%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU636^1 : 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 : n009.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:35:48 AM UTC 2026
% Result : Theorem 2.37s 1.57s
% Output : Refutation 2.37s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 10
% Syntax : Number of formulae : 62 ( 20 unt; 0 typ; 0 def)
% Number of atoms : 2714 ( 756 equ; 0 cnn)
% Maximal formula atoms : 153 ( 43 avg)
% Number of connectives : 1536 ( 66 ~; 48 |; 478 &; 319 @)
% ( 6 <=>; 619 =>; 0 <=; 0 <~>)
% Maximal formula depth : 155 ( 21 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 285 ( 282 usr; 162 con; 0-4 aty)
% Number of variables : 88 ( 0 ^; 67 !; 21 ?; 88 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_69,type,
set2prop: $i > $o ).
thf(func_def_88,type,
subset: $i > $i > $o ).
thf(func_def_89,type,
disjoint: $i > $i > $o ).
thf(func_def_90,type,
setsmeet: $i > $i > $o ).
thf(func_def_114,type,
binunion: $i > $i > $i ).
thf(func_def_122,type,
binintersect: $i > $i > $i ).
thf(func_def_135,type,
regular: $i > $o ).
thf(func_def_136,type,
setminus: $i > $i > $i ).
thf(func_def_147,type,
symdiff: $i > $i > $i ).
thf(func_def_153,type,
iskpair: $i > $o ).
thf(func_def_158,type,
kpair: $i > $i > $i ).
thf(func_def_160,type,
cartprod: $i > $i > $i ).
thf(func_def_188,type,
sP0: $i > $i > $o ).
thf(func_def_189,type,
sP1: $i > $o ).
thf(func_def_190,type,
sP2: $i > $i > $o ).
thf(func_def_191,type,
sP3: $i > $i > $o ).
thf(func_def_193,type,
sP5: $i > $i > $i > $o > $o ).
thf(func_def_201,type,
sK13: $i > $i > $i ).
thf(func_def_202,type,
sK14: $i > $i > $i ).
thf(func_def_205,type,
sK17: $i > $o ).
thf(func_def_206,type,
sK18: $i > $o ).
thf(func_def_214,type,
sK26: $i > $i > $i ).
thf(func_def_222,type,
sK34: $i > $i ).
thf(func_def_223,type,
sK35: ( $i > $o ) > $i ).
thf(func_def_224,type,
sK36: $i > $o ).
thf(func_def_225,type,
sK37: $i > $i ).
thf(func_def_228,type,
sK40: $i > $o ).
thf(func_def_230,type,
sK42: ( $i > $o ) > $i ).
thf(func_def_231,type,
sK43: $i > $o ).
thf(func_def_238,type,
sK50: $i > $o ).
thf(func_def_255,type,
sK67: $i > $o ).
thf(func_def_260,type,
sK72: $i > $o ).
thf(func_def_264,type,
sK76: $i > $i > $i ).
thf(func_def_269,type,
sK81: $i > $o ).
thf(func_def_271,type,
sK83: ( $i > $o ) > $i > $i ).
thf(func_def_273,type,
sK85: $i > $o ).
thf(func_def_277,type,
sK89: $i > $o ).
thf(func_def_299,type,
sK111: $i > $o ).
thf(func_def_301,type,
sK113: ( $i > $o ) > $i > $i ).
thf(func_def_302,type,
sK114: $i > $i > $i ).
thf(func_def_303,type,
sK115: $i > $i > $i ).
thf(func_def_304,type,
sK116: $i > $i > $i ).
thf(func_def_305,type,
sK117: $i > $i > $i ).
thf(func_def_306,type,
sK118: $i > $i > $i ).
thf(func_def_307,type,
sK119: $i > $i > $i > $i ).
thf(func_def_308,type,
sK120: $i > $i ).
thf(func_def_309,type,
sK121: $i > $i ).
thf(func_def_310,type,
sK122: $i > $i ).
thf(func_def_311,type,
sK123: $i > $i ).
thf(func_def_312,type,
sK124: $i > $i > $i > $i ).
thf(func_def_313,type,
sK125: $i > $i > $i > $i ).
thf(func_def_314,type,
sK126: $i > $i > $i ).
thf(func_def_315,type,
sK127: $i > $i > $i ).
thf(func_def_316,type,
sK128: $i > $i > $i ).
thf(func_def_317,type,
sK129: $i > $i ).
thf(func_def_319,type,
sK131: $i > $i ).
thf(func_def_320,type,
sK132: $i > $i ).
thf(func_def_321,type,
sK133: ( $i > $o ) > $i > $i ).
thf(func_def_323,type,
sK135: $i > $o ).
thf(func_def_333,type,
sK145: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_334,type,
sK146: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_335,type,
sK147: $i > $o ).
thf(func_def_336,type,
sK148: $i > $o ).
thf(func_def_340,type,
sK152: $i > $o ).
thf(func_def_343,type,
sK155: ( $i > $o ) > $i ).
thf(func_def_344,type,
sK156: $i > $o ).
thf(func_def_345,type,
sK157: $i > $i ).
thf(func_def_349,type,
sK161: $i > $o ).
thf(func_def_351,type,
sK163: ( $i > $o ) > $i > $i ).
thf(func_def_352,type,
sK164: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_353,type,
sK165: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_354,type,
sK166: $i > $o ).
thf(func_def_355,type,
sK167: $i > $o ).
thf(func_def_361,type,
sK173: $i > $i > $i ).
thf(func_def_363,type,
sK175: $i > $i > $o ).
thf(func_def_364,type,
sK176: $i > $i ).
thf(func_def_365,type,
sK177: $i > $i ).
thf(func_def_366,type,
sK178: ( $i > $i > $o ) > $i > $i ).
thf(func_def_367,type,
sK179: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_368,type,
sK180: ( $i > $i > $o ) > $i > $i ).
thf(func_def_372,type,
sK184: $i > $i > $i ).
thf(func_def_393,type,
sK205: $i > ( $i > $o ) > $i ).
thf(func_def_394,type,
sK206: $i > $o ).
thf(func_def_415,type,
sK227: $i > $o ).
thf(func_def_417,type,
sK229: $i > ( $i > $o ) > $i ).
thf(func_def_427,type,
sK239: $i > $i > $i > $i ).
thf(func_def_428,type,
sK240: $i > $i > $i > $i ).
thf(func_def_433,type,
sK245: $i > $i ).
thf(func_def_439,type,
sK251: $i > $i ).
thf(func_def_457,type,
sK269: $i > $i > $i ).
thf(func_def_463,type,
sK275: ( $i > $o ) > $i > $i ).
thf(func_def_465,type,
sK277: $i > $o ).
thf(func_def_467,type,
sK279: $i > $i ).
thf(func_def_469,type,
sK281: $i > $i ).
thf(func_def_473,type,
sK285: $i > $o ).
thf(func_def_486,type,
sK298: $i > $o ).
thf(func_def_487,type,
sK299: $i > $o ).
thf(func_def_488,type,
sK300: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_489,type,
sK301: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_490,type,
sK302: $i > $i > $i ).
thf(func_def_509,type,
sK321: $o > $i > $i > $i ).
thf(func_def_514,type,
sK326: $i > $o ).
thf(func_def_522,type,
sK334: ( $i > $o ) > $i ).
thf(func_def_523,type,
sK335: ( $i > $o ) > $i ).
thf(func_def_524,type,
sK336: $i > $o ).
thf(func_def_530,type,
sK342: $i > $i ).
thf(func_def_551,type,
sK363: $i > $i > $i ).
thf(func_def_555,type,
sK367: $i > $i > $i ).
thf(func_def_571,type,
sK383: ( $i > $o ) > $i > $i ).
thf(func_def_573,type,
sK385: $i > $o ).
thf(func_def_581,type,
sK393: $i > $i ).
thf(func_def_594,type,
sK406: $i > $o ).
thf(func_def_610,type,
ph422:
!>[X0: $tType] : X0 ).
thf(f3199,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f3198]) ).
thf(f3198,plain,
sK324 != sK324,
inference(superposition,[status(thm)],[f1899,f3190]) ).
thf(f3190,plain,
! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 ),
inference(backward_demodulation,[status(thm)],[f3179,f3189]) ).
thf(f3189,plain,
! [X0: $i] :
( ( binintersect @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) @ X0 )
= X0 ),
inference(trivial_inequality_removal,[status(thm)],[f3185]) ).
thf(f3185,plain,
! [X0: $i] :
( ( $true != $true )
| ( ( binintersect @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) @ X0 )
= X0 ) ),
inference(superposition,[status(thm)],[f2830,f2780]) ).
thf(f2780,plain,
! [X1: $i] :
( $true
= ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2456]) ).
thf(f2456,plain,
! [X1: $i] :
( ( $true
= ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) ) )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f1763,f1896]) ).
thf(f1896,plain,
setunionsingleton2 = $true,
inference(cnf_transformation,[status(thm)],[f1207]) ).
thf(f1207,plain,
( ( setunionE2 = $true )
& ( setadjoinSub2 = $true )
& ( emptyI = $true )
& ( notdallE = $true )
& ( exuE3e = $true )
& ( binintersectSubset3 = $true )
& ( kpairp = $true )
& ( binunionIL = $true )
& ( emptyE1 = $true )
& ( exuE2 = $true )
& ( notinemptyset = $true )
& ( quantDeMorgan4 = $true )
& ( ubforcartprodlem3 = $true )
& ( setminusSubset2 = $true )
& ( binintersectLsub = $true )
& ( setminusSubset1 = $true )
& ( emptyinPowerset = $true )
& ( binunionEcases = $true )
& ( setadjoinIR = $true )
& ( dsetconstr__Cong = $true )
& ( subsetE = $true )
& ( setminusLsub = $true )
& ( setukpairIR = $true )
& ( exuE3u = $true )
& ( singletoninpowunion = $true )
& ( binintersectSubset2 = $true )
& ( upairsetIL = $true )
& ( symdiffI1 = $true )
& ( sepInPowerset = $true )
& ( binintersectSubset4 = $true )
& ( setadjoin__Cong = $true )
& ( cartprodmempair1 = $true )
& ( emptysetsubset = $true )
& ( exu__Cong = $true )
& ( singletonsubset = $true )
& ( subsetI1 = $true )
& ( setadjoinAx = $true )
& ( binintersectI = $true )
& ( setminusER = $true )
& ( binunionIR = $true )
& ( wellorderingAx = $true )
& ( setbeta = $true )
& ( powersetE = $true )
& ( symdiffE = $true )
& ( sepSubset = $true )
& ( setminusERneg = $true )
& ( setextsub = $true )
& ( emptysetE = $true )
& ( quantDeMorgan1 = $true )
& ( binintersectEL = $true )
& ( dsetconstrEL = $true )
& ( subsetemptysetimpeq = $true )
& ( singletonsswitch = $true )
& ( binintersectSubset5 = $true )
& ( quantDeMorgan2 = $true )
& ( upairset2E = $true )
& ( noeltsimpempty = $true )
& ( exuI2 = $true )
& ( prop2set2propI = $true )
& ( nonemptyImpWitness = $true )
& ( quantDeMorgan3 = $true )
& ( symdiffI2 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( setadjoinE = $true )
& ( binintersectER = $true )
& ( omegaSAx = $true )
& ( ubforcartprodlem1 = $true )
& ( ( setunion @ ( setadjoin @ sK324 @ emptyset ) )
!= sK324 )
& ( omega__Cong = $true )
& ( binintersectSubset1 = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( upairsetE = $true )
& ( setminusI = $true )
& ( inCongP = $true )
& ( secondinupair = $true )
& ( emptysetimpfalse = $true )
& ( nonemptyI = $true )
& ( setunionAx = $true )
& ( powerset__Cong = $true )
& ( setextAx = $true )
& ( nonemptyE1 = $true )
& ( subsetTrans = $true )
& ( setminusELneg = $true )
& ( symdiffIneg1 = $true )
& ( vacuousDall = $true )
& ( bs114d = $true )
& ( uniqinunit = $true )
& ( setunionE = $true )
& ( powersetE1 = $true )
& ( dsetconstrER = $true )
& ( setadjoinSub = $true )
& ( setadjoinOr = $true )
& ( powersetI1 = $true )
& ( foundationAx = $true )
& ( exuI1 = $true )
& ( descr__Cong = $true )
& ( setminusIRneg = $true )
& ( binunionE = $true )
& ( prop2setE = $true )
& ( setext = $true )
& ( subsetRefl = $true )
& ( setminusILneg = $true )
& ( eqimpsubset1 = $true )
& ( upairsetIR = $true )
& ( setadjoinIL = $true )
& ( notequalI2 = $true )
& ( emptyinunitempty = $true )
& ( binunionLsub = $true )
& ( emptysetAx = $true )
& ( nonemptyI1 = $true )
& ( omega0Ax = $true )
& ( subsetI2 = $true )
& ( setunionI = $true )
& ( singletoninpowerset = $true )
& ( subset2powerset = $true )
& ( upairset2IR = $true )
& ( prop2setI = $true )
& ( disjointsetsI1 = $true )
& ( powersetsubset = $true )
& ( kpairiskpair = $true )
& ( emptyInPowerset = $true )
& ( inPowerset = $true )
& ( replAx = $true )
& ( notdexE = $true )
& ( eqimpsubset2 = $true )
& ( powersetAx = $true )
& ( subsetE2 = $true )
& ( setoftrueEq = $true )
& ( symdiffIneg2 = $true )
& ( cartprodpairin = $true )
& ( setukpairIL = $true )
& ( setunion__Cong = $true )
& ( ubforcartprodlem2 = $true )
& ( exuEu = $true )
& ( emptyset__Cong = $true )
& ( in__Cong = $true )
& ( descrp = $true )
& ( powersetI = $true )
& ( notsubsetI = $true )
& ( omegaIndAx = $true )
& ( exuE1 = $true )
& ( subPowSU = $true )
& ( binintersectRsub = $true )
& ( setminusEL = $true )
& ( exuI3 = $true )
& ( notequalI1 = $true )
& ( dsetconstrI = $true )
& ( notinsingleton = $true )
& ( binunionRsub = $true )
& ( cartprodmempair = $true )
& ( eqinunit = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK324])],[f548,f1206]) ).
thf(f1206,plain,
( ? [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
!= X0 )
=> ( ( setunion @ ( setadjoin @ sK324 @ emptyset ) )
!= sK324 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f548,plain,
( ( setunionE2 = $true )
& ( setadjoinSub2 = $true )
& ( emptyI = $true )
& ( notdallE = $true )
& ( exuE3e = $true )
& ( binintersectSubset3 = $true )
& ( kpairp = $true )
& ( binunionIL = $true )
& ( emptyE1 = $true )
& ( exuE2 = $true )
& ( notinemptyset = $true )
& ( quantDeMorgan4 = $true )
& ( ubforcartprodlem3 = $true )
& ( setminusSubset2 = $true )
& ( binintersectLsub = $true )
& ( setminusSubset1 = $true )
& ( emptyinPowerset = $true )
& ( binunionEcases = $true )
& ( setadjoinIR = $true )
& ( dsetconstr__Cong = $true )
& ( subsetE = $true )
& ( setminusLsub = $true )
& ( setukpairIR = $true )
& ( exuE3u = $true )
& ( singletoninpowunion = $true )
& ( binintersectSubset2 = $true )
& ( upairsetIL = $true )
& ( symdiffI1 = $true )
& ( sepInPowerset = $true )
& ( binintersectSubset4 = $true )
& ( setadjoin__Cong = $true )
& ( cartprodmempair1 = $true )
& ( emptysetsubset = $true )
& ( exu__Cong = $true )
& ( singletonsubset = $true )
& ( subsetI1 = $true )
& ( setadjoinAx = $true )
& ( binintersectI = $true )
& ( setminusER = $true )
& ( binunionIR = $true )
& ( wellorderingAx = $true )
& ( setbeta = $true )
& ( powersetE = $true )
& ( symdiffE = $true )
& ( sepSubset = $true )
& ( setminusERneg = $true )
& ( setextsub = $true )
& ( emptysetE = $true )
& ( quantDeMorgan1 = $true )
& ( binintersectEL = $true )
& ( dsetconstrEL = $true )
& ( subsetemptysetimpeq = $true )
& ( singletonsswitch = $true )
& ( binintersectSubset5 = $true )
& ( quantDeMorgan2 = $true )
& ( upairset2E = $true )
& ( noeltsimpempty = $true )
& ( exuI2 = $true )
& ( prop2set2propI = $true )
& ( nonemptyImpWitness = $true )
& ( quantDeMorgan3 = $true )
& ( symdiffI2 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( setadjoinE = $true )
& ( binintersectER = $true )
& ( omegaSAx = $true )
& ( ubforcartprodlem1 = $true )
& ? [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
!= X0 )
& ( omega__Cong = $true )
& ( binintersectSubset1 = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( upairsetE = $true )
& ( setminusI = $true )
& ( inCongP = $true )
& ( secondinupair = $true )
& ( emptysetimpfalse = $true )
& ( nonemptyI = $true )
& ( setunionAx = $true )
& ( powerset__Cong = $true )
& ( setextAx = $true )
& ( nonemptyE1 = $true )
& ( subsetTrans = $true )
& ( setminusELneg = $true )
& ( symdiffIneg1 = $true )
& ( vacuousDall = $true )
& ( bs114d = $true )
& ( uniqinunit = $true )
& ( setunionE = $true )
& ( powersetE1 = $true )
& ( dsetconstrER = $true )
& ( setadjoinSub = $true )
& ( setadjoinOr = $true )
& ( powersetI1 = $true )
& ( foundationAx = $true )
& ( exuI1 = $true )
& ( descr__Cong = $true )
& ( setminusIRneg = $true )
& ( binunionE = $true )
& ( prop2setE = $true )
& ( setext = $true )
& ( subsetRefl = $true )
& ( setminusILneg = $true )
& ( eqimpsubset1 = $true )
& ( upairsetIR = $true )
& ( setadjoinIL = $true )
& ( notequalI2 = $true )
& ( emptyinunitempty = $true )
& ( binunionLsub = $true )
& ( emptysetAx = $true )
& ( nonemptyI1 = $true )
& ( omega0Ax = $true )
& ( subsetI2 = $true )
& ( setunionI = $true )
& ( singletoninpowerset = $true )
& ( subset2powerset = $true )
& ( upairset2IR = $true )
& ( prop2setI = $true )
& ( disjointsetsI1 = $true )
& ( powersetsubset = $true )
& ( kpairiskpair = $true )
& ( emptyInPowerset = $true )
& ( inPowerset = $true )
& ( replAx = $true )
& ( notdexE = $true )
& ( eqimpsubset2 = $true )
& ( powersetAx = $true )
& ( subsetE2 = $true )
& ( setoftrueEq = $true )
& ( symdiffIneg2 = $true )
& ( cartprodpairin = $true )
& ( setukpairIL = $true )
& ( setunion__Cong = $true )
& ( ubforcartprodlem2 = $true )
& ( exuEu = $true )
& ( emptyset__Cong = $true )
& ( in__Cong = $true )
& ( descrp = $true )
& ( powersetI = $true )
& ( notsubsetI = $true )
& ( omegaIndAx = $true )
& ( exuE1 = $true )
& ( subPowSU = $true )
& ( binintersectRsub = $true )
& ( setminusEL = $true )
& ( exuI3 = $true )
& ( notequalI1 = $true )
& ( dsetconstrI = $true )
& ( notinsingleton = $true )
& ( binunionRsub = $true )
& ( cartprodmempair = $true )
& ( eqinunit = $true ) ),
inference(flattening,[status(thm)],[f547]) ).
thf(f547,plain,
( ? [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
!= X0 )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f461]) ).
thf(f461,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f460]) ).
thf(f460,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f158]) ).
thf(f158,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f157]) ).
thf(f157,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.EkEyqyb2Tu/DTF2THF_18509.p',setunionsingleton) ).
thf(f1763,plain,
! [X1: $i] :
( ( $true
= ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) ) )
| ( setunionsingleton2 != $true ) ),
inference(cnf_transformation,[status(thm)],[f1145]) ).
thf(f1145,plain,
( ( ( setunionsingleton2 = $true )
| ( $true
!= ( subset @ sK289 @ ( setunion @ ( setadjoin @ sK289 @ emptyset ) ) ) ) )
& ( ! [X1: $i] :
( $true
= ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) ) )
| ( setunionsingleton2 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK289])],[f1143,f1144]) ).
thf(f1144,plain,
( ? [X0: $i] :
( $true
!= ( subset @ X0 @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) ) )
=> ( $true
!= ( subset @ sK289 @ ( setunion @ ( setadjoin @ sK289 @ emptyset ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1143,plain,
( ( ( setunionsingleton2 = $true )
| ? [X0: $i] :
( $true
!= ( subset @ X0 @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) ) ) )
& ( ! [X1: $i] :
( $true
= ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) ) )
| ( setunionsingleton2 != $true ) ) ),
inference(rectify,[status(thm)],[f1142]) ).
thf(f1142,plain,
( ( ( setunionsingleton2 = $true )
| ? [X0: $i] :
( $true
!= ( subset @ X0 @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) ) ) )
& ( ! [X0: $i] :
( $true
= ( subset @ X0 @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) ) )
| ( setunionsingleton2 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f335]) ).
thf(f335,plain,
( ( setunionsingleton2 = $true )
<=> ! [X0: $i] :
( $true
= ( subset @ X0 @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f334]) ).
thf(f334,plain,
( setunionsingleton2
= ( ! [X0: $i] : ( subset @ X0 @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) ) ) ),
inference(rectify,[status(thm)],[f156]) ).
thf(f156,axiom,
( setunionsingleton2
= ( ! [X3: $i] : ( subset @ X3 @ ( setunion @ ( setadjoin @ X3 @ emptyset ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.EkEyqyb2Tu/DTF2THF_18509.p',setunionsingleton2) ).
thf(f2830,plain,
! [X0: $i,X1: $i] :
( ( $true
!= ( subset @ X1 @ X0 ) )
| ( ( binintersect @ X0 @ X1 )
= X1 ) ),
inference(trivial_inequality_removal,[status(thm)],[f2139]) ).
thf(f2139,plain,
! [X0: $i,X1: $i] :
( ( $true != $true )
| ( ( binintersect @ X0 @ X1 )
= X1 )
| ( $true
!= ( subset @ X1 @ X0 ) ) ),
inference(definition_unfolding,[status(thm)],[f1408,f1938]) ).
thf(f1938,plain,
binintersectSubset4 = $true,
inference(cnf_transformation,[status(thm)],[f1207]) ).
thf(f1408,plain,
! [X0: $i,X1: $i] :
( ( ( binintersect @ X0 @ X1 )
= X1 )
| ( $true
!= ( subset @ X1 @ X0 ) )
| ( binintersectSubset4 != $true ) ),
inference(cnf_transformation,[status(thm)],[f708]) ).
thf(f708,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( binintersect @ X0 @ X1 )
= X1 )
| ( $true
!= ( subset @ X1 @ X0 ) ) )
| ( binintersectSubset4 != $true ) )
& ( ( binintersectSubset4 = $true )
| ( ( ( binintersect @ sK30 @ sK31 )
!= sK31 )
& ( $true
= ( subset @ sK31 @ sK30 ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK30,sK31])],[f706,f707]) ).
thf(f707,plain,
( ? [X2: $i,X3: $i] :
( ( ( binintersect @ X2 @ X3 )
!= X3 )
& ( $true
= ( subset @ X3 @ X2 ) ) )
=> ( ( ( binintersect @ sK30 @ sK31 )
!= sK31 )
& ( $true
= ( subset @ sK31 @ sK30 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f706,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( binintersect @ X0 @ X1 )
= X1 )
| ( $true
!= ( subset @ X1 @ X0 ) ) )
| ( binintersectSubset4 != $true ) )
& ( ( binintersectSubset4 = $true )
| ? [X2: $i,X3: $i] :
( ( ( binintersect @ X2 @ X3 )
!= X3 )
& ( $true
= ( subset @ X3 @ X2 ) ) ) ) ),
inference(rectify,[status(thm)],[f705]) ).
thf(f705,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( binintersect @ X0 @ X1 )
= X1 )
| ( $true
!= ( subset @ X1 @ X0 ) ) )
| ( binintersectSubset4 != $true ) )
& ( ( binintersectSubset4 = $true )
| ? [X0: $i,X1: $i] :
( ( ( binintersect @ X0 @ X1 )
!= X1 )
& ( $true
= ( subset @ X1 @ X0 ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f639]) ).
thf(f639,plain,
( ! [X0: $i,X1: $i] :
( ( ( binintersect @ X0 @ X1 )
= X1 )
| ( $true
!= ( subset @ X1 @ X0 ) ) )
<=> ( binintersectSubset4 = $true ) ),
inference(ennf_transformation,[status(thm)],[f272]) ).
thf(f272,plain,
( ! [X0: $i,X1: $i] :
( ( $true
= ( subset @ X1 @ X0 ) )
=> ( ( binintersect @ X0 @ X1 )
= X1 ) )
<=> ( binintersectSubset4 = $true ) ),
inference(fool_elimination,[status(thm)],[f271]) ).
thf(f271,plain,
( binintersectSubset4
= ( ! [X0: $i,X1: $i] :
( ( subset @ X1 @ X0 )
=> ( ( binintersect @ X0 @ X1 )
= X1 ) ) ) ),
inference(rectify,[status(thm)],[f117]) ).
thf(f117,axiom,
( binintersectSubset4
= ( ! [X3: $i,X4: $i] :
( ( subset @ X4 @ X3 )
=> ( ( binintersect @ X3 @ X4 )
= X4 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.EkEyqyb2Tu/DTF2THF_18509.p',binintersectSubset4) ).
thf(f3179,plain,
! [X0: $i] :
( ( binintersect @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) @ X0 )
= ( setunion @ ( setadjoin @ X0 @ emptyset ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3176]) ).
thf(f3176,plain,
! [X0: $i] :
( ( $true != $true )
| ( ( binintersect @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) @ X0 )
= ( setunion @ ( setadjoin @ X0 @ emptyset ) ) ) ),
inference(superposition,[status(thm)],[f2737,f2683]) ).
thf(f2683,plain,
! [X1: $i] :
( $true
= ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 ) ),
inference(trivial_inequality_removal,[status(thm)],[f2190]) ).
thf(f2190,plain,
! [X1: $i] :
( ( $true != $true )
| ( $true
= ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 ) ) ),
inference(definition_unfolding,[status(thm)],[f1456,f1895]) ).
thf(f1895,plain,
setunionsingleton1 = $true,
inference(cnf_transformation,[status(thm)],[f1207]) ).
thf(f1456,plain,
! [X1: $i] :
( ( $true
= ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 ) )
| ( setunionsingleton1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f769]) ).
thf(f769,plain,
( ( ( setunionsingleton1 = $true )
| ( ( subset @ ( setunion @ ( setadjoin @ sK65 @ emptyset ) ) @ sK65 )
!= $true ) )
& ( ! [X1: $i] :
( $true
= ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 ) )
| ( setunionsingleton1 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK65])],[f767,f768]) ).
thf(f768,plain,
( ? [X0: $i] :
( $true
!= ( subset @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) @ X0 ) )
=> ( ( subset @ ( setunion @ ( setadjoin @ sK65 @ emptyset ) ) @ sK65 )
!= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f767,plain,
( ( ( setunionsingleton1 = $true )
| ? [X0: $i] :
( $true
!= ( subset @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) @ X0 ) ) )
& ( ! [X1: $i] :
( $true
= ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 ) )
| ( setunionsingleton1 != $true ) ) ),
inference(rectify,[status(thm)],[f766]) ).
