%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU716^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n010.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 09:36:37 AM UTC 2026
% Result : Theorem 2.73s 1.93s
% Output : Refutation 2.73s
% Verified :
% SZS Type : Refutation
% Derivation depth : 21
% Number of leaves : 20
% Syntax : Number of formulae : 113 ( 26 unt; 0 typ; 2 def)
% Number of atoms : 4879 (1319 equ; 0 cnn)
% Maximal formula atoms : 238 ( 43 avg)
% Number of connectives : 2999 ( 205 ~; 171 |; 779 &; 850 @)
% ( 17 <=>; 977 =>; 0 <=; 0 <~>)
% Maximal formula depth : 245 ( 19 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 439 ( 436 usr; 252 con; 0-4 aty)
% Number of variables : 211 ( 0 ^; 155 !; 56 ?; 211 :)
% 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_203,type,
ksnd: $i > $i ).
thf(func_def_213,type,
breln: $i > $i > $i > $o ).
thf(func_def_214,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_222,type,
func: $i > $i > $i > $o ).
thf(func_def_223,type,
funcSet: $i > $i > $i ).
thf(func_def_226,type,
ap: $i > $i > $i > $i > $i ).
thf(func_def_232,type,
lam: $i > $i > ( $i > $i ) > $i ).
thf(func_def_259,type,
if: $i > $o > $i > $i > $i ).
thf(func_def_284,type,
sP0: $i > $o > $i > $i > $o ).
thf(func_def_285,type,
sP1: $i > $i > $i > $o ).
thf(func_def_286,type,
sP2: $i > $i > $o ).
thf(func_def_287,type,
sP3: $i > $o ).
thf(func_def_288,type,
sP4: $i > $i > $o ).
thf(func_def_289,type,
sP5: $i > $i > $o ).
thf(func_def_293,type,
sP9: $i > $i > $i > $i > $o ).
thf(func_def_301,type,
sK17: $i > $o ).
thf(func_def_306,type,
sK22: $i > $i ).
thf(func_def_309,type,
sK25: $i > $i ).
thf(func_def_311,type,
sK27: $i > $i ).
thf(func_def_312,type,
sK28: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_313,type,
sK29: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_314,type,
sK30: $i > $o ).
thf(func_def_315,type,
sK31: $i > $o ).
thf(func_def_326,type,
sK42: $i > $i ).
thf(func_def_328,type,
sK44: ( $i > $i ) > $i > $i > $i ).
thf(func_def_329,type,
sK45: $i > ( $i > $o ) > $i ).
thf(func_def_330,type,
sK46: $i > $o ).
thf(func_def_367,type,
sK83: $i > $i > $o ).
thf(func_def_369,type,
sK85: $i > $o ).
thf(func_def_370,type,
sK86: $i > $i ).
thf(func_def_371,type,
sK87: ( $i > $o ) > $i ).
thf(func_def_374,type,
sK90: $i > $i ).
thf(func_def_383,type,
sK99: $i > $i > $i ).
thf(func_def_386,type,
sK102: $i > $i > $i > $i ).
thf(func_def_390,type,
sK106: $i > $i > $i > $i ).
thf(func_def_401,type,
sK117: $i > $i > $i ).
thf(func_def_402,type,
sK118: $i > $i > $i ).
thf(func_def_403,type,
sK119: $i > $i > $i ).
thf(func_def_404,type,
sK120: $i > $i > $i ).
thf(func_def_405,type,
sK121: $i > $i > $i > $i ).
thf(func_def_406,type,
sK122: $i > $i ).
thf(func_def_407,type,
sK123: $i > $i ).
thf(func_def_408,type,
sK124: $i > $i ).
thf(func_def_409,type,
sK125: $i > $i ).
thf(func_def_410,type,
sK126: $i > $i > $i > $i ).
thf(func_def_411,type,
sK127: $i > $i > $i > $i ).
thf(func_def_412,type,
sK128: $i > $i > $i ).
thf(func_def_413,type,
sK129: $i > $i > $i ).
thf(func_def_414,type,
sK130: $i > $i > $i ).
thf(func_def_415,type,
sK131: $i > $i ).
thf(func_def_416,type,
sK132: $i > $i > $i ).
thf(func_def_418,type,
sK134: $i > $i ).
thf(func_def_419,type,
sK135: $i > $i ).
thf(func_def_421,type,
sK137: $i > $o ).
thf(func_def_427,type,
sK143: $i > $i ).
thf(func_def_435,type,
sK151: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_436,type,
sK152: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_440,type,
sK156: $i > $o ).
thf(func_def_442,type,
sK158: $i > $i > $i ).
thf(func_def_448,type,
sK164: $i > $o ).
thf(func_def_450,type,
sK166: $i > ( $i > $o ) > $i ).
thf(func_def_456,type,
sK172: $i > $i > $o ).
thf(func_def_457,type,
sK173: $i > $i ).
thf(func_def_459,type,
sK175: $i > $o ).
thf(func_def_462,type,
sK178: $i > ( $i > $o ) > $i ).
thf(func_def_463,type,
sK179: $i > ( $i > $o ) > $i ).
thf(func_def_464,type,
sK180: $i > $o ).
thf(func_def_469,type,
sK185: $i > $o ).
thf(func_def_471,type,
sK187: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_472,type,
sK188: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_474,type,
sK190: $i > $i ).
thf(func_def_478,type,
sK194: $i > $i > $i > $i > $i ).
thf(func_def_479,type,
sK195: $i > $i > $i > $i > $i ).
thf(func_def_513,type,
sK229: $i > $o ).
thf(func_def_515,type,
sK231: ( $i > $o ) > $i > $i ).
thf(func_def_516,type,
sK232: ( $i > $o ) > $i > $i ).
thf(func_def_528,type,
sK244: $i > $o ).
thf(func_def_529,type,
sK245: ( $i > $o ) > $i ).
thf(func_def_540,type,
sK256: $i > $i > $o ).
thf(func_def_546,type,
sK262: $i > $i > $i ).
thf(func_def_552,type,
sK268: $i > $o ).
thf(func_def_554,type,
sK270: ( $i > $o ) > $i > $i ).
thf(func_def_610,type,
sK326: $i > $i > $i > $i > $i ).
thf(func_def_616,type,
sK332: $i > ( $i > $o ) > $i ).
thf(func_def_617,type,
sK333: $i > $o ).
thf(func_def_628,type,
sK344: $i > $i ).
thf(func_def_630,type,
sK346: ( $i > $i ) > $i > $i > $i ).
thf(func_def_635,type,
sK351: $i > $i > $i ).
thf(func_def_644,type,
sK360: $i > $i > $o ).
thf(func_def_650,type,
sK366: $i > $i > $o ).
thf(func_def_651,type,
sK367: $i > $i ).
thf(func_def_652,type,
sK368: $i > $i ).
thf(func_def_653,type,
sK369: ( $i > $i > $o ) > $i > $i ).
thf(func_def_654,type,
sK370: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_655,type,
sK371: ( $i > $i > $o ) > $i > $i ).
thf(func_def_664,type,
sK380: ( $i > $o ) > $i ).
thf(func_def_665,type,
sK381: ( $i > $o ) > $i ).
thf(func_def_666,type,
sK382: $i > $o ).
thf(func_def_669,type,
sK385: $i > $o ).
thf(func_def_672,type,
sK388: $i > ( $i > $o ) > $i ).
thf(func_def_673,type,
sK389: $i > $o ).
thf(func_def_677,type,
sK393: $i > $o ).
thf(func_def_678,type,
sK394: ( $i > $o ) > $i > $i > $i ).
thf(func_def_691,type,
sK407: $i > $i > $i ).
thf(func_def_696,type,
sK412: $i > $o ).
thf(func_def_714,type,
sK430: $i > $o ).
thf(func_def_740,type,
sK456: $i > ( $i > $o ) > $i ).
thf(func_def_741,type,
sK457: $i > $o ).
thf(func_def_744,type,
sK460: $i > $o ).
thf(func_def_746,type,
sK462: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_747,type,
sK463: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_748,type,
sK464: $i > $o ).
thf(func_def_749,type,
sK465: $i > $o ).
thf(func_def_765,type,
sK481: $i > $o ).
thf(func_def_775,type,
sK491: $i > $i > $i ).
thf(func_def_776,type,
sK492: $i > $i ).
thf(func_def_779,type,
sK495: $i > $i > ( $i > $i ) > $i ).
thf(func_def_784,type,
sK500: $i > $i > $o ).
thf(func_def_795,type,
sK511: $i > $i > $o ).
thf(func_def_798,type,
sK514: $i > $i > $i > $i ).
thf(func_def_799,type,
sK515: $i > $i > $i > $i ).
thf(func_def_805,type,
sK521: $i > ( $i > $i ) > $i > $i ).
thf(func_def_807,type,
sK523: $i > $i ).
thf(func_def_820,type,
sK536: $o > $i > $i > $i ).
thf(func_def_825,type,
sK541: $i > $i > $i > $i > $i ).
thf(func_def_837,type,
sK553: $i > $o ).
thf(func_def_840,type,
sK556: $i > ( $i > $o ) > $i ).
thf(func_def_841,type,
sK557: $i > ( $i > $o ) > $i ).
thf(func_def_858,type,
sK574: $i > $o ).
thf(func_def_859,type,
sK575: $i > $i ).
thf(func_def_860,type,
sK576: ( $i > $o ) > $i ).
thf(func_def_867,type,
sK583: $i > $o ).
thf(func_def_872,type,
sK588: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_873,type,
sK589: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_876,type,
sK592: $i > $o ).
thf(func_def_877,type,
sK593: $i > $o ).
thf(func_def_905,type,
sK621: $i > $i > $i > $i > $i ).
thf(func_def_906,type,
sK622: $i > $i > $i > $i > $i ).
thf(func_def_907,type,
sK623: $i > $i > $i > $i > $i ).
thf(func_def_908,type,
sK624: $i > $i > $i > $i > $i ).
thf(func_def_917,type,
sK633: $i > $o ).
thf(func_def_918,type,
sK634: $i > $o ).
thf(func_def_924,type,
sK640: $i > $o ).
thf(func_def_934,type,
sK650: $i > $o ).
thf(func_def_939,type,
sK655: $i > $i ).
thf(func_def_940,type,
sK656: ( $i > $i ) > $i > $i > $i ).
thf(func_def_945,type,
sK661: $i > $o ).
thf(func_def_954,type,
sK670: $i > $i > $i ).
thf(func_def_961,type,
sK677: $i > $i > $i > $i > $i ).
thf(func_def_962,type,
sK678: $i > $i > $i ).
thf(func_def_963,type,
sK679: $i > $i > $i ).
thf(func_def_969,type,
sK685: ( $i > $i ) > $i > $i > $i ).
thf(func_def_972,type,
sK688: $i > $i ).
thf(func_def_975,type,
sK691: $i > $i > $i ).
thf(func_def_997,type,
sK713: $i > $i > $o ).
thf(func_def_1007,type,
sK723: ( $i > $o ) > $i > $i ).
thf(func_def_1009,type,
sK725: $i > $o ).
thf(func_def_1017,type,
ph733:
!>[X0: $tType] : X0 ).
thf(f7576,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f6603,f7545,f7558]) ).
thf(f7558,plain,
( spl732_70
| ~ spl732_72 ),
inference(avatar_split_clause,[status(thm)],[f7557,f4730,f4678]) ).
thf(f4678,definition,
( spl732_70
<=> ( $true
= ( subset @ sK310 @ sK312 ) ) ),
introduced(definition,[new_symbols(naming,[spl732_70])],[avatar_definition]) ).
thf(f4730,definition,
( spl732_72
<=> ( ( in @ sK310 @ ( powerset @ sK312 ) )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl732_72])],[avatar_definition]) ).
thf(f7557,plain,
( ( $true
= ( subset @ sK310 @ sK312 ) )
| ~ spl732_72 ),
inference(trivial_inequality_removal,[status(thm)],[f7556]) ).
thf(f7556,plain,
( ( $true
= ( subset @ sK310 @ sK312 ) )
| ( $true != $true )
| ~ spl732_72 ),
inference(forward_demodulation,[status(thm)],[f7555,f2567]) ).
thf(f2567,plain,
powersetE1 = $true,
inference(cnf_transformation,[status(thm)],[f1496]) ).
thf(f1496,plain,
( ( exuE3e = $true )
& ( setextsub = $true )
& ( setminusIRneg = $true )
& ( binintersectSubset3 = $true )
& ( setadjoinE = $true )
& ( cartprodmempair1 = $true )
& ( setminusEL = $true )
& ( emptysetimpfalse = $true )
& ( emptyE1 = $true )
& ( kpairsurjEq = $true )
& ( cartprodpairmemER = $true )
& ( ex1I2 = $true )
& ( setminusILneg = $true )
& ( emptyI = $true )
& ( setunionAx = $true )
& ( setukpairIR = $true )
& ( emptysetE = $true )
& ( ubforcartprodlem3 = $true )
& ( setminusELneg = $true )
& ( setukpairinjL1 = $true )
& ( cartprodmempair = $true )
& ( binintersectSubset4 = $true )
& ( omegaIndAx = $true )
& ( complementT_lem = $true )
& ( dsetconstrI = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( exu__Cong = $true )
& ( singletonsswitch = $true )
& ( singletoninpowunion = $true )
& ( funcextLem = $true )
& ( binunionIR = $true )
& ( upairsetIR = $true )
& ( setadjoinIR = $true )
& ( setminusI = $true )
& ( setukpairIL = $true )
& ( dpsetconstrSub = $true )
& ( ( in @ sK310 @ ( powerset @ sK311 ) )
= $true )
& ( $true
= ( in @ sK312 @ ( powerset @ sK311 ) ) )
& ! [X3: $i] :
( ( $true
!= ( in @ X3 @ sK310 ) )
| ( $true
= ( in @ X3 @ sK312 ) )
| ( $true
!= ( in @ X3 @ sK311 ) ) )
& ( $true
!= ( subset @ sK310 @ sK312 ) )
& ( secondinupair = $true )
& ( nonemptyI1 = $true )
& ( singletonprop = $true )
& ( exuEu = $true )
& ( cartprodfstpairEq = $true )
& ( binintersectEL = $true )
& ( upairsetE = $true )
& ( setukpairinjL2 = $true )
& ( iftrueProp1 = $true )
& ( iftrueProp2 = $true )
& ( cartprodpairsurjEq = $true )
& ( beta1 = $true )
& ( apProp = $true )
& ( subsetI1 = $true )
& ( subsetE = $true )
& ( dpsetconstrEL1 = $true )
& ( subsetRefl = $true )
& ( setadjoinSub2 = $true )
& ( ex1E1 = $true )
& ( exuE1 = $true )
& ( theeq = $true )
& ( disjointsetsI1 = $true )
& ( binintersectI = $true )
& ( ubforcartprodlem1 = $true )
& ( funcGraphProp4 = $true )
& ( setadjoin__Cong = $true )
& ( ap2p = $true )
& ( brelnall1 = $true )
& ( symdiffIneg1 = $true )
& ( funcGraphProp2 = $true )
& ( binintersectLsub = $true )
& ( cartprodsndin = $true )
& ( nonemptyE1 = $true )
& ( notsubsetI = $true )
& ( setminusSubset2 = $true )
& ( ifp = $true )
& ( in__Cong = $true )
& ( prop2setE = $true )
& ( notdexE = $true )
& ( iffalse = $true )
& ( funcGraphProp3 = $true )
& ( setextAx = $true )
& ( setadjoinOr = $true )
& ( eqbreln = $true )
& ( powersetAx = $true )
& ( sepSubset = $true )
& ( vacuousDall = $true )
& ( emptysetAx = $true )
& ( symdiffIneg2 = $true )
& ( prop2set2propI = $true )
& ( iftrueorfalse = $true )
& ( dsetconstrEL = $true )
& ( brelnall2 = $true )
& ( exuE2 = $true )
& ( upairset2E = $true )
& ( exuE3u = $true )
& ( subset2powerset = $true )
& ( infuncsetfunc = $true )
& ( omega0Ax = $true )
& ( exuI3 = $true )
& ( emptyset__Cong = $true )
& ( setminusSubset1 = $true )
& ( powersetI = $true )
& ( funcImageSingleton = $true )
& ( quantDeMorgan2 = $true )
& ( powersetE = $true )
& ( lamProp = $true )
& ( binintersectSubset1 = $true )
& ( setukpairinjR = $true )
& ( omega__Cong = $true )
& ( binunionRsub = $true )
& ( subsetTrans = $true )
& ( eqimpsubset1 = $true )
& ( symdiffI1 = $true )
& ( kfstsingleton = $true )
& ( setukpairinjR11 = $true )
& ( binintersectT_lem = $true )
& ( binintersectSubset5 = $true )
& ( upairequniteq = $true )
& ( noeltsimpempty = $true )
& ( ifSingleton = $true )
& ( subsetI2 = $true )
& ( inCongP = $true )
& ( emptysetsubset = $true )
& ( setextT = $true )
& ( cartprodsndpairEq = $true )
& ( ap2apEq2 = $true )
& ( ex1I = $true )
& ( sepInPowerset = $true )
& ( setoftrueEq = $true )
& ( setadjoinIL = $true )
& ( symdiffE = $true )
& ( notequalI2 = $true )
& ( binunionEcases = $true )
& ( setadjoinAx = $true )
& ( ap2apEq1 = $true )
& ( setunionsingleton = $true )
& ( emptyinunitempty = $true )
& ( ksndsingleton = $true )
& ( eta1 = $true )
& ( cartprodmempaircEq = $true )
& ( descrp = $true )
& ( cartprodpairin = $true )
& ( binintersectER = $true )
& ( ex1E2 = $true )
& ( symdiffI2 = $true )
& ( setukpairinjL = $true )
& ( prop2setI = $true )
& ( powerset__Cong = $true )
& ( inPowerset = $true )
& ( quantDeMorgan4 = $true )
& ( notequalI1 = $true )
& ( upairsetIL = $true )
& ( ksndpairEq = $true )
& ( setunion__Cong = $true )
& ( setunionE2 = $true )
& ( exuI1 = $true )
& ( iftrue = $true )
& ( subbreln = $true )
& ( setext = $true )
& ( notinsingleton = $true )
& ( setminusERneg = $true )
& ( binintersectSubset2 = $true )
& ( binunionIL = $true )
& ( setminusER = $true )
& ( eqimpsubset2 = $true )
& ( funcGraphProp1 = $true )
& ( dpsetconstrER = $true )
& ( setminusLsub = $true )
& ( nonemptyImpWitness = $true )
& ( setunionI = $true )
& ( replAx = $true )
& ( app = $true )
& ( singletonsuniq = $true )
& ( omegaSAx = $true )
& ( powersetsubset = $true )
& ( exuI2 = $true )
& ( powersetI1 = $true )
& ( dsetconstrER = $true )
& ( foundationAx = $true )
& ( kpairiskpair = $true )
& ( quantDeMorgan3 = $true )
& ( nonemptyI = $true )
& ( wellorderingAx = $true )
& ( singletonsubset = $true )
& ( cartprodpairmemEL = $true )
& ( setadjoinSub = $true )
& ( setbeta = $true )
& ( bs114d = $true )
& ( theprop = $true )
& ( setukpairinjR1 = $true )
& ( emptyinPowerset = $true )
& ( emptyInPowerset = $true )
& ( subsetemptysetimpeq = $true )
& ( subsetE2 = $true )
& ( lam2lamEq = $true )
& ( funcinfuncset = $true )
& ( iffalseProp2 = $true )
& ( funcext = $true )
& ( upairset2IR = $true )
& ( lamp = $true )
& ( dpsetconstrERa = $true )
& ( beta2 = $true )
& ( binunionE = $true )
& ( iffalseProp1 = $true )
& ( binunionT_lem = $true )
& ( notinemptyset = $true )
& ( powersetE1 = $true )
& ( kpairp = $true )
& ( dpsetconstrI = $true )
& ( uniqinunit = $true )
& ( binunionLsub = $true )
& ( ubforcartprodlem2 = $true )
& ( descr__Cong = $true )
& ( dsetconstr__Cong = $true )
& ( binintersectRsub = $true )
& ( setukpairinjR2 = $true )
& ( eta2 = $true )
& ( singletoninpowerset = $true )
& ( setukpairinjR12 = $true )
& ( funcext2 = $true )
& ( eqinunit = $true )
& ( upairinpowunion = $true )
& ( dpsetconstrEL2 = $true )
& ( cartprodfstin = $true )
& ( setunionsingleton1 = $true )
& ( upairsubunion = $true )
& ( setminusT_lem = $true )
& ( kfstpairEq = $true )
& ( setOfPairsIsBReln = $true )
& ( powersetT_lem = $true )
& ( lam2p = $true )
& ( quantDeMorgan1 = $true )
& ( setunionsingleton2 = $true )
& ( notdallE = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK310,sK311,sK312])],[f755,f1495,f1494]) ).
thf(f1494,plain,
( ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) )
& ? [X2: $i] :
( ( ( in @ X2 @ ( powerset @ X1 ) )
= $true )
& ! [X3: $i] :
( ( ( in @ X3 @ X0 )
!= $true )
| ( ( in @ X3 @ X2 )
= $true )
| ( $true
!= ( in @ X3 @ X1 ) ) )
& ( ( subset @ X0 @ X2 )
!= $true ) ) )
=> ( ( ( in @ sK310 @ ( powerset @ sK311 ) )
= $true )
& ? [X2: $i] :
( ( ( in @ X2 @ ( powerset @ sK311 ) )
= $true )
& ! [X3: $i] :
( ( $true
!= ( in @ X3 @ sK310 ) )
| ( ( in @ X3 @ X2 )
= $true )
| ( $true
!= ( in @ X3 @ sK311 ) ) )
& ( ( subset @ sK310 @ X2 )
!= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1495,plain,
( ? [X2: $i] :
( ( ( in @ X2 @ ( powerset @ sK311 ) )
= $true )
& ! [X3: $i] :
( ( $true
!= ( in @ X3 @ sK310 ) )
| ( ( in @ X3 @ X2 )
= $true )
| ( $true
!= ( in @ X3 @ sK311 ) ) )
& ( ( subset @ sK310 @ X2 )
!= $true ) )
=> ( ( $true
= ( in @ sK312 @ ( powerset @ sK311 ) ) )
& ! [X3: $i] :
( ( $true
!= ( in @ X3 @ sK310 ) )
| ( $true
= ( in @ X3 @ sK312 ) )
| ( $true
!= ( in @ X3 @ sK311 ) ) )
& ( $true
!= ( subset @ sK310 @ sK312 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f755,plain,
( ( exuE3e = $true )
& ( setextsub = $true )
& ( setminusIRneg = $true )
& ( binintersectSubset3 = $true )
& ( setadjoinE = $true )
& ( cartprodmempair1 = $true )
& ( setminusEL = $true )
& ( emptysetimpfalse = $true )
& ( emptyE1 = $true )
& ( kpairsurjEq = $true )
& ( cartprodpairmemER = $true )
& ( ex1I2 = $true )
& ( setminusILneg = $true )
& ( emptyI = $true )
& ( setunionAx = $true )
& ( setukpairIR = $true )
& ( emptysetE = $true )
& ( ubforcartprodlem3 = $true )
& ( setminusELneg = $true )
& ( setukpairinjL1 = $true )
& ( cartprodmempair = $true )
& ( binintersectSubset4 = $true )
& ( omegaIndAx = $true )
& ( complementT_lem = $true )
& ( dsetconstrI = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( exu__Cong = $true )
& ( singletonsswitch = $true )
& ( singletoninpowunion = $true )
& ( funcextLem = $true )
& ( binunionIR = $true )
& ( upairsetIR = $true )
& ( setadjoinIR = $true )
& ( setminusI = $true )
& ( setukpairIL = $true )
& ( dpsetconstrSub = $true )
& ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) )
& ? [X2: $i] :
( ( ( in @ X2 @ ( powerset @ X1 ) )
= $true )
& ! [X3: $i] :
( ( ( in @ X3 @ X0 )
!= $true )
| ( ( in @ X3 @ X2 )
= $true )
| ( $true
!= ( in @ X3 @ X1 ) ) )
& ( ( subset @ X0 @ X2 )
!= $true ) ) )
& ( secondinupair = $true )
& ( nonemptyI1 = $true )
& ( singletonprop = $true )
& ( exuEu = $true )
& ( cartprodfstpairEq = $true )
& ( binintersectEL = $true )
& ( upairsetE = $true )
& ( setukpairinjL2 = $true )
& ( iftrueProp1 = $true )
& ( iftrueProp2 = $true )
& ( cartprodpairsurjEq = $true )
& ( beta1 = $true )
& ( apProp = $true )
& ( subsetI1 = $true )
& ( subsetE = $true )
& ( dpsetconstrEL1 = $true )
& ( subsetRefl = $true )
& ( setadjoinSub2 = $true )
& ( ex1E1 = $true )
& ( exuE1 = $true )
& ( theeq = $true )
& ( disjointsetsI1 = $true )
& ( binintersectI = $true )
& ( ubforcartprodlem1 = $true )
& ( funcGraphProp4 = $true )
& ( setadjoin__Cong = $true )
& ( ap2p = $true )
& ( brelnall1 = $true )
& ( symdiffIneg1 = $true )
& ( funcGraphProp2 = $true )
& ( binintersectLsub = $true )
& ( cartprodsndin = $true )
& ( nonemptyE1 = $true )
& ( notsubsetI = $true )
& ( setminusSubset2 = $true )
& ( ifp = $true )
& ( in__Cong = $true )
& ( prop2setE = $true )
& ( notdexE = $true )
& ( iffalse = $true )
& ( funcGraphProp3 = $true )
& ( setextAx = $true )
& ( setadjoinOr = $true )
& ( eqbreln = $true )
& ( powersetAx = $true )
& ( sepSubset = $true )
& ( vacuousDall = $true )
& ( emptysetAx = $true )
& ( symdiffIneg2 = $true )
& ( prop2set2propI = $true )
& ( iftrueorfalse = $true )
& ( dsetconstrEL = $true )
& ( brelnall2 = $true )
& ( exuE2 = $true )
& ( upairset2E = $true )
& ( exuE3u = $true )
& ( subset2powerset = $true )
& ( infuncsetfunc = $true )
& ( omega0Ax = $true )
& ( exuI3 = $true )
& ( emptyset__Cong = $true )
& ( setminusSubset1 = $true )
& ( powersetI = $true )
& ( funcImageSingleton = $true )
& ( quantDeMorgan2 = $true )
& ( powersetE = $true )
& ( lamProp = $true )
& ( binintersectSubset1 = $true )
& ( setukpairinjR = $true )
& ( omega__Cong = $true )
& ( binunionRsub = $true )
& ( subsetTrans = $true )
& ( eqimpsubset1 = $true )
& ( symdiffI1 = $true )
& ( kfstsingleton = $true )
& ( setukpairinjR11 = $true )
& ( binintersectT_lem = $true )
& ( binintersectSubset5 = $true )
& ( upairequniteq = $true )
& ( noeltsimpempty = $true )
& ( ifSingleton = $true )
& ( subsetI2 = $true )
& ( inCongP = $true )
& ( emptysetsubset = $true )
& ( setextT = $true )
& ( cartprodsndpairEq = $true )
& ( ap2apEq2 = $true )
& ( ex1I = $true )
& ( sepInPowerset = $true )
& ( setoftrueEq = $true )
& ( setadjoinIL = $true )
& ( symdiffE = $true )
& ( notequalI2 = $true )
& ( binunionEcases = $true )
& ( setadjoinAx = $true )
& ( ap2apEq1 = $true )
& ( setunionsingleton = $true )
& ( emptyinunitempty = $true )
& ( ksndsingleton = $true )
& ( eta1 = $true )
& ( cartprodmempaircEq = $true )
& ( descrp = $true )
& ( cartprodpairin = $true )
& ( binintersectER = $true )
& ( ex1E2 = $true )
& ( symdiffI2 = $true )
& ( setukpairinjL = $true )
& ( prop2setI = $true )
& ( powerset__Cong = $true )
& ( inPowerset = $true )
& ( quantDeMorgan4 = $true )
& ( notequalI1 = $true )
& ( upairsetIL = $true )
& ( ksndpairEq = $true )
& ( setunion__Cong = $true )
& ( setunionE2 = $true )
& ( exuI1 = $true )
& ( iftrue = $true )
& ( subbreln = $true )
& ( setext = $true )
& ( notinsingleton = $true )
& ( setminusERneg = $true )
& ( binintersectSubset2 = $true )
& ( binunionIL = $true )
& ( setminusER = $true )
& ( eqimpsubset2 = $true )
& ( funcGraphProp1 = $true )
& ( dpsetconstrER = $true )
& ( setminusLsub = $true )
& ( nonemptyImpWitness = $true )
& ( setunionI = $true )
& ( replAx = $true )
& ( app = $true )
& ( singletonsuniq = $true )
& ( omegaSAx = $true )
& ( powersetsubset = $true )
& ( exuI2 = $true )
& ( powersetI1 = $true )
& ( dsetconstrER = $true )
& ( foundationAx = $true )
& ( kpairiskpair = $true )
& ( quantDeMorgan3 = $true )
& ( nonemptyI = $true )
& ( wellorderingAx = $true )
& ( singletonsubset = $true )
& ( cartprodpairmemEL = $true )
& ( setadjoinSub = $true )
& ( setbeta = $true )
& ( bs114d = $true )
& ( theprop = $true )
& ( setukpairinjR1 = $true )
& ( emptyinPowerset = $true )
& ( emptyInPowerset = $true )
& ( subsetemptysetimpeq = $true )
& ( subsetE2 = $true )
& ( lam2lamEq = $true )
& ( funcinfuncset = $true )
& ( iffalseProp2 = $true )
& ( funcext = $true )
& ( upairset2IR = $true )
& ( lamp = $true )
& ( dpsetconstrERa = $true )
& ( beta2 = $true )
& ( binunionE = $true )
& ( iffalseProp1 = $true )
& ( binunionT_lem = $true )
& ( notinemptyset = $true )
& ( powersetE1 = $true )
& ( kpairp = $true )
& ( dpsetconstrI = $true )
& ( uniqinunit = $true )
& ( binunionLsub = $true )
& ( ubforcartprodlem2 = $true )
& ( descr__Cong = $true )
& ( dsetconstr__Cong = $true )
& ( binintersectRsub = $true )
& ( setukpairinjR2 = $true )
& ( eta2 = $true )
& ( singletoninpowerset = $true )
& ( setukpairinjR12 = $true )
& ( funcext2 = $true )
& ( eqinunit = $true )
& ( upairinpowunion = $true )
& ( dpsetconstrEL2 = $true )
& ( cartprodfstin = $true )
& ( setunionsingleton1 = $true )
& ( upairsubunion = $true )
& ( setminusT_lem = $true )
& ( kfstpairEq = $true )
& ( setOfPairsIsBReln = $true )
& ( powersetT_lem = $true )
& ( lam2p = $true )
& ( quantDeMorgan1 = $true )
& ( setunionsingleton2 = $true )
& ( notdallE = $true ) ),
inference(flattening,[status(thm)],[f754]) ).
thf(f754,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( ( subset @ X0 @ X2 )
!= $true )
& ! [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
| ( ( in @ X3 @ X0 )
!= $true )
| ( $true
!= ( in @ X3 @ X1 ) ) )
& ( ( in @ X2 @ ( powerset @ X1 ) )
= $true ) )
& ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) ) )
& ( setextT = $true )
& ( complementT_lem = $true )
& ( setminusT_lem = $true )
& ( powersetT_lem = $true )
& ( binunionT_lem = $true )
& ( binintersectT_lem = $true )
& ( iftrueorfalse = $true )
& ( iffalse = $true )
& ( iftrue = $true )
& ( theeq = $true )
& ( ifp = $true )
& ( ifSingleton = $true )
& ( iftrueProp2 = $true )
& ( iftrueProp1 = $true )
& ( iffalseProp2 = $true )
& ( iffalseProp1 = $true )
& ( eta2 = $true )
& ( beta2 = $true )
& ( lam2lamEq = $true )
& ( eta1 = $true )
& ( beta1 = $true )
& ( ap2apEq2 = $true )
& ( ap2apEq1 = $true )
& ( funcext2 = $true )
& ( funcext = $true )
& ( eqbreln = $true )
& ( subbreln = $true )
& ( funcGraphProp4 = $true )
& ( funcextLem = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp3 = $true )
& ( funcGraphProp1 = $true )
& ( ex1E2 = $true )
& ( brelnall2 = $true )
& ( brelnall1 = $true )
& ( lam2p = $true )
& ( lamp = $true )
& ( lamProp = $true )
& ( funcinfuncset = $true )
& ( ap2p = $true )
& ( infuncsetfunc = $true )
& ( app = $true )
& ( apProp = $true )
& ( funcImageSingleton = $true )
& ( dpsetconstrER = $true )
& ( dpsetconstrEL2 = $true )
& ( dpsetconstrEL1 = $true )
& ( dpsetconstrERa = $true )
& ( setOfPairsIsBReln = $true )
& ( dpsetconstrSub = $true )
& ( dpsetconstrI = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodsndin = $true )
& ( kpairsurjEq = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( upairequniteq = $true )
& ( 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)],[f650]) ).
thf(f650,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 )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp3 = $true )
=> ( ( funcGraphProp2 = $true )
=> ( ( funcextLem = $true )
=> ( ( funcGraphProp4 = $true )
=> ( ( subbreln = $true )
=> ( ( eqbreln = $true )
=> ( ( funcext = $true )
=> ( ( funcext2 = $true )
=> ( ( ap2apEq1 = $true )
=> ( ( ap2apEq2 = $true )
=> ( ( beta1 = $true )
=> ( ( eta1 = $true )
=> ( ( lam2lamEq = $true )
=> ( ( beta2 = $true )
=> ( ( eta2 = $true )
=> ( ( iffalseProp1 = $true )
=> ( ( iffalseProp2 = $true )
=> ( ( iftrueProp1 = $true )
=> ( ( iftrueProp2 = $true )
=> ( ( ifSingleton = $true )
=> ( ( ifp = $true )
=> ( ( theeq = $true )
=> ( ( iftrue = $true )
=> ( ( iffalse = $true )
=> ( ( iftrueorfalse = $true )
=> ( ( binintersectT_lem = $true )
=> ( ( binunionT_lem = $true )
=> ( ( powersetT_lem = $true )
=> ( ( setminusT_lem = $true )
=> ( ( complementT_lem = $true )
=> ( ( setextT = $true )
=> ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) )
=> ! [X2: $i] :
( ( ( in @ X2 @ ( powerset @ X1 ) )
= $true )
=> ( ! [X3: $i] :
( ( $true
= ( in @ X3 @ X1 ) )
=> ( ( ( in @ X3 @ X0 )
= $true )
=> ( ( in @ X3 @ X2 )
= $true ) ) )
=> ( ( subset @ X0 @ X2 )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f649]) ).
thf(f649,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
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( powerset @ X1 ) )
=> ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X0 )
=> ( in @ X3 @ X2 ) ) )
=> ( subset @ X0 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f238]) ).
thf(f238,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
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ! [X11: $i,X3: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ( ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ( in @ X1 @ X11 )
=> ( in @ X1 @ X16 ) ) )
=> ( subset @ X11 @ X16 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f237]) ).
thf(f237,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
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ! [X11: $i,X3: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ( ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ( in @ X1 @ X11 )
=> ( in @ X1 @ X16 ) ) )
=> ( subset @ X11 @ X16 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.DvUejQI5fj/DTF2THF_12059.p',subsetTI) ).
thf(f7555,plain,
( ( $true
= ( subset @ sK310 @ sK312 ) )
| ( powersetE1 != $true )
| ~ spl732_72 ),
inference(trivial_inequality_removal,[status(thm)],[f7548]) ).
thf(f7548,plain,
( ( $true
= ( subset @ sK310 @ sK312 ) )
| ( powersetE1 != $true )
| ( $true != $true )
| ~ spl732_72 ),
inference(superposition,[status(thm)],[f3032,f4732]) ).
thf(f4732,plain,
( ( ( in @ sK310 @ ( powerset @ sK312 ) )
= $true )
| ~ spl732_72 ),
inference(avatar_component_clause,[status(thm)],[f4730]) ).
thf(f3032,plain,
! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ ( powerset @ X1 ) ) )
| ( ( subset @ X0 @ X1 )
= $true )
| ( powersetE1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f1807]) ).
thf(f1807,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ ( powerset @ X1 ) ) )
| ( ( subset @ X0 @ X1 )
= $true ) )
| ( powersetE1 != $true ) )
& ( ( powersetE1 = $true )
| ( ( $true
= ( in @ sK519 @ ( powerset @ sK520 ) ) )
& ( ( subset @ sK519 @ sK520 )
!= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK519,sK520])],[f1805,f1806]) ).
thf(f1806,plain,
( ? [X2: $i,X3: $i] :
( ( ( in @ X2 @ ( powerset @ X3 ) )
= $true )
& ( ( subset @ X2 @ X3 )
!= $true ) )
=> ( ( $true
= ( in @ sK519 @ ( powerset @ sK520 ) ) )
& ( ( subset @ sK519 @ sK520 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1805,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ ( powerset @ X1 ) ) )
| ( ( subset @ X0 @ X1 )
= $true ) )
| ( powersetE1 != $true ) )
& ( ( powersetE1 = $true )
| ? [X2: $i,X3: $i] :
( ( ( in @ X2 @ ( powerset @ X3 ) )
= $true )
& ( ( subset @ X2 @ X3 )
!= $true ) ) ) ),
inference(rectify,[status(thm)],[f1804]) ).
thf(f1804,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ ( powerset @ X1 ) ) )
| ( ( subset @ X0 @ X1 )
= $true ) )
| ( powersetE1 != $true ) )
& ( ( powersetE1 = $true )
| ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) )
& ( ( subset @ X0 @ X1 )
!= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f962]) ).
thf(f962,plain,
( ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ ( powerset @ X1 ) ) )
| ( ( subset @ X0 @ X1 )
= $true ) )
<=> ( powersetE1 = $true ) ),
inference(ennf_transformation,[status(thm)],[f520]) ).
thf(f520,plain,
( ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) )
=> ( ( subset @ X0 @ X1 )
= $true ) )
<=> ( powersetE1 = $true ) ),
inference(fool_elimination,[status(thm)],[f519]) ).
thf(f519,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
=> ( subset @ X0 @ X1 ) )
= powersetE1 ),
inference(rectify,[status(thm)],[f96]) ).
thf(f96,axiom,
! [X4: $i,X3: $i] :
( ( ( in @ X4 @ ( powerset @ X3 ) )
=> ( subset @ X4 @ X3 ) )
= powersetE1 ),
file('/export/starexec/sandbox2/tmp/tmp.DvUejQI5fj/DTF2THF_12059.p',powersetE1) ).
