setoftrueEq,definition,
% 2.61/1.66 ( setoftrueEq
% 2.61/1.66 = ( ! [A: $i] :
% 2.61/1.66 ( ( dsetconstr @ A
% 2.61/1.66 @ ^ [Xx: $i] : $true )
% 2.61/1.66 = A ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(powersetI_type,type,
% 2.61/1.66 powersetI: $o ).
% 2.61/1.66
% 2.61/1.66 thf(powersetI,definition,
% 2.61/1.66 ( powersetI
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ! [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ B )
% 2.61/1.66 => ( in @ Xx @ A ) )
% 2.61/1.66 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(emptyinPowerset_type,type,
% 2.61/1.66 emptyinPowerset: $o ).
% 2.61/1.66
% 2.61/1.66 thf(emptyinPowerset,definition,
% 2.61/1.66 ( emptyinPowerset
% 2.61/1.66 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(emptyInPowerset_type,type,
% 2.61/1.66 emptyInPowerset: $o ).
% 2.61/1.66
% 2.61/1.66 thf(emptyInPowerset,definition,
% 2.61/1.66 ( emptyInPowerset
% 2.61/1.66 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(powersetE_type,type,
% 2.61/1.66 powersetE: $o ).
% 2.61/1.66
% 2.61/1.66 thf(powersetE,definition,
% 2.61/1.66 ( powersetE
% 2.61/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.66 ( ( in @ B @ ( powerset @ A ) )
% 2.61/1.66 => ( ( in @ Xx @ B )
% 2.61/1.66 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setunionI_type,type,
% 2.61/1.66 setunionI: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setunionI,definition,
% 2.61/1.66 ( setunionI
% 2.61/1.66 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.61/1.66 ( ( in @ Xx @ B )
% 2.61/1.66 => ( ( in @ B @ A )
% 2.61/1.66 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setunionE_type,type,
% 2.61/1.66 setunionE: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setunionE,definition,
% 2.61/1.66 ( setunionE
% 2.61/1.66 = ( ! [A: $i,Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ ( setunion @ A ) )
% 2.61/1.66 => ! [Xphi: $o] :
% 2.61/1.66 ( ! [B: $i] :
% 2.61/1.66 ( ( in @ Xx @ B )
% 2.61/1.66 => ( ( in @ B @ A )
% 2.61/1.66 => Xphi ) )
% 2.61/1.66 => Xphi ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(subPowSU_type,type,
% 2.61/1.66 subPowSU: $o ).
% 2.61/1.66
% 2.61/1.66 thf(subPowSU,definition,
% 2.61/1.66 ( subPowSU
% 2.61/1.66 = ( ! [A: $i,Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(exuE2_type,type,
% 2.61/1.66 exuE2: $o ).
% 2.61/1.66
% 2.61/1.66 thf(exuE2,definition,
% 2.61/1.66 ( exuE2
% 2.61/1.66 = ( ! [Xphi: $i > $o] :
% 2.61/1.66 ( ( exu
% 2.61/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.66 => ? [Xx: $i] :
% 2.61/1.66 ! [Xy: $i] :
% 2.61/1.66 ( ( Xphi @ Xy )
% 2.61/1.66 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(nonemptyImpWitness_type,type,
% 2.61/1.66 nonemptyImpWitness: $o ).
% 2.61/1.66
% 2.61/1.66 thf(nonemptyImpWitness,definition,
% 2.61/1.66 ( nonemptyImpWitness
% 2.61/1.66 = ( ! [A: $i] :
% 2.61/1.66 ( ( nonempty @ A )
% 2.61/1.66 => ? [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 & $true ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(uniqinunit_type,type,
% 2.61/1.66 uniqinunit: $o ).
% 2.61/1.66
% 2.61/1.66 thf(uniqinunit,definition,
% 2.61/1.66 ( uniqinunit
% 2.61/1.66 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.66 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.61/1.66 => ( Xx = Xy ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(notinsingleton_type,type,
% 2.61/1.66 notinsingleton: $o ).
% 2.61/1.66
% 2.61/1.66 thf(notinsingleton,definition,
% 2.61/1.66 ( notinsingleton
% 2.61/1.66 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.66 ( ( Xx != Xy )
% 2.61/1.66 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(eqinunit_type,type,
% 2.61/1.66 eqinunit: $o ).
% 2.61/1.66
% 2.61/1.66 thf(eqinunit,definition,
% 2.61/1.66 ( eqinunit
% 2.61/1.66 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.66 ( ( Xx = Xy )
% 2.61/1.66 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(singletonsswitch_type,type,
% 2.61/1.66 singletonsswitch: $o ).
% 2.61/1.66
% 2.61/1.66 thf(singletonsswitch,definition,
% 2.61/1.66 ( singletonsswitch
% 2.61/1.66 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.66 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.61/1.66 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(upairsetE_type,type,
% 2.61/1.66 upairsetE: $o ).
% 2.61/1.66
% 2.61/1.66 thf(upairsetE,definition,
% 2.61/1.66 ( upairsetE
% 2.61/1.66 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.61/1.66 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.61/1.66 => ( ( Xz = Xx )
% 2.61/1.66 | ( Xz = Xy ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(upairsetIL_type,type,
% 2.61/1.66 upairsetIL: $o ).
% 2.61/1.66
% 2.61/1.66 thf(upairsetIL,definition,
% 2.61/1.66 ( upairsetIL
% 2.61/1.66 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(upairsetIR_type,type,
% 2.61/1.66 upairsetIR: $o ).
% 2.61/1.66
% 2.61/1.66 thf(upairsetIR,definition,
% 2.61/1.66 ( upairsetIR
% 2.61/1.66 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(emptyE1_type,type,
% 2.61/1.66 emptyE1: $o ).
% 2.61/1.66
% 2.61/1.66 thf(emptyE1,definition,
% 2.61/1.66 ( emptyE1
% 2.61/1.66 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.66 ( ? [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 & ( Xphi @ Xx ) )
% 2.61/1.66 => ( ( ( dsetconstr @ A
% 2.61/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.66 = emptyset )
% 2.61/1.66 => $false ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(vacuousDall_type,type,
% 2.61/1.66 vacuousDall: $o ).
% 2.61/1.66
% 2.61/1.66 thf(vacuousDall,definition,
% 2.61/1.66 ( vacuousDall
% 2.61/1.66 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ emptyset )
% 2.61/1.66 => ( Xphi @ Xx ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(quantDeMorgan1_type,type,
% 2.61/1.66 quantDeMorgan1: $o ).
% 2.61/1.66
% 2.61/1.66 thf(quantDeMorgan1,definition,
% 2.61/1.66 ( quantDeMorgan1
% 2.61/1.66 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.66 ( ~ ! [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 => ( Xphi @ Xx ) )
% 2.61/1.66 => ? [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(quantDeMorgan2_type,type,
% 2.61/1.66 quantDeMorgan2: $o ).
% 2.61/1.66
% 2.61/1.66 thf(quantDeMorgan2,definition,
% 2.61/1.66 ( quantDeMorgan2
% 2.61/1.66 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.66 ( ! [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 => ~ ( Xphi @ Xx ) )
% 2.61/1.66 => ~ ? [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(quantDeMorgan3_type,type,
% 2.61/1.66 quantDeMorgan3: $o ).
% 2.61/1.66
% 2.61/1.66 thf(quantDeMorgan3,definition,
% 2.61/1.66 ( quantDeMorgan3
% 2.61/1.66 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.66 ( ~ ? [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 & ( Xphi @ Xx ) )
% 2.61/1.66 => ! [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(quantDeMorgan4_type,type,
% 2.61/1.66 quantDeMorgan4: $o ).
% 2.61/1.66
% 2.61/1.66 thf(quantDeMorgan4,definition,
% 2.61/1.66 ( quantDeMorgan4
% 2.61/1.66 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.66 ( ? [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 & ~ ( Xphi @ Xx ) )
% 2.61/1.66 => ~ ! [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(prop2setI_type,type,
% 2.61/1.66 prop2setI: $o ).
% 2.61/1.66
% 2.61/1.66 thf(prop2setI,definition,
% 2.61/1.66 ( prop2setI
% 2.61/1.66 = ( ! [Xphi: $o] :
% 2.61/1.66 ( Xphi
% 2.61/1.66 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(set2prop_type,type,
% 2.61/1.66 set2prop: $i > $o ).
% 2.61/1.66
% 2.61/1.66 thf(prop2set2propI_type,type,
% 2.61/1.66 prop2set2propI: $o ).
% 2.61/1.66
% 2.61/1.66 thf(prop2set2propI,definition,
% 2.61/1.66 ( prop2set2propI
% 2.61/1.66 = ( ! [Xphi: $o] :
% 2.61/1.66 ( Xphi
% 2.61/1.66 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(notdexE_type,type,
% 2.61/1.66 notdexE: $o ).
% 2.61/1.66
% 2.61/1.66 thf(notdexE,definition,
% 2.61/1.66 ( notdexE
% 2.61/1.66 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.66 ( ~ ? [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 & ( Xphi @ Xx ) )
% 2.61/1.66 => ! [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(notdallE_type,type,
% 2.61/1.66 notdallE: $o ).
% 2.61/1.66
% 2.61/1.66 thf(notdallE,definition,
% 2.61/1.66 ( notdallE
% 2.61/1.66 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.66 ( ~ ! [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 => ( Xphi @ Xx ) )
% 2.61/1.66 => ? [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(exuI1_type,type,
% 2.61/1.66 exuI1: $o ).