thf(f766,plain,
( ( ( setunionsingleton1 = $true )
| ? [X0: $i] :
( $true
!= ( subset @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) @ X0 ) ) )
& ( ! [X0: $i] :
( $true
= ( subset @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) @ X0 ) )
| ( setunionsingleton1 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f387]) ).
thf(f387,plain,
( ( setunionsingleton1 = $true )
<=> ! [X0: $i] :
( $true
= ( subset @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) @ X0 ) ) ),
inference(fool_elimination,[status(thm)],[f386]) ).
thf(f386,plain,
! [X0: $i] :
( ( subset @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) @ X0 )
= setunionsingleton1 ),
inference(rectify,[status(thm)],[f155]) ).
thf(f155,axiom,
! [X3: $i] :
( ( subset @ ( setunion @ ( setadjoin @ X3 @ emptyset ) ) @ X3 )
= setunionsingleton1 ),
file('/export/starexec/sandbox/tmp/tmp.EkEyqyb2Tu/DTF2THF_18509.p',setunionsingleton1) ).
thf(f2737,plain,
! [X0: $i,X1: $i] :
( ( $true
!= ( subset @ X1 @ X0 ) )
| ( ( binintersect @ X1 @ X0 )
= X1 ) ),
inference(trivial_inequality_removal,[status(thm)],[f2406]) ).
thf(f2406,plain,
! [X0: $i,X1: $i] :
( ( $true != $true )
| ( ( binintersect @ X1 @ X0 )
= X1 )
| ( $true
!= ( subset @ X1 @ X0 ) ) ),
inference(definition_unfolding,[status(thm)],[f1716,f1942]) ).
thf(f1942,plain,
binintersectSubset2 = $true,
inference(cnf_transformation,[status(thm)],[f1207]) ).
thf(f1716,plain,
! [X0: $i,X1: $i] :
( ( $true
!= ( subset @ X1 @ X0 ) )
| ( ( binintersect @ X1 @ X0 )
= X1 )
| ( binintersectSubset2 != $true ) ),
inference(cnf_transformation,[status(thm)],[f1081]) ).
thf(f1081,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
!= ( subset @ X1 @ X0 ) )
| ( ( binintersect @ X1 @ X0 )
= X1 ) )
| ( binintersectSubset2 != $true ) )
& ( ( binintersectSubset2 = $true )
| ( ( $true
= ( subset @ sK253 @ sK252 ) )
& ( ( binintersect @ sK253 @ sK252 )
!= sK253 ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK252,sK253])],[f1079,f1080]) ).
thf(f1080,plain,
( ? [X2: $i,X3: $i] :
( ( $true
= ( subset @ X3 @ X2 ) )
& ( ( binintersect @ X3 @ X2 )
!= X3 ) )
=> ( ( $true
= ( subset @ sK253 @ sK252 ) )
& ( ( binintersect @ sK253 @ sK252 )
!= sK253 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1079,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
!= ( subset @ X1 @ X0 ) )
| ( ( binintersect @ X1 @ X0 )
= X1 ) )
| ( binintersectSubset2 != $true ) )
& ( ( binintersectSubset2 = $true )
| ? [X2: $i,X3: $i] :
( ( $true
= ( subset @ X3 @ X2 ) )
& ( ( binintersect @ X3 @ X2 )
!= X3 ) ) ) ),
inference(rectify,[status(thm)],[f1078]) ).
thf(f1078,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
!= ( subset @ X1 @ X0 ) )
| ( ( binintersect @ X1 @ X0 )
= X1 ) )
| ( binintersectSubset2 != $true ) )
& ( ( binintersectSubset2 = $true )
| ? [X0: $i,X1: $i] :
( ( $true
= ( subset @ X1 @ X0 ) )
& ( ( binintersect @ X1 @ X0 )
!= X1 ) ) ) ),
inference(nnf_transformation,[status(thm)],[f632]) ).
thf(f632,plain,
( ! [X0: $i,X1: $i] :
( ( $true
!= ( subset @ X1 @ X0 ) )
| ( ( binintersect @ X1 @ X0 )
= X1 ) )
<=> ( binintersectSubset2 = $true ) ),
inference(ennf_transformation,[status(thm)],[f411]) ).
thf(f411,plain,
( ( binintersectSubset2 = $true )
<=> ! [X0: $i,X1: $i] :
( ( $true
= ( subset @ X1 @ X0 ) )
=> ( ( binintersect @ X1 @ X0 )
= X1 ) ) ),
inference(fool_elimination,[status(thm)],[f410]) ).
thf(f410,plain,
! [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
=> ( ( binintersect @ X1 @ X0 )
= X1 ) )
= binintersectSubset2 ),
inference(rectify,[status(thm)],[f112]) ).
thf(f112,axiom,
! [X4: $i,X3: $i] :
( ( ( subset @ X3 @ X4 )
=> ( ( binintersect @ X3 @ X4 )
= X3 ) )
= binintersectSubset2 ),
file('/export/starexec/sandbox/tmp/tmp.EkEyqyb2Tu/DTF2THF_18509.p',binintersectSubset2) ).
thf(f1899,plain,
( ( setunion @ ( setadjoin @ sK324 @ emptyset ) )
!= sK324 ),
inference(cnf_transformation,[status(thm)],[f1207]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU636^1 : 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
% 0.15/0.33 % Computer : n009.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 12:29:10 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
% 0.31/0.45 ---- Original DTF file ---
% 0.31/0.45 thf(spec,logic,$$dhol).
% 0.31/0.45 %------------------------------------------------------------------------------
% 0.31/0.45 % File : SEU636^1 : TPTP v9.2.1. Released v3.7.0.
% 0.31/0.45 % Domain : Set Theory
% 0.31/0.45 % Problem : Ordered Pairs - Singletons
% 0.31/0.45 % Version : Especial.
% 0.31/0.45 % English : (! x:i.setunion (setadjoin x emptyset) = x)
% 0.31/0.45
% 0.31/0.45 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.31/0.45 % Source : [Bro08]
% 0.31/0.45 % Names : ZFC138g [Bro08]
% 0.31/0.45
% 0.31/0.45 % Status : Theorem
% 0.31/0.45 % Rating : 0.22 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, 0.50 v5.4.0, 0.60 v5.3.0, 0.40 v5.1.0, 0.60 v4.1.0, 0.67 v3.7.0
% 0.31/0.45 % Syntax : Number of formulae : 332 ( 156 unt; 175 typ; 156 def)
% 0.31/0.45 % Number of atoms : 826 ( 218 equ; 0 cnn)
% 0.31/0.45 % Maximal formula atoms : 153 ( 5 avg)
% 0.31/0.45 % Number of connectives : 1418 ( 47 ~; 7 |; 33 &; 956 @)
% 0.31/0.45 % ( 14 <=>; 361 =>; 0 <=; 0 <~>)
% 0.31/0.45 % Maximal formula depth : 154 ( 2 avg)
% 0.31/0.45 % Number of types : 2 ( 0 usr)
% 0.31/0.45 % Number of type conns : 69 ( 69 >; 0 *; 0 +; 0 <<)
% 0.31/0.45 % Number of symbols : 178 ( 175 usr; 156 con; 0-2 aty)
% 0.31/0.45 % Number of variables : 449 ( 33 ^; 382 !; 34 ?; 449 :)
% 0.31/0.45 % SPC : TH0_THM_EQU_NAR
% 0.31/0.45
% 0.31/0.45 % Comments : http://mathgate.info/detsetitem.php?id=193
% 0.31/0.45 %------------------------------------------------------------------------------
% 0.31/0.45 thf(in_type,type,
% 0.31/0.45 in: $i > $i > $o ).
% 0.31/0.45
% 0.31/0.45 thf(exu_type,type,
% 0.31/0.45 exu: ( $i > $o ) > $o ).
% 0.31/0.45
% 0.31/0.45 thf(exu,definition,
% 0.31/0.45 ( exu
% 0.31/0.45 = ( ^ [Xphi: $i > $o] :
% 0.31/0.45 ? [Xx: $i] :
% 0.31/0.45 ( ( Xphi @ Xx )
% 0.31/0.45 & ! [Xy: $i] :
% 0.31/0.45 ( ( Xphi @ Xy )
% 0.31/0.45 => ( Xx = Xy ) ) ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(setextAx_type,type,
% 0.31/0.45 setextAx: $o ).
% 0.31/0.45
% 0.31/0.45 thf(setextAx,definition,
% 0.31/0.45 ( setextAx
% 0.31/0.45 = ( ! [A: $i,B: $i] :
% 0.31/0.45 ( ! [Xx: $i] :
% 0.31/0.45 ( ( in @ Xx @ A )
% 0.31/0.45 <=> ( in @ Xx @ B ) )
% 0.31/0.45 => ( A = B ) ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(emptyset_type,type,
% 0.31/0.45 emptyset: $i ).
% 0.31/0.45
% 0.31/0.45 thf(emptysetAx_type,type,
% 0.31/0.45 emptysetAx: $o ).
% 0.31/0.45
% 0.31/0.45 thf(emptysetAx,definition,
% 0.31/0.45 ( emptysetAx
% 0.31/0.45 = ( ! [Xx: $i] :
% 0.31/0.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(setadjoin_type,type,
% 0.31/0.45 setadjoin: $i > $i > $i ).
% 0.31/0.45
% 0.31/0.45 thf(setadjoinAx_type,type,
% 0.31/0.45 setadjoinAx: $o ).
% 0.31/0.45
% 0.31/0.45 thf(setadjoinAx,definition,
% 0.31/0.45 ( setadjoinAx
% 0.31/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/0.45 <=> ( ( Xy = Xx )
% 0.31/0.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(powerset_type,type,
% 0.31/0.45 powerset: $i > $i ).
% 0.31/0.45
% 0.31/0.45 thf(powersetAx_type,type,
% 0.31/0.45 powersetAx: $o ).
% 0.31/0.45
% 0.31/0.45 thf(powersetAx,definition,
% 0.31/0.45 ( powersetAx
% 0.31/0.45 = ( ! [A: $i,B: $i] :
% 0.31/0.45 ( ( in @ B @ ( powerset @ A ) )
% 0.31/0.45 <=> ! [Xx: $i] :
% 0.31/0.45 ( ( in @ Xx @ B )
% 0.31/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.45
% 0.31/0.45 thf(setunion_type,type,
% 0.31/0.45 setunion: $i > $i ).
% 0.31/0.45
% 0.31/0.45 thf(setunionAx_type,type,
% 0.31/0.45 setunionAx: $o ).
% 0.31/0.45
% 0.31/0.45 thf(setunionAx,definition,
% 0.31/0.45 ( setunionAx
% 0.31/0.45 = ( ! [A: $i,Xx: $i] :
% 0.31/0.45 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.46 <=> ? [B: $i] :
% 0.31/0.46 ( ( in @ Xx @ B )
% 0.31/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(omega_type,type,
% 0.31/0.46 omega: $i ).
% 0.31/0.46
% 0.31/0.46 thf(omega0Ax_type,type,
% 0.31/0.46 omega0Ax: $o ).
% 0.31/0.46
% 0.31/0.46 thf(omega0Ax,definition,
% 0.31/0.46 ( omega0Ax
% 0.31/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(omegaSAx_type,type,
% 0.31/0.46 omegaSAx: $o ).
% 0.31/0.46
% 0.31/0.46 thf(omegaSAx,definition,
% 0.31/0.46 ( omegaSAx
% 0.31/0.46 = ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ omega )
% 0.31/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(omegaIndAx_type,type,
% 0.31/0.46 omegaIndAx: $o ).
% 0.31/0.46
% 0.31/0.46 thf(omegaIndAx,definition,
% 0.31/0.46 ( omegaIndAx
% 0.31/0.46 = ( ! [A: $i] :
% 0.31/0.46 ( ( ( in @ emptyset @ A )
% 0.31/0.46 & ! [Xx: $i] :
% 0.31/0.46 ( ( ( in @ Xx @ omega )
% 0.31/0.46 & ( in @ Xx @ A ) )
% 0.31/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.31/0.46 => ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ omega )
% 0.31/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(replAx_type,type,
% 0.31/0.46 replAx: $o ).
% 0.31/0.46
% 0.31/0.46 thf(replAx,definition,
% 0.31/0.46 ( replAx
% 0.31/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( exu
% 0.31/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.31/0.46 => ? [B: $i] :
% 0.31/0.46 ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ B )
% 0.31/0.46 <=> ? [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ A )
% 0.31/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(foundationAx_type,type,
% 0.31/0.46 foundationAx: $o ).
% 0.31/0.46
% 0.31/0.46 thf(foundationAx,definition,
% 0.31/0.46 ( foundationAx
% 0.31/0.46 = ( ! [A: $i] :
% 0.31/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.31/0.46 => ? [B: $i] :
% 0.31/0.46 ( ( in @ B @ A )
% 0.31/0.46 & ~ ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ B )
% 0.31/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(wellorderingAx_type,type,
% 0.31/0.46 wellorderingAx: $o ).
% 0.31/0.46
% 0.31/0.46 thf(wellorderingAx,definition,
% 0.31/0.46 ( wellorderingAx
% 0.31/0.46 = ( ! [A: $i] :
% 0.31/0.46 ? [B: $i] :
% 0.31/0.46 ( ! [C: $i] :
% 0.31/0.46 ( ( in @ C @ B )
% 0.31/0.46 => ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ C )
% 0.31/0.46 => ( in @ Xx @ A ) ) )
% 0.31/0.46 & ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( ( in @ Xx @ A )
% 0.31/0.46 & ( in @ Xy @ A ) )
% 0.31/0.46 => ( ! [C: $i] :
% 0.31/0.46 ( ( in @ C @ B )
% 0.31/0.46 => ( ( in @ Xx @ C )
% 0.31/0.46 <=> ( in @ Xy @ C ) ) )
% 0.31/0.46 => ( Xx = Xy ) ) )
% 0.31/0.46 & ! [C: $i,D: $i] :
% 0.31/0.46 ( ( ( in @ C @ B )
% 0.31/0.46 & ( in @ D @ B ) )
% 0.31/0.46 => ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ C )
% 0.31/0.46 => ( in @ Xx @ D ) )
% 0.31/0.46 | ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ D )
% 0.31/0.46 => ( in @ Xx @ C ) ) ) )
% 0.31/0.46 & ! [C: $i] :
% 0.31/0.46 ( ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ C )
% 0.31/0.46 => ( in @ Xx @ A ) )
% 0.31/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.31/0.46 => ? [D: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ D @ B )
% 0.31/0.46 & ( in @ Xx @ C )
% 0.31/0.46 & ~ ? [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ D )
% 0.31/0.46 & ( in @ Xy @ C ) )
% 0.31/0.46 & ! [E: $i] :
% 0.31/0.46 ( ( in @ E @ B )
% 0.31/0.46 => ( ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ E )
% 0.31/0.46 => ( in @ Xy @ D ) )
% 0.31/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(descr_type,type,
% 0.31/0.46 descr: ( $i > $o ) > $i ).
% 0.31/0.46
% 0.31/0.46 thf(descrp_type,type,
% 0.31/0.46 descrp: $o ).
% 0.31/0.46
% 0.31/0.46 thf(descrp,definition,
% 0.31/0.46 ( descrp
% 0.31/0.46 = ( ! [Xphi: $i > $o] :
% 0.31/0.46 ( ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 => ( Xphi
% 0.31/0.46 @ ( descr
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(dsetconstr_type,type,
% 0.31/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.31/0.46
% 0.31/0.46 thf(dsetconstrI_type,type,
% 0.31/0.46 dsetconstrI: $o ).
% 0.31/0.46
% 0.31/0.46 thf(dsetconstrI,definition,
% 0.31/0.46 ( dsetconstrI
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ( Xphi @ Xx )
% 0.31/0.46 => ( in @ Xx
% 0.31/0.46 @ ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(dsetconstrEL_type,type,
% 0.31/0.46 dsetconstrEL: $o ).
% 0.31/0.46
% 0.31/0.46 thf(dsetconstrEL,definition,
% 0.31/0.46 ( dsetconstrEL
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx
% 0.31/0.46 @ ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(dsetconstrER_type,type,
% 0.31/0.46 dsetconstrER: $o ).
% 0.31/0.46
% 0.31/0.46 thf(dsetconstrER,definition,
% 0.31/0.46 ( dsetconstrER
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx
% 0.31/0.46 @ ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(exuE1_type,type,
% 0.31/0.46 exuE1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(exuE1,definition,
% 0.31/0.46 ( exuE1
% 0.31/0.46 = ( ! [Xphi: $i > $o] :
% 0.31/0.46 ( ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 => ? [Xx: $i] :
% 0.31/0.46 ( ( Xphi @ Xx )
% 0.31/0.46 & ! [Xy: $i] :
% 0.31/0.46 ( ( Xphi @ Xy )
% 0.31/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(prop2set_type,type,
% 0.31/0.46 prop2set: $o > $i ).
% 0.31/0.46
% 0.31/0.46 thf(prop2setE_type,type,
% 0.31/0.46 prop2setE: $o ).
% 0.31/0.46
% 0.31/0.46 thf(prop2setE,definition,
% 0.31/0.46 ( prop2setE
% 0.31/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.31/0.46 => Xphi ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(emptysetE_type,type,
% 0.31/0.46 emptysetE: $o ).
% 0.31/0.46
% 0.31/0.46 thf(emptysetE,definition,
% 0.31/0.46 ( emptysetE
% 0.31/0.46 = ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ emptyset )
% 0.31/0.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(emptysetimpfalse_type,type,
% 0.31/0.46 emptysetimpfalse: $o ).
% 0.31/0.46
% 0.31/0.46 thf(emptysetimpfalse,definition,
% 0.31/0.46 ( emptysetimpfalse
% 0.31/0.46 = ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ emptyset )
% 0.31/0.46 => $false ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(notinemptyset_type,type,
% 0.31/0.46 notinemptyset: $o ).
% 0.31/0.46
% 0.31/0.46 thf(notinemptyset,definition,
% 0.31/0.46 ( notinemptyset
% 0.31/0.46 = ( ! [Xx: $i] :
% 0.31/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(exuE3e_type,type,
% 0.31/0.46 exuE3e: $o ).
% 0.31/0.46
% 0.31/0.46 thf(exuE3e,definition,
% 0.31/0.46 ( exuE3e
% 0.31/0.46 = ( ! [Xphi: $i > $o] :
% 0.31/0.46 ( ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setext_type,type,
% 0.31/0.46 setext: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setext,definition,
% 0.31/0.46 ( setext
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ Xx @ B ) )
% 0.31/0.46 => ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ B )
% 0.31/0.46 => ( in @ Xx @ A ) )
% 0.31/0.46 => ( A = B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(emptyI_type,type,
% 0.31/0.46 emptyI: $o ).
% 0.31/0.46
% 0.31/0.46 thf(emptyI,definition,
% 0.31/0.46 ( emptyI
% 0.31/0.46 = ( ! [A: $i] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ~ ( in @ Xx @ A )
% 0.31/0.46 => ( A = emptyset ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(noeltsimpempty_type,type,
% 0.31/0.46 noeltsimpempty: $o ).
% 0.31/0.46
% 0.31/0.46 thf(noeltsimpempty,definition,
% 0.31/0.46 ( noeltsimpempty
% 0.31/0.46 = ( ! [A: $i] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ~ ( in @ Xx @ A )
% 0.31/0.46 => ( A = emptyset ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setbeta_type,type,
% 0.31/0.46 setbeta: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setbeta,definition,
% 0.31/0.46 ( setbeta
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ( in @ Xx
% 0.31/0.46 @ ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/0.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(nonempty_type,type,
% 0.31/0.46 nonempty: $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(nonempty,definition,
% 0.31/0.46 ( nonempty
% 0.31/0.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(nonemptyE1_type,type,
% 0.31/0.46 nonemptyE1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(nonemptyE1,definition,
% 0.31/0.46 ( nonemptyE1
% 0.31/0.46 = ( ! [A: $i] :
% 0.31/0.46 ( ( nonempty @ A )
% 0.31/0.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(nonemptyI_type,type,
% 0.31/0.46 nonemptyI: $o ).
% 0.31/0.46
% 0.31/0.46 thf(nonemptyI,definition,
% 0.31/0.46 ( nonemptyI
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ( Xphi @ Xx )
% 0.31/0.46 => ( nonempty
% 0.31/0.46 @ ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(nonemptyI1_type,type,
% 0.31/0.46 nonemptyI1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(nonemptyI1,definition,
% 0.31/0.46 ( nonemptyI1
% 0.31/0.46 = ( ! [A: $i] :
% 0.31/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.31/0.46 => ( nonempty @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinIL_type,type,
% 0.31/0.46 setadjoinIL: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinIL,definition,
% 0.31/0.46 ( setadjoinIL
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(emptyinunitempty_type,type,
% 0.31/0.46 emptyinunitempty: $o ).
% 0.31/0.46
% 0.31/0.46 thf(emptyinunitempty,definition,
% 0.31/0.46 ( emptyinunitempty
% 0.31/0.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinIR_type,type,
% 0.31/0.46 setadjoinIR: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinIR,definition,
% 0.31/0.46 ( setadjoinIR
% 0.31/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ A )
% 0.31/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinE_type,type,
% 0.31/0.46 setadjoinE: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinE,definition,
% 0.31/0.46 ( setadjoinE
% 0.31/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/0.46 => ! [Xphi: $o] :
% 0.31/0.46 ( ( ( Xy = Xx )
% 0.31/0.46 => Xphi )
% 0.31/0.46 => ( ( ( in @ Xy @ A )
% 0.31/0.46 => Xphi )
% 0.31/0.46 => Xphi ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinOr_type,type,
% 0.31/0.46 setadjoinOr: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinOr,definition,
% 0.31/0.46 ( setadjoinOr
% 0.31/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/0.46 => ( ( Xy = Xx )
% 0.31/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setoftrueEq_type,type,
% 0.31/0.46 setoftrueEq: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setoftrueEq,definition,
% 0.31/0.46 ( setoftrueEq
% 0.31/0.46 = ( ! [A: $i] :
% 0.31/0.46 ( ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xx: $i] : $true )
% 0.31/0.46 = A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(powersetI_type,type,
% 0.31/0.46 powersetI: $o ).
% 0.31/0.46
% 0.31/0.46 thf(powersetI,definition,
% 0.31/0.46 ( powersetI
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ B )
% 0.31/0.46 => ( in @ Xx @ A ) )
% 0.31/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(emptyinPowerset_type,type,
% 0.31/0.46 emptyinPowerset: $o ).
% 0.31/0.46
% 0.31/0.46 thf(emptyinPowerset,definition,
% 0.31/0.46 ( emptyinPowerset
% 0.31/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(emptyInPowerset_type,type,
% 0.31/0.46 emptyInPowerset: $o ).
% 0.31/0.46
% 0.31/0.46 thf(emptyInPowerset,definition,
% 0.31/0.46 ( emptyInPowerset
% 0.31/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(powersetE_type,type,
% 0.31/0.46 powersetE: $o ).
% 0.31/0.46
% 0.31/0.46 thf(powersetE,definition,
% 0.31/0.46 ( powersetE
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.31/0.46 => ( ( in @ Xx @ B )
% 0.31/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setunionI_type,type,
% 0.31/0.46 setunionI: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setunionI,definition,
% 0.31/0.46 ( setunionI
% 0.31/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.31/0.46 ( ( in @ Xx @ B )
% 0.31/0.46 => ( ( in @ B @ A )
% 0.31/0.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setunionE_type,type,
% 0.31/0.46 setunionE: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setunionE,definition,
% 0.31/0.46 ( setunionE
% 0.31/0.46 = ( ! [A: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.46 => ! [Xphi: $o] :
% 0.31/0.46 ( ! [B: $i] :
% 0.31/0.46 ( ( in @ Xx @ B )
% 0.31/0.46 => ( ( in @ B @ A )
% 0.31/0.46 => Xphi ) )
% 0.31/0.46 => Xphi ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(subPowSU_type,type,
% 0.31/0.46 subPowSU: $o ).
% 0.31/0.46
% 0.31/0.46 thf(subPowSU,definition,
% 0.31/0.46 ( subPowSU
% 0.31/0.46 = ( ! [A: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(exuE2_type,type,
% 0.31/0.46 exuE2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(exuE2,definition,
% 0.31/0.46 ( exuE2
% 0.31/0.46 = ( ! [Xphi: $i > $o] :
% 0.31/0.46 ( ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 => ? [Xx: $i] :
% 0.31/0.46 ! [Xy: $i] :
% 0.31/0.46 ( ( Xphi @ Xy )
% 0.31/0.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(nonemptyImpWitness_type,type,
% 0.31/0.46 nonemptyImpWitness: $o ).
% 0.31/0.46
% 0.31/0.46 thf(nonemptyImpWitness,definition,
% 0.31/0.46 ( nonemptyImpWitness
% 0.31/0.46 = ( ! [A: $i] :
% 0.31/0.46 ( ( nonempty @ A )
% 0.31/0.46 => ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & $true ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(uniqinunit_type,type,
% 0.31/0.46 uniqinunit: $o ).
% 0.31/0.46
% 0.31/0.46 thf(uniqinunit,definition,
% 0.31/0.46 ( uniqinunit
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(notinsingleton_type,type,
% 0.31/0.46 notinsingleton: $o ).
% 0.31/0.46
% 0.31/0.46 thf(notinsingleton,definition,
% 0.31/0.46 ( notinsingleton
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( Xx != Xy )
% 0.31/0.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(eqinunit_type,type,
% 0.31/0.46 eqinunit: $o ).
% 0.31/0.46
% 0.31/0.46 thf(eqinunit,definition,
% 0.31/0.46 ( eqinunit
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( Xx = Xy )
% 0.31/0.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(singletonsswitch_type,type,
% 0.31/0.46 singletonsswitch: $o ).
% 0.31/0.46
% 0.31/0.46 thf(singletonsswitch,definition,
% 0.31/0.46 ( singletonsswitch
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(upairsetE_type,type,
% 0.31/0.46 upairsetE: $o ).
% 0.31/0.46
% 0.31/0.46 thf(upairsetE,definition,
% 0.31/0.46 ( upairsetE
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.31/0.46 => ( ( Xz = Xx )
% 0.31/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(upairsetIL_type,type,
% 0.31/0.46 upairsetIL: $o ).
% 0.31/0.46
% 0.31/0.46 thf(upairsetIL,definition,
% 0.31/0.46 ( upairsetIL
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(upairsetIR_type,type,
% 0.31/0.46 upairsetIR: $o ).
% 0.31/0.46
% 0.31/0.46 thf(upairsetIR,definition,
% 0.31/0.46 ( upairsetIR
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(emptyE1_type,type,
% 0.31/0.46 emptyE1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(emptyE1,definition,
% 0.31/0.46 ( emptyE1
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ( Xphi @ Xx ) )
% 0.31/0.46 => ( ( ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 = emptyset )
% 0.31/0.46 => $false ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(vacuousDall_type,type,
% 0.31/0.46 vacuousDall: $o ).
% 0.31/0.46
% 0.31/0.46 thf(vacuousDall,definition,
% 0.31/0.46 ( vacuousDall
% 0.31/0.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ emptyset )
% 0.31/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(quantDeMorgan1_type,type,
% 0.31/0.46 quantDeMorgan1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(quantDeMorgan1,definition,
% 0.31/0.46 ( quantDeMorgan1
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( ~ ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( Xphi @ Xx ) )
% 0.31/0.46 => ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(quantDeMorgan2_type,type,
% 0.31/0.46 quantDeMorgan2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(quantDeMorgan2,definition,
% 0.31/0.46 ( quantDeMorgan2
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ~ ( Xphi @ Xx ) )
% 0.31/0.46 => ~ ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(quantDeMorgan3_type,type,
% 0.31/0.46 quantDeMorgan3: $o ).