thf(f7545,plain,
spl732_72,
inference(avatar_split_clause,[status(thm)],[f7544,f4730]) ).
thf(f7544,plain,
( ( in @ sK310 @ ( powerset @ sK312 ) )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f7543]) ).
thf(f7543,plain,
( ( $true != $true )
| ( ( in @ sK310 @ ( powerset @ sK312 ) )
= $true ) ),
inference(forward_demodulation,[status(thm)],[f7532,f2672]) ).
thf(f2672,plain,
powersetI = $true,
inference(cnf_transformation,[status(thm)],[f1496]) ).
thf(f7532,plain,
( ( ( in @ sK310 @ ( powerset @ sK312 ) )
= $true )
| ( powersetI != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f7531]) ).
thf(f7531,plain,
( ( $true != $true )
| ( ( in @ sK310 @ ( powerset @ sK312 ) )
= $true )
| ( powersetI != $true ) ),
inference(duplicate_literal_removal,[status(thm)],[f7524]) ).
thf(f7524,plain,
( ( powersetI != $true )
| ( $true != $true )
| ( ( in @ sK310 @ ( powerset @ sK312 ) )
= $true )
| ( ( in @ sK310 @ ( powerset @ sK312 ) )
= $true ) ),
inference(superposition,[status(thm)],[f4761,f2256]) ).
thf(f2256,plain,
! [X0: $i,X1: $i] :
( ( ( in @ ( sK99 @ X1 @ X0 ) @ X1 )
= $true )
| ( powersetI != $true )
| ( ( in @ X1 @ ( powerset @ X0 ) )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f1162]) ).
thf(f1162,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
| ( ( ( in @ ( sK99 @ X1 @ X0 ) @ X1 )
= $true )
& ( ( in @ ( sK99 @ X1 @ X0 ) @ X0 )
!= $true ) ) )
| ( powersetI != $true ) )
& ( ( powersetI = $true )
| ( ( ( in @ sK101 @ ( powerset @ sK100 ) )
!= $true )
& ! [X5: $i] :
( ( ( in @ X5 @ sK101 )
!= $true )
| ( $true
= ( in @ X5 @ sK100 ) ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK99,sK100,sK101])],[f1159,f1161,f1160]) ).
thf(f1160,plain,
! [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
& ( ( in @ X2 @ X0 )
!= $true ) )
=> ( ( ( in @ ( sK99 @ X1 @ X0 ) @ X1 )
= $true )
& ( ( in @ ( sK99 @ X1 @ X0 ) @ X0 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1161,plain,
( ? [X3: $i,X4: $i] :
( ( ( in @ X4 @ ( powerset @ X3 ) )
!= $true )
& ! [X5: $i] :
( ( ( in @ X5 @ X4 )
!= $true )
| ( ( in @ X5 @ X3 )
= $true ) ) )
=> ( ( ( in @ sK101 @ ( powerset @ sK100 ) )
!= $true )
& ! [X5: $i] :
( ( ( in @ X5 @ sK101 )
!= $true )
| ( $true
= ( in @ X5 @ sK100 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1159,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
| ? [X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
& ( ( in @ X2 @ X0 )
!= $true ) ) )
| ( powersetI != $true ) )
& ( ( powersetI = $true )
| ? [X3: $i,X4: $i] :
( ( ( in @ X4 @ ( powerset @ X3 ) )
!= $true )
& ! [X5: $i] :
( ( ( in @ X5 @ X4 )
!= $true )
| ( ( in @ X5 @ X3 )
= $true ) ) ) ) ),
inference(rectify,[status(thm)],[f1158]) ).
thf(f1158,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
| ? [X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
& ( ( in @ X2 @ X0 )
!= $true ) ) )
| ( powersetI != $true ) )
& ( ( powersetI = $true )
| ? [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
!= $true )
& ! [X2: $i] :
( ( ( in @ X2 @ X1 )
!= $true )
| ( ( in @ X2 @ X0 )
= $true ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f767]) ).
thf(f767,plain,
( ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
| ? [X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
& ( ( in @ X2 @ X0 )
!= $true ) ) )
<=> ( powersetI = $true ) ),
inference(ennf_transformation,[status(thm)],[f310]) ).
thf(f310,plain,
( ( powersetI = $true )
<=> ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
=> ( ( in @ X2 @ X0 )
= $true ) )
=> ( ( in @ X1 @ ( powerset @ X0 ) )
= $true ) ) ),
inference(fool_elimination,[status(thm)],[f309]) ).
thf(f309,plain,
( powersetI
= ( ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ X0 ) )
=> ( in @ X1 @ ( powerset @ X0 ) ) ) ) ),
inference(rectify,[status(thm)],[f37]) ).
thf(f37,axiom,
( powersetI
= ( ! [X3: $i,X4: $i] :
( ! [X1: $i] :
( ( in @ X1 @ X4 )
=> ( in @ X1 @ X3 ) )
=> ( in @ X4 @ ( powerset @ X3 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.DvUejQI5fj/DTF2THF_12059.p',powersetI) ).
thf(f4761,plain,
! [X0: $i] :
( ( $true
!= ( in @ ( sK99 @ X0 @ sK312 ) @ sK310 ) )
| ( ( in @ X0 @ ( powerset @ sK312 ) )
= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4760]) ).
thf(f4760,plain,
! [X0: $i] :
( ( $true != $true )
| ( $true
!= ( in @ ( sK99 @ X0 @ sK312 ) @ sK310 ) )
| ( ( in @ X0 @ ( powerset @ sK312 ) )
= $true ) ),
inference(duplicate_literal_removal,[status(thm)],[f4743]) ).
thf(f4743,plain,
! [X0: $i] :
( ( $true
!= ( in @ ( sK99 @ X0 @ sK312 ) @ sK310 ) )
| ( ( in @ X0 @ ( powerset @ sK312 ) )
= $true )
| ( $true != $true )
| ( $true
!= ( in @ ( sK99 @ X0 @ sK312 ) @ sK310 ) ) ),
inference(superposition,[status(thm)],[f3629,f3540]) ).
thf(f3540,plain,
! [X0: $i] :
( ( $true
= ( in @ X0 @ sK311 ) )
| ( $true
!= ( in @ X0 @ sK310 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3539]) ).
thf(f3539,plain,
! [X0: $i] :
( ( $true
!= ( in @ X0 @ sK310 ) )
| ( $true
= ( in @ X0 @ sK311 ) )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f3532,f2690]) ).
thf(f2690,plain,
powersetAx = $true,
inference(cnf_transformation,[status(thm)],[f1496]) ).
thf(f3532,plain,
! [X0: $i] :
( ( $true
!= ( in @ X0 @ sK310 ) )
| ( powersetAx != $true )
| ( $true
= ( in @ X0 @ sK311 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3526]) ).
thf(f3526,plain,
! [X0: $i] :
( ( $true != $true )
| ( powersetAx != $true )
| ( $true
= ( in @ X0 @ sK311 ) )
| ( $true
!= ( in @ X0 @ sK310 ) ) ),
inference(superposition,[status(thm)],[f2891,f2738]) ).
thf(f2738,plain,
( ( in @ sK310 @ ( powerset @ sK311 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f1496]) ).
thf(f2891,plain,
! [X7: $i,X4: $i,X5: $i] :
( ( ( in @ X5 @ ( powerset @ X4 ) )
!= $true )
| ( ( in @ X7 @ X4 )
= $true )
| ( powersetAx != $true )
| ( $true
!= ( in @ X7 @ X5 ) ) ),
inference(cnf_transformation,[status(thm)],[f1635]) ).
thf(f1635,plain,
( ( ( powersetAx = $true )
| ( ( ( ( ( in @ sK406 @ sK404 )
!= $true )
& ( ( in @ sK406 @ sK405 )
= $true ) )
| ( $true
!= ( in @ sK405 @ ( powerset @ sK404 ) ) ) )
& ( ! [X3: $i] :
( ( $true
= ( in @ X3 @ sK404 ) )
| ( ( in @ X3 @ sK405 )
!= $true ) )
| ( $true
= ( in @ sK405 @ ( powerset @ sK404 ) ) ) ) ) )
& ( ! [X4: $i,X5: $i] :
( ( ( ( in @ X5 @ ( powerset @ X4 ) )
= $true )
| ( ( $true
!= ( in @ ( sK407 @ X5 @ X4 ) @ X4 ) )
& ( ( in @ ( sK407 @ X5 @ X4 ) @ X5 )
= $true ) ) )
& ( ! [X7: $i] :
( ( ( in @ X7 @ X4 )
= $true )
| ( $true
!= ( in @ X7 @ X5 ) ) )
| ( ( in @ X5 @ ( powerset @ X4 ) )
!= $true ) ) )
| ( powersetAx != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK404,sK405,sK406,sK407])],[f1631,f1634,f1633,f1632]) ).
thf(f1632,plain,
( ? [X0: $i,X1: $i] :
( ( ? [X2: $i] :
( ( ( in @ X2 @ X0 )
!= $true )
& ( ( in @ X2 @ X1 )
= $true ) )
| ( ( in @ X1 @ ( powerset @ X0 ) )
!= $true ) )
& ( ! [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
| ( $true
!= ( in @ X3 @ X1 ) ) )
| ( ( in @ X1 @ ( powerset @ X0 ) )
= $true ) ) )
=> ( ( ? [X2: $i] :
( ( ( in @ X2 @ sK404 )
!= $true )
& ( $true
= ( in @ X2 @ sK405 ) ) )
| ( $true
!= ( in @ sK405 @ ( powerset @ sK404 ) ) ) )
& ( ! [X3: $i] :
( ( $true
= ( in @ X3 @ sK404 ) )
| ( ( in @ X3 @ sK405 )
!= $true ) )
| ( $true
= ( in @ sK405 @ ( powerset @ sK404 ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1633,plain,
( ? [X2: $i] :
( ( ( in @ X2 @ sK404 )
!= $true )
& ( $true
= ( in @ X2 @ sK405 ) ) )
=> ( ( ( in @ sK406 @ sK404 )
!= $true )
& ( ( in @ sK406 @ sK405 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1634,plain,
! [X4: $i,X5: $i] :
( ? [X6: $i] :
( ( ( in @ X6 @ X4 )
!= $true )
& ( ( in @ X6 @ X5 )
= $true ) )
=> ( ( $true
!= ( in @ ( sK407 @ X5 @ X4 ) @ X4 ) )
& ( ( in @ ( sK407 @ X5 @ X4 ) @ X5 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1631,plain,
( ( ( powersetAx = $true )
| ? [X0: $i,X1: $i] :
( ( ? [X2: $i] :
( ( ( in @ X2 @ X0 )
!= $true )
& ( ( in @ X2 @ X1 )
= $true ) )
| ( ( in @ X1 @ ( powerset @ X0 ) )
!= $true ) )
& ( ! [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
| ( $true
!= ( in @ X3 @ X1 ) ) )
| ( ( in @ X1 @ ( powerset @ X0 ) )
= $true ) ) ) )
& ( ! [X4: $i,X5: $i] :
( ( ( ( in @ X5 @ ( powerset @ X4 ) )
= $true )
| ? [X6: $i] :
( ( ( in @ X6 @ X4 )
!= $true )
& ( ( in @ X6 @ X5 )
= $true ) ) )
& ( ! [X7: $i] :
( ( ( in @ X7 @ X4 )
= $true )
| ( $true
!= ( in @ X7 @ X5 ) ) )
| ( ( in @ X5 @ ( powerset @ X4 ) )
!= $true ) ) )
| ( powersetAx != $true ) ) ),
inference(rectify,[status(thm)],[f1630]) ).
thf(f1630,plain,
( ( ( powersetAx = $true )
| ? [X0: $i,X1: $i] :
( ( ? [X2: $i] :
( ( ( in @ X2 @ X0 )
!= $true )
& ( ( in @ X2 @ X1 )
= $true ) )
| ( ( in @ X1 @ ( powerset @ X0 ) )
!= $true ) )
& ( ! [X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
| ( ( in @ X2 @ X1 )
!= $true ) )
| ( ( in @ X1 @ ( powerset @ X0 ) )
= $true ) ) ) )
& ( ! [X0: $i,X1: $i] :
( ( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
| ? [X2: $i] :
( ( ( in @ X2 @ X0 )
!= $true )
& ( ( in @ X2 @ X1 )
= $true ) ) )
& ( ! [X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
| ( ( in @ X2 @ X1 )
!= $true ) )
| ( ( in @ X1 @ ( powerset @ X0 ) )
!= $true ) ) )
| ( powersetAx != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f864]) ).
thf(f864,plain,
( ( powersetAx = $true )
<=> ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
<=> ! [X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
| ( ( in @ X2 @ X1 )
!= $true ) ) ) ),
inference(ennf_transformation,[status(thm)],[f604]) ).
thf(f604,plain,
( ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
=> ( ( in @ X2 @ X0 )
= $true ) )
<=> ( ( in @ X1 @ ( powerset @ X0 ) )
= $true ) )
<=> ( powersetAx = $true ) ),
inference(fool_elimination,[status(thm)],[f603]) ).
thf(f603,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
<=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ X0 ) ) )
= powersetAx ),
inference(rectify,[status(thm)],[f5]) ).
thf(f5,axiom,
! [X3: $i,X4: $i] :
( ( ( in @ X4 @ ( powerset @ X3 ) )
<=> ! [X1: $i] :
( ( in @ X1 @ X4 )
=> ( in @ X1 @ X3 ) ) )
= powersetAx ),
file('/export/starexec/sandbox2/tmp/tmp.DvUejQI5fj/DTF2THF_12059.p',powersetAx) ).
thf(f3629,plain,
! [X0: $i] :
( ( $true
!= ( in @ ( sK99 @ X0 @ sK312 ) @ sK311 ) )
| ( $true
!= ( in @ ( sK99 @ X0 @ sK312 ) @ sK310 ) )
| ( ( in @ X0 @ ( powerset @ sK312 ) )
= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3628]) ).
thf(f3628,plain,
! [X0: $i] :
( ( $true
!= ( in @ ( sK99 @ X0 @ sK312 ) @ sK310 ) )
| ( ( in @ X0 @ ( powerset @ sK312 ) )
= $true )
| ( $true != $true )
| ( $true
!= ( in @ ( sK99 @ X0 @ sK312 ) @ sK311 ) ) ),
inference(forward_demodulation,[status(thm)],[f3579,f2672]) ).
thf(f3579,plain,
! [X0: $i] :
( ( $true
!= ( in @ ( sK99 @ X0 @ sK312 ) @ sK311 ) )
| ( $true
!= ( in @ ( sK99 @ X0 @ sK312 ) @ sK310 ) )
| ( ( in @ X0 @ ( powerset @ sK312 ) )
= $true )
| ( powersetI != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3564]) ).
thf(f3564,plain,
! [X0: $i] :
( ( $true
!= ( in @ ( sK99 @ X0 @ sK312 ) @ sK311 ) )
| ( $true != $true )
| ( $true
!= ( in @ ( sK99 @ X0 @ sK312 ) @ sK310 ) )
| ( powersetI != $true )
| ( ( in @ X0 @ ( powerset @ sK312 ) )
= $true ) ),
inference(superposition,[status(thm)],[f2255,f2736]) ).
thf(f2736,plain,
! [X3: $i] :
( ( $true
= ( in @ X3 @ sK312 ) )
| ( $true
!= ( in @ X3 @ sK310 ) )
| ( $true
!= ( in @ X3 @ sK311 ) ) ),
inference(cnf_transformation,[status(thm)],[f1496]) ).
thf(f2255,plain,
! [X0: $i,X1: $i] :
( ( ( in @ ( sK99 @ X1 @ X0 ) @ X0 )
!= $true )
| ( powersetI != $true )
| ( ( in @ X1 @ ( powerset @ X0 ) )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f1162]) ).
thf(f6603,plain,
~ spl732_70,
inference(avatar_split_clause,[status(thm)],[f6602,f4678]) ).
thf(f6602,plain,
( $true
!= ( subset @ sK310 @ sK312 ) ),
inference(trivial_inequality_removal,[status(thm)],[f6601]) ).
thf(f6601,plain,
( ( $true
!= ( subset @ sK310 @ sK312 ) )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f6588,f2638]) ).
thf(f2638,plain,
setunionsingleton = $true,
inference(cnf_transformation,[status(thm)],[f1496]) ).
thf(f6588,plain,
( ( $true
!= ( subset @ sK310 @ sK312 ) )
| ( setunionsingleton != $true ) ),
inference(superposition,[status(thm)],[f3846,f2880]) ).
thf(f2880,plain,
! [X0: $i] :
( ( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
| ( setunionsingleton != $true ) ),
inference(cnf_transformation,[status(thm)],[f1617]) ).
thf(f1617,plain,
( ( ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
| ( setunionsingleton != $true ) )
& ( ( setunionsingleton = $true )
| ( sK395
!= ( setunion @ ( setadjoin @ sK395 @ emptyset ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK395])],[f1615,f1616]) ).
thf(f1616,plain,
( ? [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
!= X1 )
=> ( sK395
!= ( setunion @ ( setadjoin @ sK395 @ emptyset ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1615,plain,
( ( ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
| ( setunionsingleton != $true ) )
& ( ( setunionsingleton = $true )
| ? [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
!= X1 ) ) ),
inference(rectify,[status(thm)],[f1614]) ).
thf(f1614,plain,
( ( ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
| ( setunionsingleton != $true ) )
& ( ( setunionsingleton = $true )
| ? [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
!= X0 ) ) ),
inference(nnf_transformation,[status(thm)],[f246]) ).
thf(f246,plain,
( ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
<=> ( setunionsingleton = $true ) ),
inference(fool_elimination,[status(thm)],[f245]) ).
thf(f245,plain,
( setunionsingleton
= ( ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 ) ) ),
inference(rectify,[status(thm)],[f155]) ).
thf(f155,axiom,
( setunionsingleton
= ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.DvUejQI5fj/DTF2THF_12059.p',setunionsingleton) ).
thf(f3846,plain,
( ( subset @ ( setunion @ ( setadjoin @ sK310 @ emptyset ) ) @ sK312 )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f3845]) ).
thf(f3845,plain,
( ( $true != $true )
| ( ( subset @ ( setunion @ ( setadjoin @ sK310 @ emptyset ) ) @ sK312 )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f3792,f2541]) ).
thf(f2541,plain,
setunionsingleton2 = $true,
inference(cnf_transformation,[status(thm)],[f1496]) ).
thf(f3792,plain,
( ( setunionsingleton2 != $true )
| ( ( subset @ ( setunion @ ( setadjoin @ sK310 @ emptyset ) ) @ sK312 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3756]) ).
thf(f3756,plain,
( ( $true != $true )
| ( ( subset @ ( setunion @ ( setadjoin @ sK310 @ emptyset ) ) @ sK312 )
!= $true )
| ( setunionsingleton2 != $true ) ),
inference(superposition,[status(thm)],[f3493,f3088]) ).
thf(f3088,plain,
! [X0: $i] :
( ( ( subset @ X0 @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) )
= $true )
| ( setunionsingleton2 != $true ) ),
inference(cnf_transformation,[status(thm)],[f1870]) ).
thf(f1870,plain,
( ( ! [X0: $i] :
( ( subset @ X0 @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) )
= $true )
| ( setunionsingleton2 != $true ) )
& ( ( setunionsingleton2 = $true )
| ( $true
!= ( subset @ sK560 @ ( setunion @ ( setadjoin @ sK560 @ emptyset ) ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK560])],[f1868,f1869]) ).
thf(f1869,plain,
( ? [X1: $i] :
( ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
!= $true )
=> ( $true
!= ( subset @ sK560 @ ( setunion @ ( setadjoin @ sK560 @ emptyset ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1868,plain,
( ( ! [X0: $i] :
( ( subset @ X0 @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) )
= $true )
| ( setunionsingleton2 != $true ) )
& ( ( setunionsingleton2 = $true )
| ? [X1: $i] :
( ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) )
!= $true ) ) ),
inference(rectify,[status(thm)],[f1867]) ).
thf(f1867,plain,
( ( ! [X0: $i] :
( ( subset @ X0 @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) )
= $true )
| ( setunionsingleton2 != $true ) )
& ( ( setunionsingleton2 = $true )
| ? [X0: $i] :
( ( subset @ X0 @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) )
!= $true ) ) ),
inference(nnf_transformation,[status(thm)],[f606]) ).
thf(f606,plain,
( ! [X0: $i] :
( ( subset @ X0 @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) )
= $true )
<=> ( setunionsingleton2 = $true ) ),
inference(fool_elimination,[status(thm)],[f605]) ).
thf(f605,plain,
! [X0: $i] :
( ( subset @ X0 @ ( setunion @ ( setadjoin @ X0 @ emptyset ) ) )
= setunionsingleton2 ),
inference(rectify,[status(thm)],[f154]) ).
thf(f154,axiom,
! [X3: $i] :
( ( subset @ X3 @ ( setunion @ ( setadjoin @ X3 @ emptyset ) ) )
= setunionsingleton2 ),
file('/export/starexec/sandbox2/tmp/tmp.DvUejQI5fj/DTF2THF_12059.p',setunionsingleton2) ).
thf(f3493,plain,
! [X0: $i] :
( ( ( subset @ sK310 @ X0 )
!= $true )
| ( ( subset @ X0 @ sK312 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3492]) ).
thf(f3492,plain,
! [X0: $i] :
( ( ( subset @ X0 @ sK312 )
!= $true )
| ( ( subset @ sK310 @ X0 )
!= $true )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f3485,f2663]) ).
thf(f2663,plain,
subsetTrans = $true,
inference(cnf_transformation,[status(thm)],[f1496]) ).
thf(f3485,plain,
! [X0: $i] :
( ( ( subset @ sK310 @ X0 )
!= $true )
| ( subsetTrans != $true )
| ( ( subset @ X0 @ sK312 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3483]) ).
thf(f3483,plain,
! [X0: $i] :
( ( ( subset @ X0 @ sK312 )
!= $true )
| ( $true != $true )
| ( subsetTrans != $true )
| ( ( subset @ sK310 @ X0 )
!= $true ) ),
inference(superposition,[status(thm)],[f2735,f3274]) ).
thf(f3274,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( subset @ X0 @ X2 )
= $true )
| ( ( subset @ X0 @ X1 )
!= $true )
| ( subsetTrans != $true )
| ( ( subset @ X1 @ X2 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f2086]) ).
thf(f2086,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X0 @ X2 )
= $true )
| ( ( subset @ X1 @ X2 )
!= $true )
| ( ( subset @ X0 @ X1 )
!= $true ) )
| ( subsetTrans != $true ) )
& ( ( subsetTrans = $true )
| ( ( $true
!= ( subset @ sK696 @ sK698 ) )
& ( $true
= ( subset @ sK697 @ sK698 ) )
& ( $true
= ( subset @ sK696 @ sK697 ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK696,sK697,sK698])],[f2084,f2085]) ).
thf(f2085,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( ( subset @ X3 @ X5 )
!= $true )
& ( ( subset @ X4 @ X5 )
= $true )
& ( ( subset @ X3 @ X4 )
= $true ) )
=> ( ( $true
!= ( subset @ sK696 @ sK698 ) )
& ( $true
= ( subset @ sK697 @ sK698 ) )
& ( $true
= ( subset @ sK696 @ sK697 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2084,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X0 @ X2 )
= $true )
| ( ( subset @ X1 @ X2 )
!= $true )
| ( ( subset @ X0 @ X1 )
!= $true ) )
| ( subsetTrans != $true ) )
& ( ( subsetTrans = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( ( subset @ X3 @ X5 )
!= $true )
& ( ( subset @ X4 @ X5 )
= $true )
& ( ( subset @ X3 @ X4 )
= $true ) ) ) ),
inference(rectify,[status(thm)],[f2083]) ).
thf(f2083,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X0 @ X2 )
= $true )
| ( ( subset @ X1 @ X2 )
!= $true )
| ( ( subset @ X0 @ X1 )
!= $true ) )
| ( subsetTrans != $true ) )
& ( ( subsetTrans = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X0 @ X2 )
!= $true )
& ( ( subset @ X1 @ X2 )
= $true )
& ( ( subset @ X0 @ X1 )
= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f937]) ).
thf(f937,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X0 @ X2 )
= $true )
| ( ( subset @ X1 @ X2 )
!= $true )
| ( ( subset @ X0 @ X1 )
!= $true ) )
<=> ( subsetTrans = $true ) ),
inference(flattening,[status(thm)],[f936]) ).
thf(f936,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X0 @ X2 )
= $true )
| ( ( subset @ X1 @ X2 )
!= $true )
| ( ( subset @ X0 @ X1 )
!= $true ) )
<=> ( subsetTrans = $true ) ),
inference(ennf_transformation,[status(thm)],[f322]) ).
thf(f322,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X0 @ X1 )
= $true )
=> ( ( ( subset @ X1 @ X2 )
= $true )
=> ( ( subset @ X0 @ X2 )
= $true ) ) )
<=> ( subsetTrans = $true ) ),
inference(fool_elimination,[status(thm)],[f321]) ).
thf(f321,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X0 @ X1 )
=> ( ( subset @ X1 @ X2 )
=> ( subset @ X0 @ X2 ) ) )
= subsetTrans ),
inference(rectify,[status(thm)],[f89]) ).
thf(f89,axiom,
! [X3: $i,X4: $i,X5: $i] :
( ( ( subset @ X3 @ X4 )
=> ( ( subset @ X4 @ X5 )
=> ( subset @ X3 @ X5 ) ) )
= subsetTrans ),
file('/export/starexec/sandbox2/tmp/tmp.DvUejQI5fj/DTF2THF_12059.p',subsetTrans) ).
thf(f2735,plain,
( $true
!= ( subset @ sK310 @ sK312 ) ),
inference(cnf_transformation,[status(thm)],[f1496]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.13 % Problem : SEU716^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.14 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.35 % Computer : n010.cluster.edu
% 0.15/0.35 % Model : x86_64 x86_64
% 0.15/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.35 % Memory : 8042.1875MB
% 0.15/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.35 % CPULimit : 300
% 0.15/0.35 % WCLimit : 300
% 0.15/0.35 % DateTime : Tue Feb 24 12:31:18 EST 2026
% 0.15/0.35 % CPUTime :
% 0.15/0.35 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.26/0.47 ---- Original DTF file ---
% 0.26/0.47 thf(spec,logic,$$dhol).
% 0.26/0.47 %------------------------------------------------------------------------------
% 0.26/0.47 % File : SEU716^1 : TPTP v9.2.1. Released v3.7.0.
% 0.26/0.47 % Domain : Set Theory
% 0.26/0.47 % Problem : Typed Set Theory - Laws for Typed Sets
% 0.26/0.47 % Version : Especial.
% 0.26/0.47 % English : (! A:i.! X:i.in X (powerset A) -> (! Y:i.in Y (powerset A) ->
% 0.26/0.47 % (! x:i.in x A -> in x X -> in x Y) -> subset X Y))
% 0.26/0.47
% 0.26/0.47 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.26/0.47 % Source : [Bro08]
% 0.26/0.47 % Names : ZFC218g [Bro08]
% 0.26/0.47
% 0.26/0.47 % Status : Theorem
% 0.26/0.47 % Rating : 0.56 v9.1.0, 0.38 v9.0.0, 0.40 v8.2.0, 0.46 v8.1.0, 0.27 v7.5.0, 0.29 v7.4.0, 0.33 v7.3.0, 0.44 v7.2.0, 0.38 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.57 v6.1.0, 0.86 v6.0.0, 0.71 v5.5.0, 0.83 v5.4.0, 0.80 v5.2.0, 1.00 v3.7.0
% 0.26/0.47 % Syntax : Number of formulae : 507 ( 236 unt; 270 typ; 236 def)
% 0.26/0.47 % Number of atoms : 1330 ( 361 equ; 0 cnn)
% 0.26/0.47 % Maximal formula atoms : 238 ( 5 avg)
% 0.26/0.47 % Number of connectives : 2588 ( 55 ~; 12 |; 44 &;1832 @)
% 0.26/0.47 % ( 14 <=>; 631 =>; 0 <=; 0 <~>)
% 0.26/0.47 % Maximal formula depth : 244 ( 2 avg)
% 0.26/0.47 % Number of types : 2 ( 0 usr)
% 0.26/0.47 % Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% 0.26/0.47 % Number of symbols : 273 ( 270 usr; 236 con; 0-4 aty)
% 0.26/0.47 % Number of variables : 794 ( 74 ^; 685 !; 35 ?; 794 :)
% 0.26/0.47 % SPC : TH0_THM_EQU_NAR
% 0.26/0.47
% 0.26/0.47 % Comments : http://mathgate.info/detsetitem.php?id=274
% 0.26/0.47 %------------------------------------------------------------------------------
% 0.26/0.47 thf(in_type,type,
% 0.26/0.47 in: $i > $i > $o ).
% 0.26/0.47
% 0.26/0.47 thf(exu_type,type,
% 0.26/0.47 exu: ( $i > $o ) > $o ).
% 0.26/0.47
% 0.26/0.47 thf(exu,definition,
% 0.26/0.47 ( exu
% 0.26/0.47 = ( ^ [Xphi: $i > $o] :
% 0.26/0.47 ? [Xx: $i] :
% 0.26/0.47 ( ( Xphi @ Xx )
% 0.26/0.47 & ! [Xy: $i] :
% 0.26/0.47 ( ( Xphi @ Xy )
% 0.26/0.47 => ( Xx = Xy ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(setextAx_type,type,
% 0.26/0.47 setextAx: $o ).
% 0.26/0.47
% 0.26/0.47 thf(setextAx,definition,
% 0.26/0.47 ( setextAx
% 0.26/0.47 = ( ! [A: $i,B: $i] :
% 0.26/0.47 ( ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ A )
% 0.26/0.47 <=> ( in @ Xx @ B ) )
% 0.26/0.47 => ( A = B ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(emptyset_type,type,
% 0.26/0.47 emptyset: $i ).
% 0.26/0.47
% 0.26/0.47 thf(emptysetAx_type,type,
% 0.26/0.47 emptysetAx: $o ).
% 0.26/0.47
% 0.26/0.47 thf(emptysetAx,definition,
% 0.26/0.47 ( emptysetAx
% 0.26/0.47 = ( ! [Xx: $i] :
% 0.26/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(setadjoin_type,type,
% 0.26/0.47 setadjoin: $i > $i > $i ).
% 0.26/0.47
% 0.26/0.47 thf(setadjoinAx_type,type,
% 0.26/0.47 setadjoinAx: $o ).
% 0.26/0.47
% 0.26/0.47 thf(setadjoinAx,definition,
% 0.26/0.47 ( setadjoinAx
% 0.26/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.26/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.26/0.47 <=> ( ( Xy = Xx )
% 0.26/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(powerset_type,type,
% 0.26/0.47 powerset: $i > $i ).
% 0.26/0.47
% 0.26/0.47 thf(powersetAx_type,type,
% 0.26/0.47 powersetAx: $o ).
% 0.26/0.47
% 0.26/0.47 thf(powersetAx,definition,
% 0.26/0.47 ( powersetAx
% 0.26/0.47 = ( ! [A: $i,B: $i] :
% 0.26/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.26/0.47 <=> ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ B )
% 0.26/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(setunion_type,type,
% 0.26/0.47 setunion: $i > $i ).
% 0.26/0.47
% 0.26/0.47 thf(setunionAx_type,type,
% 0.26/0.47 setunionAx: $o ).
% 0.26/0.47
% 0.26/0.47 thf(setunionAx,definition,
% 0.26/0.47 ( setunionAx
% 0.26/0.47 = ( ! [A: $i,Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.26/0.47 <=> ? [B: $i] :
% 0.26/0.47 ( ( in @ Xx @ B )
% 0.26/0.47 & ( in @ B @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(omega_type,type,
% 0.26/0.47 omega: $i ).
% 0.26/0.47
% 0.26/0.47 thf(omega0Ax_type,type,
% 0.26/0.47 omega0Ax: $o ).
% 0.26/0.47
% 0.26/0.47 thf(omega0Ax,definition,
% 0.26/0.47 ( omega0Ax
% 0.26/0.47 = ( in @ emptyset @ omega ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(omegaSAx_type,type,
% 0.26/0.47 omegaSAx: $o ).
% 0.26/0.47
% 0.26/0.47 thf(omegaSAx,definition,
% 0.26/0.47 ( omegaSAx
% 0.26/0.47 = ( ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ omega )
% 0.26/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(omegaIndAx_type,type,
% 0.26/0.47 omegaIndAx: $o ).
% 0.26/0.47
% 0.26/0.47 thf(omegaIndAx,definition,
% 0.26/0.47 ( omegaIndAx
% 0.26/0.47 = ( ! [A: $i] :
% 0.26/0.47 ( ( ( in @ emptyset @ A )
% 0.26/0.47 & ! [Xx: $i] :
% 0.26/0.47 ( ( ( in @ Xx @ omega )
% 0.26/0.47 & ( in @ Xx @ A ) )
% 0.26/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.26/0.47 => ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ omega )
% 0.26/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(replAx_type,type,
% 0.26/0.47 replAx: $o ).
% 0.26/0.47
% 0.26/0.47 thf(replAx,definition,
% 0.26/0.47 ( replAx
% 0.26/0.47 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.26/0.47 ( ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ A )
% 0.26/0.47 => ( exu
% 0.26/0.47 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.26/0.47 => ? [B: $i] :
% 0.26/0.47 ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ B )
% 0.26/0.47 <=> ? [Xy: $i] :
% 0.26/0.47 ( ( in @ Xy @ A )
% 0.26/0.47 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(foundationAx_type,type,
% 0.26/0.47 foundationAx: $o ).
% 0.26/0.47
% 0.26/0.47 thf(foundationAx,definition,
% 0.26/0.47 ( foundationAx
% 0.26/0.47 = ( ! [A: $i] :
% 0.26/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.26/0.47 => ? [B: $i] :
% 0.26/0.47 ( ( in @ B @ A )
% 0.26/0.47 & ~ ? [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ B )
% 0.26/0.47 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(wellorderingAx_type,type,
% 0.26/0.47 wellorderingAx: $o ).
% 0.26/0.47
% 0.26/0.47 thf(wellorderingAx,definition,
% 0.26/0.47 ( wellorderingAx
% 0.26/0.47 = ( ! [A: $i] :
% 0.26/0.47 ? [B: $i] :
% 0.26/0.47 ( ! [C: $i] :
% 0.26/0.47 ( ( in @ C @ B )
% 0.26/0.47 => ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ C )
% 0.26/0.47 => ( in @ Xx @ A ) ) )
% 0.26/0.47 & ! [Xx: $i,Xy: $i] :
% 0.26/0.47 ( ( ( in @ Xx @ A )
% 0.26/0.47 & ( in @ Xy @ A ) )
% 0.26/0.47 => ( ! [C: $i] :
% 0.26/0.47 ( ( in @ C @ B )
% 0.26/0.47 => ( ( in @ Xx @ C )
% 0.26/0.47 <=> ( in @ Xy @ C ) ) )
% 0.26/0.47 => ( Xx = Xy ) ) )
% 0.26/0.47 & ! [C: $i,D: $i] :
% 0.26/0.47 ( ( ( in @ C @ B )
% 0.26/0.47 & ( in @ D @ B ) )
% 0.26/0.47 => ( ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ C )
% 0.26/0.47 => ( in @ Xx @ D ) )
% 0.26/0.47 | ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ D )
% 0.26/0.47 => ( in @ Xx @ C ) ) ) )
% 0.26/0.47 & ! [C: $i] :
% 0.26/0.47 ( ( ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ C )
% 0.26/0.47 => ( in @ Xx @ A ) )
% 0.26/0.47 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.26/0.47 => ? [D: $i,Xx: $i] :
% 0.26/0.47 ( ( in @ D @ B )
% 0.26/0.47 & ( in @ Xx @ C )
% 0.26/0.47 & ~ ? [Xy: $i] :
% 0.26/0.47 ( ( in @ Xy @ D )
% 0.26/0.47 & ( in @ Xy @ C ) )
% 0.26/0.47 & ! [E: $i] :
% 0.26/0.47 ( ( in @ E @ B )
% 0.26/0.47 => ( ! [Xy: $i] :
% 0.26/0.47 ( ( in @ Xy @ E )
% 0.26/0.47 => ( in @ Xy @ D ) )
% 0.26/0.47 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(descr_type,type,
% 0.26/0.47 descr: ( $i > $o ) > $i ).
% 0.26/0.47
% 0.26/0.47 thf(descrp_type,type,
% 0.26/0.47 descrp: $o ).
% 0.26/0.47
% 0.26/0.47 thf(descrp,definition,
% 0.26/0.47 ( descrp
% 0.26/0.47 = ( ! [Xphi: $i > $o] :
% 0.26/0.47 ( ( exu
% 0.26/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.47 => ( Xphi
% 0.26/0.47 @ ( descr
% 0.26/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(dsetconstr_type,type,
% 0.26/0.47 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.26/0.47
% 0.26/0.47 thf(dsetconstrI_type,type,
% 0.26/0.47 dsetconstrI: $o ).
% 0.26/0.47
% 0.26/0.47 thf(dsetconstrI,definition,
% 0.26/0.47 ( dsetconstrI
% 0.26/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ A )
% 0.26/0.47 => ( ( Xphi @ Xx )
% 0.26/0.47 => ( in @ Xx
% 0.26/0.47 @ ( dsetconstr @ A
% 0.26/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(dsetconstrEL_type,type,
% 0.26/0.47 dsetconstrEL: $o ).
% 0.26/0.47
% 0.26/0.47 thf(dsetconstrEL,definition,
% 0.26/0.47 ( dsetconstrEL
% 0.26/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.26/0.47 ( ( in @ Xx
% 0.26/0.47 @ ( dsetconstr @ A
% 0.26/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.26/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(dsetconstrER_type,type,
% 0.26/0.47 dsetconstrER: $o ).
% 0.26/0.47
% 0.26/0.47 thf(dsetconstrER,definition,
% 0.26/0.47 ( dsetconstrER
% 0.26/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.26/0.47 ( ( in @ Xx
% 0.26/0.47 @ ( dsetconstr @ A
% 0.26/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.26/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(exuE1_type,type,
% 0.26/0.47 exuE1: $o ).
% 0.26/0.47
% 0.26/0.47 thf(exuE1,definition,
% 0.26/0.47 ( exuE1
% 0.26/0.47 = ( ! [Xphi: $i > $o] :
% 0.26/0.47 ( ( exu
% 0.26/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.47 => ? [Xx: $i] :
% 0.26/0.47 ( ( Xphi @ Xx )
% 0.26/0.47 & ! [Xy: $i] :
% 0.26/0.47 ( ( Xphi @ Xy )
% 0.26/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(prop2set_type,type,
% 0.26/0.47 prop2set: $o > $i ).
