%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU652^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n013.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:35:58 AM UTC 2026
% Result : Theorem 4.78s 2.03s
% Output : Refutation 4.78s
% Verified :
% SZS Type : Refutation
% Derivation depth : 16
% Number of leaves : 12
% Syntax : Number of formulae : 70 ( 20 unt; 0 typ; 2 def)
% Number of atoms : 3118 ( 839 equ; 0 cnn)
% Maximal formula atoms : 170 ( 44 avg)
% Number of connectives : 1843 ( 75 ~; 56 |; 531 &; 486 @)
% ( 8 <=>; 687 =>; 0 <=; 0 <~>)
% Maximal formula depth : 174 ( 21 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 325 ( 322 usr; 185 con; 0-4 aty)
% Number of variables : 119 ( 0 ^; 83 !; 36 ?; 119 :)
% 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_177,type,
singleton: $i > $o ).
thf(func_def_179,type,
ex1: $i > ( $i > $o ) > $o ).
thf(func_def_184,type,
atmost1p: $i > $o ).
thf(func_def_185,type,
atleast2p: $i > $o ).
thf(func_def_186,type,
atmost2p: $i > $o ).
thf(func_def_187,type,
upairsetp: $i > $o ).
thf(func_def_191,type,
kfst: $i > $i ).
thf(func_def_211,type,
sP0: $i > $i > $o ).
thf(func_def_212,type,
sP1: $i > $o ).
thf(func_def_213,type,
sP2: $i > $i > $o ).
thf(func_def_214,type,
sP3: $i > $i > $o ).
thf(func_def_216,type,
sP5: $i > $o > $i > $i > $o ).
thf(func_def_223,type,
sK12: $i > $o ).
thf(func_def_226,type,
sK15: $i > $i > ( $i > $o ) > $i ).
thf(func_def_238,type,
sK27: ( $i > $o ) > $i > $i ).
thf(func_def_240,type,
sK29: $i > $o ).
thf(func_def_242,type,
sK31: $i > $i > $i ).
thf(func_def_243,type,
sK32: $i > $i > $i ).
thf(func_def_244,type,
sK33: $i > $i > $i ).
thf(func_def_245,type,
sK34: $i > $i > $i ).
thf(func_def_246,type,
sK35: $i > $i > $i > $i ).
thf(func_def_247,type,
sK36: $i > $i ).
thf(func_def_248,type,
sK37: $i > $i ).
thf(func_def_249,type,
sK38: $i > $i ).
thf(func_def_250,type,
sK39: $i > $i ).
thf(func_def_251,type,
sK40: $i > $i > $i ).
thf(func_def_252,type,
sK41: $i > $i > $i > $i ).
thf(func_def_253,type,
sK42: $i > $i > $i > $i ).
thf(func_def_254,type,
sK43: $i > $i > $i ).
thf(func_def_255,type,
sK44: $i > $i > $i ).
thf(func_def_256,type,
sK45: $i > $i ).
thf(func_def_257,type,
sK46: $i > $i > $i ).
thf(func_def_259,type,
sK48: $i > $i ).
thf(func_def_260,type,
sK49: $i > $i ).
thf(func_def_267,type,
sK56: $i > $o ).
thf(func_def_275,type,
sK64: $i > $i > $i ).
thf(func_def_279,type,
sK68: $i > $o ).
thf(func_def_288,type,
sK77: $i > $i ).
thf(func_def_297,type,
sK86: ( $i > $o ) > $i ).
thf(func_def_298,type,
sK87: ( $i > $o ) > $i ).
thf(func_def_299,type,
sK88: $i > $o ).
thf(func_def_302,type,
sK91: $i > $i ).
thf(func_def_306,type,
sK95: ( $i > $o ) > $i > $i ).
thf(func_def_308,type,
sK97: $i > $o ).
thf(func_def_324,type,
sK113: $i > $i ).
thf(func_def_331,type,
sK120: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_332,type,
sK121: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_333,type,
sK122: $i > $o ).
thf(func_def_334,type,
sK123: $i > $o ).
thf(func_def_337,type,
sK126: $i > $i > $i ).
thf(func_def_343,type,
sK132: $i > $o ).
thf(func_def_345,type,
sK134: $i > $o ).
thf(func_def_347,type,
sK136: ( $i > $o ) > $i > $i ).
thf(func_def_359,type,
sK148: $i > $i ).
thf(func_def_362,type,
sK151: $i > $i ).
thf(func_def_372,type,
sK161: $i > $i > $i ).
thf(func_def_379,type,
sK168: $i > $i > $i ).
thf(func_def_391,type,
sK180: $i > $i > $i ).
thf(func_def_412,type,
sK201: $i > $i > $i > $i ).
thf(func_def_413,type,
sK202: $i > $i > $i > $i ).
thf(func_def_414,type,
sK203: $i > $o ).
thf(func_def_415,type,
sK204: $i > $i ).
thf(func_def_416,type,
sK205: ( $i > $o ) > $i ).
thf(func_def_418,type,
sK207: $i > $o ).
thf(func_def_423,type,
sK212: $i > $o ).
thf(func_def_431,type,
sK220: $i > $i > $o ).
thf(func_def_433,type,
sK222: $i > $i ).
thf(func_def_434,type,
sK223: $i > $i ).
thf(func_def_435,type,
sK224: $i > ( $i > $i > $o ) > $i ).
thf(func_def_436,type,
sK225: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_437,type,
sK226: $i > ( $i > $i > $o ) > $i ).
thf(func_def_443,type,
sK232: $i > $i > $i ).
thf(func_def_444,type,
sK233: $i > ( $i > $o ) > $i ).
thf(func_def_445,type,
sK234: $i > $o ).
thf(func_def_447,type,
sK236: $i > $o ).
thf(func_def_460,type,
sK249: $i > $o ).
thf(func_def_466,type,
sK255: $i > $o ).
thf(func_def_488,type,
sK277: $i > $i ).
thf(func_def_492,type,
sK281: ( $i > $o ) > $i > $i ).
thf(func_def_494,type,
sK283: $i > $o ).
thf(func_def_495,type,
sK284: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_496,type,
sK285: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_499,type,
sK288: $i > $o ).
thf(func_def_500,type,
sK289: $i > $o ).
thf(func_def_508,type,
sK297: $i > $o ).
thf(func_def_511,type,
sK300: $i > ( $i > $o ) > $i ).
thf(func_def_512,type,
sK301: $i > ( $i > $o ) > $i ).
thf(func_def_513,type,
sK302: $i > ( $i > $o ) > $i ).
thf(func_def_514,type,
sK303: $i > $o ).
thf(func_def_521,type,
sK310: $i > $i ).
thf(func_def_524,type,
sK313: $i > $o ).
thf(func_def_525,type,
sK314: $i > $o ).
thf(func_def_526,type,
sK315: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_527,type,
sK316: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_529,type,
sK318: $i > $o ).
thf(func_def_530,type,
sK319: ( $i > $o ) > $i ).
thf(func_def_532,type,
sK321: ( $i > $o ) > $i > $i ).
thf(func_def_534,type,
sK323: $i > $o ).
thf(func_def_545,type,
sK334: ( $i > $o ) > $i ).
thf(func_def_546,type,
sK335: $i > $o ).
thf(func_def_547,type,
sK336: $i > $i ).
thf(func_def_560,type,
sK349: $i > $i > $i ).
thf(func_def_561,type,
sK350: $i > $i > $i ).
thf(func_def_575,type,
sK364: $i > $o ).
thf(func_def_589,type,
sK378: $i > $o ).
thf(func_def_594,type,
sK383: $i > $o ).
thf(func_def_608,type,
sK397: ( $i > $o ) > $i > $i ).
thf(func_def_610,type,
sK399: $i > $o ).
thf(func_def_614,type,
sK403: $i > $o ).
thf(func_def_641,type,
sK430: $i > $o ).
thf(func_def_643,type,
sK432: ( $i > $o ) > $i > $i ).
thf(func_def_644,type,
sK433: ( $i > $o ) > $i > $i ).
thf(func_def_652,type,
sK441: $i > $i > $i ).
thf(func_def_655,type,
sK444: $o > $i > $i > $i ).
thf(func_def_666,type,
sK455: $i > $i > $i ).
thf(func_def_677,type,
sK466: ( $i > $o ) > $i > $i ).
thf(func_def_679,type,
sK468: $i > $o ).
thf(func_def_683,type,
ph472:
!>[X0: $tType] : X0 ).
thf(f4350,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f2733,f4337,f4349]) ).
thf(f4349,plain,
( ~ spl471_6
| ~ spl471_3 ),
inference(avatar_split_clause,[status(thm)],[f4340,f2570,f2634]) ).
thf(f2634,definition,
( spl471_6
<=> ( sK271 = sK272 ) ),
introduced(definition,[new_symbols(naming,[spl471_6])],[avatar_definition]) ).
thf(f2570,definition,
( spl471_3
<=> ( sK272 = sK273 ) ),
introduced(definition,[new_symbols(naming,[spl471_3])],[avatar_definition]) ).
thf(f4340,plain,
( ( sK271 != sK272 )
| ~ spl471_3 ),
inference(superposition,[status(thm)],[f1981,f2572]) ).
thf(f2572,plain,
( ( sK272 = sK273 )
| ~ spl471_3 ),
inference(avatar_component_clause,[status(thm)],[f2570]) ).
thf(f1981,plain,
sK271 != sK273,
inference(cnf_transformation,[status(thm)],[f1176]) ).
thf(f1176,plain,
( ( setukpairinjL = $true )
& ( setadjoinSub2 = $true )
& ( singletonprop = $true )
& ( setminusIRneg = $true )
& ( emptyI = $true )
& ( subsetI1 = $true )
& ( ubforcartprodlem1 = $true )
& ( setukpairinjR1 = $true )
& ( emptysetE = $true )
& ( descrp = $true )
& ( ex1E1 = $true )
& ( upairinpowunion = $true )
& ( binunionIR = $true )
& ( powersetAx = $true )
& ( subsetI2 = $true )
& ( binintersectSubset5 = $true )
& ( exuE1 = $true )
& ( singletonsswitch = $true )
& ( eqimpsubset2 = $true )
& ( prop2setI = $true )
& ( subsetE2 = $true )
& ( setoftrueEq = $true )
& ( binintersectSubset1 = $true )
& ( singletoninpowerset = $true )
& ( setminusELneg = $true )
& ( setadjoinOr = $true )
& ( singletonsubset = $true )
& ( binintersectER = $true )
& ( exuE3e = $true )
& ( setunionE2 = $true )
& ( emptysetAx = $true )
& ( setunionsingleton = $true )
& ( dsetconstr__Cong = $true )
& ( subPowSU = $true )
& ( exuI3 = $true )
& ( cartprodmempair = $true )
& ( subsetRefl = $true )
& ( setminusEL = $true )
& ( setminusILneg = $true )
& ( upairsetE = $true )
& ( kfstpairEq = $true )
& ( setunionAx = $true )
& ( symdiffIneg2 = $true )
& ( emptyset__Cong = $true )
& ( kpairp = $true )
& ( setukpairinjL1 = $true )
& ( omegaSAx = $true )
& ( setminusSubset2 = $true )
& ( setminusLsub = $true )
& ( dsetconstrER = $true )
& ( emptyinPowerset = $true )
& ( replAx = $true )
& ( setunionE = $true )
& ( quantDeMorgan2 = $true )
& ( secondinupair = $true )
& ( exuI2 = $true )
& ( omega0Ax = $true )
& ( ubforcartprodlem2 = $true )
& ( powersetI1 = $true )
& ( singletoninpowunion = $true )
& ( setunionsingleton1 = $true )
& ( notequalI1 = $true )
& ( notinemptyset = $true )
& ( emptyE1 = $true )
& ( upairset2E = $true )
& ( singletonsuniq = $true )
& ( emptyinunitempty = $true )
& ( symdiffIneg1 = $true )
& ( symdiffE = $true )
& ( dsetconstrI = $true )
& ( ex1I2 = $true )
& ( disjointsetsI1 = $true )
& ( kfstsingleton = $true )
& ( setadjoinIL = $true )
& ( binintersectEL = $true )
& ( setukpairinjR11 = $true )
& ( setadjoinSub = $true )
& ( eqimpsubset1 = $true )
& ( exuEu = $true )
& ( notdallE = $true )
& ( prop2setE = $true )
& ( upairsetIL = $true )
& ( ( setadjoin @ sK272 @ emptyset )
= ( setadjoin @ sK273 @ ( setadjoin @ sK271 @ emptyset ) ) )
& ( sK271 != sK273 )
& ( powersetI = $true )
& ( exu__Cong = $true )
& ( powersetsubset = $true )
& ( upairsetIR = $true )
& ( cartprodpairin = $true )
& ( quantDeMorgan4 = $true )
& ( symdiffI2 = $true )
& ( setext = $true )
& ( setunionsingleton2 = $true )
& ( notequalI2 = $true )
& ( emptysetsubset = $true )
& ( binunionRsub = $true )
& ( subset2powerset = $true )
& ( powersetE = $true )
& ( nonemptyE1 = $true )
& ( cartprodfstin = $true )
& ( subsetTrans = $true )
& ( setminusI = $true )
& ( setukpairIL = $true )
& ( quantDeMorgan3 = $true )
& ( omegaIndAx = $true )
& ( symdiffI1 = $true )
& ( subsetE = $true )
& ( dsetconstrEL = $true )
& ( setminusERneg = $true )
& ( setminusER = $true )
& ( descr__Cong = $true )
& ( setukpairinjR12 = $true )
& ( exuE2 = $true )
& ( eqinunit = $true )
& ( setextAx = $true )
& ( ubforcartprodlem3 = $true )
& ( binintersectSubset2 = $true )
& ( setunion__Cong = $true )
& ( inCongP = $true )
& ( binintersectSubset3 = $true )
& ( setadjoinAx = $true )
& ( setadjoinIR = $true )
& ( binintersectLsub = $true )
& ( subsetemptysetimpeq = $true )
& ( powersetE1 = $true )
& ( kpairiskpair = $true )
& ( exuE3u = $true )
& ( exuI1 = $true )
& ( cartprodmempair1 = $true )
& ( sepSubset = $true )
& ( inPowerset = $true )
& ( binunionEcases = $true )
& ( in__Cong = $true )
& ( foundationAx = $true )
& ( binunionIL = $true )
& ( emptyInPowerset = $true )
& ( prop2set2propI = $true )
& ( omega__Cong = $true )
& ( wellorderingAx = $true )
& ( emptysetimpfalse = $true )
& ( theprop = $true )
& ( binunionLsub = $true )
& ( notsubsetI = $true )
& ( binunionE = $true )
& ( notinsingleton = $true )
& ( powerset__Cong = $true )
& ( noeltsimpempty = $true )
& ( setadjoin__Cong = $true )
& ( ex1I = $true )
& ( upairset2IR = $true )
& ( bs114d = $true )
& ( vacuousDall = $true )
& ( setukpairinjL2 = $true )
& ( setadjoinE = $true )
& ( sepInPowerset = $true )
& ( binintersectI = $true )
& ( setbeta = $true )
& ( setukpairIR = $true )
& ( quantDeMorgan1 = $true )
& ( nonemptyImpWitness = $true )
& ( nonemptyI = $true )
& ( setextsub = $true )
& ( uniqinunit = $true )
& ( upairsubunion = $true )
& ( binintersectSubset4 = $true )
& ( setminusSubset1 = $true )
& ( setunionI = $true )
& ( notdexE = $true )
& ( nonemptyI1 = $true )
& ( binintersectRsub = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK271,sK272,sK273])],[f611,f1175]) ).
thf(f1175,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ ( setadjoin @ X0 @ emptyset ) ) )
& ( X0 != X2 ) )
=> ( ( ( setadjoin @ sK272 @ emptyset )
= ( setadjoin @ sK273 @ ( setadjoin @ sK271 @ emptyset ) ) )
& ( sK271 != sK273 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f611,plain,
( ( setukpairinjL = $true )
& ( setadjoinSub2 = $true )
& ( singletonprop = $true )
& ( setminusIRneg = $true )
& ( emptyI = $true )
& ( subsetI1 = $true )
& ( ubforcartprodlem1 = $true )
& ( setukpairinjR1 = $true )
& ( emptysetE = $true )
& ( descrp = $true )
& ( ex1E1 = $true )
& ( upairinpowunion = $true )
& ( binunionIR = $true )
& ( powersetAx = $true )
& ( subsetI2 = $true )
& ( binintersectSubset5 = $true )
& ( exuE1 = $true )
& ( singletonsswitch = $true )
& ( eqimpsubset2 = $true )
& ( prop2setI = $true )
& ( subsetE2 = $true )
& ( setoftrueEq = $true )
& ( binintersectSubset1 = $true )
& ( singletoninpowerset = $true )
& ( setminusELneg = $true )
& ( setadjoinOr = $true )
& ( singletonsubset = $true )
& ( binintersectER = $true )
& ( exuE3e = $true )
& ( setunionE2 = $true )
& ( emptysetAx = $true )
& ( setunionsingleton = $true )
& ( dsetconstr__Cong = $true )
& ( subPowSU = $true )
& ( exuI3 = $true )
& ( cartprodmempair = $true )
& ( subsetRefl = $true )
& ( setminusEL = $true )
& ( setminusILneg = $true )
& ( upairsetE = $true )
& ( kfstpairEq = $true )
& ( setunionAx = $true )
& ( symdiffIneg2 = $true )
& ( emptyset__Cong = $true )
& ( kpairp = $true )
& ( setukpairinjL1 = $true )
& ( omegaSAx = $true )
& ( setminusSubset2 = $true )
& ( setminusLsub = $true )
& ( dsetconstrER = $true )
& ( emptyinPowerset = $true )
& ( replAx = $true )
& ( setunionE = $true )
& ( quantDeMorgan2 = $true )
& ( secondinupair = $true )
& ( exuI2 = $true )
& ( omega0Ax = $true )
& ( ubforcartprodlem2 = $true )
& ( powersetI1 = $true )
& ( singletoninpowunion = $true )
& ( setunionsingleton1 = $true )
& ( notequalI1 = $true )
& ( notinemptyset = $true )
& ( emptyE1 = $true )
& ( upairset2E = $true )
& ( singletonsuniq = $true )
& ( emptyinunitempty = $true )
& ( symdiffIneg1 = $true )
& ( symdiffE = $true )
& ( dsetconstrI = $true )
& ( ex1I2 = $true )
& ( disjointsetsI1 = $true )
& ( kfstsingleton = $true )
& ( setadjoinIL = $true )
& ( binintersectEL = $true )
& ( setukpairinjR11 = $true )
& ( setadjoinSub = $true )
& ( eqimpsubset1 = $true )
& ( exuEu = $true )
& ( notdallE = $true )
& ( prop2setE = $true )
& ( upairsetIL = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ ( setadjoin @ X0 @ emptyset ) ) )
& ( X0 != X2 ) )
& ( powersetI = $true )
& ( exu__Cong = $true )
& ( powersetsubset = $true )
& ( upairsetIR = $true )
& ( cartprodpairin = $true )
& ( quantDeMorgan4 = $true )
& ( symdiffI2 = $true )
& ( setext = $true )
& ( setunionsingleton2 = $true )
& ( notequalI2 = $true )
& ( emptysetsubset = $true )
& ( binunionRsub = $true )
& ( subset2powerset = $true )
& ( powersetE = $true )
& ( nonemptyE1 = $true )
& ( cartprodfstin = $true )
& ( subsetTrans = $true )
& ( setminusI = $true )
& ( setukpairIL = $true )
& ( quantDeMorgan3 = $true )
& ( omegaIndAx = $true )
& ( symdiffI1 = $true )
& ( subsetE = $true )
& ( dsetconstrEL = $true )
& ( setminusERneg = $true )
& ( setminusER = $true )
& ( descr__Cong = $true )
& ( setukpairinjR12 = $true )
& ( exuE2 = $true )
& ( eqinunit = $true )
& ( setextAx = $true )
& ( ubforcartprodlem3 = $true )
& ( binintersectSubset2 = $true )
& ( setunion__Cong = $true )
& ( inCongP = $true )
& ( binintersectSubset3 = $true )
& ( setadjoinAx = $true )
& ( setadjoinIR = $true )
& ( binintersectLsub = $true )
& ( subsetemptysetimpeq = $true )
& ( powersetE1 = $true )
& ( kpairiskpair = $true )
& ( exuE3u = $true )
& ( exuI1 = $true )
& ( cartprodmempair1 = $true )
& ( sepSubset = $true )
& ( inPowerset = $true )
& ( binunionEcases = $true )
& ( in__Cong = $true )
& ( foundationAx = $true )
& ( binunionIL = $true )
& ( emptyInPowerset = $true )
& ( prop2set2propI = $true )
& ( omega__Cong = $true )
& ( wellorderingAx = $true )
& ( emptysetimpfalse = $true )
& ( theprop = $true )
& ( binunionLsub = $true )
& ( notsubsetI = $true )
& ( binunionE = $true )
& ( notinsingleton = $true )
& ( powerset__Cong = $true )
& ( noeltsimpempty = $true )
& ( setadjoin__Cong = $true )
& ( ex1I = $true )
& ( upairset2IR = $true )
& ( bs114d = $true )
& ( vacuousDall = $true )
& ( setukpairinjL2 = $true )
& ( setadjoinE = $true )
& ( sepInPowerset = $true )
& ( binintersectI = $true )
& ( setbeta = $true )
& ( setukpairIR = $true )
& ( quantDeMorgan1 = $true )
& ( nonemptyImpWitness = $true )
& ( nonemptyI = $true )
& ( setextsub = $true )
& ( uniqinunit = $true )
& ( upairsubunion = $true )
& ( binintersectSubset4 = $true )
& ( setminusSubset1 = $true )
& ( setunionI = $true )
& ( notdexE = $true )
& ( nonemptyI1 = $true )
& ( binintersectRsub = $true ) ),
inference(flattening,[status(thm)],[f610]) ).
thf(f610,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ ( setadjoin @ X0 @ emptyset ) ) )
& ( X0 != X2 ) )
& ( setukpairinjR1 = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjL2 = $true )
& ( cartprodfstin = $true )
& ( kfstpairEq = $true )
& ( theprop = $true )
& ( kfstsingleton = $true )
& ( setukpairinjL1 = $true )
& ( singletonsuniq = $true )
& ( ex1I2 = $true )
& ( ex1I = $true )
& ( ex1E1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton = $true )
& ( 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)],[f280]) ).
thf(f280,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 )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ ( setadjoin @ X0 @ emptyset ) ) )
=> ( X0 = X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f279]) ).
thf(f279,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
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ ( setadjoin @ X0 @ emptyset ) ) )
=> ( X0 = X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f176]) ).
thf(f176,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
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ! [X2: $i,X8: $i,X1: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X8 @ emptyset ) )
=> ( X1 = X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f175]) ).
thf(f175,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
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ! [X2: $i,X8: $i,X1: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X8 @ emptyset ) )
=> ( X1 = X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.19e6KiooJw/DTF2THF_26384.p',upairequniteq) ).
thf(f4337,plain,
spl471_3,
inference(avatar_split_clause,[status(thm)],[f4336,f2570]) ).
thf(f4336,plain,
sK272 = sK273,
inference(trivial_inequality_removal,[status(thm)],[f4335]) ).
thf(f4335,plain,
( ( sK272 = sK273 )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f4312,f2010]) ).
thf(f2010,plain,
secondinupair = $true,
inference(cnf_transformation,[status(thm)],[f1176]) ).
thf(f4312,plain,
( ( sK272 = sK273 )
| ( secondinupair != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4301]) ).
thf(f4301,plain,
( ( $true != $true )
| ( sK272 = sK273 )
| ( secondinupair != $true ) ),
inference(superposition,[status(thm)],[f2522,f2181]) ).
thf(f2181,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) )
= $true )
| ( secondinupair != $true ) ),
inference(cnf_transformation,[status(thm)],[f1319]) ).
thf(f1319,plain,
( ( ! [X0: $i,X1: $i] :
( ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) )
= $true )
| ( secondinupair != $true ) )
& ( ( secondinupair = $true )
| ( ( in @ sK352 @ ( setadjoin @ sK351 @ ( setadjoin @ sK352 @ emptyset ) ) )
!= $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK351,sK352])],[f1317,f1318]) ).
thf(f1318,plain,
( ? [X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) )
!= $true )
=> ( ( in @ sK352 @ ( setadjoin @ sK351 @ ( setadjoin @ sK352 @ emptyset ) ) )
!= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1317,plain,
( ( ! [X0: $i,X1: $i] :
( ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) )
= $true )
| ( secondinupair != $true ) )
& ( ( secondinupair = $true )
| ? [X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) )
!= $true ) ) ),
inference(rectify,[status(thm)],[f1316]) ).
thf(f1316,plain,
( ( ! [X0: $i,X1: $i] :
( ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) )
= $true )
| ( secondinupair != $true ) )
& ( ( secondinupair = $true )
| ? [X0: $i,X1: $i] :
( ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) )
!= $true ) ) ),
inference(nnf_transformation,[status(thm)],[f191]) ).
thf(f191,plain,
( ! [X0: $i,X1: $i] :
( ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) )
= $true )
<=> ( secondinupair = $true ) ),
inference(fool_elimination,[status(thm)],[f190]) ).
thf(f190,plain,
( secondinupair
= ( ! [X0: $i,X1: $i] : ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) ) ),
inference(rectify,[status(thm)],[f136]) ).
thf(f136,axiom,
( secondinupair
= ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.19e6KiooJw/DTF2THF_26384.p',secondinupair) ).
thf(f2522,plain,
! [X0: $i] :
( ( ( in @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK273 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK272 @ emptyset ) @ emptyset ) ) )
!= $true )
| ( sK273 = X0 ) ),
inference(trivial_inequality_removal,[status(thm)],[f2521]) ).
thf(f2521,plain,
! [X0: $i] :
( ( sK273 = X0 )
| ( ( in @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK273 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK272 @ emptyset ) @ emptyset ) ) )
!= $true )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f2467,f2019]) ).
thf(f2019,plain,
setukpairinjL1 = $true,
inference(cnf_transformation,[status(thm)],[f1176]) ).
thf(f2467,plain,
! [X0: $i] :
( ( sK273 = X0 )
| ( ( in @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK273 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK272 @ emptyset ) @ emptyset ) ) )
!= $true )
| ( setukpairinjL1 != $true ) ),
inference(superposition,[status(thm)],[f1717,f1982]) ).
thf(f1982,plain,
( ( setadjoin @ sK272 @ emptyset )
= ( setadjoin @ sK273 @ ( setadjoin @ sK271 @ emptyset ) ) ),
inference(cnf_transformation,[status(thm)],[f1176]) ).
thf(f1717,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( in @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) )
!= $true )
| ( X0 = X2 )
| ( setukpairinjL1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f967]) ).
thf(f967,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( X0 = X2 )
| ( ( in @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) )
!= $true ) )
| ( setukpairinjL1 != $true ) )
& ( ( setukpairinjL1 = $true )
| ( ( sK143 != sK141 )
& ( ( in @ ( setadjoin @ sK141 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK143 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK143 @ ( setadjoin @ sK142 @ emptyset ) ) @ emptyset ) ) )
= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK141,sK142,sK143])],[f965,f966]) ).
thf(f966,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( X3 != X5 )
& ( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X5 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X5 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
= $true ) )
=> ( ( sK143 != sK141 )
& ( ( in @ ( setadjoin @ sK141 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK143 @ emptyset ) @ ( setadjoin @ ( setadjoin @ sK143 @ ( setadjoin @ sK142 @ emptyset ) ) @ emptyset ) ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f965,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( X0 = X2 )
| ( ( in @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) )
!= $true ) )
| ( setukpairinjL1 != $true ) )
& ( ( setukpairinjL1 = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( X3 != X5 )
& ( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X5 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X5 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
= $true ) ) ) ),
inference(rectify,[status(thm)],[f964]) ).
thf(f964,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( X0 = X2 )
| ( ( in @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) )
!= $true ) )
| ( setukpairinjL1 != $true ) )
& ( ( setukpairinjL1 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( X0 != X2 )
& ( ( in @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) )
= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f667]) ).
thf(f667,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( X0 = X2 )
| ( ( in @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) )
!= $true ) )
<=> ( setukpairinjL1 = $true ) ),
inference(ennf_transformation,[status(thm)],[f195]) ).
thf(f195,plain,
( ( setukpairinjL1 = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) )
= $true )
=> ( X0 = X2 ) ) ),
inference(fool_elimination,[status(thm)],[f194]) ).
thf(f194,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ ( setadjoin @ X0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X1 @ emptyset ) ) @ emptyset ) ) )
=> ( X0 = X2 ) )
= setukpairinjL1 ),
inference(rectify,[status(thm)],[f165]) ).
thf(f165,axiom,
! [X8: $i,X2: $i,X1: $i] :
( ( ( in @ ( setadjoin @ X8 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X8 ) )
= setukpairinjL1 ),
file('/export/starexec/sandbox/tmp/tmp.19e6KiooJw/DTF2THF_26384.p',setukpairinjL1) ).
thf(f2733,plain,
spl471_6,
inference(avatar_split_clause,[status(thm)],[f2732,f2634]) ).
thf(f2732,plain,
sK271 = sK272,
inference(trivial_inequality_removal,[status(thm)],[f2731]) ).
thf(f2731,plain,
( ( sK271 = sK272 )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f2715,f1902]) ).
thf(f1902,plain,
uniqinunit = $true,
inference(cnf_transformation,[status(thm)],[f1176]) ).
thf(f2715,plain,
( ( sK271 = sK272 )
| ( uniqinunit != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f2708]) ).
thf(f2708,plain,
( ( sK271 = sK272 )
| ( uniqinunit != $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f1891,f2539]) ).
thf(f2539,plain,
( ( in @ sK271 @ ( setadjoin @ sK272 @ emptyset ) )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f2538]) ).
thf(f2538,plain,
( ( ( in @ sK271 @ ( setadjoin @ sK272 @ emptyset ) )
= $true )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f2478,f1915]) ).
thf(f1915,plain,
upairset2IR = $true,
inference(cnf_transformation,[status(thm)],[f1176]) ).
thf(f2478,plain,
( ( ( in @ sK271 @ ( setadjoin @ sK272 @ emptyset ) )
= $true )
| ( upairset2IR != $true ) ),
inference(superposition,[status(thm)],[f2268,f1982]) ).
thf(f2268,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
= $true )
| ( upairset2IR != $true ) ),
inference(cnf_transformation,[status(thm)],[f1436]) ).
thf(f1436,plain,
( ( ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
= $true )
| ( upairset2IR != $true ) )
& ( ( upairset2IR = $true )
| ( ( in @ sK417 @ ( setadjoin @ sK418 @ ( setadjoin @ sK417 @ emptyset ) ) )
!= $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK417,sK418])],[f1434,f1435]) ).
thf(f1435,plain,
( ? [X2: $i,X3: $i] :
( ( in @ X2 @ ( setadjoin @ X3 @ ( setadjoin @ X2 @ emptyset ) ) )
!= $true )
=> ( ( in @ sK417 @ ( setadjoin @ sK418 @ ( setadjoin @ sK417 @ emptyset ) ) )
!= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1434,plain,
( ( ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
= $true )
| ( upairset2IR != $true ) )
& ( ( upairset2IR = $true )
| ? [X2: $i,X3: $i] :
( ( in @ X2 @ ( setadjoin @ X3 @ ( setadjoin @ X2 @ emptyset ) ) )
!= $true ) ) ),
inference(rectify,[status(thm)],[f1433]) ).
thf(f1433,plain,
( ( ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
= $true )
| ( upairset2IR != $true ) )
& ( ( upairset2IR = $true )
| ? [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
!= $true ) ) ),
inference(nnf_transformation,[status(thm)],[f266]) ).
thf(f266,plain,
( ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) )
= $true )
<=> ( upairset2IR = $true ) ),
inference(fool_elimination,[status(thm)],[f265]) ).
thf(f265,plain,
( upairset2IR
= ( ! [X0: $i,X1: $i] : ( in @ X0 @ ( setadjoin @ X1 @ ( setadjoin @ X0 @ emptyset ) ) ) ) ),
inference(rectify,[status(thm)],[f102]) ).
thf(f102,axiom,
( upairset2IR
= ( ! [X2: $i,X1: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.19e6KiooJw/DTF2THF_26384.p',upairset2IR) ).
thf(f1891,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) )
!= $true )
| ( X0 = X1 )
| ( uniqinunit != $true ) ),
inference(cnf_transformation,[status(thm)],[f1170]) ).
thf(f1170,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) )
!= $true )
| ( X0 = X1 ) )
| ( uniqinunit != $true ) )
& ( ( uniqinunit = $true )
| ( ( ( in @ sK268 @ ( setadjoin @ sK267 @ emptyset ) )
= $true )
& ( sK267 != sK268 ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK267,sK268])],[f1168,f1169]) ).
thf(f1169,plain,
( ? [X2: $i,X3: $i] :
( ( ( in @ X3 @ ( setadjoin @ X2 @ emptyset ) )
= $true )
& ( X2 != X3 ) )
=> ( ( ( in @ sK268 @ ( setadjoin @ sK267 @ emptyset ) )
= $true )
& ( sK267 != sK268 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1168,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) )
!= $true )
| ( X0 = X1 ) )
| ( uniqinunit != $true ) )
& ( ( uniqinunit = $true )
| ? [X2: $i,X3: $i] :
( ( ( in @ X3 @ ( setadjoin @ X2 @ emptyset ) )
= $true )
& ( X2 != X3 ) ) ) ),
inference(rectify,[status(thm)],[f1167]) ).
thf(f1167,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) )
!= $true )
| ( X0 = X1 ) )
| ( uniqinunit != $true ) )
& ( ( uniqinunit = $true )
| ? [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) )
= $true )
& ( X0 != X1 ) ) ) ),
inference(nnf_transformation,[status(thm)],[f695]) ).
thf(f695,plain,
( ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) )
!= $true )
| ( X0 = X1 ) )
<=> ( uniqinunit = $true ) ),
inference(ennf_transformation,[status(thm)],[f378]) ).
thf(f378,plain,
( ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) )
= $true )
=> ( X0 = X1 ) )
<=> ( uniqinunit = $true ) ),
inference(fool_elimination,[status(thm)],[f377]) ).
thf(f377,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) )
=> ( X0 = X1 ) )
= uniqinunit ),
inference(rectify,[status(thm)],[f46]) ).
thf(f46,axiom,
! [X2: $i,X1: $i] :
( ( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
= uniqinunit ),
file('/export/starexec/sandbox/tmp/tmp.19e6KiooJw/DTF2THF_26384.p',uniqinunit) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU652^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.18/0.34 % Computer : n013.cluster.edu
% 0.18/0.34 % Model : x86_64 x86_64
% 0.18/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.34 % Memory : 8042.1875MB
% 0.18/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.18/0.34 % CPULimit : 300
% 0.18/0.34 % WCLimit : 300
% 0.18/0.34 % DateTime : Tue Feb 24 12:29:23 EST 2026
% 0.18/0.34 % CPUTime :
% 0.18/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.34/0.49 ---- Original DTF file ---
% 0.34/0.49 thf(spec,logic,$$dhol).