% 2.61/1.66
% 2.61/1.66 thf(exuI1,definition,
% 2.61/1.66 ( exuI1
% 2.61/1.66 = ( ! [Xphi: $i > $o] :
% 2.61/1.66 ( ? [Xx: $i] :
% 2.61/1.66 ( ( Xphi @ Xx )
% 2.61/1.66 & ! [Xy: $i] :
% 2.61/1.66 ( ( Xphi @ Xy )
% 2.61/1.66 => ( Xx = Xy ) ) )
% 2.61/1.66 => ( exu
% 2.61/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(exuI3_type,type,
% 2.61/1.66 exuI3: $o ).
% 2.61/1.66
% 2.61/1.66 thf(exuI3,definition,
% 2.61/1.66 ( exuI3
% 2.61/1.66 = ( ! [Xphi: $i > $o] :
% 2.61/1.66 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.61/1.66 => ( ! [Xx: $i,Xy: $i] :
% 2.61/1.66 ( ( Xphi @ Xx )
% 2.61/1.66 => ( ( Xphi @ Xy )
% 2.61/1.66 => ( Xx = Xy ) ) )
% 2.61/1.66 => ( exu
% 2.61/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(exuI2_type,type,
% 2.61/1.66 exuI2: $o ).
% 2.61/1.66
% 2.61/1.66 thf(exuI2,definition,
% 2.61/1.66 ( exuI2
% 2.61/1.66 = ( ! [Xphi: $i > $o] :
% 2.61/1.66 ( ? [Xx: $i] :
% 2.61/1.66 ! [Xy: $i] :
% 2.61/1.66 ( ( Xphi @ Xy )
% 2.61/1.66 <=> ( Xy = Xx ) )
% 2.61/1.66 => ( exu
% 2.61/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(inCongP_type,type,
% 2.61/1.66 inCongP: $o ).
% 2.61/1.66
% 2.61/1.66 thf(inCongP,definition,
% 2.61/1.66 ( inCongP
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ( A = B )
% 2.61/1.66 => ! [Xx: $i,Xy: $i] :
% 2.61/1.66 ( ( Xx = Xy )
% 2.61/1.66 => ( ( in @ Xx @ A )
% 2.61/1.66 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(in__Cong_type,type,
% 2.61/1.66 in__Cong: $o ).
% 2.61/1.66
% 2.61/1.66 thf(in__Cong,definition,
% 2.61/1.66 ( in__Cong
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ( A = B )
% 2.61/1.66 => ! [Xx: $i,Xy: $i] :
% 2.61/1.66 ( ( Xx = Xy )
% 2.61/1.66 => ( ( in @ Xx @ A )
% 2.61/1.66 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(exuE3u_type,type,
% 2.61/1.66 exuE3u: $o ).
% 2.61/1.66
% 2.61/1.66 thf(exuE3u,definition,
% 2.61/1.66 ( exuE3u
% 2.61/1.66 = ( ! [Xphi: $i > $o] :
% 2.61/1.66 ( ( exu
% 2.61/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.66 => ! [Xx: $i,Xy: $i] :
% 2.61/1.66 ( ( Xphi @ Xx )
% 2.61/1.66 => ( ( Xphi @ Xy )
% 2.61/1.66 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(exu__Cong_type,type,
% 2.61/1.66 exu__Cong: $o ).
% 2.61/1.66
% 2.61/1.66 thf(exu__Cong,definition,
% 2.61/1.66 ( exu__Cong
% 2.61/1.66 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.61/1.66 ( ! [Xx: $i,Xy: $i] :
% 2.61/1.66 ( ( Xx = Xy )
% 2.61/1.66 => ( ( Xphi @ Xx )
% 2.61/1.66 <=> ( Xpsi @ Xy ) ) )
% 2.61/1.66 => ( ( exu
% 2.61/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.66 <=> ( exu
% 2.61/1.66 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(emptyset__Cong_type,type,
% 2.61/1.66 emptyset__Cong: $o ).
% 2.61/1.66
% 2.61/1.66 thf(emptyset__Cong,definition,
% 2.61/1.66 ( emptyset__Cong
% 2.61/1.66 = ( emptyset = emptyset ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setadjoin__Cong_type,type,
% 2.61/1.66 setadjoin__Cong: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setadjoin__Cong,definition,
% 2.61/1.66 ( setadjoin__Cong
% 2.61/1.66 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.66 ( ( Xx = Xy )
% 2.61/1.66 => ! [Xz: $i,Xu: $i] :
% 2.61/1.66 ( ( Xz = Xu )
% 2.61/1.66 => ( ( setadjoin @ Xx @ Xz )
% 2.61/1.66 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(powerset__Cong_type,type,
% 2.61/1.66 powerset__Cong: $o ).
% 2.61/1.66
% 2.61/1.66 thf(powerset__Cong,definition,
% 2.61/1.66 ( powerset__Cong
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ( A = B )
% 2.61/1.66 => ( ( powerset @ A )
% 2.61/1.66 = ( powerset @ B ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setunion__Cong_type,type,
% 2.61/1.66 setunion__Cong: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setunion__Cong,definition,
% 2.61/1.66 ( setunion__Cong
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ( A = B )
% 2.61/1.66 => ( ( setunion @ A )
% 2.61/1.66 = ( setunion @ B ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(omega__Cong_type,type,
% 2.61/1.66 omega__Cong: $o ).
% 2.61/1.66
% 2.61/1.66 thf(omega__Cong,definition,
% 2.61/1.66 ( omega__Cong
% 2.61/1.66 = ( omega = omega ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(exuEu_type,type,
% 2.61/1.66 exuEu: $o ).
% 2.61/1.66
% 2.61/1.66 thf(exuEu,definition,
% 2.61/1.66 ( exuEu
% 2.61/1.66 = ( ! [Xphi: $i > $o] :
% 2.61/1.66 ( ( exu
% 2.61/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.66 => ! [Xx: $i,Xy: $i] :
% 2.61/1.66 ( ( Xphi @ Xx )
% 2.61/1.66 => ( ( Xphi @ Xy )
% 2.61/1.66 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(descr__Cong_type,type,
% 2.61/1.66 descr__Cong: $o ).
% 2.61/1.66
% 2.61/1.66 thf(descr__Cong,definition,
% 2.61/1.66 ( descr__Cong
% 2.61/1.66 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.61/1.66 ( ! [Xx: $i,Xy: $i] :
% 2.61/1.66 ( ( Xx = Xy )
% 2.61/1.66 => ( ( Xphi @ Xx )
% 2.61/1.66 <=> ( Xpsi @ Xy ) ) )
% 2.61/1.66 => ( ( exu
% 2.61/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.66 => ( ( exu
% 2.61/1.66 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.61/1.66 => ( ( descr
% 2.61/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.66 = ( descr
% 2.61/1.66 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(dsetconstr__Cong_type,type,
% 2.61/1.66 dsetconstr__Cong: $o ).
% 2.61/1.66
% 2.61/1.66 thf(dsetconstr__Cong,definition,
% 2.61/1.66 ( dsetconstr__Cong
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ( A = B )
% 2.61/1.66 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.61/1.66 ( ! [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 => ! [Xy: $i] :
% 2.61/1.66 ( ( in @ Xy @ B )
% 2.61/1.66 => ( ( Xx = Xy )
% 2.61/1.66 => ( ( Xphi @ Xx )
% 2.61/1.66 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.61/1.66 => ( ( dsetconstr @ A
% 2.61/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.66 = ( dsetconstr @ B
% 2.61/1.66 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(subset_type,type,
% 2.61/1.66 subset: $i > $i > $o ).
% 2.61/1.66
% 2.61/1.66 thf(disjoint_type,type,
% 2.61/1.66 disjoint: $i > $i > $o ).
% 2.61/1.66
% 2.61/1.66 thf(setsmeet_type,type,
% 2.61/1.66 setsmeet: $i > $i > $o ).
% 2.61/1.66
% 2.61/1.66 thf(subsetI1_type,type,
% 2.61/1.66 subsetI1: $o ).
% 2.61/1.66
% 2.61/1.66 thf(subsetI1,definition,
% 2.61/1.66 ( subsetI1
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ! [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 => ( in @ Xx @ B ) )
% 2.61/1.66 => ( subset @ A @ B ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(eqimpsubset2_type,type,
% 2.61/1.66 eqimpsubset2: $o ).
% 2.61/1.66
% 2.61/1.66 thf(eqimpsubset2,definition,
% 2.61/1.66 ( eqimpsubset2
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ( A = B )
% 2.61/1.66 => ( subset @ B @ A ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(eqimpsubset1_type,type,
% 2.61/1.66 eqimpsubset1: $o ).
% 2.61/1.66
% 2.61/1.66 thf(eqimpsubset1,definition,
% 2.61/1.66 ( eqimpsubset1
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ( A = B )
% 2.61/1.66 => ( subset @ A @ B ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(subsetI2_type,type,
% 2.61/1.66 subsetI2: $o ).
% 2.61/1.66
% 2.61/1.66 thf(subsetI2,definition,
% 2.61/1.66 ( subsetI2
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ! [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 => ( in @ Xx @ B ) )
% 2.61/1.66 => ( subset @ A @ B ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(emptysetsubset_type,type,
% 2.61/1.66 emptysetsubset: $o ).
% 2.61/1.66
% 2.61/1.66 thf(emptysetsubset,definition,
% 2.61/1.66 ( emptysetsubset
% 2.61/1.66 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(subsetE_type,type,
% 2.61/1.66 subsetE: $o ).
% 2.61/1.66
% 2.61/1.66 thf(subsetE,definition,
% 2.61/1.66 ( subsetE
% 2.61/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.66 ( ( subset @ A @ B )
% 2.61/1.66 => ( ( in @ Xx @ A )
% 2.61/1.66 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(subsetE2_type,type,
% 2.61/1.66 subsetE2: $o ).