% 0.26/0.47
% 0.26/0.47 thf(prop2setE_type,type,
% 0.26/0.47 prop2setE: $o ).
% 0.26/0.47
% 0.26/0.47 thf(prop2setE,definition,
% 0.26/0.47 ( prop2setE
% 0.26/0.47 = ( ! [Xphi: $o,Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.26/0.47 => Xphi ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(emptysetE_type,type,
% 0.26/0.47 emptysetE: $o ).
% 0.26/0.47
% 0.26/0.47 thf(emptysetE,definition,
% 0.26/0.47 ( emptysetE
% 0.26/0.47 = ( ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ emptyset )
% 0.26/0.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(emptysetimpfalse_type,type,
% 0.26/0.47 emptysetimpfalse: $o ).
% 0.26/0.47
% 0.26/0.47 thf(emptysetimpfalse,definition,
% 0.26/0.47 ( emptysetimpfalse
% 0.26/0.47 = ( ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ emptyset )
% 0.26/0.47 => $false ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(notinemptyset_type,type,
% 0.26/0.47 notinemptyset: $o ).
% 0.26/0.47
% 0.26/0.47 thf(notinemptyset,definition,
% 0.26/0.47 ( notinemptyset
% 0.26/0.47 = ( ! [Xx: $i] :
% 0.26/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(exuE3e_type,type,
% 0.26/0.47 exuE3e: $o ).
% 0.26/0.47
% 0.26/0.47 thf(exuE3e,definition,
% 0.26/0.47 ( exuE3e
% 0.26/0.47 = ( ! [Xphi: $i > $o] :
% 0.26/0.47 ( ( exu
% 0.26/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.47 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(setext_type,type,
% 0.26/0.47 setext: $o ).
% 0.26/0.47
% 0.26/0.47 thf(setext,definition,
% 0.26/0.47 ( setext
% 0.26/0.47 = ( ! [A: $i,B: $i] :
% 0.26/0.47 ( ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ A )
% 0.26/0.47 => ( in @ Xx @ B ) )
% 0.26/0.47 => ( ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ B )
% 0.26/0.47 => ( in @ Xx @ A ) )
% 0.26/0.47 => ( A = B ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(emptyI_type,type,
% 0.26/0.47 emptyI: $o ).
% 0.26/0.47
% 0.26/0.47 thf(emptyI,definition,
% 0.26/0.47 ( emptyI
% 0.26/0.47 = ( ! [A: $i] :
% 0.26/0.47 ( ! [Xx: $i] :
% 0.26/0.47 ~ ( in @ Xx @ A )
% 0.26/0.47 => ( A = emptyset ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(noeltsimpempty_type,type,
% 0.26/0.47 noeltsimpempty: $o ).
% 0.26/0.47
% 0.26/0.47 thf(noeltsimpempty,definition,
% 0.26/0.47 ( noeltsimpempty
% 0.26/0.47 = ( ! [A: $i] :
% 0.26/0.47 ( ! [Xx: $i] :
% 0.26/0.47 ~ ( in @ Xx @ A )
% 0.26/0.47 => ( A = emptyset ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(setbeta_type,type,
% 0.26/0.47 setbeta: $o ).
% 0.26/0.47
% 0.26/0.47 thf(setbeta,definition,
% 0.26/0.47 ( setbeta
% 0.26/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ A )
% 0.26/0.47 => ( ( in @ Xx
% 0.26/0.47 @ ( dsetconstr @ A
% 0.26/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.26/0.47 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(nonempty_type,type,
% 0.26/0.47 nonempty: $i > $o ).
% 0.26/0.47
% 0.26/0.47 thf(nonempty,definition,
% 0.26/0.47 ( nonempty
% 0.26/0.47 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(nonemptyE1_type,type,
% 0.26/0.47 nonemptyE1: $o ).
% 0.26/0.47
% 0.26/0.47 thf(nonemptyE1,definition,
% 0.26/0.47 ( nonemptyE1
% 0.26/0.47 = ( ! [A: $i] :
% 0.26/0.47 ( ( nonempty @ A )
% 0.26/0.47 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(nonemptyI_type,type,
% 0.26/0.47 nonemptyI: $o ).
% 0.26/0.47
% 0.26/0.47 thf(nonemptyI,definition,
% 0.26/0.47 ( nonemptyI
% 0.26/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ A )
% 0.26/0.47 => ( ( Xphi @ Xx )
% 0.26/0.47 => ( nonempty
% 0.26/0.47 @ ( dsetconstr @ A
% 0.26/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(nonemptyI1_type,type,
% 0.26/0.47 nonemptyI1: $o ).
% 0.26/0.47
% 0.26/0.47 thf(nonemptyI1,definition,
% 0.26/0.47 ( nonemptyI1
% 0.26/0.47 = ( ! [A: $i] :
% 0.26/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.26/0.47 => ( nonempty @ A ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(setadjoinIL_type,type,
% 0.26/0.47 setadjoinIL: $o ).
% 0.26/0.47
% 0.26/0.47 thf(setadjoinIL,definition,
% 0.26/0.47 ( setadjoinIL
% 0.26/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(emptyinunitempty_type,type,
% 0.26/0.47 emptyinunitempty: $o ).
% 0.26/0.47
% 0.26/0.47 thf(emptyinunitempty,definition,
% 0.26/0.47 ( emptyinunitempty
% 0.26/0.47 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(setadjoinIR_type,type,
% 0.26/0.47 setadjoinIR: $o ).
% 0.26/0.47
% 0.26/0.47 thf(setadjoinIR,definition,
% 0.26/0.47 ( setadjoinIR
% 0.26/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.26/0.47 ( ( in @ Xy @ A )
% 0.26/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(setadjoinE_type,type,
% 0.26/0.47 setadjoinE: $o ).
% 0.26/0.47
% 0.26/0.47 thf(setadjoinE,definition,
% 0.26/0.47 ( setadjoinE
% 0.26/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.26/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.26/0.47 => ! [Xphi: $o] :
% 0.26/0.47 ( ( ( Xy = Xx )
% 0.26/0.47 => Xphi )
% 0.26/0.47 => ( ( ( in @ Xy @ A )
% 0.26/0.47 => Xphi )
% 0.26/0.47 => Xphi ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(setadjoinOr_type,type,
% 0.26/0.47 setadjoinOr: $o ).
% 0.26/0.47
% 0.26/0.47 thf(setadjoinOr,definition,
% 0.26/0.47 ( setadjoinOr
% 0.26/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.26/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.26/0.47 => ( ( Xy = Xx )
% 0.26/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(setoftrueEq_type,type,
% 0.26/0.47 setoftrueEq: $o ).
% 0.26/0.47
% 0.26/0.47 thf(setoftrueEq,definition,
% 0.26/0.47 ( setoftrueEq
% 0.26/0.47 = ( ! [A: $i] :
% 0.26/0.47 ( ( dsetconstr @ A
% 0.26/0.47 @ ^ [Xx: $i] : $true )
% 0.26/0.47 = A ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(powersetI_type,type,
% 0.26/0.47 powersetI: $o ).
% 0.26/0.47
% 0.26/0.47 thf(powersetI,definition,
% 0.26/0.47 ( powersetI
% 0.26/0.47 = ( ! [A: $i,B: $i] :
% 0.26/0.47 ( ! [Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ B )
% 0.26/0.47 => ( in @ Xx @ A ) )
% 0.26/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(emptyinPowerset_type,type,
% 0.26/0.47 emptyinPowerset: $o ).
% 0.26/0.47
% 0.26/0.47 thf(emptyinPowerset,definition,
% 0.26/0.47 ( emptyinPowerset
% 0.26/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(emptyInPowerset_type,type,
% 0.26/0.47 emptyInPowerset: $o ).
% 0.26/0.47
% 0.26/0.47 thf(emptyInPowerset,definition,
% 0.26/0.47 ( emptyInPowerset
% 0.26/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(powersetE_type,type,
% 0.26/0.47 powersetE: $o ).
% 0.26/0.47
% 0.26/0.47 thf(powersetE,definition,
% 0.26/0.47 ( powersetE
% 0.26/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.26/0.47 => ( ( in @ Xx @ B )
% 0.26/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(setunionI_type,type,
% 0.26/0.47 setunionI: $o ).
% 0.26/0.47
% 0.26/0.47 thf(setunionI,definition,
% 0.26/0.47 ( setunionI
% 0.26/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.26/0.47 ( ( in @ Xx @ B )
% 0.26/0.47 => ( ( in @ B @ A )
% 0.26/0.47 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(setunionE_type,type,
% 0.26/0.47 setunionE: $o ).
% 0.26/0.47
% 0.26/0.47 thf(setunionE,definition,
% 0.26/0.47 ( setunionE
% 0.26/0.47 = ( ! [A: $i,Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.26/0.47 => ! [Xphi: $o] :
% 0.26/0.47 ( ! [B: $i] :
% 0.26/0.47 ( ( in @ Xx @ B )
% 0.26/0.47 => ( ( in @ B @ A )
% 0.26/0.47 => Xphi ) )
% 0.26/0.47 => Xphi ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(subPowSU_type,type,
% 0.26/0.47 subPowSU: $o ).
% 0.26/0.47
% 0.26/0.47 thf(subPowSU,definition,
% 0.26/0.47 ( subPowSU
% 0.26/0.47 = ( ! [A: $i,Xx: $i] :
% 0.26/0.47 ( ( in @ Xx @ A )
% 0.26/0.47 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(exuE2_type,type,
% 0.26/0.47 exuE2: $o ).
% 0.26/0.47
% 0.26/0.47 thf(exuE2,definition,
% 0.26/0.47 ( exuE2
% 0.26/0.47 = ( ! [Xphi: $i > $o] :
% 0.26/0.47 ( ( exu
% 0.26/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.47 => ? [Xx: $i] :
% 0.26/0.47 ! [Xy: $i] :
% 0.26/0.47 ( ( Xphi @ Xy )
% 0.26/0.47 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.26/0.47
% 0.26/0.47 thf(nonemptyImpWitness_type,type,
% 0.26/0.47 nonemptyImpWitness: $o ).
% 0.26/0.47
% 0.26/0.47 thf(nonemptyImpWitness,definition,
% 0.26/0.47 ( nonemptyImpWitness
% 0.26/0.48 = ( ! [A: $i] :
% 0.26/0.48 ( ( nonempty @ A )
% 0.26/0.48 => ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & $true ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(uniqinunit_type,type,
% 0.26/0.48 uniqinunit: $o ).
% 0.26/0.48
% 0.26/0.48 thf(uniqinunit,definition,
% 0.26/0.48 ( uniqinunit
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.26/0.48 => ( Xx = Xy ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(notinsingleton_type,type,
% 0.26/0.48 notinsingleton: $o ).
% 0.26/0.48
% 0.26/0.48 thf(notinsingleton,definition,
% 0.26/0.48 ( notinsingleton
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xx != Xy )
% 0.26/0.48 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(eqinunit_type,type,
% 0.26/0.48 eqinunit: $o ).
% 0.26/0.48
% 0.26/0.48 thf(eqinunit,definition,
% 0.26/0.48 ( eqinunit
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xx = Xy )
% 0.26/0.48 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(singletonsswitch_type,type,
% 0.26/0.48 singletonsswitch: $o ).
% 0.26/0.48
% 0.26/0.48 thf(singletonsswitch,definition,
% 0.26/0.48 ( singletonsswitch
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.26/0.48 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(upairsetE_type,type,
% 0.26/0.48 upairsetE: $o ).
% 0.26/0.48
% 0.26/0.48 thf(upairsetE,definition,
% 0.26/0.48 ( upairsetE
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.26/0.48 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.26/0.48 => ( ( Xz = Xx )
% 0.26/0.48 | ( Xz = Xy ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(upairsetIL_type,type,
% 0.26/0.48 upairsetIL: $o ).
% 0.26/0.48
% 0.26/0.48 thf(upairsetIL,definition,
% 0.26/0.48 ( upairsetIL
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(upairsetIR_type,type,
% 0.26/0.48 upairsetIR: $o ).
% 0.26/0.48
% 0.26/0.48 thf(upairsetIR,definition,
% 0.26/0.48 ( upairsetIR
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(emptyE1_type,type,
% 0.26/0.48 emptyE1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(emptyE1,definition,
% 0.26/0.48 ( emptyE1
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & ( Xphi @ Xx ) )
% 0.26/0.48 => ( ( ( dsetconstr @ A
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.48 = emptyset )
% 0.26/0.48 => $false ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(vacuousDall_type,type,
% 0.26/0.48 vacuousDall: $o ).
% 0.26/0.48
% 0.26/0.48 thf(vacuousDall,definition,
% 0.26/0.48 ( vacuousDall
% 0.26/0.48 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ emptyset )
% 0.26/0.48 => ( Xphi @ Xx ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(quantDeMorgan1_type,type,
% 0.26/0.48 quantDeMorgan1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(quantDeMorgan1,definition,
% 0.26/0.48 ( quantDeMorgan1
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( ~ ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( Xphi @ Xx ) )
% 0.26/0.48 => ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(quantDeMorgan2_type,type,
% 0.26/0.48 quantDeMorgan2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(quantDeMorgan2,definition,
% 0.26/0.48 ( quantDeMorgan2
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ~ ( Xphi @ Xx ) )
% 0.26/0.48 => ~ ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(quantDeMorgan3_type,type,
% 0.26/0.48 quantDeMorgan3: $o ).
% 0.26/0.48
% 0.26/0.48 thf(quantDeMorgan3,definition,
% 0.26/0.48 ( quantDeMorgan3
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( ~ ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & ( Xphi @ Xx ) )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(quantDeMorgan4_type,type,
% 0.26/0.48 quantDeMorgan4: $o ).
% 0.26/0.48
% 0.26/0.48 thf(quantDeMorgan4,definition,
% 0.26/0.48 ( quantDeMorgan4
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & ~ ( Xphi @ Xx ) )
% 0.26/0.48 => ~ ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(prop2setI_type,type,
% 0.26/0.48 prop2setI: $o ).
% 0.26/0.48
% 0.26/0.48 thf(prop2setI,definition,
% 0.26/0.48 ( prop2setI
% 0.26/0.48 = ( ! [Xphi: $o] :
% 0.26/0.48 ( Xphi
% 0.26/0.48 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(set2prop_type,type,
% 0.26/0.48 set2prop: $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(prop2set2propI_type,type,
% 0.26/0.48 prop2set2propI: $o ).
% 0.26/0.48
% 0.26/0.48 thf(prop2set2propI,definition,
% 0.26/0.48 ( prop2set2propI
% 0.26/0.48 = ( ! [Xphi: $o] :
% 0.26/0.48 ( Xphi
% 0.26/0.48 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(notdexE_type,type,
% 0.26/0.48 notdexE: $o ).
% 0.26/0.48
% 0.26/0.48 thf(notdexE,definition,
% 0.26/0.48 ( notdexE
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( ~ ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & ( Xphi @ Xx ) )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(notdallE_type,type,
% 0.26/0.48 notdallE: $o ).
% 0.26/0.48
% 0.26/0.48 thf(notdallE,definition,
% 0.26/0.48 ( notdallE
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( ~ ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( Xphi @ Xx ) )
% 0.26/0.48 => ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(exuI1_type,type,
% 0.26/0.48 exuI1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(exuI1,definition,
% 0.26/0.48 ( exuI1
% 0.26/0.48 = ( ! [Xphi: $i > $o] :
% 0.26/0.48 ( ? [Xx: $i] :
% 0.26/0.48 ( ( Xphi @ Xx )
% 0.26/0.48 & ! [Xy: $i] :
% 0.26/0.48 ( ( Xphi @ Xy )
% 0.26/0.48 => ( Xx = Xy ) ) )
% 0.26/0.48 => ( exu
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(exuI3_type,type,
% 0.26/0.48 exuI3: $o ).
% 0.26/0.48
% 0.26/0.48 thf(exuI3,definition,
% 0.26/0.48 ( exuI3
% 0.26/0.48 = ( ! [Xphi: $i > $o] :
% 0.26/0.48 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.26/0.48 => ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xphi @ Xx )
% 0.26/0.48 => ( ( Xphi @ Xy )
% 0.26/0.48 => ( Xx = Xy ) ) )
% 0.26/0.48 => ( exu
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(exuI2_type,type,
% 0.26/0.48 exuI2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(exuI2,definition,
% 0.26/0.48 ( exuI2
% 0.26/0.48 = ( ! [Xphi: $i > $o] :
% 0.26/0.48 ( ? [Xx: $i] :
% 0.26/0.48 ! [Xy: $i] :
% 0.26/0.48 ( ( Xphi @ Xy )
% 0.26/0.48 <=> ( Xy = Xx ) )
% 0.26/0.48 => ( exu
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(inCongP_type,type,
% 0.26/0.48 inCongP: $o ).
% 0.26/0.48
% 0.26/0.48 thf(inCongP,definition,
% 0.26/0.48 ( inCongP
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( A = B )
% 0.26/0.48 => ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xx = Xy )
% 0.26/0.48 => ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(in__Cong_type,type,
% 0.26/0.48 in__Cong: $o ).
% 0.26/0.48
% 0.26/0.48 thf(in__Cong,definition,
% 0.26/0.48 ( in__Cong
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( A = B )
% 0.26/0.48 => ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xx = Xy )
% 0.26/0.48 => ( ( in @ Xx @ A )
% 0.26/0.48 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(exuE3u_type,type,
% 0.26/0.48 exuE3u: $o ).
% 0.26/0.48
% 0.26/0.48 thf(exuE3u,definition,
% 0.26/0.48 ( exuE3u
% 0.26/0.48 = ( ! [Xphi: $i > $o] :
% 0.26/0.48 ( ( exu
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.48 => ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xphi @ Xx )
% 0.26/0.48 => ( ( Xphi @ Xy )
% 0.26/0.48 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(exu__Cong_type,type,
% 0.26/0.48 exu__Cong: $o ).
% 0.26/0.48
% 0.26/0.48 thf(exu__Cong,definition,
% 0.26/0.48 ( exu__Cong
% 0.26/0.48 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.26/0.48 ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xx = Xy )
% 0.26/0.48 => ( ( Xphi @ Xx )
% 0.26/0.48 <=> ( Xpsi @ Xy ) ) )
% 0.26/0.48 => ( ( exu
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.48 <=> ( exu
% 0.26/0.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(emptyset__Cong_type,type,
% 0.26/0.48 emptyset__Cong: $o ).
% 0.26/0.48
% 0.26/0.48 thf(emptyset__Cong,definition,
% 0.26/0.48 ( emptyset__Cong
% 0.26/0.48 = ( emptyset = emptyset ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setadjoin__Cong_type,type,
% 0.26/0.48 setadjoin__Cong: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setadjoin__Cong,definition,
% 0.26/0.48 ( setadjoin__Cong
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xx = Xy )
% 0.26/0.48 => ! [Xz: $i,Xu: $i] :
% 0.26/0.48 ( ( Xz = Xu )
% 0.26/0.48 => ( ( setadjoin @ Xx @ Xz )
% 0.26/0.48 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(powerset__Cong_type,type,
% 0.26/0.48 powerset__Cong: $o ).
% 0.26/0.48
% 0.26/0.48 thf(powerset__Cong,definition,
% 0.26/0.48 ( powerset__Cong
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( A = B )
% 0.26/0.48 => ( ( powerset @ A )
% 0.26/0.48 = ( powerset @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setunion__Cong_type,type,
% 0.26/0.48 setunion__Cong: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setunion__Cong,definition,
% 0.26/0.48 ( setunion__Cong
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( A = B )
% 0.26/0.48 => ( ( setunion @ A )
% 0.26/0.48 = ( setunion @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(omega__Cong_type,type,
% 0.26/0.48 omega__Cong: $o ).
% 0.26/0.48
% 0.26/0.48 thf(omega__Cong,definition,
% 0.26/0.48 ( omega__Cong
% 0.26/0.48 = ( omega = omega ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(exuEu_type,type,
% 0.26/0.48 exuEu: $o ).
% 0.26/0.48
% 0.26/0.48 thf(exuEu,definition,
% 0.26/0.48 ( exuEu
% 0.26/0.48 = ( ! [Xphi: $i > $o] :
% 0.26/0.48 ( ( exu
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.48 => ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xphi @ Xx )
% 0.26/0.48 => ( ( Xphi @ Xy )
% 0.26/0.48 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(descr__Cong_type,type,
% 0.26/0.48 descr__Cong: $o ).
% 0.26/0.48
% 0.26/0.48 thf(descr__Cong,definition,
% 0.26/0.48 ( descr__Cong
% 0.26/0.48 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.26/0.48 ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xx = Xy )
% 0.26/0.48 => ( ( Xphi @ Xx )
% 0.26/0.48 <=> ( Xpsi @ Xy ) ) )
% 0.26/0.48 => ( ( exu
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.48 => ( ( exu
% 0.26/0.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.26/0.48 => ( ( descr
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.48 = ( descr
% 0.26/0.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(dsetconstr__Cong_type,type,
% 0.26/0.48 dsetconstr__Cong: $o ).
% 0.26/0.48
% 0.26/0.48 thf(dsetconstr__Cong,definition,
% 0.26/0.48 ( dsetconstr__Cong
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( A = B )
% 0.26/0.48 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( Xx = Xy )
% 0.26/0.48 => ( ( Xphi @ Xx )
% 0.26/0.48 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.26/0.48 => ( ( dsetconstr @ A
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.48 = ( dsetconstr @ B
% 0.26/0.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(subset_type,type,
% 0.26/0.48 subset: $i > $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(disjoint_type,type,
% 0.26/0.48 disjoint: $i > $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(setsmeet_type,type,
% 0.26/0.48 setsmeet: $i > $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(subsetI1_type,type,
% 0.26/0.48 subsetI1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(subsetI1,definition,
% 0.26/0.48 ( subsetI1
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ Xx @ B ) )
% 0.26/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(eqimpsubset2_type,type,
% 0.26/0.48 eqimpsubset2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(eqimpsubset2,definition,
% 0.26/0.48 ( eqimpsubset2
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( A = B )
% 0.26/0.48 => ( subset @ B @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(eqimpsubset1_type,type,
% 0.26/0.48 eqimpsubset1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(eqimpsubset1,definition,
% 0.26/0.48 ( eqimpsubset1
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( A = B )
% 0.26/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(subsetI2_type,type,
% 0.26/0.48 subsetI2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(subsetI2,definition,
% 0.26/0.48 ( subsetI2
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ Xx @ B ) )
% 0.26/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(emptysetsubset_type,type,
% 0.26/0.48 emptysetsubset: $o ).
% 0.26/0.48
% 0.26/0.48 thf(emptysetsubset,definition,
% 0.26/0.48 ( emptysetsubset
% 0.26/0.48 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(subsetE_type,type,
% 0.26/0.48 subsetE: $o ).
% 0.26/0.48
% 0.26/0.48 thf(subsetE,definition,
% 0.26/0.48 ( subsetE
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( subset @ A @ B )
% 0.26/0.48 => ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(subsetE2_type,type,
% 0.26/0.48 subsetE2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(subsetE2,definition,
% 0.26/0.48 ( subsetE2
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( subset @ A @ B )
% 0.26/0.48 => ( ~ ( in @ Xx @ B )
% 0.26/0.48 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(notsubsetI_type,type,
% 0.26/0.48 notsubsetI: $o ).
% 0.26/0.48
% 0.26/0.48 thf(notsubsetI,definition,
% 0.26/0.48 ( notsubsetI
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ~ ( in @ Xx @ B )
% 0.26/0.48 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(notequalI1_type,type,
% 0.26/0.48 notequalI1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(notequalI1,definition,
% 0.26/0.48 ( notequalI1
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ~ ( subset @ A @ B )
% 0.26/0.48 => ( A != B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(notequalI2_type,type,
% 0.26/0.48 notequalI2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(notequalI2,definition,
% 0.26/0.48 ( notequalI2
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ~ ( in @ Xx @ B )
% 0.26/0.48 => ( A != B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(subsetRefl_type,type,
% 0.26/0.48 subsetRefl: $o ).
% 0.26/0.48
% 0.26/0.48 thf(subsetRefl,definition,
% 0.26/0.48 ( subsetRefl
% 0.26/0.48 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(subsetTrans_type,type,
% 0.26/0.48 subsetTrans: $o ).
% 0.26/0.48
% 0.26/0.48 thf(subsetTrans,definition,
% 0.26/0.48 ( subsetTrans
% 0.26/0.48 = ( ! [A: $i,B: $i,C: $i] :
% 0.26/0.48 ( ( subset @ A @ B )
% 0.26/0.48 => ( ( subset @ B @ C )
% 0.26/0.48 => ( subset @ A @ C ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setadjoinSub_type,type,
% 0.26/0.48 setadjoinSub: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setadjoinSub,definition,
% 0.26/0.48 ( setadjoinSub
% 0.26/0.48 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setadjoinSub2_type,type,
% 0.26/0.48 setadjoinSub2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setadjoinSub2,definition,
% 0.26/0.48 ( setadjoinSub2
% 0.26/0.48 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.26/0.48 ( ( subset @ A @ B )
% 0.26/0.48 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(subset2powerset_type,type,
% 0.26/0.48 subset2powerset: $o ).
% 0.26/0.48
% 0.26/0.48 thf(subset2powerset,definition,
% 0.26/0.48 ( subset2powerset
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( subset @ A @ B )
% 0.26/0.48 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setextsub_type,type,
% 0.26/0.48 setextsub: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setextsub,definition,
% 0.26/0.48 ( setextsub
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( subset @ A @ B )
% 0.26/0.48 => ( ( subset @ B @ A )
% 0.26/0.48 => ( A = B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(subsetemptysetimpeq_type,type,
% 0.26/0.48 subsetemptysetimpeq: $o ).
% 0.26/0.48
% 0.26/0.48 thf(subsetemptysetimpeq,definition,
% 0.26/0.48 ( subsetemptysetimpeq
% 0.26/0.48 = ( ! [A: $i] :
% 0.26/0.48 ( ( subset @ A @ emptyset )
% 0.26/0.48 => ( A = emptyset ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(powersetI1_type,type,
% 0.26/0.48 powersetI1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(powersetI1,definition,
% 0.26/0.48 ( powersetI1
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( subset @ B @ A )
% 0.26/0.48 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(powersetE1_type,type,
% 0.26/0.48 powersetE1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(powersetE1,definition,
% 0.26/0.48 ( powersetE1
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( in @ B @ ( powerset @ A ) )
% 0.26/0.48 => ( subset @ B @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(inPowerset_type,type,
% 0.26/0.48 inPowerset: $o ).
% 0.26/0.48
% 0.26/0.48 thf(inPowerset,definition,
% 0.26/0.48 ( inPowerset
% 0.26/0.48 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(powersetsubset_type,type,
% 0.26/0.48 powersetsubset: $o ).
% 0.26/0.48
% 0.26/0.48 thf(powersetsubset,definition,
% 0.26/0.48 ( powersetsubset
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( subset @ A @ B )
% 0.26/0.48 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(sepInPowerset_type,type,
% 0.26/0.48 sepInPowerset: $o ).
% 0.26/0.48
% 0.26/0.48 thf(sepInPowerset,definition,
% 0.26/0.48 ( sepInPowerset
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( in
% 0.26/0.48 @ ( dsetconstr @ A
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.48 @ ( powerset @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(sepSubset_type,type,
% 0.26/0.48 sepSubset: $o ).
% 0.26/0.48
% 0.26/0.48 thf(sepSubset,definition,
% 0.26/0.48 ( sepSubset
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( subset
% 0.26/0.48 @ ( dsetconstr @ A
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.48 @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binunion_type,type,
% 0.26/0.48 binunion: $i > $i > $i ).
% 0.26/0.48
% 0.26/0.48 thf(binunionIL_type,type,
% 0.26/0.48 binunionIL: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binunionIL,definition,
% 0.26/0.48 ( binunionIL
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(upairset2IR_type,type,
% 0.26/0.48 upairset2IR: $o ).
% 0.26/0.48
% 0.26/0.48 thf(upairset2IR,definition,
% 0.26/0.48 ( upairset2IR
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binunionIR_type,type,
% 0.26/0.48 binunionIR: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binunionIR,definition,
% 0.26/0.48 ( binunionIR
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ B )
% 0.26/0.48 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binunionEcases_type,type,
% 0.26/0.48 binunionEcases: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binunionEcases,definition,
% 0.26/0.48 ( binunionEcases
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.26/0.48 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.26/0.48 => ( ( ( in @ Xx @ A )
% 0.26/0.48 => Xphi )
% 0.26/0.48 => ( ( ( in @ Xx @ B )
% 0.26/0.48 => Xphi )
% 0.26/0.48 => Xphi ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binunionE_type,type,
% 0.26/0.48 binunionE: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binunionE,definition,
% 0.26/0.48 ( binunionE
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.26/0.48 => ( ( in @ Xx @ A )
% 0.26/0.48 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binunionLsub_type,type,
% 0.26/0.48 binunionLsub: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binunionLsub,definition,
% 0.26/0.48 ( binunionLsub
% 0.26/0.48 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binunionRsub_type,type,
% 0.26/0.48 binunionRsub: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binunionRsub,definition,
% 0.26/0.48 ( binunionRsub
% 0.26/0.48 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binintersect_type,type,
% 0.26/0.48 binintersect: $i > $i > $i ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectI_type,type,
% 0.26/0.48 binintersectI: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectI,definition,
% 0.26/0.48 ( binintersectI
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ( in @ Xx @ B )
% 0.26/0.48 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectSubset5_type,type,
% 0.26/0.48 binintersectSubset5: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectSubset5,definition,
% 0.26/0.48 ( binintersectSubset5
% 0.26/0.48 = ( ! [A: $i,B: $i,C: $i] :
% 0.26/0.48 ( ( subset @ C @ A )
% 0.26/0.48 => ( ( subset @ C @ B )
% 0.26/0.48 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectEL_type,type,
% 0.26/0.48 binintersectEL: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectEL,definition,
% 0.26/0.48 ( binintersectEL
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.26/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectLsub_type,type,
% 0.26/0.48 binintersectLsub: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectLsub,definition,
% 0.26/0.48 ( binintersectLsub
% 0.26/0.48 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectSubset2_type,type,
% 0.26/0.48 binintersectSubset2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectSubset2,definition,
% 0.26/0.48 ( binintersectSubset2
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( subset @ A @ B )
% 0.26/0.48 => ( ( binintersect @ A @ B )
% 0.26/0.48 = A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectSubset3_type,type,
% 0.26/0.48 binintersectSubset3: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectSubset3,definition,
% 0.26/0.48 ( binintersectSubset3
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( ( binintersect @ A @ B )
% 0.26/0.48 = B )
% 0.26/0.48 => ( subset @ B @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectER_type,type,
% 0.26/0.48 binintersectER: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectER,definition,
% 0.26/0.48 ( binintersectER
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.26/0.48 => ( in @ Xx @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(disjointsetsI1_type,type,
% 0.26/0.48 disjointsetsI1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(disjointsetsI1,definition,
% 0.26/0.48 ( disjointsetsI1
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ~ ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & ( in @ Xx @ B ) )
% 0.26/0.48 => ( ( binintersect @ A @ B )
% 0.26/0.48 = emptyset ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectRsub_type,type,
% 0.26/0.48 binintersectRsub: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectRsub,definition,
% 0.26/0.48 ( binintersectRsub
% 0.26/0.48 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectSubset4_type,type,
% 0.26/0.48 binintersectSubset4: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectSubset4,definition,
% 0.26/0.48 ( binintersectSubset4
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( subset @ B @ A )
% 0.26/0.48 => ( ( binintersect @ A @ B )
% 0.26/0.48 = B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectSubset1_type,type,
% 0.26/0.48 binintersectSubset1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectSubset1,definition,
% 0.26/0.48 ( binintersectSubset1
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( ( binintersect @ A @ B )
% 0.26/0.48 = A )
% 0.26/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(bs114d_type,type,
% 0.26/0.48 bs114d: $o ).
% 0.26/0.48
% 0.26/0.48 thf(bs114d,definition,
% 0.26/0.48 ( bs114d
% 0.26/0.48 = ( ! [A: $i,B: $i,C: $i] :
% 0.26/0.48 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.26/0.48 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(regular_type,type,
% 0.26/0.48 regular: $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(setminus_type,type,
% 0.26/0.48 setminus: $i > $i > $i ).
% 0.26/0.48
% 0.26/0.48 thf(setminusI_type,type,
% 0.26/0.48 setminusI: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setminusI,definition,
% 0.26/0.48 ( setminusI
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ~ ( in @ Xx @ B )
% 0.26/0.48 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setminusEL_type,type,
% 0.26/0.48 setminusEL: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setminusEL,definition,
% 0.26/0.48 ( setminusEL
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.26/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setminusER_type,type,
% 0.26/0.48 setminusER: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setminusER,definition,
% 0.26/0.48 ( setminusER
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.26/0.48 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setminusSubset2_type,type,
% 0.26/0.48 setminusSubset2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setminusSubset2,definition,
% 0.26/0.48 ( setminusSubset2
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( subset @ A @ B )
% 0.26/0.48 => ( ( setminus @ A @ B )
% 0.26/0.48 = emptyset ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setminusERneg_type,type,
% 0.26/0.48 setminusERneg: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setminusERneg,definition,
% 0.26/0.48 ( setminusERneg
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.26/0.48 => ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setminusELneg_type,type,
% 0.26/0.48 setminusELneg: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setminusELneg,definition,
% 0.26/0.48 ( setminusELneg
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.26/0.48 => ( ~ ( in @ Xx @ B )
% 0.26/0.48 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setminusILneg_type,type,
% 0.26/0.48 setminusILneg: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setminusILneg,definition,
% 0.26/0.48 ( setminusILneg
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ~ ( in @ Xx @ A )
% 0.26/0.48 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setminusIRneg_type,type,
% 0.26/0.48 setminusIRneg: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setminusIRneg,definition,
% 0.26/0.48 ( setminusIRneg
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ B )
% 0.26/0.48 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setminusLsub_type,type,
% 0.26/0.48 setminusLsub: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setminusLsub,definition,
% 0.26/0.48 ( setminusLsub
% 0.26/0.48 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setminusSubset1_type,type,
% 0.26/0.48 setminusSubset1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setminusSubset1,definition,
% 0.26/0.48 ( setminusSubset1
% 0.26/0.48 = ( ! [A: $i,B: $i] :
% 0.26/0.48 ( ( ( setminus @ A @ B )
% 0.26/0.48 = emptyset )
% 0.26/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(symdiff_type,type,
% 0.26/0.48 symdiff: $i > $i > $i ).
% 0.26/0.48
% 0.26/0.48 thf(symdiffE_type,type,
% 0.26/0.48 symdiffE: $o ).
% 0.26/0.48
% 0.26/0.48 thf(symdiffE,definition,
% 0.26/0.48 ( symdiffE
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.26/0.48 => ! [Xphi: $o] :
% 0.26/0.48 ( ( ( in @ Xx @ A )
% 0.26/0.48 => ( ~ ( in @ Xx @ B )
% 0.26/0.48 => Xphi ) )
% 0.26/0.48 => ( ( ~ ( in @ Xx @ A )
% 0.26/0.48 => ( ( in @ Xx @ B )
% 0.26/0.48 => Xphi ) )
% 0.26/0.48 => Xphi ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(symdiffI1_type,type,
% 0.26/0.48 symdiffI1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(symdiffI1,definition,
% 0.26/0.48 ( symdiffI1
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ~ ( in @ Xx @ B )
% 0.26/0.48 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(symdiffI2_type,type,
% 0.26/0.48 symdiffI2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(symdiffI2,definition,
% 0.26/0.48 ( symdiffI2
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ~ ( in @ Xx @ A )
% 0.26/0.48 => ( ( in @ Xx @ B )
% 0.26/0.48 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(symdiffIneg1_type,type,
% 0.26/0.48 symdiffIneg1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(symdiffIneg1,definition,
% 0.26/0.48 ( symdiffIneg1
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ( in @ Xx @ B )
% 0.26/0.48 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(symdiffIneg2_type,type,
% 0.26/0.48 symdiffIneg2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(symdiffIneg2,definition,
% 0.26/0.48 ( symdiffIneg2
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ~ ( in @ Xx @ A )
% 0.26/0.48 => ( ~ ( in @ Xx @ B )
% 0.26/0.48 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(iskpair_type,type,
% 0.26/0.48 iskpair: $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(secondinupair_type,type,
% 0.26/0.48 secondinupair: $o ).
% 0.26/0.48
% 0.26/0.48 thf(secondinupair,definition,
% 0.26/0.48 ( secondinupair
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairIL_type,type,
% 0.26/0.48 setukpairIL: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairIL,definition,
% 0.26/0.48 ( setukpairIL
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairIR_type,type,
% 0.26/0.48 setukpairIR: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairIR,definition,
% 0.26/0.48 ( setukpairIR
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(kpairiskpair_type,type,
% 0.26/0.48 kpairiskpair: $o ).
% 0.26/0.48
% 0.26/0.48 thf(kpairiskpair,definition,
% 0.26/0.48 ( kpairiskpair
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(kpair_type,type,
% 0.26/0.48 kpair: $i > $i > $i ).
% 0.26/0.48
% 0.26/0.48 thf(kpairp_type,type,
% 0.26/0.48 kpairp: $o ).
% 0.26/0.48
% 0.26/0.48 thf(kpairp,definition,
% 0.26/0.48 ( kpairp
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(cartprod_type,type,
% 0.26/0.48 cartprod: $i > $i > $i ).
% 0.26/0.48
% 0.26/0.48 thf(singletonsubset_type,type,
% 0.26/0.48 singletonsubset: $o ).
% 0.26/0.48
% 0.26/0.48 thf(singletonsubset,definition,
% 0.26/0.48 ( singletonsubset
% 0.26/0.48 = ( ! [A: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(singletoninpowerset_type,type,
% 0.26/0.48 singletoninpowerset: $o ).
% 0.26/0.48
% 0.26/0.48 thf(singletoninpowerset,definition,
% 0.26/0.48 ( singletoninpowerset
% 0.26/0.48 = ( ! [A: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(singletoninpowunion_type,type,
% 0.26/0.48 singletoninpowunion: $o ).
% 0.26/0.48
% 0.26/0.48 thf(singletoninpowunion,definition,
% 0.26/0.48 ( singletoninpowunion
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(upairset2E_type,type,
% 0.26/0.48 upairset2E: $o ).