% 0.34/0.49 %------------------------------------------------------------------------------
% 0.34/0.49 % File : SEU652^1 : TPTP v9.2.1. Released v3.7.0.
% 0.34/0.49 % Domain : Set Theory
% 0.34/0.49 % Problem : Ordered Pairs - Properties of Pairs
% 0.34/0.49 % Version : Especial.
% 0.34/0.49 % English : (! x:i.! y:i.! z:i.setadjoin x (setadjoin y emptyset) =
% 0.34/0.49 % setadjoin z emptyset -> x = y)
% 0.34/0.49
% 0.34/0.49 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.34/0.49 % Source : [Bro08]
% 0.34/0.49 % Names : ZFC154g [Bro08]
% 0.34/0.49
% 0.34/0.49 % Status : Theorem
% 0.34/0.49 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0, 0.29 v7.4.0, 0.33 v7.2.0, 0.25 v7.1.0, 0.50 v7.0.0, 0.57 v6.4.0, 0.67 v6.3.0, 0.80 v6.2.0, 0.57 v5.5.0, 0.67 v5.4.0, 0.80 v4.1.0, 1.00 v3.7.0
% 0.34/0.49 % Syntax : Number of formulae : 373 ( 174 unt; 198 typ; 174 def)
% 0.34/0.49 % Number of atoms : 922 ( 257 equ; 0 cnn)
% 0.34/0.49 % Maximal formula atoms : 170 ( 5 avg)
% 0.34/0.49 % Number of connectives : 1627 ( 47 ~; 7 |; 37 &;1115 @)
% 0.34/0.49 % ( 14 <=>; 407 =>; 0 <=; 0 <~>)
% 0.34/0.49 % Maximal formula depth : 173 ( 2 avg)
% 0.34/0.49 % Number of types : 2 ( 0 usr)
% 0.34/0.49 % Number of type conns : 83 ( 83 >; 0 *; 0 +; 0 <<)
% 0.34/0.49 % Number of symbols : 201 ( 198 usr; 172 con; 0-2 aty)
% 0.34/0.49 % Number of variables : 507 ( 42 ^; 427 !; 38 ?; 507 :)
% 0.34/0.49 % SPC : TH0_THM_EQU_NAR
% 0.34/0.49
% 0.34/0.49 % Comments : http://mathgate.info/detsetitem.php?id=208
% 0.34/0.49 %------------------------------------------------------------------------------
% 0.34/0.49 thf(in_type,type,
% 0.34/0.49 in: $i > $i > $o ).
% 0.34/0.49
% 0.34/0.49 thf(exu_type,type,
% 0.34/0.49 exu: ( $i > $o ) > $o ).
% 0.34/0.49
% 0.34/0.49 thf(exu,definition,
% 0.34/0.49 ( exu
% 0.34/0.49 = ( ^ [Xphi: $i > $o] :
% 0.34/0.49 ? [Xx: $i] :
% 0.34/0.49 ( ( Xphi @ Xx )
% 0.34/0.49 & ! [Xy: $i] :
% 0.34/0.49 ( ( Xphi @ Xy )
% 0.34/0.49 => ( Xx = Xy ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setextAx_type,type,
% 0.34/0.49 setextAx: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setextAx,definition,
% 0.34/0.49 ( setextAx
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 <=> ( in @ Xx @ B ) )
% 0.34/0.49 => ( A = B ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(emptyset_type,type,
% 0.34/0.49 emptyset: $i ).
% 0.34/0.49
% 0.34/0.49 thf(emptysetAx_type,type,
% 0.34/0.49 emptysetAx: $o ).
% 0.34/0.49
% 0.34/0.49 thf(emptysetAx,definition,
% 0.34/0.49 ( emptysetAx
% 0.34/0.49 = ( ! [Xx: $i] :
% 0.34/0.49 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoin_type,type,
% 0.34/0.49 setadjoin: $i > $i > $i ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoinAx_type,type,
% 0.34/0.49 setadjoinAx: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoinAx,definition,
% 0.34/0.49 ( setadjoinAx
% 0.34/0.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.34/0.49 <=> ( ( Xy = Xx )
% 0.34/0.49 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(powerset_type,type,
% 0.34/0.49 powerset: $i > $i ).
% 0.34/0.49
% 0.34/0.49 thf(powersetAx_type,type,
% 0.34/0.49 powersetAx: $o ).
% 0.34/0.49
% 0.34/0.49 thf(powersetAx,definition,
% 0.34/0.49 ( powersetAx
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ( in @ B @ ( powerset @ A ) )
% 0.34/0.49 <=> ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ B )
% 0.34/0.49 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setunion_type,type,
% 0.34/0.49 setunion: $i > $i ).
% 0.34/0.49
% 0.34/0.49 thf(setunionAx_type,type,
% 0.34/0.49 setunionAx: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setunionAx,definition,
% 0.34/0.49 ( setunionAx
% 0.34/0.49 = ( ! [A: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ ( setunion @ A ) )
% 0.34/0.49 <=> ? [B: $i] :
% 0.34/0.49 ( ( in @ Xx @ B )
% 0.34/0.49 & ( in @ B @ A ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(omega_type,type,
% 0.34/0.49 omega: $i ).
% 0.34/0.49
% 0.34/0.49 thf(omega0Ax_type,type,
% 0.34/0.49 omega0Ax: $o ).
% 0.34/0.49
% 0.34/0.49 thf(omega0Ax,definition,
% 0.34/0.49 ( omega0Ax
% 0.34/0.49 = ( in @ emptyset @ omega ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(omegaSAx_type,type,
% 0.34/0.49 omegaSAx: $o ).
% 0.34/0.49
% 0.34/0.49 thf(omegaSAx,definition,
% 0.34/0.49 ( omegaSAx
% 0.34/0.49 = ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ omega )
% 0.34/0.49 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(omegaIndAx_type,type,
% 0.34/0.49 omegaIndAx: $o ).
% 0.34/0.49
% 0.34/0.49 thf(omegaIndAx,definition,
% 0.34/0.49 ( omegaIndAx
% 0.34/0.49 = ( ! [A: $i] :
% 0.34/0.49 ( ( ( in @ emptyset @ A )
% 0.34/0.49 & ! [Xx: $i] :
% 0.34/0.49 ( ( ( in @ Xx @ omega )
% 0.34/0.49 & ( in @ Xx @ A ) )
% 0.34/0.49 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.34/0.49 => ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ omega )
% 0.34/0.49 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(replAx_type,type,
% 0.34/0.49 replAx: $o ).
% 0.34/0.49
% 0.34/0.49 thf(replAx,definition,
% 0.34/0.49 ( replAx
% 0.34/0.49 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.34/0.49 ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( exu
% 0.34/0.49 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.34/0.49 => ? [B: $i] :
% 0.34/0.49 ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ B )
% 0.34/0.49 <=> ? [Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ A )
% 0.34/0.49 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(foundationAx_type,type,
% 0.34/0.49 foundationAx: $o ).
% 0.34/0.49
% 0.34/0.49 thf(foundationAx,definition,
% 0.34/0.49 ( foundationAx
% 0.34/0.49 = ( ! [A: $i] :
% 0.34/0.49 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.34/0.49 => ? [B: $i] :
% 0.34/0.49 ( ( in @ B @ A )
% 0.34/0.49 & ~ ? [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ B )
% 0.34/0.49 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(wellorderingAx_type,type,
% 0.34/0.49 wellorderingAx: $o ).
% 0.34/0.49
% 0.34/0.49 thf(wellorderingAx,definition,
% 0.34/0.49 ( wellorderingAx
% 0.34/0.49 = ( ! [A: $i] :
% 0.34/0.49 ? [B: $i] :
% 0.34/0.49 ( ! [C: $i] :
% 0.34/0.49 ( ( in @ C @ B )
% 0.34/0.49 => ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ C )
% 0.34/0.49 => ( in @ Xx @ A ) ) )
% 0.34/0.49 & ! [Xx: $i,Xy: $i] :
% 0.34/0.49 ( ( ( in @ Xx @ A )
% 0.34/0.49 & ( in @ Xy @ A ) )
% 0.34/0.49 => ( ! [C: $i] :
% 0.34/0.49 ( ( in @ C @ B )
% 0.34/0.49 => ( ( in @ Xx @ C )
% 0.34/0.49 <=> ( in @ Xy @ C ) ) )
% 0.34/0.49 => ( Xx = Xy ) ) )
% 0.34/0.49 & ! [C: $i,D: $i] :
% 0.34/0.49 ( ( ( in @ C @ B )
% 0.34/0.49 & ( in @ D @ B ) )
% 0.34/0.49 => ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ C )
% 0.34/0.49 => ( in @ Xx @ D ) )
% 0.34/0.49 | ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ D )
% 0.34/0.49 => ( in @ Xx @ C ) ) ) )
% 0.34/0.49 & ! [C: $i] :
% 0.34/0.49 ( ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ C )
% 0.34/0.49 => ( in @ Xx @ A ) )
% 0.34/0.49 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.34/0.49 => ? [D: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ D @ B )
% 0.34/0.49 & ( in @ Xx @ C )
% 0.34/0.49 & ~ ? [Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ D )
% 0.34/0.49 & ( in @ Xy @ C ) )
% 0.34/0.49 & ! [E: $i] :
% 0.34/0.49 ( ( in @ E @ B )
% 0.34/0.49 => ( ! [Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ E )
% 0.34/0.49 => ( in @ Xy @ D ) )
% 0.34/0.49 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(descr_type,type,
% 0.34/0.49 descr: ( $i > $o ) > $i ).
% 0.34/0.49
% 0.34/0.49 thf(descrp_type,type,
% 0.34/0.49 descrp: $o ).
% 0.34/0.49
% 0.34/0.49 thf(descrp,definition,
% 0.34/0.49 ( descrp
% 0.34/0.49 = ( ! [Xphi: $i > $o] :
% 0.34/0.49 ( ( exu
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.34/0.49 => ( Xphi
% 0.34/0.49 @ ( descr
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(dsetconstr_type,type,
% 0.34/0.49 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.34/0.49
% 0.34/0.49 thf(dsetconstrI_type,type,
% 0.34/0.49 dsetconstrI: $o ).
% 0.34/0.49
% 0.34/0.49 thf(dsetconstrI,definition,
% 0.34/0.49 ( dsetconstrI
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( ( Xphi @ Xx )
% 0.34/0.49 => ( in @ Xx
% 0.34/0.49 @ ( dsetconstr @ A
% 0.34/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(dsetconstrEL_type,type,
% 0.34/0.49 dsetconstrEL: $o ).
% 0.34/0.49
% 0.34/0.49 thf(dsetconstrEL,definition,
% 0.34/0.49 ( dsetconstrEL
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx
% 0.34/0.49 @ ( dsetconstr @ A
% 0.34/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.34/0.49 => ( in @ Xx @ A ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(dsetconstrER_type,type,
% 0.34/0.49 dsetconstrER: $o ).
% 0.34/0.49
% 0.34/0.49 thf(dsetconstrER,definition,
% 0.34/0.49 ( dsetconstrER
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx
% 0.34/0.49 @ ( dsetconstr @ A
% 0.34/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.34/0.49 => ( Xphi @ Xx ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(exuE1_type,type,
% 0.34/0.49 exuE1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(exuE1,definition,
% 0.34/0.49 ( exuE1
% 0.34/0.49 = ( ! [Xphi: $i > $o] :
% 0.34/0.49 ( ( exu
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.34/0.49 => ? [Xx: $i] :
% 0.34/0.49 ( ( Xphi @ Xx )
% 0.34/0.49 & ! [Xy: $i] :
% 0.34/0.49 ( ( Xphi @ Xy )
% 0.34/0.49 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(prop2set_type,type,
% 0.34/0.49 prop2set: $o > $i ).
% 0.34/0.49
% 0.34/0.49 thf(prop2setE_type,type,
% 0.34/0.49 prop2setE: $o ).
% 0.34/0.49
% 0.34/0.49 thf(prop2setE,definition,
% 0.34/0.49 ( prop2setE
% 0.34/0.49 = ( ! [Xphi: $o,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.34/0.49 => Xphi ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(emptysetE_type,type,
% 0.34/0.49 emptysetE: $o ).
% 0.34/0.49
% 0.34/0.49 thf(emptysetE,definition,
% 0.34/0.49 ( emptysetE
% 0.34/0.49 = ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ emptyset )
% 0.34/0.49 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(emptysetimpfalse_type,type,
% 0.34/0.49 emptysetimpfalse: $o ).
% 0.34/0.49
% 0.34/0.49 thf(emptysetimpfalse,definition,
% 0.34/0.49 ( emptysetimpfalse
% 0.34/0.49 = ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ emptyset )
% 0.34/0.49 => $false ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(notinemptyset_type,type,
% 0.34/0.49 notinemptyset: $o ).
% 0.34/0.49
% 0.34/0.49 thf(notinemptyset,definition,
% 0.34/0.49 ( notinemptyset
% 0.34/0.49 = ( ! [Xx: $i] :
% 0.34/0.49 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(exuE3e_type,type,
% 0.34/0.49 exuE3e: $o ).
% 0.34/0.49
% 0.34/0.49 thf(exuE3e,definition,
% 0.34/0.49 ( exuE3e
% 0.34/0.49 = ( ! [Xphi: $i > $o] :
% 0.34/0.49 ( ( exu
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.34/0.49 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setext_type,type,
% 0.34/0.49 setext: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setext,definition,
% 0.34/0.49 ( setext
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( in @ Xx @ B ) )
% 0.34/0.49 => ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ B )
% 0.34/0.49 => ( in @ Xx @ A ) )
% 0.34/0.49 => ( A = B ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(emptyI_type,type,
% 0.34/0.49 emptyI: $o ).
% 0.34/0.49
% 0.34/0.49 thf(emptyI,definition,
% 0.34/0.49 ( emptyI
% 0.34/0.49 = ( ! [A: $i] :
% 0.34/0.49 ( ! [Xx: $i] :
% 0.34/0.49 ~ ( in @ Xx @ A )
% 0.34/0.49 => ( A = emptyset ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(noeltsimpempty_type,type,
% 0.34/0.49 noeltsimpempty: $o ).
% 0.34/0.49
% 0.34/0.49 thf(noeltsimpempty,definition,
% 0.34/0.49 ( noeltsimpempty
% 0.34/0.49 = ( ! [A: $i] :
% 0.34/0.49 ( ! [Xx: $i] :
% 0.34/0.49 ~ ( in @ Xx @ A )
% 0.34/0.49 => ( A = emptyset ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setbeta_type,type,
% 0.34/0.49 setbeta: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setbeta,definition,
% 0.34/0.49 ( setbeta
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( ( in @ Xx
% 0.34/0.49 @ ( dsetconstr @ A
% 0.34/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.34/0.49 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(nonempty_type,type,
% 0.34/0.49 nonempty: $i > $o ).
% 0.34/0.49
% 0.34/0.49 thf(nonempty,definition,
% 0.34/0.49 ( nonempty
% 0.34/0.49 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(nonemptyE1_type,type,
% 0.34/0.49 nonemptyE1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(nonemptyE1,definition,
% 0.34/0.49 ( nonemptyE1
% 0.34/0.49 = ( ! [A: $i] :
% 0.34/0.49 ( ( nonempty @ A )
% 0.34/0.49 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(nonemptyI_type,type,
% 0.34/0.49 nonemptyI: $o ).
% 0.34/0.49
% 0.34/0.49 thf(nonemptyI,definition,
% 0.34/0.49 ( nonemptyI
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( ( Xphi @ Xx )
% 0.34/0.49 => ( nonempty
% 0.34/0.49 @ ( dsetconstr @ A
% 0.34/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(nonemptyI1_type,type,
% 0.34/0.49 nonemptyI1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(nonemptyI1,definition,
% 0.34/0.49 ( nonemptyI1
% 0.34/0.49 = ( ! [A: $i] :
% 0.34/0.49 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.34/0.49 => ( nonempty @ A ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoinIL_type,type,
% 0.34/0.49 setadjoinIL: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoinIL,definition,
% 0.34/0.49 ( setadjoinIL
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(emptyinunitempty_type,type,
% 0.34/0.49 emptyinunitempty: $o ).
% 0.34/0.49
% 0.34/0.49 thf(emptyinunitempty,definition,
% 0.34/0.49 ( emptyinunitempty
% 0.34/0.49 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoinIR_type,type,
% 0.34/0.49 setadjoinIR: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoinIR,definition,
% 0.34/0.49 ( setadjoinIR
% 0.34/0.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ A )
% 0.34/0.49 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoinE_type,type,
% 0.34/0.49 setadjoinE: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoinE,definition,
% 0.34/0.49 ( setadjoinE
% 0.34/0.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.34/0.49 => ! [Xphi: $o] :
% 0.34/0.49 ( ( ( Xy = Xx )
% 0.34/0.49 => Xphi )
% 0.34/0.49 => ( ( ( in @ Xy @ A )
% 0.34/0.49 => Xphi )
% 0.34/0.49 => Xphi ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoinOr_type,type,
% 0.34/0.49 setadjoinOr: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoinOr,definition,
% 0.34/0.49 ( setadjoinOr
% 0.34/0.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.34/0.49 => ( ( Xy = Xx )
% 0.34/0.49 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setoftrueEq_type,type,
% 0.34/0.49 setoftrueEq: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setoftrueEq,definition,
% 0.34/0.49 ( setoftrueEq
% 0.34/0.49 = ( ! [A: $i] :
% 0.34/0.49 ( ( dsetconstr @ A
% 0.34/0.49 @ ^ [Xx: $i] : $true )
% 0.34/0.49 = A ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(powersetI_type,type,
% 0.34/0.49 powersetI: $o ).
% 0.34/0.49
% 0.34/0.49 thf(powersetI,definition,
% 0.34/0.49 ( powersetI
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ B )
% 0.34/0.49 => ( in @ Xx @ A ) )
% 0.34/0.49 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(emptyinPowerset_type,type,
% 0.34/0.49 emptyinPowerset: $o ).
% 0.34/0.49
% 0.34/0.49 thf(emptyinPowerset,definition,
% 0.34/0.49 ( emptyinPowerset
% 0.34/0.49 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(emptyInPowerset_type,type,
% 0.34/0.49 emptyInPowerset: $o ).
% 0.34/0.49
% 0.34/0.49 thf(emptyInPowerset,definition,
% 0.34/0.49 ( emptyInPowerset
% 0.34/0.49 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(powersetE_type,type,
% 0.34/0.49 powersetE: $o ).
% 0.34/0.49
% 0.34/0.49 thf(powersetE,definition,
% 0.34/0.49 ( powersetE
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ B @ ( powerset @ A ) )
% 0.34/0.49 => ( ( in @ Xx @ B )
% 0.34/0.49 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setunionI_type,type,
% 0.34/0.49 setunionI: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setunionI,definition,
% 0.34/0.49 ( setunionI
% 0.34/0.49 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.34/0.49 ( ( in @ Xx @ B )
% 0.34/0.49 => ( ( in @ B @ A )
% 0.34/0.49 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setunionE_type,type,
% 0.34/0.49 setunionE: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setunionE,definition,
% 0.34/0.49 ( setunionE
% 0.34/0.49 = ( ! [A: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ ( setunion @ A ) )
% 0.34/0.49 => ! [Xphi: $o] :
% 0.34/0.49 ( ! [B: $i] :
% 0.34/0.49 ( ( in @ Xx @ B )
% 0.34/0.49 => ( ( in @ B @ A )
% 0.34/0.49 => Xphi ) )
% 0.34/0.49 => Xphi ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(subPowSU_type,type,
% 0.34/0.49 subPowSU: $o ).
% 0.34/0.49
% 0.34/0.49 thf(subPowSU,definition,
% 0.34/0.49 ( subPowSU
% 0.34/0.49 = ( ! [A: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(exuE2_type,type,
% 0.34/0.49 exuE2: $o ).
% 0.34/0.49
% 0.34/0.49 thf(exuE2,definition,
% 0.34/0.49 ( exuE2
% 0.34/0.49 = ( ! [Xphi: $i > $o] :
% 0.34/0.49 ( ( exu
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.34/0.49 => ? [Xx: $i] :
% 0.34/0.49 ! [Xy: $i] :
% 0.34/0.49 ( ( Xphi @ Xy )
% 0.34/0.49 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(nonemptyImpWitness_type,type,
% 0.34/0.49 nonemptyImpWitness: $o ).
% 0.34/0.49
% 0.34/0.49 thf(nonemptyImpWitness,definition,
% 0.34/0.49 ( nonemptyImpWitness
% 0.34/0.49 = ( ! [A: $i] :
% 0.34/0.49 ( ( nonempty @ A )
% 0.34/0.49 => ? [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 & $true ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(uniqinunit_type,type,
% 0.34/0.49 uniqinunit: $o ).
% 0.34/0.49
% 0.34/0.49 thf(uniqinunit,definition,
% 0.34/0.49 ( uniqinunit
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.34/0.49 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.34/0.49 => ( Xx = Xy ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(notinsingleton_type,type,
% 0.34/0.49 notinsingleton: $o ).
% 0.34/0.49
% 0.34/0.49 thf(notinsingleton,definition,
% 0.34/0.49 ( notinsingleton
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.34/0.49 ( ( Xx != Xy )
% 0.34/0.49 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(eqinunit_type,type,
% 0.34/0.49 eqinunit: $o ).
% 0.34/0.49
% 0.34/0.49 thf(eqinunit,definition,
% 0.34/0.49 ( eqinunit
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.34/0.49 ( ( Xx = Xy )
% 0.34/0.49 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(singletonsswitch_type,type,
% 0.34/0.49 singletonsswitch: $o ).
% 0.34/0.49
% 0.34/0.49 thf(singletonsswitch,definition,
% 0.34/0.49 ( singletonsswitch
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.34/0.49 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.34/0.49 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(upairsetE_type,type,
% 0.34/0.49 upairsetE: $o ).
% 0.34/0.49
% 0.34/0.49 thf(upairsetE,definition,
% 0.34/0.49 ( upairsetE
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.34/0.49 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.34/0.49 => ( ( Xz = Xx )
% 0.34/0.49 | ( Xz = Xy ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(upairsetIL_type,type,
% 0.34/0.49 upairsetIL: $o ).
% 0.34/0.49
% 0.34/0.49 thf(upairsetIL,definition,
% 0.34/0.49 ( upairsetIL
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(upairsetIR_type,type,
% 0.34/0.49 upairsetIR: $o ).
% 0.34/0.49
% 0.34/0.49 thf(upairsetIR,definition,
% 0.34/0.49 ( upairsetIR
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(emptyE1_type,type,
% 0.34/0.49 emptyE1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(emptyE1,definition,
% 0.34/0.49 ( emptyE1
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.34/0.49 ( ? [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 & ( Xphi @ Xx ) )
% 0.34/0.49 => ( ( ( dsetconstr @ A
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.34/0.49 = emptyset )
% 0.34/0.49 => $false ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(vacuousDall_type,type,
% 0.34/0.49 vacuousDall: $o ).
% 0.34/0.49
% 0.34/0.49 thf(vacuousDall,definition,
% 0.34/0.49 ( vacuousDall
% 0.34/0.49 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ emptyset )
% 0.34/0.49 => ( Xphi @ Xx ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(quantDeMorgan1_type,type,
% 0.34/0.49 quantDeMorgan1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(quantDeMorgan1,definition,
% 0.34/0.49 ( quantDeMorgan1
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.34/0.49 ( ~ ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( Xphi @ Xx ) )
% 0.34/0.49 => ? [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(quantDeMorgan2_type,type,
% 0.34/0.49 quantDeMorgan2: $o ).
% 0.34/0.49
% 0.34/0.49 thf(quantDeMorgan2,definition,
% 0.34/0.49 ( quantDeMorgan2
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.34/0.49 ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ~ ( Xphi @ Xx ) )
% 0.34/0.49 => ~ ? [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(quantDeMorgan3_type,type,
% 0.34/0.49 quantDeMorgan3: $o ).
% 0.34/0.49
% 0.34/0.49 thf(quantDeMorgan3,definition,
% 0.34/0.49 ( quantDeMorgan3
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.34/0.49 ( ~ ? [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 & ( Xphi @ Xx ) )
% 0.34/0.49 => ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(quantDeMorgan4_type,type,
% 0.34/0.49 quantDeMorgan4: $o ).
% 0.34/0.49
% 0.34/0.49 thf(quantDeMorgan4,definition,
% 0.34/0.49 ( quantDeMorgan4
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.34/0.49 ( ? [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 & ~ ( Xphi @ Xx ) )
% 0.34/0.49 => ~ ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(prop2setI_type,type,
% 0.34/0.49 prop2setI: $o ).
% 0.34/0.49
% 0.34/0.49 thf(prop2setI,definition,
% 0.34/0.49 ( prop2setI
% 0.34/0.49 = ( ! [Xphi: $o] :
% 0.34/0.49 ( Xphi
% 0.34/0.49 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(set2prop_type,type,
% 0.34/0.49 set2prop: $i > $o ).
% 0.34/0.49
% 0.34/0.49 thf(prop2set2propI_type,type,
% 0.34/0.49 prop2set2propI: $o ).
% 0.34/0.49
% 0.34/0.49 thf(prop2set2propI,definition,
% 0.34/0.49 ( prop2set2propI
% 0.34/0.49 = ( ! [Xphi: $o] :
% 0.34/0.49 ( Xphi
% 0.34/0.49 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(notdexE_type,type,
% 0.34/0.49 notdexE: $o ).
% 0.34/0.49
% 0.34/0.49 thf(notdexE,definition,
% 0.34/0.49 ( notdexE
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.34/0.49 ( ~ ? [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 & ( Xphi @ Xx ) )
% 0.34/0.49 => ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(notdallE_type,type,
% 0.34/0.49 notdallE: $o ).
% 0.34/0.49
% 0.34/0.49 thf(notdallE,definition,
% 0.34/0.49 ( notdallE
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.34/0.49 ( ~ ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( Xphi @ Xx ) )
% 0.34/0.49 => ? [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(exuI1_type,type,
% 0.34/0.49 exuI1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(exuI1,definition,
% 0.34/0.49 ( exuI1
% 0.34/0.49 = ( ! [Xphi: $i > $o] :
% 0.34/0.49 ( ? [Xx: $i] :
% 0.34/0.49 ( ( Xphi @ Xx )
% 0.34/0.49 & ! [Xy: $i] :
% 0.34/0.49 ( ( Xphi @ Xy )
% 0.34/0.49 => ( Xx = Xy ) ) )
% 0.34/0.49 => ( exu
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(exuI3_type,type,
% 0.34/0.49 exuI3: $o ).