% 0.31/0.46
% 0.31/0.46 thf(quantDeMorgan3,definition,
% 0.31/0.46 ( quantDeMorgan3
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( ~ ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ( Xphi @ Xx ) )
% 0.31/0.46 => ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(quantDeMorgan4_type,type,
% 0.31/0.46 quantDeMorgan4: $o ).
% 0.31/0.46
% 0.31/0.46 thf(quantDeMorgan4,definition,
% 0.31/0.46 ( quantDeMorgan4
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ~ ( Xphi @ Xx ) )
% 0.31/0.46 => ~ ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(prop2setI_type,type,
% 0.31/0.46 prop2setI: $o ).
% 0.31/0.46
% 0.31/0.46 thf(prop2setI,definition,
% 0.31/0.46 ( prop2setI
% 0.31/0.46 = ( ! [Xphi: $o] :
% 0.31/0.46 ( Xphi
% 0.31/0.46 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(set2prop_type,type,
% 0.31/0.46 set2prop: $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(prop2set2propI_type,type,
% 0.31/0.46 prop2set2propI: $o ).
% 0.31/0.46
% 0.31/0.46 thf(prop2set2propI,definition,
% 0.31/0.46 ( prop2set2propI
% 0.31/0.46 = ( ! [Xphi: $o] :
% 0.31/0.46 ( Xphi
% 0.31/0.46 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(notdexE_type,type,
% 0.31/0.46 notdexE: $o ).
% 0.31/0.46
% 0.31/0.46 thf(notdexE,definition,
% 0.31/0.46 ( notdexE
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( ~ ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ( Xphi @ Xx ) )
% 0.31/0.46 => ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(notdallE_type,type,
% 0.31/0.46 notdallE: $o ).
% 0.31/0.46
% 0.31/0.46 thf(notdallE,definition,
% 0.31/0.46 ( notdallE
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( ~ ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( Xphi @ Xx ) )
% 0.31/0.46 => ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(exuI1_type,type,
% 0.31/0.46 exuI1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(exuI1,definition,
% 0.31/0.46 ( exuI1
% 0.31/0.46 = ( ! [Xphi: $i > $o] :
% 0.31/0.46 ( ? [Xx: $i] :
% 0.31/0.46 ( ( Xphi @ Xx )
% 0.31/0.46 & ! [Xy: $i] :
% 0.31/0.46 ( ( Xphi @ Xy )
% 0.31/0.46 => ( Xx = Xy ) ) )
% 0.31/0.46 => ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(exuI3_type,type,
% 0.31/0.46 exuI3: $o ).
% 0.31/0.46
% 0.31/0.46 thf(exuI3,definition,
% 0.31/0.46 ( exuI3
% 0.31/0.46 = ( ! [Xphi: $i > $o] :
% 0.31/0.46 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.31/0.46 => ( ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( Xphi @ Xx )
% 0.31/0.46 => ( ( Xphi @ Xy )
% 0.31/0.46 => ( Xx = Xy ) ) )
% 0.31/0.46 => ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(exuI2_type,type,
% 0.31/0.46 exuI2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(exuI2,definition,
% 0.31/0.46 ( exuI2
% 0.31/0.46 = ( ! [Xphi: $i > $o] :
% 0.31/0.46 ( ? [Xx: $i] :
% 0.31/0.46 ! [Xy: $i] :
% 0.31/0.46 ( ( Xphi @ Xy )
% 0.31/0.46 <=> ( Xy = Xx ) )
% 0.31/0.46 => ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(inCongP_type,type,
% 0.31/0.46 inCongP: $o ).
% 0.31/0.46
% 0.31/0.46 thf(inCongP,definition,
% 0.31/0.46 ( inCongP
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( A = B )
% 0.31/0.46 => ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( Xx = Xy )
% 0.31/0.46 => ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(in__Cong_type,type,
% 0.31/0.46 in__Cong: $o ).
% 0.31/0.46
% 0.31/0.46 thf(in__Cong,definition,
% 0.31/0.46 ( in__Cong
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( A = B )
% 0.31/0.46 => ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( Xx = Xy )
% 0.31/0.46 => ( ( in @ Xx @ A )
% 0.31/0.46 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(exuE3u_type,type,
% 0.31/0.46 exuE3u: $o ).
% 0.31/0.46
% 0.31/0.46 thf(exuE3u,definition,
% 0.31/0.46 ( exuE3u
% 0.31/0.46 = ( ! [Xphi: $i > $o] :
% 0.31/0.46 ( ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 => ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( Xphi @ Xx )
% 0.31/0.46 => ( ( Xphi @ Xy )
% 0.31/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(exu__Cong_type,type,
% 0.31/0.46 exu__Cong: $o ).
% 0.31/0.46
% 0.31/0.46 thf(exu__Cong,definition,
% 0.31/0.46 ( exu__Cong
% 0.31/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( Xx = Xy )
% 0.31/0.46 => ( ( Xphi @ Xx )
% 0.31/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.31/0.46 => ( ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 <=> ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(emptyset__Cong_type,type,
% 0.31/0.46 emptyset__Cong: $o ).
% 0.31/0.46
% 0.31/0.46 thf(emptyset__Cong,definition,
% 0.31/0.46 ( emptyset__Cong
% 0.31/0.46 = ( emptyset = emptyset ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoin__Cong_type,type,
% 0.31/0.46 setadjoin__Cong: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoin__Cong,definition,
% 0.31/0.46 ( setadjoin__Cong
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( Xx = Xy )
% 0.31/0.46 => ! [Xz: $i,Xu: $i] :
% 0.31/0.46 ( ( Xz = Xu )
% 0.31/0.46 => ( ( setadjoin @ Xx @ Xz )
% 0.31/0.46 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(powerset__Cong_type,type,
% 0.31/0.46 powerset__Cong: $o ).
% 0.31/0.46
% 0.31/0.46 thf(powerset__Cong,definition,
% 0.31/0.46 ( powerset__Cong
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( A = B )
% 0.31/0.46 => ( ( powerset @ A )
% 0.31/0.46 = ( powerset @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setunion__Cong_type,type,
% 0.31/0.46 setunion__Cong: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setunion__Cong,definition,
% 0.31/0.46 ( setunion__Cong
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( A = B )
% 0.31/0.46 => ( ( setunion @ A )
% 0.31/0.46 = ( setunion @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(omega__Cong_type,type,
% 0.31/0.46 omega__Cong: $o ).
% 0.31/0.46
% 0.31/0.46 thf(omega__Cong,definition,
% 0.31/0.46 ( omega__Cong
% 0.31/0.46 = ( omega = omega ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(exuEu_type,type,
% 0.31/0.46 exuEu: $o ).
% 0.31/0.46
% 0.31/0.46 thf(exuEu,definition,
% 0.31/0.46 ( exuEu
% 0.31/0.46 = ( ! [Xphi: $i > $o] :
% 0.31/0.46 ( ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 => ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( Xphi @ Xx )
% 0.31/0.46 => ( ( Xphi @ Xy )
% 0.31/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(descr__Cong_type,type,
% 0.31/0.46 descr__Cong: $o ).
% 0.31/0.46
% 0.31/0.46 thf(descr__Cong,definition,
% 0.31/0.46 ( descr__Cong
% 0.31/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( Xx = Xy )
% 0.31/0.46 => ( ( Xphi @ Xx )
% 0.31/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.31/0.46 => ( ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 => ( ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.31/0.46 => ( ( descr
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 = ( descr
% 0.31/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(dsetconstr__Cong_type,type,
% 0.31/0.46 dsetconstr__Cong: $o ).
% 0.31/0.46
% 0.31/0.46 thf(dsetconstr__Cong,definition,
% 0.31/0.46 ( dsetconstr__Cong
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( A = B )
% 0.31/0.46 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( ( Xx = Xy )
% 0.31/0.46 => ( ( Xphi @ Xx )
% 0.31/0.46 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.31/0.46 => ( ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 = ( dsetconstr @ B
% 0.31/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(subset_type,type,
% 0.31/0.46 subset: $i > $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(disjoint_type,type,
% 0.31/0.46 disjoint: $i > $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(setsmeet_type,type,
% 0.31/0.46 setsmeet: $i > $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(subsetI1_type,type,
% 0.31/0.46 subsetI1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(subsetI1,definition,
% 0.31/0.46 ( subsetI1
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ Xx @ B ) )
% 0.31/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(eqimpsubset2_type,type,
% 0.31/0.46 eqimpsubset2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(eqimpsubset2,definition,
% 0.31/0.46 ( eqimpsubset2
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( A = B )
% 0.31/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(eqimpsubset1_type,type,
% 0.31/0.46 eqimpsubset1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(eqimpsubset1,definition,
% 0.31/0.46 ( eqimpsubset1
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( A = B )
% 0.31/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(subsetI2_type,type,
% 0.31/0.46 subsetI2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(subsetI2,definition,
% 0.31/0.46 ( subsetI2
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ Xx @ B ) )
% 0.31/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(emptysetsubset_type,type,
% 0.31/0.46 emptysetsubset: $o ).
% 0.31/0.46
% 0.31/0.46 thf(emptysetsubset,definition,
% 0.31/0.46 ( emptysetsubset
% 0.31/0.46 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(subsetE_type,type,
% 0.31/0.46 subsetE: $o ).
% 0.31/0.46
% 0.31/0.46 thf(subsetE,definition,
% 0.31/0.46 ( subsetE
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( subset @ A @ B )
% 0.31/0.46 => ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(subsetE2_type,type,
% 0.31/0.46 subsetE2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(subsetE2,definition,
% 0.31/0.46 ( subsetE2
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( subset @ A @ B )
% 0.31/0.46 => ( ~ ( in @ Xx @ B )
% 0.31/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(notsubsetI_type,type,
% 0.31/0.46 notsubsetI: $o ).
% 0.31/0.46
% 0.31/0.46 thf(notsubsetI,definition,
% 0.31/0.46 ( notsubsetI
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ~ ( in @ Xx @ B )
% 0.31/0.46 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(notequalI1_type,type,
% 0.31/0.46 notequalI1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(notequalI1,definition,
% 0.31/0.46 ( notequalI1
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ~ ( subset @ A @ B )
% 0.31/0.46 => ( A != B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(notequalI2_type,type,
% 0.31/0.46 notequalI2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(notequalI2,definition,
% 0.31/0.46 ( notequalI2
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ~ ( in @ Xx @ B )
% 0.31/0.46 => ( A != B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(subsetRefl_type,type,
% 0.31/0.46 subsetRefl: $o ).
% 0.31/0.46
% 0.31/0.46 thf(subsetRefl,definition,
% 0.31/0.46 ( subsetRefl
% 0.31/0.46 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(subsetTrans_type,type,
% 0.31/0.46 subsetTrans: $o ).
% 0.31/0.46
% 0.31/0.46 thf(subsetTrans,definition,
% 0.31/0.46 ( subsetTrans
% 0.31/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/0.46 ( ( subset @ A @ B )
% 0.31/0.46 => ( ( subset @ B @ C )
% 0.31/0.46 => ( subset @ A @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinSub_type,type,
% 0.31/0.46 setadjoinSub: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinSub,definition,
% 0.31/0.46 ( setadjoinSub
% 0.31/0.46 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinSub2_type,type,
% 0.31/0.46 setadjoinSub2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinSub2,definition,
% 0.31/0.46 ( setadjoinSub2
% 0.31/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.31/0.46 ( ( subset @ A @ B )
% 0.31/0.46 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(subset2powerset_type,type,
% 0.31/0.46 subset2powerset: $o ).
% 0.31/0.46
% 0.31/0.46 thf(subset2powerset,definition,
% 0.31/0.46 ( subset2powerset
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( subset @ A @ B )
% 0.31/0.46 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setextsub_type,type,
% 0.31/0.46 setextsub: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setextsub,definition,
% 0.31/0.46 ( setextsub
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( subset @ A @ B )
% 0.31/0.46 => ( ( subset @ B @ A )
% 0.31/0.46 => ( A = B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(subsetemptysetimpeq_type,type,
% 0.31/0.46 subsetemptysetimpeq: $o ).
% 0.31/0.46
% 0.31/0.46 thf(subsetemptysetimpeq,definition,
% 0.31/0.46 ( subsetemptysetimpeq
% 0.31/0.46 = ( ! [A: $i] :
% 0.31/0.46 ( ( subset @ A @ emptyset )
% 0.31/0.46 => ( A = emptyset ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(powersetI1_type,type,
% 0.31/0.46 powersetI1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(powersetI1,definition,
% 0.31/0.46 ( powersetI1
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( subset @ B @ A )
% 0.31/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(powersetE1_type,type,
% 0.31/0.46 powersetE1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(powersetE1,definition,
% 0.31/0.46 ( powersetE1
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.31/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(inPowerset_type,type,
% 0.31/0.46 inPowerset: $o ).
% 0.31/0.46
% 0.31/0.46 thf(inPowerset,definition,
% 0.31/0.46 ( inPowerset
% 0.31/0.46 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(powersetsubset_type,type,
% 0.31/0.46 powersetsubset: $o ).
% 0.31/0.46
% 0.31/0.46 thf(powersetsubset,definition,
% 0.31/0.46 ( powersetsubset
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( subset @ A @ B )
% 0.31/0.46 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sepInPowerset_type,type,
% 0.31/0.46 sepInPowerset: $o ).
% 0.31/0.46
% 0.31/0.46 thf(sepInPowerset,definition,
% 0.31/0.46 ( sepInPowerset
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( in
% 0.31/0.46 @ ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 @ ( powerset @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sepSubset_type,type,
% 0.31/0.46 sepSubset: $o ).
% 0.31/0.46
% 0.31/0.46 thf(sepSubset,definition,
% 0.31/0.46 ( sepSubset
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( subset
% 0.31/0.46 @ ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binunion_type,type,
% 0.31/0.46 binunion: $i > $i > $i ).
% 0.31/0.46
% 0.31/0.46 thf(binunionIL_type,type,
% 0.31/0.46 binunionIL: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binunionIL,definition,
% 0.31/0.46 ( binunionIL
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(upairset2IR_type,type,
% 0.31/0.46 upairset2IR: $o ).
% 0.31/0.46
% 0.31/0.46 thf(upairset2IR,definition,
% 0.31/0.46 ( upairset2IR
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binunionIR_type,type,
% 0.31/0.46 binunionIR: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binunionIR,definition,
% 0.31/0.46 ( binunionIR
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ B )
% 0.31/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binunionEcases_type,type,
% 0.31/0.46 binunionEcases: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binunionEcases,definition,
% 0.31/0.46 ( binunionEcases
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.31/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.31/0.46 => ( ( ( in @ Xx @ A )
% 0.31/0.46 => Xphi )
% 0.31/0.46 => ( ( ( in @ Xx @ B )
% 0.31/0.46 => Xphi )
% 0.31/0.46 => Xphi ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binunionE_type,type,
% 0.31/0.46 binunionE: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binunionE,definition,
% 0.31/0.46 ( binunionE
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.31/0.46 => ( ( in @ Xx @ A )
% 0.31/0.46 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binunionLsub_type,type,
% 0.31/0.46 binunionLsub: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binunionLsub,definition,
% 0.31/0.46 ( binunionLsub
% 0.31/0.46 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binunionRsub_type,type,
% 0.31/0.46 binunionRsub: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binunionRsub,definition,
% 0.31/0.46 ( binunionRsub
% 0.31/0.46 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binintersect_type,type,
% 0.31/0.46 binintersect: $i > $i > $i ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectI_type,type,
% 0.31/0.46 binintersectI: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectI,definition,
% 0.31/0.46 ( binintersectI
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ( in @ Xx @ B )
% 0.31/0.46 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectSubset5_type,type,
% 0.31/0.46 binintersectSubset5: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectSubset5,definition,
% 0.31/0.46 ( binintersectSubset5
% 0.31/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/0.46 ( ( subset @ C @ A )
% 0.31/0.46 => ( ( subset @ C @ B )
% 0.31/0.46 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectEL_type,type,
% 0.31/0.46 binintersectEL: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectEL,definition,
% 0.31/0.46 ( binintersectEL
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.31/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectLsub_type,type,
% 0.31/0.46 binintersectLsub: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectLsub,definition,
% 0.31/0.46 ( binintersectLsub
% 0.31/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectSubset2_type,type,
% 0.31/0.46 binintersectSubset2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectSubset2,definition,
% 0.31/0.46 ( binintersectSubset2
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( subset @ A @ B )
% 0.31/0.46 => ( ( binintersect @ A @ B )
% 0.31/0.46 = A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectSubset3_type,type,
% 0.31/0.46 binintersectSubset3: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectSubset3,definition,
% 0.31/0.46 ( binintersectSubset3
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( ( binintersect @ A @ B )
% 0.31/0.46 = B )
% 0.31/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectER_type,type,
% 0.31/0.46 binintersectER: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectER,definition,
% 0.31/0.46 ( binintersectER
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.31/0.46 => ( in @ Xx @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(disjointsetsI1_type,type,
% 0.31/0.46 disjointsetsI1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(disjointsetsI1,definition,
% 0.31/0.46 ( disjointsetsI1
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ~ ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ( in @ Xx @ B ) )
% 0.31/0.46 => ( ( binintersect @ A @ B )
% 0.31/0.46 = emptyset ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectRsub_type,type,
% 0.31/0.46 binintersectRsub: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectRsub,definition,
% 0.31/0.46 ( binintersectRsub
% 0.31/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectSubset4_type,type,
% 0.31/0.46 binintersectSubset4: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectSubset4,definition,
% 0.31/0.46 ( binintersectSubset4
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( subset @ B @ A )
% 0.31/0.46 => ( ( binintersect @ A @ B )
% 0.31/0.46 = B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectSubset1_type,type,
% 0.31/0.46 binintersectSubset1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectSubset1,definition,
% 0.31/0.46 ( binintersectSubset1
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( ( binintersect @ A @ B )
% 0.31/0.46 = A )
% 0.31/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(bs114d_type,type,
% 0.31/0.46 bs114d: $o ).
% 0.31/0.46
% 0.31/0.46 thf(bs114d,definition,
% 0.31/0.46 ( bs114d
% 0.31/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/0.46 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.31/0.46 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(regular_type,type,
% 0.31/0.46 regular: $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(setminus_type,type,
% 0.31/0.46 setminus: $i > $i > $i ).
% 0.31/0.46
% 0.31/0.46 thf(setminusI_type,type,
% 0.31/0.46 setminusI: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setminusI,definition,
% 0.31/0.46 ( setminusI
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ~ ( in @ Xx @ B )
% 0.31/0.46 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setminusEL_type,type,
% 0.31/0.46 setminusEL: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setminusEL,definition,
% 0.31/0.46 ( setminusEL
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setminusER_type,type,
% 0.31/0.46 setminusER: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setminusER,definition,
% 0.31/0.46 ( setminusER
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.46 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setminusSubset2_type,type,
% 0.31/0.46 setminusSubset2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setminusSubset2,definition,
% 0.31/0.46 ( setminusSubset2
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( subset @ A @ B )
% 0.31/0.46 => ( ( setminus @ A @ B )
% 0.31/0.46 = emptyset ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setminusERneg_type,type,
% 0.31/0.46 setminusERneg: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setminusERneg,definition,
% 0.31/0.46 ( setminusERneg
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.46 => ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setminusELneg_type,type,
% 0.31/0.46 setminusELneg: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setminusELneg,definition,
% 0.31/0.46 ( setminusELneg
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.46 => ( ~ ( in @ Xx @ B )
% 0.31/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setminusILneg_type,type,
% 0.31/0.46 setminusILneg: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setminusILneg,definition,
% 0.31/0.46 ( setminusILneg
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ~ ( in @ Xx @ A )
% 0.31/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setminusIRneg_type,type,
% 0.31/0.46 setminusIRneg: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setminusIRneg,definition,
% 0.31/0.46 ( setminusIRneg
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ B )
% 0.31/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setminusLsub_type,type,
% 0.31/0.46 setminusLsub: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setminusLsub,definition,
% 0.31/0.46 ( setminusLsub
% 0.31/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setminusSubset1_type,type,
% 0.31/0.46 setminusSubset1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setminusSubset1,definition,
% 0.31/0.46 ( setminusSubset1
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( ( setminus @ A @ B )
% 0.31/0.46 = emptyset )
% 0.31/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(symdiff_type,type,
% 0.31/0.46 symdiff: $i > $i > $i ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffE_type,type,
% 0.31/0.46 symdiffE: $o ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffE,definition,
% 0.31/0.46 ( symdiffE
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.31/0.46 => ! [Xphi: $o] :
% 0.31/0.46 ( ( ( in @ Xx @ A )
% 0.31/0.46 => ( ~ ( in @ Xx @ B )
% 0.31/0.46 => Xphi ) )
% 0.31/0.46 => ( ( ~ ( in @ Xx @ A )
% 0.31/0.46 => ( ( in @ Xx @ B )
% 0.31/0.46 => Xphi ) )
% 0.31/0.46 => Xphi ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffI1_type,type,
% 0.31/0.46 symdiffI1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffI1,definition,
% 0.31/0.46 ( symdiffI1
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ~ ( in @ Xx @ B )
% 0.31/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffI2_type,type,
% 0.31/0.46 symdiffI2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffI2,definition,
% 0.31/0.46 ( symdiffI2
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ~ ( in @ Xx @ A )
% 0.31/0.46 => ( ( in @ Xx @ B )
% 0.31/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffIneg1_type,type,
% 0.31/0.46 symdiffIneg1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffIneg1,definition,
% 0.31/0.46 ( symdiffIneg1
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ( in @ Xx @ B )
% 0.31/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffIneg2_type,type,
% 0.31/0.46 symdiffIneg2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffIneg2,definition,
% 0.31/0.46 ( symdiffIneg2
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ~ ( in @ Xx @ A )
% 0.31/0.46 => ( ~ ( in @ Xx @ B )
% 0.31/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(iskpair_type,type,
% 0.31/0.46 iskpair: $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(iskpair,definition,
% 0.31/0.46 ( iskpair
% 0.31/0.46 = ( ^ [A: $i] :
% 0.31/0.46 ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.46 & ? [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ ( setunion @ A ) )
% 0.31/0.46 & ( A
% 0.31/0.46 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(secondinupair_type,type,
% 0.31/0.46 secondinupair: $o ).
% 0.31/0.46
% 0.31/0.46 thf(secondinupair,definition,
% 0.31/0.46 ( secondinupair
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairIL_type,type,
% 0.31/0.46 setukpairIL: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairIL,definition,
% 0.31/0.46 ( setukpairIL
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairIR_type,type,
% 0.31/0.46 setukpairIR: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairIR,definition,
% 0.31/0.46 ( setukpairIR
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(kpairiskpair_type,type,
% 0.31/0.46 kpairiskpair: $o ).
% 0.31/0.46
% 0.31/0.46 thf(kpairiskpair,definition,
% 0.31/0.46 ( kpairiskpair
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(kpair_type,type,
% 0.31/0.46 kpair: $i > $i > $i ).
% 0.31/0.46
% 0.31/0.46 thf(kpair,definition,
% 0.31/0.46 ( kpair
% 0.31/0.46 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(kpairp_type,type,
% 0.31/0.46 kpairp: $o ).
% 0.31/0.46
% 0.31/0.46 thf(kpairp,definition,
% 0.31/0.46 ( kpairp
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(cartprod_type,type,
% 0.31/0.46 cartprod: $i > $i > $i ).
% 0.31/0.46
% 0.31/0.46 thf(singletonsubset_type,type,
% 0.31/0.46 singletonsubset: $o ).
% 0.31/0.46
% 0.31/0.46 thf(singletonsubset,definition,
% 0.31/0.46 ( singletonsubset
% 0.31/0.46 = ( ! [A: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(singletoninpowerset_type,type,
% 0.31/0.46 singletoninpowerset: $o ).
% 0.31/0.46
% 0.31/0.46 thf(singletoninpowerset,definition,
% 0.31/0.46 ( singletoninpowerset
% 0.31/0.46 = ( ! [A: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(singletoninpowunion_type,type,
% 0.31/0.46 singletoninpowunion: $o ).
% 0.31/0.46
% 0.31/0.46 thf(singletoninpowunion,definition,
% 0.31/0.46 ( singletoninpowunion
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(upairset2E_type,type,
% 0.31/0.46 upairset2E: $o ).
% 0.31/0.46
% 0.31/0.46 thf(upairset2E,definition,
% 0.31/0.46 ( upairset2E
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.31/0.46 => ( ( Xz = Xx )
% 0.31/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(upairsubunion_type,type,
% 0.31/0.46 upairsubunion: $o ).
% 0.31/0.46
% 0.31/0.46 thf(upairsubunion,definition,
% 0.31/0.46 ( upairsubunion
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(upairinpowunion_type,type,
% 0.31/0.46 upairinpowunion: $o ).
% 0.31/0.46
% 0.31/0.46 thf(upairinpowunion,definition,
% 0.31/0.46 ( upairinpowunion
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(ubforcartprodlem1_type,type,
% 0.31/0.46 ubforcartprodlem1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(ubforcartprodlem1,definition,
% 0.31/0.46 ( ubforcartprodlem1
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(ubforcartprodlem2_type,type,
% 0.31/0.46 ubforcartprodlem2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(ubforcartprodlem2,definition,
% 0.31/0.46 ( ubforcartprodlem2
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(ubforcartprodlem3_type,type,
% 0.31/0.46 ubforcartprodlem3: $o ).