% 0.26/0.48
% 0.26/0.48 thf(upairset2E,definition,
% 0.26/0.48 ( upairset2E
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.26/0.48 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.26/0.48 => ( ( Xz = Xx )
% 0.26/0.48 | ( Xz = Xy ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(upairsubunion_type,type,
% 0.26/0.48 upairsubunion: $o ).
% 0.26/0.48
% 0.26/0.48 thf(upairsubunion,definition,
% 0.26/0.48 ( upairsubunion
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(upairinpowunion_type,type,
% 0.26/0.48 upairinpowunion: $o ).
% 0.26/0.48
% 0.26/0.48 thf(upairinpowunion,definition,
% 0.26/0.48 ( upairinpowunion
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ubforcartprodlem1_type,type,
% 0.26/0.48 ubforcartprodlem1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ubforcartprodlem1,definition,
% 0.26/0.48 ( ubforcartprodlem1
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ubforcartprodlem2_type,type,
% 0.26/0.48 ubforcartprodlem2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ubforcartprodlem2,definition,
% 0.26/0.48 ( ubforcartprodlem2
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ubforcartprodlem3_type,type,
% 0.26/0.48 ubforcartprodlem3: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ubforcartprodlem3,definition,
% 0.26/0.48 ( ubforcartprodlem3
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodpairin_type,type,
% 0.26/0.48 cartprodpairin: $o ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodpairin,definition,
% 0.26/0.48 ( cartprodpairin
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodmempair1_type,type,
% 0.26/0.48 cartprodmempair1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodmempair1,definition,
% 0.26/0.48 ( cartprodmempair1
% 0.26/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.26/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.26/0.48 => ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & ? [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 & ( Xu
% 0.26/0.48 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodmempair_type,type,
% 0.26/0.48 cartprodmempair: $o ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodmempair,definition,
% 0.26/0.48 ( cartprodmempair
% 0.26/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.26/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.26/0.48 => ( iskpair @ Xu ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setunionE2_type,type,
% 0.26/0.48 setunionE2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setunionE2,definition,
% 0.26/0.48 ( setunionE2
% 0.26/0.48 = ( ! [A: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ ( setunion @ A ) )
% 0.26/0.48 => ? [X: $i] :
% 0.26/0.48 ( ( in @ X @ A )
% 0.26/0.48 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setunionsingleton1_type,type,
% 0.26/0.48 setunionsingleton1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setunionsingleton1,definition,
% 0.26/0.48 ( setunionsingleton1
% 0.26/0.48 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setunionsingleton2_type,type,
% 0.26/0.48 setunionsingleton2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setunionsingleton2,definition,
% 0.26/0.48 ( setunionsingleton2
% 0.26/0.48 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setunionsingleton_type,type,
% 0.26/0.48 setunionsingleton: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setunionsingleton,definition,
% 0.26/0.48 ( setunionsingleton
% 0.26/0.48 = ( ! [Xx: $i] :
% 0.26/0.48 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.26/0.48 = Xx ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(singleton_type,type,
% 0.26/0.48 singleton: $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(singletonprop_type,type,
% 0.26/0.48 singletonprop: $o ).
% 0.26/0.48
% 0.26/0.48 thf(singletonprop,definition,
% 0.26/0.48 ( singletonprop
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ A )
% 0.26/0.48 => ( ( Xphi @ Xx )
% 0.26/0.48 => ( ( Xphi @ Xy )
% 0.26/0.48 => ( Xx = Xy ) ) ) ) )
% 0.26/0.48 => ( ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & ( Xphi @ Xx ) )
% 0.26/0.48 => ( singleton
% 0.26/0.48 @ ( dsetconstr @ A
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ex1_type,type,
% 0.26/0.48 ex1: $i > ( $i > $o ) > $o ).
% 0.26/0.48
% 0.26/0.48 thf(ex1,definition,
% 0.26/0.48 ( ex1
% 0.26/0.48 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( singleton
% 0.26/0.48 @ ( dsetconstr @ A
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ex1E1_type,type,
% 0.26/0.48 ex1E1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ex1E1,definition,
% 0.26/0.48 ( ex1E1
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( ( ex1 @ A
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.48 => ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ex1I_type,type,
% 0.26/0.48 ex1I: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ex1I,definition,
% 0.26/0.48 ( ex1I
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ( Xphi @ Xx )
% 0.26/0.48 => ( ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ A )
% 0.26/0.48 => ( ( Xphi @ Xy )
% 0.26/0.48 => ( Xy = Xx ) ) )
% 0.26/0.48 => ( ex1 @ A
% 0.26/0.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ex1I2_type,type,
% 0.26/0.48 ex1I2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ex1I2,definition,
% 0.26/0.48 ( ex1I2
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ A )
% 0.26/0.48 => ( ( Xphi @ Xx )
% 0.26/0.48 => ( ( Xphi @ Xy )
% 0.26/0.48 => ( Xx = Xy ) ) ) ) )
% 0.26/0.48 => ( ? [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 & ( Xphi @ Xx ) )
% 0.26/0.48 => ( ex1 @ A
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(singletonsuniq_type,type,
% 0.26/0.48 singletonsuniq: $o ).
% 0.26/0.48
% 0.26/0.48 thf(singletonsuniq,definition,
% 0.26/0.48 ( singletonsuniq
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( ( setadjoin @ Xx @ emptyset )
% 0.26/0.48 = ( setadjoin @ Xy @ emptyset ) )
% 0.26/0.48 => ( Xx = Xy ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(atmost1p_type,type,
% 0.26/0.48 atmost1p: $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(atleast2p_type,type,
% 0.26/0.48 atleast2p: $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(atmost2p_type,type,
% 0.26/0.48 atmost2p: $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(upairsetp_type,type,
% 0.26/0.48 upairsetp: $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjL1_type,type,
% 0.26/0.48 setukpairinjL1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjL1,definition,
% 0.26/0.48 ( setukpairinjL1
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.26/0.48 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.26/0.48 => ( Xx = Xz ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(kfstsingleton_type,type,
% 0.26/0.48 kfstsingleton: $o ).
% 0.26/0.48
% 0.26/0.48 thf(kfstsingleton,definition,
% 0.26/0.48 ( kfstsingleton
% 0.26/0.48 = ( ! [Xu: $i] :
% 0.26/0.48 ( ( iskpair @ Xu )
% 0.26/0.48 => ( singleton
% 0.26/0.48 @ ( dsetconstr @ ( setunion @ Xu )
% 0.26/0.48 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(theprop_type,type,
% 0.26/0.48 theprop: $o ).
% 0.26/0.48
% 0.26/0.48 thf(theprop,definition,
% 0.26/0.48 ( theprop
% 0.26/0.48 = ( ! [X: $i] :
% 0.26/0.48 ( ( singleton @ X )
% 0.26/0.48 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(kfst_type,type,
% 0.26/0.48 kfst: $i > $i ).
% 0.26/0.48
% 0.26/0.48 thf(kfstpairEq_type,type,
% 0.26/0.48 kfstpairEq: $o ).
% 0.26/0.48
% 0.26/0.48 thf(kfstpairEq,definition,
% 0.26/0.48 ( kfstpairEq
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.26/0.48 = Xx ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodfstin_type,type,
% 0.26/0.48 cartprodfstin: $o ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodfstin,definition,
% 0.26/0.48 ( cartprodfstin
% 0.26/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.26/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.26/0.48 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjL2_type,type,
% 0.26/0.48 setukpairinjL2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjL2,definition,
% 0.26/0.48 ( setukpairinjL2
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.26/0.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.26/0.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.26/0.48 => ( Xx = Xz ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjL_type,type,
% 0.26/0.48 setukpairinjL: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjL,definition,
% 0.26/0.48 ( setukpairinjL
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.26/0.48 ( ( ( kpair @ Xx @ Xy )
% 0.26/0.48 = ( kpair @ Xz @ Xu ) )
% 0.26/0.48 => ( Xx = Xz ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjR11_type,type,
% 0.26/0.48 setukpairinjR11: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjR11,definition,
% 0.26/0.48 ( setukpairinjR11
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xx = Xy )
% 0.26/0.48 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.26/0.48 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjR12_type,type,
% 0.26/0.48 setukpairinjR12: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjR12,definition,
% 0.26/0.48 ( setukpairinjR12
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xx = Xy )
% 0.26/0.48 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.26/0.48 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjR1_type,type,
% 0.26/0.48 setukpairinjR1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjR1,definition,
% 0.26/0.48 ( setukpairinjR1
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.26/0.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.26/0.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.26/0.48 => ( ( Xz = Xu )
% 0.26/0.48 => ( Xy = Xu ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(upairequniteq_type,type,
% 0.26/0.48 upairequniteq: $o ).
% 0.26/0.48
% 0.26/0.48 thf(upairequniteq,definition,
% 0.26/0.48 ( upairequniteq
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.26/0.48 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.26/0.48 = ( setadjoin @ Xz @ emptyset ) )
% 0.26/0.48 => ( Xx = Xy ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjR2_type,type,
% 0.26/0.48 setukpairinjR2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjR2,definition,
% 0.26/0.48 ( setukpairinjR2
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.26/0.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.26/0.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.26/0.48 => ( Xy = Xu ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjR_type,type,
% 0.26/0.48 setukpairinjR: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setukpairinjR,definition,
% 0.26/0.48 ( setukpairinjR
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.26/0.48 ( ( ( kpair @ Xx @ Xy )
% 0.26/0.48 = ( kpair @ Xz @ Xu ) )
% 0.26/0.48 => ( Xy = Xu ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ksndsingleton_type,type,
% 0.26/0.48 ksndsingleton: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ksndsingleton,definition,
% 0.26/0.48 ( ksndsingleton
% 0.26/0.48 = ( ! [Xu: $i] :
% 0.26/0.48 ( ( iskpair @ Xu )
% 0.26/0.48 => ( singleton
% 0.26/0.48 @ ( dsetconstr @ ( setunion @ Xu )
% 0.26/0.48 @ ^ [Xx: $i] :
% 0.26/0.48 ( Xu
% 0.26/0.48 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ksnd_type,type,
% 0.26/0.48 ksnd: $i > $i ).
% 0.26/0.48
% 0.26/0.48 thf(ksndpairEq_type,type,
% 0.26/0.48 ksndpairEq: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ksndpairEq,definition,
% 0.26/0.48 ( ksndpairEq
% 0.26/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.26/0.48 = Xy ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(kpairsurjEq_type,type,
% 0.26/0.48 kpairsurjEq: $o ).
% 0.26/0.48
% 0.26/0.48 thf(kpairsurjEq,definition,
% 0.26/0.48 ( kpairsurjEq
% 0.26/0.48 = ( ! [Xu: $i] :
% 0.26/0.48 ( ( iskpair @ Xu )
% 0.26/0.48 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.26/0.48 = Xu ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodsndin_type,type,
% 0.26/0.48 cartprodsndin: $o ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodsndin,definition,
% 0.26/0.48 ( cartprodsndin
% 0.26/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.26/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.26/0.48 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodpairmemEL_type,type,
% 0.26/0.48 cartprodpairmemEL: $o ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodpairmemEL,definition,
% 0.26/0.48 ( cartprodpairmemEL
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.26/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodpairmemER_type,type,
% 0.26/0.48 cartprodpairmemER: $o ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodpairmemER,definition,
% 0.26/0.48 ( cartprodpairmemER
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.26/0.48 => ( in @ Xy @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodmempaircEq_type,type,
% 0.26/0.48 cartprodmempaircEq: $o ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodmempaircEq,definition,
% 0.26/0.48 ( cartprodmempaircEq
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( kpair @ Xx @ Xy )
% 0.26/0.48 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodfstpairEq_type,type,
% 0.26/0.48 cartprodfstpairEq: $o ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodfstpairEq,definition,
% 0.26/0.48 ( cartprodfstpairEq
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.26/0.48 = Xx ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodsndpairEq_type,type,
% 0.26/0.48 cartprodsndpairEq: $o ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodsndpairEq,definition,
% 0.26/0.48 ( cartprodsndpairEq
% 0.26/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.26/0.48 = Xy ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodpairsurjEq_type,type,
% 0.26/0.48 cartprodpairsurjEq: $o ).
% 0.26/0.48
% 0.26/0.48 thf(cartprodpairsurjEq,definition,
% 0.26/0.48 ( cartprodpairsurjEq
% 0.26/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.26/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.26/0.48 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.26/0.48 = Xu ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(breln_type,type,
% 0.26/0.48 breln: $i > $i > $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(breln,definition,
% 0.26/0.48 ( breln
% 0.26/0.48 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstr_type,type,
% 0.26/0.48 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstrI_type,type,
% 0.26/0.48 dpsetconstrI: $o ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstrI,definition,
% 0.26/0.48 ( dpsetconstrI
% 0.26/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( Xphi @ Xx @ Xy )
% 0.26/0.48 => ( in @ ( kpair @ Xx @ Xy )
% 0.26/0.48 @ ( dpsetconstr @ A @ B
% 0.26/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstrSub_type,type,
% 0.26/0.48 dpsetconstrSub: $o ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstrSub,definition,
% 0.26/0.48 ( dpsetconstrSub
% 0.26/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.26/0.48 ( subset
% 0.26/0.48 @ ( dpsetconstr @ A @ B
% 0.26/0.48 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.26/0.48 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setOfPairsIsBReln_type,type,
% 0.26/0.48 setOfPairsIsBReln: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setOfPairsIsBReln,definition,
% 0.26/0.48 ( setOfPairsIsBReln
% 0.26/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.26/0.48 ( breln @ A @ B
% 0.26/0.48 @ ( dpsetconstr @ A @ B
% 0.26/0.48 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstrERa_type,type,
% 0.26/0.48 dpsetconstrERa: $o ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstrERa,definition,
% 0.26/0.48 ( dpsetconstrERa
% 0.26/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.26/0.48 @ ( dpsetconstr @ A @ B
% 0.26/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.26/0.48 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstrEL1_type,type,
% 0.26/0.48 dpsetconstrEL1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstrEL1,definition,
% 0.26/0.48 ( dpsetconstrEL1
% 0.26/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( in @ ( kpair @ Xx @ Xy )
% 0.26/0.48 @ ( dpsetconstr @ A @ B
% 0.26/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.26/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstrEL2_type,type,
% 0.26/0.48 dpsetconstrEL2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstrEL2,definition,
% 0.26/0.48 ( dpsetconstrEL2
% 0.26/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( in @ ( kpair @ Xx @ Xy )
% 0.26/0.48 @ ( dpsetconstr @ A @ B
% 0.26/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.26/0.48 => ( in @ Xy @ B ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstrER_type,type,
% 0.26/0.48 dpsetconstrER: $o ).
% 0.26/0.48
% 0.26/0.48 thf(dpsetconstrER,definition,
% 0.26/0.48 ( dpsetconstrER
% 0.26/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( in @ ( kpair @ Xx @ Xy )
% 0.26/0.48 @ ( dpsetconstr @ A @ B
% 0.26/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.26/0.48 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(func_type,type,
% 0.26/0.48 func: $i > $i > $i > $o ).
% 0.26/0.48
% 0.26/0.48 thf(funcSet_type,type,
% 0.26/0.48 funcSet: $i > $i > $i ).
% 0.26/0.48
% 0.26/0.48 thf(funcImageSingleton_type,type,
% 0.26/0.48 funcImageSingleton: $o ).
% 0.26/0.48
% 0.26/0.48 thf(funcImageSingleton,definition,
% 0.26/0.48 ( funcImageSingleton
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( func @ A @ B @ Xf )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( singleton
% 0.26/0.48 @ ( dsetconstr @ B
% 0.26/0.48 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(apProp_type,type,
% 0.26/0.48 apProp: $o ).
% 0.26/0.48
% 0.26/0.48 thf(apProp,definition,
% 0.26/0.48 ( apProp
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( func @ A @ B @ Xf )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in
% 0.26/0.48 @ ( setunion
% 0.26/0.48 @ ( dsetconstr @ B
% 0.26/0.48 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.26/0.48 @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ap_type,type,
% 0.26/0.48 ap: $i > $i > $i > $i > $i ).
% 0.26/0.48
% 0.26/0.48 thf(app_type,type,
% 0.26/0.48 app: $o ).
% 0.26/0.48
% 0.26/0.48 thf(app,definition,
% 0.26/0.48 ( app
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( func @ A @ B @ Xf )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(infuncsetfunc_type,type,
% 0.26/0.48 infuncsetfunc: $o ).
% 0.26/0.48
% 0.26/0.48 thf(infuncsetfunc,definition,
% 0.26/0.48 ( infuncsetfunc
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.26/0.48 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ap2p_type,type,
% 0.26/0.48 ap2p: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ap2p,definition,
% 0.26/0.48 ( ap2p
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(funcinfuncset_type,type,
% 0.26/0.48 funcinfuncset: $o ).
% 0.26/0.48
% 0.26/0.48 thf(funcinfuncset,definition,
% 0.26/0.48 ( funcinfuncset
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( func @ A @ B @ Xf )
% 0.26/0.48 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(lamProp_type,type,
% 0.26/0.48 lamProp: $o ).
% 0.26/0.48
% 0.26/0.48 thf(lamProp,definition,
% 0.26/0.48 ( lamProp
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.26/0.48 => ( func @ A @ B
% 0.26/0.48 @ ( dpsetconstr @ A @ B
% 0.26/0.48 @ ^ [Xx: $i,Xy: $i] :
% 0.26/0.48 ( ( Xf @ Xx )
% 0.26/0.48 = Xy ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(lam_type,type,
% 0.26/0.48 lam: $i > $i > ( $i > $i ) > $i ).
% 0.26/0.48
% 0.26/0.48 thf(lamp_type,type,
% 0.26/0.48 lamp: $o ).
% 0.26/0.48
% 0.26/0.48 thf(lamp,definition,
% 0.26/0.48 ( lamp
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.26/0.48 => ( func @ A @ B
% 0.26/0.48 @ ( lam @ A @ B
% 0.26/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(lam2p_type,type,
% 0.26/0.48 lam2p: $o ).
% 0.26/0.48
% 0.26/0.48 thf(lam2p,definition,
% 0.26/0.48 ( lam2p
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.26/0.48 => ( in
% 0.26/0.48 @ ( lam @ A @ B
% 0.26/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.26/0.48 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(brelnall1_type,type,
% 0.26/0.48 brelnall1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(brelnall1,definition,
% 0.26/0.48 ( brelnall1
% 0.26/0.48 = ( ! [A: $i,B: $i,R: $i] :
% 0.26/0.48 ( ( breln @ A @ B @ R )
% 0.26/0.48 => ! [Xphi: $i > $o] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.26/0.48 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ R )
% 0.26/0.48 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(brelnall2_type,type,
% 0.26/0.48 brelnall2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(brelnall2,definition,
% 0.26/0.48 ( brelnall2
% 0.26/0.48 = ( ! [A: $i,B: $i,R: $i] :
% 0.26/0.48 ( ( breln @ A @ B @ R )
% 0.26/0.48 => ! [Xphi: $i > $o] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.26/0.48 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ R )
% 0.26/0.48 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ex1E2_type,type,
% 0.26/0.48 ex1E2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ex1E2,definition,
% 0.26/0.48 ( ex1E2
% 0.26/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.48 ( ( ex1 @ A
% 0.26/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ A )
% 0.26/0.48 => ( ( Xphi @ Xx )
% 0.26/0.48 => ( ( Xphi @ Xy )
% 0.26/0.48 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(funcGraphProp1_type,type,
% 0.26/0.48 funcGraphProp1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(funcGraphProp1,definition,
% 0.26/0.48 ( funcGraphProp1
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( func @ A @ B @ Xf )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(funcGraphProp3_type,type,
% 0.26/0.48 funcGraphProp3: $o ).
% 0.26/0.48
% 0.26/0.48 thf(funcGraphProp3,definition,
% 0.26/0.48 ( funcGraphProp3
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(funcGraphProp2_type,type,
% 0.26/0.48 funcGraphProp2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(funcGraphProp2,definition,
% 0.26/0.48 ( funcGraphProp2
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( func @ A @ B @ Xf )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.26/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.26/0.48 = Xy ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(funcextLem_type,type,
% 0.26/0.48 funcextLem: $o ).
% 0.26/0.48
% 0.26/0.48 thf(funcextLem,definition,
% 0.26/0.48 ( funcextLem
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( func @ A @ B @ Xf )
% 0.26/0.48 => ! [Xg: $i] :
% 0.26/0.48 ( ( func @ A @ B @ Xg )
% 0.26/0.48 => ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.26/0.48 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.26/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(funcGraphProp4_type,type,
% 0.26/0.48 funcGraphProp4: $o ).
% 0.26/0.48
% 0.26/0.48 thf(funcGraphProp4,definition,
% 0.26/0.48 ( funcGraphProp4
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.26/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.26/0.48 = Xy ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(subbreln_type,type,
% 0.26/0.48 subbreln: $o ).
% 0.26/0.48
% 0.26/0.48 thf(subbreln,definition,
% 0.26/0.48 ( subbreln
% 0.26/0.48 = ( ! [A: $i,B: $i,R: $i] :
% 0.26/0.48 ( ( breln @ A @ B @ R )
% 0.26/0.48 => ! [S: $i] :
% 0.26/0.48 ( ( breln @ A @ B @ S )
% 0.26/0.48 => ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.26/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.26/0.48 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(eqbreln_type,type,
% 0.26/0.48 eqbreln: $o ).
% 0.26/0.48
% 0.26/0.48 thf(eqbreln,definition,
% 0.26/0.48 ( eqbreln
% 0.26/0.48 = ( ! [A: $i,B: $i,R: $i] :
% 0.26/0.48 ( ( breln @ A @ B @ R )
% 0.26/0.48 => ! [S: $i] :
% 0.26/0.48 ( ( breln @ A @ B @ S )
% 0.26/0.48 => ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.26/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.26/0.48 => ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ B )
% 0.26/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.26/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.26/0.48 => ( R = S ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(funcext_type,type,
% 0.26/0.48 funcext: $o ).
% 0.26/0.48
% 0.26/0.48 thf(funcext,definition,
% 0.26/0.48 ( funcext
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( func @ A @ B @ Xf )
% 0.26/0.48 => ! [Xg: $i] :
% 0.26/0.48 ( ( func @ A @ B @ Xg )
% 0.26/0.48 => ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.26/0.48 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.26/0.48 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(funcext2_type,type,
% 0.26/0.48 funcext2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(funcext2,definition,
% 0.26/0.48 ( funcext2
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.26/0.48 => ! [Xg: $i] :
% 0.26/0.48 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.26/0.48 => ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.26/0.48 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.26/0.48 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ap2apEq1_type,type,
% 0.26/0.48 ap2apEq1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ap2apEq1,definition,
% 0.26/0.48 ( ap2apEq1
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.26/0.48 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ap2apEq2_type,type,
% 0.26/0.48 ap2apEq2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ap2apEq2,definition,
% 0.26/0.48 ( ap2apEq2
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( func @ A @ B @ Xf )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.26/0.48 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(beta1_type,type,
% 0.26/0.48 beta1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(beta1,definition,
% 0.26/0.48 ( beta1
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ( ap @ A @ B
% 0.26/0.48 @ ( lam @ A @ B
% 0.26/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.26/0.48 @ Xx )
% 0.26/0.48 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(eta1_type,type,
% 0.26/0.48 eta1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(eta1,definition,
% 0.26/0.48 ( eta1
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( func @ A @ B @ Xf )
% 0.26/0.48 => ( ( lam @ A @ B
% 0.26/0.48 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.26/0.48 = Xf ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(lam2lamEq_type,type,
% 0.26/0.48 lam2lamEq: $o ).
% 0.26/0.48
% 0.26/0.48 thf(lam2lamEq,definition,
% 0.26/0.48 ( lam2lamEq
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.26/0.48 => ( ( lam @ A @ B
% 0.26/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.26/0.48 = ( lam @ A @ B
% 0.26/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(beta2_type,type,
% 0.26/0.48 beta2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(beta2,definition,
% 0.26/0.48 ( beta2
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.26/0.48 ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ( ap @ A @ B
% 0.26/0.48 @ ( lam @ A @ B
% 0.26/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.26/0.48 @ Xx )
% 0.26/0.48 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(eta2_type,type,
% 0.26/0.48 eta2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(eta2,definition,
% 0.26/0.48 ( eta2
% 0.26/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.26/0.48 => ( ( lam @ A @ B
% 0.26/0.48 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.26/0.48 = Xf ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(iffalseProp1_type,type,
% 0.26/0.48 iffalseProp1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(iffalseProp1,definition,
% 0.26/0.48 ( iffalseProp1
% 0.26/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ A )
% 0.26/0.48 => ( ~ Xphi
% 0.26/0.48 => ( in @ Xy
% 0.26/0.48 @ ( dsetconstr @ A
% 0.26/0.48 @ ^ [Xz: $i] :
% 0.26/0.48 ( ( Xphi
% 0.26/0.48 & ( Xz = Xx ) )
% 0.26/0.48 | ( ~ Xphi
% 0.26/0.48 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(iffalseProp2_type,type,
% 0.26/0.48 iffalseProp2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(iffalseProp2,definition,
% 0.26/0.48 ( iffalseProp2
% 0.26/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ A )
% 0.26/0.48 => ( ~ Xphi
% 0.26/0.48 => ( ( dsetconstr @ A
% 0.26/0.48 @ ^ [Xz: $i] :
% 0.26/0.48 ( ( Xphi
% 0.26/0.48 & ( Xz = Xx ) )
% 0.26/0.48 | ( ~ Xphi
% 0.26/0.48 & ( Xz = Xy ) ) ) )
% 0.26/0.48 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(iftrueProp1_type,type,
% 0.26/0.48 iftrueProp1: $o ).
% 0.26/0.48
% 0.26/0.48 thf(iftrueProp1,definition,
% 0.26/0.48 ( iftrueProp1
% 0.26/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ A )
% 0.26/0.48 => ( Xphi
% 0.26/0.48 => ( in @ Xx
% 0.26/0.48 @ ( dsetconstr @ A
% 0.26/0.48 @ ^ [Xz: $i] :
% 0.26/0.48 ( ( Xphi
% 0.26/0.48 & ( Xz = Xx ) )
% 0.26/0.48 | ( ~ Xphi
% 0.26/0.48 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(iftrueProp2_type,type,
% 0.26/0.48 iftrueProp2: $o ).
% 0.26/0.48
% 0.26/0.48 thf(iftrueProp2,definition,
% 0.26/0.48 ( iftrueProp2
% 0.26/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ A )
% 0.26/0.48 => ( Xphi
% 0.26/0.48 => ( ( dsetconstr @ A
% 0.26/0.48 @ ^ [Xz: $i] :
% 0.26/0.48 ( ( Xphi
% 0.26/0.48 & ( Xz = Xx ) )
% 0.26/0.48 | ( ~ Xphi
% 0.26/0.48 & ( Xz = Xy ) ) ) )
% 0.26/0.48 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(ifSingleton_type,type,
% 0.26/0.48 ifSingleton: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ifSingleton,definition,
% 0.26/0.48 ( ifSingleton
% 0.26/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ A )
% 0.26/0.48 => ( singleton
% 0.26/0.48 @ ( dsetconstr @ A
% 0.26/0.48 @ ^ [Xz: $i] :
% 0.26/0.48 ( ( Xphi
% 0.26/0.48 & ( Xz = Xx ) )
% 0.26/0.48 | ( ~ Xphi
% 0.26/0.48 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(if_type,type,
% 0.26/0.48 if: $i > $o > $i > $i > $i ).
% 0.26/0.48
% 0.26/0.48 thf(ifp_type,type,
% 0.26/0.48 ifp: $o ).
% 0.26/0.48
% 0.26/0.48 thf(ifp,definition,
% 0.26/0.48 ( ifp
% 0.26/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ A )
% 0.26/0.48 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(theeq_type,type,
% 0.26/0.48 theeq: $o ).
% 0.26/0.48
% 0.26/0.48 thf(theeq,definition,
% 0.26/0.48 ( theeq
% 0.26/0.48 = ( ! [X: $i] :
% 0.26/0.48 ( ( singleton @ X )
% 0.26/0.48 => ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ X )
% 0.26/0.48 => ( ( setunion @ X )
% 0.26/0.48 = Xx ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(iftrue_type,type,
% 0.26/0.48 iftrue: $o ).
% 0.26/0.48
% 0.26/0.48 thf(iftrue,definition,
% 0.26/0.48 ( iftrue
% 0.26/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ A )
% 0.26/0.48 => ( Xphi
% 0.26/0.48 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.26/0.48 = Xx ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(iffalse_type,type,
% 0.26/0.48 iffalse: $o ).
% 0.26/0.48
% 0.26/0.48 thf(iffalse,definition,
% 0.26/0.48 ( iffalse
% 0.26/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ A )
% 0.26/0.48 => ( ~ Xphi
% 0.26/0.48 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.26/0.48 = Xy ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(iftrueorfalse_type,type,
% 0.26/0.48 iftrueorfalse: $o ).
% 0.26/0.48
% 0.26/0.48 thf(iftrueorfalse,definition,
% 0.26/0.48 ( iftrueorfalse
% 0.26/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ! [Xy: $i] :
% 0.26/0.48 ( ( in @ Xy @ A )
% 0.26/0.48 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectT_lem_type,type,
% 0.26/0.48 binintersectT_lem: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binintersectT_lem,definition,
% 0.26/0.48 ( binintersectT_lem
% 0.26/0.48 = ( ! [A: $i,X: $i] :
% 0.26/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.26/0.48 => ! [Y: $i] :
% 0.26/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.26/0.48 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(binunionT_lem_type,type,
% 0.26/0.48 binunionT_lem: $o ).
% 0.26/0.48
% 0.26/0.48 thf(binunionT_lem,definition,
% 0.26/0.48 ( binunionT_lem
% 0.26/0.48 = ( ! [A: $i,X: $i] :
% 0.26/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.26/0.48 => ! [Y: $i] :
% 0.26/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.26/0.48 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(powersetT_lem_type,type,
% 0.26/0.48 powersetT_lem: $o ).
% 0.26/0.48
% 0.26/0.48 thf(powersetT_lem,definition,
% 0.26/0.48 ( powersetT_lem
% 0.26/0.48 = ( ! [A: $i,X: $i] :
% 0.26/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.26/0.48 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setminusT_lem_type,type,
% 0.26/0.48 setminusT_lem: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setminusT_lem,definition,
% 0.26/0.48 ( setminusT_lem
% 0.26/0.48 = ( ! [A: $i,X: $i] :
% 0.26/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.26/0.48 => ! [Y: $i] :
% 0.26/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.26/0.48 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(complementT_lem_type,type,
% 0.26/0.48 complementT_lem: $o ).
% 0.26/0.48
% 0.26/0.48 thf(complementT_lem,definition,
% 0.26/0.48 ( complementT_lem
% 0.26/0.48 = ( ! [A: $i,X: $i] :
% 0.26/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.26/0.48 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(setextT_type,type,
% 0.26/0.48 setextT: $o ).
% 0.26/0.48
% 0.26/0.48 thf(setextT,definition,
% 0.26/0.48 ( setextT
% 0.26/0.48 = ( ! [A: $i,X: $i] :
% 0.26/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.26/0.48 => ! [Y: $i] :
% 0.26/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.26/0.48 => ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ( in @ Xx @ X )
% 0.26/0.48 => ( in @ Xx @ Y ) ) )
% 0.26/0.48 => ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ( in @ Xx @ Y )
% 0.26/0.48 => ( in @ Xx @ X ) ) )
% 0.26/0.48 => ( X = Y ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 thf(subsetTI,conjecture,
% 0.26/0.48 ( setextAx
% 0.26/0.48 => ( emptysetAx
% 0.26/0.48 => ( setadjoinAx
% 0.26/0.48 => ( powersetAx
% 0.26/0.48 => ( setunionAx
% 0.26/0.48 => ( omega0Ax
% 0.26/0.48 => ( omegaSAx
% 0.26/0.48 => ( omegaIndAx
% 0.26/0.48 => ( replAx
% 0.26/0.48 => ( foundationAx
% 0.26/0.48 => ( wellorderingAx
% 0.26/0.48 => ( descrp
% 0.26/0.48 => ( dsetconstrI
% 0.26/0.48 => ( dsetconstrEL
% 0.26/0.48 => ( dsetconstrER
% 0.26/0.48 => ( exuE1
% 0.26/0.48 => ( prop2setE
% 0.26/0.48 => ( emptysetE
% 0.26/0.48 => ( emptysetimpfalse
% 0.26/0.48 => ( notinemptyset
% 0.26/0.48 => ( exuE3e
% 0.26/0.48 => ( setext
% 0.26/0.48 => ( emptyI
% 0.26/0.48 => ( noeltsimpempty
% 0.26/0.48 => ( setbeta
% 0.26/0.48 => ( nonemptyE1
% 0.26/0.48 => ( nonemptyI
% 0.26/0.48 => ( nonemptyI1
% 0.26/0.48 => ( setadjoinIL
% 0.26/0.48 => ( emptyinunitempty
% 0.26/0.48 => ( setadjoinIR
% 0.26/0.48 => ( setadjoinE
% 0.26/0.48 => ( setadjoinOr
% 0.26/0.48 => ( setoftrueEq
% 0.26/0.48 => ( powersetI
% 0.26/0.48 => ( emptyinPowerset
% 0.26/0.48 => ( emptyInPowerset
% 0.26/0.48 => ( powersetE
% 0.26/0.48 => ( setunionI
% 0.26/0.48 => ( setunionE
% 0.26/0.48 => ( subPowSU
% 0.26/0.48 => ( exuE2
% 0.26/0.48 => ( nonemptyImpWitness
% 0.26/0.48 => ( uniqinunit
% 0.26/0.48 => ( notinsingleton
% 0.26/0.48 => ( eqinunit
% 0.26/0.48 => ( singletonsswitch
% 0.26/0.48 => ( upairsetE
% 0.26/0.48 => ( upairsetIL
% 0.26/0.48 => ( upairsetIR
% 0.26/0.48 => ( emptyE1
% 0.26/0.48 => ( vacuousDall
% 0.26/0.48 => ( quantDeMorgan1
% 0.26/0.48 => ( quantDeMorgan2
% 0.26/0.48 => ( quantDeMorgan3
% 0.26/0.48 => ( quantDeMorgan4
% 0.26/0.48 => ( prop2setI
% 0.26/0.48 => ( prop2set2propI
% 0.26/0.48 => ( notdexE
% 0.26/0.48 => ( notdallE
% 0.26/0.48 => ( exuI1
% 0.26/0.48 => ( exuI3
% 0.26/0.48 => ( exuI2
% 0.26/0.48 => ( inCongP
% 0.26/0.48 => ( in__Cong
% 0.26/0.48 => ( exuE3u
% 0.26/0.48 => ( exu__Cong
% 0.26/0.48 => ( emptyset__Cong
% 0.26/0.48 => ( setadjoin__Cong
% 0.26/0.48 => ( powerset__Cong
% 0.26/0.48 => ( setunion__Cong
% 0.26/0.48 => ( omega__Cong
% 0.26/0.48 => ( exuEu
% 0.26/0.48 => ( descr__Cong
% 0.26/0.48 => ( dsetconstr__Cong
% 0.26/0.48 => ( subsetI1
% 0.26/0.48 => ( eqimpsubset2
% 0.26/0.48 => ( eqimpsubset1
% 0.26/0.48 => ( subsetI2
% 0.26/0.48 => ( emptysetsubset
% 0.26/0.48 => ( subsetE
% 0.26/0.48 => ( subsetE2
% 0.26/0.48 => ( notsubsetI
% 0.26/0.48 => ( notequalI1
% 0.26/0.48 => ( notequalI2
% 0.26/0.48 => ( subsetRefl
% 0.26/0.48 => ( subsetTrans
% 0.26/0.48 => ( setadjoinSub
% 0.26/0.48 => ( setadjoinSub2
% 0.26/0.48 => ( subset2powerset
% 0.26/0.48 => ( setextsub
% 0.26/0.48 => ( subsetemptysetimpeq
% 0.26/0.48 => ( powersetI1
% 0.26/0.48 => ( powersetE1
% 0.26/0.48 => ( inPowerset
% 0.26/0.48 => ( powersetsubset
% 0.26/0.48 => ( sepInPowerset
% 0.26/0.48 => ( sepSubset
% 0.26/0.48 => ( binunionIL
% 0.26/0.48 => ( upairset2IR
% 0.26/0.48 => ( binunionIR
% 0.26/0.48 => ( binunionEcases
% 0.26/0.48 => ( binunionE
% 0.26/0.48 => ( binunionLsub
% 0.26/0.48 => ( binunionRsub
% 0.26/0.48 => ( binintersectI
% 0.26/0.48 => ( binintersectSubset5
% 0.26/0.48 => ( binintersectEL
% 0.26/0.48 => ( binintersectLsub
% 0.26/0.48 => ( binintersectSubset2
% 0.26/0.48 => ( binintersectSubset3
% 0.26/0.48 => ( binintersectER
% 0.26/0.48 => ( disjointsetsI1
% 0.26/0.48 => ( binintersectRsub
% 0.26/0.48 => ( binintersectSubset4
% 0.26/0.48 => ( binintersectSubset1
% 0.26/0.48 => ( bs114d
% 0.26/0.48 => ( setminusI
% 0.26/0.48 => ( setminusEL
% 0.26/0.48 => ( setminusER
% 0.26/0.48 => ( setminusSubset2
% 0.26/0.48 => ( setminusERneg
% 0.26/0.48 => ( setminusELneg
% 0.26/0.48 => ( setminusILneg
% 0.26/0.48 => ( setminusIRneg
% 0.26/0.48 => ( setminusLsub
% 0.26/0.48 => ( setminusSubset1
% 0.26/0.48 => ( symdiffE
% 0.26/0.48 => ( symdiffI1
% 0.26/0.48 => ( symdiffI2
% 0.26/0.48 => ( symdiffIneg1
% 0.26/0.48 => ( symdiffIneg2
% 0.26/0.48 => ( secondinupair
% 0.26/0.48 => ( setukpairIL
% 0.26/0.48 => ( setukpairIR
% 0.26/0.48 => ( kpairiskpair
% 0.26/0.48 => ( kpairp
% 0.26/0.48 => ( singletonsubset
% 0.26/0.48 => ( singletoninpowerset
% 0.26/0.48 => ( singletoninpowunion
% 0.26/0.48 => ( upairset2E
% 0.26/0.48 => ( upairsubunion
% 0.26/0.48 => ( upairinpowunion
% 0.26/0.48 => ( ubforcartprodlem1
% 0.26/0.48 => ( ubforcartprodlem2
% 0.26/0.48 => ( ubforcartprodlem3
% 0.26/0.48 => ( cartprodpairin
% 0.26/0.48 => ( cartprodmempair1
% 0.26/0.48 => ( cartprodmempair
% 0.26/0.48 => ( setunionE2
% 0.26/0.48 => ( setunionsingleton1
% 0.26/0.48 => ( setunionsingleton2
% 0.26/0.48 => ( setunionsingleton
% 0.26/0.48 => ( singletonprop
% 0.26/0.48 => ( ex1E1
% 0.26/0.48 => ( ex1I
% 0.26/0.48 => ( ex1I2
% 0.26/0.48 => ( singletonsuniq
% 0.26/0.48 => ( setukpairinjL1
% 0.26/0.48 => ( kfstsingleton
% 0.26/0.48 => ( theprop
% 0.26/0.48 => ( kfstpairEq
% 0.26/0.48 => ( cartprodfstin
% 0.26/0.48 => ( setukpairinjL2
% 0.26/0.48 => ( setukpairinjL
% 0.26/0.48 => ( setukpairinjR11
% 0.26/0.48 => ( setukpairinjR12
% 0.26/0.48 => ( setukpairinjR1
% 0.26/0.48 => ( upairequniteq
% 0.26/0.48 => ( setukpairinjR2
% 0.26/0.48 => ( setukpairinjR
% 0.26/0.48 => ( ksndsingleton
% 0.26/0.48 => ( ksndpairEq
% 0.26/0.48 => ( kpairsurjEq
% 0.26/0.48 => ( cartprodsndin
% 0.26/0.48 => ( cartprodpairmemEL
% 0.26/0.48 => ( cartprodpairmemER
% 0.26/0.48 => ( cartprodmempaircEq
% 0.26/0.48 => ( cartprodfstpairEq
% 0.26/0.48 => ( cartprodsndpairEq
% 0.26/0.48 => ( cartprodpairsurjEq
% 0.26/0.48 => ( dpsetconstrI
% 0.26/0.48 => ( dpsetconstrSub
% 0.26/0.48 => ( setOfPairsIsBReln
% 0.26/0.48 => ( dpsetconstrERa
% 0.26/0.48 => ( dpsetconstrEL1
% 0.26/0.48 => ( dpsetconstrEL2
% 0.26/0.48 => ( dpsetconstrER
% 0.26/0.48 => ( funcImageSingleton
% 0.26/0.48 => ( apProp
% 0.26/0.48 => ( app
% 0.26/0.48 => ( infuncsetfunc
% 0.26/0.48 => ( ap2p
% 0.26/0.48 => ( funcinfuncset
% 0.26/0.48 => ( lamProp
% 0.26/0.48 => ( lamp
% 0.26/0.48 => ( lam2p
% 0.26/0.48 => ( brelnall1
% 0.26/0.48 => ( brelnall2
% 0.26/0.48 => ( ex1E2
% 0.26/0.48 => ( funcGraphProp1
% 0.26/0.48 => ( funcGraphProp3
% 0.26/0.48 => ( funcGraphProp2
% 0.26/0.48 => ( funcextLem
% 0.26/0.48 => ( funcGraphProp4
% 0.26/0.48 => ( subbreln
% 0.26/0.48 => ( eqbreln
% 0.26/0.48 => ( funcext
% 0.26/0.48 => ( funcext2
% 0.26/0.48 => ( ap2apEq1
% 0.26/0.48 => ( ap2apEq2
% 0.26/0.48 => ( beta1
% 0.26/0.48 => ( eta1
% 0.26/0.48 => ( lam2lamEq
% 0.26/0.48 => ( beta2
% 0.26/0.48 => ( eta2
% 0.26/0.48 => ( iffalseProp1
% 0.26/0.48 => ( iffalseProp2
% 0.26/0.48 => ( iftrueProp1
% 0.26/0.48 => ( iftrueProp2
% 0.26/0.48 => ( ifSingleton
% 0.26/0.48 => ( ifp
% 0.26/0.48 => ( theeq
% 0.26/0.48 => ( iftrue
% 0.26/0.48 => ( iffalse
% 0.26/0.48 => ( iftrueorfalse
% 0.26/0.48 => ( binintersectT_lem
% 0.26/0.48 => ( binunionT_lem
% 0.26/0.48 => ( powersetT_lem
% 0.26/0.48 => ( setminusT_lem
% 0.26/0.48 => ( complementT_lem
% 0.26/0.48 => ( setextT
% 0.26/0.48 => ! [A: $i,X: $i] :
% 0.26/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.26/0.48 => ! [Y: $i] :
% 0.26/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.26/0.48 => ( ! [Xx: $i] :
% 0.26/0.48 ( ( in @ Xx @ A )
% 0.26/0.48 => ( ( in @ Xx @ X )
% 0.26/0.48 => ( in @ Xx @ Y ) ) )
% 0.26/0.48 => ( subset @ X @ Y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.26/0.48
% 0.26/0.48 %------------------------------------------------------------------------------
% 0.26/0.48 ------------------------
% 2.65/1.51 ---- Embedded in THF ---
% 2.65/1.51 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.65/1.51 %%% Generated on Tue Feb 24 12:31:19 EST 2026
% 2.65/1.51 %%% using '$$dhol' embedding, version 1.3.0.