% 2.61/1.66
% 2.61/1.66 thf(subsetE2,definition,
% 2.61/1.66 ( subsetE2
% 2.61/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.66 ( ( subset @ A @ B )
% 2.61/1.66 => ( ~ ( in @ Xx @ B )
% 2.61/1.66 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(notsubsetI_type,type,
% 2.61/1.66 notsubsetI: $o ).
% 2.61/1.66
% 2.61/1.66 thf(notsubsetI,definition,
% 2.61/1.66 ( notsubsetI
% 2.61/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 => ( ~ ( in @ Xx @ B )
% 2.61/1.66 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(notequalI1_type,type,
% 2.61/1.66 notequalI1: $o ).
% 2.61/1.66
% 2.61/1.66 thf(notequalI1,definition,
% 2.61/1.66 ( notequalI1
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ~ ( subset @ A @ B )
% 2.61/1.66 => ( A != B ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(notequalI2_type,type,
% 2.61/1.66 notequalI2: $o ).
% 2.61/1.66
% 2.61/1.66 thf(notequalI2,definition,
% 2.61/1.66 ( notequalI2
% 2.61/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 => ( ~ ( in @ Xx @ B )
% 2.61/1.66 => ( A != B ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(subsetRefl_type,type,
% 2.61/1.66 subsetRefl: $o ).
% 2.61/1.66
% 2.61/1.66 thf(subsetRefl,definition,
% 2.61/1.66 ( subsetRefl
% 2.61/1.66 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(subsetTrans_type,type,
% 2.61/1.66 subsetTrans: $o ).
% 2.61/1.66
% 2.61/1.66 thf(subsetTrans,definition,
% 2.61/1.66 ( subsetTrans
% 2.61/1.66 = ( ! [A: $i,B: $i,C: $i] :
% 2.61/1.66 ( ( subset @ A @ B )
% 2.61/1.66 => ( ( subset @ B @ C )
% 2.61/1.66 => ( subset @ A @ C ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setadjoinSub_type,type,
% 2.61/1.66 setadjoinSub: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setadjoinSub,definition,
% 2.61/1.66 ( setadjoinSub
% 2.61/1.66 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setadjoinSub2_type,type,
% 2.61/1.66 setadjoinSub2: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setadjoinSub2,definition,
% 2.61/1.66 ( setadjoinSub2
% 2.61/1.66 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.61/1.66 ( ( subset @ A @ B )
% 2.61/1.66 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(subset2powerset_type,type,
% 2.61/1.66 subset2powerset: $o ).
% 2.61/1.66
% 2.61/1.66 thf(subset2powerset,definition,
% 2.61/1.66 ( subset2powerset
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ( subset @ A @ B )
% 2.61/1.66 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setextsub_type,type,
% 2.61/1.66 setextsub: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setextsub,definition,
% 2.61/1.66 ( setextsub
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ( subset @ A @ B )
% 2.61/1.66 => ( ( subset @ B @ A )
% 2.61/1.66 => ( A = B ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(subsetemptysetimpeq_type,type,
% 2.61/1.66 subsetemptysetimpeq: $o ).
% 2.61/1.66
% 2.61/1.66 thf(subsetemptysetimpeq,definition,
% 2.61/1.66 ( subsetemptysetimpeq
% 2.61/1.66 = ( ! [A: $i] :
% 2.61/1.66 ( ( subset @ A @ emptyset )
% 2.61/1.66 => ( A = emptyset ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(powersetI1_type,type,
% 2.61/1.66 powersetI1: $o ).
% 2.61/1.66
% 2.61/1.66 thf(powersetI1,definition,
% 2.61/1.66 ( powersetI1
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ( subset @ B @ A )
% 2.61/1.66 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(powersetE1_type,type,
% 2.61/1.66 powersetE1: $o ).
% 2.61/1.66
% 2.61/1.66 thf(powersetE1,definition,
% 2.61/1.66 ( powersetE1
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ( in @ B @ ( powerset @ A ) )
% 2.61/1.66 => ( subset @ B @ A ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(inPowerset_type,type,
% 2.61/1.66 inPowerset: $o ).
% 2.61/1.66
% 2.61/1.66 thf(inPowerset,definition,
% 2.61/1.66 ( inPowerset
% 2.61/1.66 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(powersetsubset_type,type,
% 2.61/1.66 powersetsubset: $o ).
% 2.61/1.66
% 2.61/1.66 thf(powersetsubset,definition,
% 2.61/1.66 ( powersetsubset
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ( subset @ A @ B )
% 2.61/1.66 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(sepInPowerset_type,type,
% 2.61/1.66 sepInPowerset: $o ).
% 2.61/1.66
% 2.61/1.66 thf(sepInPowerset,definition,
% 2.61/1.66 ( sepInPowerset
% 2.61/1.66 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.66 ( in
% 2.61/1.66 @ ( dsetconstr @ A
% 2.61/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.66 @ ( powerset @ A ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(sepSubset_type,type,
% 2.61/1.66 sepSubset: $o ).
% 2.61/1.66
% 2.61/1.66 thf(sepSubset,definition,
% 2.61/1.66 ( sepSubset
% 2.61/1.66 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.66 ( subset
% 2.61/1.66 @ ( dsetconstr @ A
% 2.61/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.66 @ A ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(binunion_type,type,
% 2.61/1.66 binunion: $i > $i > $i ).
% 2.61/1.66
% 2.61/1.66 thf(binunionIL_type,type,
% 2.61/1.66 binunionIL: $o ).
% 2.61/1.66
% 2.61/1.66 thf(binunionIL,definition,
% 2.61/1.66 ( binunionIL
% 2.61/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(upairset2IR_type,type,
% 2.61/1.66 upairset2IR: $o ).
% 2.61/1.66
% 2.61/1.66 thf(upairset2IR,definition,
% 2.61/1.66 ( upairset2IR
% 2.61/1.66 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(binunionIR_type,type,
% 2.61/1.66 binunionIR: $o ).
% 2.61/1.66
% 2.61/1.66 thf(binunionIR,definition,
% 2.61/1.66 ( binunionIR
% 2.61/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ B )
% 2.61/1.66 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(binunionEcases_type,type,
% 2.61/1.66 binunionEcases: $o ).
% 2.61/1.66
% 2.61/1.66 thf(binunionEcases,definition,
% 2.61/1.66 ( binunionEcases
% 2.61/1.66 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.61/1.66 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.61/1.66 => ( ( ( in @ Xx @ A )
% 2.61/1.66 => Xphi )
% 2.61/1.66 => ( ( ( in @ Xx @ B )
% 2.61/1.66 => Xphi )
% 2.61/1.66 => Xphi ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(binunionE_type,type,
% 2.61/1.66 binunionE: $o ).
% 2.61/1.66
% 2.61/1.66 thf(binunionE,definition,
% 2.61/1.66 ( binunionE
% 2.61/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.61/1.66 => ( ( in @ Xx @ A )
% 2.61/1.66 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(binunionLsub_type,type,
% 2.61/1.66 binunionLsub: $o ).
% 2.61/1.66
% 2.61/1.66 thf(binunionLsub,definition,
% 2.61/1.66 ( binunionLsub
% 2.61/1.66 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(binunionRsub_type,type,
% 2.61/1.66 binunionRsub: $o ).
% 2.61/1.66
% 2.61/1.66 thf(binunionRsub,definition,
% 2.61/1.66 ( binunionRsub
% 2.61/1.66 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(binintersect_type,type,
% 2.61/1.66 binintersect: $i > $i > $i ).
% 2.61/1.66
% 2.61/1.66 thf(binintersectI_type,type,
% 2.61/1.66 binintersectI: $o ).
% 2.61/1.66
% 2.61/1.66 thf(binintersectI,definition,
% 2.61/1.66 ( binintersectI
% 2.61/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 => ( ( in @ Xx @ B )
% 2.61/1.66 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(binintersectSubset5_type,type,
% 2.61/1.66 binintersectSubset5: $o ).
% 2.61/1.66
% 2.61/1.66 thf(binintersectSubset5,definition,
% 2.61/1.66 ( binintersectSubset5
% 2.61/1.66 = ( ! [A: $i,B: $i,C: $i] :
% 2.61/1.66 ( ( subset @ C @ A )
% 2.61/1.66 => ( ( subset @ C @ B )
% 2.61/1.66 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(binintersectEL_type,type,
% 2.61/1.66 binintersectEL: $o ).
% 2.61/1.66
% 2.61/1.66 thf(binintersectEL,definition,
% 2.61/1.66 ( binintersectEL
% 2.61/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.61/1.66 => ( in @ Xx @ A ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(binintersectLsub_type,type,
% 2.61/1.66 binintersectLsub: $o ).
% 2.61/1.66
% 2.61/1.66 thf(binintersectLsub,definition,
% 2.61/1.66 ( binintersectLsub
% 2.61/1.66 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(binintersectSubset2_type,type,
% 2.61/1.66 binintersectSubset2: $o ).
% 2.61/1.66
% 2.61/1.66 thf(binintersectSubset2,definition,
% 2.61/1.66 ( binintersectSubset2
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ( subset @ A @ B )
% 2.61/1.66 => ( ( binintersect @ A @ B )
% 2.61/1.66 = A ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(binintersectSubset3_type,type,
% 2.61/1.66 binintersectSubset3: $o ).
% 2.61/1.66
% 2.61/1.66 thf(binintersectSubset3,definition,
% 2.61/1.66 ( binintersectSubset3
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ( ( binintersect @ A @ B )
% 2.61/1.66 = B )
% 2.61/1.66 => ( subset @ B @ A ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(binintersectER_type,type,
% 2.61/1.66 binintersectER: $o ).
% 2.61/1.66
% 2.61/1.66 thf(binintersectER,definition,
% 2.61/1.66 ( binintersectER
% 2.61/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.61/1.66 => ( in @ Xx @ B ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(disjointsetsI1_type,type,
% 2.61/1.66 disjointsetsI1: $o ).