% 0.31/0.46
% 0.31/0.46 thf(ubforcartprodlem3,definition,
% 0.31/0.46 ( ubforcartprodlem3
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodpairin_type,type,
% 0.31/0.46 cartprodpairin: $o ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodpairin,definition,
% 0.31/0.46 ( cartprodpairin
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodmempair1_type,type,
% 0.31/0.46 cartprodmempair1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodmempair1,definition,
% 0.31/0.46 ( cartprodmempair1
% 0.31/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.46 => ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ? [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 & ( Xu
% 0.31/0.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodmempair_type,type,
% 0.31/0.46 cartprodmempair: $o ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodmempair,definition,
% 0.31/0.46 ( cartprodmempair
% 0.31/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.46 => ( iskpair @ Xu ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setunionE2_type,type,
% 0.31/0.46 setunionE2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setunionE2,definition,
% 0.31/0.46 ( setunionE2
% 0.31/0.46 = ( ! [A: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.46 => ? [X: $i] :
% 0.31/0.46 ( ( in @ X @ A )
% 0.31/0.46 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setunionsingleton1_type,type,
% 0.31/0.46 setunionsingleton1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setunionsingleton1,definition,
% 0.31/0.46 ( setunionsingleton1
% 0.31/0.46 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setunionsingleton2_type,type,
% 0.31/0.46 setunionsingleton2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setunionsingleton2,definition,
% 0.31/0.46 ( setunionsingleton2
% 0.31/0.46 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setunionsingleton,conjecture,
% 0.31/0.46 ( setextAx
% 0.31/0.46 => ( emptysetAx
% 0.31/0.46 => ( setadjoinAx
% 0.31/0.46 => ( powersetAx
% 0.31/0.46 => ( setunionAx
% 0.31/0.46 => ( omega0Ax
% 0.31/0.46 => ( omegaSAx
% 0.31/0.46 => ( omegaIndAx
% 0.31/0.46 => ( replAx
% 0.31/0.46 => ( foundationAx
% 0.31/0.46 => ( wellorderingAx
% 0.31/0.46 => ( descrp
% 0.31/0.46 => ( dsetconstrI
% 0.31/0.46 => ( dsetconstrEL
% 0.31/0.46 => ( dsetconstrER
% 0.31/0.46 => ( exuE1
% 0.31/0.46 => ( prop2setE
% 0.31/0.46 => ( emptysetE
% 0.31/0.46 => ( emptysetimpfalse
% 0.31/0.46 => ( notinemptyset
% 0.31/0.46 => ( exuE3e
% 0.31/0.46 => ( setext
% 0.31/0.46 => ( emptyI
% 0.31/0.46 => ( noeltsimpempty
% 0.31/0.46 => ( setbeta
% 0.31/0.46 => ( nonemptyE1
% 0.31/0.46 => ( nonemptyI
% 0.31/0.46 => ( nonemptyI1
% 0.31/0.46 => ( setadjoinIL
% 0.31/0.46 => ( emptyinunitempty
% 0.31/0.46 => ( setadjoinIR
% 0.31/0.46 => ( setadjoinE
% 0.31/0.46 => ( setadjoinOr
% 0.31/0.46 => ( setoftrueEq
% 0.31/0.46 => ( powersetI
% 0.31/0.46 => ( emptyinPowerset
% 0.31/0.46 => ( emptyInPowerset
% 0.31/0.46 => ( powersetE
% 0.31/0.46 => ( setunionI
% 0.31/0.46 => ( setunionE
% 0.31/0.46 => ( subPowSU
% 0.31/0.46 => ( exuE2
% 0.31/0.46 => ( nonemptyImpWitness
% 0.31/0.46 => ( uniqinunit
% 0.31/0.46 => ( notinsingleton
% 0.31/0.46 => ( eqinunit
% 0.31/0.46 => ( singletonsswitch
% 0.31/0.46 => ( upairsetE
% 0.31/0.46 => ( upairsetIL
% 0.31/0.46 => ( upairsetIR
% 0.31/0.46 => ( emptyE1
% 0.31/0.46 => ( vacuousDall
% 0.31/0.46 => ( quantDeMorgan1
% 0.31/0.46 => ( quantDeMorgan2
% 0.31/0.46 => ( quantDeMorgan3
% 0.31/0.46 => ( quantDeMorgan4
% 0.31/0.46 => ( prop2setI
% 0.31/0.46 => ( prop2set2propI
% 0.31/0.46 => ( notdexE
% 0.31/0.46 => ( notdallE
% 0.31/0.46 => ( exuI1
% 0.31/0.46 => ( exuI3
% 0.31/0.46 => ( exuI2
% 0.31/0.46 => ( inCongP
% 0.31/0.46 => ( in__Cong
% 0.31/0.46 => ( exuE3u
% 0.31/0.46 => ( exu__Cong
% 0.31/0.46 => ( emptyset__Cong
% 0.31/0.46 => ( setadjoin__Cong
% 0.31/0.46 => ( powerset__Cong
% 0.31/0.46 => ( setunion__Cong
% 0.31/0.46 => ( omega__Cong
% 0.31/0.46 => ( exuEu
% 0.31/0.46 => ( descr__Cong
% 0.31/0.46 => ( dsetconstr__Cong
% 0.31/0.46 => ( subsetI1
% 0.31/0.46 => ( eqimpsubset2
% 0.31/0.46 => ( eqimpsubset1
% 0.31/0.46 => ( subsetI2
% 0.31/0.46 => ( emptysetsubset
% 0.31/0.46 => ( subsetE
% 0.31/0.46 => ( subsetE2
% 0.31/0.46 => ( notsubsetI
% 0.31/0.46 => ( notequalI1
% 0.31/0.46 => ( notequalI2
% 0.31/0.46 => ( subsetRefl
% 0.31/0.46 => ( subsetTrans
% 0.31/0.46 => ( setadjoinSub
% 0.31/0.46 => ( setadjoinSub2
% 0.31/0.46 => ( subset2powerset
% 0.31/0.46 => ( setextsub
% 0.31/0.46 => ( subsetemptysetimpeq
% 0.31/0.46 => ( powersetI1
% 0.31/0.46 => ( powersetE1
% 0.31/0.46 => ( inPowerset
% 0.31/0.46 => ( powersetsubset
% 0.31/0.46 => ( sepInPowerset
% 0.31/0.46 => ( sepSubset
% 0.31/0.46 => ( binunionIL
% 0.31/0.46 => ( upairset2IR
% 0.31/0.46 => ( binunionIR
% 0.31/0.46 => ( binunionEcases
% 0.31/0.46 => ( binunionE
% 0.31/0.46 => ( binunionLsub
% 0.31/0.46 => ( binunionRsub
% 0.31/0.46 => ( binintersectI
% 0.31/0.46 => ( binintersectSubset5
% 0.31/0.46 => ( binintersectEL
% 0.31/0.46 => ( binintersectLsub
% 0.31/0.46 => ( binintersectSubset2
% 0.31/0.46 => ( binintersectSubset3
% 0.31/0.46 => ( binintersectER
% 0.31/0.46 => ( disjointsetsI1
% 0.31/0.46 => ( binintersectRsub
% 0.31/0.46 => ( binintersectSubset4
% 0.31/0.46 => ( binintersectSubset1
% 0.31/0.46 => ( bs114d
% 0.31/0.46 => ( setminusI
% 0.31/0.46 => ( setminusEL
% 0.31/0.46 => ( setminusER
% 0.31/0.46 => ( setminusSubset2
% 0.31/0.46 => ( setminusERneg
% 0.31/0.46 => ( setminusELneg
% 0.31/0.46 => ( setminusILneg
% 0.31/0.46 => ( setminusIRneg
% 0.31/0.46 => ( setminusLsub
% 0.31/0.46 => ( setminusSubset1
% 0.31/0.46 => ( symdiffE
% 0.31/0.46 => ( symdiffI1
% 0.31/0.46 => ( symdiffI2
% 0.31/0.46 => ( symdiffIneg1
% 0.31/0.46 => ( symdiffIneg2
% 0.31/0.46 => ( secondinupair
% 0.31/0.46 => ( setukpairIL
% 0.31/0.46 => ( setukpairIR
% 0.31/0.46 => ( kpairiskpair
% 0.31/0.46 => ( kpairp
% 0.31/0.46 => ( singletonsubset
% 0.31/0.46 => ( singletoninpowerset
% 0.31/0.46 => ( singletoninpowunion
% 0.31/0.46 => ( upairset2E
% 0.31/0.46 => ( upairsubunion
% 0.31/0.46 => ( upairinpowunion
% 0.31/0.46 => ( ubforcartprodlem1
% 0.31/0.46 => ( ubforcartprodlem2
% 0.31/0.46 => ( ubforcartprodlem3
% 0.31/0.46 => ( cartprodpairin
% 0.31/0.46 => ( cartprodmempair1
% 0.31/0.46 => ( cartprodmempair
% 0.31/0.46 => ( setunionE2
% 0.31/0.46 => ( setunionsingleton1
% 0.31/0.46 => ( setunionsingleton2
% 0.31/0.46 => ! [Xx: $i] :
% 0.31/0.46 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.31/0.46 = Xx ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 %------------------------------------------------------------------------------
% 0.31/0.46 ------------------------
% 2.13/1.36 ---- Embedded in THF ---
% 2.13/1.36 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.13/1.36 %%% Generated on Tue Feb 24 12:29:11 EST 2026
% 2.13/1.36 %%% using '$$dhol' embedding, version 1.3.0.
% 2.13/1.36 %%% Logic specification used:
% 2.13/1.36 %%% thf(spec, logic, $$dhol).
% 2.13/1.36
% 2.13/1.36 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.EkEyqyb2Tu/DTF2DTF18509.p
% See solution above
% 2.13/1.36 ------------------------
% 2.13/1.38 ---- Cleaned THF ---
% 2.13/1.38 thf(in_type,type,
% 2.13/1.38 in: $i > $i > $o ).
% 2.13/1.38
% 2.13/1.38 thf(exu_type,type,
% 2.13/1.38 exu: ( $i > $o ) > $o ).
% 2.13/1.38
% 2.13/1.38 thf(exu,definition,
% 2.13/1.38 ( exu
% 2.13/1.38 = ( ^ [Xphi: $i > $o] :
% 2.13/1.38 ? [Xx: $i] :
% 2.13/1.38 ( ( Xphi @ Xx )
% 2.13/1.38 & ! [Xy: $i] :
% 2.13/1.38 ( ( Xphi @ Xy )
% 2.13/1.38 => ( Xx = Xy ) ) ) ) ) ).
% 2.13/1.38
% 2.13/1.38 thf(setextAx_type,type,
% 2.13/1.38 setextAx: $o ).
% 2.13/1.38
% 2.13/1.38 thf(setextAx,definition,
% 2.13/1.38 ( setextAx
% 2.13/1.38 = ( ! [A: $i,B: $i] :
% 2.13/1.38 ( ! [Xx: $i] :
% 2.13/1.38 ( ( in @ Xx @ A )
% 2.13/1.38 <=> ( in @ Xx @ B ) )
% 2.13/1.38 => ( A = B ) ) ) ) ).
% 2.13/1.38
% 2.13/1.38 thf(emptyset_type,type,
% 2.13/1.38 emptyset: $i ).
% 2.13/1.38
% 2.13/1.38 thf(emptysetAx_type,type,
% 2.13/1.38 emptysetAx: $o ).
% 2.13/1.38
% 2.13/1.38 thf(emptysetAx,definition,
% 2.13/1.38 ( emptysetAx
% 2.13/1.38 = ( ! [Xx: $i] :
% 2.13/1.38 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.13/1.38
% 2.13/1.38 thf(setadjoin_type,type,
% 2.13/1.38 setadjoin: $i > $i > $i ).
% 2.13/1.38
% 2.13/1.38 thf(setadjoinAx_type,type,
% 2.13/1.38 setadjoinAx: $o ).
% 2.13/1.38
% 2.13/1.38 thf(setadjoinAx,definition,
% 2.13/1.38 ( setadjoinAx
% 2.13/1.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.13/1.38 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.13/1.38 <=> ( ( Xy = Xx )
% 2.13/1.38 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(powerset_type,type,
% 2.13/1.39 powerset: $i > $i ).
% 2.13/1.39
% 2.13/1.39 thf(powersetAx_type,type,
% 2.13/1.39 powersetAx: $o ).
% 2.13/1.39
% 2.13/1.39 thf(powersetAx,definition,
% 2.13/1.39 ( powersetAx
% 2.13/1.39 = ( ! [A: $i,B: $i] :
% 2.13/1.39 ( ( in @ B @ ( powerset @ A ) )
% 2.13/1.39 <=> ! [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ B )
% 2.13/1.39 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setunion_type,type,
% 2.13/1.39 setunion: $i > $i ).
% 2.13/1.39
% 2.13/1.39 thf(setunionAx_type,type,
% 2.13/1.39 setunionAx: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setunionAx,definition,
% 2.13/1.39 ( setunionAx
% 2.13/1.39 = ( ! [A: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ ( setunion @ A ) )
% 2.13/1.39 <=> ? [B: $i] :
% 2.13/1.39 ( ( in @ Xx @ B )
% 2.13/1.39 & ( in @ B @ A ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(omega_type,type,
% 2.13/1.39 omega: $i ).
% 2.13/1.39
% 2.13/1.39 thf(omega0Ax_type,type,
% 2.13/1.39 omega0Ax: $o ).
% 2.13/1.39
% 2.13/1.39 thf(omega0Ax,definition,
% 2.13/1.39 ( omega0Ax
% 2.13/1.39 = ( in @ emptyset @ omega ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(omegaSAx_type,type,
% 2.13/1.39 omegaSAx: $o ).
% 2.13/1.39
% 2.13/1.39 thf(omegaSAx,definition,
% 2.13/1.39 ( omegaSAx
% 2.13/1.39 = ( ! [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ omega )
% 2.13/1.39 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(omegaIndAx_type,type,
% 2.13/1.39 omegaIndAx: $o ).
% 2.13/1.39
% 2.13/1.39 thf(omegaIndAx,definition,
% 2.13/1.39 ( omegaIndAx
% 2.13/1.39 = ( ! [A: $i] :
% 2.13/1.39 ( ( ( in @ emptyset @ A )
% 2.13/1.39 & ! [Xx: $i] :
% 2.13/1.39 ( ( ( in @ Xx @ omega )
% 2.13/1.39 & ( in @ Xx @ A ) )
% 2.13/1.39 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.13/1.39 => ! [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ omega )
% 2.13/1.39 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(replAx_type,type,
% 2.13/1.39 replAx: $o ).
% 2.13/1.39
% 2.13/1.39 thf(replAx,definition,
% 2.13/1.39 ( replAx
% 2.13/1.39 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.13/1.39 ( ! [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ( exu
% 2.13/1.39 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.13/1.39 => ? [B: $i] :
% 2.13/1.39 ! [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ B )
% 2.13/1.39 <=> ? [Xy: $i] :
% 2.13/1.39 ( ( in @ Xy @ A )
% 2.13/1.39 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(foundationAx_type,type,
% 2.13/1.39 foundationAx: $o ).
% 2.13/1.39
% 2.13/1.39 thf(foundationAx,definition,
% 2.13/1.39 ( foundationAx
% 2.13/1.39 = ( ! [A: $i] :
% 2.13/1.39 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.13/1.39 => ? [B: $i] :
% 2.13/1.39 ( ( in @ B @ A )
% 2.13/1.39 & ~ ? [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ B )
% 2.13/1.39 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(wellorderingAx_type,type,
% 2.13/1.39 wellorderingAx: $o ).
% 2.13/1.39
% 2.13/1.39 thf(wellorderingAx,definition,
% 2.13/1.39 ( wellorderingAx
% 2.13/1.39 = ( ! [A: $i] :
% 2.13/1.39 ? [B: $i] :
% 2.13/1.39 ( ! [C: $i] :
% 2.13/1.39 ( ( in @ C @ B )
% 2.13/1.39 => ! [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ C )
% 2.13/1.39 => ( in @ Xx @ A ) ) )
% 2.13/1.39 & ! [Xx: $i,Xy: $i] :
% 2.13/1.39 ( ( ( in @ Xx @ A )
% 2.13/1.39 & ( in @ Xy @ A ) )
% 2.13/1.39 => ( ! [C: $i] :
% 2.13/1.39 ( ( in @ C @ B )
% 2.13/1.39 => ( ( in @ Xx @ C )
% 2.13/1.39 <=> ( in @ Xy @ C ) ) )
% 2.13/1.39 => ( Xx = Xy ) ) )
% 2.13/1.39 & ! [C: $i,D: $i] :
% 2.13/1.39 ( ( ( in @ C @ B )
% 2.13/1.39 & ( in @ D @ B ) )
% 2.13/1.39 => ( ! [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ C )
% 2.13/1.39 => ( in @ Xx @ D ) )
% 2.13/1.39 | ! [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ D )
% 2.13/1.39 => ( in @ Xx @ C ) ) ) )
% 2.13/1.39 & ! [C: $i] :
% 2.13/1.39 ( ( ! [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ C )
% 2.13/1.39 => ( in @ Xx @ A ) )
% 2.13/1.39 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.13/1.39 => ? [D: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ D @ B )
% 2.13/1.39 & ( in @ Xx @ C )
% 2.13/1.39 & ~ ? [Xy: $i] :
% 2.13/1.39 ( ( in @ Xy @ D )
% 2.13/1.39 & ( in @ Xy @ C ) )
% 2.13/1.39 & ! [E: $i] :
% 2.13/1.39 ( ( in @ E @ B )
% 2.13/1.39 => ( ! [Xy: $i] :
% 2.13/1.39 ( ( in @ Xy @ E )
% 2.13/1.39 => ( in @ Xy @ D ) )
% 2.13/1.39 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(descr_type,type,
% 2.13/1.39 descr: ( $i > $o ) > $i ).
% 2.13/1.39
% 2.13/1.39 thf(descrp_type,type,
% 2.13/1.39 descrp: $o ).
% 2.13/1.39
% 2.13/1.39 thf(descrp,definition,
% 2.13/1.39 ( descrp
% 2.13/1.39 = ( ! [Xphi: $i > $o] :
% 2.13/1.39 ( ( exu
% 2.13/1.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.13/1.39 => ( Xphi
% 2.13/1.39 @ ( descr
% 2.13/1.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(dsetconstr_type,type,
% 2.13/1.39 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.13/1.39
% 2.13/1.39 thf(dsetconstrI_type,type,
% 2.13/1.39 dsetconstrI: $o ).
% 2.13/1.39
% 2.13/1.39 thf(dsetconstrI,definition,
% 2.13/1.39 ( dsetconstrI
% 2.13/1.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ( ( Xphi @ Xx )
% 2.13/1.39 => ( in @ Xx
% 2.13/1.39 @ ( dsetconstr @ A
% 2.13/1.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(dsetconstrEL_type,type,
% 2.13/1.39 dsetconstrEL: $o ).
% 2.13/1.39
% 2.13/1.39 thf(dsetconstrEL,definition,
% 2.13/1.39 ( dsetconstrEL
% 2.13/1.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx
% 2.13/1.39 @ ( dsetconstr @ A
% 2.13/1.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.13/1.39 => ( in @ Xx @ A ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(dsetconstrER_type,type,
% 2.13/1.39 dsetconstrER: $o ).
% 2.13/1.39
% 2.13/1.39 thf(dsetconstrER,definition,
% 2.13/1.39 ( dsetconstrER
% 2.13/1.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx
% 2.13/1.39 @ ( dsetconstr @ A
% 2.13/1.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.13/1.39 => ( Xphi @ Xx ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(exuE1_type,type,
% 2.13/1.39 exuE1: $o ).
% 2.13/1.39
% 2.13/1.39 thf(exuE1,definition,
% 2.13/1.39 ( exuE1
% 2.13/1.39 = ( ! [Xphi: $i > $o] :
% 2.13/1.39 ( ( exu
% 2.13/1.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.13/1.39 => ? [Xx: $i] :
% 2.13/1.39 ( ( Xphi @ Xx )
% 2.13/1.39 & ! [Xy: $i] :
% 2.13/1.39 ( ( Xphi @ Xy )
% 2.13/1.39 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(prop2set_type,type,
% 2.13/1.39 prop2set: $o > $i ).
% 2.13/1.39
% 2.13/1.39 thf(prop2setE_type,type,
% 2.13/1.39 prop2setE: $o ).
% 2.13/1.39
% 2.13/1.39 thf(prop2setE,definition,
% 2.13/1.39 ( prop2setE
% 2.13/1.39 = ( ! [Xphi: $o,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.13/1.39 => Xphi ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(emptysetE_type,type,
% 2.13/1.39 emptysetE: $o ).
% 2.13/1.39
% 2.13/1.39 thf(emptysetE,definition,
% 2.13/1.39 ( emptysetE
% 2.13/1.39 = ( ! [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ emptyset )
% 2.13/1.39 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(emptysetimpfalse_type,type,
% 2.13/1.39 emptysetimpfalse: $o ).
% 2.13/1.39
% 2.13/1.39 thf(emptysetimpfalse,definition,
% 2.13/1.39 ( emptysetimpfalse
% 2.13/1.39 = ( ! [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ emptyset )
% 2.13/1.39 => $false ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(notinemptyset_type,type,
% 2.13/1.39 notinemptyset: $o ).
% 2.13/1.39
% 2.13/1.39 thf(notinemptyset,definition,
% 2.13/1.39 ( notinemptyset
% 2.13/1.39 = ( ! [Xx: $i] :
% 2.13/1.39 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(exuE3e_type,type,
% 2.13/1.39 exuE3e: $o ).
% 2.13/1.39
% 2.13/1.39 thf(exuE3e,definition,
% 2.13/1.39 ( exuE3e
% 2.13/1.39 = ( ! [Xphi: $i > $o] :
% 2.13/1.39 ( ( exu
% 2.13/1.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.13/1.39 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setext_type,type,
% 2.13/1.39 setext: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setext,definition,
% 2.13/1.39 ( setext
% 2.13/1.39 = ( ! [A: $i,B: $i] :
% 2.13/1.39 ( ! [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ( in @ Xx @ B ) )
% 2.13/1.39 => ( ! [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ B )
% 2.13/1.39 => ( in @ Xx @ A ) )
% 2.13/1.39 => ( A = B ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(emptyI_type,type,
% 2.13/1.39 emptyI: $o ).
% 2.13/1.39
% 2.13/1.39 thf(emptyI,definition,
% 2.13/1.39 ( emptyI
% 2.13/1.39 = ( ! [A: $i] :
% 2.13/1.39 ( ! [Xx: $i] :
% 2.13/1.39 ~ ( in @ Xx @ A )
% 2.13/1.39 => ( A = emptyset ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(noeltsimpempty_type,type,
% 2.13/1.39 noeltsimpempty: $o ).
% 2.13/1.39
% 2.13/1.39 thf(noeltsimpempty,definition,
% 2.13/1.39 ( noeltsimpempty
% 2.13/1.39 = ( ! [A: $i] :
% 2.13/1.39 ( ! [Xx: $i] :
% 2.13/1.39 ~ ( in @ Xx @ A )
% 2.13/1.39 => ( A = emptyset ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setbeta_type,type,
% 2.13/1.39 setbeta: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setbeta,definition,
% 2.13/1.39 ( setbeta
% 2.13/1.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ( ( in @ Xx
% 2.13/1.39 @ ( dsetconstr @ A
% 2.13/1.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.13/1.39 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(nonempty_type,type,
% 2.13/1.39 nonempty: $i > $o ).
% 2.13/1.39
% 2.13/1.39 thf(nonempty,definition,
% 2.13/1.39 ( nonempty
% 2.13/1.39 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(nonemptyE1_type,type,
% 2.13/1.39 nonemptyE1: $o ).
% 2.13/1.39
% 2.13/1.39 thf(nonemptyE1,definition,
% 2.13/1.39 ( nonemptyE1
% 2.13/1.39 = ( ! [A: $i] :
% 2.13/1.39 ( ( nonempty @ A )
% 2.13/1.39 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(nonemptyI_type,type,
% 2.13/1.39 nonemptyI: $o ).
% 2.13/1.39
% 2.13/1.39 thf(nonemptyI,definition,
% 2.13/1.39 ( nonemptyI
% 2.13/1.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ( ( Xphi @ Xx )
% 2.13/1.39 => ( nonempty
% 2.13/1.39 @ ( dsetconstr @ A
% 2.13/1.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(nonemptyI1_type,type,
% 2.13/1.39 nonemptyI1: $o ).
% 2.13/1.39
% 2.13/1.39 thf(nonemptyI1,definition,
% 2.13/1.39 ( nonemptyI1
% 2.13/1.39 = ( ! [A: $i] :
% 2.13/1.39 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.13/1.39 => ( nonempty @ A ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setadjoinIL_type,type,
% 2.13/1.39 setadjoinIL: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setadjoinIL,definition,
% 2.13/1.39 ( setadjoinIL
% 2.13/1.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(emptyinunitempty_type,type,
% 2.13/1.39 emptyinunitempty: $o ).
% 2.13/1.39
% 2.13/1.39 thf(emptyinunitempty,definition,
% 2.13/1.39 ( emptyinunitempty
% 2.13/1.39 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setadjoinIR_type,type,
% 2.13/1.39 setadjoinIR: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setadjoinIR,definition,
% 2.13/1.39 ( setadjoinIR
% 2.13/1.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.13/1.39 ( ( in @ Xy @ A )
% 2.13/1.39 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setadjoinE_type,type,
% 2.13/1.39 setadjoinE: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setadjoinE,definition,
% 2.13/1.39 ( setadjoinE
% 2.13/1.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.13/1.39 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.13/1.39 => ! [Xphi: $o] :
% 2.13/1.39 ( ( ( Xy = Xx )
% 2.13/1.39 => Xphi )
% 2.13/1.39 => ( ( ( in @ Xy @ A )
% 2.13/1.39 => Xphi )
% 2.13/1.39 => Xphi ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setadjoinOr_type,type,
% 2.13/1.39 setadjoinOr: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setadjoinOr,definition,
% 2.13/1.39 ( setadjoinOr
% 2.13/1.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.13/1.39 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.13/1.39 => ( ( Xy = Xx )
% 2.13/1.39 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setoftrueEq_type,type,
% 2.13/1.39 setoftrueEq: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setoftrueEq,definition,
% 2.13/1.39 ( setoftrueEq
% 2.13/1.39 = ( ! [A: $i] :
% 2.13/1.39 ( ( dsetconstr @ A
% 2.13/1.39 @ ^ [Xx: $i] : $true )
% 2.13/1.39 = A ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(powersetI_type,type,
% 2.13/1.39 powersetI: $o ).
% 2.13/1.39
% 2.13/1.39 thf(powersetI,definition,
% 2.13/1.39 ( powersetI
% 2.13/1.39 = ( ! [A: $i,B: $i] :
% 2.13/1.39 ( ! [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ B )
% 2.13/1.39 => ( in @ Xx @ A ) )
% 2.13/1.39 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(emptyinPowerset_type,type,
% 2.13/1.39 emptyinPowerset: $o ).
% 2.13/1.39
% 2.13/1.39 thf(emptyinPowerset,definition,
% 2.13/1.39 ( emptyinPowerset
% 2.13/1.39 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(emptyInPowerset_type,type,
% 2.13/1.39 emptyInPowerset: $o ).
% 2.13/1.39
% 2.13/1.39 thf(emptyInPowerset,definition,
% 2.13/1.39 ( emptyInPowerset
% 2.13/1.39 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(powersetE_type,type,
% 2.13/1.39 powersetE: $o ).
% 2.13/1.39
% 2.13/1.39 thf(powersetE,definition,
% 2.13/1.39 ( powersetE
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ B @ ( powerset @ A ) )
% 2.13/1.39 => ( ( in @ Xx @ B )
% 2.13/1.39 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setunionI_type,type,
% 2.13/1.39 setunionI: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setunionI,definition,
% 2.13/1.39 ( setunionI
% 2.13/1.39 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.13/1.39 ( ( in @ Xx @ B )
% 2.13/1.39 => ( ( in @ B @ A )
% 2.13/1.39 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setunionE_type,type,
% 2.13/1.39 setunionE: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setunionE,definition,
% 2.13/1.39 ( setunionE
% 2.13/1.39 = ( ! [A: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ ( setunion @ A ) )
% 2.13/1.39 => ! [Xphi: $o] :
% 2.13/1.39 ( ! [B: $i] :
% 2.13/1.39 ( ( in @ Xx @ B )
% 2.13/1.39 => ( ( in @ B @ A )
% 2.13/1.39 => Xphi ) )
% 2.13/1.39 => Xphi ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(subPowSU_type,type,
% 2.13/1.39 subPowSU: $o ).
% 2.13/1.39
% 2.13/1.39 thf(subPowSU,definition,
% 2.13/1.39 ( subPowSU
% 2.13/1.39 = ( ! [A: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(exuE2_type,type,
% 2.13/1.39 exuE2: $o ).
% 2.13/1.39
% 2.13/1.39 thf(exuE2,definition,
% 2.13/1.39 ( exuE2
% 2.13/1.39 = ( ! [Xphi: $i > $o] :
% 2.13/1.39 ( ( exu
% 2.13/1.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.13/1.39 => ? [Xx: $i] :
% 2.13/1.39 ! [Xy: $i] :
% 2.13/1.39 ( ( Xphi @ Xy )
% 2.13/1.39 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(nonemptyImpWitness_type,type,
% 2.13/1.39 nonemptyImpWitness: $o ).