% 2.65/1.51 %%% Logic specification used:
% 2.65/1.51 %%% thf(spec, logic, $$dhol).
% 2.65/1.51
% 2.65/1.51 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.DvUejQI5fj/DTF2DTF12059.p
% See solution above
% 2.65/1.52 ------------------------
% 2.65/1.56 ---- Cleaned THF ---
% 2.65/1.56 thf(in_type,type,
% 2.65/1.56 in: $i > $i > $o ).
% 2.65/1.56
% 2.65/1.56 thf(exu_type,type,
% 2.65/1.56 exu: ( $i > $o ) > $o ).
% 2.65/1.56
% 2.65/1.56 thf(exu,definition,
% 2.65/1.56 ( exu
% 2.65/1.56 = ( ^ [Xphi: $i > $o] :
% 2.65/1.56 ? [Xx: $i] :
% 2.65/1.56 ( ( Xphi @ Xx )
% 2.65/1.56 & ! [Xy: $i] :
% 2.65/1.56 ( ( Xphi @ Xy )
% 2.65/1.56 => ( Xx = Xy ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(setextAx_type,type,
% 2.65/1.56 setextAx: $o ).
% 2.65/1.56
% 2.65/1.56 thf(setextAx,definition,
% 2.65/1.56 ( setextAx
% 2.65/1.56 = ( ! [A: $i,B: $i] :
% 2.65/1.56 ( ! [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 <=> ( in @ Xx @ B ) )
% 2.65/1.56 => ( A = B ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(emptyset_type,type,
% 2.65/1.56 emptyset: $i ).
% 2.65/1.56
% 2.65/1.56 thf(emptysetAx_type,type,
% 2.65/1.56 emptysetAx: $o ).
% 2.65/1.56
% 2.65/1.56 thf(emptysetAx,definition,
% 2.65/1.56 ( emptysetAx
% 2.65/1.56 = ( ! [Xx: $i] :
% 2.65/1.56 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(setadjoin_type,type,
% 2.65/1.56 setadjoin: $i > $i > $i ).
% 2.65/1.56
% 2.65/1.56 thf(setadjoinAx_type,type,
% 2.65/1.56 setadjoinAx: $o ).
% 2.65/1.56
% 2.65/1.56 thf(setadjoinAx,definition,
% 2.65/1.56 ( setadjoinAx
% 2.65/1.56 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.65/1.56 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.65/1.56 <=> ( ( Xy = Xx )
% 2.65/1.56 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(powerset_type,type,
% 2.65/1.56 powerset: $i > $i ).
% 2.65/1.56
% 2.65/1.56 thf(powersetAx_type,type,
% 2.65/1.56 powersetAx: $o ).
% 2.65/1.56
% 2.65/1.56 thf(powersetAx,definition,
% 2.65/1.56 ( powersetAx
% 2.65/1.56 = ( ! [A: $i,B: $i] :
% 2.65/1.56 ( ( in @ B @ ( powerset @ A ) )
% 2.65/1.56 <=> ! [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ B )
% 2.65/1.56 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(setunion_type,type,
% 2.65/1.56 setunion: $i > $i ).
% 2.65/1.56
% 2.65/1.56 thf(setunionAx_type,type,
% 2.65/1.56 setunionAx: $o ).
% 2.65/1.56
% 2.65/1.56 thf(setunionAx,definition,
% 2.65/1.56 ( setunionAx
% 2.65/1.56 = ( ! [A: $i,Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ ( setunion @ A ) )
% 2.65/1.56 <=> ? [B: $i] :
% 2.65/1.56 ( ( in @ Xx @ B )
% 2.65/1.56 & ( in @ B @ A ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(omega_type,type,
% 2.65/1.56 omega: $i ).
% 2.65/1.56
% 2.65/1.56 thf(omega0Ax_type,type,
% 2.65/1.56 omega0Ax: $o ).
% 2.65/1.56
% 2.65/1.56 thf(omega0Ax,definition,
% 2.65/1.56 ( omega0Ax
% 2.65/1.56 = ( in @ emptyset @ omega ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(omegaSAx_type,type,
% 2.65/1.56 omegaSAx: $o ).
% 2.65/1.56
% 2.65/1.56 thf(omegaSAx,definition,
% 2.65/1.56 ( omegaSAx
% 2.65/1.56 = ( ! [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ omega )
% 2.65/1.56 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(omegaIndAx_type,type,
% 2.65/1.56 omegaIndAx: $o ).
% 2.65/1.56
% 2.65/1.56 thf(omegaIndAx,definition,
% 2.65/1.56 ( omegaIndAx
% 2.65/1.56 = ( ! [A: $i] :
% 2.65/1.56 ( ( ( in @ emptyset @ A )
% 2.65/1.56 & ! [Xx: $i] :
% 2.65/1.56 ( ( ( in @ Xx @ omega )
% 2.65/1.56 & ( in @ Xx @ A ) )
% 2.65/1.56 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.65/1.56 => ! [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ omega )
% 2.65/1.56 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(replAx_type,type,
% 2.65/1.56 replAx: $o ).
% 2.65/1.56
% 2.65/1.56 thf(replAx,definition,
% 2.65/1.56 ( replAx
% 2.65/1.56 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.65/1.56 ( ! [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 => ( exu
% 2.65/1.56 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.65/1.56 => ? [B: $i] :
% 2.65/1.56 ! [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ B )
% 2.65/1.56 <=> ? [Xy: $i] :
% 2.65/1.56 ( ( in @ Xy @ A )
% 2.65/1.56 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(foundationAx_type,type,
% 2.65/1.56 foundationAx: $o ).
% 2.65/1.56
% 2.65/1.56 thf(foundationAx,definition,
% 2.65/1.56 ( foundationAx
% 2.65/1.56 = ( ! [A: $i] :
% 2.65/1.56 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.65/1.56 => ? [B: $i] :
% 2.65/1.56 ( ( in @ B @ A )
% 2.65/1.56 & ~ ? [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ B )
% 2.65/1.56 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(wellorderingAx_type,type,
% 2.65/1.56 wellorderingAx: $o ).
% 2.65/1.56
% 2.65/1.56 thf(wellorderingAx,definition,
% 2.65/1.56 ( wellorderingAx
% 2.65/1.56 = ( ! [A: $i] :
% 2.65/1.56 ? [B: $i] :
% 2.65/1.56 ( ! [C: $i] :
% 2.65/1.56 ( ( in @ C @ B )
% 2.65/1.56 => ! [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ C )
% 2.65/1.56 => ( in @ Xx @ A ) ) )
% 2.65/1.56 & ! [Xx: $i,Xy: $i] :
% 2.65/1.56 ( ( ( in @ Xx @ A )
% 2.65/1.56 & ( in @ Xy @ A ) )
% 2.65/1.56 => ( ! [C: $i] :
% 2.65/1.56 ( ( in @ C @ B )
% 2.65/1.56 => ( ( in @ Xx @ C )
% 2.65/1.56 <=> ( in @ Xy @ C ) ) )
% 2.65/1.56 => ( Xx = Xy ) ) )
% 2.65/1.56 & ! [C: $i,D: $i] :
% 2.65/1.56 ( ( ( in @ C @ B )
% 2.65/1.56 & ( in @ D @ B ) )
% 2.65/1.56 => ( ! [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ C )
% 2.65/1.56 => ( in @ Xx @ D ) )
% 2.65/1.56 | ! [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ D )
% 2.65/1.56 => ( in @ Xx @ C ) ) ) )
% 2.65/1.56 & ! [C: $i] :
% 2.65/1.56 ( ( ! [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ C )
% 2.65/1.56 => ( in @ Xx @ A ) )
% 2.65/1.56 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.65/1.56 => ? [D: $i,Xx: $i] :
% 2.65/1.56 ( ( in @ D @ B )
% 2.65/1.56 & ( in @ Xx @ C )
% 2.65/1.56 & ~ ? [Xy: $i] :
% 2.65/1.56 ( ( in @ Xy @ D )
% 2.65/1.56 & ( in @ Xy @ C ) )
% 2.65/1.56 & ! [E: $i] :
% 2.65/1.56 ( ( in @ E @ B )
% 2.65/1.56 => ( ! [Xy: $i] :
% 2.65/1.56 ( ( in @ Xy @ E )
% 2.65/1.56 => ( in @ Xy @ D ) )
% 2.65/1.56 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(descr_type,type,
% 2.65/1.56 descr: ( $i > $o ) > $i ).
% 2.65/1.56
% 2.65/1.56 thf(descrp_type,type,
% 2.65/1.56 descrp: $o ).
% 2.65/1.56
% 2.65/1.56 thf(descrp,definition,
% 2.65/1.56 ( descrp
% 2.65/1.56 = ( ! [Xphi: $i > $o] :
% 2.65/1.56 ( ( exu
% 2.65/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.56 => ( Xphi
% 2.65/1.56 @ ( descr
% 2.65/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(dsetconstr_type,type,
% 2.65/1.56 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.65/1.56
% 2.65/1.56 thf(dsetconstrI_type,type,
% 2.65/1.56 dsetconstrI: $o ).
% 2.65/1.56
% 2.65/1.56 thf(dsetconstrI,definition,
% 2.65/1.56 ( dsetconstrI
% 2.65/1.56 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 => ( ( Xphi @ Xx )
% 2.65/1.56 => ( in @ Xx
% 2.65/1.56 @ ( dsetconstr @ A
% 2.65/1.56 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(dsetconstrEL_type,type,
% 2.65/1.56 dsetconstrEL: $o ).
% 2.65/1.56
% 2.65/1.56 thf(dsetconstrEL,definition,
% 2.65/1.56 ( dsetconstrEL
% 2.65/1.56 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.65/1.56 ( ( in @ Xx
% 2.65/1.56 @ ( dsetconstr @ A
% 2.65/1.56 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.65/1.56 => ( in @ Xx @ A ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(dsetconstrER_type,type,
% 2.65/1.56 dsetconstrER: $o ).
% 2.65/1.56
% 2.65/1.56 thf(dsetconstrER,definition,
% 2.65/1.56 ( dsetconstrER
% 2.65/1.56 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.65/1.56 ( ( in @ Xx
% 2.65/1.56 @ ( dsetconstr @ A
% 2.65/1.56 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.65/1.56 => ( Xphi @ Xx ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(exuE1_type,type,
% 2.65/1.56 exuE1: $o ).
% 2.65/1.56
% 2.65/1.56 thf(exuE1,definition,
% 2.65/1.56 ( exuE1
% 2.65/1.56 = ( ! [Xphi: $i > $o] :
% 2.65/1.56 ( ( exu
% 2.65/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.56 => ? [Xx: $i] :
% 2.65/1.56 ( ( Xphi @ Xx )
% 2.65/1.56 & ! [Xy: $i] :
% 2.65/1.56 ( ( Xphi @ Xy )
% 2.65/1.56 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(prop2set_type,type,
% 2.65/1.56 prop2set: $o > $i ).
% 2.65/1.56
% 2.65/1.56 thf(prop2setE_type,type,
% 2.65/1.56 prop2setE: $o ).
% 2.65/1.56
% 2.65/1.56 thf(prop2setE,definition,
% 2.65/1.56 ( prop2setE
% 2.65/1.56 = ( ! [Xphi: $o,Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.65/1.56 => Xphi ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(emptysetE_type,type,
% 2.65/1.56 emptysetE: $o ).
% 2.65/1.56
% 2.65/1.56 thf(emptysetE,definition,
% 2.65/1.56 ( emptysetE
% 2.65/1.56 = ( ! [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ emptyset )
% 2.65/1.56 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(emptysetimpfalse_type,type,
% 2.65/1.56 emptysetimpfalse: $o ).
% 2.65/1.56
% 2.65/1.56 thf(emptysetimpfalse,definition,
% 2.65/1.56 ( emptysetimpfalse
% 2.65/1.56 = ( ! [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ emptyset )
% 2.65/1.56 => $false ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(notinemptyset_type,type,
% 2.65/1.56 notinemptyset: $o ).
% 2.65/1.56
% 2.65/1.56 thf(notinemptyset,definition,
% 2.65/1.56 ( notinemptyset
% 2.65/1.56 = ( ! [Xx: $i] :
% 2.65/1.56 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(exuE3e_type,type,
% 2.65/1.56 exuE3e: $o ).
% 2.65/1.56
% 2.65/1.56 thf(exuE3e,definition,
% 2.65/1.56 ( exuE3e
% 2.65/1.56 = ( ! [Xphi: $i > $o] :
% 2.65/1.56 ( ( exu
% 2.65/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.56 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(setext_type,type,
% 2.65/1.56 setext: $o ).
% 2.65/1.56
% 2.65/1.56 thf(setext,definition,
% 2.65/1.56 ( setext
% 2.65/1.56 = ( ! [A: $i,B: $i] :
% 2.65/1.56 ( ! [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 => ( in @ Xx @ B ) )
% 2.65/1.56 => ( ! [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ B )
% 2.65/1.56 => ( in @ Xx @ A ) )
% 2.65/1.56 => ( A = B ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(emptyI_type,type,
% 2.65/1.56 emptyI: $o ).
% 2.65/1.56
% 2.65/1.56 thf(emptyI,definition,
% 2.65/1.56 ( emptyI
% 2.65/1.56 = ( ! [A: $i] :
% 2.65/1.56 ( ! [Xx: $i] :
% 2.65/1.56 ~ ( in @ Xx @ A )
% 2.65/1.56 => ( A = emptyset ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(noeltsimpempty_type,type,
% 2.65/1.56 noeltsimpempty: $o ).
% 2.65/1.56
% 2.65/1.56 thf(noeltsimpempty,definition,
% 2.65/1.56 ( noeltsimpempty
% 2.65/1.56 = ( ! [A: $i] :
% 2.65/1.56 ( ! [Xx: $i] :
% 2.65/1.56 ~ ( in @ Xx @ A )
% 2.65/1.56 => ( A = emptyset ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(setbeta_type,type,
% 2.65/1.56 setbeta: $o ).
% 2.65/1.56
% 2.65/1.56 thf(setbeta,definition,
% 2.65/1.56 ( setbeta
% 2.65/1.56 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 => ( ( in @ Xx
% 2.65/1.56 @ ( dsetconstr @ A
% 2.65/1.56 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.65/1.56 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(nonempty_type,type,
% 2.65/1.56 nonempty: $i > $o ).
% 2.65/1.56
% 2.65/1.56 thf(nonempty,definition,
% 2.65/1.56 ( nonempty
% 2.65/1.56 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(nonemptyE1_type,type,
% 2.65/1.56 nonemptyE1: $o ).
% 2.65/1.56
% 2.65/1.56 thf(nonemptyE1,definition,
% 2.65/1.56 ( nonemptyE1
% 2.65/1.56 = ( ! [A: $i] :
% 2.65/1.56 ( ( nonempty @ A )
% 2.65/1.56 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(nonemptyI_type,type,
% 2.65/1.56 nonemptyI: $o ).
% 2.65/1.56
% 2.65/1.56 thf(nonemptyI,definition,
% 2.65/1.56 ( nonemptyI
% 2.65/1.56 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 => ( ( Xphi @ Xx )
% 2.65/1.56 => ( nonempty
% 2.65/1.56 @ ( dsetconstr @ A
% 2.65/1.56 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(nonemptyI1_type,type,
% 2.65/1.56 nonemptyI1: $o ).
% 2.65/1.56
% 2.65/1.56 thf(nonemptyI1,definition,
% 2.65/1.56 ( nonemptyI1
% 2.65/1.56 = ( ! [A: $i] :
% 2.65/1.56 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.65/1.56 => ( nonempty @ A ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(setadjoinIL_type,type,
% 2.65/1.56 setadjoinIL: $o ).
% 2.65/1.56
% 2.65/1.56 thf(setadjoinIL,definition,
% 2.65/1.56 ( setadjoinIL
% 2.65/1.56 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(emptyinunitempty_type,type,
% 2.65/1.56 emptyinunitempty: $o ).
% 2.65/1.56
% 2.65/1.56 thf(emptyinunitempty,definition,
% 2.65/1.56 ( emptyinunitempty
% 2.65/1.56 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(setadjoinIR_type,type,
% 2.65/1.56 setadjoinIR: $o ).
% 2.65/1.56
% 2.65/1.56 thf(setadjoinIR,definition,
% 2.65/1.56 ( setadjoinIR
% 2.65/1.56 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.65/1.56 ( ( in @ Xy @ A )
% 2.65/1.56 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(setadjoinE_type,type,
% 2.65/1.56 setadjoinE: $o ).
% 2.65/1.56
% 2.65/1.56 thf(setadjoinE,definition,
% 2.65/1.56 ( setadjoinE
% 2.65/1.56 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.65/1.56 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.65/1.56 => ! [Xphi: $o] :
% 2.65/1.56 ( ( ( Xy = Xx )
% 2.65/1.56 => Xphi )
% 2.65/1.56 => ( ( ( in @ Xy @ A )
% 2.65/1.56 => Xphi )
% 2.65/1.56 => Xphi ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(setadjoinOr_type,type,
% 2.65/1.56 setadjoinOr: $o ).
% 2.65/1.56
% 2.65/1.56 thf(setadjoinOr,definition,
% 2.65/1.56 ( setadjoinOr
% 2.65/1.56 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.65/1.56 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.65/1.56 => ( ( Xy = Xx )
% 2.65/1.56 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(setoftrueEq_type,type,
% 2.65/1.56 setoftrueEq: $o ).
% 2.65/1.56
% 2.65/1.56 thf(setoftrueEq,definition,
% 2.65/1.56 ( setoftrueEq
% 2.65/1.56 = ( ! [A: $i] :
% 2.65/1.56 ( ( dsetconstr @ A
% 2.65/1.56 @ ^ [Xx: $i] : $true )
% 2.65/1.56 = A ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(powersetI_type,type,
% 2.65/1.56 powersetI: $o ).
% 2.65/1.56
% 2.65/1.56 thf(powersetI,definition,
% 2.65/1.56 ( powersetI
% 2.65/1.56 = ( ! [A: $i,B: $i] :
% 2.65/1.56 ( ! [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ B )
% 2.65/1.56 => ( in @ Xx @ A ) )
% 2.65/1.56 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(emptyinPowerset_type,type,
% 2.65/1.56 emptyinPowerset: $o ).
% 2.65/1.56
% 2.65/1.56 thf(emptyinPowerset,definition,
% 2.65/1.56 ( emptyinPowerset
% 2.65/1.56 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(emptyInPowerset_type,type,
% 2.65/1.56 emptyInPowerset: $o ).
% 2.65/1.56
% 2.65/1.56 thf(emptyInPowerset,definition,
% 2.65/1.56 ( emptyInPowerset
% 2.65/1.56 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(powersetE_type,type,
% 2.65/1.56 powersetE: $o ).
% 2.65/1.56
% 2.65/1.56 thf(powersetE,definition,
% 2.65/1.56 ( powersetE
% 2.65/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.56 ( ( in @ B @ ( powerset @ A ) )
% 2.65/1.56 => ( ( in @ Xx @ B )
% 2.65/1.56 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(setunionI_type,type,
% 2.65/1.56 setunionI: $o ).
% 2.65/1.56
% 2.65/1.56 thf(setunionI,definition,
% 2.65/1.56 ( setunionI
% 2.65/1.56 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.65/1.56 ( ( in @ Xx @ B )
% 2.65/1.56 => ( ( in @ B @ A )
% 2.65/1.56 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(setunionE_type,type,
% 2.65/1.56 setunionE: $o ).
% 2.65/1.56
% 2.65/1.56 thf(setunionE,definition,
% 2.65/1.56 ( setunionE
% 2.65/1.56 = ( ! [A: $i,Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ ( setunion @ A ) )
% 2.65/1.56 => ! [Xphi: $o] :
% 2.65/1.56 ( ! [B: $i] :
% 2.65/1.56 ( ( in @ Xx @ B )
% 2.65/1.56 => ( ( in @ B @ A )
% 2.65/1.56 => Xphi ) )
% 2.65/1.56 => Xphi ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(subPowSU_type,type,
% 2.65/1.56 subPowSU: $o ).
% 2.65/1.56
% 2.65/1.56 thf(subPowSU,definition,
% 2.65/1.56 ( subPowSU
% 2.65/1.56 = ( ! [A: $i,Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(exuE2_type,type,
% 2.65/1.56 exuE2: $o ).
% 2.65/1.56
% 2.65/1.56 thf(exuE2,definition,
% 2.65/1.56 ( exuE2
% 2.65/1.56 = ( ! [Xphi: $i > $o] :
% 2.65/1.56 ( ( exu
% 2.65/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.56 => ? [Xx: $i] :
% 2.65/1.56 ! [Xy: $i] :
% 2.65/1.56 ( ( Xphi @ Xy )
% 2.65/1.56 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(nonemptyImpWitness_type,type,
% 2.65/1.56 nonemptyImpWitness: $o ).
% 2.65/1.56
% 2.65/1.56 thf(nonemptyImpWitness,definition,
% 2.65/1.56 ( nonemptyImpWitness
% 2.65/1.56 = ( ! [A: $i] :
% 2.65/1.56 ( ( nonempty @ A )
% 2.65/1.56 => ? [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 & $true ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(uniqinunit_type,type,
% 2.65/1.56 uniqinunit: $o ).
% 2.65/1.56
% 2.65/1.56 thf(uniqinunit,definition,
% 2.65/1.56 ( uniqinunit
% 2.65/1.56 = ( ! [Xx: $i,Xy: $i] :
% 2.65/1.56 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.65/1.56 => ( Xx = Xy ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(notinsingleton_type,type,
% 2.65/1.56 notinsingleton: $o ).
% 2.65/1.56
% 2.65/1.56 thf(notinsingleton,definition,
% 2.65/1.56 ( notinsingleton
% 2.65/1.56 = ( ! [Xx: $i,Xy: $i] :
% 2.65/1.56 ( ( Xx != Xy )
% 2.65/1.56 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(eqinunit_type,type,
% 2.65/1.56 eqinunit: $o ).
% 2.65/1.56
% 2.65/1.56 thf(eqinunit,definition,
% 2.65/1.56 ( eqinunit
% 2.65/1.56 = ( ! [Xx: $i,Xy: $i] :
% 2.65/1.56 ( ( Xx = Xy )
% 2.65/1.56 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(singletonsswitch_type,type,
% 2.65/1.56 singletonsswitch: $o ).
% 2.65/1.56
% 2.65/1.56 thf(singletonsswitch,definition,
% 2.65/1.56 ( singletonsswitch
% 2.65/1.56 = ( ! [Xx: $i,Xy: $i] :
% 2.65/1.56 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.65/1.56 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(upairsetE_type,type,
% 2.65/1.56 upairsetE: $o ).
% 2.65/1.56
% 2.65/1.56 thf(upairsetE,definition,
% 2.65/1.56 ( upairsetE
% 2.65/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.65/1.56 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.65/1.56 => ( ( Xz = Xx )
% 2.65/1.56 | ( Xz = Xy ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(upairsetIL_type,type,
% 2.65/1.56 upairsetIL: $o ).
% 2.65/1.56
% 2.65/1.56 thf(upairsetIL,definition,
% 2.65/1.56 ( upairsetIL
% 2.65/1.56 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(upairsetIR_type,type,
% 2.65/1.56 upairsetIR: $o ).
% 2.65/1.56
% 2.65/1.56 thf(upairsetIR,definition,
% 2.65/1.56 ( upairsetIR
% 2.65/1.56 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(emptyE1_type,type,
% 2.65/1.56 emptyE1: $o ).
% 2.65/1.56
% 2.65/1.56 thf(emptyE1,definition,
% 2.65/1.56 ( emptyE1
% 2.65/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.56 ( ? [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 & ( Xphi @ Xx ) )
% 2.65/1.56 => ( ( ( dsetconstr @ A
% 2.65/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.56 = emptyset )
% 2.65/1.56 => $false ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(vacuousDall_type,type,
% 2.65/1.56 vacuousDall: $o ).
% 2.65/1.56
% 2.65/1.56 thf(vacuousDall,definition,
% 2.65/1.56 ( vacuousDall
% 2.65/1.56 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ emptyset )
% 2.65/1.56 => ( Xphi @ Xx ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(quantDeMorgan1_type,type,
% 2.65/1.56 quantDeMorgan1: $o ).
% 2.65/1.56
% 2.65/1.56 thf(quantDeMorgan1,definition,
% 2.65/1.56 ( quantDeMorgan1
% 2.65/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.56 ( ~ ! [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 => ( Xphi @ Xx ) )
% 2.65/1.56 => ? [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(quantDeMorgan2_type,type,
% 2.65/1.56 quantDeMorgan2: $o ).
% 2.65/1.56
% 2.65/1.56 thf(quantDeMorgan2,definition,
% 2.65/1.56 ( quantDeMorgan2
% 2.65/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.56 ( ! [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 => ~ ( Xphi @ Xx ) )
% 2.65/1.56 => ~ ? [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(quantDeMorgan3_type,type,
% 2.65/1.56 quantDeMorgan3: $o ).
% 2.65/1.56
% 2.65/1.56 thf(quantDeMorgan3,definition,
% 2.65/1.56 ( quantDeMorgan3
% 2.65/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.56 ( ~ ? [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 & ( Xphi @ Xx ) )
% 2.65/1.56 => ! [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(quantDeMorgan4_type,type,
% 2.65/1.56 quantDeMorgan4: $o ).
% 2.65/1.56
% 2.65/1.56 thf(quantDeMorgan4,definition,
% 2.65/1.56 ( quantDeMorgan4
% 2.65/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.56 ( ? [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 & ~ ( Xphi @ Xx ) )
% 2.65/1.56 => ~ ! [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(prop2setI_type,type,
% 2.65/1.56 prop2setI: $o ).
% 2.65/1.56
% 2.65/1.56 thf(prop2setI,definition,
% 2.65/1.56 ( prop2setI
% 2.65/1.56 = ( ! [Xphi: $o] :
% 2.65/1.56 ( Xphi
% 2.65/1.56 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(set2prop_type,type,
% 2.65/1.56 set2prop: $i > $o ).
% 2.65/1.56
% 2.65/1.56 thf(prop2set2propI_type,type,
% 2.65/1.56 prop2set2propI: $o ).
% 2.65/1.56
% 2.65/1.56 thf(prop2set2propI,definition,
% 2.65/1.56 ( prop2set2propI
% 2.65/1.56 = ( ! [Xphi: $o] :
% 2.65/1.56 ( Xphi
% 2.65/1.56 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(notdexE_type,type,
% 2.65/1.56 notdexE: $o ).
% 2.65/1.56
% 2.65/1.56 thf(notdexE,definition,
% 2.65/1.56 ( notdexE
% 2.65/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.56 ( ~ ? [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 & ( Xphi @ Xx ) )
% 2.65/1.56 => ! [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(notdallE_type,type,
% 2.65/1.56 notdallE: $o ).
% 2.65/1.56
% 2.65/1.56 thf(notdallE,definition,
% 2.65/1.56 ( notdallE
% 2.65/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.56 ( ~ ! [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 => ( Xphi @ Xx ) )
% 2.65/1.56 => ? [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(exuI1_type,type,
% 2.65/1.56 exuI1: $o ).
% 2.65/1.56
% 2.65/1.56 thf(exuI1,definition,
% 2.65/1.56 ( exuI1
% 2.65/1.56 = ( ! [Xphi: $i > $o] :
% 2.65/1.56 ( ? [Xx: $i] :
% 2.65/1.56 ( ( Xphi @ Xx )
% 2.65/1.56 & ! [Xy: $i] :
% 2.65/1.56 ( ( Xphi @ Xy )
% 2.65/1.56 => ( Xx = Xy ) ) )
% 2.65/1.56 => ( exu
% 2.65/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(exuI3_type,type,
% 2.65/1.56 exuI3: $o ).
% 2.65/1.56
% 2.65/1.56 thf(exuI3,definition,
% 2.65/1.56 ( exuI3
% 2.65/1.56 = ( ! [Xphi: $i > $o] :
% 2.65/1.56 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.65/1.56 => ( ! [Xx: $i,Xy: $i] :
% 2.65/1.56 ( ( Xphi @ Xx )
% 2.65/1.56 => ( ( Xphi @ Xy )
% 2.65/1.56 => ( Xx = Xy ) ) )
% 2.65/1.56 => ( exu
% 2.65/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(exuI2_type,type,
% 2.65/1.56 exuI2: $o ).
% 2.65/1.56
% 2.65/1.56 thf(exuI2,definition,
% 2.65/1.56 ( exuI2
% 2.65/1.56 = ( ! [Xphi: $i > $o] :
% 2.65/1.56 ( ? [Xx: $i] :
% 2.65/1.56 ! [Xy: $i] :
% 2.65/1.56 ( ( Xphi @ Xy )
% 2.65/1.56 <=> ( Xy = Xx ) )
% 2.65/1.56 => ( exu
% 2.65/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(inCongP_type,type,
% 2.65/1.56 inCongP: $o ).
% 2.65/1.56
% 2.65/1.56 thf(inCongP,definition,
% 2.65/1.56 ( inCongP
% 2.65/1.56 = ( ! [A: $i,B: $i] :
% 2.65/1.56 ( ( A = B )
% 2.65/1.56 => ! [Xx: $i,Xy: $i] :
% 2.65/1.56 ( ( Xx = Xy )
% 2.65/1.56 => ( ( in @ Xx @ A )
% 2.65/1.56 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(in__Cong_type,type,
% 2.65/1.56 in__Cong: $o ).
% 2.65/1.56
% 2.65/1.56 thf(in__Cong,definition,
% 2.65/1.56 ( in__Cong
% 2.65/1.56 = ( ! [A: $i,B: $i] :
% 2.65/1.56 ( ( A = B )
% 2.65/1.56 => ! [Xx: $i,Xy: $i] :
% 2.65/1.56 ( ( Xx = Xy )
% 2.65/1.56 => ( ( in @ Xx @ A )
% 2.65/1.56 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(exuE3u_type,type,
% 2.65/1.56 exuE3u: $o ).
% 2.65/1.56
% 2.65/1.56 thf(exuE3u,definition,
% 2.65/1.56 ( exuE3u
% 2.65/1.56 = ( ! [Xphi: $i > $o] :
% 2.65/1.56 ( ( exu
% 2.65/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.56 => ! [Xx: $i,Xy: $i] :
% 2.65/1.56 ( ( Xphi @ Xx )
% 2.65/1.56 => ( ( Xphi @ Xy )
% 2.65/1.56 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(exu__Cong_type,type,
% 2.65/1.56 exu__Cong: $o ).
% 2.65/1.56
% 2.65/1.56 thf(exu__Cong,definition,
% 2.65/1.56 ( exu__Cong
% 2.65/1.56 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.65/1.56 ( ! [Xx: $i,Xy: $i] :
% 2.65/1.56 ( ( Xx = Xy )
% 2.65/1.56 => ( ( Xphi @ Xx )
% 2.65/1.56 <=> ( Xpsi @ Xy ) ) )
% 2.65/1.56 => ( ( exu
% 2.65/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.56 <=> ( exu
% 2.65/1.56 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(emptyset__Cong_type,type,
% 2.65/1.56 emptyset__Cong: $o ).
% 2.65/1.56
% 2.65/1.56 thf(emptyset__Cong,definition,
% 2.65/1.56 ( emptyset__Cong
% 2.65/1.56 = ( emptyset = emptyset ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(setadjoin__Cong_type,type,
% 2.65/1.56 setadjoin__Cong: $o ).
% 2.65/1.56
% 2.65/1.56 thf(setadjoin__Cong,definition,
% 2.65/1.56 ( setadjoin__Cong
% 2.65/1.56 = ( ! [Xx: $i,Xy: $i] :
% 2.65/1.56 ( ( Xx = Xy )
% 2.65/1.56 => ! [Xz: $i,Xu: $i] :
% 2.65/1.56 ( ( Xz = Xu )
% 2.65/1.56 => ( ( setadjoin @ Xx @ Xz )
% 2.65/1.56 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(powerset__Cong_type,type,
% 2.65/1.56 powerset__Cong: $o ).
% 2.65/1.56
% 2.65/1.56 thf(powerset__Cong,definition,
% 2.65/1.56 ( powerset__Cong
% 2.65/1.56 = ( ! [A: $i,B: $i] :
% 2.65/1.56 ( ( A = B )
% 2.65/1.56 => ( ( powerset @ A )
% 2.65/1.56 = ( powerset @ B ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(setunion__Cong_type,type,
% 2.65/1.56 setunion__Cong: $o ).
% 2.65/1.56
% 2.65/1.56 thf(setunion__Cong,definition,
% 2.65/1.56 ( setunion__Cong
% 2.65/1.56 = ( ! [A: $i,B: $i] :
% 2.65/1.56 ( ( A = B )
% 2.65/1.56 => ( ( setunion @ A )
% 2.65/1.56 = ( setunion @ B ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(omega__Cong_type,type,
% 2.65/1.56 omega__Cong: $o ).
% 2.65/1.56
% 2.65/1.56 thf(omega__Cong,definition,
% 2.65/1.56 ( omega__Cong
% 2.65/1.56 = ( omega = omega ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(exuEu_type,type,
% 2.65/1.56 exuEu: $o ).
% 2.65/1.56
% 2.65/1.56 thf(exuEu,definition,
% 2.65/1.56 ( exuEu
% 2.65/1.56 = ( ! [Xphi: $i > $o] :
% 2.65/1.56 ( ( exu
% 2.65/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.56 => ! [Xx: $i,Xy: $i] :
% 2.65/1.56 ( ( Xphi @ Xx )
% 2.65/1.56 => ( ( Xphi @ Xy )
% 2.65/1.56 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(descr__Cong_type,type,
% 2.65/1.56 descr__Cong: $o ).