% 2.61/1.66
% 2.61/1.66 thf(disjointsetsI1,definition,
% 2.61/1.66 ( disjointsetsI1
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ~ ? [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 & ( in @ Xx @ B ) )
% 2.61/1.66 => ( ( binintersect @ A @ B )
% 2.61/1.66 = emptyset ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(binintersectRsub_type,type,
% 2.61/1.66 binintersectRsub: $o ).
% 2.61/1.66
% 2.61/1.66 thf(binintersectRsub,definition,
% 2.61/1.66 ( binintersectRsub
% 2.61/1.66 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(binintersectSubset4_type,type,
% 2.61/1.66 binintersectSubset4: $o ).
% 2.61/1.66
% 2.61/1.66 thf(binintersectSubset4,definition,
% 2.61/1.66 ( binintersectSubset4
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ( subset @ B @ A )
% 2.61/1.66 => ( ( binintersect @ A @ B )
% 2.61/1.66 = B ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(binintersectSubset1_type,type,
% 2.61/1.66 binintersectSubset1: $o ).
% 2.61/1.66
% 2.61/1.66 thf(binintersectSubset1,definition,
% 2.61/1.66 ( binintersectSubset1
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ( ( binintersect @ A @ B )
% 2.61/1.66 = A )
% 2.61/1.66 => ( subset @ A @ B ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(bs114d_type,type,
% 2.61/1.66 bs114d: $o ).
% 2.61/1.66
% 2.61/1.66 thf(bs114d,definition,
% 2.61/1.66 ( bs114d
% 2.61/1.66 = ( ! [A: $i,B: $i,C: $i] :
% 2.61/1.66 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.61/1.66 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(regular_type,type,
% 2.61/1.66 regular: $i > $o ).
% 2.61/1.66
% 2.61/1.66 thf(setminus_type,type,
% 2.61/1.66 setminus: $i > $i > $i ).
% 2.61/1.66
% 2.61/1.66 thf(setminusI_type,type,
% 2.61/1.66 setminusI: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setminusI,definition,
% 2.61/1.66 ( setminusI
% 2.61/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 => ( ~ ( in @ Xx @ B )
% 2.61/1.66 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setminusEL_type,type,
% 2.61/1.66 setminusEL: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setminusEL,definition,
% 2.61/1.66 ( setminusEL
% 2.61/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.61/1.66 => ( in @ Xx @ A ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setminusER_type,type,
% 2.61/1.66 setminusER: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setminusER,definition,
% 2.61/1.66 ( setminusER
% 2.61/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.61/1.66 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setminusSubset2_type,type,
% 2.61/1.66 setminusSubset2: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setminusSubset2,definition,
% 2.61/1.66 ( setminusSubset2
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ( subset @ A @ B )
% 2.61/1.66 => ( ( setminus @ A @ B )
% 2.61/1.66 = emptyset ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setminusERneg_type,type,
% 2.61/1.66 setminusERneg: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setminusERneg,definition,
% 2.61/1.66 ( setminusERneg
% 2.61/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.66 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.61/1.66 => ( ( in @ Xx @ A )
% 2.61/1.66 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setminusELneg_type,type,
% 2.61/1.66 setminusELneg: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setminusELneg,definition,
% 2.61/1.66 ( setminusELneg
% 2.61/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.66 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.61/1.66 => ( ~ ( in @ Xx @ B )
% 2.61/1.66 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setminusILneg_type,type,
% 2.61/1.66 setminusILneg: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setminusILneg,definition,
% 2.61/1.66 ( setminusILneg
% 2.61/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.66 ( ~ ( in @ Xx @ A )
% 2.61/1.66 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setminusIRneg_type,type,
% 2.61/1.66 setminusIRneg: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setminusIRneg,definition,
% 2.61/1.66 ( setminusIRneg
% 2.61/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ B )
% 2.61/1.66 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setminusLsub_type,type,
% 2.61/1.66 setminusLsub: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setminusLsub,definition,
% 2.61/1.66 ( setminusLsub
% 2.61/1.66 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(setminusSubset1_type,type,
% 2.61/1.66 setminusSubset1: $o ).
% 2.61/1.66
% 2.61/1.66 thf(setminusSubset1,definition,
% 2.61/1.66 ( setminusSubset1
% 2.61/1.66 = ( ! [A: $i,B: $i] :
% 2.61/1.66 ( ( ( setminus @ A @ B )
% 2.61/1.66 = emptyset )
% 2.61/1.66 => ( subset @ A @ B ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(symdiff_type,type,
% 2.61/1.66 symdiff: $i > $i > $i ).
% 2.61/1.66
% 2.61/1.66 thf(symdiffE_type,type,
% 2.61/1.66 symdiffE: $o ).
% 2.61/1.66
% 2.61/1.66 thf(symdiffE,definition,
% 2.61/1.66 ( symdiffE
% 2.61/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.61/1.66 => ! [Xphi: $o] :
% 2.61/1.66 ( ( ( in @ Xx @ A )
% 2.61/1.66 => ( ~ ( in @ Xx @ B )
% 2.61/1.66 => Xphi ) )
% 2.61/1.66 => ( ( ~ ( in @ Xx @ A )
% 2.61/1.66 => ( ( in @ Xx @ B )
% 2.61/1.66 => Xphi ) )
% 2.61/1.66 => Xphi ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(symdiffI1_type,type,
% 2.61/1.66 symdiffI1: $o ).
% 2.61/1.66
% 2.61/1.66 thf(symdiffI1,definition,
% 2.61/1.66 ( symdiffI1
% 2.61/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 => ( ~ ( in @ Xx @ B )
% 2.61/1.66 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(symdiffI2_type,type,
% 2.61/1.66 symdiffI2: $o ).
% 2.61/1.66
% 2.61/1.66 thf(symdiffI2,definition,
% 2.61/1.66 ( symdiffI2
% 2.61/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.66 ( ~ ( in @ Xx @ A )
% 2.61/1.66 => ( ( in @ Xx @ B )
% 2.61/1.66 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(symdiffIneg1_type,type,
% 2.61/1.66 symdiffIneg1: $o ).
% 2.61/1.66
% 2.61/1.66 thf(symdiffIneg1,definition,
% 2.61/1.66 ( symdiffIneg1
% 2.61/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ A )
% 2.61/1.66 => ( ( in @ Xx @ B )
% 2.61/1.66 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(symdiffIneg2_type,type,
% 2.61/1.66 symdiffIneg2: $o ).
% 2.61/1.66
% 2.61/1.66 thf(symdiffIneg2,definition,
% 2.61/1.66 ( symdiffIneg2
% 2.61/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.66 ( ~ ( in @ Xx @ A )
% 2.61/1.66 => ( ~ ( in @ Xx @ B )
% 2.61/1.66 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(iskpair_type,type,
% 2.61/1.66 iskpair: $i > $o ).