% 0.34/0.49
% 0.34/0.49 thf(exuI3,definition,
% 0.34/0.49 ( exuI3
% 0.34/0.49 = ( ! [Xphi: $i > $o] :
% 0.34/0.49 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.34/0.49 => ( ! [Xx: $i,Xy: $i] :
% 0.34/0.49 ( ( Xphi @ Xx )
% 0.34/0.49 => ( ( Xphi @ Xy )
% 0.34/0.49 => ( Xx = Xy ) ) )
% 0.34/0.49 => ( exu
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(exuI2_type,type,
% 0.34/0.49 exuI2: $o ).
% 0.34/0.49
% 0.34/0.49 thf(exuI2,definition,
% 0.34/0.49 ( exuI2
% 0.34/0.49 = ( ! [Xphi: $i > $o] :
% 0.34/0.49 ( ? [Xx: $i] :
% 0.34/0.49 ! [Xy: $i] :
% 0.34/0.49 ( ( Xphi @ Xy )
% 0.34/0.49 <=> ( Xy = Xx ) )
% 0.34/0.49 => ( exu
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(inCongP_type,type,
% 0.34/0.49 inCongP: $o ).
% 0.34/0.49
% 0.34/0.49 thf(inCongP,definition,
% 0.34/0.49 ( inCongP
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ( A = B )
% 0.34/0.49 => ! [Xx: $i,Xy: $i] :
% 0.34/0.49 ( ( Xx = Xy )
% 0.34/0.49 => ( ( in @ Xx @ A )
% 0.34/0.49 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(in__Cong_type,type,
% 0.34/0.49 in__Cong: $o ).
% 0.34/0.49
% 0.34/0.49 thf(in__Cong,definition,
% 0.34/0.49 ( in__Cong
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ( A = B )
% 0.34/0.49 => ! [Xx: $i,Xy: $i] :
% 0.34/0.49 ( ( Xx = Xy )
% 0.34/0.49 => ( ( in @ Xx @ A )
% 0.34/0.49 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(exuE3u_type,type,
% 0.34/0.49 exuE3u: $o ).
% 0.34/0.49
% 0.34/0.49 thf(exuE3u,definition,
% 0.34/0.49 ( exuE3u
% 0.34/0.49 = ( ! [Xphi: $i > $o] :
% 0.34/0.49 ( ( exu
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.34/0.49 => ! [Xx: $i,Xy: $i] :
% 0.34/0.49 ( ( Xphi @ Xx )
% 0.34/0.49 => ( ( Xphi @ Xy )
% 0.34/0.49 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(exu__Cong_type,type,
% 0.34/0.49 exu__Cong: $o ).
% 0.34/0.49
% 0.34/0.49 thf(exu__Cong,definition,
% 0.34/0.49 ( exu__Cong
% 0.34/0.49 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.34/0.49 ( ! [Xx: $i,Xy: $i] :
% 0.34/0.49 ( ( Xx = Xy )
% 0.34/0.49 => ( ( Xphi @ Xx )
% 0.34/0.49 <=> ( Xpsi @ Xy ) ) )
% 0.34/0.49 => ( ( exu
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.34/0.49 <=> ( exu
% 0.34/0.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(emptyset__Cong_type,type,
% 0.34/0.49 emptyset__Cong: $o ).
% 0.34/0.49
% 0.34/0.49 thf(emptyset__Cong,definition,
% 0.34/0.49 ( emptyset__Cong
% 0.34/0.49 = ( emptyset = emptyset ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoin__Cong_type,type,
% 0.34/0.49 setadjoin__Cong: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoin__Cong,definition,
% 0.34/0.49 ( setadjoin__Cong
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.34/0.49 ( ( Xx = Xy )
% 0.34/0.49 => ! [Xz: $i,Xu: $i] :
% 0.34/0.49 ( ( Xz = Xu )
% 0.34/0.49 => ( ( setadjoin @ Xx @ Xz )
% 0.34/0.49 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(powerset__Cong_type,type,
% 0.34/0.49 powerset__Cong: $o ).
% 0.34/0.49
% 0.34/0.49 thf(powerset__Cong,definition,
% 0.34/0.49 ( powerset__Cong
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ( A = B )
% 0.34/0.49 => ( ( powerset @ A )
% 0.34/0.49 = ( powerset @ B ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setunion__Cong_type,type,
% 0.34/0.49 setunion__Cong: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setunion__Cong,definition,
% 0.34/0.49 ( setunion__Cong
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ( A = B )
% 0.34/0.49 => ( ( setunion @ A )
% 0.34/0.49 = ( setunion @ B ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(omega__Cong_type,type,
% 0.34/0.49 omega__Cong: $o ).
% 0.34/0.49
% 0.34/0.49 thf(omega__Cong,definition,
% 0.34/0.49 ( omega__Cong
% 0.34/0.49 = ( omega = omega ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(exuEu_type,type,
% 0.34/0.49 exuEu: $o ).
% 0.34/0.49
% 0.34/0.49 thf(exuEu,definition,
% 0.34/0.49 ( exuEu
% 0.34/0.49 = ( ! [Xphi: $i > $o] :
% 0.34/0.49 ( ( exu
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.34/0.49 => ! [Xx: $i,Xy: $i] :
% 0.34/0.49 ( ( Xphi @ Xx )
% 0.34/0.49 => ( ( Xphi @ Xy )
% 0.34/0.49 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(descr__Cong_type,type,
% 0.34/0.49 descr__Cong: $o ).
% 0.34/0.49
% 0.34/0.49 thf(descr__Cong,definition,
% 0.34/0.49 ( descr__Cong
% 0.34/0.49 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.34/0.49 ( ! [Xx: $i,Xy: $i] :
% 0.34/0.49 ( ( Xx = Xy )
% 0.34/0.49 => ( ( Xphi @ Xx )
% 0.34/0.49 <=> ( Xpsi @ Xy ) ) )
% 0.34/0.49 => ( ( exu
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.34/0.49 => ( ( exu
% 0.34/0.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.34/0.49 => ( ( descr
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.34/0.49 = ( descr
% 0.34/0.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(dsetconstr__Cong_type,type,
% 0.34/0.49 dsetconstr__Cong: $o ).
% 0.34/0.49
% 0.34/0.49 thf(dsetconstr__Cong,definition,
% 0.34/0.49 ( dsetconstr__Cong
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ( A = B )
% 0.34/0.49 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.34/0.49 ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ! [Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ B )
% 0.34/0.49 => ( ( Xx = Xy )
% 0.34/0.49 => ( ( Xphi @ Xx )
% 0.34/0.49 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.34/0.49 => ( ( dsetconstr @ A
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.34/0.49 = ( dsetconstr @ B
% 0.34/0.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(subset_type,type,
% 0.34/0.49 subset: $i > $i > $o ).
% 0.34/0.49
% 0.34/0.49 thf(disjoint_type,type,
% 0.34/0.49 disjoint: $i > $i > $o ).
% 0.34/0.49
% 0.34/0.49 thf(setsmeet_type,type,
% 0.34/0.49 setsmeet: $i > $i > $o ).
% 0.34/0.49
% 0.34/0.49 thf(subsetI1_type,type,
% 0.34/0.49 subsetI1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(subsetI1,definition,
% 0.34/0.49 ( subsetI1
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( in @ Xx @ B ) )
% 0.34/0.49 => ( subset @ A @ B ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(eqimpsubset2_type,type,
% 0.34/0.49 eqimpsubset2: $o ).
% 0.34/0.49
% 0.34/0.49 thf(eqimpsubset2,definition,
% 0.34/0.49 ( eqimpsubset2
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ( A = B )
% 0.34/0.49 => ( subset @ B @ A ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(eqimpsubset1_type,type,
% 0.34/0.49 eqimpsubset1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(eqimpsubset1,definition,
% 0.34/0.49 ( eqimpsubset1
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ( A = B )
% 0.34/0.49 => ( subset @ A @ B ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(subsetI2_type,type,
% 0.34/0.49 subsetI2: $o ).
% 0.34/0.49
% 0.34/0.49 thf(subsetI2,definition,
% 0.34/0.49 ( subsetI2
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( in @ Xx @ B ) )
% 0.34/0.49 => ( subset @ A @ B ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(emptysetsubset_type,type,
% 0.34/0.49 emptysetsubset: $o ).
% 0.34/0.49
% 0.34/0.49 thf(emptysetsubset,definition,
% 0.34/0.49 ( emptysetsubset
% 0.34/0.49 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(subsetE_type,type,
% 0.34/0.49 subsetE: $o ).
% 0.34/0.49
% 0.34/0.49 thf(subsetE,definition,
% 0.34/0.49 ( subsetE
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( subset @ A @ B )
% 0.34/0.49 => ( ( in @ Xx @ A )
% 0.34/0.49 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(subsetE2_type,type,
% 0.34/0.49 subsetE2: $o ).
% 0.34/0.49
% 0.34/0.49 thf(subsetE2,definition,
% 0.34/0.49 ( subsetE2
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( subset @ A @ B )
% 0.34/0.49 => ( ~ ( in @ Xx @ B )
% 0.34/0.49 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(notsubsetI_type,type,
% 0.34/0.49 notsubsetI: $o ).
% 0.34/0.49
% 0.34/0.49 thf(notsubsetI,definition,
% 0.34/0.49 ( notsubsetI
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( ~ ( in @ Xx @ B )
% 0.34/0.49 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(notequalI1_type,type,
% 0.34/0.49 notequalI1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(notequalI1,definition,
% 0.34/0.49 ( notequalI1
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ~ ( subset @ A @ B )
% 0.34/0.49 => ( A != B ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(notequalI2_type,type,
% 0.34/0.49 notequalI2: $o ).
% 0.34/0.49
% 0.34/0.49 thf(notequalI2,definition,
% 0.34/0.49 ( notequalI2
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( ~ ( in @ Xx @ B )
% 0.34/0.49 => ( A != B ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(subsetRefl_type,type,
% 0.34/0.49 subsetRefl: $o ).
% 0.34/0.49
% 0.34/0.49 thf(subsetRefl,definition,
% 0.34/0.49 ( subsetRefl
% 0.34/0.49 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(subsetTrans_type,type,
% 0.34/0.49 subsetTrans: $o ).
% 0.34/0.49
% 0.34/0.49 thf(subsetTrans,definition,
% 0.34/0.49 ( subsetTrans
% 0.34/0.49 = ( ! [A: $i,B: $i,C: $i] :
% 0.34/0.49 ( ( subset @ A @ B )
% 0.34/0.49 => ( ( subset @ B @ C )
% 0.34/0.49 => ( subset @ A @ C ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoinSub_type,type,
% 0.34/0.49 setadjoinSub: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoinSub,definition,
% 0.34/0.49 ( setadjoinSub
% 0.34/0.49 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoinSub2_type,type,
% 0.34/0.49 setadjoinSub2: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoinSub2,definition,
% 0.34/0.49 ( setadjoinSub2
% 0.34/0.49 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.34/0.49 ( ( subset @ A @ B )
% 0.34/0.49 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(subset2powerset_type,type,
% 0.34/0.49 subset2powerset: $o ).
% 0.34/0.49
% 0.34/0.49 thf(subset2powerset,definition,
% 0.34/0.49 ( subset2powerset
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ( subset @ A @ B )
% 0.34/0.49 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setextsub_type,type,
% 0.34/0.49 setextsub: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setextsub,definition,
% 0.34/0.49 ( setextsub
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ( subset @ A @ B )
% 0.34/0.49 => ( ( subset @ B @ A )
% 0.34/0.49 => ( A = B ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(subsetemptysetimpeq_type,type,
% 0.34/0.49 subsetemptysetimpeq: $o ).
% 0.34/0.49
% 0.34/0.49 thf(subsetemptysetimpeq,definition,
% 0.34/0.49 ( subsetemptysetimpeq
% 0.34/0.49 = ( ! [A: $i] :
% 0.34/0.49 ( ( subset @ A @ emptyset )
% 0.34/0.49 => ( A = emptyset ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(powersetI1_type,type,
% 0.34/0.49 powersetI1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(powersetI1,definition,
% 0.34/0.49 ( powersetI1
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ( subset @ B @ A )
% 0.34/0.49 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(powersetE1_type,type,
% 0.34/0.49 powersetE1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(powersetE1,definition,
% 0.34/0.49 ( powersetE1
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ( in @ B @ ( powerset @ A ) )
% 0.34/0.49 => ( subset @ B @ A ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(inPowerset_type,type,
% 0.34/0.49 inPowerset: $o ).
% 0.34/0.49
% 0.34/0.49 thf(inPowerset,definition,
% 0.34/0.49 ( inPowerset
% 0.34/0.49 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(powersetsubset_type,type,
% 0.34/0.49 powersetsubset: $o ).
% 0.34/0.49
% 0.34/0.49 thf(powersetsubset,definition,
% 0.34/0.49 ( powersetsubset
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ( subset @ A @ B )
% 0.34/0.49 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(sepInPowerset_type,type,
% 0.34/0.49 sepInPowerset: $o ).
% 0.34/0.49
% 0.34/0.49 thf(sepInPowerset,definition,
% 0.34/0.49 ( sepInPowerset
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.34/0.49 ( in
% 0.34/0.49 @ ( dsetconstr @ A
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.34/0.49 @ ( powerset @ A ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(sepSubset_type,type,
% 0.34/0.49 sepSubset: $o ).
% 0.34/0.49
% 0.34/0.49 thf(sepSubset,definition,
% 0.34/0.49 ( sepSubset
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.34/0.49 ( subset
% 0.34/0.49 @ ( dsetconstr @ A
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.34/0.49 @ A ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(binunion_type,type,
% 0.34/0.49 binunion: $i > $i > $i ).
% 0.34/0.49
% 0.34/0.49 thf(binunionIL_type,type,
% 0.34/0.49 binunionIL: $o ).
% 0.34/0.49
% 0.34/0.49 thf(binunionIL,definition,
% 0.34/0.49 ( binunionIL
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(upairset2IR_type,type,
% 0.34/0.49 upairset2IR: $o ).
% 0.34/0.49
% 0.34/0.49 thf(upairset2IR,definition,
% 0.34/0.49 ( upairset2IR
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(binunionIR_type,type,
% 0.34/0.49 binunionIR: $o ).
% 0.34/0.49
% 0.34/0.49 thf(binunionIR,definition,
% 0.34/0.49 ( binunionIR
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ B )
% 0.34/0.49 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(binunionEcases_type,type,
% 0.34/0.49 binunionEcases: $o ).
% 0.34/0.49
% 0.34/0.49 thf(binunionEcases,definition,
% 0.34/0.49 ( binunionEcases
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.34/0.49 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.34/0.49 => ( ( ( in @ Xx @ A )
% 0.34/0.49 => Xphi )
% 0.34/0.49 => ( ( ( in @ Xx @ B )
% 0.34/0.49 => Xphi )
% 0.34/0.49 => Xphi ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(binunionE_type,type,
% 0.34/0.49 binunionE: $o ).
% 0.34/0.49
% 0.34/0.49 thf(binunionE,definition,
% 0.34/0.49 ( binunionE
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.34/0.49 => ( ( in @ Xx @ A )
% 0.34/0.49 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(binunionLsub_type,type,
% 0.34/0.49 binunionLsub: $o ).
% 0.34/0.49
% 0.34/0.49 thf(binunionLsub,definition,
% 0.34/0.49 ( binunionLsub
% 0.34/0.49 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(binunionRsub_type,type,
% 0.34/0.49 binunionRsub: $o ).
% 0.34/0.49
% 0.34/0.49 thf(binunionRsub,definition,
% 0.34/0.49 ( binunionRsub
% 0.34/0.49 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(binintersect_type,type,
% 0.34/0.49 binintersect: $i > $i > $i ).
% 0.34/0.49
% 0.34/0.49 thf(binintersectI_type,type,
% 0.34/0.49 binintersectI: $o ).
% 0.34/0.49
% 0.34/0.49 thf(binintersectI,definition,
% 0.34/0.49 ( binintersectI
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( ( in @ Xx @ B )
% 0.34/0.49 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(binintersectSubset5_type,type,
% 0.34/0.49 binintersectSubset5: $o ).
% 0.34/0.49
% 0.34/0.49 thf(binintersectSubset5,definition,
% 0.34/0.49 ( binintersectSubset5
% 0.34/0.49 = ( ! [A: $i,B: $i,C: $i] :
% 0.34/0.49 ( ( subset @ C @ A )
% 0.34/0.49 => ( ( subset @ C @ B )
% 0.34/0.49 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(binintersectEL_type,type,
% 0.34/0.49 binintersectEL: $o ).
% 0.34/0.49
% 0.34/0.49 thf(binintersectEL,definition,
% 0.34/0.49 ( binintersectEL
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.34/0.49 => ( in @ Xx @ A ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(binintersectLsub_type,type,
% 0.34/0.49 binintersectLsub: $o ).
% 0.34/0.49
% 0.34/0.49 thf(binintersectLsub,definition,
% 0.34/0.49 ( binintersectLsub
% 0.34/0.49 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(binintersectSubset2_type,type,
% 0.34/0.49 binintersectSubset2: $o ).
% 0.34/0.49
% 0.34/0.49 thf(binintersectSubset2,definition,
% 0.34/0.49 ( binintersectSubset2
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ( subset @ A @ B )
% 0.34/0.49 => ( ( binintersect @ A @ B )
% 0.34/0.49 = A ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(binintersectSubset3_type,type,
% 0.34/0.49 binintersectSubset3: $o ).
% 0.34/0.49
% 0.34/0.49 thf(binintersectSubset3,definition,
% 0.34/0.49 ( binintersectSubset3
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ( ( binintersect @ A @ B )
% 0.34/0.49 = B )
% 0.34/0.49 => ( subset @ B @ A ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(binintersectER_type,type,
% 0.34/0.49 binintersectER: $o ).
% 0.34/0.49
% 0.34/0.49 thf(binintersectER,definition,
% 0.34/0.49 ( binintersectER
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.34/0.49 => ( in @ Xx @ B ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(disjointsetsI1_type,type,
% 0.34/0.49 disjointsetsI1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(disjointsetsI1,definition,
% 0.34/0.49 ( disjointsetsI1
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ~ ? [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 & ( in @ Xx @ B ) )
% 0.34/0.49 => ( ( binintersect @ A @ B )
% 0.34/0.49 = emptyset ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(binintersectRsub_type,type,
% 0.34/0.49 binintersectRsub: $o ).
% 0.34/0.49
% 0.34/0.49 thf(binintersectRsub,definition,
% 0.34/0.49 ( binintersectRsub
% 0.34/0.49 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(binintersectSubset4_type,type,
% 0.34/0.49 binintersectSubset4: $o ).
% 0.34/0.49
% 0.34/0.49 thf(binintersectSubset4,definition,
% 0.34/0.49 ( binintersectSubset4
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ( subset @ B @ A )
% 0.34/0.49 => ( ( binintersect @ A @ B )
% 0.34/0.49 = B ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(binintersectSubset1_type,type,
% 0.34/0.49 binintersectSubset1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(binintersectSubset1,definition,
% 0.34/0.49 ( binintersectSubset1
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ( ( binintersect @ A @ B )
% 0.34/0.49 = A )
% 0.34/0.49 => ( subset @ A @ B ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(bs114d_type,type,
% 0.34/0.49 bs114d: $o ).
% 0.34/0.49
% 0.34/0.49 thf(bs114d,definition,
% 0.34/0.49 ( bs114d
% 0.34/0.49 = ( ! [A: $i,B: $i,C: $i] :
% 0.34/0.49 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.34/0.49 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(regular_type,type,
% 0.34/0.49 regular: $i > $o ).
% 0.34/0.49
% 0.34/0.49 thf(setminus_type,type,
% 0.34/0.49 setminus: $i > $i > $i ).
% 0.34/0.49
% 0.34/0.49 thf(setminusI_type,type,
% 0.34/0.49 setminusI: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setminusI,definition,
% 0.34/0.49 ( setminusI
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( ~ ( in @ Xx @ B )
% 0.34/0.49 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setminusEL_type,type,
% 0.34/0.49 setminusEL: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setminusEL,definition,
% 0.34/0.49 ( setminusEL
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.34/0.49 => ( in @ Xx @ A ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setminusER_type,type,
% 0.34/0.49 setminusER: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setminusER,definition,
% 0.34/0.49 ( setminusER
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.34/0.49 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setminusSubset2_type,type,
% 0.34/0.49 setminusSubset2: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setminusSubset2,definition,
% 0.34/0.49 ( setminusSubset2
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ( subset @ A @ B )
% 0.34/0.49 => ( ( setminus @ A @ B )
% 0.34/0.49 = emptyset ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setminusERneg_type,type,
% 0.34/0.49 setminusERneg: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setminusERneg,definition,
% 0.34/0.49 ( setminusERneg
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.34/0.49 => ( ( in @ Xx @ A )
% 0.34/0.49 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setminusELneg_type,type,
% 0.34/0.49 setminusELneg: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setminusELneg,definition,
% 0.34/0.49 ( setminusELneg
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.34/0.49 => ( ~ ( in @ Xx @ B )
% 0.34/0.49 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setminusILneg_type,type,
% 0.34/0.49 setminusILneg: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setminusILneg,definition,
% 0.34/0.49 ( setminusILneg
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ~ ( in @ Xx @ A )
% 0.34/0.49 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setminusIRneg_type,type,
% 0.34/0.49 setminusIRneg: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setminusIRneg,definition,
% 0.34/0.49 ( setminusIRneg
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ B )
% 0.34/0.49 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setminusLsub_type,type,
% 0.34/0.49 setminusLsub: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setminusLsub,definition,
% 0.34/0.49 ( setminusLsub
% 0.34/0.49 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setminusSubset1_type,type,
% 0.34/0.49 setminusSubset1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setminusSubset1,definition,
% 0.34/0.49 ( setminusSubset1
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ( ( setminus @ A @ B )
% 0.34/0.49 = emptyset )
% 0.34/0.49 => ( subset @ A @ B ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(symdiff_type,type,
% 0.34/0.49 symdiff: $i > $i > $i ).
% 0.34/0.49
% 0.34/0.49 thf(symdiffE_type,type,
% 0.34/0.49 symdiffE: $o ).
% 0.34/0.49
% 0.34/0.49 thf(symdiffE,definition,
% 0.34/0.49 ( symdiffE
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.34/0.49 => ! [Xphi: $o] :
% 0.34/0.49 ( ( ( in @ Xx @ A )
% 0.34/0.49 => ( ~ ( in @ Xx @ B )
% 0.34/0.49 => Xphi ) )
% 0.34/0.49 => ( ( ~ ( in @ Xx @ A )
% 0.34/0.49 => ( ( in @ Xx @ B )
% 0.34/0.49 => Xphi ) )
% 0.34/0.49 => Xphi ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(symdiffI1_type,type,
% 0.34/0.49 symdiffI1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(symdiffI1,definition,
% 0.34/0.49 ( symdiffI1
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( ~ ( in @ Xx @ B )
% 0.34/0.49 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(symdiffI2_type,type,
% 0.34/0.49 symdiffI2: $o ).
% 0.34/0.49
% 0.34/0.49 thf(symdiffI2,definition,
% 0.34/0.49 ( symdiffI2
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ~ ( in @ Xx @ A )
% 0.34/0.49 => ( ( in @ Xx @ B )
% 0.34/0.49 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(symdiffIneg1_type,type,
% 0.34/0.49 symdiffIneg1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(symdiffIneg1,definition,
% 0.34/0.49 ( symdiffIneg1
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( ( in @ Xx @ B )
% 0.34/0.49 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(symdiffIneg2_type,type,
% 0.34/0.49 symdiffIneg2: $o ).
% 0.34/0.49
% 0.34/0.49 thf(symdiffIneg2,definition,
% 0.34/0.49 ( symdiffIneg2
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ~ ( in @ Xx @ A )
% 0.34/0.49 => ( ~ ( in @ Xx @ B )
% 0.34/0.49 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(iskpair_type,type,
% 0.34/0.49 iskpair: $i > $o ).
% 0.34/0.49
% 0.34/0.49 thf(iskpair,definition,
% 0.34/0.49 ( iskpair
% 0.34/0.49 = ( ^ [A: $i] :
% 0.34/0.49 ? [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ ( setunion @ A ) )
% 0.34/0.49 & ? [Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ ( setunion @ A ) )
% 0.34/0.49 & ( A
% 0.34/0.49 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(secondinupair_type,type,
% 0.34/0.49 secondinupair: $o ).
% 0.34/0.49
% 0.34/0.49 thf(secondinupair,definition,
% 0.34/0.49 ( secondinupair
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setukpairIL_type,type,
% 0.34/0.49 setukpairIL: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setukpairIL,definition,
% 0.34/0.49 ( setukpairIL
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setukpairIR_type,type,
% 0.34/0.49 setukpairIR: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setukpairIR,definition,
% 0.34/0.49 ( setukpairIR
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(kpairiskpair_type,type,
% 0.34/0.49 kpairiskpair: $o ).
% 0.34/0.49
% 0.34/0.49 thf(kpairiskpair,definition,
% 0.34/0.49 ( kpairiskpair
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(kpair_type,type,
% 0.34/0.49 kpair: $i > $i > $i ).
% 0.34/0.49
% 0.34/0.49 thf(kpair,definition,
% 0.34/0.49 ( kpair
% 0.34/0.49 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(kpairp_type,type,
% 0.34/0.49 kpairp: $o ).
% 0.34/0.49
% 0.34/0.49 thf(kpairp,definition,
% 0.34/0.49 ( kpairp
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(cartprod_type,type,
% 0.34/0.49 cartprod: $i > $i > $i ).
% 0.34/0.49
% 0.34/0.49 thf(singletonsubset_type,type,
% 0.34/0.49 singletonsubset: $o ).
% 0.34/0.49
% 0.34/0.49 thf(singletonsubset,definition,
% 0.34/0.49 ( singletonsubset
% 0.34/0.49 = ( ! [A: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(singletoninpowerset_type,type,
% 0.34/0.49 singletoninpowerset: $o ).
% 0.34/0.49
% 0.34/0.49 thf(singletoninpowerset,definition,
% 0.34/0.49 ( singletoninpowerset
% 0.34/0.49 = ( ! [A: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(singletoninpowunion_type,type,
% 0.34/0.49 singletoninpowunion: $o ).
% 0.34/0.49
% 0.34/0.49 thf(singletoninpowunion,definition,
% 0.34/0.49 ( singletoninpowunion
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(upairset2E_type,type,
% 0.34/0.49 upairset2E: $o ).
% 0.34/0.49
% 0.34/0.49 thf(upairset2E,definition,
% 0.34/0.49 ( upairset2E
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.34/0.49 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.34/0.49 => ( ( Xz = Xx )
% 0.34/0.49 | ( Xz = Xy ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(upairsubunion_type,type,
% 0.34/0.49 upairsubunion: $o ).
% 0.34/0.49
% 0.34/0.49 thf(upairsubunion,definition,
% 0.34/0.49 ( upairsubunion
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ! [Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ B )
% 0.34/0.49 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(upairinpowunion_type,type,
% 0.34/0.49 upairinpowunion: $o ).
% 0.34/0.49
% 0.34/0.49 thf(upairinpowunion,definition,
% 0.34/0.49 ( upairinpowunion
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ! [Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ B )
% 0.34/0.49 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(ubforcartprodlem1_type,type,
% 0.34/0.49 ubforcartprodlem1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(ubforcartprodlem1,definition,
% 0.34/0.49 ( ubforcartprodlem1
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ! [Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ B )
% 0.34/0.49 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(ubforcartprodlem2_type,type,
% 0.34/0.49 ubforcartprodlem2: $o ).
% 0.34/0.49
% 0.34/0.49 thf(ubforcartprodlem2,definition,
% 0.34/0.49 ( ubforcartprodlem2
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ! [Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ B )
% 0.34/0.49 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(ubforcartprodlem3_type,type,
% 0.34/0.49 ubforcartprodlem3: $o ).
% 0.34/0.49
% 0.34/0.49 thf(ubforcartprodlem3,definition,
% 0.34/0.49 ( ubforcartprodlem3
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ! [Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ B )
% 0.34/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(cartprodpairin_type,type,
% 0.34/0.49 cartprodpairin: $o ).
% 0.34/0.49
% 0.34/0.49 thf(cartprodpairin,definition,
% 0.34/0.49 ( cartprodpairin
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ! [Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ B )
% 0.34/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(cartprodmempair1_type,type,
% 0.34/0.49 cartprodmempair1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(cartprodmempair1,definition,
% 0.34/0.49 ( cartprodmempair1
% 0.34/0.49 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.34/0.49 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.34/0.49 => ? [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 & ? [Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ B )
% 0.34/0.49 & ( Xu
% 0.34/0.49 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(cartprodmempair_type,type,
% 0.34/0.49 cartprodmempair: $o ).
% 0.34/0.49
% 0.34/0.49 thf(cartprodmempair,definition,
% 0.34/0.49 ( cartprodmempair
% 0.34/0.49 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.34/0.49 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.34/0.49 => ( iskpair @ Xu ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setunionE2_type,type,
% 0.34/0.49 setunionE2: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setunionE2,definition,
% 0.34/0.49 ( setunionE2
% 0.34/0.49 = ( ! [A: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ ( setunion @ A ) )
% 0.34/0.49 => ? [X: $i] :
% 0.34/0.49 ( ( in @ X @ A )
% 0.34/0.49 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setunionsingleton1_type,type,
% 0.34/0.49 setunionsingleton1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setunionsingleton1,definition,
% 0.34/0.49 ( setunionsingleton1
% 0.34/0.49 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setunionsingleton2_type,type,
% 0.34/0.49 setunionsingleton2: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setunionsingleton2,definition,
% 0.34/0.49 ( setunionsingleton2
% 0.34/0.49 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setunionsingleton_type,type,
% 0.34/0.49 setunionsingleton: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setunionsingleton,definition,
% 0.34/0.49 ( setunionsingleton
% 0.34/0.49 = ( ! [Xx: $i] :
% 0.34/0.49 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.34/0.49 = Xx ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(singleton_type,type,
% 0.34/0.49 singleton: $i > $o ).
% 0.34/0.49
% 0.34/0.49 thf(singleton,definition,
% 0.34/0.49 ( singleton
% 0.34/0.49 = ( ^ [A: $i] :
% 0.34/0.49 ? [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 & ( A
% 0.34/0.49 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(singletonprop_type,type,
% 0.34/0.49 singletonprop: $o ).
% 0.34/0.49
% 0.34/0.49 thf(singletonprop,definition,
% 0.34/0.49 ( singletonprop
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.34/0.49 ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ! [Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ A )
% 0.34/0.49 => ( ( Xphi @ Xx )
% 0.34/0.49 => ( ( Xphi @ Xy )
% 0.34/0.49 => ( Xx = Xy ) ) ) ) )
% 0.34/0.49 => ( ? [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 & ( Xphi @ Xx ) )
% 0.34/0.49 => ( singleton
% 0.34/0.49 @ ( dsetconstr @ A
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(ex1_type,type,
% 0.34/0.49 ex1: $i > ( $i > $o ) > $o ).