% 2.65/1.56
% 2.65/1.56 thf(descr__Cong,definition,
% 2.65/1.56 ( descr__Cong
% 2.65/1.56 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.65/1.56 ( ! [Xx: $i,Xy: $i] :
% 2.65/1.56 ( ( Xx = Xy )
% 2.65/1.56 => ( ( Xphi @ Xx )
% 2.65/1.56 <=> ( Xpsi @ Xy ) ) )
% 2.65/1.56 => ( ( exu
% 2.65/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.56 => ( ( exu
% 2.65/1.56 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.65/1.56 => ( ( descr
% 2.65/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.56 = ( descr
% 2.65/1.56 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(dsetconstr__Cong_type,type,
% 2.65/1.56 dsetconstr__Cong: $o ).
% 2.65/1.56
% 2.65/1.56 thf(dsetconstr__Cong,definition,
% 2.65/1.56 ( dsetconstr__Cong
% 2.65/1.56 = ( ! [A: $i,B: $i] :
% 2.65/1.56 ( ( A = B )
% 2.65/1.56 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.65/1.56 ( ! [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 => ! [Xy: $i] :
% 2.65/1.56 ( ( in @ Xy @ B )
% 2.65/1.56 => ( ( Xx = Xy )
% 2.65/1.56 => ( ( Xphi @ Xx )
% 2.65/1.56 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.65/1.56 => ( ( dsetconstr @ A
% 2.65/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.56 = ( dsetconstr @ B
% 2.65/1.56 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(subset_type,type,
% 2.65/1.56 subset: $i > $i > $o ).
% 2.65/1.56
% 2.65/1.56 thf(disjoint_type,type,
% 2.65/1.56 disjoint: $i > $i > $o ).
% 2.65/1.56
% 2.65/1.56 thf(setsmeet_type,type,
% 2.65/1.56 setsmeet: $i > $i > $o ).
% 2.65/1.56
% 2.65/1.56 thf(subsetI1_type,type,
% 2.65/1.56 subsetI1: $o ).
% 2.65/1.56
% 2.65/1.56 thf(subsetI1,definition,
% 2.65/1.56 ( subsetI1
% 2.65/1.56 = ( ! [A: $i,B: $i] :
% 2.65/1.56 ( ! [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 => ( in @ Xx @ B ) )
% 2.65/1.56 => ( subset @ A @ B ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(eqimpsubset2_type,type,
% 2.65/1.56 eqimpsubset2: $o ).
% 2.65/1.56
% 2.65/1.56 thf(eqimpsubset2,definition,
% 2.65/1.56 ( eqimpsubset2
% 2.65/1.56 = ( ! [A: $i,B: $i] :
% 2.65/1.56 ( ( A = B )
% 2.65/1.56 => ( subset @ B @ A ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(eqimpsubset1_type,type,
% 2.65/1.56 eqimpsubset1: $o ).
% 2.65/1.56
% 2.65/1.56 thf(eqimpsubset1,definition,
% 2.65/1.56 ( eqimpsubset1
% 2.65/1.56 = ( ! [A: $i,B: $i] :
% 2.65/1.56 ( ( A = B )
% 2.65/1.56 => ( subset @ A @ B ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(subsetI2_type,type,
% 2.65/1.56 subsetI2: $o ).
% 2.65/1.56
% 2.65/1.56 thf(subsetI2,definition,
% 2.65/1.56 ( subsetI2
% 2.65/1.56 = ( ! [A: $i,B: $i] :
% 2.65/1.56 ( ! [Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 => ( in @ Xx @ B ) )
% 2.65/1.56 => ( subset @ A @ B ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(emptysetsubset_type,type,
% 2.65/1.56 emptysetsubset: $o ).
% 2.65/1.56
% 2.65/1.56 thf(emptysetsubset,definition,
% 2.65/1.56 ( emptysetsubset
% 2.65/1.56 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(subsetE_type,type,
% 2.65/1.56 subsetE: $o ).
% 2.65/1.56
% 2.65/1.56 thf(subsetE,definition,
% 2.65/1.56 ( subsetE
% 2.65/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.56 ( ( subset @ A @ B )
% 2.65/1.56 => ( ( in @ Xx @ A )
% 2.65/1.56 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(subsetE2_type,type,
% 2.65/1.56 subsetE2: $o ).
% 2.65/1.56
% 2.65/1.56 thf(subsetE2,definition,
% 2.65/1.56 ( subsetE2
% 2.65/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.56 ( ( subset @ A @ B )
% 2.65/1.56 => ( ~ ( in @ Xx @ B )
% 2.65/1.56 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(notsubsetI_type,type,
% 2.65/1.56 notsubsetI: $o ).
% 2.65/1.56
% 2.65/1.56 thf(notsubsetI,definition,
% 2.65/1.56 ( notsubsetI
% 2.65/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 => ( ~ ( in @ Xx @ B )
% 2.65/1.56 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(notequalI1_type,type,
% 2.65/1.56 notequalI1: $o ).
% 2.65/1.56
% 2.65/1.56 thf(notequalI1,definition,
% 2.65/1.56 ( notequalI1
% 2.65/1.56 = ( ! [A: $i,B: $i] :
% 2.65/1.56 ( ~ ( subset @ A @ B )
% 2.65/1.56 => ( A != B ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(notequalI2_type,type,
% 2.65/1.56 notequalI2: $o ).
% 2.65/1.56
% 2.65/1.56 thf(notequalI2,definition,
% 2.65/1.56 ( notequalI2
% 2.65/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.56 ( ( in @ Xx @ A )
% 2.65/1.56 => ( ~ ( in @ Xx @ B )
% 2.65/1.56 => ( A != B ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(subsetRefl_type,type,
% 2.65/1.56 subsetRefl: $o ).
% 2.65/1.56
% 2.65/1.56 thf(subsetRefl,definition,
% 2.65/1.56 ( subsetRefl
% 2.65/1.56 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(subsetTrans_type,type,
% 2.65/1.56 subsetTrans: $o ).
% 2.65/1.56
% 2.65/1.56 thf(subsetTrans,definition,
% 2.65/1.56 ( subsetTrans
% 2.65/1.56 = ( ! [A: $i,B: $i,C: $i] :
% 2.65/1.56 ( ( subset @ A @ B )
% 2.65/1.56 => ( ( subset @ B @ C )
% 2.65/1.56 => ( subset @ A @ C ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(setadjoinSub_type,type,
% 2.65/1.56 setadjoinSub: $o ).
% 2.65/1.56
% 2.65/1.56 thf(setadjoinSub,definition,
% 2.65/1.56 ( setadjoinSub
% 2.65/1.56 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(setadjoinSub2_type,type,
% 2.65/1.56 setadjoinSub2: $o ).
% 2.65/1.56
% 2.65/1.56 thf(setadjoinSub2,definition,
% 2.65/1.56 ( setadjoinSub2
% 2.65/1.56 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.65/1.56 ( ( subset @ A @ B )
% 2.65/1.56 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(subset2powerset_type,type,
% 2.65/1.56 subset2powerset: $o ).
% 2.65/1.56
% 2.65/1.56 thf(subset2powerset,definition,
% 2.65/1.56 ( subset2powerset
% 2.65/1.56 = ( ! [A: $i,B: $i] :
% 2.65/1.56 ( ( subset @ A @ B )
% 2.65/1.56 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.65/1.56
% 2.65/1.56 thf(setextsub_type,type,
% 2.65/1.56 setextsub: $o ).
% 2.65/1.56
% 2.65/1.56 thf(setextsub,definition,
% 2.65/1.56 ( setextsub
% 2.65/1.56 = ( ! [A: $i,B: $i] :
% 2.65/1.56 ( ( subset @ A @ B )
% 2.65/1.57 => ( ( subset @ B @ A )
% 2.65/1.57 => ( A = B ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(subsetemptysetimpeq_type,type,
% 2.65/1.57 subsetemptysetimpeq: $o ).
% 2.65/1.57
% 2.65/1.57 thf(subsetemptysetimpeq,definition,
% 2.65/1.57 ( subsetemptysetimpeq
% 2.65/1.57 = ( ! [A: $i] :
% 2.65/1.57 ( ( subset @ A @ emptyset )
% 2.65/1.57 => ( A = emptyset ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(powersetI1_type,type,
% 2.65/1.57 powersetI1: $o ).
% 2.65/1.57
% 2.65/1.57 thf(powersetI1,definition,
% 2.65/1.57 ( powersetI1
% 2.65/1.57 = ( ! [A: $i,B: $i] :
% 2.65/1.57 ( ( subset @ B @ A )
% 2.65/1.57 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(powersetE1_type,type,
% 2.65/1.57 powersetE1: $o ).
% 2.65/1.57
% 2.65/1.57 thf(powersetE1,definition,
% 2.65/1.57 ( powersetE1
% 2.65/1.57 = ( ! [A: $i,B: $i] :
% 2.65/1.57 ( ( in @ B @ ( powerset @ A ) )
% 2.65/1.57 => ( subset @ B @ A ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(inPowerset_type,type,
% 2.65/1.57 inPowerset: $o ).
% 2.65/1.57
% 2.65/1.57 thf(inPowerset,definition,
% 2.65/1.57 ( inPowerset
% 2.65/1.57 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(powersetsubset_type,type,
% 2.65/1.57 powersetsubset: $o ).
% 2.65/1.57
% 2.65/1.57 thf(powersetsubset,definition,
% 2.65/1.57 ( powersetsubset
% 2.65/1.57 = ( ! [A: $i,B: $i] :
% 2.65/1.57 ( ( subset @ A @ B )
% 2.65/1.57 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(sepInPowerset_type,type,
% 2.65/1.57 sepInPowerset: $o ).
% 2.65/1.57
% 2.65/1.57 thf(sepInPowerset,definition,
% 2.65/1.57 ( sepInPowerset
% 2.65/1.57 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.57 ( in
% 2.65/1.57 @ ( dsetconstr @ A
% 2.65/1.57 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.57 @ ( powerset @ A ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(sepSubset_type,type,
% 2.65/1.57 sepSubset: $o ).
% 2.65/1.57
% 2.65/1.57 thf(sepSubset,definition,
% 2.65/1.57 ( sepSubset
% 2.65/1.57 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.57 ( subset
% 2.65/1.57 @ ( dsetconstr @ A
% 2.65/1.57 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.57 @ A ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(binunion_type,type,
% 2.65/1.57 binunion: $i > $i > $i ).
% 2.65/1.57
% 2.65/1.57 thf(binunionIL_type,type,
% 2.65/1.57 binunionIL: $o ).
% 2.65/1.57
% 2.65/1.57 thf(binunionIL,definition,
% 2.65/1.57 ( binunionIL
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(upairset2IR_type,type,
% 2.65/1.57 upairset2IR: $o ).
% 2.65/1.57
% 2.65/1.57 thf(upairset2IR,definition,
% 2.65/1.57 ( upairset2IR
% 2.65/1.57 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(binunionIR_type,type,
% 2.65/1.57 binunionIR: $o ).
% 2.65/1.57
% 2.65/1.57 thf(binunionIR,definition,
% 2.65/1.57 ( binunionIR
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ B )
% 2.65/1.57 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(binunionEcases_type,type,
% 2.65/1.57 binunionEcases: $o ).
% 2.65/1.57
% 2.65/1.57 thf(binunionEcases,definition,
% 2.65/1.57 ( binunionEcases
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.65/1.57 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.65/1.57 => ( ( ( in @ Xx @ A )
% 2.65/1.57 => Xphi )
% 2.65/1.57 => ( ( ( in @ Xx @ B )
% 2.65/1.57 => Xphi )
% 2.65/1.57 => Xphi ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(binunionE_type,type,
% 2.65/1.57 binunionE: $o ).
% 2.65/1.57
% 2.65/1.57 thf(binunionE,definition,
% 2.65/1.57 ( binunionE
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.65/1.57 => ( ( in @ Xx @ A )
% 2.65/1.57 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(binunionLsub_type,type,
% 2.65/1.57 binunionLsub: $o ).
% 2.65/1.57
% 2.65/1.57 thf(binunionLsub,definition,
% 2.65/1.57 ( binunionLsub
% 2.65/1.57 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(binunionRsub_type,type,
% 2.65/1.57 binunionRsub: $o ).
% 2.65/1.57
% 2.65/1.57 thf(binunionRsub,definition,
% 2.65/1.57 ( binunionRsub
% 2.65/1.57 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(binintersect_type,type,
% 2.65/1.57 binintersect: $i > $i > $i ).
% 2.65/1.57
% 2.65/1.57 thf(binintersectI_type,type,
% 2.65/1.57 binintersectI: $o ).
% 2.65/1.57
% 2.65/1.57 thf(binintersectI,definition,
% 2.65/1.57 ( binintersectI
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( ( in @ Xx @ B )
% 2.65/1.57 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(binintersectSubset5_type,type,
% 2.65/1.57 binintersectSubset5: $o ).
% 2.65/1.57
% 2.65/1.57 thf(binintersectSubset5,definition,
% 2.65/1.57 ( binintersectSubset5
% 2.65/1.57 = ( ! [A: $i,B: $i,C: $i] :
% 2.65/1.57 ( ( subset @ C @ A )
% 2.65/1.57 => ( ( subset @ C @ B )
% 2.65/1.57 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(binintersectEL_type,type,
% 2.65/1.57 binintersectEL: $o ).
% 2.65/1.57
% 2.65/1.57 thf(binintersectEL,definition,
% 2.65/1.57 ( binintersectEL
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.65/1.57 => ( in @ Xx @ A ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(binintersectLsub_type,type,
% 2.65/1.57 binintersectLsub: $o ).
% 2.65/1.57
% 2.65/1.57 thf(binintersectLsub,definition,
% 2.65/1.57 ( binintersectLsub
% 2.65/1.57 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(binintersectSubset2_type,type,
% 2.65/1.57 binintersectSubset2: $o ).
% 2.65/1.57
% 2.65/1.57 thf(binintersectSubset2,definition,
% 2.65/1.57 ( binintersectSubset2
% 2.65/1.57 = ( ! [A: $i,B: $i] :
% 2.65/1.57 ( ( subset @ A @ B )
% 2.65/1.57 => ( ( binintersect @ A @ B )
% 2.65/1.57 = A ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(binintersectSubset3_type,type,
% 2.65/1.57 binintersectSubset3: $o ).
% 2.65/1.57
% 2.65/1.57 thf(binintersectSubset3,definition,
% 2.65/1.57 ( binintersectSubset3
% 2.65/1.57 = ( ! [A: $i,B: $i] :
% 2.65/1.57 ( ( ( binintersect @ A @ B )
% 2.65/1.57 = B )
% 2.65/1.57 => ( subset @ B @ A ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(binintersectER_type,type,
% 2.65/1.57 binintersectER: $o ).
% 2.65/1.57
% 2.65/1.57 thf(binintersectER,definition,
% 2.65/1.57 ( binintersectER
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.65/1.57 => ( in @ Xx @ B ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(disjointsetsI1_type,type,
% 2.65/1.57 disjointsetsI1: $o ).
% 2.65/1.57
% 2.65/1.57 thf(disjointsetsI1,definition,
% 2.65/1.57 ( disjointsetsI1
% 2.65/1.57 = ( ! [A: $i,B: $i] :
% 2.65/1.57 ( ~ ? [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 & ( in @ Xx @ B ) )
% 2.65/1.57 => ( ( binintersect @ A @ B )
% 2.65/1.57 = emptyset ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(binintersectRsub_type,type,
% 2.65/1.57 binintersectRsub: $o ).
% 2.65/1.57
% 2.65/1.57 thf(binintersectRsub,definition,
% 2.65/1.57 ( binintersectRsub
% 2.65/1.57 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(binintersectSubset4_type,type,
% 2.65/1.57 binintersectSubset4: $o ).
% 2.65/1.57
% 2.65/1.57 thf(binintersectSubset4,definition,
% 2.65/1.57 ( binintersectSubset4
% 2.65/1.57 = ( ! [A: $i,B: $i] :
% 2.65/1.57 ( ( subset @ B @ A )
% 2.65/1.57 => ( ( binintersect @ A @ B )
% 2.65/1.57 = B ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(binintersectSubset1_type,type,
% 2.65/1.57 binintersectSubset1: $o ).
% 2.65/1.57
% 2.65/1.57 thf(binintersectSubset1,definition,
% 2.65/1.57 ( binintersectSubset1
% 2.65/1.57 = ( ! [A: $i,B: $i] :
% 2.65/1.57 ( ( ( binintersect @ A @ B )
% 2.65/1.57 = A )
% 2.65/1.57 => ( subset @ A @ B ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(bs114d_type,type,
% 2.65/1.57 bs114d: $o ).
% 2.65/1.57
% 2.65/1.57 thf(bs114d,definition,
% 2.65/1.57 ( bs114d
% 2.65/1.57 = ( ! [A: $i,B: $i,C: $i] :
% 2.65/1.57 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.65/1.57 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(regular_type,type,
% 2.65/1.57 regular: $i > $o ).
% 2.65/1.57
% 2.65/1.57 thf(setminus_type,type,
% 2.65/1.57 setminus: $i > $i > $i ).
% 2.65/1.57
% 2.65/1.57 thf(setminusI_type,type,
% 2.65/1.57 setminusI: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setminusI,definition,
% 2.65/1.57 ( setminusI
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( ~ ( in @ Xx @ B )
% 2.65/1.57 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(setminusEL_type,type,
% 2.65/1.57 setminusEL: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setminusEL,definition,
% 2.65/1.57 ( setminusEL
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.65/1.57 => ( in @ Xx @ A ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(setminusER_type,type,
% 2.65/1.57 setminusER: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setminusER,definition,
% 2.65/1.57 ( setminusER
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.65/1.57 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(setminusSubset2_type,type,
% 2.65/1.57 setminusSubset2: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setminusSubset2,definition,
% 2.65/1.57 ( setminusSubset2
% 2.65/1.57 = ( ! [A: $i,B: $i] :
% 2.65/1.57 ( ( subset @ A @ B )
% 2.65/1.57 => ( ( setminus @ A @ B )
% 2.65/1.57 = emptyset ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(setminusERneg_type,type,
% 2.65/1.57 setminusERneg: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setminusERneg,definition,
% 2.65/1.57 ( setminusERneg
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.65/1.57 => ( ( in @ Xx @ A )
% 2.65/1.57 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(setminusELneg_type,type,
% 2.65/1.57 setminusELneg: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setminusELneg,definition,
% 2.65/1.57 ( setminusELneg
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.65/1.57 => ( ~ ( in @ Xx @ B )
% 2.65/1.57 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(setminusILneg_type,type,
% 2.65/1.57 setminusILneg: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setminusILneg,definition,
% 2.65/1.57 ( setminusILneg
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ~ ( in @ Xx @ A )
% 2.65/1.57 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(setminusIRneg_type,type,
% 2.65/1.57 setminusIRneg: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setminusIRneg,definition,
% 2.65/1.57 ( setminusIRneg
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ B )
% 2.65/1.57 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(setminusLsub_type,type,
% 2.65/1.57 setminusLsub: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setminusLsub,definition,
% 2.65/1.57 ( setminusLsub
% 2.65/1.57 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(setminusSubset1_type,type,
% 2.65/1.57 setminusSubset1: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setminusSubset1,definition,
% 2.65/1.57 ( setminusSubset1
% 2.65/1.57 = ( ! [A: $i,B: $i] :
% 2.65/1.57 ( ( ( setminus @ A @ B )
% 2.65/1.57 = emptyset )
% 2.65/1.57 => ( subset @ A @ B ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(symdiff_type,type,
% 2.65/1.57 symdiff: $i > $i > $i ).
% 2.65/1.57
% 2.65/1.57 thf(symdiffE_type,type,
% 2.65/1.57 symdiffE: $o ).
% 2.65/1.57
% 2.65/1.57 thf(symdiffE,definition,
% 2.65/1.57 ( symdiffE
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.65/1.57 => ! [Xphi: $o] :
% 2.65/1.57 ( ( ( in @ Xx @ A )
% 2.65/1.57 => ( ~ ( in @ Xx @ B )
% 2.65/1.57 => Xphi ) )
% 2.65/1.57 => ( ( ~ ( in @ Xx @ A )
% 2.65/1.57 => ( ( in @ Xx @ B )
% 2.65/1.57 => Xphi ) )
% 2.65/1.57 => Xphi ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(symdiffI1_type,type,
% 2.65/1.57 symdiffI1: $o ).
% 2.65/1.57
% 2.65/1.57 thf(symdiffI1,definition,
% 2.65/1.57 ( symdiffI1
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( ~ ( in @ Xx @ B )
% 2.65/1.57 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(symdiffI2_type,type,
% 2.65/1.57 symdiffI2: $o ).
% 2.65/1.57
% 2.65/1.57 thf(symdiffI2,definition,
% 2.65/1.57 ( symdiffI2
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ~ ( in @ Xx @ A )
% 2.65/1.57 => ( ( in @ Xx @ B )
% 2.65/1.57 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(symdiffIneg1_type,type,
% 2.65/1.57 symdiffIneg1: $o ).
% 2.65/1.57
% 2.65/1.57 thf(symdiffIneg1,definition,
% 2.65/1.57 ( symdiffIneg1
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( ( in @ Xx @ B )
% 2.65/1.57 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(symdiffIneg2_type,type,
% 2.65/1.57 symdiffIneg2: $o ).
% 2.65/1.57
% 2.65/1.57 thf(symdiffIneg2,definition,
% 2.65/1.57 ( symdiffIneg2
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ~ ( in @ Xx @ A )
% 2.65/1.57 => ( ~ ( in @ Xx @ B )
% 2.65/1.57 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(iskpair_type,type,
% 2.65/1.57 iskpair: $i > $o ).
% 2.65/1.57
% 2.65/1.57 thf(secondinupair_type,type,
% 2.65/1.57 secondinupair: $o ).
% 2.65/1.57
% 2.65/1.57 thf(secondinupair,definition,
% 2.65/1.57 ( secondinupair
% 2.65/1.57 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(setukpairIL_type,type,
% 2.65/1.57 setukpairIL: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setukpairIL,definition,
% 2.65/1.57 ( setukpairIL
% 2.65/1.57 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(setukpairIR_type,type,
% 2.65/1.57 setukpairIR: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setukpairIR,definition,
% 2.65/1.57 ( setukpairIR
% 2.65/1.57 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(kpairiskpair_type,type,
% 2.65/1.57 kpairiskpair: $o ).
% 2.65/1.57
% 2.65/1.57 thf(kpairiskpair,definition,
% 2.65/1.57 ( kpairiskpair
% 2.65/1.57 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(kpair_type,type,
% 2.65/1.57 kpair: $i > $i > $i ).
% 2.65/1.57
% 2.65/1.57 thf(kpairp_type,type,
% 2.65/1.57 kpairp: $o ).
% 2.65/1.57
% 2.65/1.57 thf(kpairp,definition,
% 2.65/1.57 ( kpairp
% 2.65/1.57 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(cartprod_type,type,
% 2.65/1.57 cartprod: $i > $i > $i ).
% 2.65/1.57
% 2.65/1.57 thf(singletonsubset_type,type,
% 2.65/1.57 singletonsubset: $o ).
% 2.65/1.57
% 2.65/1.57 thf(singletonsubset,definition,
% 2.65/1.57 ( singletonsubset
% 2.65/1.57 = ( ! [A: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(singletoninpowerset_type,type,
% 2.65/1.57 singletoninpowerset: $o ).
% 2.65/1.57
% 2.65/1.57 thf(singletoninpowerset,definition,
% 2.65/1.57 ( singletoninpowerset
% 2.65/1.57 = ( ! [A: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(singletoninpowunion_type,type,
% 2.65/1.57 singletoninpowunion: $o ).
% 2.65/1.57
% 2.65/1.57 thf(singletoninpowunion,definition,
% 2.65/1.57 ( singletoninpowunion
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(upairset2E_type,type,
% 2.65/1.57 upairset2E: $o ).
% 2.65/1.57
% 2.65/1.57 thf(upairset2E,definition,
% 2.65/1.57 ( upairset2E
% 2.65/1.57 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.65/1.57 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.65/1.57 => ( ( Xz = Xx )
% 2.65/1.57 | ( Xz = Xy ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(upairsubunion_type,type,
% 2.65/1.57 upairsubunion: $o ).
% 2.65/1.57
% 2.65/1.57 thf(upairsubunion,definition,
% 2.65/1.57 ( upairsubunion
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ B )
% 2.65/1.57 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(upairinpowunion_type,type,
% 2.65/1.57 upairinpowunion: $o ).
% 2.65/1.57
% 2.65/1.57 thf(upairinpowunion,definition,
% 2.65/1.57 ( upairinpowunion
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ B )
% 2.65/1.57 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(ubforcartprodlem1_type,type,
% 2.65/1.57 ubforcartprodlem1: $o ).
% 2.65/1.57
% 2.65/1.57 thf(ubforcartprodlem1,definition,
% 2.65/1.57 ( ubforcartprodlem1
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ B )
% 2.65/1.57 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(ubforcartprodlem2_type,type,
% 2.65/1.57 ubforcartprodlem2: $o ).
% 2.65/1.57
% 2.65/1.57 thf(ubforcartprodlem2,definition,
% 2.65/1.57 ( ubforcartprodlem2
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ B )
% 2.65/1.57 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(ubforcartprodlem3_type,type,
% 2.65/1.57 ubforcartprodlem3: $o ).
% 2.65/1.57
% 2.65/1.57 thf(ubforcartprodlem3,definition,
% 2.65/1.57 ( ubforcartprodlem3
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ B )
% 2.65/1.57 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(cartprodpairin_type,type,
% 2.65/1.57 cartprodpairin: $o ).
% 2.65/1.57
% 2.65/1.57 thf(cartprodpairin,definition,
% 2.65/1.57 ( cartprodpairin
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ B )
% 2.65/1.57 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(cartprodmempair1_type,type,
% 2.65/1.57 cartprodmempair1: $o ).
% 2.65/1.57
% 2.65/1.57 thf(cartprodmempair1,definition,
% 2.65/1.57 ( cartprodmempair1
% 2.65/1.57 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.65/1.57 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.65/1.57 => ? [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 & ? [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ B )
% 2.65/1.57 & ( Xu
% 2.65/1.57 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(cartprodmempair_type,type,
% 2.65/1.57 cartprodmempair: $o ).
% 2.65/1.57
% 2.65/1.57 thf(cartprodmempair,definition,
% 2.65/1.57 ( cartprodmempair
% 2.65/1.57 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.65/1.57 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.65/1.57 => ( iskpair @ Xu ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(setunionE2_type,type,
% 2.65/1.57 setunionE2: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setunionE2,definition,
% 2.65/1.57 ( setunionE2
% 2.65/1.57 = ( ! [A: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ ( setunion @ A ) )
% 2.65/1.57 => ? [X: $i] :
% 2.65/1.57 ( ( in @ X @ A )
% 2.65/1.57 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(setunionsingleton1_type,type,
% 2.65/1.57 setunionsingleton1: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setunionsingleton1,definition,
% 2.65/1.57 ( setunionsingleton1
% 2.65/1.57 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(setunionsingleton2_type,type,
% 2.65/1.57 setunionsingleton2: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setunionsingleton2,definition,
% 2.65/1.57 ( setunionsingleton2
% 2.65/1.57 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(setunionsingleton_type,type,
% 2.65/1.57 setunionsingleton: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setunionsingleton,definition,
% 2.65/1.57 ( setunionsingleton
% 2.65/1.57 = ( ! [Xx: $i] :
% 2.65/1.57 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.65/1.57 = Xx ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(singleton_type,type,
% 2.65/1.57 singleton: $i > $o ).
% 2.65/1.57
% 2.65/1.57 thf(singletonprop_type,type,
% 2.65/1.57 singletonprop: $o ).
% 2.65/1.57
% 2.65/1.57 thf(singletonprop,definition,
% 2.65/1.57 ( singletonprop
% 2.65/1.57 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.57 ( ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ A )
% 2.65/1.57 => ( ( Xphi @ Xx )
% 2.65/1.57 => ( ( Xphi @ Xy )
% 2.65/1.57 => ( Xx = Xy ) ) ) ) )
% 2.65/1.57 => ( ? [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 & ( Xphi @ Xx ) )
% 2.65/1.57 => ( singleton
% 2.65/1.57 @ ( dsetconstr @ A
% 2.65/1.57 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(ex1_type,type,
% 2.65/1.57 ex1: $i > ( $i > $o ) > $o ).
% 2.65/1.57
% 2.65/1.57 thf(ex1,definition,
% 2.65/1.57 ( ex1
% 2.65/1.57 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.65/1.57 ( singleton
% 2.65/1.57 @ ( dsetconstr @ A
% 2.65/1.57 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(ex1E1_type,type,
% 2.65/1.57 ex1E1: $o ).
% 2.65/1.57
% 2.65/1.57 thf(ex1E1,definition,
% 2.65/1.57 ( ex1E1
% 2.65/1.57 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.57 ( ( ex1 @ A
% 2.65/1.57 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.57 => ? [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(ex1I_type,type,
% 2.65/1.57 ex1I: $o ).
% 2.65/1.57
% 2.65/1.57 thf(ex1I,definition,
% 2.65/1.57 ( ex1I
% 2.65/1.57 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( ( Xphi @ Xx )
% 2.65/1.57 => ( ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ A )
% 2.65/1.57 => ( ( Xphi @ Xy )
% 2.65/1.57 => ( Xy = Xx ) ) )
% 2.65/1.57 => ( ex1 @ A
% 2.65/1.57 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(ex1I2_type,type,
% 2.65/1.57 ex1I2: $o ).
% 2.65/1.57
% 2.65/1.57 thf(ex1I2,definition,
% 2.65/1.57 ( ex1I2
% 2.65/1.57 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.57 ( ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ A )
% 2.65/1.57 => ( ( Xphi @ Xx )
% 2.65/1.57 => ( ( Xphi @ Xy )
% 2.65/1.57 => ( Xx = Xy ) ) ) ) )
% 2.65/1.57 => ( ? [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 & ( Xphi @ Xx ) )
% 2.65/1.57 => ( ex1 @ A
% 2.65/1.57 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(singletonsuniq_type,type,
% 2.65/1.57 singletonsuniq: $o ).
% 2.65/1.57
% 2.65/1.57 thf(singletonsuniq,definition,
% 2.65/1.57 ( singletonsuniq
% 2.65/1.57 = ( ! [Xx: $i,Xy: $i] :
% 2.65/1.57 ( ( ( setadjoin @ Xx @ emptyset )
% 2.65/1.57 = ( setadjoin @ Xy @ emptyset ) )
% 2.65/1.57 => ( Xx = Xy ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(atmost1p_type,type,
% 2.65/1.57 atmost1p: $i > $o ).
% 2.65/1.57
% 2.65/1.57 thf(atleast2p_type,type,
% 2.65/1.57 atleast2p: $i > $o ).
% 2.65/1.57
% 2.65/1.57 thf(atmost2p_type,type,
% 2.65/1.57 atmost2p: $i > $o ).
% 2.65/1.57
% 2.65/1.57 thf(upairsetp_type,type,
% 2.65/1.57 upairsetp: $i > $o ).
% 2.65/1.57
% 2.65/1.57 thf(setukpairinjL1_type,type,
% 2.65/1.57 setukpairinjL1: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setukpairinjL1,definition,
% 2.65/1.57 ( setukpairinjL1
% 2.65/1.57 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.65/1.57 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.65/1.57 => ( Xx = Xz ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(kfstsingleton_type,type,
% 2.65/1.57 kfstsingleton: $o ).
% 2.65/1.57
% 2.65/1.57 thf(kfstsingleton,definition,
% 2.65/1.57 ( kfstsingleton
% 2.65/1.57 = ( ! [Xu: $i] :
% 2.65/1.57 ( ( iskpair @ Xu )
% 2.65/1.57 => ( singleton
% 2.65/1.57 @ ( dsetconstr @ ( setunion @ Xu )
% 2.65/1.57 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(theprop_type,type,
% 2.65/1.57 theprop: $o ).
% 2.65/1.57
% 2.65/1.57 thf(theprop,definition,
% 2.65/1.57 ( theprop
% 2.65/1.57 = ( ! [X: $i] :
% 2.65/1.57 ( ( singleton @ X )
% 2.65/1.57 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(kfst_type,type,
% 2.65/1.57 kfst: $i > $i ).
% 2.65/1.57
% 2.65/1.57 thf(kfstpairEq_type,type,
% 2.65/1.57 kfstpairEq: $o ).
% 2.65/1.57
% 2.65/1.57 thf(kfstpairEq,definition,
% 2.65/1.57 ( kfstpairEq
% 2.65/1.57 = ( ! [Xx: $i,Xy: $i] :
% 2.65/1.57 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.65/1.57 = Xx ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(cartprodfstin_type,type,
% 2.65/1.57 cartprodfstin: $o ).
% 2.65/1.57
% 2.65/1.57 thf(cartprodfstin,definition,
% 2.65/1.57 ( cartprodfstin
% 2.65/1.57 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.65/1.57 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.65/1.57 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(setukpairinjL2_type,type,
% 2.65/1.57 setukpairinjL2: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setukpairinjL2,definition,
% 2.65/1.57 ( setukpairinjL2
% 2.65/1.57 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.65/1.57 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.65/1.57 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.65/1.57 => ( Xx = Xz ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(setukpairinjL_type,type,
% 2.65/1.57 setukpairinjL: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setukpairinjL,definition,
% 2.65/1.57 ( setukpairinjL
% 2.65/1.57 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.65/1.57 ( ( ( kpair @ Xx @ Xy )
% 2.65/1.57 = ( kpair @ Xz @ Xu ) )
% 2.65/1.57 => ( Xx = Xz ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(setukpairinjR11_type,type,
% 2.65/1.57 setukpairinjR11: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setukpairinjR11,definition,
% 2.65/1.57 ( setukpairinjR11
% 2.65/1.57 = ( ! [Xx: $i,Xy: $i] :
% 2.65/1.57 ( ( Xx = Xy )
% 2.65/1.57 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.65/1.57 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(setukpairinjR12_type,type,
% 2.65/1.57 setukpairinjR12: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setukpairinjR12,definition,
% 2.65/1.57 ( setukpairinjR12
% 2.65/1.57 = ( ! [Xx: $i,Xy: $i] :
% 2.65/1.57 ( ( Xx = Xy )
% 2.65/1.57 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.65/1.57 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(setukpairinjR1_type,type,
% 2.65/1.57 setukpairinjR1: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setukpairinjR1,definition,
% 2.65/1.57 ( setukpairinjR1
% 2.65/1.57 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.65/1.57 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.65/1.57 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.65/1.57 => ( ( Xz = Xu )
% 2.65/1.57 => ( Xy = Xu ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(upairequniteq_type,type,
% 2.65/1.57 upairequniteq: $o ).
% 2.65/1.57
% 2.65/1.57 thf(upairequniteq,definition,
% 2.65/1.57 ( upairequniteq
% 2.65/1.57 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.65/1.57 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.65/1.57 = ( setadjoin @ Xz @ emptyset ) )
% 2.65/1.57 => ( Xx = Xy ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(setukpairinjR2_type,type,
% 2.65/1.57 setukpairinjR2: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setukpairinjR2,definition,
% 2.65/1.57 ( setukpairinjR2
% 2.65/1.57 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.65/1.57 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.65/1.57 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.65/1.57 => ( Xy = Xu ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(setukpairinjR_type,type,
% 2.65/1.57 setukpairinjR: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setukpairinjR,definition,
% 2.65/1.57 ( setukpairinjR
% 2.65/1.57 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.65/1.57 ( ( ( kpair @ Xx @ Xy )
% 2.65/1.57 = ( kpair @ Xz @ Xu ) )
% 2.65/1.57 => ( Xy = Xu ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(ksndsingleton_type,type,
% 2.65/1.57 ksndsingleton: $o ).
% 2.65/1.57
% 2.65/1.57 thf(ksndsingleton,definition,
% 2.65/1.57 ( ksndsingleton
% 2.65/1.57 = ( ! [Xu: $i] :
% 2.65/1.57 ( ( iskpair @ Xu )
% 2.65/1.57 => ( singleton
% 2.65/1.57 @ ( dsetconstr @ ( setunion @ Xu )
% 2.65/1.57 @ ^ [Xx: $i] :
% 2.65/1.57 ( Xu
% 2.65/1.57 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(ksnd_type,type,
% 2.65/1.57 ksnd: $i > $i ).
% 2.65/1.57
% 2.65/1.57 thf(ksndpairEq_type,type,
% 2.65/1.57 ksndpairEq: $o ).
% 2.65/1.57
% 2.65/1.57 thf(ksndpairEq,definition,
% 2.65/1.57 ( ksndpairEq
% 2.65/1.57 = ( ! [Xx: $i,Xy: $i] :
% 2.65/1.57 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.65/1.57 = Xy ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(kpairsurjEq_type,type,
% 2.65/1.57 kpairsurjEq: $o ).
% 2.65/1.57
% 2.65/1.57 thf(kpairsurjEq,definition,
% 2.65/1.57 ( kpairsurjEq
% 2.65/1.57 = ( ! [Xu: $i] :
% 2.65/1.57 ( ( iskpair @ Xu )
% 2.65/1.57 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.65/1.57 = Xu ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(cartprodsndin_type,type,
% 2.65/1.57 cartprodsndin: $o ).
% 2.65/1.57
% 2.65/1.57 thf(cartprodsndin,definition,
% 2.65/1.57 ( cartprodsndin
% 2.65/1.57 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.65/1.57 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.65/1.57 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(cartprodpairmemEL_type,type,
% 2.65/1.57 cartprodpairmemEL: $o ).
% 2.65/1.57
% 2.65/1.57 thf(cartprodpairmemEL,definition,
% 2.65/1.57 ( cartprodpairmemEL
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.65/1.57 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.65/1.57 => ( in @ Xx @ A ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(cartprodpairmemER_type,type,
% 2.65/1.57 cartprodpairmemER: $o ).
% 2.65/1.57
% 2.65/1.57 thf(cartprodpairmemER,definition,
% 2.65/1.57 ( cartprodpairmemER
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.65/1.57 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.65/1.57 => ( in @ Xy @ B ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(cartprodmempaircEq_type,type,
% 2.65/1.57 cartprodmempaircEq: $o ).
% 2.65/1.57
% 2.65/1.57 thf(cartprodmempaircEq,definition,
% 2.65/1.57 ( cartprodmempaircEq
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ B )
% 2.65/1.57 => ( ( kpair @ Xx @ Xy )
% 2.65/1.57 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(cartprodfstpairEq_type,type,
% 2.65/1.57 cartprodfstpairEq: $o ).
% 2.65/1.57
% 2.65/1.57 thf(cartprodfstpairEq,definition,
% 2.65/1.57 ( cartprodfstpairEq
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ B )
% 2.65/1.57 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.65/1.57 = Xx ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(cartprodsndpairEq_type,type,
% 2.65/1.57 cartprodsndpairEq: $o ).