% 2.13/1.39
% 2.13/1.39 thf(nonemptyImpWitness,definition,
% 2.13/1.39 ( nonemptyImpWitness
% 2.13/1.39 = ( ! [A: $i] :
% 2.13/1.39 ( ( nonempty @ A )
% 2.13/1.39 => ? [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 & $true ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(uniqinunit_type,type,
% 2.13/1.39 uniqinunit: $o ).
% 2.13/1.39
% 2.13/1.39 thf(uniqinunit,definition,
% 2.13/1.39 ( uniqinunit
% 2.13/1.39 = ( ! [Xx: $i,Xy: $i] :
% 2.13/1.39 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.13/1.39 => ( Xx = Xy ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(notinsingleton_type,type,
% 2.13/1.39 notinsingleton: $o ).
% 2.13/1.39
% 2.13/1.39 thf(notinsingleton,definition,
% 2.13/1.39 ( notinsingleton
% 2.13/1.39 = ( ! [Xx: $i,Xy: $i] :
% 2.13/1.39 ( ( Xx != Xy )
% 2.13/1.39 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(eqinunit_type,type,
% 2.13/1.39 eqinunit: $o ).
% 2.13/1.39
% 2.13/1.39 thf(eqinunit,definition,
% 2.13/1.39 ( eqinunit
% 2.13/1.39 = ( ! [Xx: $i,Xy: $i] :
% 2.13/1.39 ( ( Xx = Xy )
% 2.13/1.39 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(singletonsswitch_type,type,
% 2.13/1.39 singletonsswitch: $o ).
% 2.13/1.39
% 2.13/1.39 thf(singletonsswitch,definition,
% 2.13/1.39 ( singletonsswitch
% 2.13/1.39 = ( ! [Xx: $i,Xy: $i] :
% 2.13/1.39 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.13/1.39 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(upairsetE_type,type,
% 2.13/1.39 upairsetE: $o ).
% 2.13/1.39
% 2.13/1.39 thf(upairsetE,definition,
% 2.13/1.39 ( upairsetE
% 2.13/1.39 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.13/1.39 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.13/1.39 => ( ( Xz = Xx )
% 2.13/1.39 | ( Xz = Xy ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(upairsetIL_type,type,
% 2.13/1.39 upairsetIL: $o ).
% 2.13/1.39
% 2.13/1.39 thf(upairsetIL,definition,
% 2.13/1.39 ( upairsetIL
% 2.13/1.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(upairsetIR_type,type,
% 2.13/1.39 upairsetIR: $o ).
% 2.13/1.39
% 2.13/1.39 thf(upairsetIR,definition,
% 2.13/1.39 ( upairsetIR
% 2.13/1.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(emptyE1_type,type,
% 2.13/1.39 emptyE1: $o ).
% 2.13/1.39
% 2.13/1.39 thf(emptyE1,definition,
% 2.13/1.39 ( emptyE1
% 2.13/1.39 = ( ! [A: $i,Xphi: $i > $o] :
% 2.13/1.39 ( ? [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 & ( Xphi @ Xx ) )
% 2.13/1.39 => ( ( ( dsetconstr @ A
% 2.13/1.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.13/1.39 = emptyset )
% 2.13/1.39 => $false ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(vacuousDall_type,type,
% 2.13/1.39 vacuousDall: $o ).
% 2.13/1.39
% 2.13/1.39 thf(vacuousDall,definition,
% 2.13/1.39 ( vacuousDall
% 2.13/1.39 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ emptyset )
% 2.13/1.39 => ( Xphi @ Xx ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(quantDeMorgan1_type,type,
% 2.13/1.39 quantDeMorgan1: $o ).
% 2.13/1.39
% 2.13/1.39 thf(quantDeMorgan1,definition,
% 2.13/1.39 ( quantDeMorgan1
% 2.13/1.39 = ( ! [A: $i,Xphi: $i > $o] :
% 2.13/1.39 ( ~ ! [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ( Xphi @ Xx ) )
% 2.13/1.39 => ? [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(quantDeMorgan2_type,type,
% 2.13/1.39 quantDeMorgan2: $o ).
% 2.13/1.39
% 2.13/1.39 thf(quantDeMorgan2,definition,
% 2.13/1.39 ( quantDeMorgan2
% 2.13/1.39 = ( ! [A: $i,Xphi: $i > $o] :
% 2.13/1.39 ( ! [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ~ ( Xphi @ Xx ) )
% 2.13/1.39 => ~ ? [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(quantDeMorgan3_type,type,
% 2.13/1.39 quantDeMorgan3: $o ).
% 2.13/1.39
% 2.13/1.39 thf(quantDeMorgan3,definition,
% 2.13/1.39 ( quantDeMorgan3
% 2.13/1.39 = ( ! [A: $i,Xphi: $i > $o] :
% 2.13/1.39 ( ~ ? [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 & ( Xphi @ Xx ) )
% 2.13/1.39 => ! [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(quantDeMorgan4_type,type,
% 2.13/1.39 quantDeMorgan4: $o ).
% 2.13/1.39
% 2.13/1.39 thf(quantDeMorgan4,definition,
% 2.13/1.39 ( quantDeMorgan4
% 2.13/1.39 = ( ! [A: $i,Xphi: $i > $o] :
% 2.13/1.39 ( ? [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 & ~ ( Xphi @ Xx ) )
% 2.13/1.39 => ~ ! [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(prop2setI_type,type,
% 2.13/1.39 prop2setI: $o ).
% 2.13/1.39
% 2.13/1.39 thf(prop2setI,definition,
% 2.13/1.39 ( prop2setI
% 2.13/1.39 = ( ! [Xphi: $o] :
% 2.13/1.39 ( Xphi
% 2.13/1.39 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(set2prop_type,type,
% 2.13/1.39 set2prop: $i > $o ).
% 2.13/1.39
% 2.13/1.39 thf(prop2set2propI_type,type,
% 2.13/1.39 prop2set2propI: $o ).
% 2.13/1.39
% 2.13/1.39 thf(prop2set2propI,definition,
% 2.13/1.39 ( prop2set2propI
% 2.13/1.39 = ( ! [Xphi: $o] :
% 2.13/1.39 ( Xphi
% 2.13/1.39 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(notdexE_type,type,
% 2.13/1.39 notdexE: $o ).
% 2.13/1.39
% 2.13/1.39 thf(notdexE,definition,
% 2.13/1.39 ( notdexE
% 2.13/1.39 = ( ! [A: $i,Xphi: $i > $o] :
% 2.13/1.39 ( ~ ? [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 & ( Xphi @ Xx ) )
% 2.13/1.39 => ! [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(notdallE_type,type,
% 2.13/1.39 notdallE: $o ).
% 2.13/1.39
% 2.13/1.39 thf(notdallE,definition,
% 2.13/1.39 ( notdallE
% 2.13/1.39 = ( ! [A: $i,Xphi: $i > $o] :
% 2.13/1.39 ( ~ ! [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ( Xphi @ Xx ) )
% 2.13/1.39 => ? [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(exuI1_type,type,
% 2.13/1.39 exuI1: $o ).
% 2.13/1.39
% 2.13/1.39 thf(exuI1,definition,
% 2.13/1.39 ( exuI1
% 2.13/1.39 = ( ! [Xphi: $i > $o] :
% 2.13/1.39 ( ? [Xx: $i] :
% 2.13/1.39 ( ( Xphi @ Xx )
% 2.13/1.39 & ! [Xy: $i] :
% 2.13/1.39 ( ( Xphi @ Xy )
% 2.13/1.39 => ( Xx = Xy ) ) )
% 2.13/1.39 => ( exu
% 2.13/1.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(exuI3_type,type,
% 2.13/1.39 exuI3: $o ).
% 2.13/1.39
% 2.13/1.39 thf(exuI3,definition,
% 2.13/1.39 ( exuI3
% 2.13/1.39 = ( ! [Xphi: $i > $o] :
% 2.13/1.39 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.13/1.39 => ( ! [Xx: $i,Xy: $i] :
% 2.13/1.39 ( ( Xphi @ Xx )
% 2.13/1.39 => ( ( Xphi @ Xy )
% 2.13/1.39 => ( Xx = Xy ) ) )
% 2.13/1.39 => ( exu
% 2.13/1.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(exuI2_type,type,
% 2.13/1.39 exuI2: $o ).
% 2.13/1.39
% 2.13/1.39 thf(exuI2,definition,
% 2.13/1.39 ( exuI2
% 2.13/1.39 = ( ! [Xphi: $i > $o] :
% 2.13/1.39 ( ? [Xx: $i] :
% 2.13/1.39 ! [Xy: $i] :
% 2.13/1.39 ( ( Xphi @ Xy )
% 2.13/1.39 <=> ( Xy = Xx ) )
% 2.13/1.39 => ( exu
% 2.13/1.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(inCongP_type,type,
% 2.13/1.39 inCongP: $o ).
% 2.13/1.39
% 2.13/1.39 thf(inCongP,definition,
% 2.13/1.39 ( inCongP
% 2.13/1.39 = ( ! [A: $i,B: $i] :
% 2.13/1.39 ( ( A = B )
% 2.13/1.39 => ! [Xx: $i,Xy: $i] :
% 2.13/1.39 ( ( Xx = Xy )
% 2.13/1.39 => ( ( in @ Xx @ A )
% 2.13/1.39 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(in__Cong_type,type,
% 2.13/1.39 in__Cong: $o ).
% 2.13/1.39
% 2.13/1.39 thf(in__Cong,definition,
% 2.13/1.39 ( in__Cong
% 2.13/1.39 = ( ! [A: $i,B: $i] :
% 2.13/1.39 ( ( A = B )
% 2.13/1.39 => ! [Xx: $i,Xy: $i] :
% 2.13/1.39 ( ( Xx = Xy )
% 2.13/1.39 => ( ( in @ Xx @ A )
% 2.13/1.39 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(exuE3u_type,type,
% 2.13/1.39 exuE3u: $o ).
% 2.13/1.39
% 2.13/1.39 thf(exuE3u,definition,
% 2.13/1.39 ( exuE3u
% 2.13/1.39 = ( ! [Xphi: $i > $o] :
% 2.13/1.39 ( ( exu
% 2.13/1.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.13/1.39 => ! [Xx: $i,Xy: $i] :
% 2.13/1.39 ( ( Xphi @ Xx )
% 2.13/1.39 => ( ( Xphi @ Xy )
% 2.13/1.39 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(exu__Cong_type,type,
% 2.13/1.39 exu__Cong: $o ).
% 2.13/1.39
% 2.13/1.39 thf(exu__Cong,definition,
% 2.13/1.39 ( exu__Cong
% 2.13/1.39 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.13/1.39 ( ! [Xx: $i,Xy: $i] :
% 2.13/1.39 ( ( Xx = Xy )
% 2.13/1.39 => ( ( Xphi @ Xx )
% 2.13/1.39 <=> ( Xpsi @ Xy ) ) )
% 2.13/1.39 => ( ( exu
% 2.13/1.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.13/1.39 <=> ( exu
% 2.13/1.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(emptyset__Cong_type,type,
% 2.13/1.39 emptyset__Cong: $o ).
% 2.13/1.39
% 2.13/1.39 thf(emptyset__Cong,definition,
% 2.13/1.39 ( emptyset__Cong
% 2.13/1.39 = ( emptyset = emptyset ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setadjoin__Cong_type,type,
% 2.13/1.39 setadjoin__Cong: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setadjoin__Cong,definition,
% 2.13/1.39 ( setadjoin__Cong
% 2.13/1.39 = ( ! [Xx: $i,Xy: $i] :
% 2.13/1.39 ( ( Xx = Xy )
% 2.13/1.39 => ! [Xz: $i,Xu: $i] :
% 2.13/1.39 ( ( Xz = Xu )
% 2.13/1.39 => ( ( setadjoin @ Xx @ Xz )
% 2.13/1.39 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(powerset__Cong_type,type,
% 2.13/1.39 powerset__Cong: $o ).
% 2.13/1.39
% 2.13/1.39 thf(powerset__Cong,definition,
% 2.13/1.39 ( powerset__Cong
% 2.13/1.39 = ( ! [A: $i,B: $i] :
% 2.13/1.39 ( ( A = B )
% 2.13/1.39 => ( ( powerset @ A )
% 2.13/1.39 = ( powerset @ B ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setunion__Cong_type,type,
% 2.13/1.39 setunion__Cong: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setunion__Cong,definition,
% 2.13/1.39 ( setunion__Cong
% 2.13/1.39 = ( ! [A: $i,B: $i] :
% 2.13/1.39 ( ( A = B )
% 2.13/1.39 => ( ( setunion @ A )
% 2.13/1.39 = ( setunion @ B ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(omega__Cong_type,type,
% 2.13/1.39 omega__Cong: $o ).
% 2.13/1.39
% 2.13/1.39 thf(omega__Cong,definition,
% 2.13/1.39 ( omega__Cong
% 2.13/1.39 = ( omega = omega ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(exuEu_type,type,
% 2.13/1.39 exuEu: $o ).
% 2.13/1.39
% 2.13/1.39 thf(exuEu,definition,
% 2.13/1.39 ( exuEu
% 2.13/1.39 = ( ! [Xphi: $i > $o] :
% 2.13/1.39 ( ( exu
% 2.13/1.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.13/1.39 => ! [Xx: $i,Xy: $i] :
% 2.13/1.39 ( ( Xphi @ Xx )
% 2.13/1.39 => ( ( Xphi @ Xy )
% 2.13/1.39 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(descr__Cong_type,type,
% 2.13/1.39 descr__Cong: $o ).
% 2.13/1.39
% 2.13/1.39 thf(descr__Cong,definition,
% 2.13/1.39 ( descr__Cong
% 2.13/1.39 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.13/1.39 ( ! [Xx: $i,Xy: $i] :
% 2.13/1.39 ( ( Xx = Xy )
% 2.13/1.39 => ( ( Xphi @ Xx )
% 2.13/1.39 <=> ( Xpsi @ Xy ) ) )
% 2.13/1.39 => ( ( exu
% 2.13/1.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.13/1.39 => ( ( exu
% 2.13/1.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.13/1.39 => ( ( descr
% 2.13/1.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.13/1.39 = ( descr
% 2.13/1.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(dsetconstr__Cong_type,type,
% 2.13/1.39 dsetconstr__Cong: $o ).
% 2.13/1.39
% 2.13/1.39 thf(dsetconstr__Cong,definition,
% 2.13/1.39 ( dsetconstr__Cong
% 2.13/1.39 = ( ! [A: $i,B: $i] :
% 2.13/1.39 ( ( A = B )
% 2.13/1.39 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.13/1.39 ( ! [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ! [Xy: $i] :
% 2.13/1.39 ( ( in @ Xy @ B )
% 2.13/1.39 => ( ( Xx = Xy )
% 2.13/1.39 => ( ( Xphi @ Xx )
% 2.13/1.39 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.13/1.39 => ( ( dsetconstr @ A
% 2.13/1.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.13/1.39 = ( dsetconstr @ B
% 2.13/1.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(subset_type,type,
% 2.13/1.39 subset: $i > $i > $o ).
% 2.13/1.39
% 2.13/1.39 thf(disjoint_type,type,
% 2.13/1.39 disjoint: $i > $i > $o ).
% 2.13/1.39
% 2.13/1.39 thf(setsmeet_type,type,
% 2.13/1.39 setsmeet: $i > $i > $o ).
% 2.13/1.39
% 2.13/1.39 thf(subsetI1_type,type,
% 2.13/1.39 subsetI1: $o ).
% 2.13/1.39
% 2.13/1.39 thf(subsetI1,definition,
% 2.13/1.39 ( subsetI1
% 2.13/1.39 = ( ! [A: $i,B: $i] :
% 2.13/1.39 ( ! [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ( in @ Xx @ B ) )
% 2.13/1.39 => ( subset @ A @ B ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(eqimpsubset2_type,type,
% 2.13/1.39 eqimpsubset2: $o ).
% 2.13/1.39
% 2.13/1.39 thf(eqimpsubset2,definition,
% 2.13/1.39 ( eqimpsubset2
% 2.13/1.39 = ( ! [A: $i,B: $i] :
% 2.13/1.39 ( ( A = B )
% 2.13/1.39 => ( subset @ B @ A ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(eqimpsubset1_type,type,
% 2.13/1.39 eqimpsubset1: $o ).
% 2.13/1.39
% 2.13/1.39 thf(eqimpsubset1,definition,
% 2.13/1.39 ( eqimpsubset1
% 2.13/1.39 = ( ! [A: $i,B: $i] :
% 2.13/1.39 ( ( A = B )
% 2.13/1.39 => ( subset @ A @ B ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(subsetI2_type,type,
% 2.13/1.39 subsetI2: $o ).
% 2.13/1.39
% 2.13/1.39 thf(subsetI2,definition,
% 2.13/1.39 ( subsetI2
% 2.13/1.39 = ( ! [A: $i,B: $i] :
% 2.13/1.39 ( ! [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ( in @ Xx @ B ) )
% 2.13/1.39 => ( subset @ A @ B ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(emptysetsubset_type,type,
% 2.13/1.39 emptysetsubset: $o ).
% 2.13/1.39
% 2.13/1.39 thf(emptysetsubset,definition,
% 2.13/1.39 ( emptysetsubset
% 2.13/1.39 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(subsetE_type,type,
% 2.13/1.39 subsetE: $o ).
% 2.13/1.39
% 2.13/1.39 thf(subsetE,definition,
% 2.13/1.39 ( subsetE
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ( subset @ A @ B )
% 2.13/1.39 => ( ( in @ Xx @ A )
% 2.13/1.39 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(subsetE2_type,type,
% 2.13/1.39 subsetE2: $o ).
% 2.13/1.39
% 2.13/1.39 thf(subsetE2,definition,
% 2.13/1.39 ( subsetE2
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ( subset @ A @ B )
% 2.13/1.39 => ( ~ ( in @ Xx @ B )
% 2.13/1.39 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(notsubsetI_type,type,
% 2.13/1.39 notsubsetI: $o ).
% 2.13/1.39
% 2.13/1.39 thf(notsubsetI,definition,
% 2.13/1.39 ( notsubsetI
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ( ~ ( in @ Xx @ B )
% 2.13/1.39 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(notequalI1_type,type,
% 2.13/1.39 notequalI1: $o ).
% 2.13/1.39
% 2.13/1.39 thf(notequalI1,definition,
% 2.13/1.39 ( notequalI1
% 2.13/1.39 = ( ! [A: $i,B: $i] :
% 2.13/1.39 ( ~ ( subset @ A @ B )
% 2.13/1.39 => ( A != B ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(notequalI2_type,type,
% 2.13/1.39 notequalI2: $o ).
% 2.13/1.39
% 2.13/1.39 thf(notequalI2,definition,
% 2.13/1.39 ( notequalI2
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ( ~ ( in @ Xx @ B )
% 2.13/1.39 => ( A != B ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(subsetRefl_type,type,
% 2.13/1.39 subsetRefl: $o ).
% 2.13/1.39
% 2.13/1.39 thf(subsetRefl,definition,
% 2.13/1.39 ( subsetRefl
% 2.13/1.39 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(subsetTrans_type,type,
% 2.13/1.39 subsetTrans: $o ).
% 2.13/1.39
% 2.13/1.39 thf(subsetTrans,definition,
% 2.13/1.39 ( subsetTrans
% 2.13/1.39 = ( ! [A: $i,B: $i,C: $i] :
% 2.13/1.39 ( ( subset @ A @ B )
% 2.13/1.39 => ( ( subset @ B @ C )
% 2.13/1.39 => ( subset @ A @ C ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setadjoinSub_type,type,
% 2.13/1.39 setadjoinSub: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setadjoinSub,definition,
% 2.13/1.39 ( setadjoinSub
% 2.13/1.39 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setadjoinSub2_type,type,
% 2.13/1.39 setadjoinSub2: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setadjoinSub2,definition,
% 2.13/1.39 ( setadjoinSub2
% 2.13/1.39 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.13/1.39 ( ( subset @ A @ B )
% 2.13/1.39 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(subset2powerset_type,type,
% 2.13/1.39 subset2powerset: $o ).
% 2.13/1.39
% 2.13/1.39 thf(subset2powerset,definition,
% 2.13/1.39 ( subset2powerset
% 2.13/1.39 = ( ! [A: $i,B: $i] :
% 2.13/1.39 ( ( subset @ A @ B )
% 2.13/1.39 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setextsub_type,type,
% 2.13/1.39 setextsub: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setextsub,definition,
% 2.13/1.39 ( setextsub
% 2.13/1.39 = ( ! [A: $i,B: $i] :
% 2.13/1.39 ( ( subset @ A @ B )
% 2.13/1.39 => ( ( subset @ B @ A )
% 2.13/1.39 => ( A = B ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(subsetemptysetimpeq_type,type,
% 2.13/1.39 subsetemptysetimpeq: $o ).
% 2.13/1.39
% 2.13/1.39 thf(subsetemptysetimpeq,definition,
% 2.13/1.39 ( subsetemptysetimpeq
% 2.13/1.39 = ( ! [A: $i] :
% 2.13/1.39 ( ( subset @ A @ emptyset )
% 2.13/1.39 => ( A = emptyset ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(powersetI1_type,type,
% 2.13/1.39 powersetI1: $o ).
% 2.13/1.39
% 2.13/1.39 thf(powersetI1,definition,
% 2.13/1.39 ( powersetI1
% 2.13/1.39 = ( ! [A: $i,B: $i] :
% 2.13/1.39 ( ( subset @ B @ A )
% 2.13/1.39 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(powersetE1_type,type,
% 2.13/1.39 powersetE1: $o ).
% 2.13/1.39
% 2.13/1.39 thf(powersetE1,definition,
% 2.13/1.39 ( powersetE1
% 2.13/1.39 = ( ! [A: $i,B: $i] :
% 2.13/1.39 ( ( in @ B @ ( powerset @ A ) )
% 2.13/1.39 => ( subset @ B @ A ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(inPowerset_type,type,
% 2.13/1.39 inPowerset: $o ).
% 2.13/1.39
% 2.13/1.39 thf(inPowerset,definition,
% 2.13/1.39 ( inPowerset
% 2.13/1.39 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(powersetsubset_type,type,
% 2.13/1.39 powersetsubset: $o ).
% 2.13/1.39
% 2.13/1.39 thf(powersetsubset,definition,
% 2.13/1.39 ( powersetsubset
% 2.13/1.39 = ( ! [A: $i,B: $i] :
% 2.13/1.39 ( ( subset @ A @ B )
% 2.13/1.39 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(sepInPowerset_type,type,
% 2.13/1.39 sepInPowerset: $o ).
% 2.13/1.39
% 2.13/1.39 thf(sepInPowerset,definition,
% 2.13/1.39 ( sepInPowerset
% 2.13/1.39 = ( ! [A: $i,Xphi: $i > $o] :
% 2.13/1.39 ( in
% 2.13/1.39 @ ( dsetconstr @ A
% 2.13/1.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.13/1.39 @ ( powerset @ A ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(sepSubset_type,type,
% 2.13/1.39 sepSubset: $o ).
% 2.13/1.39
% 2.13/1.39 thf(sepSubset,definition,
% 2.13/1.39 ( sepSubset
% 2.13/1.39 = ( ! [A: $i,Xphi: $i > $o] :
% 2.13/1.39 ( subset
% 2.13/1.39 @ ( dsetconstr @ A
% 2.13/1.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.13/1.39 @ A ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(binunion_type,type,
% 2.13/1.39 binunion: $i > $i > $i ).
% 2.13/1.39
% 2.13/1.39 thf(binunionIL_type,type,
% 2.13/1.39 binunionIL: $o ).
% 2.13/1.39
% 2.13/1.39 thf(binunionIL,definition,
% 2.13/1.39 ( binunionIL
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(upairset2IR_type,type,
% 2.13/1.39 upairset2IR: $o ).
% 2.13/1.39
% 2.13/1.39 thf(upairset2IR,definition,
% 2.13/1.39 ( upairset2IR
% 2.13/1.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(binunionIR_type,type,
% 2.13/1.39 binunionIR: $o ).
% 2.13/1.39
% 2.13/1.39 thf(binunionIR,definition,
% 2.13/1.39 ( binunionIR
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ B )
% 2.13/1.39 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(binunionEcases_type,type,
% 2.13/1.39 binunionEcases: $o ).
% 2.13/1.39
% 2.13/1.39 thf(binunionEcases,definition,
% 2.13/1.39 ( binunionEcases
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.13/1.39 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.13/1.39 => ( ( ( in @ Xx @ A )
% 2.13/1.39 => Xphi )
% 2.13/1.39 => ( ( ( in @ Xx @ B )
% 2.13/1.39 => Xphi )
% 2.13/1.39 => Xphi ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(binunionE_type,type,
% 2.13/1.39 binunionE: $o ).
% 2.13/1.39
% 2.13/1.39 thf(binunionE,definition,
% 2.13/1.39 ( binunionE
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.13/1.39 => ( ( in @ Xx @ A )
% 2.13/1.39 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(binunionLsub_type,type,
% 2.13/1.39 binunionLsub: $o ).
% 2.13/1.39
% 2.13/1.39 thf(binunionLsub,definition,
% 2.13/1.39 ( binunionLsub
% 2.13/1.39 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(binunionRsub_type,type,
% 2.13/1.39 binunionRsub: $o ).
% 2.13/1.39
% 2.13/1.39 thf(binunionRsub,definition,
% 2.13/1.39 ( binunionRsub
% 2.13/1.39 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(binintersect_type,type,
% 2.13/1.39 binintersect: $i > $i > $i ).
% 2.13/1.39
% 2.13/1.39 thf(binintersectI_type,type,
% 2.13/1.39 binintersectI: $o ).
% 2.13/1.39
% 2.13/1.39 thf(binintersectI,definition,
% 2.13/1.39 ( binintersectI
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ( ( in @ Xx @ B )
% 2.13/1.39 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(binintersectSubset5_type,type,
% 2.13/1.39 binintersectSubset5: $o ).
% 2.13/1.39
% 2.13/1.39 thf(binintersectSubset5,definition,
% 2.13/1.39 ( binintersectSubset5
% 2.13/1.39 = ( ! [A: $i,B: $i,C: $i] :
% 2.13/1.39 ( ( subset @ C @ A )
% 2.13/1.39 => ( ( subset @ C @ B )
% 2.13/1.39 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(binintersectEL_type,type,
% 2.13/1.39 binintersectEL: $o ).
% 2.13/1.39
% 2.13/1.39 thf(binintersectEL,definition,
% 2.13/1.39 ( binintersectEL
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.13/1.39 => ( in @ Xx @ A ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(binintersectLsub_type,type,
% 2.13/1.39 binintersectLsub: $o ).
% 2.13/1.39
% 2.13/1.39 thf(binintersectLsub,definition,
% 2.13/1.39 ( binintersectLsub
% 2.13/1.39 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(binintersectSubset2_type,type,
% 2.13/1.39 binintersectSubset2: $o ).
% 2.13/1.39
% 2.13/1.39 thf(binintersectSubset2,definition,
% 2.13/1.39 ( binintersectSubset2
% 2.13/1.39 = ( ! [A: $i,B: $i] :
% 2.13/1.39 ( ( subset @ A @ B )
% 2.13/1.39 => ( ( binintersect @ A @ B )
% 2.13/1.39 = A ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(binintersectSubset3_type,type,
% 2.13/1.39 binintersectSubset3: $o ).