% 2.61/1.66
% 2.61/1.66 thf(iskpair,definition,
% 2.61/1.66 ( iskpair
% 2.61/1.66 = ( ^ [A: $i] :
% 2.61/1.66 ? [Xx: $i] :
% 2.61/1.66 ( ( in @ Xx @ ( setunion @ A ) )
% 2.61/1.66 & ? [Xy: $i] :
% 2.61/1.66 ( ( in @ Xy @ ( setunion @ A ) )
% 2.61/1.66 & ( A
% 2.61/1.66 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 2.61/1.66
% 2.61/1.66 thf(secondinupair,conjecture,
% 2.61/1.66 ( setextAx
% 2.61/1.66 => ( emptysetAx
% 2.61/1.66 => ( setadjoinAx
% 2.61/1.66 => ( powersetAx
% 2.61/1.66 => ( setunionAx
% 2.61/1.66 => ( omega0Ax
% 2.61/1.66 => ( omegaSAx
% 2.61/1.66 => ( omegaIndAx
% 2.61/1.66 => ( replAx
% 2.61/1.66 => ( foundationAx
% 2.61/1.66 => ( wellorderingAx
% 2.61/1.66 => ( descrp
% 2.61/1.66 => ( dsetconstrI
% 2.61/1.66 => ( dsetconstrEL
% 2.61/1.66 => ( dsetconstrER
% 2.61/1.66 => ( exuE1
% 2.61/1.66 => ( prop2setE
% 2.61/1.66 => ( emptysetE
% 2.61/1.66 => ( emptysetimpfalse
% 2.61/1.66 => ( notinemptyset
% 2.61/1.66 => ( exuE3e
% 2.61/1.66 => ( setext
% 2.61/1.66 => ( emptyI
% 2.61/1.66 => ( noeltsimpempty
% 2.61/1.66 => ( setbeta
% 2.61/1.66 => ( nonemptyE1
% 2.61/1.66 => ( nonemptyI
% 2.61/1.66 => ( nonemptyI1
% 2.61/1.66 => ( setadjoinIL
% 2.61/1.66 => ( emptyinunitempty
% 2.61/1.66 => ( setadjoinIR
% 2.61/1.66 => ( setadjoinE
% 2.61/1.66 => ( setadjoinOr
% 2.61/1.66 => ( setoftrueEq
% 2.61/1.66 => ( powersetI
% 2.61/1.66 => ( emptyinPowerset
% 2.61/1.66 => ( emptyInPowerset
% 2.61/1.66 => ( powersetE
% 2.61/1.66 => ( setunionI
% 2.61/1.66 => ( setunionE
% 2.61/1.66 => ( subPowSU
% 2.61/1.66 => ( exuE2
% 2.61/1.66 => ( nonemptyImpWitness
% 2.61/1.66 => ( uniqinunit
% 2.61/1.66 => ( notinsingleton
% 2.61/1.66 => ( eqinunit
% 2.61/1.66 => ( singletonsswitch
% 2.61/1.66 => ( upairsetE
% 2.61/1.66 => ( upairsetIL
% 2.61/1.66 => ( upairsetIR
% 2.61/1.66 => ( emptyE1
% 2.61/1.66 => ( vacuousDall
% 2.61/1.66 => ( quantDeMorgan1
% 2.61/1.66 => ( quantDeMorgan2
% 2.61/1.66 => ( quantDeMorgan3
% 2.61/1.66 => ( quantDeMorgan4
% 2.61/1.66 => ( prop2setI
% 2.61/1.66 => ( prop2set2propI
% 2.61/1.66 => ( notdexE
% 2.61/1.66 => ( notdallE
% 2.61/1.66 => ( exuI1
% 2.61/1.66 => ( exuI3
% 2.61/1.66 => ( exuI2
% 2.61/1.66 => ( inCongP
% 2.61/1.66 => ( in__Cong
% 2.61/1.66 => ( exuE3u
% 2.61/1.66 => ( exu__Cong
% 2.61/1.66 => ( emptyset__Cong
% 2.61/1.66 => ( setadjoin__Cong
% 2.61/1.66 => ( powerset__Cong
% 2.61/1.66 => ( setunion__Cong
% 2.61/1.66 => ( omega__Cong
% 2.61/1.66 => ( exuEu
% 2.61/1.66 => ( descr__Cong
% 2.61/1.66 => ( dsetconstr__Cong
% 2.61/1.66 => ( subsetI1
% 2.61/1.66 => ( eqimpsubset2
% 2.61/1.66 => ( eqimpsubset1
% 2.61/1.66 => ( subsetI2
% 2.61/1.66 => ( emptysetsubset
% 2.61/1.66 => ( subsetE
% 2.61/1.66 => ( subsetE2
% 2.61/1.66 => ( notsubsetI
% 2.61/1.66 => ( notequalI1
% 2.61/1.66 => ( notequalI2
% 2.61/1.66 => ( subsetRefl
% 2.61/1.66 => ( subsetTrans
% 2.61/1.66 => ( setadjoinSub
% 2.61/1.66 => ( setadjoinSub2
% 2.61/1.66 => ( subset2powerset
% 2.61/1.66 => ( setextsub
% 2.61/1.66 => ( subsetemptysetimpeq
% 2.61/1.66 => ( powersetI1
% 2.61/1.66 => ( powersetE1
% 2.61/1.66 => ( inPowerset
% 2.61/1.66 => ( powersetsubset
% 2.61/1.66 => ( sepInPowerset
% 2.61/1.66 => ( sepSubset
% 2.61/1.66 => ( binunionIL
% 2.61/1.66 => ( upairset2IR
% 2.61/1.66 => ( binunionIR
% 2.61/1.66 => ( binunionEcases
% 2.61/1.66 => ( binunionE
% 2.61/1.66 => ( binunionLsub
% 2.61/1.66 => ( binunionRsub
% 2.61/1.66 => ( binintersectI
% 2.61/1.66 => ( binintersectSubset5
% 2.61/1.66 => ( binintersectEL
% 2.61/1.66 => ( binintersectLsub
% 2.61/1.66 => ( binintersectSubset2
% 2.61/1.66 => ( binintersectSubset3
% 2.61/1.66 => ( binintersectER
% 2.61/1.66 => ( disjointsetsI1
% 2.61/1.66 => ( binintersectRsub
% 2.61/1.66 => ( binintersectSubset4
% 2.61/1.66 => ( binintersectSubset1
% 2.61/1.66 => ( bs114d
% 2.61/1.66 => ( setminusI
% 2.61/1.66 => ( setminusEL
% 2.61/1.66 => ( setminusER
% 2.61/1.66 => ( setminusSubset2
% 2.61/1.66 => ( setminusERneg
% 2.61/1.66 => ( setminusELneg
% 2.61/1.66 => ( setminusILneg
% 2.61/1.66 => ( setminusIRneg
% 2.61/1.66 => ( setminusLsub
% 2.61/1.66 => ( setminusSubset1
% 2.61/1.66 => ( symdiffE
% 2.61/1.66 => ( symdiffI1
% 2.61/1.66 => ( symdiffI2
% 2.61/1.66 => ( symdiffIneg1
% 2.61/1.66 => ( symdiffIneg2
% 2.61/1.66 => ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.61/1.66 ------------------------
% 2.61/1.69 % (13104)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_13000 for (2999ds/18Mi)
% 2.61/1.69 % (13102)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_13000 for (2999ds/2Mi)
% 2.61/1.69 % (13101)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_13000 for (2999ds/2Mi)
% 2.61/1.69 % (13100)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_13000 for (2999ds/27Mi)
% 2.61/1.69 % (13103)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_13000 for (2999ds/275Mi)
% 2.61/1.69 % (13099)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_13000 for (2999ds/4Mi)
% 2.61/1.69 % (13098)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_13000 for (2999ds/183Mi)
% 2.61/1.69 % (13101)Instruction limit reached!
% 2.61/1.69 % (13101)------------------------------
% 2.61/1.69 % (13101)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.69 % (13101)Termination reason: Unknown
% 2.61/1.69 % (13101)Termination phase: shuffling
% 2.61/1.69 % (13102)Instruction limit reached!
% 2.61/1.69 % (13102)------------------------------
% 2.61/1.69 % (13102)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.69 % (13102)Termination reason: Unknown
% 2.61/1.69 % (13102)Termination phase: shuffling
% 2.61/1.69
% 2.61/1.69 % (13102)Memory used [KB]: 1279
% 2.61/1.69 % (13102)Time elapsed: 0.004 s
% 2.61/1.69 % (13102)Instructions burned: 4 (million)
% 2.61/1.69 % (13102)------------------------------
% 2.61/1.69 % (13102)------------------------------
% 2.61/1.69
% 2.61/1.69 % (13101)Memory used [KB]: 1279
% 2.61/1.69 % (13101)Time elapsed: 0.004 s
% 2.61/1.69 % (13101)Instructions burned: 3 (million)
% 2.61/1.69 % (13101)------------------------------
% 2.61/1.69 % (13101)------------------------------
% 2.61/1.69 % (13099)Instruction limit reached!
% 2.61/1.69 % (13099)------------------------------
% 2.61/1.69 % (13099)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.69 % (13099)Termination reason: Unknown
% 2.61/1.69 % (13099)Termination phase: shuffling
% 2.61/1.69
% 2.61/1.69 % (13099)Memory used [KB]: 1279
% 2.61/1.69 % (13099)Time elapsed: 0.004 s
% 2.61/1.69 % (13099)Instructions burned: 5 (million)
% 2.61/1.69 % (13099)------------------------------
% 2.61/1.69 % (13099)------------------------------
% 2.61/1.69 % (13104)Instruction limit reached!
% 2.61/1.69 % (13104)------------------------------
% 2.61/1.69 % (13104)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.61/1.69 % (13104)Termination reason: Unknown
% 2.61/1.69 % (13104)Termination phase: Preprocessing 2
% 2.61/1.69
% 2.61/1.69 % (13104)Memory used [KB]: 1407
% 2.61/1.69 % (13104)Time elapsed: 0.011 s
% 2.61/1.69 % (13104)Instructions burned: 18 (million)
% 2.61/1.69 % (13104)------------------------------
% 2.61/1.69 % (13104)------------------------------
% 2.66/1.70 % (13100)Instruction limit reached!
% 2.66/1.70 % (13100)------------------------------
% 2.66/1.70 % (13100)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.70 % (13100)Termination reason: Unknown
% 2.66/1.70 % (13100)Termination phase: Property scanning
% 2.66/1.70
% 2.66/1.70 % (13100)Memory used [KB]: 1663
% 2.66/1.70 % (13100)Time elapsed: 0.014 s
% 2.66/1.70 % (13100)Instructions burned: 27 (million)
% 2.66/1.70 % (13100)------------------------------
% 2.66/1.70 % (13100)------------------------------
% 2.66/1.70 % (13106)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_13000 for (2999ds/37Mi)
% 2.66/1.70 % (13105)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_13000 for (2999ds/3Mi)
% 2.66/1.70 % (13107)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_13000 for (2999ds/15Mi)
% 2.66/1.71 % (13105)Instruction limit reached!
% 2.66/1.71 % (13105)------------------------------
% 2.66/1.71 % (13105)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.71 % (13105)Termination reason: Unknown
% 2.66/1.71 % (13105)Termination phase: shuffling
% 2.66/1.71
% 2.66/1.71 % (13105)Memory used [KB]: 1279
% 2.66/1.71 % (13105)Time elapsed: 0.004 s
% 2.66/1.71 % (13105)Instructions burned: 4 (million)
% 2.66/1.71 % (13105)------------------------------
% 2.66/1.71 % (13105)------------------------------
% 2.66/1.71 % (13108)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_13000 for (2999ds/3Mi)
% 2.66/1.71 % (13107)Instruction limit reached!
% 2.66/1.71 % (13107)------------------------------
% 2.66/1.71 % (13107)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.71 % (13107)Termination reason: Unknown
% 2.66/1.71 % (13107)Termination phase: Property scanning
% 2.66/1.71
% 2.66/1.71 % (13107)Memory used [KB]: 1407
% 2.66/1.71 % (13107)Time elapsed: 0.009 s
% 2.66/1.71 % (13107)Instructions burned: 15 (million)
% 2.66/1.71 % (13107)------------------------------
% 2.66/1.71 % (13107)------------------------------
% 2.66/1.71 % (13108)Instruction limit reached!