% 2.65/1.57
% 2.65/1.57 thf(cartprodsndpairEq,definition,
% 2.65/1.57 ( cartprodsndpairEq
% 2.65/1.57 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ B )
% 2.65/1.57 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.65/1.57 = Xy ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(cartprodpairsurjEq_type,type,
% 2.65/1.57 cartprodpairsurjEq: $o ).
% 2.65/1.57
% 2.65/1.57 thf(cartprodpairsurjEq,definition,
% 2.65/1.57 ( cartprodpairsurjEq
% 2.65/1.57 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.65/1.57 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.65/1.57 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.65/1.57 = Xu ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(breln_type,type,
% 2.65/1.57 breln: $i > $i > $i > $o ).
% 2.65/1.57
% 2.65/1.57 thf(breln,definition,
% 2.65/1.57 ( breln
% 2.65/1.57 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(dpsetconstr_type,type,
% 2.65/1.57 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.65/1.57
% 2.65/1.57 thf(dpsetconstrI_type,type,
% 2.65/1.57 dpsetconstrI: $o ).
% 2.65/1.57
% 2.65/1.57 thf(dpsetconstrI,definition,
% 2.65/1.57 ( dpsetconstrI
% 2.65/1.57 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ B )
% 2.65/1.57 => ( ( Xphi @ Xx @ Xy )
% 2.65/1.57 => ( in @ ( kpair @ Xx @ Xy )
% 2.65/1.57 @ ( dpsetconstr @ A @ B
% 2.65/1.57 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(dpsetconstrSub_type,type,
% 2.65/1.57 dpsetconstrSub: $o ).
% 2.65/1.57
% 2.65/1.57 thf(dpsetconstrSub,definition,
% 2.65/1.57 ( dpsetconstrSub
% 2.65/1.57 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.65/1.57 ( subset
% 2.65/1.57 @ ( dpsetconstr @ A @ B
% 2.65/1.57 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.65/1.57 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(setOfPairsIsBReln_type,type,
% 2.65/1.57 setOfPairsIsBReln: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setOfPairsIsBReln,definition,
% 2.65/1.57 ( setOfPairsIsBReln
% 2.65/1.57 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.65/1.57 ( breln @ A @ B
% 2.65/1.57 @ ( dpsetconstr @ A @ B
% 2.65/1.57 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(dpsetconstrERa_type,type,
% 2.65/1.57 dpsetconstrERa: $o ).
% 2.65/1.57
% 2.65/1.57 thf(dpsetconstrERa,definition,
% 2.65/1.57 ( dpsetconstrERa
% 2.65/1.57 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ B )
% 2.65/1.57 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.65/1.57 @ ( dpsetconstr @ A @ B
% 2.65/1.57 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.65/1.57 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(dpsetconstrEL1_type,type,
% 2.65/1.57 dpsetconstrEL1: $o ).
% 2.65/1.57
% 2.65/1.57 thf(dpsetconstrEL1,definition,
% 2.65/1.57 ( dpsetconstrEL1
% 2.65/1.57 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.65/1.57 ( ( in @ ( kpair @ Xx @ Xy )
% 2.65/1.57 @ ( dpsetconstr @ A @ B
% 2.65/1.57 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.65/1.57 => ( in @ Xx @ A ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(dpsetconstrEL2_type,type,
% 2.65/1.57 dpsetconstrEL2: $o ).
% 2.65/1.57
% 2.65/1.57 thf(dpsetconstrEL2,definition,
% 2.65/1.57 ( dpsetconstrEL2
% 2.65/1.57 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.65/1.57 ( ( in @ ( kpair @ Xx @ Xy )
% 2.65/1.57 @ ( dpsetconstr @ A @ B
% 2.65/1.57 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.65/1.57 => ( in @ Xy @ B ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(dpsetconstrER_type,type,
% 2.65/1.57 dpsetconstrER: $o ).
% 2.65/1.57
% 2.65/1.57 thf(dpsetconstrER,definition,
% 2.65/1.57 ( dpsetconstrER
% 2.65/1.57 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.65/1.57 ( ( in @ ( kpair @ Xx @ Xy )
% 2.65/1.57 @ ( dpsetconstr @ A @ B
% 2.65/1.57 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.65/1.57 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(func_type,type,
% 2.65/1.57 func: $i > $i > $i > $o ).
% 2.65/1.57
% 2.65/1.57 thf(funcSet_type,type,
% 2.65/1.57 funcSet: $i > $i > $i ).
% 2.65/1.57
% 2.65/1.57 thf(funcImageSingleton_type,type,
% 2.65/1.57 funcImageSingleton: $o ).
% 2.65/1.57
% 2.65/1.57 thf(funcImageSingleton,definition,
% 2.65/1.57 ( funcImageSingleton
% 2.65/1.57 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.57 ( ( func @ A @ B @ Xf )
% 2.65/1.57 => ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( singleton
% 2.65/1.57 @ ( dsetconstr @ B
% 2.65/1.57 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(apProp_type,type,
% 2.65/1.57 apProp: $o ).
% 2.65/1.57
% 2.65/1.57 thf(apProp,definition,
% 2.65/1.57 ( apProp
% 2.65/1.57 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.57 ( ( func @ A @ B @ Xf )
% 2.65/1.57 => ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( in
% 2.65/1.57 @ ( setunion
% 2.65/1.57 @ ( dsetconstr @ B
% 2.65/1.57 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.65/1.57 @ B ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(ap_type,type,
% 2.65/1.57 ap: $i > $i > $i > $i > $i ).
% 2.65/1.57
% 2.65/1.57 thf(app_type,type,
% 2.65/1.57 app: $o ).
% 2.65/1.57
% 2.65/1.57 thf(app,definition,
% 2.65/1.57 ( app
% 2.65/1.57 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.57 ( ( func @ A @ B @ Xf )
% 2.65/1.57 => ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(infuncsetfunc_type,type,
% 2.65/1.57 infuncsetfunc: $o ).
% 2.65/1.57
% 2.65/1.57 thf(infuncsetfunc,definition,
% 2.65/1.57 ( infuncsetfunc
% 2.65/1.57 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.57 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.65/1.57 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(ap2p_type,type,
% 2.65/1.57 ap2p: $o ).
% 2.65/1.57
% 2.65/1.57 thf(ap2p,definition,
% 2.65/1.57 ( ap2p
% 2.65/1.57 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.57 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.65/1.57 => ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(funcinfuncset_type,type,
% 2.65/1.57 funcinfuncset: $o ).
% 2.65/1.57
% 2.65/1.57 thf(funcinfuncset,definition,
% 2.65/1.57 ( funcinfuncset
% 2.65/1.57 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.57 ( ( func @ A @ B @ Xf )
% 2.65/1.57 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(lamProp_type,type,
% 2.65/1.57 lamProp: $o ).
% 2.65/1.57
% 2.65/1.57 thf(lamProp,definition,
% 2.65/1.57 ( lamProp
% 2.65/1.57 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.65/1.57 ( ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.65/1.57 => ( func @ A @ B
% 2.65/1.57 @ ( dpsetconstr @ A @ B
% 2.65/1.57 @ ^ [Xx: $i,Xy: $i] :
% 2.65/1.57 ( ( Xf @ Xx )
% 2.65/1.57 = Xy ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(lam_type,type,
% 2.65/1.57 lam: $i > $i > ( $i > $i ) > $i ).
% 2.65/1.57
% 2.65/1.57 thf(lamp_type,type,
% 2.65/1.57 lamp: $o ).
% 2.65/1.57
% 2.65/1.57 thf(lamp,definition,
% 2.65/1.57 ( lamp
% 2.65/1.57 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.65/1.57 ( ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.65/1.57 => ( func @ A @ B
% 2.65/1.57 @ ( lam @ A @ B
% 2.65/1.57 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(lam2p_type,type,
% 2.65/1.57 lam2p: $o ).
% 2.65/1.57
% 2.65/1.57 thf(lam2p,definition,
% 2.65/1.57 ( lam2p
% 2.65/1.57 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.65/1.57 ( ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.65/1.57 => ( in
% 2.65/1.57 @ ( lam @ A @ B
% 2.65/1.57 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.65/1.57 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(brelnall1_type,type,
% 2.65/1.57 brelnall1: $o ).
% 2.65/1.57
% 2.65/1.57 thf(brelnall1,definition,
% 2.65/1.57 ( brelnall1
% 2.65/1.57 = ( ! [A: $i,B: $i,R: $i] :
% 2.65/1.57 ( ( breln @ A @ B @ R )
% 2.65/1.57 => ! [Xphi: $i > $o] :
% 2.65/1.57 ( ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ B )
% 2.65/1.57 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.65/1.57 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.65/1.57 => ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ R )
% 2.65/1.57 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(brelnall2_type,type,
% 2.65/1.57 brelnall2: $o ).
% 2.65/1.57
% 2.65/1.57 thf(brelnall2,definition,
% 2.65/1.57 ( brelnall2
% 2.65/1.57 = ( ! [A: $i,B: $i,R: $i] :
% 2.65/1.57 ( ( breln @ A @ B @ R )
% 2.65/1.57 => ! [Xphi: $i > $o] :
% 2.65/1.57 ( ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ B )
% 2.65/1.57 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.65/1.57 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.65/1.57 => ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ R )
% 2.65/1.57 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(ex1E2_type,type,
% 2.65/1.57 ex1E2: $o ).
% 2.65/1.57
% 2.65/1.57 thf(ex1E2,definition,
% 2.65/1.57 ( ex1E2
% 2.65/1.57 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.57 ( ( ex1 @ A
% 2.65/1.57 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.57 => ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ A )
% 2.65/1.57 => ( ( Xphi @ Xx )
% 2.65/1.57 => ( ( Xphi @ Xy )
% 2.65/1.57 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(funcGraphProp1_type,type,
% 2.65/1.57 funcGraphProp1: $o ).
% 2.65/1.57
% 2.65/1.57 thf(funcGraphProp1,definition,
% 2.65/1.57 ( funcGraphProp1
% 2.65/1.57 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.57 ( ( func @ A @ B @ Xf )
% 2.65/1.57 => ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(funcGraphProp3_type,type,
% 2.65/1.57 funcGraphProp3: $o ).
% 2.65/1.57
% 2.65/1.57 thf(funcGraphProp3,definition,
% 2.65/1.57 ( funcGraphProp3
% 2.65/1.57 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.57 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.65/1.57 => ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(funcGraphProp2_type,type,
% 2.65/1.57 funcGraphProp2: $o ).
% 2.65/1.57
% 2.65/1.57 thf(funcGraphProp2,definition,
% 2.65/1.57 ( funcGraphProp2
% 2.65/1.57 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.57 ( ( func @ A @ B @ Xf )
% 2.65/1.57 => ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ B )
% 2.65/1.57 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.65/1.57 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.65/1.57 = Xy ) ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(funcextLem_type,type,
% 2.65/1.57 funcextLem: $o ).
% 2.65/1.57
% 2.65/1.57 thf(funcextLem,definition,
% 2.65/1.57 ( funcextLem
% 2.65/1.57 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.57 ( ( func @ A @ B @ Xf )
% 2.65/1.57 => ! [Xg: $i] :
% 2.65/1.57 ( ( func @ A @ B @ Xg )
% 2.65/1.57 => ( ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.65/1.57 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.65/1.57 => ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ B )
% 2.65/1.57 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.65/1.57 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(funcGraphProp4_type,type,
% 2.65/1.57 funcGraphProp4: $o ).
% 2.65/1.57
% 2.65/1.57 thf(funcGraphProp4,definition,
% 2.65/1.57 ( funcGraphProp4
% 2.65/1.57 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.57 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.65/1.57 => ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ B )
% 2.65/1.57 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.65/1.57 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.65/1.57 = Xy ) ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(subbreln_type,type,
% 2.65/1.57 subbreln: $o ).
% 2.65/1.57
% 2.65/1.57 thf(subbreln,definition,
% 2.65/1.57 ( subbreln
% 2.65/1.57 = ( ! [A: $i,B: $i,R: $i] :
% 2.65/1.57 ( ( breln @ A @ B @ R )
% 2.65/1.57 => ! [S: $i] :
% 2.65/1.57 ( ( breln @ A @ B @ S )
% 2.65/1.57 => ( ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ B )
% 2.65/1.57 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.65/1.57 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.65/1.57 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(eqbreln_type,type,
% 2.65/1.57 eqbreln: $o ).
% 2.65/1.57
% 2.65/1.57 thf(eqbreln,definition,
% 2.65/1.57 ( eqbreln
% 2.65/1.57 = ( ! [A: $i,B: $i,R: $i] :
% 2.65/1.57 ( ( breln @ A @ B @ R )
% 2.65/1.57 => ! [S: $i] :
% 2.65/1.57 ( ( breln @ A @ B @ S )
% 2.65/1.57 => ( ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ B )
% 2.65/1.57 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.65/1.57 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.65/1.57 => ( ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ B )
% 2.65/1.57 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.65/1.57 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.65/1.57 => ( R = S ) ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(funcext_type,type,
% 2.65/1.57 funcext: $o ).
% 2.65/1.57
% 2.65/1.57 thf(funcext,definition,
% 2.65/1.57 ( funcext
% 2.65/1.57 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.57 ( ( func @ A @ B @ Xf )
% 2.65/1.57 => ! [Xg: $i] :
% 2.65/1.57 ( ( func @ A @ B @ Xg )
% 2.65/1.57 => ( ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.65/1.57 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.65/1.57 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(funcext2_type,type,
% 2.65/1.57 funcext2: $o ).
% 2.65/1.57
% 2.65/1.57 thf(funcext2,definition,
% 2.65/1.57 ( funcext2
% 2.65/1.57 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.57 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.65/1.57 => ! [Xg: $i] :
% 2.65/1.57 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.65/1.57 => ( ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.65/1.57 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.65/1.57 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(ap2apEq1_type,type,
% 2.65/1.57 ap2apEq1: $o ).
% 2.65/1.57
% 2.65/1.57 thf(ap2apEq1,definition,
% 2.65/1.57 ( ap2apEq1
% 2.65/1.57 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.57 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.65/1.57 => ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.65/1.57 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(ap2apEq2_type,type,
% 2.65/1.57 ap2apEq2: $o ).
% 2.65/1.57
% 2.65/1.57 thf(ap2apEq2,definition,
% 2.65/1.57 ( ap2apEq2
% 2.65/1.57 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.57 ( ( func @ A @ B @ Xf )
% 2.65/1.57 => ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.65/1.57 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(beta1_type,type,
% 2.65/1.57 beta1: $o ).
% 2.65/1.57
% 2.65/1.57 thf(beta1,definition,
% 2.65/1.57 ( beta1
% 2.65/1.57 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.65/1.57 ( ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.65/1.57 => ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( ( ap @ A @ B
% 2.65/1.57 @ ( lam @ A @ B
% 2.65/1.57 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.65/1.57 @ Xx )
% 2.65/1.57 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(eta1_type,type,
% 2.65/1.57 eta1: $o ).
% 2.65/1.57
% 2.65/1.57 thf(eta1,definition,
% 2.65/1.57 ( eta1
% 2.65/1.57 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.57 ( ( func @ A @ B @ Xf )
% 2.65/1.57 => ( ( lam @ A @ B
% 2.65/1.57 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.65/1.57 = Xf ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(lam2lamEq_type,type,
% 2.65/1.57 lam2lamEq: $o ).
% 2.65/1.57
% 2.65/1.57 thf(lam2lamEq,definition,
% 2.65/1.57 ( lam2lamEq
% 2.65/1.57 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.65/1.57 ( ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.65/1.57 => ( ( lam @ A @ B
% 2.65/1.57 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.65/1.57 = ( lam @ A @ B
% 2.65/1.57 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(beta2_type,type,
% 2.65/1.57 beta2: $o ).
% 2.65/1.57
% 2.65/1.57 thf(beta2,definition,
% 2.65/1.57 ( beta2
% 2.65/1.57 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.65/1.57 ( ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.65/1.57 => ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( ( ap @ A @ B
% 2.65/1.57 @ ( lam @ A @ B
% 2.65/1.57 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.65/1.57 @ Xx )
% 2.65/1.57 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(eta2_type,type,
% 2.65/1.57 eta2: $o ).
% 2.65/1.57
% 2.65/1.57 thf(eta2,definition,
% 2.65/1.57 ( eta2
% 2.65/1.57 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.57 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.65/1.57 => ( ( lam @ A @ B
% 2.65/1.57 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.65/1.57 = Xf ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(iffalseProp1_type,type,
% 2.65/1.57 iffalseProp1: $o ).
% 2.65/1.57
% 2.65/1.57 thf(iffalseProp1,definition,
% 2.65/1.57 ( iffalseProp1
% 2.65/1.57 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ A )
% 2.65/1.57 => ( ~ Xphi
% 2.65/1.57 => ( in @ Xy
% 2.65/1.57 @ ( dsetconstr @ A
% 2.65/1.57 @ ^ [Xz: $i] :
% 2.65/1.57 ( ( Xphi
% 2.65/1.57 & ( Xz = Xx ) )
% 2.65/1.57 | ( ~ Xphi
% 2.65/1.57 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(iffalseProp2_type,type,
% 2.65/1.57 iffalseProp2: $o ).
% 2.65/1.57
% 2.65/1.57 thf(iffalseProp2,definition,
% 2.65/1.57 ( iffalseProp2
% 2.65/1.57 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ A )
% 2.65/1.57 => ( ~ Xphi
% 2.65/1.57 => ( ( dsetconstr @ A
% 2.65/1.57 @ ^ [Xz: $i] :
% 2.65/1.57 ( ( Xphi
% 2.65/1.57 & ( Xz = Xx ) )
% 2.65/1.57 | ( ~ Xphi
% 2.65/1.57 & ( Xz = Xy ) ) ) )
% 2.65/1.57 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(iftrueProp1_type,type,
% 2.65/1.57 iftrueProp1: $o ).
% 2.65/1.57
% 2.65/1.57 thf(iftrueProp1,definition,
% 2.65/1.57 ( iftrueProp1
% 2.65/1.57 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ A )
% 2.65/1.57 => ( Xphi
% 2.65/1.57 => ( in @ Xx
% 2.65/1.57 @ ( dsetconstr @ A
% 2.65/1.57 @ ^ [Xz: $i] :
% 2.65/1.57 ( ( Xphi
% 2.65/1.57 & ( Xz = Xx ) )
% 2.65/1.57 | ( ~ Xphi
% 2.65/1.57 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(iftrueProp2_type,type,
% 2.65/1.57 iftrueProp2: $o ).
% 2.65/1.57
% 2.65/1.57 thf(iftrueProp2,definition,
% 2.65/1.57 ( iftrueProp2
% 2.65/1.57 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ A )
% 2.65/1.57 => ( Xphi
% 2.65/1.57 => ( ( dsetconstr @ A
% 2.65/1.57 @ ^ [Xz: $i] :
% 2.65/1.57 ( ( Xphi
% 2.65/1.57 & ( Xz = Xx ) )
% 2.65/1.57 | ( ~ Xphi
% 2.65/1.57 & ( Xz = Xy ) ) ) )
% 2.65/1.57 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(ifSingleton_type,type,
% 2.65/1.57 ifSingleton: $o ).
% 2.65/1.57
% 2.65/1.57 thf(ifSingleton,definition,
% 2.65/1.57 ( ifSingleton
% 2.65/1.57 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ A )
% 2.65/1.57 => ( singleton
% 2.65/1.57 @ ( dsetconstr @ A
% 2.65/1.57 @ ^ [Xz: $i] :
% 2.65/1.57 ( ( Xphi
% 2.65/1.57 & ( Xz = Xx ) )
% 2.65/1.57 | ( ~ Xphi
% 2.65/1.57 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(if_type,type,
% 2.65/1.57 if: $i > $o > $i > $i > $i ).
% 2.65/1.57
% 2.65/1.57 thf(ifp_type,type,
% 2.65/1.57 ifp: $o ).
% 2.65/1.57
% 2.65/1.57 thf(ifp,definition,
% 2.65/1.57 ( ifp
% 2.65/1.57 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ A )
% 2.65/1.57 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(theeq_type,type,
% 2.65/1.57 theeq: $o ).
% 2.65/1.57
% 2.65/1.57 thf(theeq,definition,
% 2.65/1.57 ( theeq
% 2.65/1.57 = ( ! [X: $i] :
% 2.65/1.57 ( ( singleton @ X )
% 2.65/1.57 => ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ X )
% 2.65/1.57 => ( ( setunion @ X )
% 2.65/1.57 = Xx ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(iftrue_type,type,
% 2.65/1.57 iftrue: $o ).
% 2.65/1.57
% 2.65/1.57 thf(iftrue,definition,
% 2.65/1.57 ( iftrue
% 2.65/1.57 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ A )
% 2.65/1.57 => ( Xphi
% 2.65/1.57 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.65/1.57 = Xx ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(iffalse_type,type,
% 2.65/1.57 iffalse: $o ).
% 2.65/1.57
% 2.65/1.57 thf(iffalse,definition,
% 2.65/1.57 ( iffalse
% 2.65/1.57 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ A )
% 2.65/1.57 => ( ~ Xphi
% 2.65/1.57 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.65/1.57 = Xy ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(iftrueorfalse_type,type,
% 2.65/1.57 iftrueorfalse: $o ).
% 2.65/1.57
% 2.65/1.57 thf(iftrueorfalse,definition,
% 2.65/1.57 ( iftrueorfalse
% 2.65/1.57 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ! [Xy: $i] :
% 2.65/1.57 ( ( in @ Xy @ A )
% 2.65/1.57 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(binintersectT_lem_type,type,
% 2.65/1.57 binintersectT_lem: $o ).
% 2.65/1.57
% 2.65/1.57 thf(binintersectT_lem,definition,
% 2.65/1.57 ( binintersectT_lem
% 2.65/1.57 = ( ! [A: $i,X: $i] :
% 2.65/1.57 ( ( in @ X @ ( powerset @ A ) )
% 2.65/1.57 => ! [Y: $i] :
% 2.65/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 2.65/1.57 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(binunionT_lem_type,type,
% 2.65/1.57 binunionT_lem: $o ).
% 2.65/1.57
% 2.65/1.57 thf(binunionT_lem,definition,
% 2.65/1.57 ( binunionT_lem
% 2.65/1.57 = ( ! [A: $i,X: $i] :
% 2.65/1.57 ( ( in @ X @ ( powerset @ A ) )
% 2.65/1.57 => ! [Y: $i] :
% 2.65/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 2.65/1.57 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(powersetT_lem_type,type,
% 2.65/1.57 powersetT_lem: $o ).
% 2.65/1.57
% 2.65/1.57 thf(powersetT_lem,definition,
% 2.65/1.57 ( powersetT_lem
% 2.65/1.57 = ( ! [A: $i,X: $i] :
% 2.65/1.57 ( ( in @ X @ ( powerset @ A ) )
% 2.65/1.57 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(setminusT_lem_type,type,
% 2.65/1.57 setminusT_lem: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setminusT_lem,definition,
% 2.65/1.57 ( setminusT_lem
% 2.65/1.57 = ( ! [A: $i,X: $i] :
% 2.65/1.57 ( ( in @ X @ ( powerset @ A ) )
% 2.65/1.57 => ! [Y: $i] :
% 2.65/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 2.65/1.57 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(complementT_lem_type,type,
% 2.65/1.57 complementT_lem: $o ).
% 2.65/1.57
% 2.65/1.57 thf(complementT_lem,definition,
% 2.65/1.57 ( complementT_lem
% 2.65/1.57 = ( ! [A: $i,X: $i] :
% 2.65/1.57 ( ( in @ X @ ( powerset @ A ) )
% 2.65/1.57 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(setextT_type,type,
% 2.65/1.57 setextT: $o ).
% 2.65/1.57
% 2.65/1.57 thf(setextT,definition,
% 2.65/1.57 ( setextT
% 2.65/1.57 = ( ! [A: $i,X: $i] :
% 2.65/1.57 ( ( in @ X @ ( powerset @ A ) )
% 2.65/1.57 => ! [Y: $i] :
% 2.65/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 2.65/1.57 => ( ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( ( in @ Xx @ X )
% 2.65/1.57 => ( in @ Xx @ Y ) ) )
% 2.65/1.57 => ( ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( ( in @ Xx @ Y )
% 2.65/1.57 => ( in @ Xx @ X ) ) )
% 2.65/1.57 => ( X = Y ) ) ) ) ) ) ) ).
% 2.65/1.57
% 2.65/1.57 thf(subsetTI,conjecture,
% 2.65/1.57 ( setextAx
% 2.65/1.57 => ( emptysetAx
% 2.65/1.57 => ( setadjoinAx
% 2.65/1.57 => ( powersetAx
% 2.65/1.57 => ( setunionAx
% 2.65/1.57 => ( omega0Ax
% 2.65/1.57 => ( omegaSAx
% 2.65/1.57 => ( omegaIndAx
% 2.65/1.57 => ( replAx
% 2.65/1.57 => ( foundationAx
% 2.65/1.57 => ( wellorderingAx
% 2.65/1.57 => ( descrp
% 2.65/1.57 => ( dsetconstrI
% 2.65/1.57 => ( dsetconstrEL
% 2.65/1.57 => ( dsetconstrER
% 2.65/1.57 => ( exuE1
% 2.65/1.57 => ( prop2setE
% 2.65/1.57 => ( emptysetE
% 2.65/1.57 => ( emptysetimpfalse
% 2.65/1.57 => ( notinemptyset
% 2.65/1.57 => ( exuE3e
% 2.65/1.57 => ( setext
% 2.65/1.57 => ( emptyI
% 2.65/1.57 => ( noeltsimpempty
% 2.65/1.57 => ( setbeta
% 2.65/1.57 => ( nonemptyE1
% 2.65/1.57 => ( nonemptyI
% 2.65/1.57 => ( nonemptyI1
% 2.65/1.57 => ( setadjoinIL
% 2.65/1.57 => ( emptyinunitempty
% 2.65/1.57 => ( setadjoinIR
% 2.65/1.57 => ( setadjoinE
% 2.65/1.57 => ( setadjoinOr
% 2.65/1.57 => ( setoftrueEq
% 2.65/1.57 => ( powersetI
% 2.65/1.57 => ( emptyinPowerset
% 2.65/1.57 => ( emptyInPowerset
% 2.65/1.57 => ( powersetE
% 2.65/1.57 => ( setunionI
% 2.65/1.57 => ( setunionE
% 2.65/1.57 => ( subPowSU
% 2.65/1.57 => ( exuE2
% 2.65/1.57 => ( nonemptyImpWitness
% 2.65/1.57 => ( uniqinunit
% 2.65/1.57 => ( notinsingleton
% 2.65/1.57 => ( eqinunit
% 2.65/1.57 => ( singletonsswitch
% 2.65/1.57 => ( upairsetE
% 2.65/1.57 => ( upairsetIL
% 2.65/1.57 => ( upairsetIR
% 2.65/1.57 => ( emptyE1
% 2.65/1.57 => ( vacuousDall
% 2.65/1.57 => ( quantDeMorgan1
% 2.65/1.57 => ( quantDeMorgan2
% 2.65/1.57 => ( quantDeMorgan3
% 2.65/1.57 => ( quantDeMorgan4
% 2.65/1.57 => ( prop2setI
% 2.65/1.57 => ( prop2set2propI
% 2.65/1.57 => ( notdexE
% 2.65/1.57 => ( notdallE
% 2.65/1.57 => ( exuI1
% 2.65/1.57 => ( exuI3
% 2.65/1.57 => ( exuI2
% 2.65/1.57 => ( inCongP
% 2.65/1.57 => ( in__Cong
% 2.65/1.57 => ( exuE3u
% 2.65/1.57 => ( exu__Cong
% 2.65/1.57 => ( emptyset__Cong
% 2.65/1.57 => ( setadjoin__Cong
% 2.65/1.57 => ( powerset__Cong
% 2.65/1.57 => ( setunion__Cong
% 2.65/1.57 => ( omega__Cong
% 2.65/1.57 => ( exuEu
% 2.65/1.57 => ( descr__Cong
% 2.65/1.57 => ( dsetconstr__Cong
% 2.65/1.57 => ( subsetI1
% 2.65/1.57 => ( eqimpsubset2
% 2.65/1.57 => ( eqimpsubset1
% 2.65/1.57 => ( subsetI2
% 2.65/1.57 => ( emptysetsubset
% 2.65/1.57 => ( subsetE
% 2.65/1.57 => ( subsetE2
% 2.65/1.57 => ( notsubsetI
% 2.65/1.57 => ( notequalI1
% 2.65/1.57 => ( notequalI2
% 2.65/1.57 => ( subsetRefl
% 2.65/1.57 => ( subsetTrans
% 2.65/1.57 => ( setadjoinSub
% 2.65/1.57 => ( setadjoinSub2
% 2.65/1.57 => ( subset2powerset
% 2.65/1.57 => ( setextsub
% 2.65/1.57 => ( subsetemptysetimpeq
% 2.65/1.57 => ( powersetI1
% 2.65/1.57 => ( powersetE1
% 2.65/1.57 => ( inPowerset
% 2.65/1.57 => ( powersetsubset
% 2.65/1.57 => ( sepInPowerset
% 2.65/1.57 => ( sepSubset
% 2.65/1.57 => ( binunionIL
% 2.65/1.57 => ( upairset2IR
% 2.65/1.57 => ( binunionIR
% 2.65/1.57 => ( binunionEcases
% 2.65/1.57 => ( binunionE
% 2.65/1.57 => ( binunionLsub
% 2.65/1.57 => ( binunionRsub
% 2.65/1.57 => ( binintersectI
% 2.65/1.57 => ( binintersectSubset5
% 2.65/1.57 => ( binintersectEL
% 2.65/1.57 => ( binintersectLsub
% 2.65/1.57 => ( binintersectSubset2
% 2.65/1.57 => ( binintersectSubset3
% 2.65/1.57 => ( binintersectER
% 2.65/1.57 => ( disjointsetsI1
% 2.65/1.57 => ( binintersectRsub
% 2.65/1.57 => ( binintersectSubset4
% 2.65/1.57 => ( binintersectSubset1
% 2.65/1.57 => ( bs114d
% 2.65/1.57 => ( setminusI
% 2.65/1.57 => ( setminusEL
% 2.65/1.57 => ( setminusER
% 2.65/1.57 => ( setminusSubset2
% 2.65/1.57 => ( setminusERneg
% 2.65/1.57 => ( setminusELneg
% 2.65/1.57 => ( setminusILneg
% 2.65/1.57 => ( setminusIRneg
% 2.65/1.57 => ( setminusLsub
% 2.65/1.57 => ( setminusSubset1
% 2.65/1.57 => ( symdiffE
% 2.65/1.57 => ( symdiffI1
% 2.65/1.57 => ( symdiffI2
% 2.65/1.57 => ( symdiffIneg1
% 2.65/1.57 => ( symdiffIneg2
% 2.65/1.57 => ( secondinupair
% 2.65/1.57 => ( setukpairIL
% 2.65/1.57 => ( setukpairIR
% 2.65/1.57 => ( kpairiskpair
% 2.65/1.57 => ( kpairp
% 2.65/1.57 => ( singletonsubset
% 2.65/1.57 => ( singletoninpowerset
% 2.65/1.57 => ( singletoninpowunion
% 2.65/1.57 => ( upairset2E
% 2.65/1.57 => ( upairsubunion
% 2.65/1.57 => ( upairinpowunion
% 2.65/1.57 => ( ubforcartprodlem1
% 2.65/1.57 => ( ubforcartprodlem2
% 2.65/1.57 => ( ubforcartprodlem3
% 2.65/1.57 => ( cartprodpairin
% 2.65/1.57 => ( cartprodmempair1
% 2.65/1.57 => ( cartprodmempair
% 2.65/1.57 => ( setunionE2
% 2.65/1.57 => ( setunionsingleton1
% 2.65/1.57 => ( setunionsingleton2
% 2.65/1.57 => ( setunionsingleton
% 2.65/1.57 => ( singletonprop
% 2.65/1.57 => ( ex1E1
% 2.65/1.57 => ( ex1I
% 2.65/1.57 => ( ex1I2
% 2.65/1.57 => ( singletonsuniq
% 2.65/1.57 => ( setukpairinjL1
% 2.65/1.57 => ( kfstsingleton
% 2.65/1.57 => ( theprop
% 2.65/1.57 => ( kfstpairEq
% 2.65/1.57 => ( cartprodfstin
% 2.65/1.57 => ( setukpairinjL2
% 2.65/1.57 => ( setukpairinjL
% 2.65/1.57 => ( setukpairinjR11
% 2.65/1.57 => ( setukpairinjR12
% 2.65/1.57 => ( setukpairinjR1
% 2.65/1.57 => ( upairequniteq
% 2.65/1.57 => ( setukpairinjR2
% 2.65/1.57 => ( setukpairinjR
% 2.65/1.57 => ( ksndsingleton
% 2.65/1.57 => ( ksndpairEq
% 2.65/1.57 => ( kpairsurjEq
% 2.65/1.57 => ( cartprodsndin
% 2.65/1.57 => ( cartprodpairmemEL
% 2.65/1.57 => ( cartprodpairmemER
% 2.65/1.57 => ( cartprodmempaircEq
% 2.65/1.57 => ( cartprodfstpairEq
% 2.65/1.57 => ( cartprodsndpairEq
% 2.65/1.57 => ( cartprodpairsurjEq
% 2.65/1.57 => ( dpsetconstrI
% 2.65/1.57 => ( dpsetconstrSub
% 2.65/1.57 => ( setOfPairsIsBReln
% 2.65/1.57 => ( dpsetconstrERa
% 2.65/1.57 => ( dpsetconstrEL1
% 2.65/1.57 => ( dpsetconstrEL2
% 2.65/1.57 => ( dpsetconstrER
% 2.65/1.57 => ( funcImageSingleton
% 2.65/1.57 => ( apProp
% 2.65/1.57 => ( app
% 2.65/1.57 => ( infuncsetfunc
% 2.65/1.57 => ( ap2p
% 2.65/1.57 => ( funcinfuncset
% 2.65/1.57 => ( lamProp
% 2.65/1.57 => ( lamp
% 2.65/1.57 => ( lam2p
% 2.65/1.57 => ( brelnall1
% 2.65/1.57 => ( brelnall2
% 2.65/1.57 => ( ex1E2
% 2.65/1.57 => ( funcGraphProp1
% 2.65/1.57 => ( funcGraphProp3
% 2.65/1.57 => ( funcGraphProp2
% 2.65/1.57 => ( funcextLem
% 2.65/1.57 => ( funcGraphProp4
% 2.65/1.57 => ( subbreln
% 2.65/1.57 => ( eqbreln
% 2.65/1.57 => ( funcext
% 2.65/1.57 => ( funcext2
% 2.65/1.57 => ( ap2apEq1
% 2.65/1.57 => ( ap2apEq2
% 2.65/1.57 => ( beta1
% 2.65/1.57 => ( eta1
% 2.65/1.57 => ( lam2lamEq
% 2.65/1.57 => ( beta2
% 2.65/1.57 => ( eta2
% 2.65/1.57 => ( iffalseProp1
% 2.65/1.57 => ( iffalseProp2
% 2.65/1.57 => ( iftrueProp1
% 2.65/1.57 => ( iftrueProp2
% 2.65/1.57 => ( ifSingleton
% 2.65/1.57 => ( ifp
% 2.65/1.57 => ( theeq
% 2.65/1.57 => ( iftrue
% 2.65/1.57 => ( iffalse
% 2.65/1.57 => ( iftrueorfalse
% 2.65/1.57 => ( binintersectT_lem
% 2.65/1.57 => ( binunionT_lem
% 2.65/1.57 => ( powersetT_lem
% 2.65/1.57 => ( setminusT_lem
% 2.65/1.57 => ( complementT_lem
% 2.65/1.57 => ( setextT
% 2.65/1.57 => ! [A: $i,X: $i] :
% 2.65/1.57 ( ( in @ X @ ( powerset @ A ) )
% 2.65/1.57 => ! [Y: $i] :
% 2.65/1.57 ( ( in @ Y @ ( powerset @ A ) )
% 2.65/1.57 => ( ! [Xx: $i] :
% 2.65/1.57 ( ( in @ Xx @ A )
% 2.65/1.57 => ( ( in @ Xx @ X )
% 2.65/1.57 => ( in @ Xx @ Y ) ) )
% 2.65/1.57 => ( subset @ X @ Y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.65/1.57 ------------------------
% 2.65/1.59 % (12299)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_12059 for (2999ds/27Mi)
% 2.65/1.59 % (12301)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_12059 for (2999ds/2Mi)
% 2.65/1.60 % (12301)Instruction limit reached!
% 2.65/1.60 % (12301)------------------------------
% 2.65/1.60 % (12301)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.60 % (12301)Termination reason: Unknown
% 2.65/1.60 % (12301)Termination phase: shuffling
% 2.65/1.60
% 2.65/1.60 % (12301)Memory used [KB]: 1407
% 2.65/1.60 % (12301)Time elapsed: 0.002 s
% 2.65/1.60 % (12301)Instructions burned: 3 (million)
% 2.65/1.60 % (12301)------------------------------
% 2.65/1.60 % (12301)------------------------------
% 2.73/1.60 % (12297)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_12059 for (2999ds/183Mi)
% 2.73/1.60 % (12298)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_12059 for (2999ds/4Mi)
% 2.73/1.60 % (12300)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_12059 for (2999ds/2Mi)
% 2.73/1.60 % (12302)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_12059 for (2999ds/275Mi)
% 2.73/1.60 % (12299)Instruction limit reached!
% 2.73/1.60 % (12299)------------------------------
% 2.73/1.60 % (12299)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.60 % (12299)Termination reason: Unknown
% 2.73/1.60 % (12299)Termination phase: shuffling
% 2.73/1.60
% 2.73/1.60 % (12299)Memory used [KB]: 1918
% 2.73/1.60 % (12299)Time elapsed: 0.009 s
% 2.73/1.60 % (12299)Instructions burned: 27 (million)
% 2.73/1.60 % (12299)------------------------------
% 2.73/1.60 % (12299)------------------------------
% 2.73/1.60 % (12303)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_12059 for (2999ds/18Mi)
% 2.73/1.60 % (12300)Instruction limit reached!
% 2.73/1.60 % (12300)------------------------------
% 2.73/1.60 % (12300)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.60 % (12300)Termination reason: Unknown
% 2.73/1.60 % (12300)Termination phase: shuffling
% 2.73/1.60
% 2.73/1.60 % (12300)Memory used [KB]: 1407
% 2.73/1.60 % (12300)Time elapsed: 0.004 s
% 2.73/1.60 % (12300)Instructions burned: 3 (million)
% 2.73/1.60 % (12300)------------------------------
% 2.73/1.60 % (12300)------------------------------
% 2.73/1.60 % (12298)Instruction limit reached!