% 2.13/1.39
% 2.13/1.39 thf(binintersectSubset3,definition,
% 2.13/1.39 ( binintersectSubset3
% 2.13/1.39 = ( ! [A: $i,B: $i] :
% 2.13/1.39 ( ( ( binintersect @ A @ B )
% 2.13/1.39 = B )
% 2.13/1.39 => ( subset @ B @ A ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(binintersectER_type,type,
% 2.13/1.39 binintersectER: $o ).
% 2.13/1.39
% 2.13/1.39 thf(binintersectER,definition,
% 2.13/1.39 ( binintersectER
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.13/1.39 => ( in @ Xx @ B ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(disjointsetsI1_type,type,
% 2.13/1.39 disjointsetsI1: $o ).
% 2.13/1.39
% 2.13/1.39 thf(disjointsetsI1,definition,
% 2.13/1.39 ( disjointsetsI1
% 2.13/1.39 = ( ! [A: $i,B: $i] :
% 2.13/1.39 ( ~ ? [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 & ( in @ Xx @ B ) )
% 2.13/1.39 => ( ( binintersect @ A @ B )
% 2.13/1.39 = emptyset ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(binintersectRsub_type,type,
% 2.13/1.39 binintersectRsub: $o ).
% 2.13/1.39
% 2.13/1.39 thf(binintersectRsub,definition,
% 2.13/1.39 ( binintersectRsub
% 2.13/1.39 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(binintersectSubset4_type,type,
% 2.13/1.39 binintersectSubset4: $o ).
% 2.13/1.39
% 2.13/1.39 thf(binintersectSubset4,definition,
% 2.13/1.39 ( binintersectSubset4
% 2.13/1.39 = ( ! [A: $i,B: $i] :
% 2.13/1.39 ( ( subset @ B @ A )
% 2.13/1.39 => ( ( binintersect @ A @ B )
% 2.13/1.39 = B ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(binintersectSubset1_type,type,
% 2.13/1.39 binintersectSubset1: $o ).
% 2.13/1.39
% 2.13/1.39 thf(binintersectSubset1,definition,
% 2.13/1.39 ( binintersectSubset1
% 2.13/1.39 = ( ! [A: $i,B: $i] :
% 2.13/1.39 ( ( ( binintersect @ A @ B )
% 2.13/1.39 = A )
% 2.13/1.39 => ( subset @ A @ B ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(bs114d_type,type,
% 2.13/1.39 bs114d: $o ).
% 2.13/1.39
% 2.13/1.39 thf(bs114d,definition,
% 2.13/1.39 ( bs114d
% 2.13/1.39 = ( ! [A: $i,B: $i,C: $i] :
% 2.13/1.39 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.13/1.39 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(regular_type,type,
% 2.13/1.39 regular: $i > $o ).
% 2.13/1.39
% 2.13/1.39 thf(setminus_type,type,
% 2.13/1.39 setminus: $i > $i > $i ).
% 2.13/1.39
% 2.13/1.39 thf(setminusI_type,type,
% 2.13/1.39 setminusI: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setminusI,definition,
% 2.13/1.39 ( setminusI
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ( ~ ( in @ Xx @ B )
% 2.13/1.39 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setminusEL_type,type,
% 2.13/1.39 setminusEL: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setminusEL,definition,
% 2.13/1.39 ( setminusEL
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.13/1.39 => ( in @ Xx @ A ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setminusER_type,type,
% 2.13/1.39 setminusER: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setminusER,definition,
% 2.13/1.39 ( setminusER
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.13/1.39 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setminusSubset2_type,type,
% 2.13/1.39 setminusSubset2: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setminusSubset2,definition,
% 2.13/1.39 ( setminusSubset2
% 2.13/1.39 = ( ! [A: $i,B: $i] :
% 2.13/1.39 ( ( subset @ A @ B )
% 2.13/1.39 => ( ( setminus @ A @ B )
% 2.13/1.39 = emptyset ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setminusERneg_type,type,
% 2.13/1.39 setminusERneg: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setminusERneg,definition,
% 2.13/1.39 ( setminusERneg
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.13/1.39 => ( ( in @ Xx @ A )
% 2.13/1.39 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setminusELneg_type,type,
% 2.13/1.39 setminusELneg: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setminusELneg,definition,
% 2.13/1.39 ( setminusELneg
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.13/1.39 => ( ~ ( in @ Xx @ B )
% 2.13/1.39 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setminusILneg_type,type,
% 2.13/1.39 setminusILneg: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setminusILneg,definition,
% 2.13/1.39 ( setminusILneg
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ~ ( in @ Xx @ A )
% 2.13/1.39 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setminusIRneg_type,type,
% 2.13/1.39 setminusIRneg: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setminusIRneg,definition,
% 2.13/1.39 ( setminusIRneg
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ B )
% 2.13/1.39 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setminusLsub_type,type,
% 2.13/1.39 setminusLsub: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setminusLsub,definition,
% 2.13/1.39 ( setminusLsub
% 2.13/1.39 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setminusSubset1_type,type,
% 2.13/1.39 setminusSubset1: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setminusSubset1,definition,
% 2.13/1.39 ( setminusSubset1
% 2.13/1.39 = ( ! [A: $i,B: $i] :
% 2.13/1.39 ( ( ( setminus @ A @ B )
% 2.13/1.39 = emptyset )
% 2.13/1.39 => ( subset @ A @ B ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(symdiff_type,type,
% 2.13/1.39 symdiff: $i > $i > $i ).
% 2.13/1.39
% 2.13/1.39 thf(symdiffE_type,type,
% 2.13/1.39 symdiffE: $o ).
% 2.13/1.39
% 2.13/1.39 thf(symdiffE,definition,
% 2.13/1.39 ( symdiffE
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.13/1.39 => ! [Xphi: $o] :
% 2.13/1.39 ( ( ( in @ Xx @ A )
% 2.13/1.39 => ( ~ ( in @ Xx @ B )
% 2.13/1.39 => Xphi ) )
% 2.13/1.39 => ( ( ~ ( in @ Xx @ A )
% 2.13/1.39 => ( ( in @ Xx @ B )
% 2.13/1.39 => Xphi ) )
% 2.13/1.39 => Xphi ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(symdiffI1_type,type,
% 2.13/1.39 symdiffI1: $o ).
% 2.13/1.39
% 2.13/1.39 thf(symdiffI1,definition,
% 2.13/1.39 ( symdiffI1
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ( ~ ( in @ Xx @ B )
% 2.13/1.39 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(symdiffI2_type,type,
% 2.13/1.39 symdiffI2: $o ).
% 2.13/1.39
% 2.13/1.39 thf(symdiffI2,definition,
% 2.13/1.39 ( symdiffI2
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ~ ( in @ Xx @ A )
% 2.13/1.39 => ( ( in @ Xx @ B )
% 2.13/1.39 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(symdiffIneg1_type,type,
% 2.13/1.39 symdiffIneg1: $o ).
% 2.13/1.39
% 2.13/1.39 thf(symdiffIneg1,definition,
% 2.13/1.39 ( symdiffIneg1
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ( ( in @ Xx @ B )
% 2.13/1.39 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(symdiffIneg2_type,type,
% 2.13/1.39 symdiffIneg2: $o ).
% 2.13/1.39
% 2.13/1.39 thf(symdiffIneg2,definition,
% 2.13/1.39 ( symdiffIneg2
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ~ ( in @ Xx @ A )
% 2.13/1.39 => ( ~ ( in @ Xx @ B )
% 2.13/1.39 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(iskpair_type,type,
% 2.13/1.39 iskpair: $i > $o ).
% 2.13/1.39
% 2.13/1.39 thf(iskpair,definition,
% 2.13/1.39 ( iskpair
% 2.13/1.39 = ( ^ [A: $i] :
% 2.13/1.39 ? [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ ( setunion @ A ) )
% 2.13/1.39 & ? [Xy: $i] :
% 2.13/1.39 ( ( in @ Xy @ ( setunion @ A ) )
% 2.13/1.39 & ( A
% 2.13/1.39 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(secondinupair_type,type,
% 2.13/1.39 secondinupair: $o ).
% 2.13/1.39
% 2.13/1.39 thf(secondinupair,definition,
% 2.13/1.39 ( secondinupair
% 2.13/1.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setukpairIL_type,type,
% 2.13/1.39 setukpairIL: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setukpairIL,definition,
% 2.13/1.39 ( setukpairIL
% 2.13/1.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setukpairIR_type,type,
% 2.13/1.39 setukpairIR: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setukpairIR,definition,
% 2.13/1.39 ( setukpairIR
% 2.13/1.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(kpairiskpair_type,type,
% 2.13/1.39 kpairiskpair: $o ).
% 2.13/1.39
% 2.13/1.39 thf(kpairiskpair,definition,
% 2.13/1.39 ( kpairiskpair
% 2.13/1.39 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(kpair_type,type,
% 2.13/1.39 kpair: $i > $i > $i ).
% 2.13/1.39
% 2.13/1.39 thf(kpair,definition,
% 2.13/1.39 ( kpair
% 2.13/1.39 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(kpairp_type,type,
% 2.13/1.39 kpairp: $o ).
% 2.13/1.39
% 2.13/1.39 thf(kpairp,definition,
% 2.13/1.39 ( kpairp
% 2.13/1.39 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(cartprod_type,type,
% 2.13/1.39 cartprod: $i > $i > $i ).
% 2.13/1.39
% 2.13/1.39 thf(singletonsubset_type,type,
% 2.13/1.39 singletonsubset: $o ).
% 2.13/1.39
% 2.13/1.39 thf(singletonsubset,definition,
% 2.13/1.39 ( singletonsubset
% 2.13/1.39 = ( ! [A: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(singletoninpowerset_type,type,
% 2.13/1.39 singletoninpowerset: $o ).
% 2.13/1.39
% 2.13/1.39 thf(singletoninpowerset,definition,
% 2.13/1.39 ( singletoninpowerset
% 2.13/1.39 = ( ! [A: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(singletoninpowunion_type,type,
% 2.13/1.39 singletoninpowunion: $o ).
% 2.13/1.39
% 2.13/1.39 thf(singletoninpowunion,definition,
% 2.13/1.39 ( singletoninpowunion
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(upairset2E_type,type,
% 2.13/1.39 upairset2E: $o ).
% 2.13/1.39
% 2.13/1.39 thf(upairset2E,definition,
% 2.13/1.39 ( upairset2E
% 2.13/1.39 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.13/1.39 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.13/1.39 => ( ( Xz = Xx )
% 2.13/1.39 | ( Xz = Xy ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(upairsubunion_type,type,
% 2.13/1.39 upairsubunion: $o ).
% 2.13/1.39
% 2.13/1.39 thf(upairsubunion,definition,
% 2.13/1.39 ( upairsubunion
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ! [Xy: $i] :
% 2.13/1.39 ( ( in @ Xy @ B )
% 2.13/1.39 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(upairinpowunion_type,type,
% 2.13/1.39 upairinpowunion: $o ).
% 2.13/1.39
% 2.13/1.39 thf(upairinpowunion,definition,
% 2.13/1.39 ( upairinpowunion
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ! [Xy: $i] :
% 2.13/1.39 ( ( in @ Xy @ B )
% 2.13/1.39 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(ubforcartprodlem1_type,type,
% 2.13/1.39 ubforcartprodlem1: $o ).
% 2.13/1.39
% 2.13/1.39 thf(ubforcartprodlem1,definition,
% 2.13/1.39 ( ubforcartprodlem1
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ! [Xy: $i] :
% 2.13/1.39 ( ( in @ Xy @ B )
% 2.13/1.39 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(ubforcartprodlem2_type,type,
% 2.13/1.39 ubforcartprodlem2: $o ).
% 2.13/1.39
% 2.13/1.39 thf(ubforcartprodlem2,definition,
% 2.13/1.39 ( ubforcartprodlem2
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ! [Xy: $i] :
% 2.13/1.39 ( ( in @ Xy @ B )
% 2.13/1.39 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(ubforcartprodlem3_type,type,
% 2.13/1.39 ubforcartprodlem3: $o ).
% 2.13/1.39
% 2.13/1.39 thf(ubforcartprodlem3,definition,
% 2.13/1.39 ( ubforcartprodlem3
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ! [Xy: $i] :
% 2.13/1.39 ( ( in @ Xy @ B )
% 2.13/1.39 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(cartprodpairin_type,type,
% 2.13/1.39 cartprodpairin: $o ).
% 2.13/1.39
% 2.13/1.39 thf(cartprodpairin,definition,
% 2.13/1.39 ( cartprodpairin
% 2.13/1.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 => ! [Xy: $i] :
% 2.13/1.39 ( ( in @ Xy @ B )
% 2.13/1.39 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(cartprodmempair1_type,type,
% 2.13/1.39 cartprodmempair1: $o ).
% 2.13/1.39
% 2.13/1.39 thf(cartprodmempair1,definition,
% 2.13/1.39 ( cartprodmempair1
% 2.13/1.39 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.13/1.39 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.13/1.39 => ? [Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ A )
% 2.13/1.39 & ? [Xy: $i] :
% 2.13/1.39 ( ( in @ Xy @ B )
% 2.13/1.39 & ( Xu
% 2.13/1.39 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(cartprodmempair_type,type,
% 2.13/1.39 cartprodmempair: $o ).
% 2.13/1.39
% 2.13/1.39 thf(cartprodmempair,definition,
% 2.13/1.39 ( cartprodmempair
% 2.13/1.39 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.13/1.39 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.13/1.39 => ( iskpair @ Xu ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setunionE2_type,type,
% 2.13/1.39 setunionE2: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setunionE2,definition,
% 2.13/1.39 ( setunionE2
% 2.13/1.39 = ( ! [A: $i,Xx: $i] :
% 2.13/1.39 ( ( in @ Xx @ ( setunion @ A ) )
% 2.13/1.39 => ? [X: $i] :
% 2.13/1.39 ( ( in @ X @ A )
% 2.13/1.39 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setunionsingleton1_type,type,
% 2.13/1.39 setunionsingleton1: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setunionsingleton1,definition,
% 2.13/1.39 ( setunionsingleton1
% 2.13/1.39 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setunionsingleton2_type,type,
% 2.13/1.39 setunionsingleton2: $o ).
% 2.13/1.39
% 2.13/1.39 thf(setunionsingleton2,definition,
% 2.13/1.39 ( setunionsingleton2
% 2.13/1.39 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.13/1.39
% 2.13/1.39 thf(setunionsingleton,conjecture,
% 2.13/1.39 ( setextAx
% 2.13/1.39 => ( emptysetAx
% 2.13/1.39 => ( setadjoinAx
% 2.13/1.39 => ( powersetAx
% 2.13/1.39 => ( setunionAx
% 2.13/1.39 => ( omega0Ax
% 2.13/1.39 => ( omegaSAx
% 2.13/1.39 => ( omegaIndAx
% 2.13/1.39 => ( replAx
% 2.13/1.39 => ( foundationAx
% 2.13/1.39 => ( wellorderingAx
% 2.13/1.39 => ( descrp
% 2.13/1.39 => ( dsetconstrI
% 2.13/1.39 => ( dsetconstrEL
% 2.13/1.39 => ( dsetconstrER
% 2.13/1.39 => ( exuE1
% 2.13/1.39 => ( prop2setE
% 2.13/1.39 => ( emptysetE
% 2.13/1.39 => ( emptysetimpfalse
% 2.13/1.39 => ( notinemptyset
% 2.13/1.39 => ( exuE3e
% 2.13/1.39 => ( setext
% 2.13/1.39 => ( emptyI
% 2.13/1.39 => ( noeltsimpempty
% 2.13/1.39 => ( setbeta
% 2.13/1.39 => ( nonemptyE1
% 2.13/1.39 => ( nonemptyI
% 2.13/1.39 => ( nonemptyI1
% 2.13/1.39 => ( setadjoinIL
% 2.13/1.39 => ( emptyinunitempty
% 2.13/1.39 => ( setadjoinIR
% 2.13/1.39 => ( setadjoinE
% 2.13/1.39 => ( setadjoinOr
% 2.13/1.39 => ( setoftrueEq
% 2.13/1.39 => ( powersetI
% 2.13/1.39 => ( emptyinPowerset
% 2.13/1.39 => ( emptyInPowerset
% 2.13/1.39 => ( powersetE
% 2.13/1.39 => ( setunionI
% 2.13/1.39 => ( setunionE
% 2.13/1.39 => ( subPowSU
% 2.13/1.39 => ( exuE2
% 2.13/1.39 => ( nonemptyImpWitness
% 2.13/1.39 => ( uniqinunit
% 2.13/1.39 => ( notinsingleton
% 2.13/1.39 => ( eqinunit
% 2.13/1.39 => ( singletonsswitch
% 2.13/1.39 => ( upairsetE
% 2.13/1.39 => ( upairsetIL
% 2.13/1.39 => ( upairsetIR
% 2.13/1.39 => ( emptyE1
% 2.13/1.39 => ( vacuousDall
% 2.13/1.39 => ( quantDeMorgan1
% 2.13/1.39 => ( quantDeMorgan2
% 2.13/1.39 => ( quantDeMorgan3
% 2.13/1.39 => ( quantDeMorgan4
% 2.13/1.39 => ( prop2setI
% 2.13/1.39 => ( prop2set2propI
% 2.13/1.39 => ( notdexE
% 2.13/1.39 => ( notdallE
% 2.13/1.39 => ( exuI1
% 2.13/1.39 => ( exuI3
% 2.13/1.39 => ( exuI2
% 2.13/1.39 => ( inCongP
% 2.13/1.39 => ( in__Cong
% 2.13/1.39 => ( exuE3u
% 2.13/1.39 => ( exu__Cong
% 2.13/1.39 => ( emptyset__Cong
% 2.13/1.39 => ( setadjoin__Cong
% 2.13/1.39 => ( powerset__Cong
% 2.13/1.39 => ( setunion__Cong
% 2.13/1.39 => ( omega__Cong
% 2.13/1.39 => ( exuEu
% 2.13/1.39 => ( descr__Cong
% 2.13/1.39 => ( dsetconstr__Cong
% 2.13/1.39 => ( subsetI1
% 2.13/1.39 => ( eqimpsubset2
% 2.13/1.39 => ( eqimpsubset1
% 2.13/1.39 => ( subsetI2
% 2.13/1.39 => ( emptysetsubset
% 2.13/1.39 => ( subsetE
% 2.13/1.39 => ( subsetE2
% 2.13/1.39 => ( notsubsetI
% 2.13/1.39 => ( notequalI1
% 2.13/1.39 => ( notequalI2
% 2.13/1.39 => ( subsetRefl
% 2.13/1.39 => ( subsetTrans
% 2.13/1.39 => ( setadjoinSub
% 2.13/1.39 => ( setadjoinSub2
% 2.13/1.39 => ( subset2powerset
% 2.13/1.39 => ( setextsub
% 2.13/1.39 => ( subsetemptysetimpeq
% 2.13/1.39 => ( powersetI1
% 2.13/1.39 => ( powersetE1
% 2.13/1.39 => ( inPowerset
% 2.13/1.39 => ( powersetsubset
% 2.13/1.39 => ( sepInPowerset
% 2.13/1.39 => ( sepSubset
% 2.13/1.39 => ( binunionIL
% 2.13/1.39 => ( upairset2IR
% 2.13/1.39 => ( binunionIR
% 2.13/1.39 => ( binunionEcases
% 2.13/1.39 => ( binunionE
% 2.13/1.39 => ( binunionLsub
% 2.13/1.39 => ( binunionRsub
% 2.13/1.39 => ( binintersectI
% 2.13/1.39 => ( binintersectSubset5
% 2.13/1.39 => ( binintersectEL
% 2.13/1.39 => ( binintersectLsub
% 2.13/1.39 => ( binintersectSubset2
% 2.13/1.39 => ( binintersectSubset3
% 2.13/1.39 => ( binintersectER
% 2.13/1.39 => ( disjointsetsI1
% 2.13/1.39 => ( binintersectRsub
% 2.13/1.39 => ( binintersectSubset4
% 2.13/1.39 => ( binintersectSubset1
% 2.13/1.39 => ( bs114d
% 2.13/1.39 => ( setminusI
% 2.13/1.39 => ( setminusEL
% 2.13/1.39 => ( setminusER
% 2.13/1.39 => ( setminusSubset2
% 2.13/1.39 => ( setminusERneg
% 2.13/1.39 => ( setminusELneg
% 2.13/1.39 => ( setminusILneg
% 2.13/1.39 => ( setminusIRneg
% 2.13/1.39 => ( setminusLsub
% 2.13/1.39 => ( setminusSubset1
% 2.13/1.39 => ( symdiffE
% 2.13/1.39 => ( symdiffI1
% 2.13/1.39 => ( symdiffI2
% 2.13/1.39 => ( symdiffIneg1
% 2.13/1.39 => ( symdiffIneg2
% 2.13/1.39 => ( secondinupair
% 2.13/1.39 => ( setukpairIL
% 2.13/1.39 => ( setukpairIR
% 2.13/1.39 => ( kpairiskpair
% 2.13/1.39 => ( kpairp
% 2.13/1.39 => ( singletonsubset
% 2.13/1.39 => ( singletoninpowerset
% 2.13/1.39 => ( singletoninpowunion
% 2.13/1.39 => ( upairset2E
% 2.13/1.39 => ( upairsubunion
% 2.13/1.39 => ( upairinpowunion
% 2.13/1.39 => ( ubforcartprodlem1
% 2.13/1.39 => ( ubforcartprodlem2
% 2.13/1.39 => ( ubforcartprodlem3
% 2.13/1.39 => ( cartprodpairin
% 2.13/1.39 => ( cartprodmempair1
% 2.13/1.39 => ( cartprodmempair
% 2.13/1.39 => ( setunionE2
% 2.13/1.39 => ( setunionsingleton1
% 2.13/1.39 => ( setunionsingleton2
% 2.13/1.39 => ! [Xx: $i] :
% 2.13/1.39 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.13/1.39 = Xx ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.13/1.39 ------------------------
% 2.37/1.41 % (18702)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_18509 for (2999ds/183Mi)
% 2.37/1.41 % (18707)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_18509 for (2999ds/275Mi)
% 2.37/1.41 % (18704)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_18509 for (2999ds/27Mi)
% 2.37/1.42 % (18706)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_18509 for (2999ds/2Mi)
% 2.37/1.42 % (18706)Instruction limit reached!
% 2.37/1.42 % (18706)------------------------------
% 2.37/1.42 % (18706)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.42 % (18706)Termination reason: Unknown
% 2.37/1.42 % (18706)Termination phase: shuffling
% 2.37/1.42
% 2.37/1.42 % (18706)Memory used [KB]: 1279
% 2.37/1.42 % (18706)Time elapsed: 0.002 s
% 2.37/1.42 % (18706)Instructions burned: 3 (million)
% 2.37/1.42 % (18706)------------------------------
% 2.37/1.42 % (18706)------------------------------
% 2.37/1.42 % (18708)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_18509 for (2999ds/18Mi)
% 2.37/1.42 % (18703)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_18509 for (2999ds/4Mi)
% 2.37/1.42 % (18703)Instruction limit reached!
% 2.37/1.42 % (18703)------------------------------
% 2.37/1.42 % (18703)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.42 % (18703)Termination reason: Unknown
% 2.37/1.42 % (18703)Termination phase: shuffling
% 2.37/1.42
% 2.37/1.42 % (18703)Memory used [KB]: 1279
% 2.37/1.42 % (18703)Time elapsed: 0.004 s
% 2.37/1.42 % (18703)Instructions burned: 5 (million)
% 2.37/1.42 % (18703)------------------------------
% 2.37/1.42 % (18703)------------------------------
% 2.37/1.43 % (18709)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_18509 for (2999ds/3Mi)
% 2.37/1.43 % (18705)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_18509 for (2999ds/2Mi)
% 2.37/1.43 % (18705)Instruction limit reached!
% 2.37/1.43 % (18705)------------------------------
% 2.37/1.43 % (18705)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.43 % (18705)Termination reason: Unknown
% 2.37/1.43 % (18705)Termination phase: shuffling
% 2.37/1.43
% 2.37/1.43 % (18705)Memory used [KB]: 1279
% 2.37/1.43 % (18705)Time elapsed: 0.002 s
% 2.37/1.43 % (18705)Instructions burned: 2 (million)
% 2.37/1.43 % (18705)------------------------------
% 2.37/1.43 % (18705)------------------------------
% 2.37/1.43 % (18709)Instruction limit reached!
% 2.37/1.43 % (18709)------------------------------
% 2.37/1.43 % (18709)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.43 % (18709)Termination reason: Unknown
% 2.37/1.43 % (18709)Termination phase: shuffling
% 2.37/1.43
% 2.37/1.43 % (18709)Memory used [KB]: 1279
% 2.37/1.43 % (18709)Time elapsed: 0.002 s
% 2.37/1.43 % (18709)Instructions burned: 3 (million)
% 2.37/1.43 % (18709)------------------------------
% 2.37/1.43 % (18709)------------------------------
% 2.37/1.43 % (18708)Instruction limit reached!
% 2.37/1.43 % (18708)------------------------------
% 2.37/1.43 % (18708)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.43 % (18708)Termination reason: Unknown
% 2.37/1.43 % (18708)Termination phase: Property scanning
% 2.37/1.43
% 2.37/1.43 % (18708)Memory used [KB]: 1535
% 2.37/1.43 % (18708)Time elapsed: 0.013 s
% 2.37/1.43 % (18708)Instructions burned: 19 (million)
% 2.37/1.43 % (18708)------------------------------
% 2.37/1.43 % (18708)------------------------------
% 2.37/1.44 % (18710)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_18509 for (2999ds/37Mi)
% 2.37/1.44 % (18704)Instruction limit reached!
% 2.37/1.44 % (18704)------------------------------
% 2.37/1.44 % (18704)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.44 % (18704)Termination reason: Unknown
% 2.37/1.44 % (18704)Termination phase: Property scanning
% 2.37/1.44
% 2.37/1.44 % (18704)Memory used [KB]: 1791
% 2.37/1.44 % (18704)Time elapsed: 0.016 s
% 2.37/1.44 % (18704)Instructions burned: 28 (million)
% 2.37/1.44 % (18704)------------------------------
% 2.37/1.44 % (18704)------------------------------
% 2.37/1.44 % (18712)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_18509 for (2999ds/3Mi)
% 2.37/1.44 % (18711)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_18509 for (2999ds/15Mi)
% 2.37/1.44 % (18712)Instruction limit reached!
% 2.37/1.44 % (18712)------------------------------
% 2.37/1.44 % (18712)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.44 % (18712)Termination reason: Unknown
% 2.37/1.44 % (18712)Termination phase: shuffling
% 2.37/1.44
% 2.37/1.44 % (18712)Memory used [KB]: 1279
% 2.37/1.44 % (18712)Time elapsed: 0.002 s
% 2.37/1.44 % (18712)Instructions burned: 3 (million)
% 2.37/1.44 % (18712)------------------------------
% 2.37/1.44 % (18712)------------------------------
% 2.37/1.44 % (18711)Instruction limit reached!