% 2.66/1.71 % (13108)------------------------------
% 2.66/1.71 % (13108)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.71 % (13108)Termination reason: Unknown
% 2.66/1.71 % (13108)Termination phase: shuffling
% 2.66/1.71
% 2.66/1.71 % (13108)Memory used [KB]: 1279
% 2.66/1.71 % (13108)Time elapsed: 0.004 s
% 2.66/1.71 % (13108)Instructions burned: 4 (million)
% 2.66/1.71 % (13108)------------------------------
% 2.66/1.71 % (13108)------------------------------
% 2.66/1.71 % (13109)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_13000 for (2999ds/1041Mi)
% 2.66/1.72 % (13106)Instruction limit reached!
% 2.66/1.72 % (13106)------------------------------
% 2.66/1.72 % (13106)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.72 % (13106)Termination reason: Unknown
% 2.66/1.72 % (13106)Termination phase: Property scanning
% 2.66/1.72
% 2.66/1.72 % (13106)Memory used [KB]: 1663
% 2.66/1.72 % (13106)Time elapsed: 0.019 s
% 2.66/1.72 % (13106)Instructions burned: 38 (million)
% 2.66/1.72 % (13106)------------------------------
% 2.66/1.72 % (13106)------------------------------
% 2.66/1.72 % (13110)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_13000 for (2999ds/7Mi)
% 2.66/1.73 % (13110)Instruction limit reached!
% 2.66/1.73 % (13110)------------------------------
% 2.66/1.73 % (13110)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.73 % (13110)Termination reason: Unknown
% 2.66/1.73 % (13110)Termination phase: shuffling
% 2.66/1.73
% 2.66/1.73 % (13110)Memory used [KB]: 1279
% 2.66/1.73 % (13110)Time elapsed: 0.006 s
% 2.66/1.73 % (13110)Instructions burned: 8 (million)
% 2.66/1.73 % (13110)------------------------------
% 2.66/1.73 % (13110)------------------------------
% 2.66/1.73 % (13111)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_13000 for (2999ds/16Mi)
% 2.66/1.73 % (13112)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_13000 for (2999ds/3Mi)
% 2.66/1.73 % (13112)Instruction limit reached!
% 2.66/1.73 % (13112)------------------------------
% 2.66/1.73 % (13112)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.73 % (13112)Termination reason: Unknown
% 2.66/1.73 % (13112)Termination phase: shuffling
% 2.66/1.73
% 2.66/1.73 % (13112)Memory used [KB]: 1279
% 2.66/1.73 % (13112)Time elapsed: 0.003 s
% 2.66/1.73 % (13112)Instructions burned: 3 (million)
% 2.66/1.73 % (13112)------------------------------
% 2.66/1.73 % (13112)------------------------------
% 2.66/1.73 % (13111)Instruction limit reached!
% 2.66/1.73 % (13111)------------------------------
% 2.66/1.73 % (13111)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.73 % (13111)Termination reason: Unknown
% 2.66/1.73 % (13111)Termination phase: shuffling
% 2.66/1.73
% 2.66/1.73 % (13111)Memory used [KB]: 1535
% 2.66/1.73 % (13111)Time elapsed: 0.009 s
% 2.66/1.73 % (13111)Instructions burned: 16 (million)
% 2.66/1.73 % (13111)------------------------------
% 2.66/1.73 % (13111)------------------------------
% 2.66/1.74 % (13113)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_13000 for (2999ds/3Mi)
% 2.66/1.74 % (13113)Instruction limit reached!
% 2.66/1.74 % (13113)------------------------------
% 2.66/1.74 % (13113)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.74 % (13113)Termination reason: Unknown
% 2.66/1.74 % (13113)Termination phase: shuffling
% 2.66/1.74
% 2.66/1.74 % (13113)Memory used [KB]: 1151
% 2.66/1.74 % (13113)Time elapsed: 0.003 s
% 2.66/1.74 % (13113)Instructions burned: 3 (million)
% 2.66/1.74 % (13113)------------------------------
% 2.66/1.74 % (13113)------------------------------
% 2.66/1.74 % (13114)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_13000 for (2999ds/7Mi)
% 2.66/1.74 % (13103)First to succeed.
% 2.66/1.75 % (13114)Instruction limit reached!
% 2.66/1.75 % (13114)------------------------------
% 2.66/1.75 % (13114)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.75 % (13114)Termination reason: Unknown
% 2.66/1.75 % (13114)Termination phase: shuffling
% 2.66/1.75
% 2.66/1.75 % (13114)Memory used [KB]: 1279
% 2.66/1.75 % (13114)Time elapsed: 0.005 s
% 2.66/1.75 % (13114)Instructions burned: 7 (million)
% 2.66/1.75 % (13114)------------------------------
% 2.66/1.75 % (13114)------------------------------
% 2.66/1.75 % (13115)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_13000 for (2999ds/3Mi)
% 2.66/1.75 % (13115)Instruction limit reached!
% 2.66/1.75 % (13115)------------------------------
% 2.66/1.75 % (13115)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.75 % (13115)Termination reason: Unknown
% 2.66/1.75 % (13115)Termination phase: shuffling
% 2.66/1.75
% 2.66/1.75 % (13115)Memory used [KB]: 1279
% 2.66/1.75 % (13115)Time elapsed: 0.003 s
% 2.66/1.75 % (13115)Instructions burned: 4 (million)
% 2.66/1.75 % (13115)------------------------------
% 2.66/1.75 % (13115)------------------------------
% 2.66/1.75 % (13103)Refutation found. Thanks to Tanya!
% 2.66/1.75 % SZS status Theorem for DTF2THF_13000
% 2.66/1.75 % SZS output start Proof for DTF2THF_13000
% 2.66/1.75 thf(func_def_0, type, in: $i > $i > $o).
% 2.66/1.75 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.66/1.75 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.66/1.75 thf(func_def_8, type, powerset: $i > $i).
% 2.66/1.75 thf(func_def_10, type, setunion: $i > $i).
% 2.66/1.75 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.66/1.75 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.66/1.75 thf(func_def_26, type, prop2set: $o > $i).
% 2.66/1.75 thf(func_def_36, type, nonempty: $i > $o).
% 2.66/1.75 thf(func_def_69, type, set2prop: $i > $o).
% 2.66/1.75 thf(func_def_88, type, subset: $i > $i > $o).
% 2.66/1.75 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.66/1.75 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.66/1.75 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.66/1.75 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.66/1.75 thf(func_def_135, type, regular: $i > $o).
% 2.66/1.75 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.66/1.75 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.66/1.75 thf(func_def_153, type, iskpair: $i > $o).
% 2.66/1.75 thf(func_def_167, type, sP1: $o > $i > $i > $i > $o).
% 2.66/1.75 thf(func_def_168, type, sP2: $i > $i > $o).
% 2.66/1.75 thf(func_def_169, type, sP3: $i > $o).
% 2.66/1.75 thf(func_def_170, type, sP4: $i > $i > $o).
% 2.66/1.75 thf(func_def_171, type, sP5: $i > $i > $o).
% 2.66/1.75 thf(func_def_175, type, sK9: $i > $i > $i).
% 2.66/1.75 thf(func_def_181, type, sK15: $i > $o).
% 2.66/1.75 thf(func_def_191, type, sK25: ($i > $o) > $i > $i).
% 2.66/1.75 thf(func_def_193, type, sK27: $i > $o).
% 2.66/1.75 thf(func_def_197, type, sK31: ($i > $o) > ($i > $o) > $i).
% 2.66/1.75 thf(func_def_198, type, sK32: ($i > $o) > ($i > $o) > $i).
% 2.66/1.75 thf(func_def_199, type, sK33: $i > $o).
% 2.66/1.75 thf(func_def_200, type, sK34: $i > $o).
% 2.66/1.75 thf(func_def_205, type, sK39: $i > $o).
% 2.66/1.75 thf(func_def_207, type, sK41: $i > $i).
% 2.66/1.75 thf(func_def_210, type, sK44: $i > $o).
% 2.66/1.75 thf(func_def_215, type, sK49: ($i > $o) > $i).
% 2.66/1.75 thf(func_def_216, type, sK50: $i > $o).
% 2.66/1.75 thf(func_def_217, type, sK51: $i > $i).
% 2.66/1.75 thf(func_def_230, type, sK64: $i > $i > $o).
% 2.66/1.75 thf(func_def_231, type, sK65: $i > $i).
% 2.66/1.75 thf(func_def_232, type, sK66: $i > $i).
% 2.66/1.75 thf(func_def_233, type, sK67: ($i > $i > $o) > $i > $i).
% 2.66/1.75 thf(func_def_234, type, sK68: $i > ($i > $i > $o) > $i > $i).
% 2.66/1.75 thf(func_def_235, type, sK69: ($i > $i > $o) > $i > $i).
% 2.66/1.75 thf(func_def_243, type, sK77: $i > $o).
% 2.66/1.75 thf(func_def_245, type, sK79: ($i > $o) > $i > $i).
% 2.66/1.75 thf(func_def_246, type, sK80: ($i > $o) > $i > $i).
% 2.66/1.75 thf(func_def_248, type, sK82: $i > $o).
% 2.66/1.75 thf(func_def_255, type, sK89: $i > $i > $i).
% 2.66/1.75 thf(func_def_258, type, sK92: $i > $o).
% 2.66/1.75 thf(func_def_262, type, sK96: $i > $i).
% 2.66/1.75 thf(func_def_265, type, sK99: $i > $i).
% 2.66/1.75 thf(func_def_278, type, sK112: $i > $i > $i).
% 2.66/1.75 thf(func_def_281, type, sK115: $i > $o).
% 2.66/1.75 thf(func_def_283, type, sK117: $i > ($i > $o) > $i).
% 2.66/1.75 thf(func_def_291, type, sK125: $i > $i).