% 0.34/0.49
% 0.34/0.49 thf(ex1,definition,
% 0.34/0.49 ( ex1
% 0.34/0.49 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.34/0.49 ( singleton
% 0.34/0.49 @ ( dsetconstr @ A
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(ex1E1_type,type,
% 0.34/0.49 ex1E1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(ex1E1,definition,
% 0.34/0.49 ( ex1E1
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.34/0.49 ( ( ex1 @ A
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.34/0.49 => ? [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(ex1I_type,type,
% 0.34/0.49 ex1I: $o ).
% 0.34/0.49
% 0.34/0.49 thf(ex1I,definition,
% 0.34/0.49 ( ex1I
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( ( Xphi @ Xx )
% 0.34/0.49 => ( ! [Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ A )
% 0.34/0.49 => ( ( Xphi @ Xy )
% 0.34/0.49 => ( Xy = Xx ) ) )
% 0.34/0.49 => ( ex1 @ A
% 0.34/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(ex1I2_type,type,
% 0.34/0.49 ex1I2: $o ).
% 0.34/0.49
% 0.34/0.49 thf(ex1I2,definition,
% 0.34/0.49 ( ex1I2
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.34/0.49 ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ! [Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ A )
% 0.34/0.49 => ( ( Xphi @ Xx )
% 0.34/0.49 => ( ( Xphi @ Xy )
% 0.34/0.49 => ( Xx = Xy ) ) ) ) )
% 0.34/0.49 => ( ? [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 & ( Xphi @ Xx ) )
% 0.34/0.49 => ( ex1 @ A
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(singletonsuniq_type,type,
% 0.34/0.49 singletonsuniq: $o ).
% 0.34/0.49
% 0.34/0.49 thf(singletonsuniq,definition,
% 0.34/0.49 ( singletonsuniq
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.34/0.49 ( ( ( setadjoin @ Xx @ emptyset )
% 0.34/0.49 = ( setadjoin @ Xy @ emptyset ) )
% 0.34/0.49 => ( Xx = Xy ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(atmost1p_type,type,
% 0.34/0.49 atmost1p: $i > $o ).
% 0.34/0.49
% 0.34/0.49 thf(atleast2p_type,type,
% 0.34/0.49 atleast2p: $i > $o ).
% 0.34/0.49
% 0.34/0.49 thf(atmost2p_type,type,
% 0.34/0.49 atmost2p: $i > $o ).
% 0.34/0.49
% 0.34/0.49 thf(upairsetp_type,type,
% 0.34/0.49 upairsetp: $i > $o ).
% 0.34/0.49
% 0.34/0.49 thf(setukpairinjL1_type,type,
% 0.34/0.49 setukpairinjL1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setukpairinjL1,definition,
% 0.34/0.49 ( setukpairinjL1
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.34/0.49 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.34/0.49 => ( Xx = Xz ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(kfstsingleton_type,type,
% 0.34/0.49 kfstsingleton: $o ).
% 0.34/0.49
% 0.34/0.49 thf(kfstsingleton,definition,
% 0.34/0.49 ( kfstsingleton
% 0.34/0.49 = ( ! [Xu: $i] :
% 0.34/0.49 ( ( iskpair @ Xu )
% 0.34/0.49 => ( singleton
% 0.34/0.49 @ ( dsetconstr @ ( setunion @ Xu )
% 0.34/0.49 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(theprop_type,type,
% 0.34/0.49 theprop: $o ).
% 0.34/0.49
% 0.34/0.49 thf(theprop,definition,
% 0.34/0.49 ( theprop
% 0.34/0.49 = ( ! [X: $i] :
% 0.34/0.49 ( ( singleton @ X )
% 0.34/0.49 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(kfst_type,type,
% 0.34/0.49 kfst: $i > $i ).
% 0.34/0.49
% 0.34/0.49 thf(kfstpairEq_type,type,
% 0.34/0.49 kfstpairEq: $o ).
% 0.34/0.49
% 0.34/0.49 thf(kfstpairEq,definition,
% 0.34/0.49 ( kfstpairEq
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.34/0.49 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.34/0.49 = Xx ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(cartprodfstin_type,type,
% 0.34/0.49 cartprodfstin: $o ).
% 0.34/0.49
% 0.34/0.49 thf(cartprodfstin,definition,
% 0.34/0.49 ( cartprodfstin
% 0.34/0.49 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.34/0.49 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.34/0.49 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setukpairinjL2_type,type,
% 0.34/0.49 setukpairinjL2: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setukpairinjL2,definition,
% 0.34/0.49 ( setukpairinjL2
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.34/0.49 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.34/0.49 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.34/0.49 => ( Xx = Xz ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setukpairinjL_type,type,
% 0.34/0.49 setukpairinjL: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setukpairinjL,definition,
% 0.34/0.49 ( setukpairinjL
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.34/0.49 ( ( ( kpair @ Xx @ Xy )
% 0.34/0.49 = ( kpair @ Xz @ Xu ) )
% 0.34/0.49 => ( Xx = Xz ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setukpairinjR11_type,type,
% 0.34/0.49 setukpairinjR11: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setukpairinjR11,definition,
% 0.34/0.49 ( setukpairinjR11
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.34/0.49 ( ( Xx = Xy )
% 0.34/0.49 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.34/0.49 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setukpairinjR12_type,type,
% 0.34/0.49 setukpairinjR12: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setukpairinjR12,definition,
% 0.34/0.49 ( setukpairinjR12
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.34/0.49 ( ( Xx = Xy )
% 0.34/0.49 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.34/0.49 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setukpairinjR1_type,type,
% 0.34/0.49 setukpairinjR1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setukpairinjR1,definition,
% 0.34/0.49 ( setukpairinjR1
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.34/0.49 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.34/0.49 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.34/0.49 => ( ( Xz = Xu )
% 0.34/0.49 => ( Xy = Xu ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(upairequniteq,conjecture,
% 0.34/0.49 ( setextAx
% 0.34/0.49 => ( emptysetAx
% 0.34/0.49 => ( setadjoinAx
% 0.34/0.49 => ( powersetAx
% 0.34/0.49 => ( setunionAx
% 0.34/0.49 => ( omega0Ax
% 0.34/0.49 => ( omegaSAx
% 0.34/0.49 => ( omegaIndAx
% 0.34/0.49 => ( replAx
% 0.34/0.49 => ( foundationAx
% 0.34/0.49 => ( wellorderingAx
% 0.34/0.49 => ( descrp
% 0.34/0.49 => ( dsetconstrI
% 0.34/0.49 => ( dsetconstrEL
% 0.34/0.49 => ( dsetconstrER
% 0.34/0.49 => ( exuE1
% 0.34/0.49 => ( prop2setE
% 0.34/0.49 => ( emptysetE
% 0.34/0.49 => ( emptysetimpfalse
% 0.34/0.49 => ( notinemptyset
% 0.34/0.49 => ( exuE3e
% 0.34/0.49 => ( setext
% 0.34/0.49 => ( emptyI
% 0.34/0.49 => ( noeltsimpempty
% 0.34/0.49 => ( setbeta
% 0.34/0.49 => ( nonemptyE1
% 0.34/0.49 => ( nonemptyI
% 0.34/0.49 => ( nonemptyI1
% 0.34/0.49 => ( setadjoinIL
% 0.34/0.49 => ( emptyinunitempty
% 0.34/0.49 => ( setadjoinIR
% 0.34/0.49 => ( setadjoinE
% 0.34/0.49 => ( setadjoinOr
% 0.34/0.49 => ( setoftrueEq
% 0.34/0.49 => ( powersetI
% 0.34/0.49 => ( emptyinPowerset
% 0.34/0.49 => ( emptyInPowerset
% 0.34/0.49 => ( powersetE
% 0.34/0.49 => ( setunionI
% 0.34/0.49 => ( setunionE
% 0.34/0.49 => ( subPowSU
% 0.34/0.49 => ( exuE2
% 0.34/0.49 => ( nonemptyImpWitness
% 0.34/0.49 => ( uniqinunit
% 0.34/0.49 => ( notinsingleton
% 0.34/0.49 => ( eqinunit
% 0.34/0.49 => ( singletonsswitch
% 0.34/0.49 => ( upairsetE
% 0.34/0.49 => ( upairsetIL
% 0.34/0.49 => ( upairsetIR
% 0.34/0.49 => ( emptyE1
% 0.34/0.49 => ( vacuousDall
% 0.34/0.49 => ( quantDeMorgan1
% 0.34/0.49 => ( quantDeMorgan2
% 0.34/0.49 => ( quantDeMorgan3
% 0.34/0.49 => ( quantDeMorgan4
% 0.34/0.49 => ( prop2setI
% 0.34/0.49 => ( prop2set2propI
% 0.34/0.49 => ( notdexE
% 0.34/0.49 => ( notdallE
% 0.34/0.49 => ( exuI1
% 0.34/0.49 => ( exuI3
% 0.34/0.49 => ( exuI2
% 0.34/0.49 => ( inCongP
% 0.34/0.49 => ( in__Cong
% 0.34/0.49 => ( exuE3u
% 0.34/0.49 => ( exu__Cong
% 0.34/0.49 => ( emptyset__Cong
% 0.34/0.49 => ( setadjoin__Cong
% 0.34/0.49 => ( powerset__Cong
% 0.34/0.49 => ( setunion__Cong
% 0.34/0.49 => ( omega__Cong
% 0.34/0.49 => ( exuEu
% 0.34/0.49 => ( descr__Cong
% 0.34/0.49 => ( dsetconstr__Cong
% 0.34/0.49 => ( subsetI1
% 0.34/0.49 => ( eqimpsubset2
% 0.34/0.49 => ( eqimpsubset1
% 0.34/0.49 => ( subsetI2
% 0.34/0.49 => ( emptysetsubset
% 0.34/0.49 => ( subsetE
% 0.34/0.49 => ( subsetE2
% 0.34/0.49 => ( notsubsetI
% 0.34/0.49 => ( notequalI1
% 0.34/0.49 => ( notequalI2
% 0.34/0.49 => ( subsetRefl
% 0.34/0.49 => ( subsetTrans
% 0.34/0.49 => ( setadjoinSub
% 0.34/0.49 => ( setadjoinSub2
% 0.34/0.49 => ( subset2powerset
% 0.34/0.49 => ( setextsub
% 0.34/0.49 => ( subsetemptysetimpeq
% 0.34/0.49 => ( powersetI1
% 0.34/0.49 => ( powersetE1
% 0.34/0.49 => ( inPowerset
% 0.34/0.49 => ( powersetsubset
% 0.34/0.49 => ( sepInPowerset
% 0.34/0.49 => ( sepSubset
% 0.34/0.49 => ( binunionIL
% 0.34/0.49 => ( upairset2IR
% 0.34/0.49 => ( binunionIR
% 0.34/0.49 => ( binunionEcases
% 0.34/0.49 => ( binunionE
% 0.34/0.49 => ( binunionLsub
% 0.34/0.49 => ( binunionRsub
% 0.34/0.49 => ( binintersectI
% 0.34/0.49 => ( binintersectSubset5
% 0.34/0.49 => ( binintersectEL
% 0.34/0.49 => ( binintersectLsub
% 0.34/0.49 => ( binintersectSubset2
% 0.34/0.49 => ( binintersectSubset3
% 0.34/0.49 => ( binintersectER
% 0.34/0.49 => ( disjointsetsI1
% 0.34/0.49 => ( binintersectRsub
% 0.34/0.49 => ( binintersectSubset4
% 0.34/0.49 => ( binintersectSubset1
% 0.34/0.49 => ( bs114d
% 0.34/0.49 => ( setminusI
% 0.34/0.49 => ( setminusEL
% 0.34/0.49 => ( setminusER
% 0.34/0.49 => ( setminusSubset2
% 0.34/0.49 => ( setminusERneg
% 0.34/0.49 => ( setminusELneg
% 0.34/0.49 => ( setminusILneg
% 0.34/0.49 => ( setminusIRneg
% 0.34/0.49 => ( setminusLsub
% 0.34/0.49 => ( setminusSubset1
% 0.34/0.49 => ( symdiffE
% 0.34/0.49 => ( symdiffI1
% 0.34/0.49 => ( symdiffI2
% 0.34/0.49 => ( symdiffIneg1
% 0.34/0.49 => ( symdiffIneg2
% 0.34/0.49 => ( secondinupair
% 0.34/0.49 => ( setukpairIL
% 0.34/0.49 => ( setukpairIR
% 0.34/0.49 => ( kpairiskpair
% 0.34/0.49 => ( kpairp
% 0.34/0.49 => ( singletonsubset
% 0.34/0.49 => ( singletoninpowerset
% 0.34/0.49 => ( singletoninpowunion
% 0.34/0.49 => ( upairset2E
% 0.34/0.49 => ( upairsubunion
% 0.34/0.49 => ( upairinpowunion
% 0.34/0.49 => ( ubforcartprodlem1
% 0.34/0.49 => ( ubforcartprodlem2
% 0.34/0.49 => ( ubforcartprodlem3
% 0.34/0.49 => ( cartprodpairin
% 0.34/0.49 => ( cartprodmempair1
% 0.34/0.49 => ( cartprodmempair
% 0.34/0.49 => ( setunionE2
% 0.34/0.49 => ( setunionsingleton1
% 0.34/0.49 => ( setunionsingleton2
% 0.34/0.49 => ( setunionsingleton
% 0.34/0.49 => ( singletonprop
% 0.34/0.49 => ( ex1E1
% 0.34/0.49 => ( ex1I
% 0.34/0.49 => ( ex1I2
% 0.34/0.49 => ( singletonsuniq
% 0.34/0.49 => ( setukpairinjL1
% 0.34/0.49 => ( kfstsingleton
% 0.34/0.49 => ( theprop
% 0.34/0.49 => ( kfstpairEq
% 0.34/0.49 => ( cartprodfstin
% 0.34/0.49 => ( setukpairinjL2
% 0.34/0.49 => ( setukpairinjL
% 0.34/0.49 => ( setukpairinjR11
% 0.34/0.49 => ( setukpairinjR12
% 0.34/0.49 => ( setukpairinjR1
% 0.34/0.49 => ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.34/0.49 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.34/0.49 = ( setadjoin @ Xz @ emptyset ) )
% 0.34/0.49 => ( Xx = Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 %------------------------------------------------------------------------------
% 0.34/0.49 ------------------------
% 2.60/1.69 ---- Embedded in THF ---
% 2.60/1.69 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.60/1.69 %%% Generated on Tue Feb 24 12:29:24 EST 2026
% 2.60/1.69 %%% using '$$dhol' embedding, version 1.3.0.
% 2.60/1.69 %%% Logic specification used:
% 2.60/1.69 %%% thf(spec, logic, $$dhol).
% 2.60/1.69
% 2.60/1.69 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.19e6KiooJw/DTF2DTF26384.p
% See solution above
% 2.60/1.69 ------------------------
% 2.87/1.73 ---- Cleaned THF ---
% 2.87/1.73 thf(in_type,type,
% 2.87/1.73 in: $i > $i > $o ).
% 2.87/1.73
% 2.87/1.73 thf(exu_type,type,
% 2.87/1.73 exu: ( $i > $o ) > $o ).
% 2.87/1.73
% 2.87/1.73 thf(exu,definition,
% 2.87/1.73 ( exu
% 2.87/1.73 = ( ^ [Xphi: $i > $o] :
% 2.87/1.73 ? [Xx: $i] :
% 2.87/1.73 ( ( Xphi @ Xx )
% 2.87/1.73 & ! [Xy: $i] :
% 2.87/1.73 ( ( Xphi @ Xy )
% 2.87/1.73 => ( Xx = Xy ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setextAx_type,type,
% 2.87/1.73 setextAx: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setextAx,definition,
% 2.87/1.73 ( setextAx
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 <=> ( in @ Xx @ B ) )
% 2.87/1.73 => ( A = B ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(emptyset_type,type,
% 2.87/1.73 emptyset: $i ).
% 2.87/1.73
% 2.87/1.73 thf(emptysetAx_type,type,
% 2.87/1.73 emptysetAx: $o ).
% 2.87/1.73
% 2.87/1.73 thf(emptysetAx,definition,
% 2.87/1.73 ( emptysetAx
% 2.87/1.73 = ( ! [Xx: $i] :
% 2.87/1.73 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setadjoin_type,type,
% 2.87/1.73 setadjoin: $i > $i > $i ).
% 2.87/1.73
% 2.87/1.73 thf(setadjoinAx_type,type,
% 2.87/1.73 setadjoinAx: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setadjoinAx,definition,
% 2.87/1.73 ( setadjoinAx
% 2.87/1.73 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.87/1.73 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.87/1.73 <=> ( ( Xy = Xx )
% 2.87/1.73 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(powerset_type,type,
% 2.87/1.73 powerset: $i > $i ).
% 2.87/1.73
% 2.87/1.73 thf(powersetAx_type,type,
% 2.87/1.73 powersetAx: $o ).
% 2.87/1.73
% 2.87/1.73 thf(powersetAx,definition,
% 2.87/1.73 ( powersetAx
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ( in @ B @ ( powerset @ A ) )
% 2.87/1.73 <=> ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ B )
% 2.87/1.73 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setunion_type,type,
% 2.87/1.73 setunion: $i > $i ).
% 2.87/1.73
% 2.87/1.73 thf(setunionAx_type,type,
% 2.87/1.73 setunionAx: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setunionAx,definition,
% 2.87/1.73 ( setunionAx
% 2.87/1.73 = ( ! [A: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ ( setunion @ A ) )
% 2.87/1.73 <=> ? [B: $i] :
% 2.87/1.73 ( ( in @ Xx @ B )
% 2.87/1.73 & ( in @ B @ A ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(omega_type,type,
% 2.87/1.73 omega: $i ).
% 2.87/1.73
% 2.87/1.73 thf(omega0Ax_type,type,
% 2.87/1.73 omega0Ax: $o ).
% 2.87/1.73
% 2.87/1.73 thf(omega0Ax,definition,
% 2.87/1.73 ( omega0Ax
% 2.87/1.73 = ( in @ emptyset @ omega ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(omegaSAx_type,type,
% 2.87/1.73 omegaSAx: $o ).
% 2.87/1.73
% 2.87/1.73 thf(omegaSAx,definition,
% 2.87/1.73 ( omegaSAx
% 2.87/1.73 = ( ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ omega )
% 2.87/1.73 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(omegaIndAx_type,type,
% 2.87/1.73 omegaIndAx: $o ).
% 2.87/1.73
% 2.87/1.73 thf(omegaIndAx,definition,
% 2.87/1.73 ( omegaIndAx
% 2.87/1.73 = ( ! [A: $i] :
% 2.87/1.73 ( ( ( in @ emptyset @ A )
% 2.87/1.73 & ! [Xx: $i] :
% 2.87/1.73 ( ( ( in @ Xx @ omega )
% 2.87/1.73 & ( in @ Xx @ A ) )
% 2.87/1.73 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.87/1.73 => ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ omega )
% 2.87/1.73 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(replAx_type,type,
% 2.87/1.73 replAx: $o ).
% 2.87/1.73
% 2.87/1.73 thf(replAx,definition,
% 2.87/1.73 ( replAx
% 2.87/1.73 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.87/1.73 ( ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ( exu
% 2.87/1.73 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.87/1.73 => ? [B: $i] :
% 2.87/1.73 ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ B )
% 2.87/1.73 <=> ? [Xy: $i] :
% 2.87/1.73 ( ( in @ Xy @ A )
% 2.87/1.73 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(foundationAx_type,type,
% 2.87/1.73 foundationAx: $o ).
% 2.87/1.73
% 2.87/1.73 thf(foundationAx,definition,
% 2.87/1.73 ( foundationAx
% 2.87/1.73 = ( ! [A: $i] :
% 2.87/1.73 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.87/1.73 => ? [B: $i] :
% 2.87/1.73 ( ( in @ B @ A )
% 2.87/1.73 & ~ ? [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ B )
% 2.87/1.73 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(wellorderingAx_type,type,
% 2.87/1.73 wellorderingAx: $o ).
% 2.87/1.73
% 2.87/1.73 thf(wellorderingAx,definition,
% 2.87/1.73 ( wellorderingAx
% 2.87/1.73 = ( ! [A: $i] :
% 2.87/1.73 ? [B: $i] :
% 2.87/1.73 ( ! [C: $i] :
% 2.87/1.73 ( ( in @ C @ B )
% 2.87/1.73 => ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ C )
% 2.87/1.73 => ( in @ Xx @ A ) ) )
% 2.87/1.73 & ! [Xx: $i,Xy: $i] :
% 2.87/1.73 ( ( ( in @ Xx @ A )
% 2.87/1.73 & ( in @ Xy @ A ) )
% 2.87/1.73 => ( ! [C: $i] :
% 2.87/1.73 ( ( in @ C @ B )
% 2.87/1.73 => ( ( in @ Xx @ C )
% 2.87/1.73 <=> ( in @ Xy @ C ) ) )
% 2.87/1.73 => ( Xx = Xy ) ) )
% 2.87/1.73 & ! [C: $i,D: $i] :
% 2.87/1.73 ( ( ( in @ C @ B )
% 2.87/1.73 & ( in @ D @ B ) )
% 2.87/1.73 => ( ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ C )
% 2.87/1.73 => ( in @ Xx @ D ) )
% 2.87/1.73 | ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ D )
% 2.87/1.73 => ( in @ Xx @ C ) ) ) )
% 2.87/1.73 & ! [C: $i] :
% 2.87/1.73 ( ( ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ C )
% 2.87/1.73 => ( in @ Xx @ A ) )
% 2.87/1.73 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.87/1.73 => ? [D: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ D @ B )
% 2.87/1.73 & ( in @ Xx @ C )
% 2.87/1.73 & ~ ? [Xy: $i] :
% 2.87/1.73 ( ( in @ Xy @ D )
% 2.87/1.73 & ( in @ Xy @ C ) )
% 2.87/1.73 & ! [E: $i] :
% 2.87/1.73 ( ( in @ E @ B )
% 2.87/1.73 => ( ! [Xy: $i] :
% 2.87/1.73 ( ( in @ Xy @ E )
% 2.87/1.73 => ( in @ Xy @ D ) )
% 2.87/1.73 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(descr_type,type,
% 2.87/1.73 descr: ( $i > $o ) > $i ).
% 2.87/1.73
% 2.87/1.73 thf(descrp_type,type,
% 2.87/1.73 descrp: $o ).
% 2.87/1.73
% 2.87/1.73 thf(descrp,definition,
% 2.87/1.73 ( descrp
% 2.87/1.73 = ( ! [Xphi: $i > $o] :
% 2.87/1.73 ( ( exu
% 2.87/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.87/1.73 => ( Xphi
% 2.87/1.73 @ ( descr
% 2.87/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(dsetconstr_type,type,
% 2.87/1.73 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.87/1.73
% 2.87/1.73 thf(dsetconstrI_type,type,
% 2.87/1.73 dsetconstrI: $o ).
% 2.87/1.73
% 2.87/1.73 thf(dsetconstrI,definition,
% 2.87/1.73 ( dsetconstrI
% 2.87/1.73 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ( ( Xphi @ Xx )
% 2.87/1.73 => ( in @ Xx
% 2.87/1.73 @ ( dsetconstr @ A
% 2.87/1.73 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(dsetconstrEL_type,type,
% 2.87/1.73 dsetconstrEL: $o ).
% 2.87/1.73
% 2.87/1.73 thf(dsetconstrEL,definition,
% 2.87/1.73 ( dsetconstrEL
% 2.87/1.73 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx
% 2.87/1.73 @ ( dsetconstr @ A
% 2.87/1.73 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.87/1.73 => ( in @ Xx @ A ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(dsetconstrER_type,type,
% 2.87/1.73 dsetconstrER: $o ).
% 2.87/1.73
% 2.87/1.73 thf(dsetconstrER,definition,
% 2.87/1.73 ( dsetconstrER
% 2.87/1.73 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx
% 2.87/1.73 @ ( dsetconstr @ A
% 2.87/1.73 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.87/1.73 => ( Xphi @ Xx ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(exuE1_type,type,
% 2.87/1.73 exuE1: $o ).
% 2.87/1.73
% 2.87/1.73 thf(exuE1,definition,
% 2.87/1.73 ( exuE1
% 2.87/1.73 = ( ! [Xphi: $i > $o] :
% 2.87/1.73 ( ( exu
% 2.87/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.87/1.73 => ? [Xx: $i] :
% 2.87/1.73 ( ( Xphi @ Xx )
% 2.87/1.73 & ! [Xy: $i] :
% 2.87/1.73 ( ( Xphi @ Xy )
% 2.87/1.73 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(prop2set_type,type,
% 2.87/1.73 prop2set: $o > $i ).
% 2.87/1.73
% 2.87/1.73 thf(prop2setE_type,type,
% 2.87/1.73 prop2setE: $o ).
% 2.87/1.73
% 2.87/1.73 thf(prop2setE,definition,
% 2.87/1.73 ( prop2setE
% 2.87/1.73 = ( ! [Xphi: $o,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.87/1.73 => Xphi ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(emptysetE_type,type,
% 2.87/1.73 emptysetE: $o ).
% 2.87/1.73
% 2.87/1.73 thf(emptysetE,definition,
% 2.87/1.73 ( emptysetE
% 2.87/1.73 = ( ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ emptyset )
% 2.87/1.73 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(emptysetimpfalse_type,type,
% 2.87/1.73 emptysetimpfalse: $o ).
% 2.87/1.73
% 2.87/1.73 thf(emptysetimpfalse,definition,
% 2.87/1.73 ( emptysetimpfalse
% 2.87/1.73 = ( ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ emptyset )
% 2.87/1.73 => $false ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(notinemptyset_type,type,
% 2.87/1.73 notinemptyset: $o ).
% 2.87/1.73
% 2.87/1.73 thf(notinemptyset,definition,
% 2.87/1.73 ( notinemptyset
% 2.87/1.73 = ( ! [Xx: $i] :
% 2.87/1.73 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(exuE3e_type,type,
% 2.87/1.73 exuE3e: $o ).
% 2.87/1.73
% 2.87/1.73 thf(exuE3e,definition,
% 2.87/1.73 ( exuE3e
% 2.87/1.73 = ( ! [Xphi: $i > $o] :
% 2.87/1.73 ( ( exu
% 2.87/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.87/1.73 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setext_type,type,
% 2.87/1.73 setext: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setext,definition,
% 2.87/1.73 ( setext
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ( in @ Xx @ B ) )
% 2.87/1.73 => ( ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ B )
% 2.87/1.73 => ( in @ Xx @ A ) )
% 2.87/1.73 => ( A = B ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(emptyI_type,type,
% 2.87/1.73 emptyI: $o ).
% 2.87/1.73
% 2.87/1.73 thf(emptyI,definition,
% 2.87/1.73 ( emptyI
% 2.87/1.73 = ( ! [A: $i] :
% 2.87/1.73 ( ! [Xx: $i] :
% 2.87/1.73 ~ ( in @ Xx @ A )
% 2.87/1.73 => ( A = emptyset ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(noeltsimpempty_type,type,
% 2.87/1.73 noeltsimpempty: $o ).
% 2.87/1.73
% 2.87/1.73 thf(noeltsimpempty,definition,
% 2.87/1.73 ( noeltsimpempty
% 2.87/1.73 = ( ! [A: $i] :
% 2.87/1.73 ( ! [Xx: $i] :
% 2.87/1.73 ~ ( in @ Xx @ A )
% 2.87/1.73 => ( A = emptyset ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setbeta_type,type,
% 2.87/1.73 setbeta: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setbeta,definition,
% 2.87/1.73 ( setbeta
% 2.87/1.73 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ( ( in @ Xx
% 2.87/1.73 @ ( dsetconstr @ A
% 2.87/1.73 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.87/1.73 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(nonempty_type,type,
% 2.87/1.73 nonempty: $i > $o ).
% 2.87/1.73
% 2.87/1.73 thf(nonempty,definition,
% 2.87/1.73 ( nonempty
% 2.87/1.73 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(nonemptyE1_type,type,
% 2.87/1.73 nonemptyE1: $o ).
% 2.87/1.73
% 2.87/1.73 thf(nonemptyE1,definition,
% 2.87/1.73 ( nonemptyE1
% 2.87/1.73 = ( ! [A: $i] :
% 2.87/1.73 ( ( nonempty @ A )
% 2.87/1.73 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(nonemptyI_type,type,
% 2.87/1.73 nonemptyI: $o ).
% 2.87/1.73
% 2.87/1.73 thf(nonemptyI,definition,
% 2.87/1.73 ( nonemptyI
% 2.87/1.73 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ( ( Xphi @ Xx )
% 2.87/1.73 => ( nonempty
% 2.87/1.73 @ ( dsetconstr @ A
% 2.87/1.73 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(nonemptyI1_type,type,
% 2.87/1.73 nonemptyI1: $o ).
% 2.87/1.73
% 2.87/1.73 thf(nonemptyI1,definition,
% 2.87/1.73 ( nonemptyI1
% 2.87/1.73 = ( ! [A: $i] :
% 2.87/1.73 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.87/1.73 => ( nonempty @ A ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setadjoinIL_type,type,
% 2.87/1.73 setadjoinIL: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setadjoinIL,definition,
% 2.87/1.73 ( setadjoinIL
% 2.87/1.73 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(emptyinunitempty_type,type,
% 2.87/1.73 emptyinunitempty: $o ).
% 2.87/1.73
% 2.87/1.73 thf(emptyinunitempty,definition,
% 2.87/1.73 ( emptyinunitempty
% 2.87/1.73 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setadjoinIR_type,type,
% 2.87/1.73 setadjoinIR: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setadjoinIR,definition,
% 2.87/1.73 ( setadjoinIR
% 2.87/1.73 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.87/1.73 ( ( in @ Xy @ A )
% 2.87/1.73 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setadjoinE_type,type,
% 2.87/1.73 setadjoinE: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setadjoinE,definition,
% 2.87/1.73 ( setadjoinE
% 2.87/1.73 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.87/1.73 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.87/1.73 => ! [Xphi: $o] :
% 2.87/1.73 ( ( ( Xy = Xx )
% 2.87/1.73 => Xphi )
% 2.87/1.73 => ( ( ( in @ Xy @ A )
% 2.87/1.73 => Xphi )
% 2.87/1.73 => Xphi ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setadjoinOr_type,type,
% 2.87/1.73 setadjoinOr: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setadjoinOr,definition,
% 2.87/1.73 ( setadjoinOr
% 2.87/1.73 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.87/1.73 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.87/1.73 => ( ( Xy = Xx )
% 2.87/1.73 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setoftrueEq_type,type,
% 2.87/1.73 setoftrueEq: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setoftrueEq,definition,
% 2.87/1.73 ( setoftrueEq
% 2.87/1.73 = ( ! [A: $i] :
% 2.87/1.73 ( ( dsetconstr @ A
% 2.87/1.73 @ ^ [Xx: $i] : $true )
% 2.87/1.73 = A ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(powersetI_type,type,
% 2.87/1.73 powersetI: $o ).