% 2.37/1.44 % (18711)------------------------------
% 2.37/1.44 % (18711)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.44 % (18711)Termination reason: Unknown
% 2.37/1.44 % (18711)Termination phase: Property scanning
% 2.37/1.44
% 2.37/1.44 % (18711)Memory used [KB]: 1535
% 2.37/1.44 % (18711)Time elapsed: 0.007 s
% 2.37/1.44 % (18711)Instructions burned: 15 (million)
% 2.37/1.44 % (18711)------------------------------
% 2.37/1.44 % (18711)------------------------------
% 2.37/1.45 % (18714)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_18509 for (2999ds/1041Mi)
% 2.37/1.45 % (18710)Instruction limit reached!
% 2.37/1.45 % (18710)------------------------------
% 2.37/1.45 % (18710)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.45 % (18716)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_18509 for (2999ds/16Mi)
% 2.37/1.45 % (18710)Termination reason: Unknown
% 2.37/1.45 % (18710)Termination phase: Property scanning
% 2.37/1.45
% 2.37/1.45 % (18710)Memory used [KB]: 1663
% 2.37/1.45 % (18710)Time elapsed: 0.013 s
% 2.37/1.45 % (18710)Instructions burned: 38 (million)
% 2.37/1.45 % (18710)------------------------------
% 2.37/1.45 % (18710)------------------------------
% 2.37/1.45 % (18716)Instruction limit reached!
% 2.37/1.45 % (18716)------------------------------
% 2.37/1.45 % (18716)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.45 % (18716)Termination reason: Unknown
% 2.37/1.45 % (18716)Termination phase: shuffling
% 2.37/1.45
% 2.37/1.45 % (18716)Memory used [KB]: 1663
% 2.37/1.45 % (18716)Time elapsed: 0.006 s
% 2.37/1.45 % (18716)Instructions burned: 18 (million)
% 2.37/1.45 % (18716)------------------------------
% 2.37/1.45 % (18716)------------------------------
% 2.37/1.45 % (18715)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_18509 for (2999ds/7Mi)
% 2.37/1.46 % (18715)Instruction limit reached!
% 2.37/1.46 % (18715)------------------------------
% 2.37/1.46 % (18715)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.46 % (18715)Termination reason: Unknown
% 2.37/1.46 % (18715)Termination phase: shuffling
% 2.37/1.46
% 2.37/1.46 % (18715)Memory used [KB]: 1407
% 2.37/1.46 % (18715)Time elapsed: 0.005 s
% 2.37/1.46 % (18715)Instructions burned: 7 (million)
% 2.37/1.46 % (18715)------------------------------
% 2.37/1.46 % (18715)------------------------------
% 2.37/1.46 % (18717)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_18509 for (2999ds/3Mi)
% 2.37/1.46 % (18717)Instruction limit reached!
% 2.37/1.46 % (18717)------------------------------
% 2.37/1.46 % (18717)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.46 % (18717)Termination reason: Unknown
% 2.37/1.46 % (18717)Termination phase: shuffling
% 2.37/1.46
% 2.37/1.46 % (18717)Memory used [KB]: 1279
% 2.37/1.46 % (18717)Time elapsed: 0.002 s
% 2.37/1.46 % (18717)Instructions burned: 3 (million)
% 2.37/1.46 % (18717)------------------------------
% 2.37/1.46 % (18717)------------------------------
% 2.37/1.46 % (18718)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_18509 for (2999ds/3Mi)
% 2.37/1.46 % (18702)Instruction limit reached!
% 2.37/1.46 % (18702)------------------------------
% 2.37/1.46 % (18702)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.46 % (18702)Termination reason: Unknown
% 2.37/1.46 % (18702)Termination phase: Saturation
% 2.37/1.46
% 2.37/1.46 % (18702)Memory used [KB]: 7547
% 2.37/1.46 % (18702)Time elapsed: 0.053 s
% 2.37/1.46 % (18702)Instructions burned: 185 (million)
% 2.37/1.46 % (18702)------------------------------
% 2.37/1.46 % (18702)------------------------------
% 2.37/1.46 % (18719)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_18509 for (2999ds/7Mi)
% 2.37/1.46 % (18718)Instruction limit reached!
% 2.37/1.46 % (18718)------------------------------
% 2.37/1.46 % (18718)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.46 % (18718)Termination reason: Unknown
% 2.37/1.46 % (18718)Termination phase: shuffling
% 2.37/1.46
% 2.37/1.46 % (18718)Memory used [KB]: 1279
% 2.37/1.46 % (18718)Time elapsed: 0.003 s
% 2.37/1.46 % (18718)Instructions burned: 4 (million)
% 2.37/1.46 % (18718)------------------------------
% 2.37/1.46 % (18718)------------------------------
% 2.37/1.46 % (18719)Instruction limit reached!
% 2.37/1.46 % (18719)------------------------------
% 2.37/1.46 % (18719)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.46 % (18719)Termination reason: Unknown
% 2.37/1.46 % (18719)Termination phase: shuffling
% 2.37/1.46
% 2.37/1.46 % (18719)Memory used [KB]: 1407
% 2.37/1.46 % (18719)Time elapsed: 0.003 s
% 2.37/1.46 % (18719)Instructions burned: 7 (million)
% 2.37/1.46 % (18719)------------------------------
% 2.37/1.46 % (18719)------------------------------
% 2.37/1.47 % (18720)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_18509 for (2999ds/3Mi)
% 2.37/1.47 % (18720)Instruction limit reached!
% 2.37/1.47 % (18720)------------------------------
% 2.37/1.47 % (18720)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.47 % (18720)Termination reason: Unknown
% 2.37/1.47 % (18720)Termination phase: shuffling
% 2.37/1.47
% 2.37/1.47 % (18720)Memory used [KB]: 1279
% 2.37/1.47 % (18720)Time elapsed: 0.003 s
% 2.37/1.47 % (18720)Instructions burned: 6 (million)
% 2.37/1.47 % (18720)------------------------------
% 2.37/1.47 % (18720)------------------------------
% 2.37/1.47 % (18721)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_18509 for (2999ds/4Mi)
% 2.37/1.47 % (18722)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_18509 for (2999ds/18Mi)
% 2.37/1.47 % (18721)Instruction limit reached!
% 2.37/1.47 % (18721)------------------------------
% 2.37/1.47 % (18721)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.47 % (18721)Termination reason: Unknown
% 2.37/1.47 % (18721)Termination phase: shuffling
% 2.37/1.47
% 2.37/1.47 % (18721)Memory used [KB]: 1279
% 2.37/1.47 % (18721)Time elapsed: 0.004 s
% 2.37/1.47 % (18721)Instructions burned: 5 (million)
% 2.37/1.47 % (18721)------------------------------
% 2.37/1.47 % (18721)------------------------------
% 2.37/1.48 % (18722)Instruction limit reached!
% 2.37/1.48 % (18722)------------------------------
% 2.37/1.48 % (18722)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.48 % (18722)Termination reason: Unknown
% 2.37/1.48 % (18722)Termination phase: shuffling
% 2.37/1.48
% 2.37/1.48 % (18722)Memory used [KB]: 1663
% 2.37/1.48 % (18722)Time elapsed: 0.006 s
% 2.37/1.48 % (18722)Instructions burned: 19 (million)
% 2.37/1.48 % (18722)------------------------------
% 2.37/1.48 % (18722)------------------------------
% 2.37/1.48 % (18724)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_18509 for (2999ds/6Mi)
% 2.37/1.48 % (18723)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_18509 for (2999ds/710Mi)
% 2.37/1.48 % (18724)Instruction limit reached!
% 2.37/1.48 % (18724)------------------------------
% 2.37/1.48 % (18724)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.48 % (18724)Termination reason: Unknown
% 2.37/1.48 % (18724)Termination phase: shuffling
% 2.37/1.48
% 2.37/1.48 % (18724)Memory used [KB]: 1279
% 2.37/1.48 % (18724)Time elapsed: 0.003 s
% 2.37/1.48 % (18724)Instructions burned: 6 (million)
% 2.37/1.48 % (18724)------------------------------
% 2.37/1.48 % (18724)------------------------------
% 2.37/1.48 % (18725)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_18509 for (2999ds/902Mi)
% 2.37/1.49 % (18726)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_18509 for (2999ds/21Mi)
% 2.37/1.49 % (18728)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_18509 for (2999ds/6Mi)
% 2.37/1.49 % (18727)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_18509 for (2999ds/5Mi)
% 2.37/1.49 % (18727)Instruction limit reached!
% 2.37/1.49 % (18727)------------------------------
% 2.37/1.49 % (18727)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.49 % (18727)Termination reason: Unknown
% 2.37/1.49 % (18727)Termination phase: shuffling
% 2.37/1.49
% 2.37/1.49 % (18727)Memory used [KB]: 1279
% 2.37/1.49 % (18727)Time elapsed: 0.003 s
% 2.37/1.49 % (18727)Instructions burned: 5 (million)
% 2.37/1.49 % (18727)------------------------------
% 2.37/1.49 % (18727)------------------------------
% 2.37/1.49 % (18728)Instruction limit reached!
% 2.37/1.49 % (18728)------------------------------
% 2.37/1.49 % (18728)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.49 % (18728)Termination reason: Unknown
% 2.37/1.49 % (18728)Termination phase: shuffling
% 2.37/1.49
% 2.37/1.49 % (18728)Memory used [KB]: 1407
% 2.37/1.49 % (18728)Time elapsed: 0.005 s
% 2.37/1.49 % (18728)Instructions burned: 6 (million)
% 2.37/1.49 % (18728)------------------------------
% 2.37/1.49 % (18728)------------------------------
% 2.37/1.49 % (18726)Instruction limit reached!
% 2.37/1.49 % (18726)------------------------------
% 2.37/1.49 % (18726)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.49 % (18726)Termination reason: Unknown
% 2.37/1.49 % (18726)Termination phase: shuffling
% 2.37/1.49
% 2.37/1.49 % (18726)Memory used [KB]: 1791
% 2.37/1.49 % (18726)Time elapsed: 0.010 s
% 2.37/1.49 % (18726)Instructions burned: 22 (million)
% 2.37/1.49 % (18726)------------------------------
% 2.37/1.49 % (18726)------------------------------
% 2.37/1.50 % (18729)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_18509 for (2999ds/377Mi)
% 2.37/1.50 % (18730)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_18509 for (2999ds/779Mi)
% 2.37/1.50 % (18707)Instruction limit reached!
% 2.37/1.50 % (18707)------------------------------
% 2.37/1.50 % (18707)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.50 % (18707)Termination reason: Unknown
% 2.37/1.50 % (18707)Termination phase: Saturation
% 2.37/1.50
% 2.37/1.50 % (18707)Memory used [KB]: 8059
% 2.37/1.50 % (18707)Time elapsed: 0.093 s
% 2.37/1.50 % (18707)Instructions burned: 276 (million)
% 2.37/1.50 % (18707)------------------------------
% 2.37/1.50 % (18707)------------------------------
% 2.37/1.51 % (18731)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_18509 for (2999ds/19Mi)
% 2.37/1.51 % (18731)Instruction limit reached!
% 2.37/1.51 % (18731)------------------------------
% 2.37/1.51 % (18731)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.51 % (18731)Termination reason: Unknown
% 2.37/1.51 % (18731)Termination phase: shuffling
% 2.37/1.51
% 2.37/1.51 % (18731)Memory used [KB]: 1663
% 2.37/1.51 % (18731)Time elapsed: 0.007 s
% 2.37/1.51 % (18731)Instructions burned: 21 (million)
% 2.37/1.51 % (18731)------------------------------
% 2.37/1.51 % (18731)------------------------------
% 2.37/1.51 % (18732)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_18509 for (2999ds/879Mi)
% 2.37/1.52 % (18733)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_18509 for (2998ds/17Mi)
% 2.37/1.53 % (18733)Instruction limit reached!
% 2.37/1.53 % (18733)------------------------------
% 2.37/1.53 % (18733)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.53 % (18733)Termination reason: Unknown
% 2.37/1.53 % (18733)Termination phase: shuffling
% 2.37/1.53
% 2.37/1.53 % (18733)Memory used [KB]: 1535
% 2.37/1.53 % (18733)Time elapsed: 0.009 s
% 2.37/1.53 % (18733)Instructions burned: 17 (million)
% 2.37/1.53 % (18733)------------------------------
% 2.37/1.53 % (18733)------------------------------
% 2.37/1.53 % (18725)Refutation not found, incomplete strategy
% 2.37/1.53 % (18725)------------------------------
% 2.37/1.53 % (18725)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.53 % (18725)Termination reason: Refutation not found, incomplete strategy
% 2.37/1.53
% 2.37/1.53
% 2.37/1.53 % (18725)Memory used [KB]: 7675
% 2.37/1.53 % (18725)Time elapsed: 0.054 s
% 2.37/1.53 % (18725)Instructions burned: 159 (million)
% 2.37/1.53 % (18725)------------------------------
% 2.37/1.53 % (18725)------------------------------
% 2.37/1.54 % (18736)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_18509 for (2998ds/3Mi)
% 2.37/1.54 % (18736)Instruction limit reached!
% 2.37/1.54 % (18736)------------------------------
% 2.37/1.54 % (18736)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.54 % (18736)Termination reason: Unknown
% 2.37/1.54 % (18736)Termination phase: shuffling
% 2.37/1.54
% 2.37/1.54 % (18736)Memory used [KB]: 1279
% 2.37/1.54 % (18736)Time elapsed: 0.002 s
% 2.37/1.54 % (18736)Instructions burned: 4 (million)
% 2.37/1.54 % (18736)------------------------------
% 2.37/1.54 % (18736)------------------------------
% 2.37/1.54 % (18739)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_18509 for (2998ds/30Mi)
% 2.37/1.55 % (18739)Instruction limit reached!
% 2.37/1.55 % (18739)------------------------------
% 2.37/1.55 % (18739)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.55 % (18739)Termination reason: Unknown
% 2.37/1.55 % (18739)Termination phase: Property scanning
% 2.37/1.55
% 2.37/1.55 % (18739)Memory used [KB]: 1791
% 2.37/1.55 % (18739)Time elapsed: 0.010 s
% 2.37/1.55 % (18739)Instructions burned: 31 (million)
% 2.37/1.55 % (18739)------------------------------
% 2.37/1.55 % (18739)------------------------------
% 2.37/1.55 % (18741)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_18509 for (2998ds/127Mi)
% 2.37/1.56 % (18742)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_18509 for (2998ds/100Mi)
% 2.37/1.57 % (18723)First to succeed.
% 2.37/1.57 % (18723)Refutation found. Thanks to Tanya!
% 2.37/1.57 % SZS status Theorem for DTF2THF_18509
% 2.37/1.57 % SZS output start Proof for DTF2THF_18509
% 2.37/1.57 thf(func_def_0, type, in: $i > $i > $o).
% 2.37/1.57 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.37/1.57 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.37/1.57 thf(func_def_8, type, powerset: $i > $i).
% 2.37/1.57 thf(func_def_10, type, setunion: $i > $i).
% 2.37/1.57 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.37/1.57 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.37/1.57 thf(func_def_26, type, prop2set: $o > $i).
% 2.37/1.57 thf(func_def_36, type, nonempty: $i > $o).
% 2.37/1.57 thf(func_def_69, type, set2prop: $i > $o).
% 2.37/1.57 thf(func_def_88, type, subset: $i > $i > $o).
% 2.37/1.57 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.37/1.57 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.37/1.57 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.37/1.57 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.37/1.57 thf(func_def_135, type, regular: $i > $o).
% 2.37/1.57 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.37/1.57 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.37/1.57 thf(func_def_153, type, iskpair: $i > $o).
% 2.37/1.57 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.37/1.57 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.37/1.57 thf(func_def_188, type, sP0: $i > $i > $o).
% 2.37/1.57 thf(func_def_189, type, sP1: $i > $o).
% 2.37/1.57 thf(func_def_190, type, sP2: $i > $i > $o).
% 2.37/1.57 thf(func_def_191, type, sP3: $i > $i > $o).
% 2.37/1.57 thf(func_def_193, type, sP5: $i > $i > $i > $o > $o).
% 2.37/1.57 thf(func_def_201, type, sK13: $i > $i > $i).
% 2.37/1.57 thf(func_def_202, type, sK14: $i > $i > $i).
% 2.37/1.57 thf(func_def_205, type, sK17: $i > $o).
% 2.37/1.57 thf(func_def_206, type, sK18: $i > $o).
% 2.37/1.57 thf(func_def_214, type, sK26: $i > $i > $i).
% 2.37/1.57 thf(func_def_222, type, sK34: $i > $i).
% 2.37/1.57 thf(func_def_223, type, sK35: ($i > $o) > $i).
% 2.37/1.57 thf(func_def_224, type, sK36: $i > $o).
% 2.37/1.57 thf(func_def_225, type, sK37: $i > $i).
% 2.37/1.57 thf(func_def_228, type, sK40: $i > $o).
% 2.37/1.57 thf(func_def_230, type, sK42: ($i > $o) > $i).
% 2.37/1.57 thf(func_def_231, type, sK43: $i > $o).
% 2.37/1.57 thf(func_def_238, type, sK50: $i > $o).
% 2.37/1.57 thf(func_def_255, type, sK67: $i > $o).
% 2.37/1.57 thf(func_def_260, type, sK72: $i > $o).
% 2.37/1.57 thf(func_def_264, type, sK76: $i > $i > $i).
% 2.37/1.57 thf(func_def_269, type, sK81: $i > $o).
% 2.37/1.57 thf(func_def_271, type, sK83: ($i > $o) > $i > $i).
% 2.37/1.57 thf(func_def_273, type, sK85: $i > $o).
% 2.37/1.57 thf(func_def_277, type, sK89: $i > $o).
% 2.37/1.57 thf(func_def_299, type, sK111: $i > $o).
% 2.37/1.57 thf(func_def_301, type, sK113: ($i > $o) > $i > $i).
% 2.37/1.57 thf(func_def_302, type, sK114: $i > $i > $i).
% 2.37/1.57 thf(func_def_303, type, sK115: $i > $i > $i).
% 2.37/1.57 thf(func_def_304, type, sK116: $i > $i > $i).
% 2.37/1.57 thf(func_def_305, type, sK117: $i > $i > $i).
% 2.37/1.57 thf(func_def_306, type, sK118: $i > $i > $i).
% 2.37/1.57 thf(func_def_307, type, sK119: $i > $i > $i > $i).
% 2.37/1.57 thf(func_def_308, type, sK120: $i > $i).
% 2.37/1.57 thf(func_def_309, type, sK121: $i > $i).
% 2.37/1.57 thf(func_def_310, type, sK122: $i > $i).
% 2.37/1.57 thf(func_def_311, type, sK123: $i > $i).
% 2.37/1.57 thf(func_def_312, type, sK124: $i > $i > $i > $i).
% 2.37/1.57 thf(func_def_313, type, sK125: $i > $i > $i > $i).
% 2.37/1.57 thf(func_def_314, type, sK126: $i > $i > $i).
% 2.37/1.57 thf(func_def_315, type, sK127: $i > $i > $i).
% 2.37/1.57 thf(func_def_316, type, sK128: $i > $i > $i).
% 2.37/1.57 thf(func_def_317, type, sK129: $i > $i).
% 2.37/1.57 thf(func_def_319, type, sK131: $i > $i).
% 2.37/1.57 thf(func_def_320, type, sK132: $i > $i).
% 2.37/1.57 thf(func_def_321, type, sK133: ($i > $o) > $i > $i).
% 2.37/1.57 thf(func_def_323, type, sK135: $i > $o).
% 2.37/1.57 thf(func_def_333, type, sK145: ($i > $o) > ($i > $o) > $i).
% 2.37/1.57 thf(func_def_334, type, sK146: ($i > $o) > ($i > $o) > $i).
% 2.37/1.57 thf(func_def_335, type, sK147: $i > $o).
% 2.37/1.57 thf(func_def_336, type, sK148: $i > $o).
% 2.37/1.57 thf(func_def_340, type, sK152: $i > $o).
% 2.37/1.57 thf(func_def_343, type, sK155: ($i > $o) > $i).
% 2.37/1.57 thf(func_def_344, type, sK156: $i > $o).
% 2.37/1.57 thf(func_def_345, type, sK157: $i > $i).
% 2.37/1.57 thf(func_def_349, type, sK161: $i > $o).
% 2.37/1.57 thf(func_def_351, type, sK163: ($i > $o) > $i > $i).
% 2.37/1.57 thf(func_def_352, type, sK164: ($i > $o) > ($i > $o) > $i).
% 2.37/1.57 thf(func_def_353, type, sK165: ($i > $o) > ($i > $o) > $i).
% 2.37/1.57 thf(func_def_354, type, sK166: $i > $o).
% 2.37/1.57 thf(func_def_355, type, sK167: $i > $o).
% 2.37/1.57 thf(func_def_361, type, sK173: $i > $i > $i).
% 2.37/1.57 thf(func_def_363, type, sK175: $i > $i > $o).
% 2.37/1.57 thf(func_def_364, type, sK176: $i > $i).
% 2.37/1.57 thf(func_def_365, type, sK177: $i > $i).
% 2.37/1.57 thf(func_def_366, type, sK178: ($i > $i > $o) > $i > $i).
% 2.37/1.57 thf(func_def_367, type, sK179: $i > ($i > $i > $o) > $i > $i).
% 2.37/1.57 thf(func_def_368, type, sK180: ($i > $i > $o) > $i > $i).
% 2.37/1.57 thf(func_def_372, type, sK184: $i > $i > $i).
% 2.37/1.57 thf(func_def_393, type, sK205: $i > ($i > $o) > $i).
% 2.37/1.57 thf(func_def_394, type, sK206: $i > $o).
% 2.37/1.57 thf(func_def_415, type, sK227: $i > $o).
% 2.37/1.57 thf(func_def_417, type, sK229: $i > ($i > $o) > $i).
% 2.37/1.57 thf(func_def_427, type, sK239: $i > $i > $i > $i).
% 2.37/1.57 thf(func_def_428, type, sK240: $i > $i > $i > $i).
% 2.37/1.57 thf(func_def_433, type, sK245: $i > $i).
% 2.37/1.57 thf(func_def_439, type, sK251: $i > $i).
% 2.37/1.57 thf(func_def_457, type, sK269: $i > $i > $i).
% 2.37/1.57 thf(func_def_463, type, sK275: ($i > $o) > $i > $i).
% 2.37/1.57 thf(func_def_465, type, sK277: $i > $o).
% 2.37/1.57 thf(func_def_467, type, sK279: $i > $i).
% 2.37/1.57 thf(func_def_469, type, sK281: $i > $i).
% 2.37/1.57 thf(func_def_473, type, sK285: $i > $o).
% 2.37/1.57 thf(func_def_486, type, sK298: $i > $o).
% 2.37/1.57 thf(func_def_487, type, sK299: $i > $o).
% 2.37/1.57 thf(func_def_488, type, sK300: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.37/1.57 thf(func_def_489, type, sK301: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.37/1.57 thf(func_def_490, type, sK302: $i > $i > $i).
% 2.37/1.57 thf(func_def_509, type, sK321: $o > $i > $i > $i).
% 2.37/1.57 thf(func_def_514, type, sK326: $i > $o).
% 2.37/1.57 thf(func_def_522, type, sK334: ($i > $o) > $i).
% 2.37/1.57 thf(func_def_523, type, sK335: ($i > $o) > $i).
% 2.37/1.57 thf(func_def_524, type, sK336: $i > $o).
% 2.37/1.57 thf(func_def_530, type, sK342: $i > $i).
% 2.37/1.57 thf(func_def_551, type, sK363: $i > $i > $i).
% 2.37/1.57 thf(func_def_555, type, sK367: $i > $i > $i).
% 2.37/1.57 thf(func_def_571, type, sK383: ($i > $o) > $i > $i).
% 2.37/1.57 thf(func_def_573, type, sK385: $i > $o).
% 2.37/1.57 thf(func_def_581, type, sK393: $i > $i).
% 2.37/1.57 thf(func_def_594, type, sK406: $i > $o).
% 2.37/1.57 thf(func_def_610, type, ph422: !>[X0: $tType]:(X0)).
% 2.37/1.57 thf(f3199,plain,(
% 2.37/1.57 $false),
% 2.37/1.57 inference(trivial_inequality_removal,[status(thm)],[f3198])).
% 2.37/1.57 thf(f3198,plain,(
% 2.37/1.57 (sK324 != sK324)),
% 2.37/1.57 inference(superposition,[status(thm)],[f1899,f3190])).
% 2.37/1.57 thf(f3190,plain,(
% 2.37/1.57 ( ! [X0 : $i] : (((setunion @ (setadjoin @ X0 @ emptyset)) = X0)) )),
% 2.37/1.57 inference(backward_demodulation,[status(thm)],[f3179,f3189])).
% 2.37/1.57 thf(f3189,plain,(
% 2.37/1.57 ( ! [X0 : $i] : (((binintersect @ (setunion @ (setadjoin @ X0 @ emptyset)) @ X0) = X0)) )),
% 2.37/1.57 inference(trivial_inequality_removal,[status(thm)],[f3185])).
% 2.37/1.57 thf(f3185,plain,(
% 2.37/1.57 ( ! [X0 : $i] : (($true != $true) | ((binintersect @ (setunion @ (setadjoin @ X0 @ emptyset)) @ X0) = X0)) )),
% 2.37/1.57 inference(superposition,[status(thm)],[f2830,f2780])).
% 2.37/1.57 thf(f2780,plain,(
% 2.37/1.57 ( ! [X1 : $i] : (($true = (subset @ X1 @ (setunion @ (setadjoin @ X1 @ emptyset))))) )),
% 2.37/1.57 inference(trivial_inequality_removal,[status(thm)],[f2456])).
% 2.37/1.57 thf(f2456,plain,(
% 2.37/1.57 ( ! [X1 : $i] : (($true = (subset @ X1 @ (setunion @ (setadjoin @ X1 @ emptyset)))) | ($true != $true)) )),
% 2.37/1.57 inference(definition_unfolding,[status(thm)],[f1763,f1896])).
% 2.37/1.57 thf(f1896,plain,(
% 2.37/1.57 (setunionsingleton2 = $true)),
% 2.37/1.57 inference(cnf_transformation,[status(thm)],[f1207])).