% 2.66/1.75 thf(func_def_293, type, sK127: $i > $i > $i).
% 2.66/1.75 thf(func_def_310, type, sK144: $i > $o).
% 2.66/1.75 thf(func_def_314, type, sK148: $i > $o).
% 2.66/1.75 thf(func_def_319, type, sK153: ($i > $o) > ($i > $o) > $i).
% 2.66/1.75 thf(func_def_320, type, sK154: ($i > $o) > ($i > $o) > $i).
% 2.66/1.75 thf(func_def_321, type, sK155: $i > $o).
% 2.66/1.75 thf(func_def_322, type, sK156: $i > $o).
% 2.66/1.75 thf(func_def_325, type, sK159: $i > $i > $i).
% 2.66/1.75 thf(func_def_336, type, sK170: $i > $o).
% 2.66/1.75 thf(func_def_337, type, sK171: $i > $o).
% 2.66/1.75 thf(func_def_338, type, sK172: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.66/1.75 thf(func_def_339, type, sK173: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.66/1.75 thf(func_def_347, type, sK181: $i > $i).
% 2.66/1.75 thf(func_def_365, type, sK199: $i > $i > $i).
% 2.66/1.75 thf(func_def_369, type, sK203: $i > $o).
% 2.66/1.75 thf(func_def_384, type, sK218: $i > $o).
% 2.66/1.75 thf(func_def_392, type, sK226: $i > $o).
% 2.66/1.75 thf(func_def_393, type, sK227: ($i > $o) > $i).
% 2.66/1.75 thf(func_def_394, type, sK228: $i > $i).
% 2.66/1.75 thf(func_def_396, type, sK230: $i > ($i > $o) > $i).
% 2.66/1.75 thf(func_def_397, type, sK231: $i > $o).
% 2.66/1.75 thf(func_def_400, type, sK234: $i > $o).
% 2.66/1.75 thf(func_def_402, type, sK236: $i > ($i > $o) > $i).
% 2.66/1.75 thf(func_def_418, type, sK252: $i > $i > $i).
% 2.66/1.75 thf(func_def_419, type, sK253: $i > $i > $i).
% 2.66/1.75 thf(func_def_420, type, sK254: $i > $o).
% 2.66/1.75 thf(func_def_423, type, sK257: $i > ($i > $o) > $i).
% 2.66/1.75 thf(func_def_425, type, sK259: ($i > $o) > $i).
% 2.66/1.75 thf(func_def_426, type, sK260: $i > $o).
% 2.66/1.75 thf(func_def_427, type, sK261: $i > $i).
% 2.66/1.75 thf(func_def_433, type, sK267: $i > $o).
% 2.66/1.75 thf(func_def_441, type, sK275: ($i > $o) > $i > $i).
% 2.66/1.75 thf(func_def_443, type, sK277: $i > $o).
% 2.66/1.75 thf(func_def_447, type, sK281: $i > $i > $i).
% 2.66/1.75 thf(func_def_451, type, sK285: $o > $i > $i > $i).
% 2.66/1.75 thf(func_def_452, type, sK286: $i > $i > $i).
% 2.66/1.75 thf(func_def_453, type, sK287: $i > $i > $i).
% 2.66/1.75 thf(func_def_454, type, sK288: $i > $i > $i).
% 2.66/1.75 thf(func_def_455, type, sK289: $i > $i > $i).
% 2.66/1.75 thf(func_def_456, type, sK290: $i > $i > $i).
% 2.66/1.75 thf(func_def_457, type, sK291: $i > $i > $i > $i).
% 2.66/1.75 thf(func_def_458, type, sK292: $i > $i).
% 2.66/1.75 thf(func_def_459, type, sK293: $i > $i).
% 2.66/1.75 thf(func_def_460, type, sK294: $i > $i).
% 2.66/1.75 thf(func_def_461, type, sK295: $i > $i).
% 2.66/1.75 thf(func_def_462, type, sK296: $i > $i > $i > $i).
% 2.66/1.75 thf(func_def_463, type, sK297: $i > $i > $i > $i).
% 2.66/1.75 thf(func_def_464, type, sK298: $i > $i > $i).
% 2.66/1.75 thf(func_def_465, type, sK299: $i > $i > $i).
% 2.66/1.75 thf(func_def_466, type, sK300: $i > $i > $i).
% 2.66/1.75 thf(func_def_468, type, sK302: $i > $i).
% 2.66/1.75 thf(func_def_469, type, sK303: $i > $i).
% 2.66/1.75 thf(func_def_470, type, sK304: $i > $i).
% 2.66/1.75 thf(func_def_473, type, sK307: $i > $o).
% 2.66/1.75 thf(func_def_490, type, sK324: $i > $o).
% 2.66/1.75 thf(func_def_495, type, sK329: ($i > $o) > $i).
% 2.66/1.75 thf(func_def_496, type, sK330: ($i > $o) > $i).
% 2.66/1.75 thf(func_def_497, type, sK331: $i > $o).
% 2.66/1.75 thf(func_def_501, type, sK335: $i > $o).
% 2.66/1.75 thf(func_def_503, type, sK337: $i > $i).
% 2.66/1.75 thf(func_def_531, type, ph365: !>[X0: $tType]:(X0)).
% 2.66/1.75 thf(f2561,plain,(
% 2.66/1.75 $false),
% 2.66/1.75 inference(subsumption_resolution,[status(thm)],[f1499,f2496])).
% 2.66/1.75 thf(f2496,plain,(
% 2.66/1.75 ( ! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset))))) )),
% 2.66/1.75 inference(trivial_inequality_removal,[status(thm)],[f2065])).
% 2.66/1.75 thf(f2065,plain,(
% 2.66/1.75 ( ! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)))) | ($true != $true)) )),
% 2.66/1.75 inference(definition_unfolding,[status(thm)],[f1562,f1515])).
% 2.66/1.75 thf(f1515,plain,(
% 2.66/1.75 (upairset2IR = $true)),
% 2.66/1.75 inference(cnf_transformation,[status(thm)],[f807])).
% 2.66/1.75 thf(f807,plain,(
% 2.66/1.75 (dsetconstrI = $true) & (binintersectI = $true) & (upairset2IR = $true) & (setminusI = $true) & (singletonsswitch = $true) & (binunionLsub = $true) & (exuE2 = $true) & (binintersectSubset2 = $true) & (emptysetAx = $true) & (emptyE1 = $true) & (setadjoinIR = $true) & (setunionAx = $true) & (emptyinPowerset = $true) & (setunionI = $true) & (subsetTrans = $true) & (nonemptyE1 = $true) & (noeltsimpempty = $true) & (eqinunit = $true) & ((in @ sK134 @ (setadjoin @ sK135 @ (setadjoin @ sK134 @ emptyset))) != $true) & (emptyInPowerset = $true) & (binunionEcases = $true) & (symdiffE = $true) & (notsubsetI = $true) & (omega0Ax = $true) & (upairsetIR = $true) & (foundationAx = $true) & (omega__Cong = $true) & (dsetconstr__Cong = $true) & (emptyI = $true) & (binintersectER = $true) & (emptyinunitempty = $true) & (binintersectSubset5 = $true) & (nonemptyI1 = $true) & (exuE3e = $true) & (setminusILneg = $true) & (prop2setE = $true) & (setext = $true) & (setunion__Cong = $true) & (notequalI2 = $true) & (powersetI = $true) & (notinsingleton = $true) & (powerset__Cong = $true) & (setminusELneg = $true) & (setadjoinSub = $true) & (setoftrueEq = $true) & (setminusSubset1 = $true) & (dsetconstrER = $true) & (exuE3u = $true) & (setminusLsub = $true) & (powersetAx = $true) & (disjointsetsI1 = $true) & (setminusER = $true) & (upairsetIL = $true) & (powersetE1 = $true) & (quantDeMorgan2 = $true) & (powersetE = $true) & (descr__Cong = $true) & (exuEu = $true) & (notinemptyset = $true) & (inPowerset = $true) & (descrp = $true) & (binunionIL = $true) & (inCongP = $true) & (upairsetE = $true) & (exuI1 = $true) & (symdiffIneg2 = $true) & (emptysetE = $true) & (subsetE = $true) & (subsetI2 = $true) & (binintersectLsub = $true) & (symdiffIneg1 = $true) & (quantDeMorgan4 = $true) & (quantDeMorgan3 = $true) & (exu__Cong = $true) & (dsetconstrEL = $true) & (setadjoin__Cong = $true) & (eqimpsubset2 = $true) & (setminusSubset2 = $true) & (replAx = $true) & (binunionRsub = $true) & (omegaIndAx = $true) & (subsetE2 = $true) & (setadjoinAx = $true) & (symdiffI2 = $true) & (sepInPowerset = $true) & (bs114d = $true) & (notequalI1 = $true) & (setadjoinOr = $true) & (nonemptyI = $true) & (setbeta = $true) & (exuI3 = $true) & (powersetsubset = $true) & (uniqinunit = $true) & (binintersectEL = $true) & (in__Cong = $true) & (setunionE = $true) & (vacuousDall = $true) & (subsetI1 = $true) & (exuE1 = $true) & (wellorderingAx = $true) & (binunionE = $true) & (powersetI1 = $true) & (setminusIRneg = $true) & (binintersectRsub = $true) & (subset2powerset = $true) & (emptysetimpfalse = $true) & (binintersectSubset3 = $true) & (setminusERneg = $true) & (eqimpsubset1 = $true) & (sepSubset = $true) & (nonemptyImpWitness = $true) & (quantDeMorgan1 = $true) & (subPowSU = $true) & (binintersectSubset1 = $true) & (setadjoinE = $true) & (subsetemptysetimpeq = $true) & (exuI2 = $true) & (setadjoinSub2 = $true) & (setadjoinIL = $true) & (setextAx = $true) & (omegaSAx = $true) & (subsetRefl = $true) & (prop2setI = $true) & (setminusEL = $true) & (binintersectSubset4 = $true) & (emptyset__Cong = $true) & (setextsub = $true) & (prop2set2propI = $true) & (symdiffI1 = $true) & (notdallE = $true) & (emptysetsubset = $true) & (binunionIR = $true) & (notdexE = $true)),
% 2.66/1.75 inference(skolemisation,[status(esa),new_symbols(skolem,[sK134,sK135])],[f554,f806])).