% 2.87/1.73
% 2.87/1.73 thf(powersetI,definition,
% 2.87/1.73 ( powersetI
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ B )
% 2.87/1.73 => ( in @ Xx @ A ) )
% 2.87/1.73 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(emptyinPowerset_type,type,
% 2.87/1.73 emptyinPowerset: $o ).
% 2.87/1.73
% 2.87/1.73 thf(emptyinPowerset,definition,
% 2.87/1.73 ( emptyinPowerset
% 2.87/1.73 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(emptyInPowerset_type,type,
% 2.87/1.73 emptyInPowerset: $o ).
% 2.87/1.73
% 2.87/1.73 thf(emptyInPowerset,definition,
% 2.87/1.73 ( emptyInPowerset
% 2.87/1.73 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(powersetE_type,type,
% 2.87/1.73 powersetE: $o ).
% 2.87/1.73
% 2.87/1.73 thf(powersetE,definition,
% 2.87/1.73 ( powersetE
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ B @ ( powerset @ A ) )
% 2.87/1.73 => ( ( in @ Xx @ B )
% 2.87/1.73 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setunionI_type,type,
% 2.87/1.73 setunionI: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setunionI,definition,
% 2.87/1.73 ( setunionI
% 2.87/1.73 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.87/1.73 ( ( in @ Xx @ B )
% 2.87/1.73 => ( ( in @ B @ A )
% 2.87/1.73 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setunionE_type,type,
% 2.87/1.73 setunionE: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setunionE,definition,
% 2.87/1.73 ( setunionE
% 2.87/1.73 = ( ! [A: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ ( setunion @ A ) )
% 2.87/1.73 => ! [Xphi: $o] :
% 2.87/1.73 ( ! [B: $i] :
% 2.87/1.73 ( ( in @ Xx @ B )
% 2.87/1.73 => ( ( in @ B @ A )
% 2.87/1.73 => Xphi ) )
% 2.87/1.73 => Xphi ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(subPowSU_type,type,
% 2.87/1.73 subPowSU: $o ).
% 2.87/1.73
% 2.87/1.73 thf(subPowSU,definition,
% 2.87/1.73 ( subPowSU
% 2.87/1.73 = ( ! [A: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(exuE2_type,type,
% 2.87/1.73 exuE2: $o ).
% 2.87/1.73
% 2.87/1.73 thf(exuE2,definition,
% 2.87/1.73 ( exuE2
% 2.87/1.73 = ( ! [Xphi: $i > $o] :
% 2.87/1.73 ( ( exu
% 2.87/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.87/1.73 => ? [Xx: $i] :
% 2.87/1.73 ! [Xy: $i] :
% 2.87/1.73 ( ( Xphi @ Xy )
% 2.87/1.73 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(nonemptyImpWitness_type,type,
% 2.87/1.73 nonemptyImpWitness: $o ).
% 2.87/1.73
% 2.87/1.73 thf(nonemptyImpWitness,definition,
% 2.87/1.73 ( nonemptyImpWitness
% 2.87/1.73 = ( ! [A: $i] :
% 2.87/1.73 ( ( nonempty @ A )
% 2.87/1.73 => ? [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 & $true ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(uniqinunit_type,type,
% 2.87/1.73 uniqinunit: $o ).
% 2.87/1.73
% 2.87/1.73 thf(uniqinunit,definition,
% 2.87/1.73 ( uniqinunit
% 2.87/1.73 = ( ! [Xx: $i,Xy: $i] :
% 2.87/1.73 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.87/1.73 => ( Xx = Xy ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(notinsingleton_type,type,
% 2.87/1.73 notinsingleton: $o ).
% 2.87/1.73
% 2.87/1.73 thf(notinsingleton,definition,
% 2.87/1.73 ( notinsingleton
% 2.87/1.73 = ( ! [Xx: $i,Xy: $i] :
% 2.87/1.73 ( ( Xx != Xy )
% 2.87/1.73 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(eqinunit_type,type,
% 2.87/1.73 eqinunit: $o ).
% 2.87/1.73
% 2.87/1.73 thf(eqinunit,definition,
% 2.87/1.73 ( eqinunit
% 2.87/1.73 = ( ! [Xx: $i,Xy: $i] :
% 2.87/1.73 ( ( Xx = Xy )
% 2.87/1.73 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(singletonsswitch_type,type,
% 2.87/1.73 singletonsswitch: $o ).
% 2.87/1.73
% 2.87/1.73 thf(singletonsswitch,definition,
% 2.87/1.73 ( singletonsswitch
% 2.87/1.73 = ( ! [Xx: $i,Xy: $i] :
% 2.87/1.73 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.87/1.73 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(upairsetE_type,type,
% 2.87/1.73 upairsetE: $o ).
% 2.87/1.73
% 2.87/1.73 thf(upairsetE,definition,
% 2.87/1.73 ( upairsetE
% 2.87/1.73 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.87/1.73 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.87/1.73 => ( ( Xz = Xx )
% 2.87/1.73 | ( Xz = Xy ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(upairsetIL_type,type,
% 2.87/1.73 upairsetIL: $o ).
% 2.87/1.73
% 2.87/1.73 thf(upairsetIL,definition,
% 2.87/1.73 ( upairsetIL
% 2.87/1.73 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(upairsetIR_type,type,
% 2.87/1.73 upairsetIR: $o ).
% 2.87/1.73
% 2.87/1.73 thf(upairsetIR,definition,
% 2.87/1.73 ( upairsetIR
% 2.87/1.73 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(emptyE1_type,type,
% 2.87/1.73 emptyE1: $o ).
% 2.87/1.73
% 2.87/1.73 thf(emptyE1,definition,
% 2.87/1.73 ( emptyE1
% 2.87/1.73 = ( ! [A: $i,Xphi: $i > $o] :
% 2.87/1.73 ( ? [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 & ( Xphi @ Xx ) )
% 2.87/1.73 => ( ( ( dsetconstr @ A
% 2.87/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.87/1.73 = emptyset )
% 2.87/1.73 => $false ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(vacuousDall_type,type,
% 2.87/1.73 vacuousDall: $o ).
% 2.87/1.73
% 2.87/1.73 thf(vacuousDall,definition,
% 2.87/1.73 ( vacuousDall
% 2.87/1.73 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ emptyset )
% 2.87/1.73 => ( Xphi @ Xx ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(quantDeMorgan1_type,type,
% 2.87/1.73 quantDeMorgan1: $o ).
% 2.87/1.73
% 2.87/1.73 thf(quantDeMorgan1,definition,
% 2.87/1.73 ( quantDeMorgan1
% 2.87/1.73 = ( ! [A: $i,Xphi: $i > $o] :
% 2.87/1.73 ( ~ ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ( Xphi @ Xx ) )
% 2.87/1.73 => ? [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(quantDeMorgan2_type,type,
% 2.87/1.73 quantDeMorgan2: $o ).
% 2.87/1.73
% 2.87/1.73 thf(quantDeMorgan2,definition,
% 2.87/1.73 ( quantDeMorgan2
% 2.87/1.73 = ( ! [A: $i,Xphi: $i > $o] :
% 2.87/1.73 ( ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ~ ( Xphi @ Xx ) )
% 2.87/1.73 => ~ ? [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(quantDeMorgan3_type,type,
% 2.87/1.73 quantDeMorgan3: $o ).
% 2.87/1.73
% 2.87/1.73 thf(quantDeMorgan3,definition,
% 2.87/1.73 ( quantDeMorgan3
% 2.87/1.73 = ( ! [A: $i,Xphi: $i > $o] :
% 2.87/1.73 ( ~ ? [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 & ( Xphi @ Xx ) )
% 2.87/1.73 => ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(quantDeMorgan4_type,type,
% 2.87/1.73 quantDeMorgan4: $o ).
% 2.87/1.73
% 2.87/1.73 thf(quantDeMorgan4,definition,
% 2.87/1.73 ( quantDeMorgan4
% 2.87/1.73 = ( ! [A: $i,Xphi: $i > $o] :
% 2.87/1.73 ( ? [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 & ~ ( Xphi @ Xx ) )
% 2.87/1.73 => ~ ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(prop2setI_type,type,
% 2.87/1.73 prop2setI: $o ).
% 2.87/1.73
% 2.87/1.73 thf(prop2setI,definition,
% 2.87/1.73 ( prop2setI
% 2.87/1.73 = ( ! [Xphi: $o] :
% 2.87/1.73 ( Xphi
% 2.87/1.73 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(set2prop_type,type,
% 2.87/1.73 set2prop: $i > $o ).
% 2.87/1.73
% 2.87/1.73 thf(prop2set2propI_type,type,
% 2.87/1.73 prop2set2propI: $o ).
% 2.87/1.73
% 2.87/1.73 thf(prop2set2propI,definition,
% 2.87/1.73 ( prop2set2propI
% 2.87/1.73 = ( ! [Xphi: $o] :
% 2.87/1.73 ( Xphi
% 2.87/1.73 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(notdexE_type,type,
% 2.87/1.73 notdexE: $o ).
% 2.87/1.73
% 2.87/1.73 thf(notdexE,definition,
% 2.87/1.73 ( notdexE
% 2.87/1.73 = ( ! [A: $i,Xphi: $i > $o] :
% 2.87/1.73 ( ~ ? [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 & ( Xphi @ Xx ) )
% 2.87/1.73 => ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(notdallE_type,type,
% 2.87/1.73 notdallE: $o ).
% 2.87/1.73
% 2.87/1.73 thf(notdallE,definition,
% 2.87/1.73 ( notdallE
% 2.87/1.73 = ( ! [A: $i,Xphi: $i > $o] :
% 2.87/1.73 ( ~ ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ( Xphi @ Xx ) )
% 2.87/1.73 => ? [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(exuI1_type,type,
% 2.87/1.73 exuI1: $o ).
% 2.87/1.73
% 2.87/1.73 thf(exuI1,definition,
% 2.87/1.73 ( exuI1
% 2.87/1.73 = ( ! [Xphi: $i > $o] :
% 2.87/1.73 ( ? [Xx: $i] :
% 2.87/1.73 ( ( Xphi @ Xx )
% 2.87/1.73 & ! [Xy: $i] :
% 2.87/1.73 ( ( Xphi @ Xy )
% 2.87/1.73 => ( Xx = Xy ) ) )
% 2.87/1.73 => ( exu
% 2.87/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(exuI3_type,type,
% 2.87/1.73 exuI3: $o ).
% 2.87/1.73
% 2.87/1.73 thf(exuI3,definition,
% 2.87/1.73 ( exuI3
% 2.87/1.73 = ( ! [Xphi: $i > $o] :
% 2.87/1.73 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.87/1.73 => ( ! [Xx: $i,Xy: $i] :
% 2.87/1.73 ( ( Xphi @ Xx )
% 2.87/1.73 => ( ( Xphi @ Xy )
% 2.87/1.73 => ( Xx = Xy ) ) )
% 2.87/1.73 => ( exu
% 2.87/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(exuI2_type,type,
% 2.87/1.73 exuI2: $o ).
% 2.87/1.73
% 2.87/1.73 thf(exuI2,definition,
% 2.87/1.73 ( exuI2
% 2.87/1.73 = ( ! [Xphi: $i > $o] :
% 2.87/1.73 ( ? [Xx: $i] :
% 2.87/1.73 ! [Xy: $i] :
% 2.87/1.73 ( ( Xphi @ Xy )
% 2.87/1.73 <=> ( Xy = Xx ) )
% 2.87/1.73 => ( exu
% 2.87/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(inCongP_type,type,
% 2.87/1.73 inCongP: $o ).
% 2.87/1.73
% 2.87/1.73 thf(inCongP,definition,
% 2.87/1.73 ( inCongP
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ( A = B )
% 2.87/1.73 => ! [Xx: $i,Xy: $i] :
% 2.87/1.73 ( ( Xx = Xy )
% 2.87/1.73 => ( ( in @ Xx @ A )
% 2.87/1.73 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(in__Cong_type,type,
% 2.87/1.73 in__Cong: $o ).
% 2.87/1.73
% 2.87/1.73 thf(in__Cong,definition,
% 2.87/1.73 ( in__Cong
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ( A = B )
% 2.87/1.73 => ! [Xx: $i,Xy: $i] :
% 2.87/1.73 ( ( Xx = Xy )
% 2.87/1.73 => ( ( in @ Xx @ A )
% 2.87/1.73 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(exuE3u_type,type,
% 2.87/1.73 exuE3u: $o ).
% 2.87/1.73
% 2.87/1.73 thf(exuE3u,definition,
% 2.87/1.73 ( exuE3u
% 2.87/1.73 = ( ! [Xphi: $i > $o] :
% 2.87/1.73 ( ( exu
% 2.87/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.87/1.73 => ! [Xx: $i,Xy: $i] :
% 2.87/1.73 ( ( Xphi @ Xx )
% 2.87/1.73 => ( ( Xphi @ Xy )
% 2.87/1.73 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(exu__Cong_type,type,
% 2.87/1.73 exu__Cong: $o ).
% 2.87/1.73
% 2.87/1.73 thf(exu__Cong,definition,
% 2.87/1.73 ( exu__Cong
% 2.87/1.73 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.87/1.73 ( ! [Xx: $i,Xy: $i] :
% 2.87/1.73 ( ( Xx = Xy )
% 2.87/1.73 => ( ( Xphi @ Xx )
% 2.87/1.73 <=> ( Xpsi @ Xy ) ) )
% 2.87/1.73 => ( ( exu
% 2.87/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.87/1.73 <=> ( exu
% 2.87/1.73 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(emptyset__Cong_type,type,
% 2.87/1.73 emptyset__Cong: $o ).
% 2.87/1.73
% 2.87/1.73 thf(emptyset__Cong,definition,
% 2.87/1.73 ( emptyset__Cong
% 2.87/1.73 = ( emptyset = emptyset ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setadjoin__Cong_type,type,
% 2.87/1.73 setadjoin__Cong: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setadjoin__Cong,definition,
% 2.87/1.73 ( setadjoin__Cong
% 2.87/1.73 = ( ! [Xx: $i,Xy: $i] :
% 2.87/1.73 ( ( Xx = Xy )
% 2.87/1.73 => ! [Xz: $i,Xu: $i] :
% 2.87/1.73 ( ( Xz = Xu )
% 2.87/1.73 => ( ( setadjoin @ Xx @ Xz )
% 2.87/1.73 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(powerset__Cong_type,type,
% 2.87/1.73 powerset__Cong: $o ).
% 2.87/1.73
% 2.87/1.73 thf(powerset__Cong,definition,
% 2.87/1.73 ( powerset__Cong
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ( A = B )
% 2.87/1.73 => ( ( powerset @ A )
% 2.87/1.73 = ( powerset @ B ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setunion__Cong_type,type,
% 2.87/1.73 setunion__Cong: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setunion__Cong,definition,
% 2.87/1.73 ( setunion__Cong
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ( A = B )
% 2.87/1.73 => ( ( setunion @ A )
% 2.87/1.73 = ( setunion @ B ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(omega__Cong_type,type,
% 2.87/1.73 omega__Cong: $o ).
% 2.87/1.73
% 2.87/1.73 thf(omega__Cong,definition,
% 2.87/1.73 ( omega__Cong
% 2.87/1.73 = ( omega = omega ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(exuEu_type,type,
% 2.87/1.73 exuEu: $o ).
% 2.87/1.73
% 2.87/1.73 thf(exuEu,definition,
% 2.87/1.73 ( exuEu
% 2.87/1.73 = ( ! [Xphi: $i > $o] :
% 2.87/1.73 ( ( exu
% 2.87/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.87/1.73 => ! [Xx: $i,Xy: $i] :
% 2.87/1.73 ( ( Xphi @ Xx )
% 2.87/1.73 => ( ( Xphi @ Xy )
% 2.87/1.73 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(descr__Cong_type,type,
% 2.87/1.73 descr__Cong: $o ).
% 2.87/1.73
% 2.87/1.73 thf(descr__Cong,definition,
% 2.87/1.73 ( descr__Cong
% 2.87/1.73 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.87/1.73 ( ! [Xx: $i,Xy: $i] :
% 2.87/1.73 ( ( Xx = Xy )
% 2.87/1.73 => ( ( Xphi @ Xx )
% 2.87/1.73 <=> ( Xpsi @ Xy ) ) )
% 2.87/1.73 => ( ( exu
% 2.87/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.87/1.73 => ( ( exu
% 2.87/1.73 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.87/1.73 => ( ( descr
% 2.87/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.87/1.73 = ( descr
% 2.87/1.73 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(dsetconstr__Cong_type,type,
% 2.87/1.73 dsetconstr__Cong: $o ).
% 2.87/1.73
% 2.87/1.73 thf(dsetconstr__Cong,definition,
% 2.87/1.73 ( dsetconstr__Cong
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ( A = B )
% 2.87/1.73 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.87/1.73 ( ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ! [Xy: $i] :
% 2.87/1.73 ( ( in @ Xy @ B )
% 2.87/1.73 => ( ( Xx = Xy )
% 2.87/1.73 => ( ( Xphi @ Xx )
% 2.87/1.73 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.87/1.73 => ( ( dsetconstr @ A
% 2.87/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.87/1.73 = ( dsetconstr @ B
% 2.87/1.73 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(subset_type,type,
% 2.87/1.73 subset: $i > $i > $o ).
% 2.87/1.73
% 2.87/1.73 thf(disjoint_type,type,
% 2.87/1.73 disjoint: $i > $i > $o ).
% 2.87/1.73
% 2.87/1.73 thf(setsmeet_type,type,
% 2.87/1.73 setsmeet: $i > $i > $o ).
% 2.87/1.73
% 2.87/1.73 thf(subsetI1_type,type,
% 2.87/1.73 subsetI1: $o ).
% 2.87/1.73
% 2.87/1.73 thf(subsetI1,definition,
% 2.87/1.73 ( subsetI1
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ( in @ Xx @ B ) )
% 2.87/1.73 => ( subset @ A @ B ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(eqimpsubset2_type,type,
% 2.87/1.73 eqimpsubset2: $o ).
% 2.87/1.73
% 2.87/1.73 thf(eqimpsubset2,definition,
% 2.87/1.73 ( eqimpsubset2
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ( A = B )
% 2.87/1.73 => ( subset @ B @ A ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(eqimpsubset1_type,type,
% 2.87/1.73 eqimpsubset1: $o ).
% 2.87/1.73
% 2.87/1.73 thf(eqimpsubset1,definition,
% 2.87/1.73 ( eqimpsubset1
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ( A = B )
% 2.87/1.73 => ( subset @ A @ B ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(subsetI2_type,type,
% 2.87/1.73 subsetI2: $o ).
% 2.87/1.73
% 2.87/1.73 thf(subsetI2,definition,
% 2.87/1.73 ( subsetI2
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ( in @ Xx @ B ) )
% 2.87/1.73 => ( subset @ A @ B ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(emptysetsubset_type,type,
% 2.87/1.73 emptysetsubset: $o ).
% 2.87/1.73
% 2.87/1.73 thf(emptysetsubset,definition,
% 2.87/1.73 ( emptysetsubset
% 2.87/1.73 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(subsetE_type,type,
% 2.87/1.73 subsetE: $o ).
% 2.87/1.73
% 2.87/1.73 thf(subsetE,definition,
% 2.87/1.73 ( subsetE
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ( subset @ A @ B )
% 2.87/1.73 => ( ( in @ Xx @ A )
% 2.87/1.73 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(subsetE2_type,type,
% 2.87/1.73 subsetE2: $o ).
% 2.87/1.73
% 2.87/1.73 thf(subsetE2,definition,
% 2.87/1.73 ( subsetE2
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ( subset @ A @ B )
% 2.87/1.73 => ( ~ ( in @ Xx @ B )
% 2.87/1.73 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(notsubsetI_type,type,
% 2.87/1.73 notsubsetI: $o ).
% 2.87/1.73
% 2.87/1.73 thf(notsubsetI,definition,
% 2.87/1.73 ( notsubsetI
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ( ~ ( in @ Xx @ B )
% 2.87/1.73 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(notequalI1_type,type,
% 2.87/1.73 notequalI1: $o ).
% 2.87/1.73
% 2.87/1.73 thf(notequalI1,definition,
% 2.87/1.73 ( notequalI1
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ~ ( subset @ A @ B )
% 2.87/1.73 => ( A != B ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(notequalI2_type,type,
% 2.87/1.73 notequalI2: $o ).
% 2.87/1.73
% 2.87/1.73 thf(notequalI2,definition,
% 2.87/1.73 ( notequalI2
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ( ~ ( in @ Xx @ B )
% 2.87/1.73 => ( A != B ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(subsetRefl_type,type,
% 2.87/1.73 subsetRefl: $o ).
% 2.87/1.73
% 2.87/1.73 thf(subsetRefl,definition,
% 2.87/1.73 ( subsetRefl
% 2.87/1.73 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(subsetTrans_type,type,
% 2.87/1.73 subsetTrans: $o ).
% 2.87/1.73
% 2.87/1.73 thf(subsetTrans,definition,
% 2.87/1.73 ( subsetTrans
% 2.87/1.73 = ( ! [A: $i,B: $i,C: $i] :
% 2.87/1.73 ( ( subset @ A @ B )
% 2.87/1.73 => ( ( subset @ B @ C )
% 2.87/1.73 => ( subset @ A @ C ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setadjoinSub_type,type,
% 2.87/1.73 setadjoinSub: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setadjoinSub,definition,
% 2.87/1.73 ( setadjoinSub
% 2.87/1.73 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setadjoinSub2_type,type,
% 2.87/1.73 setadjoinSub2: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setadjoinSub2,definition,
% 2.87/1.73 ( setadjoinSub2
% 2.87/1.73 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.87/1.73 ( ( subset @ A @ B )
% 2.87/1.73 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(subset2powerset_type,type,
% 2.87/1.73 subset2powerset: $o ).
% 2.87/1.73
% 2.87/1.73 thf(subset2powerset,definition,
% 2.87/1.73 ( subset2powerset
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ( subset @ A @ B )
% 2.87/1.73 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setextsub_type,type,
% 2.87/1.73 setextsub: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setextsub,definition,
% 2.87/1.73 ( setextsub
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ( subset @ A @ B )
% 2.87/1.73 => ( ( subset @ B @ A )
% 2.87/1.73 => ( A = B ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(subsetemptysetimpeq_type,type,
% 2.87/1.73 subsetemptysetimpeq: $o ).
% 2.87/1.73
% 2.87/1.73 thf(subsetemptysetimpeq,definition,
% 2.87/1.73 ( subsetemptysetimpeq
% 2.87/1.73 = ( ! [A: $i] :
% 2.87/1.73 ( ( subset @ A @ emptyset )
% 2.87/1.73 => ( A = emptyset ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(powersetI1_type,type,
% 2.87/1.73 powersetI1: $o ).
% 2.87/1.73
% 2.87/1.73 thf(powersetI1,definition,
% 2.87/1.73 ( powersetI1
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ( subset @ B @ A )
% 2.87/1.73 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(powersetE1_type,type,
% 2.87/1.73 powersetE1: $o ).
% 2.87/1.73
% 2.87/1.73 thf(powersetE1,definition,
% 2.87/1.73 ( powersetE1
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ( in @ B @ ( powerset @ A ) )
% 2.87/1.73 => ( subset @ B @ A ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(inPowerset_type,type,
% 2.87/1.73 inPowerset: $o ).
% 2.87/1.73
% 2.87/1.73 thf(inPowerset,definition,
% 2.87/1.73 ( inPowerset
% 2.87/1.73 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(powersetsubset_type,type,
% 2.87/1.73 powersetsubset: $o ).
% 2.87/1.73
% 2.87/1.73 thf(powersetsubset,definition,
% 2.87/1.73 ( powersetsubset
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ( subset @ A @ B )
% 2.87/1.73 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(sepInPowerset_type,type,
% 2.87/1.73 sepInPowerset: $o ).
% 2.87/1.73
% 2.87/1.73 thf(sepInPowerset,definition,
% 2.87/1.73 ( sepInPowerset
% 2.87/1.73 = ( ! [A: $i,Xphi: $i > $o] :
% 2.87/1.73 ( in
% 2.87/1.73 @ ( dsetconstr @ A
% 2.87/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.87/1.73 @ ( powerset @ A ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(sepSubset_type,type,
% 2.87/1.73 sepSubset: $o ).
% 2.87/1.73
% 2.87/1.73 thf(sepSubset,definition,
% 2.87/1.73 ( sepSubset
% 2.87/1.73 = ( ! [A: $i,Xphi: $i > $o] :
% 2.87/1.73 ( subset
% 2.87/1.73 @ ( dsetconstr @ A
% 2.87/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.87/1.73 @ A ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(binunion_type,type,
% 2.87/1.73 binunion: $i > $i > $i ).
% 2.87/1.73
% 2.87/1.73 thf(binunionIL_type,type,
% 2.87/1.73 binunionIL: $o ).
% 2.87/1.73
% 2.87/1.73 thf(binunionIL,definition,
% 2.87/1.73 ( binunionIL
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(upairset2IR_type,type,
% 2.87/1.73 upairset2IR: $o ).
% 2.87/1.73
% 2.87/1.73 thf(upairset2IR,definition,
% 2.87/1.73 ( upairset2IR
% 2.87/1.73 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(binunionIR_type,type,
% 2.87/1.73 binunionIR: $o ).
% 2.87/1.73
% 2.87/1.73 thf(binunionIR,definition,
% 2.87/1.73 ( binunionIR
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ B )
% 2.87/1.73 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(binunionEcases_type,type,
% 2.87/1.73 binunionEcases: $o ).
% 2.87/1.73
% 2.87/1.73 thf(binunionEcases,definition,
% 2.87/1.73 ( binunionEcases
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.87/1.73 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.87/1.73 => ( ( ( in @ Xx @ A )
% 2.87/1.73 => Xphi )
% 2.87/1.73 => ( ( ( in @ Xx @ B )
% 2.87/1.73 => Xphi )
% 2.87/1.73 => Xphi ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(binunionE_type,type,
% 2.87/1.73 binunionE: $o ).
% 2.87/1.73
% 2.87/1.73 thf(binunionE,definition,
% 2.87/1.73 ( binunionE
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.87/1.73 => ( ( in @ Xx @ A )
% 2.87/1.73 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(binunionLsub_type,type,
% 2.87/1.73 binunionLsub: $o ).
% 2.87/1.73
% 2.87/1.73 thf(binunionLsub,definition,
% 2.87/1.73 ( binunionLsub
% 2.87/1.73 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(binunionRsub_type,type,
% 2.87/1.73 binunionRsub: $o ).
% 2.87/1.73
% 2.87/1.73 thf(binunionRsub,definition,
% 2.87/1.73 ( binunionRsub
% 2.87/1.73 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(binintersect_type,type,
% 2.87/1.73 binintersect: $i > $i > $i ).
% 2.87/1.73
% 2.87/1.73 thf(binintersectI_type,type,
% 2.87/1.73 binintersectI: $o ).
% 2.87/1.73
% 2.87/1.73 thf(binintersectI,definition,
% 2.87/1.73 ( binintersectI
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ( ( in @ Xx @ B )
% 2.87/1.73 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(binintersectSubset5_type,type,
% 2.87/1.73 binintersectSubset5: $o ).
% 2.87/1.73
% 2.87/1.73 thf(binintersectSubset5,definition,
% 2.87/1.73 ( binintersectSubset5
% 2.87/1.73 = ( ! [A: $i,B: $i,C: $i] :
% 2.87/1.73 ( ( subset @ C @ A )
% 2.87/1.73 => ( ( subset @ C @ B )
% 2.87/1.73 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(binintersectEL_type,type,
% 2.87/1.73 binintersectEL: $o ).
% 2.87/1.73
% 2.87/1.73 thf(binintersectEL,definition,
% 2.87/1.73 ( binintersectEL
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.87/1.73 => ( in @ Xx @ A ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(binintersectLsub_type,type,
% 2.87/1.73 binintersectLsub: $o ).
% 2.87/1.73
% 2.87/1.73 thf(binintersectLsub,definition,
% 2.87/1.73 ( binintersectLsub
% 2.87/1.73 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(binintersectSubset2_type,type,
% 2.87/1.73 binintersectSubset2: $o ).
% 2.87/1.73
% 2.87/1.73 thf(binintersectSubset2,definition,
% 2.87/1.73 ( binintersectSubset2
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ( subset @ A @ B )
% 2.87/1.73 => ( ( binintersect @ A @ B )
% 2.87/1.73 = A ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(binintersectSubset3_type,type,
% 2.87/1.73 binintersectSubset3: $o ).
% 2.87/1.73
% 2.87/1.73 thf(binintersectSubset3,definition,
% 2.87/1.73 ( binintersectSubset3
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ( ( binintersect @ A @ B )
% 2.87/1.73 = B )
% 2.87/1.73 => ( subset @ B @ A ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(binintersectER_type,type,
% 2.87/1.73 binintersectER: $o ).
% 2.87/1.73
% 2.87/1.73 thf(binintersectER,definition,
% 2.87/1.73 ( binintersectER
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.87/1.73 => ( in @ Xx @ B ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(disjointsetsI1_type,type,
% 2.87/1.73 disjointsetsI1: $o ).
% 2.87/1.73
% 2.87/1.73 thf(disjointsetsI1,definition,
% 2.87/1.73 ( disjointsetsI1
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ~ ? [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 & ( in @ Xx @ B ) )
% 2.87/1.73 => ( ( binintersect @ A @ B )
% 2.87/1.73 = emptyset ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(binintersectRsub_type,type,
% 2.87/1.73 binintersectRsub: $o ).
% 2.87/1.73
% 2.87/1.73 thf(binintersectRsub,definition,
% 2.87/1.73 ( binintersectRsub
% 2.87/1.73 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(binintersectSubset4_type,type,
% 2.87/1.73 binintersectSubset4: $o ).
% 2.87/1.73
% 2.87/1.73 thf(binintersectSubset4,definition,
% 2.87/1.73 ( binintersectSubset4
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ( subset @ B @ A )
% 2.87/1.73 => ( ( binintersect @ A @ B )
% 2.87/1.73 = B ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(binintersectSubset1_type,type,
% 2.87/1.73 binintersectSubset1: $o ).
% 2.87/1.73
% 2.87/1.73 thf(binintersectSubset1,definition,
% 2.87/1.73 ( binintersectSubset1
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ( ( binintersect @ A @ B )
% 2.87/1.73 = A )
% 2.87/1.73 => ( subset @ A @ B ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(bs114d_type,type,
% 2.87/1.73 bs114d: $o ).
% 2.87/1.73
% 2.87/1.73 thf(bs114d,definition,
% 2.87/1.73 ( bs114d
% 2.87/1.73 = ( ! [A: $i,B: $i,C: $i] :
% 2.87/1.73 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.87/1.73 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(regular_type,type,
% 2.87/1.73 regular: $i > $o ).
% 2.87/1.73
% 2.87/1.73 thf(setminus_type,type,
% 2.87/1.73 setminus: $i > $i > $i ).