% 2.37/1.57 thf(f1207,plain,(
% 2.37/1.57 (setunionE2 = $true) & (setadjoinSub2 = $true) & (emptyI = $true) & (notdallE = $true) & (exuE3e = $true) & (binintersectSubset3 = $true) & (kpairp = $true) & (binunionIL = $true) & (emptyE1 = $true) & (exuE2 = $true) & (notinemptyset = $true) & (quantDeMorgan4 = $true) & (ubforcartprodlem3 = $true) & (setminusSubset2 = $true) & (binintersectLsub = $true) & (setminusSubset1 = $true) & (emptyinPowerset = $true) & (binunionEcases = $true) & (setadjoinIR = $true) & (dsetconstr__Cong = $true) & (subsetE = $true) & (setminusLsub = $true) & (setukpairIR = $true) & (exuE3u = $true) & (singletoninpowunion = $true) & (binintersectSubset2 = $true) & (upairsetIL = $true) & (symdiffI1 = $true) & (sepInPowerset = $true) & (binintersectSubset4 = $true) & (setadjoin__Cong = $true) & (cartprodmempair1 = $true) & (emptysetsubset = $true) & (exu__Cong = $true) & (singletonsubset = $true) & (subsetI1 = $true) & (setadjoinAx = $true) & (binintersectI = $true) & (setminusER = $true) & (binunionIR = $true) & (wellorderingAx = $true) & (setbeta = $true) & (powersetE = $true) & (symdiffE = $true) & (sepSubset = $true) & (setminusERneg = $true) & (setextsub = $true) & (emptysetE = $true) & (quantDeMorgan1 = $true) & (binintersectEL = $true) & (dsetconstrEL = $true) & (subsetemptysetimpeq = $true) & (singletonsswitch = $true) & (binintersectSubset5 = $true) & (quantDeMorgan2 = $true) & (upairset2E = $true) & (noeltsimpempty = $true) & (exuI2 = $true) & (prop2set2propI = $true) & (nonemptyImpWitness = $true) & (quantDeMorgan3 = $true) & (symdiffI2 = $true) & (upairinpowunion = $true) & (upairsubunion = $true) & (setadjoinE = $true) & (binintersectER = $true) & (omegaSAx = $true) & (ubforcartprodlem1 = $true) & ((setunion @ (setadjoin @ sK324 @ emptyset)) != sK324) & (omega__Cong = $true) & (binintersectSubset1 = $true) & (setunionsingleton2 = $true) & (setunionsingleton1 = $true) & (upairsetE = $true) & (setminusI = $true) & (inCongP = $true) & (secondinupair = $true) & (emptysetimpfalse = $true) & (nonemptyI = $true) & (setunionAx = $true) & (powerset__Cong = $true) & (setextAx = $true) & (nonemptyE1 = $true) & (subsetTrans = $true) & (setminusELneg = $true) & (symdiffIneg1 = $true) & (vacuousDall = $true) & (bs114d = $true) & (uniqinunit = $true) & (setunionE = $true) & (powersetE1 = $true) & (dsetconstrER = $true) & (setadjoinSub = $true) & (setadjoinOr = $true) & (powersetI1 = $true) & (foundationAx = $true) & (exuI1 = $true) & (descr__Cong = $true) & (setminusIRneg = $true) & (binunionE = $true) & (prop2setE = $true) & (setext = $true) & (subsetRefl = $true) & (setminusILneg = $true) & (eqimpsubset1 = $true) & (upairsetIR = $true) & (setadjoinIL = $true) & (notequalI2 = $true) & (emptyinunitempty = $true) & (binunionLsub = $true) & (emptysetAx = $true) & (nonemptyI1 = $true) & (omega0Ax = $true) & (subsetI2 = $true) & (setunionI = $true) & (singletoninpowerset = $true) & (subset2powerset = $true) & (upairset2IR = $true) & (prop2setI = $true) & (disjointsetsI1 = $true) & (powersetsubset = $true) & (kpairiskpair = $true) & (emptyInPowerset = $true) & (inPowerset = $true) & (replAx = $true) & (notdexE = $true) & (eqimpsubset2 = $true) & (powersetAx = $true) & (subsetE2 = $true) & (setoftrueEq = $true) & (symdiffIneg2 = $true) & (cartprodpairin = $true) & (setukpairIL = $true) & (setunion__Cong = $true) & (ubforcartprodlem2 = $true) & (exuEu = $true) & (emptyset__Cong = $true) & (in__Cong = $true) & (descrp = $true) & (powersetI = $true) & (notsubsetI = $true) & (omegaIndAx = $true) & (exuE1 = $true) & (subPowSU = $true) & (binintersectRsub = $true) & (setminusEL = $true) & (exuI3 = $true) & (notequalI1 = $true) & (dsetconstrI = $true) & (notinsingleton = $true) & (binunionRsub = $true) & (cartprodmempair = $true) & (eqinunit = $true)),
% 2.37/1.57 inference(skolemisation,[status(esa),new_symbols(skolem,[sK324])],[f548,f1206])).
% 2.37/1.57 thf(f1206,plain,(
% 2.37/1.57 ? [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) != X0) => ((setunion @ (setadjoin @ sK324 @ emptyset)) != sK324)),
% 2.37/1.57 introduced(definition,[],[choice_axiom])).
% 2.37/1.57 thf(f548,plain,(
% 2.37/1.57 (setunionE2 = $true) & (setadjoinSub2 = $true) & (emptyI = $true) & (notdallE = $true) & (exuE3e = $true) & (binintersectSubset3 = $true) & (kpairp = $true) & (binunionIL = $true) & (emptyE1 = $true) & (exuE2 = $true) & (notinemptyset = $true) & (quantDeMorgan4 = $true) & (ubforcartprodlem3 = $true) & (setminusSubset2 = $true) & (binintersectLsub = $true) & (setminusSubset1 = $true) & (emptyinPowerset = $true) & (binunionEcases = $true) & (setadjoinIR = $true) & (dsetconstr__Cong = $true) & (subsetE = $true) & (setminusLsub = $true) & (setukpairIR = $true) & (exuE3u = $true) & (singletoninpowunion = $true) & (binintersectSubset2 = $true) & (upairsetIL = $true) & (symdiffI1 = $true) & (sepInPowerset = $true) & (binintersectSubset4 = $true) & (setadjoin__Cong = $true) & (cartprodmempair1 = $true) & (emptysetsubset = $true) & (exu__Cong = $true) & (singletonsubset = $true) & (subsetI1 = $true) & (setadjoinAx = $true) & (binintersectI = $true) & (setminusER = $true) & (binunionIR = $true) & (wellorderingAx = $true) & (setbeta = $true) & (powersetE = $true) & (symdiffE = $true) & (sepSubset = $true) & (setminusERneg = $true) & (setextsub = $true) & (emptysetE = $true) & (quantDeMorgan1 = $true) & (binintersectEL = $true) & (dsetconstrEL = $true) & (subsetemptysetimpeq = $true) & (singletonsswitch = $true) & (binintersectSubset5 = $true) & (quantDeMorgan2 = $true) & (upairset2E = $true) & (noeltsimpempty = $true) & (exuI2 = $true) & (prop2set2propI = $true) & (nonemptyImpWitness = $true) & (quantDeMorgan3 = $true) & (symdiffI2 = $true) & (upairinpowunion = $true) & (upairsubunion = $true) & (setadjoinE = $true) & (binintersectER = $true) & (omegaSAx = $true) & (ubforcartprodlem1 = $true) & ? [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) != X0) & (omega__Cong = $true) & (binintersectSubset1 = $true) & (setunionsingleton2 = $true) & (setunionsingleton1 = $true) & (upairsetE = $true) & (setminusI = $true) & (inCongP = $true) & (secondinupair = $true) & (emptysetimpfalse = $true) & (nonemptyI = $true) & (setunionAx = $true) & (powerset__Cong = $true) & (setextAx = $true) & (nonemptyE1 = $true) & (subsetTrans = $true) & (setminusELneg = $true) & (symdiffIneg1 = $true) & (vacuousDall = $true) & (bs114d = $true) & (uniqinunit = $true) & (setunionE = $true) & (powersetE1 = $true) & (dsetconstrER = $true) & (setadjoinSub = $true) & (setadjoinOr = $true) & (powersetI1 = $true) & (foundationAx = $true) & (exuI1 = $true) & (descr__Cong = $true) & (setminusIRneg = $true) & (binunionE = $true) & (prop2setE = $true) & (setext = $true) & (subsetRefl = $true) & (setminusILneg = $true) & (eqimpsubset1 = $true) & (upairsetIR = $true) & (setadjoinIL = $true) & (notequalI2 = $true) & (emptyinunitempty = $true) & (binunionLsub = $true) & (emptysetAx = $true) & (nonemptyI1 = $true) & (omega0Ax = $true) & (subsetI2 = $true) & (setunionI = $true) & (singletoninpowerset = $true) & (subset2powerset = $true) & (upairset2IR = $true) & (prop2setI = $true) & (disjointsetsI1 = $true) & (powersetsubset = $true) & (kpairiskpair = $true) & (emptyInPowerset = $true) & (inPowerset = $true) & (replAx = $true) & (notdexE = $true) & (eqimpsubset2 = $true) & (powersetAx = $true) & (subsetE2 = $true) & (setoftrueEq = $true) & (symdiffIneg2 = $true) & (cartprodpairin = $true) & (setukpairIL = $true) & (setunion__Cong = $true) & (ubforcartprodlem2 = $true) & (exuEu = $true) & (emptyset__Cong = $true) & (in__Cong = $true) & (descrp = $true) & (powersetI = $true) & (notsubsetI = $true) & (omegaIndAx = $true) & (exuE1 = $true) & (subPowSU = $true) & (binintersectRsub = $true) & (setminusEL = $true) & (exuI3 = $true) & (notequalI1 = $true) & (dsetconstrI = $true) & (notinsingleton = $true) & (binunionRsub = $true) & (cartprodmempair = $true) & (eqinunit = $true)),
% 2.37/1.57 inference(flattening,[status(thm)],[f547])).
% 2.37/1.57 thf(f547,plain,(
% 2.37/1.57 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) != X0) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.37/1.57 inference(ennf_transformation,[status(thm)],[f461])).
% 2.37/1.57 thf(f461,plain,(
% 2.37/1.57 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.37/1.57 inference(fool_elimination,[status(thm)],[f460])).
% 2.37/1.57 thf(f460,plain,(
% 2.37/1.57 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => ! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.37/1.57 inference(rectify,[status(thm)],[f158])).
% 2.37/1.57 thf(f158,negated_conjecture,(
% 2.37/1.57 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => ! [X1 : $i] : ((setunion @ (setadjoin @ X1 @ emptyset)) = X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.37/1.57 inference(negated_conjecture,[status(cth)],[f157])).
% 2.37/1.57 thf(f157,conjecture,(
% 2.37/1.57 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => ! [X1 : $i] : ((setunion @ (setadjoin @ X1 @ emptyset)) = X1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.37/1.57 file('/export/starexec/sandbox/tmp/tmp.EkEyqyb2Tu/DTF2THF_18509.p',setunionsingleton)).
% 2.37/1.57 thf(f1763,plain,(
% 2.37/1.57 ( ! [X1 : $i] : (($true = (subset @ X1 @ (setunion @ (setadjoin @ X1 @ emptyset)))) | (setunionsingleton2 != $true)) )),
% 2.37/1.57 inference(cnf_transformation,[status(thm)],[f1145])).
% 2.37/1.57 thf(f1145,plain,(
% 2.37/1.57 ((setunionsingleton2 = $true) | ($true != (subset @ sK289 @ (setunion @ (setadjoin @ sK289 @ emptyset))))) & (! [X1 : $i] : ($true = (subset @ X1 @ (setunion @ (setadjoin @ X1 @ emptyset)))) | (setunionsingleton2 != $true))),
% 2.37/1.57 inference(skolemisation,[status(esa),new_symbols(skolem,[sK289])],[f1143,f1144])).
% 2.37/1.57 thf(f1144,plain,(
% 2.37/1.57 ? [X0 : $i] : ($true != (subset @ X0 @ (setunion @ (setadjoin @ X0 @ emptyset)))) => ($true != (subset @ sK289 @ (setunion @ (setadjoin @ sK289 @ emptyset))))),
% 2.37/1.57 introduced(definition,[],[choice_axiom])).
% 2.37/1.57 thf(f1143,plain,(
% 2.37/1.57 ((setunionsingleton2 = $true) | ? [X0 : $i] : ($true != (subset @ X0 @ (setunion @ (setadjoin @ X0 @ emptyset))))) & (! [X1 : $i] : ($true = (subset @ X1 @ (setunion @ (setadjoin @ X1 @ emptyset)))) | (setunionsingleton2 != $true))),
% 2.37/1.57 inference(rectify,[status(thm)],[f1142])).
% 2.37/1.57 thf(f1142,plain,(
% 2.37/1.57 ((setunionsingleton2 = $true) | ? [X0 : $i] : ($true != (subset @ X0 @ (setunion @ (setadjoin @ X0 @ emptyset))))) & (! [X0 : $i] : ($true = (subset @ X0 @ (setunion @ (setadjoin @ X0 @ emptyset)))) | (setunionsingleton2 != $true))),
% 2.37/1.57 inference(nnf_transformation,[status(thm)],[f335])).
% 2.37/1.57 thf(f335,plain,(
% 2.37/1.57 (setunionsingleton2 = $true) <=> ! [X0 : $i] : ($true = (subset @ X0 @ (setunion @ (setadjoin @ X0 @ emptyset))))),
% 2.37/1.57 inference(fool_elimination,[status(thm)],[f334])).
% 2.37/1.57 thf(f334,plain,(
% 2.37/1.57 (setunionsingleton2 = ! [X0 : $i] : (subset @ X0 @ (setunion @ (setadjoin @ X0 @ emptyset))))),
% 2.37/1.57 inference(rectify,[status(thm)],[f156])).
% 2.37/1.57 thf(f156,axiom,(
% 2.37/1.57 (setunionsingleton2 = ! [X3 : $i] : (subset @ X3 @ (setunion @ (setadjoin @ X3 @ emptyset))))),
% 2.37/1.57 file('/export/starexec/sandbox/tmp/tmp.EkEyqyb2Tu/DTF2THF_18509.p',setunionsingleton2)).
% 2.37/1.57 thf(f2830,plain,(
% 2.37/1.57 ( ! [X0 : $i,X1 : $i] : (($true != (subset @ X1 @ X0)) | ((binintersect @ X0 @ X1) = X1)) )),
% 2.37/1.57 inference(trivial_inequality_removal,[status(thm)],[f2139])).
% 2.37/1.57 thf(f2139,plain,(
% 2.37/1.57 ( ! [X0 : $i,X1 : $i] : (($true != $true) | ((binintersect @ X0 @ X1) = X1) | ($true != (subset @ X1 @ X0))) )),
% 2.37/1.57 inference(definition_unfolding,[status(thm)],[f1408,f1938])).
% 2.37/1.57 thf(f1938,plain,(
% 2.37/1.57 (binintersectSubset4 = $true)),
% 2.37/1.57 inference(cnf_transformation,[status(thm)],[f1207])).
% 2.37/1.57 thf(f1408,plain,(
% 2.37/1.57 ( ! [X0 : $i,X1 : $i] : (((binintersect @ X0 @ X1) = X1) | ($true != (subset @ X1 @ X0)) | (binintersectSubset4 != $true)) )),
% 2.37/1.57 inference(cnf_transformation,[status(thm)],[f708])).
% 2.37/1.57 thf(f708,plain,(
% 2.37/1.57 (! [X0 : $i,X1 : $i] : (((binintersect @ X0 @ X1) = X1) | ($true != (subset @ X1 @ X0))) | (binintersectSubset4 != $true)) & ((binintersectSubset4 = $true) | (((binintersect @ sK30 @ sK31) != sK31) & ($true = (subset @ sK31 @ sK30))))),
% 2.37/1.57 inference(skolemisation,[status(esa),new_symbols(skolem,[sK30,sK31])],[f706,f707])).
% 2.37/1.57 thf(f707,plain,(
% 2.37/1.57 ? [X2 : $i,X3 : $i] : (((binintersect @ X2 @ X3) != X3) & ($true = (subset @ X3 @ X2))) => (((binintersect @ sK30 @ sK31) != sK31) & ($true = (subset @ sK31 @ sK30)))),
% 2.37/1.57 introduced(definition,[],[choice_axiom])).
% 2.37/1.57 thf(f706,plain,(
% 2.37/1.57 (! [X0 : $i,X1 : $i] : (((binintersect @ X0 @ X1) = X1) | ($true != (subset @ X1 @ X0))) | (binintersectSubset4 != $true)) & ((binintersectSubset4 = $true) | ? [X2 : $i,X3 : $i] : (((binintersect @ X2 @ X3) != X3) & ($true = (subset @ X3 @ X2))))),
% 2.37/1.57 inference(rectify,[status(thm)],[f705])).
% 2.37/1.57 thf(f705,plain,(
% 2.37/1.57 (! [X0 : $i,X1 : $i] : (((binintersect @ X0 @ X1) = X1) | ($true != (subset @ X1 @ X0))) | (binintersectSubset4 != $true)) & ((binintersectSubset4 = $true) | ? [X0 : $i,X1 : $i] : (((binintersect @ X0 @ X1) != X1) & ($true = (subset @ X1 @ X0))))),
% 2.37/1.57 inference(nnf_transformation,[status(thm)],[f639])).
% 2.37/1.57 thf(f639,plain,(
% 2.37/1.57 ! [X0 : $i,X1 : $i] : (((binintersect @ X0 @ X1) = X1) | ($true != (subset @ X1 @ X0))) <=> (binintersectSubset4 = $true)),
% 2.37/1.57 inference(ennf_transformation,[status(thm)],[f272])).
% 2.37/1.57 thf(f272,plain,(
% 2.37/1.57 ! [X0 : $i,X1 : $i] : (($true = (subset @ X1 @ X0)) => ((binintersect @ X0 @ X1) = X1)) <=> (binintersectSubset4 = $true)),
% 2.37/1.57 inference(fool_elimination,[status(thm)],[f271])).
% 2.37/1.57 thf(f271,plain,(
% 2.37/1.57 (binintersectSubset4 = ! [X0 : $i,X1 : $i] : ((subset @ X1 @ X0) => ((binintersect @ X0 @ X1) = X1)))),
% 2.37/1.57 inference(rectify,[status(thm)],[f117])).
% 2.37/1.57 thf(f117,axiom,(
% 2.37/1.57 (binintersectSubset4 = ! [X3 : $i,X4 : $i] : ((subset @ X4 @ X3) => ((binintersect @ X3 @ X4) = X4)))),
% 2.37/1.57 file('/export/starexec/sandbox/tmp/tmp.EkEyqyb2Tu/DTF2THF_18509.p',binintersectSubset4)).
% 2.37/1.57 thf(f3179,plain,(
% 2.37/1.57 ( ! [X0 : $i] : (((binintersect @ (setunion @ (setadjoin @ X0 @ emptyset)) @ X0) = (setunion @ (setadjoin @ X0 @ emptyset)))) )),
% 2.37/1.57 inference(trivial_inequality_removal,[status(thm)],[f3176])).
% 2.37/1.57 thf(f3176,plain,(
% 2.37/1.57 ( ! [X0 : $i] : (($true != $true) | ((binintersect @ (setunion @ (setadjoin @ X0 @ emptyset)) @ X0) = (setunion @ (setadjoin @ X0 @ emptyset)))) )),
% 2.37/1.57 inference(superposition,[status(thm)],[f2737,f2683])).
% 2.37/1.57 thf(f2683,plain,(
% 2.37/1.57 ( ! [X1 : $i] : (($true = (subset @ (setunion @ (setadjoin @ X1 @ emptyset)) @ X1))) )),
% 2.37/1.57 inference(trivial_inequality_removal,[status(thm)],[f2190])).
% 2.37/1.57 thf(f2190,plain,(
% 2.37/1.57 ( ! [X1 : $i] : (($true != $true) | ($true = (subset @ (setunion @ (setadjoin @ X1 @ emptyset)) @ X1))) )),
% 2.37/1.57 inference(definition_unfolding,[status(thm)],[f1456,f1895])).
% 2.37/1.57 thf(f1895,plain,(
% 2.37/1.57 (setunionsingleton1 = $true)),
% 2.37/1.57 inference(cnf_transformation,[status(thm)],[f1207])).
% 2.37/1.57 thf(f1456,plain,(
% 2.37/1.57 ( ! [X1 : $i] : (($true = (subset @ (setunion @ (setadjoin @ X1 @ emptyset)) @ X1)) | (setunionsingleton1 != $true)) )),
% 2.37/1.57 inference(cnf_transformation,[status(thm)],[f769])).
% 2.37/1.57 thf(f769,plain,(
% 2.37/1.57 ((setunionsingleton1 = $true) | ((subset @ (setunion @ (setadjoin @ sK65 @ emptyset)) @ sK65) != $true)) & (! [X1 : $i] : ($true = (subset @ (setunion @ (setadjoin @ X1 @ emptyset)) @ X1)) | (setunionsingleton1 != $true))),
% 2.37/1.57 inference(skolemisation,[status(esa),new_symbols(skolem,[sK65])],[f767,f768])).
% 2.37/1.57 thf(f768,plain,(
% 2.37/1.57 ? [X0 : $i] : ($true != (subset @ (setunion @ (setadjoin @ X0 @ emptyset)) @ X0)) => ((subset @ (setunion @ (setadjoin @ sK65 @ emptyset)) @ sK65) != $true)),
% 2.37/1.57 introduced(definition,[],[choice_axiom])).
% 2.37/1.57 thf(f767,plain,(
% 2.37/1.57 ((setunionsingleton1 = $true) | ? [X0 : $i] : ($true != (subset @ (setunion @ (setadjoin @ X0 @ emptyset)) @ X0))) & (! [X1 : $i] : ($true = (subset @ (setunion @ (setadjoin @ X1 @ emptyset)) @ X1)) | (setunionsingleton1 != $true))),
% 2.37/1.57 inference(rectify,[status(thm)],[f766])).
% 2.37/1.57 thf(f766,plain,(
% 2.37/1.57 ((setunionsingleton1 = $true) | ? [X0 : $i] : ($true != (subset @ (setunion @ (setadjoin @ X0 @ emptyset)) @ X0))) & (! [X0 : $i] : ($true = (subset @ (setunion @ (setadjoin @ X0 @ emptyset)) @ X0)) | (setunionsingleton1 != $true))),
% 2.37/1.57 inference(nnf_transformation,[status(thm)],[f387])).
% 2.37/1.57 thf(f387,plain,(
% 2.37/1.57 (setunionsingleton1 = $true) <=> ! [X0 : $i] : ($true = (subset @ (setunion @ (setadjoin @ X0 @ emptyset)) @ X0))),
% 2.37/1.57 inference(fool_elimination,[status(thm)],[f386])).
% 2.37/1.57 thf(f386,plain,(
% 2.37/1.57 (! [X0 : $i] : (subset @ (setunion @ (setadjoin @ X0 @ emptyset)) @ X0) = setunionsingleton1)),
% 2.37/1.57 inference(rectify,[status(thm)],[f155])).
% 2.37/1.57 thf(f155,axiom,(
% 2.37/1.57 (! [X3 : $i] : (subset @ (setunion @ (setadjoin @ X3 @ emptyset)) @ X3) = setunionsingleton1)),
% 2.37/1.57 file('/export/starexec/sandbox/tmp/tmp.EkEyqyb2Tu/DTF2THF_18509.p',setunionsingleton1)).
% 2.37/1.57 thf(f2737,plain,(
% 2.37/1.57 ( ! [X0 : $i,X1 : $i] : (($true != (subset @ X1 @ X0)) | ((binintersect @ X1 @ X0) = X1)) )),
% 2.37/1.57 inference(trivial_inequality_removal,[status(thm)],[f2406])).
% 2.37/1.57 thf(f2406,plain,(
% 2.37/1.57 ( ! [X0 : $i,X1 : $i] : (($true != $true) | ((binintersect @ X1 @ X0) = X1) | ($true != (subset @ X1 @ X0))) )),
% 2.37/1.57 inference(definition_unfolding,[status(thm)],[f1716,f1942])).
% 2.37/1.57 thf(f1942,plain,(
% 2.37/1.57 (binintersectSubset2 = $true)),
% 2.37/1.57 inference(cnf_transformation,[status(thm)],[f1207])).
% 2.37/1.57 thf(f1716,plain,(
% 2.37/1.57 ( ! [X0 : $i,X1 : $i] : (($true != (subset @ X1 @ X0)) | ((binintersect @ X1 @ X0) = X1) | (binintersectSubset2 != $true)) )),
% 2.37/1.57 inference(cnf_transformation,[status(thm)],[f1081])).
% 2.37/1.57 thf(f1081,plain,(
% 2.37/1.57 (! [X0 : $i,X1 : $i] : (($true != (subset @ X1 @ X0)) | ((binintersect @ X1 @ X0) = X1)) | (binintersectSubset2 != $true)) & ((binintersectSubset2 = $true) | (($true = (subset @ sK253 @ sK252)) & ((binintersect @ sK253 @ sK252) != sK253)))),
% 2.37/1.57 inference(skolemisation,[status(esa),new_symbols(skolem,[sK252,sK253])],[f1079,f1080])).
% 2.37/1.57 thf(f1080,plain,(
% 2.37/1.57 ? [X2 : $i,X3 : $i] : (($true = (subset @ X3 @ X2)) & ((binintersect @ X3 @ X2) != X3)) => (($true = (subset @ sK253 @ sK252)) & ((binintersect @ sK253 @ sK252) != sK253))),
% 2.37/1.57 introduced(definition,[],[choice_axiom])).
% 2.37/1.57 thf(f1079,plain,(
% 2.37/1.57 (! [X0 : $i,X1 : $i] : (($true != (subset @ X1 @ X0)) | ((binintersect @ X1 @ X0) = X1)) | (binintersectSubset2 != $true)) & ((binintersectSubset2 = $true) | ? [X2 : $i,X3 : $i] : (($true = (subset @ X3 @ X2)) & ((binintersect @ X3 @ X2) != X3)))),
% 2.37/1.57 inference(rectify,[status(thm)],[f1078])).
% 2.37/1.57 thf(f1078,plain,(
% 2.37/1.57 (! [X0 : $i,X1 : $i] : (($true != (subset @ X1 @ X0)) | ((binintersect @ X1 @ X0) = X1)) | (binintersectSubset2 != $true)) & ((binintersectSubset2 = $true) | ? [X0 : $i,X1 : $i] : (($true = (subset @ X1 @ X0)) & ((binintersect @ X1 @ X0) != X1)))),
% 2.37/1.57 inference(nnf_transformation,[status(thm)],[f632])).
% 2.37/1.57 thf(f632,plain,(
% 2.37/1.57 ! [X0 : $i,X1 : $i] : (($true != (subset @ X1 @ X0)) | ((binintersect @ X1 @ X0) = X1)) <=> (binintersectSubset2 = $true)),
% 2.37/1.57 inference(ennf_transformation,[status(thm)],[f411])).
% 2.37/1.57 thf(f411,plain,(
% 2.37/1.57 (binintersectSubset2 = $true) <=> ! [X0 : $i,X1 : $i] : (($true = (subset @ X1 @ X0)) => ((binintersect @ X1 @ X0) = X1))),
% 2.37/1.57 inference(fool_elimination,[status(thm)],[f410])).
% 2.37/1.57 thf(f410,plain,(
% 2.37/1.57 (! [X0 : $i,X1 : $i] : ((subset @ X1 @ X0) => ((binintersect @ X1 @ X0) = X1)) = binintersectSubset2)),
% 2.37/1.57 inference(rectify,[status(thm)],[f112])).
% 2.37/1.57 thf(f112,axiom,(
% 2.37/1.57 (! [X4 : $i,X3 : $i] : ((subset @ X3 @ X4) => ((binintersect @ X3 @ X4) = X3)) = binintersectSubset2)),
% 2.37/1.57 file('/export/starexec/sandbox/tmp/tmp.EkEyqyb2Tu/DTF2THF_18509.p',binintersectSubset2)).
% 2.37/1.57 thf(f1899,plain,(
% 2.37/1.57 ((setunion @ (setadjoin @ sK324 @ emptyset)) != sK324)),
% 2.37/1.57 inference(cnf_transformation,[status(thm)],[f1207])).
% 2.37/1.57 % SZS output end Proof for DTF2THF_18509
% 2.37/1.57 % (18723)------------------------------
% 2.37/1.57 % (18723)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.57 % (18723)Termination reason: Refutation
% 2.37/1.57
% 2.37/1.57 % (18723)Memory used [KB]: 8187
% 2.37/1.57 % (18723)Time elapsed: 0.090 s
% 2.37/1.57 % (18723)Instructions burned: 235 (million)
% 2.37/1.57 % (18723)------------------------------
% 2.37/1.57 % (18723)------------------------------
% 2.37/1.57 % (18701)Success in time 0.161 s
%------------------------------------------------------------------------------