% 2.66/1.75 thf(f806,plain,(
% 2.66/1.75 ? [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) != $true) => ((in @ sK134 @ (setadjoin @ sK135 @ (setadjoin @ sK134 @ emptyset))) != $true)),
% 2.66/1.75 introduced(definition,[],[choice_axiom])).
% 2.66/1.75 thf(f554,plain,(
% 2.66/1.75 (dsetconstrI = $true) & (binintersectI = $true) & (upairset2IR = $true) & (setminusI = $true) & (singletonsswitch = $true) & (binunionLsub = $true) & (exuE2 = $true) & (binintersectSubset2 = $true) & (emptysetAx = $true) & (emptyE1 = $true) & (setadjoinIR = $true) & (setunionAx = $true) & (emptyinPowerset = $true) & (setunionI = $true) & (subsetTrans = $true) & (nonemptyE1 = $true) & (noeltsimpempty = $true) & (eqinunit = $true) & ? [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) != $true) & (emptyInPowerset = $true) & (binunionEcases = $true) & (symdiffE = $true) & (notsubsetI = $true) & (omega0Ax = $true) & (upairsetIR = $true) & (foundationAx = $true) & (omega__Cong = $true) & (dsetconstr__Cong = $true) & (emptyI = $true) & (binintersectER = $true) & (emptyinunitempty = $true) & (binintersectSubset5 = $true) & (nonemptyI1 = $true) & (exuE3e = $true) & (setminusILneg = $true) & (prop2setE = $true) & (setext = $true) & (setunion__Cong = $true) & (notequalI2 = $true) & (powersetI = $true) & (notinsingleton = $true) & (powerset__Cong = $true) & (setminusELneg = $true) & (setadjoinSub = $true) & (setoftrueEq = $true) & (setminusSubset1 = $true) & (dsetconstrER = $true) & (exuE3u = $true) & (setminusLsub = $true) & (powersetAx = $true) & (disjointsetsI1 = $true) & (setminusER = $true) & (upairsetIL = $true) & (powersetE1 = $true) & (quantDeMorgan2 = $true) & (powersetE = $true) & (descr__Cong = $true) & (exuEu = $true) & (notinemptyset = $true) & (inPowerset = $true) & (descrp = $true) & (binunionIL = $true) & (inCongP = $true) & (upairsetE = $true) & (exuI1 = $true) & (symdiffIneg2 = $true) & (emptysetE = $true) & (subsetE = $true) & (subsetI2 = $true) & (binintersectLsub = $true) & (symdiffIneg1 = $true) & (quantDeMorgan4 = $true) & (quantDeMorgan3 = $true) & (exu__Cong = $true) & (dsetconstrEL = $true) & (setadjoin__Cong = $true) & (eqimpsubset2 = $true) & (setminusSubset2 = $true) & (replAx = $true) & (binunionRsub = $true) & (omegaIndAx = $true) & (subsetE2 = $true) & (setadjoinAx = $true) & (symdiffI2 = $true) & (sepInPowerset = $true) & (bs114d = $true) & (notequalI1 = $true) & (setadjoinOr = $true) & (nonemptyI = $true) & (setbeta = $true) & (exuI3 = $true) & (powersetsubset = $true) & (uniqinunit = $true) & (binintersectEL = $true) & (in__Cong = $true) & (setunionE = $true) & (vacuousDall = $true) & (subsetI1 = $true) & (exuE1 = $true) & (wellorderingAx = $true) & (binunionE = $true) & (powersetI1 = $true) & (setminusIRneg = $true) & (binintersectRsub = $true) & (subset2powerset = $true) & (emptysetimpfalse = $true) & (binintersectSubset3 = $true) & (setminusERneg = $true) & (eqimpsubset1 = $true) & (sepSubset = $true) & (nonemptyImpWitness = $true) & (quantDeMorgan1 = $true) & (subPowSU = $true) & (binintersectSubset1 = $true) & (setadjoinE = $true) & (subsetemptysetimpeq = $true) & (exuI2 = $true) & (setadjoinSub2 = $true) & (setadjoinIL = $true) & (setextAx = $true) & (omegaSAx = $true) & (subsetRefl = $true) & (prop2setI = $true) & (setminusEL = $true) & (binintersectSubset4 = $true) & (emptyset__Cong = $true) & (setextsub = $true) & (prop2set2propI = $true) & (symdiffI1 = $true) & (notdallE = $true) & (emptysetsubset = $true) & (binunionIR = $true) & (notdexE = $true)),
% 2.66/1.75 inference(flattening,[status(thm)],[f553])).
% 2.66/1.75 thf(f553,plain,(
% 2.66/1.75 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) != $true) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.66/1.75 inference(ennf_transformation,[status(thm)],[f256])).
% 2.66/1.75 thf(f256,plain,(
% 2.66/1.75 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ! [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.66/1.75 inference(fool_elimination,[status(thm)],[f255])).
% 2.66/1.75 thf(f255,plain,(
% 2.66/1.75 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => ! [X0 : $i,X1 : $i] : (in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.66/1.75 inference(rectify,[status(thm)],[f137])).
% 2.66/1.75 thf(f137,negated_conjecture,(
% 2.66/1.75 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => ! [X2 : $i,X1 : $i] : (in @ X2 @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.66/1.75 inference(negated_conjecture,[status(cth)],[f136])).
% 2.66/1.75 thf(f136,conjecture,(
% 2.66/1.75 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => ! [X2 : $i,X1 : $i] : (in @ X2 @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.66/1.75 file('/export/starexec/sandbox/tmp/tmp.KCJafwTrbB/DTF2THF_13000.p',secondinupair)).
% 2.66/1.75 thf(f1562,plain,(
% 2.66/1.75 ( ! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)))) | (upairset2IR != $true)) )),
% 2.66/1.75 inference(cnf_transformation,[status(thm)],[f866])).
% 2.66/1.75 thf(f866,plain,(
% 2.66/1.75 (! [X0 : $i,X1 : $i] : ($true = (in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)))) | (upairset2IR != $true)) & ((upairset2IR = $true) | ((in @ sK167 @ (setadjoin @ sK166 @ (setadjoin @ sK167 @ emptyset))) != $true))),
% 2.66/1.75 inference(skolemisation,[status(esa),new_symbols(skolem,[sK166,sK167])],[f864,f865])).
% 2.66/1.75 thf(f865,plain,(
% 2.66/1.75 ? [X2 : $i,X3 : $i] : ((in @ X3 @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset))) != $true) => ((in @ sK167 @ (setadjoin @ sK166 @ (setadjoin @ sK167 @ emptyset))) != $true)),
% 2.66/1.75 introduced(definition,[],[choice_axiom])).
% 2.66/1.75 thf(f864,plain,(
% 2.66/1.75 (! [X0 : $i,X1 : $i] : ($true = (in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)))) | (upairset2IR != $true)) & ((upairset2IR = $true) | ? [X2 : $i,X3 : $i] : ((in @ X3 @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset))) != $true))),
% 2.66/1.75 inference(rectify,[status(thm)],[f863])).
% 2.66/1.75 thf(f863,plain,(
% 2.66/1.75 (! [X0 : $i,X1 : $i] : ($true = (in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)))) | (upairset2IR != $true)) & ((upairset2IR = $true) | ? [X0 : $i,X1 : $i] : ($true != (in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)))))),
% 2.66/1.75 inference(nnf_transformation,[status(thm)],[f168])).
% 2.66/1.75 thf(f168,plain,(
% 2.66/1.75 ! [X0 : $i,X1 : $i] : ($true = (in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)))) <=> (upairset2IR = $true)),
% 2.66/1.75 inference(fool_elimination,[status(thm)],[f167])).
% 2.66/1.75 thf(f167,plain,(
% 2.66/1.75 (upairset2IR = ! [X0 : $i,X1 : $i] : (in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset))))),
% 2.66/1.75 inference(rectify,[status(thm)],[f102])).
% 2.66/1.75 thf(f102,axiom,(
% 2.66/1.75 (upairset2IR = ! [X1 : $i,X2 : $i] : (in @ X2 @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset))))),
% 2.66/1.75 file('/export/starexec/sandbox/tmp/tmp.KCJafwTrbB/DTF2THF_13000.p',upairset2IR)).
% 2.66/1.75 thf(f1499,plain,(
% 2.66/1.75 ((in @ sK134 @ (setadjoin @ sK135 @ (setadjoin @ sK134 @ emptyset))) != $true)),
% 2.66/1.75 inference(cnf_transformation,[status(thm)],[f807])).
% 2.66/1.75 % SZS output end Proof for DTF2THF_13000
% 2.66/1.75 % (13103)------------------------------
% 2.66/1.75 % (13103)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.66/1.75 % (13103)Termination reason: Refutation
% 2.66/1.75
% 2.66/1.75 % (13103)Memory used [KB]: 7547
% 2.66/1.75 % (13103)Time elapsed: 0.064 s
% 2.66/1.75 % (13103)Instructions burned: 120 (million)
% 2.66/1.75 % (13103)------------------------------
% 2.66/1.75 % (13103)------------------------------
% 2.66/1.75 % (13097)Success in time 0.079 s
%------------------------------------------------------------------------------