% 2.87/1.73
% 2.87/1.73 thf(setminusI_type,type,
% 2.87/1.73 setminusI: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setminusI,definition,
% 2.87/1.73 ( setminusI
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ( ~ ( in @ Xx @ B )
% 2.87/1.73 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setminusEL_type,type,
% 2.87/1.73 setminusEL: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setminusEL,definition,
% 2.87/1.73 ( setminusEL
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.87/1.73 => ( in @ Xx @ A ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setminusER_type,type,
% 2.87/1.73 setminusER: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setminusER,definition,
% 2.87/1.73 ( setminusER
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.87/1.73 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setminusSubset2_type,type,
% 2.87/1.73 setminusSubset2: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setminusSubset2,definition,
% 2.87/1.73 ( setminusSubset2
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ( subset @ A @ B )
% 2.87/1.73 => ( ( setminus @ A @ B )
% 2.87/1.73 = emptyset ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setminusERneg_type,type,
% 2.87/1.73 setminusERneg: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setminusERneg,definition,
% 2.87/1.73 ( setminusERneg
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.87/1.73 => ( ( in @ Xx @ A )
% 2.87/1.73 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setminusELneg_type,type,
% 2.87/1.73 setminusELneg: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setminusELneg,definition,
% 2.87/1.73 ( setminusELneg
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.87/1.73 => ( ~ ( in @ Xx @ B )
% 2.87/1.73 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setminusILneg_type,type,
% 2.87/1.73 setminusILneg: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setminusILneg,definition,
% 2.87/1.73 ( setminusILneg
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ~ ( in @ Xx @ A )
% 2.87/1.73 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setminusIRneg_type,type,
% 2.87/1.73 setminusIRneg: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setminusIRneg,definition,
% 2.87/1.73 ( setminusIRneg
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ B )
% 2.87/1.73 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setminusLsub_type,type,
% 2.87/1.73 setminusLsub: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setminusLsub,definition,
% 2.87/1.73 ( setminusLsub
% 2.87/1.73 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setminusSubset1_type,type,
% 2.87/1.73 setminusSubset1: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setminusSubset1,definition,
% 2.87/1.73 ( setminusSubset1
% 2.87/1.73 = ( ! [A: $i,B: $i] :
% 2.87/1.73 ( ( ( setminus @ A @ B )
% 2.87/1.73 = emptyset )
% 2.87/1.73 => ( subset @ A @ B ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(symdiff_type,type,
% 2.87/1.73 symdiff: $i > $i > $i ).
% 2.87/1.73
% 2.87/1.73 thf(symdiffE_type,type,
% 2.87/1.73 symdiffE: $o ).
% 2.87/1.73
% 2.87/1.73 thf(symdiffE,definition,
% 2.87/1.73 ( symdiffE
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.87/1.73 => ! [Xphi: $o] :
% 2.87/1.73 ( ( ( in @ Xx @ A )
% 2.87/1.73 => ( ~ ( in @ Xx @ B )
% 2.87/1.73 => Xphi ) )
% 2.87/1.73 => ( ( ~ ( in @ Xx @ A )
% 2.87/1.73 => ( ( in @ Xx @ B )
% 2.87/1.73 => Xphi ) )
% 2.87/1.73 => Xphi ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(symdiffI1_type,type,
% 2.87/1.73 symdiffI1: $o ).
% 2.87/1.73
% 2.87/1.73 thf(symdiffI1,definition,
% 2.87/1.73 ( symdiffI1
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ( ~ ( in @ Xx @ B )
% 2.87/1.73 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(symdiffI2_type,type,
% 2.87/1.73 symdiffI2: $o ).
% 2.87/1.73
% 2.87/1.73 thf(symdiffI2,definition,
% 2.87/1.73 ( symdiffI2
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ~ ( in @ Xx @ A )
% 2.87/1.73 => ( ( in @ Xx @ B )
% 2.87/1.73 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(symdiffIneg1_type,type,
% 2.87/1.73 symdiffIneg1: $o ).
% 2.87/1.73
% 2.87/1.73 thf(symdiffIneg1,definition,
% 2.87/1.73 ( symdiffIneg1
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ( ( in @ Xx @ B )
% 2.87/1.73 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(symdiffIneg2_type,type,
% 2.87/1.73 symdiffIneg2: $o ).
% 2.87/1.73
% 2.87/1.73 thf(symdiffIneg2,definition,
% 2.87/1.73 ( symdiffIneg2
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ~ ( in @ Xx @ A )
% 2.87/1.73 => ( ~ ( in @ Xx @ B )
% 2.87/1.73 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(iskpair_type,type,
% 2.87/1.73 iskpair: $i > $o ).
% 2.87/1.73
% 2.87/1.73 thf(iskpair,definition,
% 2.87/1.73 ( iskpair
% 2.87/1.73 = ( ^ [A: $i] :
% 2.87/1.73 ? [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ ( setunion @ A ) )
% 2.87/1.73 & ? [Xy: $i] :
% 2.87/1.73 ( ( in @ Xy @ ( setunion @ A ) )
% 2.87/1.73 & ( A
% 2.87/1.73 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(secondinupair_type,type,
% 2.87/1.73 secondinupair: $o ).
% 2.87/1.73
% 2.87/1.73 thf(secondinupair,definition,
% 2.87/1.73 ( secondinupair
% 2.87/1.73 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setukpairIL_type,type,
% 2.87/1.73 setukpairIL: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setukpairIL,definition,
% 2.87/1.73 ( setukpairIL
% 2.87/1.73 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setukpairIR_type,type,
% 2.87/1.73 setukpairIR: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setukpairIR,definition,
% 2.87/1.73 ( setukpairIR
% 2.87/1.73 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(kpairiskpair_type,type,
% 2.87/1.73 kpairiskpair: $o ).
% 2.87/1.73
% 2.87/1.73 thf(kpairiskpair,definition,
% 2.87/1.73 ( kpairiskpair
% 2.87/1.73 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(kpair_type,type,
% 2.87/1.73 kpair: $i > $i > $i ).
% 2.87/1.73
% 2.87/1.73 thf(kpair,definition,
% 2.87/1.73 ( kpair
% 2.87/1.73 = ( ^ [Xx: $i,Xy: $i] : ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(kpairp_type,type,
% 2.87/1.73 kpairp: $o ).
% 2.87/1.73
% 2.87/1.73 thf(kpairp,definition,
% 2.87/1.73 ( kpairp
% 2.87/1.73 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(cartprod_type,type,
% 2.87/1.73 cartprod: $i > $i > $i ).
% 2.87/1.73
% 2.87/1.73 thf(singletonsubset_type,type,
% 2.87/1.73 singletonsubset: $o ).
% 2.87/1.73
% 2.87/1.73 thf(singletonsubset,definition,
% 2.87/1.73 ( singletonsubset
% 2.87/1.73 = ( ! [A: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(singletoninpowerset_type,type,
% 2.87/1.73 singletoninpowerset: $o ).
% 2.87/1.73
% 2.87/1.73 thf(singletoninpowerset,definition,
% 2.87/1.73 ( singletoninpowerset
% 2.87/1.73 = ( ! [A: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(singletoninpowunion_type,type,
% 2.87/1.73 singletoninpowunion: $o ).
% 2.87/1.73
% 2.87/1.73 thf(singletoninpowunion,definition,
% 2.87/1.73 ( singletoninpowunion
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(upairset2E_type,type,
% 2.87/1.73 upairset2E: $o ).
% 2.87/1.73
% 2.87/1.73 thf(upairset2E,definition,
% 2.87/1.73 ( upairset2E
% 2.87/1.73 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.87/1.73 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.87/1.73 => ( ( Xz = Xx )
% 2.87/1.73 | ( Xz = Xy ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(upairsubunion_type,type,
% 2.87/1.73 upairsubunion: $o ).
% 2.87/1.73
% 2.87/1.73 thf(upairsubunion,definition,
% 2.87/1.73 ( upairsubunion
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ! [Xy: $i] :
% 2.87/1.73 ( ( in @ Xy @ B )
% 2.87/1.73 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(upairinpowunion_type,type,
% 2.87/1.73 upairinpowunion: $o ).
% 2.87/1.73
% 2.87/1.73 thf(upairinpowunion,definition,
% 2.87/1.73 ( upairinpowunion
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ! [Xy: $i] :
% 2.87/1.73 ( ( in @ Xy @ B )
% 2.87/1.73 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(ubforcartprodlem1_type,type,
% 2.87/1.73 ubforcartprodlem1: $o ).
% 2.87/1.73
% 2.87/1.73 thf(ubforcartprodlem1,definition,
% 2.87/1.73 ( ubforcartprodlem1
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ! [Xy: $i] :
% 2.87/1.73 ( ( in @ Xy @ B )
% 2.87/1.73 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(ubforcartprodlem2_type,type,
% 2.87/1.73 ubforcartprodlem2: $o ).
% 2.87/1.73
% 2.87/1.73 thf(ubforcartprodlem2,definition,
% 2.87/1.73 ( ubforcartprodlem2
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ! [Xy: $i] :
% 2.87/1.73 ( ( in @ Xy @ B )
% 2.87/1.73 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(ubforcartprodlem3_type,type,
% 2.87/1.73 ubforcartprodlem3: $o ).
% 2.87/1.73
% 2.87/1.73 thf(ubforcartprodlem3,definition,
% 2.87/1.73 ( ubforcartprodlem3
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ! [Xy: $i] :
% 2.87/1.73 ( ( in @ Xy @ B )
% 2.87/1.73 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(cartprodpairin_type,type,
% 2.87/1.73 cartprodpairin: $o ).
% 2.87/1.73
% 2.87/1.73 thf(cartprodpairin,definition,
% 2.87/1.73 ( cartprodpairin
% 2.87/1.73 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ! [Xy: $i] :
% 2.87/1.73 ( ( in @ Xy @ B )
% 2.87/1.73 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(cartprodmempair1_type,type,
% 2.87/1.73 cartprodmempair1: $o ).
% 2.87/1.73
% 2.87/1.73 thf(cartprodmempair1,definition,
% 2.87/1.73 ( cartprodmempair1
% 2.87/1.73 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.87/1.73 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.87/1.73 => ? [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 & ? [Xy: $i] :
% 2.87/1.73 ( ( in @ Xy @ B )
% 2.87/1.73 & ( Xu
% 2.87/1.73 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(cartprodmempair_type,type,
% 2.87/1.73 cartprodmempair: $o ).
% 2.87/1.73
% 2.87/1.73 thf(cartprodmempair,definition,
% 2.87/1.73 ( cartprodmempair
% 2.87/1.73 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.87/1.73 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.87/1.73 => ( iskpair @ Xu ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setunionE2_type,type,
% 2.87/1.73 setunionE2: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setunionE2,definition,
% 2.87/1.73 ( setunionE2
% 2.87/1.73 = ( ! [A: $i,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ ( setunion @ A ) )
% 2.87/1.73 => ? [X: $i] :
% 2.87/1.73 ( ( in @ X @ A )
% 2.87/1.73 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setunionsingleton1_type,type,
% 2.87/1.73 setunionsingleton1: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setunionsingleton1,definition,
% 2.87/1.73 ( setunionsingleton1
% 2.87/1.73 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setunionsingleton2_type,type,
% 2.87/1.73 setunionsingleton2: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setunionsingleton2,definition,
% 2.87/1.73 ( setunionsingleton2
% 2.87/1.73 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setunionsingleton_type,type,
% 2.87/1.73 setunionsingleton: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setunionsingleton,definition,
% 2.87/1.73 ( setunionsingleton
% 2.87/1.73 = ( ! [Xx: $i] :
% 2.87/1.73 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.87/1.73 = Xx ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(singleton_type,type,
% 2.87/1.73 singleton: $i > $o ).
% 2.87/1.73
% 2.87/1.73 thf(singleton,definition,
% 2.87/1.73 ( singleton
% 2.87/1.73 = ( ^ [A: $i] :
% 2.87/1.73 ? [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 & ( A
% 2.87/1.73 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(singletonprop_type,type,
% 2.87/1.73 singletonprop: $o ).
% 2.87/1.73
% 2.87/1.73 thf(singletonprop,definition,
% 2.87/1.73 ( singletonprop
% 2.87/1.73 = ( ! [A: $i,Xphi: $i > $o] :
% 2.87/1.73 ( ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ! [Xy: $i] :
% 2.87/1.73 ( ( in @ Xy @ A )
% 2.87/1.73 => ( ( Xphi @ Xx )
% 2.87/1.73 => ( ( Xphi @ Xy )
% 2.87/1.73 => ( Xx = Xy ) ) ) ) )
% 2.87/1.73 => ( ? [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 & ( Xphi @ Xx ) )
% 2.87/1.73 => ( singleton
% 2.87/1.73 @ ( dsetconstr @ A
% 2.87/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(ex1_type,type,
% 2.87/1.73 ex1: $i > ( $i > $o ) > $o ).
% 2.87/1.73
% 2.87/1.73 thf(ex1,definition,
% 2.87/1.73 ( ex1
% 2.87/1.73 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.87/1.73 ( singleton
% 2.87/1.73 @ ( dsetconstr @ A
% 2.87/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(ex1E1_type,type,
% 2.87/1.73 ex1E1: $o ).
% 2.87/1.73
% 2.87/1.73 thf(ex1E1,definition,
% 2.87/1.73 ( ex1E1
% 2.87/1.73 = ( ! [A: $i,Xphi: $i > $o] :
% 2.87/1.73 ( ( ex1 @ A
% 2.87/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.87/1.73 => ? [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(ex1I_type,type,
% 2.87/1.73 ex1I: $o ).
% 2.87/1.73
% 2.87/1.73 thf(ex1I,definition,
% 2.87/1.73 ( ex1I
% 2.87/1.73 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ( ( Xphi @ Xx )
% 2.87/1.73 => ( ! [Xy: $i] :
% 2.87/1.73 ( ( in @ Xy @ A )
% 2.87/1.73 => ( ( Xphi @ Xy )
% 2.87/1.73 => ( Xy = Xx ) ) )
% 2.87/1.73 => ( ex1 @ A
% 2.87/1.73 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(ex1I2_type,type,
% 2.87/1.73 ex1I2: $o ).
% 2.87/1.73
% 2.87/1.73 thf(ex1I2,definition,
% 2.87/1.73 ( ex1I2
% 2.87/1.73 = ( ! [A: $i,Xphi: $i > $o] :
% 2.87/1.73 ( ! [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 => ! [Xy: $i] :
% 2.87/1.73 ( ( in @ Xy @ A )
% 2.87/1.73 => ( ( Xphi @ Xx )
% 2.87/1.73 => ( ( Xphi @ Xy )
% 2.87/1.73 => ( Xx = Xy ) ) ) ) )
% 2.87/1.73 => ( ? [Xx: $i] :
% 2.87/1.73 ( ( in @ Xx @ A )
% 2.87/1.73 & ( Xphi @ Xx ) )
% 2.87/1.73 => ( ex1 @ A
% 2.87/1.73 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(singletonsuniq_type,type,
% 2.87/1.73 singletonsuniq: $o ).
% 2.87/1.73
% 2.87/1.73 thf(singletonsuniq,definition,
% 2.87/1.73 ( singletonsuniq
% 2.87/1.73 = ( ! [Xx: $i,Xy: $i] :
% 2.87/1.73 ( ( ( setadjoin @ Xx @ emptyset )
% 2.87/1.73 = ( setadjoin @ Xy @ emptyset ) )
% 2.87/1.73 => ( Xx = Xy ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(atmost1p_type,type,
% 2.87/1.73 atmost1p: $i > $o ).
% 2.87/1.73
% 2.87/1.73 thf(atleast2p_type,type,
% 2.87/1.73 atleast2p: $i > $o ).
% 2.87/1.73
% 2.87/1.73 thf(atmost2p_type,type,
% 2.87/1.73 atmost2p: $i > $o ).
% 2.87/1.73
% 2.87/1.73 thf(upairsetp_type,type,
% 2.87/1.73 upairsetp: $i > $o ).
% 2.87/1.73
% 2.87/1.73 thf(setukpairinjL1_type,type,
% 2.87/1.73 setukpairinjL1: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setukpairinjL1,definition,
% 2.87/1.73 ( setukpairinjL1
% 2.87/1.73 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.87/1.73 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.87/1.73 => ( Xx = Xz ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(kfstsingleton_type,type,
% 2.87/1.73 kfstsingleton: $o ).
% 2.87/1.73
% 2.87/1.73 thf(kfstsingleton,definition,
% 2.87/1.73 ( kfstsingleton
% 2.87/1.73 = ( ! [Xu: $i] :
% 2.87/1.73 ( ( iskpair @ Xu )
% 2.87/1.73 => ( singleton
% 2.87/1.73 @ ( dsetconstr @ ( setunion @ Xu )
% 2.87/1.73 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(theprop_type,type,
% 2.87/1.73 theprop: $o ).
% 2.87/1.73
% 2.87/1.73 thf(theprop,definition,
% 2.87/1.73 ( theprop
% 2.87/1.73 = ( ! [X: $i] :
% 2.87/1.73 ( ( singleton @ X )
% 2.87/1.73 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(kfst_type,type,
% 2.87/1.73 kfst: $i > $i ).
% 2.87/1.73
% 2.87/1.73 thf(kfstpairEq_type,type,
% 2.87/1.73 kfstpairEq: $o ).
% 2.87/1.73
% 2.87/1.73 thf(kfstpairEq,definition,
% 2.87/1.73 ( kfstpairEq
% 2.87/1.73 = ( ! [Xx: $i,Xy: $i] :
% 2.87/1.73 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.87/1.73 = Xx ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(cartprodfstin_type,type,
% 2.87/1.73 cartprodfstin: $o ).
% 2.87/1.73
% 2.87/1.73 thf(cartprodfstin,definition,
% 2.87/1.73 ( cartprodfstin
% 2.87/1.73 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.87/1.73 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.87/1.73 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setukpairinjL2_type,type,
% 2.87/1.73 setukpairinjL2: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setukpairinjL2,definition,
% 2.87/1.73 ( setukpairinjL2
% 2.87/1.73 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.87/1.73 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.87/1.73 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.87/1.73 => ( Xx = Xz ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setukpairinjL_type,type,
% 2.87/1.73 setukpairinjL: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setukpairinjL,definition,
% 2.87/1.73 ( setukpairinjL
% 2.87/1.73 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.87/1.73 ( ( ( kpair @ Xx @ Xy )
% 2.87/1.73 = ( kpair @ Xz @ Xu ) )
% 2.87/1.73 => ( Xx = Xz ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setukpairinjR11_type,type,
% 2.87/1.73 setukpairinjR11: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setukpairinjR11,definition,
% 2.87/1.73 ( setukpairinjR11
% 2.87/1.73 = ( ! [Xx: $i,Xy: $i] :
% 2.87/1.73 ( ( Xx = Xy )
% 2.87/1.73 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.87/1.73 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setukpairinjR12_type,type,
% 2.87/1.73 setukpairinjR12: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setukpairinjR12,definition,
% 2.87/1.73 ( setukpairinjR12
% 2.87/1.73 = ( ! [Xx: $i,Xy: $i] :
% 2.87/1.73 ( ( Xx = Xy )
% 2.87/1.73 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.87/1.73 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(setukpairinjR1_type,type,
% 2.87/1.73 setukpairinjR1: $o ).
% 2.87/1.73
% 2.87/1.73 thf(setukpairinjR1,definition,
% 2.87/1.73 ( setukpairinjR1
% 2.87/1.73 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.87/1.73 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.87/1.73 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.87/1.73 => ( ( Xz = Xu )
% 2.87/1.73 => ( Xy = Xu ) ) ) ) ) ).
% 2.87/1.73
% 2.87/1.73 thf(upairequniteq,conjecture,
% 2.87/1.73 ( setextAx
% 2.87/1.73 => ( emptysetAx
% 2.87/1.73 => ( setadjoinAx
% 2.87/1.73 => ( powersetAx
% 2.87/1.73 => ( setunionAx
% 2.87/1.73 => ( omega0Ax
% 2.87/1.73 => ( omegaSAx
% 2.87/1.73 => ( omegaIndAx
% 2.87/1.73 => ( replAx
% 2.87/1.73 => ( foundationAx
% 2.87/1.73 => ( wellorderingAx
% 2.87/1.73 => ( descrp
% 2.87/1.73 => ( dsetconstrI
% 2.87/1.73 => ( dsetconstrEL
% 2.87/1.73 => ( dsetconstrER
% 2.87/1.73 => ( exuE1
% 2.87/1.73 => ( prop2setE
% 2.87/1.73 => ( emptysetE
% 2.87/1.73 => ( emptysetimpfalse
% 2.87/1.73 => ( notinemptyset
% 2.87/1.73 => ( exuE3e
% 2.87/1.73 => ( setext
% 2.87/1.73 => ( emptyI
% 2.87/1.73 => ( noeltsimpempty
% 2.87/1.73 => ( setbeta
% 2.87/1.73 => ( nonemptyE1
% 2.87/1.73 => ( nonemptyI
% 2.87/1.73 => ( nonemptyI1
% 2.87/1.73 => ( setadjoinIL
% 2.87/1.73 => ( emptyinunitempty
% 2.87/1.73 => ( setadjoinIR
% 2.87/1.73 => ( setadjoinE
% 2.87/1.73 => ( setadjoinOr
% 2.87/1.73 => ( setoftrueEq
% 2.87/1.73 => ( powersetI
% 2.87/1.73 => ( emptyinPowerset
% 2.87/1.73 => ( emptyInPowerset
% 2.87/1.73 => ( powersetE
% 2.87/1.73 => ( setunionI
% 2.87/1.73 => ( setunionE
% 2.87/1.73 => ( subPowSU
% 2.87/1.73 => ( exuE2
% 2.87/1.73 => ( nonemptyImpWitness
% 2.87/1.73 => ( uniqinunit
% 2.87/1.73 => ( notinsingleton
% 2.87/1.73 => ( eqinunit
% 2.87/1.73 => ( singletonsswitch
% 2.87/1.73 => ( upairsetE
% 2.87/1.73 => ( upairsetIL
% 2.87/1.73 => ( upairsetIR
% 2.87/1.73 => ( emptyE1
% 2.87/1.73 => ( vacuousDall
% 2.87/1.73 => ( quantDeMorgan1
% 2.87/1.73 => ( quantDeMorgan2
% 2.87/1.73 => ( quantDeMorgan3
% 2.87/1.73 => ( quantDeMorgan4
% 2.87/1.73 => ( prop2setI
% 2.87/1.73 => ( prop2set2propI
% 2.87/1.73 => ( notdexE
% 2.87/1.73 => ( notdallE
% 2.87/1.73 => ( exuI1
% 2.87/1.73 => ( exuI3
% 2.87/1.73 => ( exuI2
% 2.87/1.73 => ( inCongP
% 2.87/1.73 => ( in__Cong
% 2.87/1.73 => ( exuE3u
% 2.87/1.73 => ( exu__Cong
% 2.87/1.73 => ( emptyset__Cong
% 2.87/1.73 => ( setadjoin__Cong
% 2.87/1.73 => ( powerset__Cong
% 2.87/1.73 => ( setunion__Cong
% 2.87/1.73 => ( omega__Cong
% 2.87/1.73 => ( exuEu
% 2.87/1.73 => ( descr__Cong
% 2.87/1.73 => ( dsetconstr__Cong
% 2.87/1.73 => ( subsetI1
% 2.87/1.73 => ( eqimpsubset2
% 2.87/1.73 => ( eqimpsubset1
% 2.87/1.73 => ( subsetI2
% 2.87/1.73 => ( emptysetsubset
% 2.87/1.73 => ( subsetE
% 2.87/1.73 => ( subsetE2
% 2.87/1.73 => ( notsubsetI
% 2.87/1.73 => ( notequalI1
% 2.87/1.73 => ( notequalI2
% 2.87/1.73 => ( subsetRefl
% 2.87/1.73 => ( subsetTrans
% 2.87/1.73 => ( setadjoinSub
% 2.87/1.73 => ( setadjoinSub2
% 2.87/1.73 => ( subset2powerset
% 2.87/1.73 => ( setextsub
% 2.87/1.73 => ( subsetemptysetimpeq
% 2.87/1.73 => ( powersetI1
% 2.87/1.73 => ( powersetE1
% 2.87/1.73 => ( inPowerset
% 2.87/1.73 => ( powersetsubset
% 2.87/1.73 => ( sepInPowerset
% 2.87/1.73 => ( sepSubset
% 2.87/1.73 => ( binunionIL
% 2.87/1.73 => ( upairset2IR
% 2.87/1.73 => ( binunionIR
% 2.87/1.73 => ( binunionEcases
% 2.87/1.73 => ( binunionE
% 2.87/1.73 => ( binunionLsub
% 2.87/1.73 => ( binunionRsub
% 2.87/1.73 => ( binintersectI
% 2.87/1.73 => ( binintersectSubset5
% 2.87/1.73 => ( binintersectEL
% 2.87/1.73 => ( binintersectLsub
% 2.87/1.73 => ( binintersectSubset2
% 2.87/1.73 => ( binintersectSubset3
% 2.87/1.73 => ( binintersectER
% 2.87/1.73 => ( disjointsetsI1
% 2.87/1.73 => ( binintersectRsub
% 2.87/1.73 => ( binintersectSubset4
% 2.87/1.73 => ( binintersectSubset1
% 2.87/1.73 => ( bs114d
% 2.87/1.73 => ( setminusI
% 2.87/1.73 => ( setminusEL
% 2.87/1.73 => ( setminusER
% 2.87/1.73 => ( setminusSubset2
% 2.87/1.73 => ( setminusERneg
% 2.87/1.73 => ( setminusELneg
% 2.87/1.73 => ( setminusILneg
% 2.87/1.73 => ( setminusIRneg
% 2.87/1.73 => ( setminusLsub
% 2.87/1.73 => ( setminusSubset1
% 2.87/1.73 => ( symdiffE
% 2.87/1.73 => ( symdiffI1
% 2.87/1.73 => ( symdiffI2
% 2.87/1.73 => ( symdiffIneg1
% 2.87/1.73 => ( symdiffIneg2
% 2.87/1.73 => ( secondinupair
% 2.87/1.73 => ( setukpairIL
% 2.87/1.73 => ( setukpairIR
% 2.87/1.73 => ( kpairiskpair
% 2.87/1.73 => ( kpairp
% 2.87/1.73 => ( singletonsubset
% 2.87/1.73 => ( singletoninpowerset
% 2.87/1.73 => ( singletoninpowunion
% 2.87/1.73 => ( upairset2E
% 2.87/1.73 => ( upairsubunion
% 2.87/1.73 => ( upairinpowunion
% 2.87/1.73 => ( ubforcartprodlem1
% 2.87/1.73 => ( ubforcartprodlem2
% 2.87/1.73 => ( ubforcartprodlem3
% 2.87/1.73 => ( cartprodpairin
% 2.87/1.73 => ( cartprodmempair1
% 2.87/1.73 => ( cartprodmempair
% 2.87/1.73 => ( setunionE2
% 2.87/1.73 => ( setunionsingleton1
% 2.87/1.73 => ( setunionsingleton2
% 2.87/1.73 => ( setunionsingleton
% 2.87/1.73 => ( singletonprop
% 2.87/1.73 => ( ex1E1
% 2.87/1.73 => ( ex1I
% 2.87/1.73 => ( ex1I2
% 2.87/1.73 => ( singletonsuniq
% 2.87/1.73 => ( setukpairinjL1
% 2.87/1.73 => ( kfstsingleton
% 2.87/1.73 => ( theprop
% 2.87/1.73 => ( kfstpairEq
% 2.87/1.73 => ( cartprodfstin
% 2.87/1.73 => ( setukpairinjL2
% 2.87/1.73 => ( setukpairinjL
% 2.87/1.73 => ( setukpairinjR11
% 2.87/1.73 => ( setukpairinjR12
% 2.87/1.73 => ( setukpairinjR1
% 2.87/1.73 => ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.87/1.73 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.87/1.73 = ( setadjoin @ Xz @ emptyset ) )
% 2.87/1.73 => ( Xx = Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.87/1.73 ------------------------
% 2.87/1.76 % (26484)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_26384 for (2999ds/27Mi)
% 2.87/1.76 % (26486)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_26384 for (2999ds/2Mi)
% 2.87/1.76 % (26485)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_26384 for (2999ds/2Mi)
% 2.87/1.76 % (26488)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_26384 for (2999ds/18Mi)
% 2.87/1.76 % (26487)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_26384 for (2999ds/275Mi)
% 2.87/1.76 % (26483)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_26384 for (2999ds/4Mi)
% 2.87/1.76 % (26482)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_26384 for (2999ds/183Mi)
% 2.87/1.76 % (26485)Instruction limit reached!
% 2.87/1.76 % (26485)------------------------------
% 2.87/1.76 % (26485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.76 % (26485)Termination reason: Unknown
% 2.87/1.76 % (26485)Termination phase: shuffling
% 2.87/1.76
% 2.87/1.76 % (26486)Instruction limit reached!
% 2.87/1.76 % (26486)------------------------------
% 2.87/1.76 % (26486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.76 % (26485)Memory used [KB]: 1279
% 2.87/1.76 % (26485)Time elapsed: 0.003 s
% 2.87/1.76 % (26485)Instructions burned: 2 (million)
% 2.87/1.76 % (26485)------------------------------
% 2.87/1.76 % (26485)------------------------------
% 2.87/1.76 % (26486)Termination reason: Unknown
% 2.87/1.76 % (26486)Termination phase: shuffling
% 2.87/1.76
% 2.87/1.76 % (26486)Memory used [KB]: 1279
% 2.87/1.76 % (26486)Time elapsed: 0.003 s
% 2.87/1.76 % (26486)Instructions burned: 2 (million)
% 2.87/1.76 % (26486)------------------------------
% 2.87/1.76 % (26486)------------------------------
% 2.87/1.76 % (26483)Instruction limit reached!
% 2.87/1.76 % (26483)------------------------------
% 2.87/1.76 % (26483)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.76 % (26483)Termination reason: Unknown
% 2.87/1.76 % (26483)Termination phase: shuffling
% 2.87/1.76
% 2.87/1.76 % (26483)Memory used [KB]: 1279
% 2.87/1.76 % (26483)Time elapsed: 0.003 s
% 2.87/1.76 % (26483)Instructions burned: 4 (million)
% 2.87/1.76 % (26483)------------------------------
% 2.87/1.76 % (26483)------------------------------
% 2.87/1.77 % (26488)Instruction limit reached!
% 2.87/1.77 % (26488)------------------------------
% 2.87/1.77 % (26488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.77 % (26488)Termination reason: Unknown
% 2.87/1.77 % (26488)Termination phase: Property scanning
% 2.87/1.77
% 2.87/1.77 % (26488)Memory used [KB]: 1535
% 2.87/1.77 % (26488)Time elapsed: 0.011 s
% 2.87/1.77 % (26488)Instructions burned: 19 (million)
% 2.87/1.77 % (26488)------------------------------
% 2.87/1.77 % (26488)------------------------------
% 2.87/1.77 % (26484)Instruction limit reached!
% 2.87/1.77 % (26484)------------------------------
% 2.87/1.77 % (26484)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.77 % (26484)Termination reason: Unknown
% 2.87/1.77 % (26484)Termination phase: Property scanning
% 2.87/1.77
% 2.87/1.77 % (26484)Memory used [KB]: 1791
% 2.87/1.77 % (26484)Time elapsed: 0.015 s
% 2.87/1.77 % (26484)Instructions burned: 28 (million)
% 2.87/1.77 % (26484)------------------------------
% 2.87/1.77 % (26484)------------------------------
% 2.87/1.78 % (26489)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_26384 for (2999ds/3Mi)
% 2.87/1.78 % (26490)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_26384 for (2999ds/37Mi)
% 2.87/1.78 % (26491)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_26384 for (2999ds/15Mi)
% 2.87/1.78 % (26489)Instruction limit reached!
% 2.87/1.78 % (26489)------------------------------
% 2.87/1.78 % (26489)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.78 % (26489)Termination reason: Unknown
% 2.87/1.78 % (26489)Termination phase: shuffling
% 2.87/1.78
% 2.87/1.78 % (26489)Memory used [KB]: 1279
% 2.87/1.78 % (26489)Time elapsed: 0.003 s
% 2.87/1.78 % (26489)Instructions burned: 3 (million)
% 2.87/1.78 % (26489)------------------------------
% 2.87/1.78 % (26489)------------------------------
% 2.87/1.79 % (26491)Instruction limit reached!
% 2.87/1.79 % (26491)------------------------------
% 2.87/1.79 % (26491)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.79 % (26491)Termination reason: Unknown
% 2.87/1.79 % (26491)Termination phase: shuffling
% 2.87/1.79 % (26492)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_26384 for (2999ds/3Mi)
% 2.87/1.79
% 2.87/1.79 % (26491)Memory used [KB]: 1535
% 2.87/1.79 % (26491)Time elapsed: 0.009 s
% 2.87/1.79 % (26491)Instructions burned: 15 (million)
% 2.87/1.79 % (26491)------------------------------
% 2.87/1.79 % (26491)------------------------------
% 2.87/1.79 % (26492)Instruction limit reached!
% 2.87/1.79 % (26492)------------------------------
% 2.87/1.79 % (26492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.79 % (26492)Termination reason: Unknown
% 2.87/1.79 % (26492)Termination phase: shuffling
% 2.87/1.79
% 2.87/1.79 % (26492)Memory used [KB]: 1279
% 2.87/1.79 % (26492)Time elapsed: 0.003 s
% 2.87/1.79 % (26492)Instructions burned: 3 (million)
% 2.87/1.79 % (26492)------------------------------
% 2.87/1.79 % (26492)------------------------------
% 2.87/1.79 % (26493)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_26384 for (2999ds/1041Mi)
% 2.87/1.79 % (26494)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_26384 for (2999ds/7Mi)
% 2.87/1.79 % (26490)Instruction limit reached!
% 2.87/1.79 % (26490)------------------------------
% 2.87/1.79 % (26490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.79 % (26490)Termination reason: Unknown
% 2.87/1.79 % (26490)Termination phase: Property scanning
% 2.87/1.79
% 2.87/1.79 % (26490)Memory used [KB]: 1791
% 2.87/1.80 % (26490)Time elapsed: 0.018 s
% 2.87/1.80 % (26490)Instructions burned: 37 (million)
% 2.87/1.80 % (26490)------------------------------
% 2.87/1.80 % (26490)------------------------------
% 2.87/1.80 % (26494)Instruction limit reached!
% 2.87/1.80 % (26494)------------------------------
% 2.87/1.80 % (26494)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.80 % (26494)Termination reason: Unknown
% 2.87/1.80 % (26494)Termination phase: shuffling
% 2.87/1.80
% 2.87/1.80 % (26494)Memory used [KB]: 1407
% 2.87/1.80 % (26494)Time elapsed: 0.004 s
% 2.87/1.80 % (26494)Instructions burned: 8 (million)
% 2.87/1.80 % (26494)------------------------------
% 2.87/1.80 % (26494)------------------------------
% 2.90/1.80 % (26495)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_26384 for (2999ds/16Mi)
% 2.90/1.80 % (26496)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_26384 for (2999ds/3Mi)
% 2.90/1.80 % (26498)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_26384 for (2999ds/7Mi)
% 2.90/1.81 % (26496)Instruction limit reached!
% 2.90/1.81 % (26496)------------------------------
% 2.90/1.81 % (26496)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.81 % (26496)Termination reason: Unknown
% 2.90/1.81 % (26496)Termination phase: shuffling
% 2.90/1.81
% 2.90/1.81 % (26496)Memory used [KB]: 1279
% 2.90/1.81 % (26496)Time elapsed: 0.003 s
% 2.90/1.81 % (26496)Instructions burned: 4 (million)
% 2.90/1.81 % (26496)------------------------------
% 2.90/1.81 % (26496)------------------------------
% 2.90/1.81 % (26498)Instruction limit reached!
% 2.90/1.81 % (26498)------------------------------
% 2.90/1.81 % (26498)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.81 % (26498)Termination reason: Unknown
% 2.90/1.81 % (26498)Termination phase: shuffling
% 2.90/1.81
% 2.90/1.81 % (26498)Memory used [KB]: 1407
% 2.90/1.81 % (26498)Time elapsed: 0.004 s
% 2.90/1.81 % (26498)Instructions burned: 10 (million)
% 2.90/1.81 % (26498)------------------------------
% 2.90/1.81 % (26498)------------------------------
% 2.90/1.81 % (26495)Instruction limit reached!
% 2.90/1.81 % (26495)------------------------------
% 2.90/1.81 % (26495)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.81 % (26495)Termination reason: Unknown
% 2.90/1.81 % (26495)Termination phase: shuffling
% 2.90/1.81
% 2.90/1.81 % (26495)Memory used [KB]: 1663
% 2.90/1.81 % (26495)Time elapsed: 0.009 s
% 2.90/1.81 % (26495)Instructions burned: 16 (million)
% 2.90/1.81 % (26495)------------------------------
% 2.90/1.81 % (26495)------------------------------
% 2.90/1.81 % (26497)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_26384 for (2999ds/3Mi)
% 2.90/1.81 % (26497)Instruction limit reached!
% 2.90/1.81 % (26497)------------------------------
% 2.90/1.81 % (26497)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.81 % (26497)Termination reason: Unknown
% 2.90/1.81 % (26497)Termination phase: shuffling
% 2.90/1.81
% 2.90/1.81 % (26497)Memory used [KB]: 1279
% 2.90/1.81 % (26497)Time elapsed: 0.003 s
% 2.90/1.81 % (26497)Instructions burned: 3 (million)
% 2.90/1.81 % (26497)------------------------------
% 2.90/1.81 % (26497)------------------------------
% 2.90/1.82 % (26499)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_26384 for (2999ds/3Mi)
% 2.90/1.82 % (26499)Instruction limit reached!
% 2.90/1.82 % (26499)------------------------------
% 2.90/1.82 % (26499)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.82 % (26499)Termination reason: Unknown
% 2.90/1.82 % (26499)Termination phase: shuffling
% 2.90/1.82
% 2.90/1.82 % (26499)Memory used [KB]: 1279
% 2.90/1.82 % (26499)Time elapsed: 0.002 s
% 2.90/1.82 % (26499)Instructions burned: 4 (million)
% 2.90/1.82 % (26499)------------------------------
% 2.90/1.82 % (26499)------------------------------
% 2.90/1.82 % (26500)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_26384 for (2999ds/4Mi)
% 2.90/1.83 % (26500)Instruction limit reached!
% 2.90/1.83 % (26500)------------------------------
% 2.90/1.83 % (26500)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.83 % (26501)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_26384 for (2999ds/18Mi)
% 2.90/1.83 % (26500)Termination reason: Unknown
% 2.90/1.83 % (26500)Termination phase: shuffling
% 2.90/1.83
% 2.90/1.83 % (26500)Memory used [KB]: 1279
% 2.90/1.83 % (26500)Time elapsed: 0.004 s
% 2.90/1.83 % (26500)Instructions burned: 5 (million)
% 2.90/1.83 % (26500)------------------------------
% 2.90/1.83 % (26500)------------------------------
% 2.90/1.83 % (26502)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_26384 for (2999ds/710Mi)
% 2.90/1.83 % (26503)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_26384 for (2999ds/6Mi)
% 2.90/1.83 % (26501)Instruction limit reached!
% 2.90/1.83 % (26501)------------------------------
% 2.90/1.83 % (26501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.83 % (26501)Termination reason: Unknown
% 2.90/1.83 % (26501)Termination phase: shuffling
% 2.90/1.83
% 2.90/1.83 % (26501)Memory used [KB]: 1663
% 2.90/1.83 % (26501)Time elapsed: 0.010 s
% 2.90/1.83 % (26501)Instructions burned: 19 (million)
% 2.90/1.83 % (26501)------------------------------
% 2.90/1.83 % (26501)------------------------------
% 2.90/1.84 % (26503)Instruction limit reached!
% 2.90/1.84 % (26503)------------------------------
% 2.90/1.84 % (26503)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.84 % (26503)Termination reason: Unknown
% 2.90/1.84 % (26503)Termination phase: shuffling
% 2.90/1.84
% 2.90/1.84 % (26503)Memory used [KB]: 1407
% 2.90/1.84 % (26503)Time elapsed: 0.005 s
% 2.90/1.84 % (26503)Instructions burned: 8 (million)
% 2.90/1.84 % (26503)------------------------------
% 2.90/1.84 % (26503)------------------------------
% 2.90/1.84 % (26482)Instruction limit reached!
% 2.90/1.84 % (26482)------------------------------
% 2.90/1.84 % (26482)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.84 % (26482)Termination reason: Unknown
% 2.90/1.84 % (26482)Termination phase: Saturation
% 2.90/1.84
% 2.90/1.84 % (26482)Memory used [KB]: 7419
% 2.90/1.84 % (26482)Time elapsed: 0.085 s
% 2.90/1.84 % (26482)Instructions burned: 183 (million)
% 2.90/1.84 % (26482)------------------------------
% 2.90/1.84 % (26482)------------------------------
% 2.90/1.85 % (26504)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_26384 for (2999ds/902Mi)
% 2.90/1.85 % (26505)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_26384 for (2999ds/21Mi)
% 2.90/1.86 % (26506)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_26384 for (2999ds/5Mi)
% 2.90/1.86 % (26506)Instruction limit reached!
% 2.90/1.86 % (26506)------------------------------
% 2.90/1.86 % (26506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.86 % (26506)Termination reason: Unknown
% 2.90/1.86 % (26506)Termination phase: shuffling
% 2.90/1.86
% 2.90/1.86 % (26506)Memory used [KB]: 1279
% 2.90/1.86 % (26506)Time elapsed: 0.004 s
% 2.90/1.86 % (26506)Instructions burned: 5 (million)
% 2.90/1.86 % (26506)------------------------------
% 2.90/1.86 % (26506)------------------------------
% 2.90/1.86 % (26505)Instruction limit reached!
% 2.90/1.86 % (26505)------------------------------
% 2.90/1.86 % (26505)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.86 % (26505)Termination reason: Unknown
% 2.90/1.86 % (26505)Termination phase: shuffling
% 2.90/1.86
% 2.90/1.86 % (26505)Memory used [KB]: 1791
% 2.90/1.86 % (26505)Time elapsed: 0.011 s
% 2.90/1.86 % (26505)Instructions burned: 21 (million)
% 2.90/1.86 % (26505)------------------------------
% 2.90/1.86 % (26505)------------------------------
% 2.90/1.88 % (26507)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_26384 for (2998ds/6Mi)
% 2.90/1.88 % (26509)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_26384 for (2998ds/779Mi)
% 2.90/1.88 % (26508)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_26384 for (2998ds/377Mi)
% 2.90/1.88 % (26507)Instruction limit reached!
% 2.90/1.88 % (26507)------------------------------
% 2.90/1.88 % (26507)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.88 % (26507)Termination reason: Unknown
% 2.90/1.88 % (26507)Termination phase: shuffling
% 2.90/1.88
% 2.90/1.88 % (26507)Memory used [KB]: 1407
% 2.90/1.88 % (26507)Time elapsed: 0.005 s
% 2.90/1.88 % (26507)Instructions burned: 7 (million)
% 2.90/1.88 % (26507)------------------------------
% 2.90/1.88 % (26507)------------------------------
% 2.90/1.90 % (26510)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_26384 for (2998ds/19Mi)
% 2.90/1.90 % (26487)Instruction limit reached!
% 2.90/1.90 % (26487)------------------------------
% 2.90/1.90 % (26487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.90 % (26487)Termination reason: Unknown
% 2.90/1.90 % (26487)Termination phase: Saturation
% 2.90/1.90
% 2.90/1.90 % (26487)Memory used [KB]: 8187
% 2.90/1.90 % (26487)Time elapsed: 0.137 s
% 2.90/1.90 % (26487)Instructions burned: 275 (million)
% 2.90/1.90 % (26487)------------------------------
% 2.90/1.90 % (26487)------------------------------
% 2.90/1.91 % (26510)Instruction limit reached!
% 2.90/1.91 % (26510)------------------------------
% 2.90/1.91 % (26510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.91 % (26510)Termination reason: Unknown
% 2.90/1.91 % (26510)Termination phase: shuffling
% 2.90/1.91
% 2.90/1.91 % (26510)Memory used [KB]: 1663
% 2.90/1.91 % (26510)Time elapsed: 0.011 s
% 2.90/1.91 % (26510)Instructions burned: 20 (million)
% 2.90/1.91 % (26510)------------------------------
% 2.90/1.91 % (26510)------------------------------
% 2.90/1.91 % (26511)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_26384 for (2998ds/879Mi)
% 2.90/1.92 % (26512)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_26384 for (2998ds/17Mi)
% 2.90/1.93 % (26512)Instruction limit reached!
% 2.90/1.93 % (26512)------------------------------
% 2.90/1.93 % (26512)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.93 % (26512)Termination reason: Unknown
% 2.90/1.93 % (26512)Termination phase: shuffling
% 2.90/1.93
% 2.90/1.93 % (26512)Memory used [KB]: 1663
% 2.90/1.93 % (26512)Time elapsed: 0.010 s
% 2.90/1.93 % (26512)Instructions burned: 18 (million)
% 2.90/1.93 % (26512)------------------------------
% 2.90/1.93 % (26512)------------------------------
% 2.90/1.95 % (26513)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_26384 for (2998ds/3Mi)
% 2.90/1.95 % (26513)Instruction limit reached!
% 2.90/1.95 % (26513)------------------------------
% 2.90/1.95 % (26513)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.95 % (26513)Termination reason: Unknown
% 2.90/1.95 % (26513)Termination phase: shuffling
% 2.90/1.95
% 2.90/1.95 % (26513)Memory used [KB]: 1279
% 2.90/1.95 % (26513)Time elapsed: 0.004 s
% 2.90/1.95 % (26513)Instructions burned: 5 (million)
% 2.90/1.95 % (26513)------------------------------
% 2.90/1.95 % (26513)------------------------------
% 2.90/1.96 % (26514)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_26384 for (2998ds/30Mi)
% 2.90/1.98 % (26514)Instruction limit reached!
% 2.90/1.98 % (26514)------------------------------
% 2.90/1.98 % (26514)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.98 % (26514)Termination reason: Unknown
% 2.90/1.98 % (26514)Termination phase: Property scanning
% 2.90/1.98
% 2.90/1.98 % (26514)Memory used [KB]: 1791
% 2.90/1.98 % (26514)Time elapsed: 0.016 s
% 2.90/1.98 % (26514)Instructions burned: 31 (million)
% 2.90/1.98 % (26514)------------------------------
% 2.90/1.98 % (26514)------------------------------
% 2.90/2.00 % (26515)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_26384 for (2997ds/127Mi)
% 2.90/2.03 % (26504)First to succeed.
% 4.78/2.03 % (26504)Refutation found. Thanks to Tanya!
% 4.78/2.03 % SZS status Theorem for DTF2THF_26384
% 4.78/2.03 % SZS output start Proof for DTF2THF_26384
% 4.78/2.03 thf(func_def_0, type, in: $i > $i > $o).
% 4.78/2.03 thf(func_def_1, type, exu: ($i > $o) > $o).
% 4.78/2.03 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 4.78/2.03 thf(func_def_8, type, powerset: $i > $i).
% 4.78/2.03 thf(func_def_10, type, setunion: $i > $i).
% 4.78/2.03 thf(func_def_19, type, descr: ($i > $o) > $i).
% 4.78/2.03 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 4.78/2.03 thf(func_def_26, type, prop2set: $o > $i).
% 4.78/2.03 thf(func_def_36, type, nonempty: $i > $o).
% 4.78/2.03 thf(func_def_69, type, set2prop: $i > $o).
% 4.78/2.03 thf(func_def_88, type, subset: $i > $i > $o).
% 4.78/2.03 thf(func_def_89, type, disjoint: $i > $i > $o).
% 4.78/2.03 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 4.78/2.03 thf(func_def_114, type, binunion: $i > $i > $i).
% 4.78/2.03 thf(func_def_122, type, binintersect: $i > $i > $i).
% 4.78/2.03 thf(func_def_135, type, regular: $i > $o).
% 4.78/2.03 thf(func_def_136, type, setminus: $i > $i > $i).
% 4.78/2.03 thf(func_def_147, type, symdiff: $i > $i > $i).
% 4.78/2.03 thf(func_def_153, type, iskpair: $i > $o).
% 4.78/2.03 thf(func_def_158, type, kpair: $i > $i > $i).
% 4.78/2.03 thf(func_def_160, type, cartprod: $i > $i > $i).
% 4.78/2.03 thf(func_def_177, type, singleton: $i > $o).
% 4.78/2.03 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 4.78/2.03 thf(func_def_184, type, atmost1p: $i > $o).
% 4.78/2.03 thf(func_def_185, type, atleast2p: $i > $o).
% 4.78/2.03 thf(func_def_186, type, atmost2p: $i > $o).
% 4.78/2.03 thf(func_def_187, type, upairsetp: $i > $o).
% 4.78/2.03 thf(func_def_191, type, kfst: $i > $i).
% 4.78/2.03 thf(func_def_211, type, sP0: $i > $i > $o).
% 4.78/2.03 thf(func_def_212, type, sP1: $i > $o).
% 4.78/2.03 thf(func_def_213, type, sP2: $i > $i > $o).
% 4.78/2.03 thf(func_def_214, type, sP3: $i > $i > $o).
% 4.78/2.03 thf(func_def_216, type, sP5: $i > $o > $i > $i > $o).
% 4.78/2.03 thf(func_def_223, type, sK12: $i > $o).
% 4.78/2.03 thf(func_def_226, type, sK15: $i > $i > ($i > $o) > $i).
% 4.78/2.03 thf(func_def_238, type, sK27: ($i > $o) > $i > $i).
% 4.78/2.03 thf(func_def_240, type, sK29: $i > $o).
% 4.78/2.03 thf(func_def_242, type, sK31: $i > $i > $i).
% 4.78/2.03 thf(func_def_243, type, sK32: $i > $i > $i).
% 4.78/2.03 thf(func_def_244, type, sK33: $i > $i > $i).
% 4.78/2.03 thf(func_def_245, type, sK34: $i > $i > $i).
% 4.78/2.03 thf(func_def_246, type, sK35: $i > $i > $i > $i).
% 4.78/2.03 thf(func_def_247, type, sK36: $i > $i).
% 4.78/2.03 thf(func_def_248, type, sK37: $i > $i).
% 4.78/2.03 thf(func_def_249, type, sK38: $i > $i).
% 4.78/2.03 thf(func_def_250, type, sK39: $i > $i).
% 4.78/2.03 thf(func_def_251, type, sK40: $i > $i > $i).
% 4.78/2.03 thf(func_def_252, type, sK41: $i > $i > $i > $i).
% 4.78/2.03 thf(func_def_253, type, sK42: $i > $i > $i > $i).
% 4.78/2.03 thf(func_def_254, type, sK43: $i > $i > $i).
% 4.78/2.03 thf(func_def_255, type, sK44: $i > $i > $i).
% 4.78/2.03 thf(func_def_256, type, sK45: $i > $i).
% 4.78/2.03 thf(func_def_257, type, sK46: $i > $i > $i).
% 4.78/2.03 thf(func_def_259, type, sK48: $i > $i).
% 4.78/2.03 thf(func_def_260, type, sK49: $i > $i).
% 4.78/2.03 thf(func_def_267, type, sK56: $i > $o).
% 4.78/2.03 thf(func_def_275, type, sK64: $i > $i > $i).
% 4.78/2.03 thf(func_def_279, type, sK68: $i > $o).
% 4.78/2.03 thf(func_def_288, type, sK77: $i > $i).
% 4.78/2.03 thf(func_def_297, type, sK86: ($i > $o) > $i).
% 4.78/2.03 thf(func_def_298, type, sK87: ($i > $o) > $i).
% 4.78/2.03 thf(func_def_299, type, sK88: $i > $o).
% 4.78/2.03 thf(func_def_302, type, sK91: $i > $i).
% 4.78/2.03 thf(func_def_306, type, sK95: ($i > $o) > $i > $i).
% 4.78/2.03 thf(func_def_308, type, sK97: $i > $o).
% 4.78/2.03 thf(func_def_324, type, sK113: $i > $i).
% 4.78/2.03 thf(func_def_331, type, sK120: ($i > $o) > ($i > $o) > $i).
% 4.78/2.03 thf(func_def_332, type, sK121: ($i > $o) > ($i > $o) > $i).
% 4.78/2.03 thf(func_def_333, type, sK122: $i > $o).
% 4.78/2.03 thf(func_def_334, type, sK123: $i > $o).
% 4.78/2.03 thf(func_def_337, type, sK126: $i > $i > $i).
% 4.78/2.03 thf(func_def_343, type, sK132: $i > $o).
% 4.78/2.03 thf(func_def_345, type, sK134: $i > $o).
% 4.78/2.03 thf(func_def_347, type, sK136: ($i > $o) > $i > $i).
% 4.78/2.03 thf(func_def_359, type, sK148: $i > $i).
% 4.78/2.03 thf(func_def_362, type, sK151: $i > $i).
% 4.78/2.03 thf(func_def_372, type, sK161: $i > $i > $i).
% 4.78/2.03 thf(func_def_379, type, sK168: $i > $i > $i).
% 4.78/2.03 thf(func_def_391, type, sK180: $i > $i > $i).
% 4.78/2.03 thf(func_def_412, type, sK201: $i > $i > $i > $i).
% 4.78/2.03 thf(func_def_413, type, sK202: $i > $i > $i > $i).
% 4.78/2.03 thf(func_def_414, type, sK203: $i > $o).
% 4.78/2.03 thf(func_def_415, type, sK204: $i > $i).
% 4.78/2.03 thf(func_def_416, type, sK205: ($i > $o) > $i).
% 4.78/2.03 thf(func_def_418, type, sK207: $i > $o).
% 4.78/2.03 thf(func_def_423, type, sK212: $i > $o).
% 4.78/2.03 thf(func_def_431, type, sK220: $i > $i > $o).
% 4.78/2.03 thf(func_def_433, type, sK222: $i > $i).
% 4.78/2.03 thf(func_def_434, type, sK223: $i > $i).
% 4.78/2.03 thf(func_def_435, type, sK224: $i > ($i > $i > $o) > $i).
% 4.78/2.03 thf(func_def_436, type, sK225: $i > $i > ($i > $i > $o) > $i).
% 4.78/2.03 thf(func_def_437, type, sK226: $i > ($i > $i > $o) > $i).
% 4.78/2.03 thf(func_def_443, type, sK232: $i > $i > $i).
% 4.78/2.03 thf(func_def_444, type, sK233: $i > ($i > $o) > $i).
% 4.78/2.03 thf(func_def_445, type, sK234: $i > $o).
% 4.78/2.03 thf(func_def_447, type, sK236: $i > $o).
% 4.78/2.03 thf(func_def_460, type, sK249: $i > $o).
% 4.78/2.03 thf(func_def_466, type, sK255: $i > $o).
% 4.78/2.03 thf(func_def_488, type, sK277: $i > $i).
% 4.78/2.03 thf(func_def_492, type, sK281: ($i > $o) > $i > $i).
% 4.78/2.03 thf(func_def_494, type, sK283: $i > $o).
% 4.78/2.03 thf(func_def_495, type, sK284: ($i > $o) > ($i > $o) > $i > $i > $i).
% 4.78/2.03 thf(func_def_496, type, sK285: ($i > $o) > ($i > $o) > $i > $i > $i).
% 4.78/2.03 thf(func_def_499, type, sK288: $i > $o).
% 4.78/2.03 thf(func_def_500, type, sK289: $i > $o).
% 4.78/2.03 thf(func_def_508, type, sK297: $i > $o).
% 4.78/2.03 thf(func_def_511, type, sK300: $i > ($i > $o) > $i).
% 4.78/2.03 thf(func_def_512, type, sK301: $i > ($i > $o) > $i).
% 4.78/2.03 thf(func_def_513, type, sK302: $i > ($i > $o) > $i).
% 4.78/2.03 thf(func_def_514, type, sK303: $i > $o).
% 4.78/2.03 thf(func_def_521, type, sK310: $i > $i).
% 4.78/2.03 thf(func_def_524, type, sK313: $i > $o).
% 4.78/2.03 thf(func_def_525, type, sK314: $i > $o).
% 4.78/2.03 thf(func_def_526, type, sK315: ($i > $o) > ($i > $o) > $i).
% 4.78/2.03 thf(func_def_527, type, sK316: ($i > $o) > ($i > $o) > $i).
% 4.78/2.03 thf(func_def_529, type, sK318: $i > $o).
% 4.78/2.03 thf(func_def_530, type, sK319: ($i > $o) > $i).
% 4.78/2.03 thf(func_def_532, type, sK321: ($i > $o) > $i > $i).
% 4.78/2.03 thf(func_def_534, type, sK323: $i > $o).
% 4.78/2.03 thf(func_def_545, type, sK334: ($i > $o) > $i).
% 4.78/2.03 thf(func_def_546, type, sK335: $i > $o).
% 4.78/2.03 thf(func_def_547, type, sK336: $i > $i).
% 4.78/2.03 thf(func_def_560, type, sK349: $i > $i > $i).
% 4.78/2.03 thf(func_def_561, type, sK350: $i > $i > $i).
% 4.78/2.03 thf(func_def_575, type, sK364: $i > $o).
% 4.78/2.03 thf(func_def_589, type, sK378: $i > $o).
% 4.78/2.03 thf(func_def_594, type, sK383: $i > $o).
% 4.78/2.03 thf(func_def_608, type, sK397: ($i > $o) > $i > $i).
% 4.78/2.03 thf(func_def_610, type, sK399: $i > $o).
% 4.78/2.03 thf(func_def_614, type, sK403: $i > $o).
% 4.78/2.03 thf(func_def_641, type, sK430: $i > $o).
% 4.78/2.03 thf(func_def_643, type, sK432: ($i > $o) > $i > $i).
% 4.78/2.03 thf(func_def_644, type, sK433: ($i > $o) > $i > $i).
% 4.78/2.03 thf(func_def_652, type, sK441: $i > $i > $i).
% 4.78/2.03 thf(func_def_655, type, sK444: $o > $i > $i > $i).
% 4.78/2.03 thf(func_def_666, type, sK455: $i > $i > $i).
% 4.78/2.03 thf(func_def_677, type, sK466: ($i > $o) > $i > $i).
% 4.78/2.03 thf(func_def_679, type, sK468: $i > $o).
% 4.78/2.03 thf(func_def_683, type, ph472: !>[X0: $tType]:(X0)).
% 4.78/2.03 thf(f4350,plain,(
% 4.78/2.03 $false),
% 4.78/2.03 inference(avatar_sat_refutation,[status(thm)],[f2733,f4337,f4349])).
% 4.78/2.03 thf(f4349,plain,(
% 4.78/2.03 ~spl471_6 | ~spl471_3),
% 4.78/2.03 inference(avatar_split_clause,[status(thm)],[f4340,f2570,f2634])).
% 4.78/2.03 thf(f2634,definition,(
% 4.78/2.03 spl471_6 <=> (sK271 = sK272)),
% 4.78/2.03 introduced(definition,[new_symbols(naming,[spl471_6])],[avatar_definition])).
% 4.78/2.03 thf(f2570,definition,(
% 4.78/2.03 spl471_3 <=> (sK272 = sK273)),
% 4.78/2.03 introduced(definition,[new_symbols(naming,[spl471_3])],[avatar_definition])).
% 4.78/2.03 thf(f4340,plain,(
% 4.78/2.03 (sK271 != sK272) | ~spl471_3),
% 4.78/2.03 inference(superposition,[status(thm)],[f1981,f2572])).
% 4.78/2.03 thf(f2572,plain,(
% 4.78/2.03 (sK272 = sK273) | ~spl471_3),
% 4.78/2.03 inference(avatar_component_clause,[status(thm)],[f2570])).
% 4.78/2.03 thf(f1981,plain,(
% 4.78/2.03 (sK271 != sK273)),
% 4.78/2.03 inference(cnf_transformation,[status(thm)],[f1176])).
% 4.78/2.03 thf(f1176,plain,(
% 4.78/2.03 (setukpairinjL = $true) & (setadjoinSub2 = $true) & (singletonprop = $true) & (setminusIRneg = $true) & (emptyI = $true) & (subsetI1 = $true) & (ubforcartprodlem1 = $true) & (setukpairinjR1 = $true) & (emptysetE = $true) & (descrp = $true) & (ex1E1 = $true) & (upairinpowunion = $true) & (binunionIR = $true) & (powersetAx = $true) & (subsetI2 = $true) & (binintersectSubset5 = $true) & (exuE1 = $true) & (singletonsswitch = $true) & (eqimpsubset2 = $true) & (prop2setI = $true) & (subsetE2 = $true) & (setoftrueEq = $true) & (binintersectSubset1 = $true) & (singletoninpowerset = $true) & (setminusELneg = $true) & (setadjoinOr = $true) & (singletonsubset = $true) & (binintersectER = $true) & (exuE3e = $true) & (setunionE2 = $true) & (emptysetAx = $true) & (setunionsingleton = $true) & (dsetconstr__Cong = $true) & (subPowSU = $true) & (exuI3 = $true) & (cartprodmempair = $true) & (subsetRefl = $true) & (setminusEL = $true) & (setminusILneg = $true) & (upairsetE = $true) & (kfstpairEq = $true) & (setunionAx = $true) & (symdiffIneg2 = $true) & (emptyset__Cong = $true) & (kpairp = $true) & (setukpairinjL1 = $true) & (omegaSAx = $true) & (setminusSubset2 = $true) & (setminusLsub = $true) & (dsetconstrER = $true) & (emptyinPowerset = $true) & (replAx = $true) & (setunionE = $true) & (quantDeMorgan2 = $true) & (secondinupair = $true) & (exuI2 = $true) & (omega0Ax = $true) & (ubforcartprodlem2 = $true) & (powersetI1 = $true) & (singletoninpowunion = $true) & (setunionsingleton1 = $true) & (notequalI1 = $true) & (notinemptyset = $true) & (emptyE1 = $true) & (upairset2E = $true) & (singletonsuniq = $true) & (emptyinunitempty = $true) & (symdiffIneg1 = $true) & (symdiffE = $true) & (dsetconstrI = $true) & (ex1I2 = $true) & (disjointsetsI1 = $true) & (kfstsingleton = $true) & (setadjoinIL = $true) & (binintersectEL = $true) & (setukpairinjR11 = $true) & (setadjoinSub = $true) & (eqimpsubset1 = $true) & (exuEu = $true) & (notdallE = $true) & (prop2setE = $true) & (upairsetIL = $true) & (((setadjoin @ sK272 @ emptyset) = (setadjoin @ sK273 @ (setadjoin @ sK271 @ emptyset))) & (sK271 != sK273)) & (powersetI = $true) & (exu__Cong = $true) & (powersetsubset = $true) & (upairsetIR = $true) & (cartprodpairin = $true) & (quantDeMorgan4 = $true) & (symdiffI2 = $true) & (setext = $true) & (setunionsingleton2 = $true) & (notequalI2 = $true) & (emptysetsubset = $true) & (binunionRsub = $true) & (subset2powerset = $true) & (powersetE = $true) & (nonemptyE1 = $true) & (cartprodfstin = $true) & (subsetTrans = $true) & (setminusI = $true) & (setukpairIL = $true) & (quantDeMorgan3 = $true) & (omegaIndAx = $true) & (symdiffI1 = $true) & (subsetE = $true) & (dsetconstrEL = $true) & (setminusERneg = $true) & (setminusER = $true) & (descr__Cong = $true) & (setukpairinjR12 = $true) & (exuE2 = $true) & (eqinunit = $true) & (setextAx = $true) & (ubforcartprodlem3 = $true) & (binintersectSubset2 = $true) & (setunion__Cong = $true) & (inCongP = $true) & (binintersectSubset3 = $true) & (setadjoinAx = $true) & (setadjoinIR = $true) & (binintersectLsub = $true) & (subsetemptysetimpeq = $true) & (powersetE1 = $true) & (kpairiskpair = $true) & (exuE3u = $true) & (exuI1 = $true) & (cartprodmempair1 = $true) & (sepSubset = $true) & (inPowerset = $true) & (binunionEcases = $true) & (in__Cong = $true) & (foundationAx = $true) & (binunionIL = $true) & (emptyInPowerset = $true) & (prop2set2propI = $true) & (omega__Cong = $true) & (wellorderingAx = $true) & (emptysetimpfalse = $true) & (theprop = $true) & (binunionLsub = $true) & (notsubsetI = $true) & (binunionE = $true) & (notinsingleton = $true) & (powerset__Cong = $true) & (noeltsimpempty = $true) & (setadjoin__Cong = $true) & (ex1I = $true) & (upairset2IR = $true) & (bs114d = $true) & (vacuousDall = $true) & (setukpairinjL2 = $true) & (setadjoinE = $true) & (sepInPowerset = $true) & (binintersectI = $true) & (setbeta = $true) & (setukpairIR = $true) & (quantDeMorgan1 = $true) & (nonemptyImpWitness = $true) & (nonemptyI = $true) & (setextsub = $true) & (uniqinunit = $true) & (upairsubunion = $true) & (binintersectSubset4 = $true) & (setminusSubset1 = $true) & (setunionI = $true) & (notdexE = $true) & (nonemptyI1 = $true) & (binintersectRsub = $true)),
% 4.78/2.03 inference(skolemisation,[status(esa),new_symbols(skolem,[sK271,sK272,sK273])],[f611,f1175])).
% 4.78/2.03 thf(f1175,plain,(
% 4.78/2.03 ? [X0 : $i,X1 : $i,X2 : $i] : (((setadjoin @ X1 @ emptyset) = (setadjoin @ X2 @ (setadjoin @ X0 @ emptyset))) & (X0 != X2)) => (((setadjoin @ sK272 @ emptyset) = (setadjoin @ sK273 @ (setadjoin @ sK271 @ emptyset))) & (sK271 != sK273))),
% 4.78/2.03 introduced(definition,[],[choice_axiom])).
% 4.78/2.03 thf(f611,plain,(
% 4.78/2.03 (setukpairinjL = $true) & (setadjoinSub2 = $true) & (singletonprop = $true) & (setminusIRneg = $true) & (emptyI = $true) & (subsetI1 = $true) & (ubforcartprodlem1 = $true) & (setukpairinjR1 = $true) & (emptysetE = $true) & (descrp = $true) & (ex1E1 = $true) & (upairinpowunion = $true) & (binunionIR = $true) & (powersetAx = $true) & (subsetI2 = $true) & (binintersectSubset5 = $true) & (exuE1 = $true) & (singletonsswitch = $true) & (eqimpsubset2 = $true) & (prop2setI = $true) & (subsetE2 = $true) & (setoftrueEq = $true) & (binintersectSubset1 = $true) & (singletoninpowerset = $true) & (setminusELneg = $true) & (setadjoinOr = $true) & (singletonsubset = $true) & (binintersectER = $true) & (exuE3e = $true) & (setunionE2 = $true) & (emptysetAx = $true) & (setunionsingleton = $true) & (dsetconstr__Cong = $true) & (subPowSU = $true) & (exuI3 = $true) & (cartprodmempair = $true) & (subsetRefl = $true) & (setminusEL = $true) & (setminusILneg = $true) & (upairsetE = $true) & (kfstpairEq = $true) & (setunionAx = $true) & (symdiffIneg2 = $true) & (emptyset__Cong = $true) & (kpairp = $true) & (setukpairinjL1 = $true) & (omegaSAx = $true) & (setminusSubset2 = $true) & (setminusLsub = $true) & (dsetconstrER = $true) & (emptyinPowerset = $true) & (replAx = $true) & (setunionE = $true) & (quantDeMorgan2 = $true) & (secondinupair = $true) & (exuI2 = $true) & (omega0Ax = $true) & (ubforcartprodlem2 = $true) & (powersetI1 = $true) & (singletoninpowunion = $true) & (setunionsingleton1 = $true) & (notequalI1 = $true) & (notinemptyset = $true) & (emptyE1 = $true) & (upairset2E = $true) & (singletonsuniq = $true) & (emptyinunitempty = $true) & (symdiffIneg1 = $true) & (symdiffE = $true) & (dsetconstrI = $true) & (ex1I2 = $true) & (disjointsetsI1 = $true) & (kfstsingleton = $true) & (setadjoinIL = $true) & (binintersectEL = $true) & (setukpairinjR11 = $true) & (setadjoinSub = $true) & (eqimpsubset1 = $true) & (exuEu = $true) & (notdallE = $true) & (prop2setE = $true) & (upairsetIL = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (((setadjoin @ X1 @ emptyset) = (setadjoin @ X2 @ (setadjoin @ X0 @ emptyset))) & (X0 != X2)) & (powersetI = $true) & (exu__Cong = $true) & (powersetsubset = $true) & (upairsetIR = $true) & (cartprodpairin = $true) & (quantDeMorgan4 = $true) & (symdiffI2 = $true) & (setext = $true) & (setunionsingleton2 = $true) & (notequalI2 = $true) & (emptysetsubset = $true) & (binunionRsub = $true) & (subset2powerset = $true) & (powersetE = $true) & (nonemptyE1 = $true) & (cartprodfstin = $true) & (subsetTrans = $true) & (setminusI = $true) & (setukpairIL = $true) & (quantDeMorgan3 = $true) & (omegaIndAx = $true) & (symdiffI1 = $true) & (subsetE = $true) & (dsetconstrEL = $true) & (setminusERneg = $true) & (setminusER = $true) & (descr__Cong = $true) & (setukpairinjR12 = $true) & (exuE2 = $true) & (eqinunit = $true) & (setextAx = $true) & (ubforcartprodlem3 = $true) & (binintersectSubset2 = $true) & (setunion__Cong = $true) & (inCongP = $true) & (binintersectSubset3 = $true) & (setadjoinAx = $true) & (setadjoinIR = $true) & (binintersectLsub = $true) & (subsetemptysetimpeq = $true) & (powersetE1 = $true) & (kpairiskpair = $true) & (exuE3u = $true) & (exuI1 = $true) & (cartprodmempair1 = $true) & (sepSubset = $true) & (inPowerset = $true) & (binunionEcases = $true) & (in__Cong = $true) & (foundationAx = $true) & (binunionIL = $true) & (emptyInPowerset = $true) & (prop2set2propI = $true) & (omega__Cong = $true) & (wellorderingAx = $true) & (emptysetimpfalse = $true) & (theprop = $true) & (binunionLsub = $true) & (notsubsetI = $true) & (binunionE = $true) & (notinsingleton = $true) & (powerset__Cong = $true) & (noeltsimpempty = $true) & (setadjoin__Cong = $true) & (ex1I = $true) & (upairset2IR = $true) & (bs114d = $true) & (vacuousDall = $true) & (setukpairinjL2 = $true) & (setadjoinE = $true) & (sepInPowerset = $true) & (binintersectI = $true) & (setbeta = $true) & (setukpairIR = $true) & (quantDeMorgan1 = $true) & (nonemptyImpWitness = $true) & (nonemptyI = $true) & (setextsub = $true) & (uniqinunit = $true) & (upairsubunion = $true) & (binintersectSubset4 = $true) & (setminusSubset1 = $true) & (setunionI = $true) & (notdexE = $true) & (nonemptyI1 = $true) & (binintersectRsub = $true)),
% 4.78/2.03 inference(flattening,[status(thm)],[f610])).
% 4.78/2.03 thf(f610,plain,(
% 4.78/2.03 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i,X2 : $i] : (((setadjoin @ X1 @ emptyset) = (setadjoin @ X2 @ (setadjoin @ X0 @ emptyset))) & (X0 != X2)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (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)),
% 4.78/2.03 inference(ennf_transformation,[status(thm)],[f280])).
% 4.78/2.03 thf(f280,plain,(
% 4.78/2.03 ~((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) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ! [X0 : $i,X1 : $i,X2 : $i] : (((setadjoin @ X1 @ emptyset) = (setadjoin @ X2 @ (setadjoin @ X0 @ emptyset))) => (X0 = X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 4.78/2.03 inference(fool_elimination,[status(thm)],[f279])).
% 4.78/2.03 thf(f279,plain,(
% 4.78/2.03 ~(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 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => ! [X0 : $i,X1 : $i,X2 : $i] : (((setadjoin @ X1 @ emptyset) = (setadjoin @ X2 @ (setadjoin @ X0 @ emptyset))) => (X0 = X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 4.78/2.03 inference(rectify,[status(thm)],[f176])).
% 4.78/2.03 thf(f176,negated_conjecture,(
% 4.78/2.03 ~(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 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => ! [X2 : $i,X8 : $i,X1 : $i] : (((setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)) = (setadjoin @ X8 @ emptyset)) => (X1 = X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 4.78/2.03 inference(negated_conjecture,[status(cth)],[f175])).
% 4.78/2.03 thf(f175,conjecture,(
% 4.78/2.03 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 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => ! [X2 : $i,X8 : $i,X1 : $i] : (((setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)) = (setadjoin @ X8 @ emptyset)) => (X1 = X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 4.78/2.03 file('/export/starexec/sandbox/tmp/tmp.19e6KiooJw/DTF2THF_26384.p',upairequniteq)).
% 4.78/2.03 thf(f4337,plain,(
% 4.78/2.03 spl471_3),
% 4.78/2.03 inference(avatar_split_clause,[status(thm)],[f4336,f2570])).
% 4.78/2.03 thf(f4336,plain,(
% 4.78/2.03 (sK272 = sK273)),
% 4.78/2.03 inference(trivial_inequality_removal,[status(thm)],[f4335])).
% 4.78/2.03 thf(f4335,plain,(
% 4.78/2.03 (sK272 = sK273) | ($true != $true)),
% 4.78/2.03 inference(forward_demodulation,[status(thm)],[f4312,f2010])).
% 4.78/2.03 thf(f2010,plain,(
% 4.78/2.03 (secondinupair = $true)),
% 4.78/2.03 inference(cnf_transformation,[status(thm)],[f1176])).
% 4.78/2.03 thf(f4312,plain,(
% 4.78/2.03 (sK272 = sK273) | (secondinupair != $true)),
% 4.78/2.03 inference(trivial_inequality_removal,[status(thm)],[f4301])).
% 4.78/2.03 thf(f4301,plain,(
% 4.78/2.03 ($true != $true) | (sK272 = sK273) | (secondinupair != $true)),
% 4.78/2.03 inference(superposition,[status(thm)],[f2522,f2181])).
% 4.78/2.03 thf(f2181,plain,(
% 4.78/2.03 ( ! [X0 : $i,X1 : $i] : (((in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset))) = $true) | (secondinupair != $true)) )),
% 4.78/2.03 inference(cnf_transformation,[status(thm)],[f1319])).
% 4.78/2.03 thf(f1319,plain,(
% 4.78/2.03 (! [X0 : $i,X1 : $i] : ((in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset))) = $true) | (secondinupair != $true)) & ((secondinupair = $true) | ((in @ sK352 @ (setadjoin @ sK351 @ (setadjoin @ sK352 @ emptyset))) != $true))),
% 4.78/2.03 inference(skolemisation,[status(esa),new_symbols(skolem,[sK351,sK352])],[f1317,f1318])).
% 4.78/2.03 thf(f1318,plain,(
% 4.78/2.03 ? [X2 : $i,X3 : $i] : ((in @ X3 @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset))) != $true) => ((in @ sK352 @ (setadjoin @ sK351 @ (setadjoin @ sK352 @ emptyset))) != $true)),
% 4.78/2.03 introduced(definition,[],[choice_axiom])).
% 4.78/2.03 thf(f1317,plain,(
% 4.78/2.03 (! [X0 : $i,X1 : $i] : ((in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset))) = $true) | (secondinupair != $true)) & ((secondinupair = $true) | ? [X2 : $i,X3 : $i] : ((in @ X3 @ (setadjoin @ X2 @ (setadjoin @ X3 @ emptyset))) != $true))),
% 4.78/2.03 inference(rectify,[status(thm)],[f1316])).
% 4.78/2.03 thf(f1316,plain,(
% 4.78/2.03 (! [X0 : $i,X1 : $i] : ((in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset))) = $true) | (secondinupair != $true)) & ((secondinupair = $true) | ? [X0 : $i,X1 : $i] : ((in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset))) != $true))),
% 4.78/2.03 inference(nnf_transformation,[status(thm)],[f191])).
% 4.78/2.03 thf(f191,plain,(
% 4.78/2.03 ! [X0 : $i,X1 : $i] : ((in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset))) = $true) <=> (secondinupair = $true)),
% 4.78/2.03 inference(fool_elimination,[status(thm)],[f190])).
% 4.78/2.03 thf(f190,plain,(
% 4.78/2.03 (secondinupair = ! [X0 : $i,X1 : $i] : (in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset))))),
% 4.78/2.03 inference(rectify,[status(thm)],[f136])).
% 4.78/2.03 thf(f136,axiom,(
% 4.78/2.03 (secondinupair = ! [X1 : $i,X2 : $i] : (in @ X2 @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset))))),
% 4.78/2.03 file('/export/starexec/sandbox/tmp/tmp.19e6KiooJw/DTF2THF_26384.p',secondinupair)).
% 4.78/2.03 thf(f2522,plain,(
% 4.78/2.03 ( ! [X0 : $i] : (((in @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ sK273 @ emptyset) @ (setadjoin @ (setadjoin @ sK272 @ emptyset) @ emptyset))) != $true) | (sK273 = X0)) )),
% 4.78/2.03 inference(trivial_inequality_removal,[status(thm)],[f2521])).
% 4.78/2.03 thf(f2521,plain,(
% 4.78/2.03 ( ! [X0 : $i] : ((sK273 = X0) | ((in @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ sK273 @ emptyset) @ (setadjoin @ (setadjoin @ sK272 @ emptyset) @ emptyset))) != $true) | ($true != $true)) )),
% 4.78/2.03 inference(forward_demodulation,[status(thm)],[f2467,f2019])).
% 4.78/2.03 thf(f2019,plain,(
% 4.78/2.03 (setukpairinjL1 = $true)),
% 4.78/2.03 inference(cnf_transformation,[status(thm)],[f1176])).
% 4.78/2.03 thf(f2467,plain,(
% 4.78/2.03 ( ! [X0 : $i] : ((sK273 = X0) | ((in @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ sK273 @ emptyset) @ (setadjoin @ (setadjoin @ sK272 @ emptyset) @ emptyset))) != $true) | (setukpairinjL1 != $true)) )),
% 4.78/2.03 inference(superposition,[status(thm)],[f1717,f1982])).
% 4.78/2.03 thf(f1982,plain,(
% 4.78/2.03 ((setadjoin @ sK272 @ emptyset) = (setadjoin @ sK273 @ (setadjoin @ sK271 @ emptyset)))),
% 4.78/2.03 inference(cnf_transformation,[status(thm)],[f1176])).
% 4.78/2.03 thf(f1717,plain,(
% 4.78/2.03 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((in @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X1 @ emptyset)) @ emptyset))) != $true) | (X0 = X2) | (setukpairinjL1 != $true)) )),
% 4.78/2.03 inference(cnf_transformation,[status(thm)],[f967])).
% 4.78/2.03 thf(f967,plain,(
% 4.78/2.03 (! [X0 : $i,X1 : $i,X2 : $i] : ((X0 = X2) | ((in @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X1 @ emptyset)) @ emptyset))) != $true)) | (setukpairinjL1 != $true)) & ((setukpairinjL1 = $true) | ((sK143 != sK141) & ((in @ (setadjoin @ sK141 @ emptyset) @ (setadjoin @ (setadjoin @ sK143 @ emptyset) @ (setadjoin @ (setadjoin @ sK143 @ (setadjoin @ sK142 @ emptyset)) @ emptyset))) = $true)))),
% 4.78/2.03 inference(skolemisation,[status(esa),new_symbols(skolem,[sK141,sK142,sK143])],[f965,f966])).
% 4.78/2.03 thf(f966,plain,(
% 4.78/2.03 ? [X3 : $i,X4 : $i,X5 : $i] : ((X3 != X5) & ((in @ (setadjoin @ X3 @ emptyset) @ (setadjoin @ (setadjoin @ X5 @ emptyset) @ (setadjoin @ (setadjoin @ X5 @ (setadjoin @ X4 @ emptyset)) @ emptyset))) = $true)) => ((sK143 != sK141) & ((in @ (setadjoin @ sK141 @ emptyset) @ (setadjoin @ (setadjoin @ sK143 @ emptyset) @ (setadjoin @ (setadjoin @ sK143 @ (setadjoin @ sK142 @ emptyset)) @ emptyset))) = $true))),
% 4.78/2.03 introduced(definition,[],[choice_axiom])).
% 4.78/2.03 thf(f965,plain,(
% 4.78/2.03 (! [X0 : $i,X1 : $i,X2 : $i] : ((X0 = X2) | ((in @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X1 @ emptyset)) @ emptyset))) != $true)) | (setukpairinjL1 != $true)) & ((setukpairinjL1 = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : ((X3 != X5) & ((in @ (setadjoin @ X3 @ emptyset) @ (setadjoin @ (setadjoin @ X5 @ emptyset) @ (setadjoin @ (setadjoin @ X5 @ (setadjoin @ X4 @ emptyset)) @ emptyset))) = $true)))),
% 4.78/2.03 inference(rectify,[status(thm)],[f964])).
% 4.78/2.03 thf(f964,plain,(
% 4.78/2.03 (! [X0 : $i,X1 : $i,X2 : $i] : ((X0 = X2) | ((in @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X1 @ emptyset)) @ emptyset))) != $true)) | (setukpairinjL1 != $true)) & ((setukpairinjL1 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : ((X0 != X2) & ((in @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X1 @ emptyset)) @ emptyset))) = $true)))),
% 4.78/2.03 inference(nnf_transformation,[status(thm)],[f667])).
% 4.78/2.03 thf(f667,plain,(
% 4.78/2.03 ! [X0 : $i,X1 : $i,X2 : $i] : ((X0 = X2) | ((in @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X1 @ emptyset)) @ emptyset))) != $true)) <=> (setukpairinjL1 = $true)),
% 4.78/2.03 inference(ennf_transformation,[status(thm)],[f195])).
% 4.78/2.03 thf(f195,plain,(
% 4.78/2.03 (setukpairinjL1 = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X1 @ emptyset)) @ emptyset))) = $true) => (X0 = X2))),
% 4.78/2.03 inference(fool_elimination,[status(thm)],[f194])).
% 4.78/2.03 thf(f194,plain,(
% 4.78/2.03 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ (setadjoin @ X0 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ emptyset) @ (setadjoin @ (setadjoin @ X2 @ (setadjoin @ X1 @ emptyset)) @ emptyset))) => (X0 = X2)) = setukpairinjL1)),
% 4.78/2.03 inference(rectify,[status(thm)],[f165])).
% 4.78/2.03 thf(f165,axiom,(
% 4.78/2.03 (! [X8 : $i,X2 : $i,X1 : $i] : ((in @ (setadjoin @ X8 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ emptyset) @ (setadjoin @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)) @ emptyset))) => (X1 = X8)) = setukpairinjL1)),
% 4.78/2.03 file('/export/starexec/sandbox/tmp/tmp.19e6KiooJw/DTF2THF_26384.p',setukpairinjL1)).
% 4.78/2.03 thf(f2733,plain,(
% 4.78/2.03 spl471_6),
% 4.78/2.03 inference(avatar_split_clause,[status(thm)],[f2732,f2634])).
% 4.78/2.03 thf(f2732,plain,(
% 4.78/2.03 (sK271 = sK272)),
% 4.78/2.03 inference(trivial_inequality_removal,[status(thm)],[f2731])).
% 4.78/2.03 thf(f2731,plain,(
% 4.78/2.03 (sK271 = sK272) | ($true != $true)),
% 4.78/2.03 inference(forward_demodulation,[status(thm)],[f2715,f1902])).
% 4.78/2.03 thf(f1902,plain,(
% 4.78/2.03 (uniqinunit = $true)),
% 4.78/2.03 inference(cnf_transformation,[status(thm)],[f1176])).
% 4.78/2.03 thf(f2715,plain,(
% 4.78/2.03 (sK271 = sK272) | (uniqinunit != $true)),
% 4.78/2.03 inference(trivial_inequality_removal,[status(thm)],[f2708])).
% 4.78/2.03 thf(f2708,plain,(
% 4.78/2.03 (sK271 = sK272) | (uniqinunit != $true) | ($true != $true)),
% 4.78/2.03 inference(superposition,[status(thm)],[f1891,f2539])).
% 4.78/2.03 thf(f2539,plain,(
% 4.78/2.03 ((in @ sK271 @ (setadjoin @ sK272 @ emptyset)) = $true)),
% 4.78/2.03 inference(trivial_inequality_removal,[status(thm)],[f2538])).
% 4.78/2.03 thf(f2538,plain,(
% 4.78/2.03 ((in @ sK271 @ (setadjoin @ sK272 @ emptyset)) = $true) | ($true != $true)),
% 4.78/2.03 inference(forward_demodulation,[status(thm)],[f2478,f1915])).
% 4.78/2.03 thf(f1915,plain,(
% 4.78/2.03 (upairset2IR = $true)),
% 4.78/2.03 inference(cnf_transformation,[status(thm)],[f1176])).
% 4.78/2.03 thf(f2478,plain,(
% 4.78/2.03 ((in @ sK271 @ (setadjoin @ sK272 @ emptyset)) = $true) | (upairset2IR != $true)),
% 4.78/2.03 inference(superposition,[status(thm)],[f2268,f1982])).
% 4.78/2.03 thf(f2268,plain,(
% 4.78/2.03 ( ! [X0 : $i,X1 : $i] : (((in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) = $true) | (upairset2IR != $true)) )),
% 4.78/2.03 inference(cnf_transformation,[status(thm)],[f1436])).
% 4.78/2.03 thf(f1436,plain,(
% 4.78/2.03 (! [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) = $true) | (upairset2IR != $true)) & ((upairset2IR = $true) | ((in @ sK417 @ (setadjoin @ sK418 @ (setadjoin @ sK417 @ emptyset))) != $true))),
% 4.78/2.03 inference(skolemisation,[status(esa),new_symbols(skolem,[sK417,sK418])],[f1434,f1435])).
% 4.78/2.03 thf(f1435,plain,(
% 4.78/2.03 ? [X2 : $i,X3 : $i] : ((in @ X2 @ (setadjoin @ X3 @ (setadjoin @ X2 @ emptyset))) != $true) => ((in @ sK417 @ (setadjoin @ sK418 @ (setadjoin @ sK417 @ emptyset))) != $true)),
% 4.78/2.03 introduced(definition,[],[choice_axiom])).
% 4.78/2.03 thf(f1434,plain,(
% 4.78/2.03 (! [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) = $true) | (upairset2IR != $true)) & ((upairset2IR = $true) | ? [X2 : $i,X3 : $i] : ((in @ X2 @ (setadjoin @ X3 @ (setadjoin @ X2 @ emptyset))) != $true))),
% 4.78/2.03 inference(rectify,[status(thm)],[f1433])).
% 4.78/2.03 thf(f1433,plain,(
% 4.78/2.03 (! [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) = $true) | (upairset2IR != $true)) & ((upairset2IR = $true) | ? [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) != $true))),
% 4.78/2.03 inference(nnf_transformation,[status(thm)],[f266])).
% 4.78/2.03 thf(f266,plain,(
% 4.78/2.03 ! [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))) = $true) <=> (upairset2IR = $true)),
% 4.78/2.03 inference(fool_elimination,[status(thm)],[f265])).
% 4.78/2.03 thf(f265,plain,(
% 4.78/2.03 (upairset2IR = ! [X0 : $i,X1 : $i] : (in @ X0 @ (setadjoin @ X1 @ (setadjoin @ X0 @ emptyset))))),
% 4.78/2.03 inference(rectify,[status(thm)],[f102])).
% 4.78/2.03 thf(f102,axiom,(
% 4.78/2.03 (upairset2IR = ! [X2 : $i,X1 : $i] : (in @ X2 @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset))))),
% 4.78/2.03 file('/export/starexec/sandbox/tmp/tmp.19e6KiooJw/DTF2THF_26384.p',upairset2IR)).
% 4.78/2.03 thf(f1891,plain,(
% 4.78/2.03 ( ! [X0 : $i,X1 : $i] : (((in @ X1 @ (setadjoin @ X0 @ emptyset)) != $true) | (X0 = X1) | (uniqinunit != $true)) )),
% 4.78/2.03 inference(cnf_transformation,[status(thm)],[f1170])).
% 4.78/2.03 thf(f1170,plain,(
% 4.78/2.03 (! [X0 : $i,X1 : $i] : (((in @ X1 @ (setadjoin @ X0 @ emptyset)) != $true) | (X0 = X1)) | (uniqinunit != $true)) & ((uniqinunit = $true) | (((in @ sK268 @ (setadjoin @ sK267 @ emptyset)) = $true) & (sK267 != sK268)))),
% 4.78/2.03 inference(skolemisation,[status(esa),new_symbols(skolem,[sK267,sK268])],[f1168,f1169])).
% 4.78/2.03 thf(f1169,plain,(
% 4.78/2.03 ? [X2 : $i,X3 : $i] : (((in @ X3 @ (setadjoin @ X2 @ emptyset)) = $true) & (X2 != X3)) => (((in @ sK268 @ (setadjoin @ sK267 @ emptyset)) = $true) & (sK267 != sK268))),
% 4.78/2.03 introduced(definition,[],[choice_axiom])).
% 4.78/2.03 thf(f1168,plain,(
% 4.78/2.03 (! [X0 : $i,X1 : $i] : (((in @ X1 @ (setadjoin @ X0 @ emptyset)) != $true) | (X0 = X1)) | (uniqinunit != $true)) & ((uniqinunit = $true) | ? [X2 : $i,X3 : $i] : (((in @ X3 @ (setadjoin @ X2 @ emptyset)) = $true) & (X2 != X3)))),
% 4.78/2.03 inference(rectify,[status(thm)],[f1167])).
% 4.78/2.03 thf(f1167,plain,(
% 4.78/2.03 (! [X0 : $i,X1 : $i] : (((in @ X1 @ (setadjoin @ X0 @ emptyset)) != $true) | (X0 = X1)) | (uniqinunit != $true)) & ((uniqinunit = $true) | ? [X0 : $i,X1 : $i] : (((in @ X1 @ (setadjoin @ X0 @ emptyset)) = $true) & (X0 != X1)))),
% 4.78/2.03 inference(nnf_transformation,[status(thm)],[f695])).
% 4.78/2.03 thf(f695,plain,(
% 4.78/2.03 ! [X0 : $i,X1 : $i] : (((in @ X1 @ (setadjoin @ X0 @ emptyset)) != $true) | (X0 = X1)) <=> (uniqinunit = $true)),
% 4.78/2.03 inference(ennf_transformation,[status(thm)],[f378])).
% 4.78/2.03 thf(f378,plain,(
% 4.78/2.03 ! [X0 : $i,X1 : $i] : (((in @ X1 @ (setadjoin @ X0 @ emptyset)) = $true) => (X0 = X1)) <=> (uniqinunit = $true)),
% 4.78/2.03 inference(fool_elimination,[status(thm)],[f377])).
% 4.78/2.03 thf(f377,plain,(
% 4.78/2.03 (! [X0 : $i,X1 : $i] : ((in @ X1 @ (setadjoin @ X0 @ emptyset)) => (X0 = X1)) = uniqinunit)),
% 4.78/2.03 inference(rectify,[status(thm)],[f46])).
% 4.78/2.03 thf(f46,axiom,(
% 4.78/2.03 (! [X2 : $i,X1 : $i] : ((in @ X1 @ (setadjoin @ X2 @ emptyset)) => (X1 = X2)) = uniqinunit)),
% 4.78/2.03 file('/export/starexec/sandbox/tmp/tmp.19e6KiooJw/DTF2THF_26384.p',uniqinunit)).
% 4.78/2.03 % SZS output end Proof for DTF2THF_26384
% 4.78/2.03 % (26504)------------------------------
% 4.78/2.03 % (26504)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.78/2.03 % (26504)Termination reason: Refutation
% 4.78/2.03
% 4.78/2.03 % (26504)Memory used [KB]: 8955
% 4.78/2.03 % (26504)Time elapsed: 0.207 s
% 4.78/2.03 % (26504)Instructions burned: 374 (million)
% 4.78/2.03 % (26504)------------------------------
% 4.78/2.03 % (26504)------------------------------
% 4.78/2.03 % (26481)Success in time 0.27 s
%------------------------------------------------------------------------------