% 2.73/1.60 % (12298)------------------------------
% 2.73/1.60 % (12298)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.60 % (12298)Termination reason: Unknown
% 2.73/1.60 % (12298)Termination phase: shuffling
% 2.73/1.60
% 2.73/1.60 % (12298)Memory used [KB]: 1407
% 2.73/1.60 % (12298)Time elapsed: 0.005 s
% 2.73/1.60 % (12298)Instructions burned: 5 (million)
% 2.73/1.60 % (12298)------------------------------
% 2.73/1.60 % (12298)------------------------------
% 2.73/1.61 % (12309)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_12059 for (2999ds/3Mi)
% 2.73/1.61 % (12309)Instruction limit reached!
% 2.73/1.61 % (12309)------------------------------
% 2.73/1.61 % (12309)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.61 % (12309)Termination reason: Unknown
% 2.73/1.61 % (12309)Termination phase: shuffling
% 2.73/1.61
% 2.73/1.61 % (12309)Memory used [KB]: 1407
% 2.73/1.61 % (12309)Time elapsed: 0.003 s
% 2.73/1.61 % (12309)Instructions burned: 4 (million)
% 2.73/1.61 % (12309)------------------------------
% 2.73/1.61 % (12309)------------------------------
% 2.73/1.61 % (12315)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_12059 for (2999ds/37Mi)
% 2.73/1.61 % (12303)Instruction limit reached!
% 2.73/1.61 % (12303)------------------------------
% 2.73/1.61 % (12303)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.61 % (12303)Termination reason: Unknown
% 2.73/1.61 % (12303)Termination phase: shuffling
% 2.73/1.61
% 2.73/1.61 % (12303)Memory used [KB]: 1791
% 2.73/1.61 % (12303)Time elapsed: 0.013 s
% 2.73/1.61 % (12303)Instructions burned: 19 (million)
% 2.73/1.61 % (12303)------------------------------
% 2.73/1.61 % (12303)------------------------------
% 2.73/1.62 % (12318)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_12059 for (2999ds/1041Mi)
% 2.73/1.62 % (12316)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_12059 for (2999ds/15Mi)
% 2.73/1.62 % (12317)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_12059 for (2999ds/3Mi)
% 2.73/1.62 % (12315)Instruction limit reached!
% 2.73/1.62 % (12315)------------------------------
% 2.73/1.62 % (12315)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.62 % (12315)Termination reason: Unknown
% 2.73/1.62 % (12315)Termination phase: shuffling
% 2.73/1.62
% 2.73/1.62 % (12315)Memory used [KB]: 2174
% 2.73/1.62 % (12315)Time elapsed: 0.013 s
% 2.73/1.62 % (12315)Instructions burned: 39 (million)
% 2.73/1.62 % (12315)------------------------------
% 2.73/1.62 % (12315)------------------------------
% 2.73/1.62 % (12317)Instruction limit reached!
% 2.73/1.62 % (12317)------------------------------
% 2.73/1.62 % (12317)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.62 % (12317)Termination reason: Unknown
% 2.73/1.62 % (12317)Termination phase: shuffling
% 2.73/1.62
% 2.73/1.62 % (12317)Memory used [KB]: 1407
% 2.73/1.62 % (12317)Time elapsed: 0.004 s
% 2.73/1.62 % (12317)Instructions burned: 4 (million)
% 2.73/1.62 % (12317)------------------------------
% 2.73/1.62 % (12317)------------------------------
% 2.73/1.63 % (12316)Instruction limit reached!
% 2.73/1.63 % (12316)------------------------------
% 2.73/1.63 % (12316)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.63 % (12316)Termination reason: Unknown
% 2.73/1.63 % (12316)Termination phase: shuffling
% 2.73/1.63
% 2.73/1.63 % (12316)Memory used [KB]: 1663
% 2.73/1.63 % (12316)Time elapsed: 0.011 s
% 2.73/1.63 % (12316)Instructions burned: 15 (million)
% 2.73/1.63 % (12316)------------------------------
% 2.73/1.63 % (12316)------------------------------
% 2.73/1.63 % (12319)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_12059 for (2999ds/7Mi)
% 2.73/1.63 % (12326)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_12059 for (2999ds/16Mi)
% 2.73/1.64 % (12319)Instruction limit reached!
% 2.73/1.64 % (12319)------------------------------
% 2.73/1.64 % (12319)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.64 % (12319)Termination reason: Unknown
% 2.73/1.64 % (12319)Termination phase: shuffling
% 2.73/1.64
% 2.73/1.64 % (12319)Memory used [KB]: 1535
% 2.73/1.64 % (12319)Time elapsed: 0.006 s
% 2.73/1.64 % (12319)Instructions burned: 8 (million)
% 2.73/1.64 % (12319)------------------------------
% 2.73/1.64 % (12319)------------------------------
% 2.73/1.64 % (12328)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_12059 for (2999ds/3Mi)
% 2.73/1.64 % (12328)Instruction limit reached!
% 2.73/1.64 % (12328)------------------------------
% 2.73/1.64 % (12328)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.64 % (12328)Termination reason: Unknown
% 2.73/1.64 % (12328)Termination phase: shuffling
% 2.73/1.64
% 2.73/1.64 % (12328)Memory used [KB]: 1407
% 2.73/1.64 % (12328)Time elapsed: 0.002 s
% 2.73/1.64 % (12328)Instructions burned: 3 (million)
% 2.73/1.64 % (12328)------------------------------
% 2.73/1.64 % (12328)------------------------------
% 2.73/1.64 % (12326)Instruction limit reached!
% 2.73/1.64 % (12326)------------------------------
% 2.73/1.64 % (12326)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.64 % (12326)Termination reason: Unknown
% 2.73/1.64 % (12326)Termination phase: shuffling
% 2.73/1.64
% 2.73/1.64 % (12326)Memory used [KB]: 1791
% 2.73/1.64 % (12326)Time elapsed: 0.007 s
% 2.73/1.64 % (12326)Instructions burned: 18 (million)
% 2.73/1.64 % (12326)------------------------------
% 2.73/1.64 % (12326)------------------------------
% 2.73/1.64 % (12334)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_12059 for (2999ds/3Mi)
% 2.73/1.64 % (12334)Instruction limit reached!
% 2.73/1.64 % (12334)------------------------------
% 2.73/1.64 % (12334)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.64 % (12334)Termination reason: Unknown
% 2.73/1.64 % (12334)Termination phase: shuffling
% 2.73/1.64
% 2.73/1.64 % (12334)Memory used [KB]: 1535
% 2.73/1.64 % (12334)Time elapsed: 0.003 s
% 2.73/1.64 % (12334)Instructions burned: 5 (million)
% 2.73/1.64 % (12334)------------------------------
% 2.73/1.64 % (12334)------------------------------
% 2.73/1.65 % (12341)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_12059 for (2999ds/3Mi)
% 2.73/1.65 % (12341)Instruction limit reached!
% 2.73/1.65 % (12341)------------------------------
% 2.73/1.65 % (12341)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.65 % (12341)Termination reason: Unknown
% 2.73/1.65 % (12341)Termination phase: shuffling
% 2.73/1.65
% 2.73/1.65 % (12341)Memory used [KB]: 1407
% 2.73/1.65 % (12341)Time elapsed: 0.003 s
% 2.73/1.65 % (12341)Instructions burned: 4 (million)
% 2.73/1.65 % (12341)------------------------------
% 2.73/1.65 % (12341)------------------------------
% 2.73/1.65 % (12342)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_12059 for (2999ds/4Mi)
% 2.73/1.65 % (12342)Instruction limit reached!
% 2.73/1.65 % (12342)------------------------------
% 2.73/1.65 % (12342)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.65 % (12342)Termination reason: Unknown
% 2.73/1.65 % (12342)Termination phase: shuffling
% 2.73/1.65
% 2.73/1.65 % (12342)Memory used [KB]: 1407
% 2.73/1.65 % (12342)Time elapsed: 0.003 s
% 2.73/1.65 % (12342)Instructions burned: 4 (million)
% 2.73/1.65 % (12342)------------------------------
% 2.73/1.65 % (12342)------------------------------
% 2.73/1.65 % (12339)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_12059 for (2999ds/7Mi)
% 2.73/1.65 % (12346)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_12059 for (2999ds/18Mi)
% 2.73/1.66 % (12339)Instruction limit reached!
% 2.73/1.66 % (12339)------------------------------
% 2.73/1.66 % (12339)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.66 % (12339)Termination reason: Unknown
% 2.73/1.66 % (12339)Termination phase: shuffling
% 2.73/1.66
% 2.73/1.66 % (12339)Memory used [KB]: 1535
% 2.73/1.66 % (12339)Time elapsed: 0.006 s
% 2.73/1.66 % (12339)Instructions burned: 8 (million)
% 2.73/1.66 % (12339)------------------------------
% 2.73/1.66 % (12339)------------------------------
% 2.73/1.66 % (12346)Instruction limit reached!
% 2.73/1.66 % (12346)------------------------------
% 2.73/1.66 % (12346)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.66 % (12346)Termination reason: Unknown
% 2.73/1.66 % (12346)Termination phase: shuffling
% 2.73/1.66
% 2.73/1.66 % (12346)Memory used [KB]: 1791
% 2.73/1.66 % (12346)Time elapsed: 0.006 s
% 2.73/1.66 % (12346)Instructions burned: 18 (million)
% 2.73/1.66 % (12346)------------------------------
% 2.73/1.66 % (12346)------------------------------
% 2.73/1.66 % (12350)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_12059 for (2999ds/710Mi)
% 2.73/1.66 % (12353)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_12059 for (2999ds/6Mi)
% 2.73/1.66 % (12353)Instruction limit reached!
% 2.73/1.66 % (12353)------------------------------
% 2.73/1.66 % (12353)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.66 % (12353)Termination reason: Unknown
% 2.73/1.66 % (12353)Termination phase: shuffling
% 2.73/1.66
% 2.73/1.66 % (12353)Memory used [KB]: 1535
% 2.73/1.66 % (12353)Time elapsed: 0.004 s
% 2.73/1.66 % (12353)Instructions burned: 8 (million)
% 2.73/1.66 % (12353)------------------------------
% 2.73/1.66 % (12353)------------------------------
% 2.73/1.67 % (12355)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_12059 for (2999ds/21Mi)
% 2.73/1.67 % (12297)Instruction limit reached!
% 2.73/1.67 % (12297)------------------------------
% 2.73/1.67 % (12297)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.67 % (12297)Termination reason: Unknown
% 2.73/1.67 % (12297)Termination phase: Saturation
% 2.73/1.67
% 2.73/1.67 % (12297)Memory used [KB]: 7547
% 2.73/1.67 % (12297)Time elapsed: 0.072 s
% 2.73/1.67 % (12297)Instructions burned: 183 (million)
% 2.73/1.67 % (12297)------------------------------
% 2.73/1.67 % (12297)------------------------------
% 2.73/1.67 % (12356)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_12059 for (2999ds/5Mi)
% 2.73/1.67 % (12355)Instruction limit reached!
% 2.73/1.67 % (12355)------------------------------
% 2.73/1.67 % (12355)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.67 % (12355)Termination reason: Unknown
% 2.73/1.67 % (12355)Termination phase: shuffling
% 2.73/1.67
% 2.73/1.67 % (12355)Memory used [KB]: 1791
% 2.73/1.67 % (12355)Time elapsed: 0.007 s
% 2.73/1.67 % (12355)Instructions burned: 21 (million)
% 2.73/1.67 % (12355)------------------------------
% 2.73/1.67 % (12355)------------------------------
% 2.73/1.67 % (12354)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_12059 for (2999ds/902Mi)
% 2.73/1.68 % (12356)Instruction limit reached!
% 2.73/1.68 % (12356)------------------------------
% 2.73/1.68 % (12356)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.68 % (12356)Termination reason: Unknown
% 2.73/1.68 % (12356)Termination phase: shuffling
% 2.73/1.68
% 2.73/1.68 % (12356)Memory used [KB]: 1535
% 2.73/1.68 % (12356)Time elapsed: 0.004 s
% 2.73/1.68 % (12356)Instructions burned: 8 (million)
% 2.73/1.68 % (12356)------------------------------
% 2.73/1.68 % (12356)------------------------------
% 2.73/1.68 % (12362)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_12059 for (2999ds/6Mi)
% 2.73/1.68 % (12366)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_12059 for (2999ds/377Mi)
% 2.73/1.68 % (12362)Instruction limit reached!
% 2.73/1.68 % (12362)------------------------------
% 2.73/1.68 % (12362)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.68 % (12362)Termination reason: Unknown
% 2.73/1.68 % (12362)Termination phase: shuffling
% 2.73/1.68
% 2.73/1.68 % (12362)Memory used [KB]: 1535
% 2.73/1.68 % (12362)Time elapsed: 0.003 s
% 2.73/1.68 % (12362)Instructions burned: 7 (million)
% 2.73/1.68 % (12362)------------------------------
% 2.73/1.68 % (12362)------------------------------
% 2.73/1.69 % (12369)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_12059 for (2999ds/779Mi)
% 2.73/1.69 % (12375)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_12059 for (2999ds/19Mi)
% 2.73/1.70 % (12375)Instruction limit reached!
% 2.73/1.70 % (12375)------------------------------
% 2.73/1.70 % (12375)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.70 % (12375)Termination reason: Unknown
% 2.73/1.70 % (12375)Termination phase: shuffling
% 2.73/1.70
% 2.73/1.70 % (12375)Memory used [KB]: 1791
% 2.73/1.70 % (12375)Time elapsed: 0.007 s
% 2.73/1.70 % (12375)Instructions burned: 22 (million)
% 2.73/1.70 % (12375)------------------------------
% 2.73/1.70 % (12375)------------------------------
% 2.73/1.71 % (12379)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_12059 for (2998ds/879Mi)
% 2.73/1.73 % (12302)Instruction limit reached!
% 2.73/1.73 % (12302)------------------------------
% 2.73/1.73 % (12302)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.73 % (12302)Termination reason: Unknown
% 2.73/1.73 % (12302)Termination phase: Saturation
% 2.73/1.73
% 2.73/1.73 % (12302)Memory used [KB]: 9338
% 2.73/1.73 % (12302)Time elapsed: 0.131 s
% 2.73/1.73 % (12302)Instructions burned: 277 (million)
% 2.73/1.73 % (12302)------------------------------
% 2.73/1.73 % (12302)------------------------------
% 2.73/1.74 % (12393)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_12059 for (2998ds/17Mi)
% 2.73/1.75 % (12393)Instruction limit reached!
% 2.73/1.75 % (12393)------------------------------
% 2.73/1.75 % (12393)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.75 % (12393)Termination reason: Unknown
% 2.73/1.75 % (12393)Termination phase: shuffling
% 2.73/1.75
% 2.73/1.75 % (12393)Memory used [KB]: 1791
% 2.73/1.75 % (12393)Time elapsed: 0.007 s
% 2.73/1.75 % (12393)Instructions burned: 20 (million)
% 2.73/1.75 % (12393)------------------------------
% 2.73/1.75 % (12393)------------------------------
% 2.73/1.76 % (12401)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_12059 for (2998ds/3Mi)
% 2.73/1.76 % (12401)Instruction limit reached!
% 2.73/1.76 % (12401)------------------------------
% 2.73/1.76 % (12401)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.76 % (12401)Termination reason: Unknown
% 2.73/1.76 % (12401)Termination phase: shuffling
% 2.73/1.76
% 2.73/1.76 % (12401)Memory used [KB]: 1407
% 2.73/1.76 % (12401)Time elapsed: 0.002 s
% 2.73/1.76 % (12401)Instructions burned: 3 (million)
% 2.73/1.76 % (12401)------------------------------
% 2.73/1.76 % (12401)------------------------------
% 2.73/1.77 % (12405)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_12059 for (2998ds/30Mi)
% 2.73/1.78 % (12405)Instruction limit reached!
% 2.73/1.78 % (12405)------------------------------
% 2.73/1.78 % (12405)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.78 % (12405)Termination reason: Unknown
% 2.73/1.78 % (12405)Termination phase: shuffling
% 2.73/1.78
% 2.73/1.78 % (12405)Memory used [KB]: 1918
% 2.73/1.78 % (12405)Time elapsed: 0.011 s
% 2.73/1.78 % (12405)Instructions burned: 31 (million)
% 2.73/1.78 % (12405)------------------------------
% 2.73/1.78 % (12405)------------------------------
% 2.73/1.79 % (12366)Instruction limit reached!
% 2.73/1.79 % (12366)------------------------------
% 2.73/1.79 % (12366)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.79 % (12366)Termination reason: Unknown
% 2.73/1.79 % (12366)Termination phase: Saturation
% 2.73/1.79
% 2.73/1.79 % (12366)Memory used [KB]: 8699
% 2.73/1.79 % (12366)Time elapsed: 0.105 s
% 2.73/1.79 % (12366)Instructions burned: 378 (million)
% 2.73/1.79 % (12366)------------------------------
% 2.73/1.79 % (12366)------------------------------
% 2.73/1.79 % (12414)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_12059 for (2998ds/127Mi)
% 2.73/1.80 % (12419)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_12059 for (2998ds/100Mi)
% 2.73/1.82 % (12419)Instruction limit reached!
% 2.73/1.82 % (12419)------------------------------
% 2.73/1.82 % (12419)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.82 % (12419)Termination reason: Unknown
% 2.73/1.82 % (12419)Termination phase: Property scanning
% 2.73/1.82
% 2.73/1.82 % (12419)Memory used [KB]: 2430
% 2.73/1.82 % (12419)Time elapsed: 0.028 s
% 2.73/1.82 % (12419)Instructions burned: 102 (million)
% 2.73/1.82 % (12419)------------------------------
% 2.73/1.82 % (12419)------------------------------
% 2.73/1.82 % (12414)Instruction limit reached!
% 2.73/1.82 % (12414)------------------------------
% 2.73/1.82 % (12414)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.82 % (12414)Termination reason: Unknown
% 2.73/1.82 % (12414)Termination phase: Property scanning
% 2.73/1.82
% 2.73/1.82 % (12414)Memory used [KB]: 2558
% 2.73/1.82 % (12414)Time elapsed: 0.037 s
% 2.73/1.82 % (12414)Instructions burned: 129 (million)
% 2.73/1.82 % (12414)------------------------------
% 2.73/1.82 % (12414)------------------------------
% 2.73/1.83 % (12431)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_12059 for (2997ds/3Mi)
% 2.73/1.83 % (12432)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_12059 for (2997ds/20Mi)
% 2.73/1.83 % (12431)Instruction limit reached!
% 2.73/1.83 % (12431)------------------------------
% 2.73/1.83 % (12431)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.83 % (12431)Termination reason: Unknown
% 2.73/1.83 % (12431)Termination phase: shuffling
% 2.73/1.83
% 2.73/1.83 % (12431)Memory used [KB]: 1535
% 2.73/1.83 % (12431)Time elapsed: 0.003 s
% 2.73/1.83 % (12431)Instructions burned: 6 (million)
% 2.73/1.83 % (12431)------------------------------
% 2.73/1.83 % (12431)------------------------------
% 2.73/1.84 % (12432)Instruction limit reached!
% 2.73/1.84 % (12432)------------------------------
% 2.73/1.84 % (12432)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.84 % (12432)Termination reason: Unknown
% 2.73/1.84 % (12432)Termination phase: shuffling
% 2.73/1.84
% 2.73/1.84 % (12432)Memory used [KB]: 1791
% 2.73/1.84 % (12432)Time elapsed: 0.008 s
% 2.73/1.84 % (12432)Instructions burned: 21 (million)
% 2.73/1.84 % (12432)------------------------------
% 2.73/1.84 % (12432)------------------------------
% 2.73/1.84 % (12434)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_12059 for (2997ds/86Mi)
% 2.73/1.85 % (12438)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_12059 for (2997ds/107Mi)
% 2.73/1.87 % (12434)Instruction limit reached!
% 2.73/1.87 % (12434)------------------------------
% 2.73/1.87 % (12434)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.87 % (12434)Termination reason: Unknown
% 2.73/1.87 % (12434)Termination phase: Property scanning
% 2.73/1.87
% 2.73/1.87 % (12434)Memory used [KB]: 2430
% 2.73/1.87 % (12434)Time elapsed: 0.023 s
% 2.73/1.87 % (12434)Instructions burned: 86 (million)
% 2.73/1.87 % (12434)------------------------------
% 2.73/1.87 % (12434)------------------------------
% 2.73/1.88 % (12448)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_12059 for (2997ds/39Mi)
% 2.73/1.88 % (12438)Instruction limit reached!
% 2.73/1.88 % (12438)------------------------------
% 2.73/1.88 % (12438)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.88 % (12438)Termination reason: Unknown
% 2.73/1.88 % (12438)Termination phase: Function definition elimination
% 2.73/1.88
% 2.73/1.88 % (12438)Memory used [KB]: 2430
% 2.73/1.88 % (12438)Time elapsed: 0.032 s
% 2.73/1.88 % (12438)Instructions burned: 110 (million)
% 2.73/1.88 % (12438)------------------------------
% 2.73/1.88 % (12438)------------------------------
% 2.73/1.89 % (12448)Instruction limit reached!
% 2.73/1.89 % (12448)------------------------------
% 2.73/1.89 % (12448)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.89 % (12448)Termination reason: Unknown
% 2.73/1.89 % (12448)Termination phase: shuffling
% 2.73/1.89
% 2.73/1.89 % (12448)Memory used [KB]: 2302
% 2.73/1.89 % (12448)Time elapsed: 0.012 s
% 2.73/1.89 % (12448)Instructions burned: 41 (million)
% 2.73/1.89 % (12448)------------------------------
% 2.73/1.89 % (12448)------------------------------
% 2.73/1.89 % (12350)Instruction limit reached!
% 2.73/1.89 % (12350)------------------------------
% 2.73/1.89 % (12350)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.89 % (12350)Termination reason: Unknown
% 2.73/1.89 % (12350)Termination phase: Saturation
% 2.73/1.89
% 2.73/1.89 % (12350)Memory used [KB]: 10234
% 2.73/1.89 % (12350)Time elapsed: 0.233 s
% 2.73/1.89 % (12350)Instructions burned: 712 (million)
% 2.73/1.89 % (12350)------------------------------
% 2.73/1.89 % (12350)------------------------------
% 2.73/1.89 % (12453)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_12059 for (2997ds/448Mi)
% 2.73/1.90 % (12454)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_12059 for (2997ds/46Mi)
% 2.73/1.90 % (12318)Instruction limit reached!
% 2.73/1.90 % (12318)------------------------------
% 2.73/1.90 % (12318)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.90 % (12318)Termination reason: Unknown
% 2.73/1.90 % (12318)Termination phase: Saturation
% 2.73/1.90
% 2.73/1.90 % (12318)Memory used [KB]: 13432
% 2.73/1.90 % (12318)Time elapsed: 0.281 s
% 2.73/1.90 % (12318)Instructions burned: 1041 (million)
% 2.73/1.90 % (12318)------------------------------
% 2.73/1.90 % (12318)------------------------------
% 2.73/1.90 % (12369)Instruction limit reached!
% 2.73/1.90 % (12369)------------------------------
% 2.73/1.90 % (12369)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.90 % (12369)Termination reason: Unknown
% 2.73/1.90 % (12369)Termination phase: Saturation
% 2.73/1.90
% 2.73/1.90 % (12369)Memory used [KB]: 7419
% 2.73/1.90 % (12369)Time elapsed: 0.214 s
% 2.73/1.90 % (12369)Instructions burned: 779 (million)
% 2.73/1.90 % (12369)------------------------------
% 2.73/1.90 % (12369)------------------------------
% 2.73/1.90 % (12455)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_12059 for (2996ds/98Mi)
% 2.73/1.91 % (12459)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_12059 for (2996ds/507Mi)
% 2.73/1.91 % (12461)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_12059 for (2996ds/149Mi)
% 2.73/1.91 % (12454)Instruction limit reached!
% 2.73/1.91 % (12454)------------------------------
% 2.73/1.91 % (12454)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.91 % (12454)Termination reason: Unknown
% 2.73/1.91 % (12454)Termination phase: Property scanning
% 2.73/1.91
% 2.73/1.91 % (12454)Memory used [KB]: 2302
% 2.73/1.91 % (12454)Time elapsed: 0.014 s
% 2.73/1.91 % (12454)Instructions burned: 47 (million)
% 2.73/1.91 % (12454)------------------------------
% 2.73/1.91 % (12454)------------------------------
% 2.73/1.92 % (12468)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_12059 for (2996ds/39Mi)
% 2.73/1.92 % (12354)First to succeed.
% 2.73/1.93 % (12455)Instruction limit reached!
% 2.73/1.93 % (12455)------------------------------
% 2.73/1.93 % (12455)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.93 % (12455)Termination reason: Unknown
% 2.73/1.93 % (12455)Termination phase: Function definition elimination
% 2.73/1.93
% 2.73/1.93 % (12455)Memory used [KB]: 2430
% 2.73/1.93 % (12455)Time elapsed: 0.027 s
% 2.73/1.93 % (12455)Instructions burned: 100 (million)
% 2.73/1.93 % (12455)------------------------------
% 2.73/1.93 % (12455)------------------------------
% 2.73/1.93 % (12468)Instruction limit reached!
% 2.73/1.93 % (12468)------------------------------
% 2.73/1.93 % (12468)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.93 % (12468)Termination reason: Unknown
% 2.73/1.93 % (12468)Termination phase: Property scanning
% 2.73/1.93
% 2.73/1.93 % (12468)Memory used [KB]: 2302
% 2.73/1.93 % (12468)Time elapsed: 0.013 s
% 2.73/1.93 % (12468)Instructions burned: 42 (million)
% 2.73/1.93 % (12468)------------------------------
% 2.73/1.93 % (12468)------------------------------
% 2.73/1.93 % (12354)Refutation found. Thanks to Tanya!
% 2.73/1.93 % SZS status Theorem for DTF2THF_12059
% 2.73/1.93 % SZS output start Proof for DTF2THF_12059
% 2.73/1.93 thf(func_def_0, type, in: $i > $i > $o).
% 2.73/1.93 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.73/1.93 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.73/1.93 thf(func_def_8, type, powerset: $i > $i).
% 2.73/1.93 thf(func_def_10, type, setunion: $i > $i).
% 2.73/1.93 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.73/1.93 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.73/1.93 thf(func_def_26, type, prop2set: $o > $i).
% 2.73/1.93 thf(func_def_36, type, nonempty: $i > $o).
% 2.73/1.93 thf(func_def_69, type, set2prop: $i > $o).
% 2.73/1.93 thf(func_def_88, type, subset: $i > $i > $o).
% 2.73/1.93 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.73/1.93 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.73/1.93 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.73/1.93 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.73/1.93 thf(func_def_135, type, regular: $i > $o).
% 2.73/1.93 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.73/1.93 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.73/1.93 thf(func_def_153, type, iskpair: $i > $o).
% 2.73/1.93 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.73/1.93 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.73/1.93 thf(func_def_177, type, singleton: $i > $o).
% 2.73/1.93 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.73/1.93 thf(func_def_184, type, atmost1p: $i > $o).
% 2.73/1.93 thf(func_def_185, type, atleast2p: $i > $o).
% 2.73/1.93 thf(func_def_186, type, atmost2p: $i > $o).
% 2.73/1.93 thf(func_def_187, type, upairsetp: $i > $o).
% 2.73/1.93 thf(func_def_191, type, kfst: $i > $i).
% 2.73/1.93 thf(func_def_203, type, ksnd: $i > $i).
% 2.73/1.93 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.73/1.93 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.73/1.93 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.73/1.93 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.73/1.93 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.73/1.93 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 2.73/1.93 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 2.73/1.93 thf(func_def_284, type, sP0: $i > $o > $i > $i > $o).
% 2.73/1.93 thf(func_def_285, type, sP1: $i > $i > $i > $o).
% 2.73/1.93 thf(func_def_286, type, sP2: $i > $i > $o).
% 2.73/1.93 thf(func_def_287, type, sP3: $i > $o).
% 2.73/1.93 thf(func_def_288, type, sP4: $i > $i > $o).
% 2.73/1.93 thf(func_def_289, type, sP5: $i > $i > $o).
% 2.73/1.93 thf(func_def_293, type, sP9: $i > $i > $i > $i > $o).
% 2.73/1.93 thf(func_def_301, type, sK17: $i > $o).
% 2.73/1.93 thf(func_def_306, type, sK22: $i > $i).
% 2.73/1.93 thf(func_def_309, type, sK25: $i > $i).
% 2.73/1.93 thf(func_def_311, type, sK27: $i > $i).
% 2.73/1.93 thf(func_def_312, type, sK28: ($i > $o) > ($i > $o) > $i).
% 2.73/1.93 thf(func_def_313, type, sK29: ($i > $o) > ($i > $o) > $i).
% 2.73/1.93 thf(func_def_314, type, sK30: $i > $o).
% 2.73/1.93 thf(func_def_315, type, sK31: $i > $o).
% 2.73/1.93 thf(func_def_326, type, sK42: $i > $i).
% 2.73/1.93 thf(func_def_328, type, sK44: ($i > $i) > $i > $i > $i).
% 2.73/1.93 thf(func_def_329, type, sK45: $i > ($i > $o) > $i).
% 2.73/1.93 thf(func_def_330, type, sK46: $i > $o).
% 2.73/1.93 thf(func_def_367, type, sK83: $i > $i > $o).
% 2.73/1.93 thf(func_def_369, type, sK85: $i > $o).
% 2.73/1.93 thf(func_def_370, type, sK86: $i > $i).
% 2.73/1.93 thf(func_def_371, type, sK87: ($i > $o) > $i).
% 2.73/1.93 thf(func_def_374, type, sK90: $i > $i).
% 2.73/1.93 thf(func_def_383, type, sK99: $i > $i > $i).
% 2.73/1.93 thf(func_def_386, type, sK102: $i > $i > $i > $i).
% 2.73/1.93 thf(func_def_390, type, sK106: $i > $i > $i > $i).
% 2.73/1.93 thf(func_def_401, type, sK117: $i > $i > $i).
% 2.73/1.93 thf(func_def_402, type, sK118: $i > $i > $i).
% 2.73/1.93 thf(func_def_403, type, sK119: $i > $i > $i).
% 2.73/1.93 thf(func_def_404, type, sK120: $i > $i > $i).
% 2.73/1.93 thf(func_def_405, type, sK121: $i > $i > $i > $i).
% 2.73/1.93 thf(func_def_406, type, sK122: $i > $i).
% 2.73/1.93 thf(func_def_407, type, sK123: $i > $i).
% 2.73/1.93 thf(func_def_408, type, sK124: $i > $i).
% 2.73/1.93 thf(func_def_409, type, sK125: $i > $i).
% 2.73/1.93 thf(func_def_410, type, sK126: $i > $i > $i > $i).
% 2.73/1.93 thf(func_def_411, type, sK127: $i > $i > $i > $i).
% 2.73/1.93 thf(func_def_412, type, sK128: $i > $i > $i).
% 2.73/1.93 thf(func_def_413, type, sK129: $i > $i > $i).
% 2.73/1.93 thf(func_def_414, type, sK130: $i > $i > $i).
% 2.73/1.93 thf(func_def_415, type, sK131: $i > $i).
% 2.73/1.93 thf(func_def_416, type, sK132: $i > $i > $i).
% 2.73/1.93 thf(func_def_418, type, sK134: $i > $i).
% 2.73/1.93 thf(func_def_419, type, sK135: $i > $i).
% 2.73/1.93 thf(func_def_421, type, sK137: $i > $o).
% 2.73/1.93 thf(func_def_427, type, sK143: $i > $i).
% 2.73/1.93 thf(func_def_435, type, sK151: ($i > $o) > $i > $i > $i > $i).
% 2.73/1.93 thf(func_def_436, type, sK152: ($i > $o) > $i > $i > $i > $i).
% 2.73/1.93 thf(func_def_440, type, sK156: $i > $o).
% 2.73/1.93 thf(func_def_442, type, sK158: $i > $i > $i).
% 2.73/1.93 thf(func_def_448, type, sK164: $i > $o).
% 2.73/1.93 thf(func_def_450, type, sK166: $i > ($i > $o) > $i).
% 2.73/1.93 thf(func_def_456, type, sK172: $i > $i > $o).
% 2.73/1.93 thf(func_def_457, type, sK173: $i > $i).
% 2.73/1.93 thf(func_def_459, type, sK175: $i > $o).
% 2.73/1.93 thf(func_def_462, type, sK178: $i > ($i > $o) > $i).
% 2.73/1.93 thf(func_def_463, type, sK179: $i > ($i > $o) > $i).
% 2.73/1.93 thf(func_def_464, type, sK180: $i > $o).
% 2.73/1.93 thf(func_def_469, type, sK185: $i > $o).
% 2.73/1.93 thf(func_def_471, type, sK187: ($i > $o) > $i > $i > $i > $i).
% 2.73/1.93 thf(func_def_472, type, sK188: ($i > $o) > $i > $i > $i > $i).
% 2.73/1.93 thf(func_def_474, type, sK190: $i > $i).
% 2.73/1.93 thf(func_def_478, type, sK194: $i > $i > $i > $i > $i).
% 2.73/1.93 thf(func_def_479, type, sK195: $i > $i > $i > $i > $i).
% 2.73/1.93 thf(func_def_513, type, sK229: $i > $o).
% 2.73/1.93 thf(func_def_515, type, sK231: ($i > $o) > $i > $i).
% 2.73/1.93 thf(func_def_516, type, sK232: ($i > $o) > $i > $i).
% 2.73/1.93 thf(func_def_528, type, sK244: $i > $o).
% 2.73/1.93 thf(func_def_529, type, sK245: ($i > $o) > $i).
% 2.73/1.93 thf(func_def_540, type, sK256: $i > $i > $o).
% 2.73/1.93 thf(func_def_546, type, sK262: $i > $i > $i).
% 2.73/1.93 thf(func_def_552, type, sK268: $i > $o).
% 2.73/1.93 thf(func_def_554, type, sK270: ($i > $o) > $i > $i).
% 2.73/1.93 thf(func_def_610, type, sK326: $i > $i > $i > $i > $i).
% 2.73/1.93 thf(func_def_616, type, sK332: $i > ($i > $o) > $i).
% 2.73/1.93 thf(func_def_617, type, sK333: $i > $o).
% 2.73/1.93 thf(func_def_628, type, sK344: $i > $i).
% 2.73/1.93 thf(func_def_630, type, sK346: ($i > $i) > $i > $i > $i).
% 2.73/1.93 thf(func_def_635, type, sK351: $i > $i > $i).
% 2.73/1.93 thf(func_def_644, type, sK360: $i > $i > $o).
% 2.73/1.93 thf(func_def_650, type, sK366: $i > $i > $o).
% 2.73/1.93 thf(func_def_651, type, sK367: $i > $i).
% 2.73/1.93 thf(func_def_652, type, sK368: $i > $i).
% 2.73/1.93 thf(func_def_653, type, sK369: ($i > $i > $o) > $i > $i).
% 2.73/1.93 thf(func_def_654, type, sK370: $i > ($i > $i > $o) > $i > $i).
% 2.73/1.93 thf(func_def_655, type, sK371: ($i > $i > $o) > $i > $i).
% 2.73/1.93 thf(func_def_664, type, sK380: ($i > $o) > $i).
% 2.73/1.93 thf(func_def_665, type, sK381: ($i > $o) > $i).
% 2.73/1.93 thf(func_def_666, type, sK382: $i > $o).
% 2.73/1.93 thf(func_def_669, type, sK385: $i > $o).
% 2.73/1.93 thf(func_def_672, type, sK388: $i > ($i > $o) > $i).
% 2.73/1.93 thf(func_def_673, type, sK389: $i > $o).
% 2.73/1.93 thf(func_def_677, type, sK393: $i > $o).
% 2.73/1.93 thf(func_def_678, type, sK394: ($i > $o) > $i > $i > $i).
% 2.73/1.93 thf(func_def_691, type, sK407: $i > $i > $i).
% 2.73/1.93 thf(func_def_696, type, sK412: $i > $o).
% 2.73/1.93 thf(func_def_714, type, sK430: $i > $o).
% 2.73/1.93 thf(func_def_740, type, sK456: $i > ($i > $o) > $i).
% 2.73/1.93 thf(func_def_741, type, sK457: $i > $o).
% 2.73/1.93 thf(func_def_744, type, sK460: $i > $o).
% 2.73/1.93 thf(func_def_746, type, sK462: ($i > $o) > ($i > $o) > $i).
% 2.73/1.93 thf(func_def_747, type, sK463: ($i > $o) > ($i > $o) > $i).
% 2.73/1.93 thf(func_def_748, type, sK464: $i > $o).
% 2.73/1.93 thf(func_def_749, type, sK465: $i > $o).
% 2.73/1.93 thf(func_def_765, type, sK481: $i > $o).
% 2.73/1.93 thf(func_def_775, type, sK491: $i > $i > $i).
% 2.73/1.93 thf(func_def_776, type, sK492: $i > $i).
% 2.73/1.93 thf(func_def_779, type, sK495: $i > $i > ($i > $i) > $i).
% 2.73/1.93 thf(func_def_784, type, sK500: $i > $i > $o).
% 2.73/1.93 thf(func_def_795, type, sK511: $i > $i > $o).
% 2.73/1.93 thf(func_def_798, type, sK514: $i > $i > $i > $i).
% 2.73/1.93 thf(func_def_799, type, sK515: $i > $i > $i > $i).
% 2.73/1.93 thf(func_def_805, type, sK521: $i > ($i > $i) > $i > $i).
% 2.73/1.93 thf(func_def_807, type, sK523: $i > $i).
% 2.73/1.93 thf(func_def_820, type, sK536: $o > $i > $i > $i).
% 2.73/1.93 thf(func_def_825, type, sK541: $i > $i > $i > $i > $i).
% 2.73/1.93 thf(func_def_837, type, sK553: $i > $o).
% 2.73/1.93 thf(func_def_840, type, sK556: $i > ($i > $o) > $i).
% 2.73/1.93 thf(func_def_841, type, sK557: $i > ($i > $o) > $i).
% 2.73/1.93 thf(func_def_858, type, sK574: $i > $o).
% 2.73/1.93 thf(func_def_859, type, sK575: $i > $i).
% 2.73/1.93 thf(func_def_860, type, sK576: ($i > $o) > $i).
% 2.73/1.93 thf(func_def_867, type, sK583: $i > $o).
% 2.73/1.93 thf(func_def_872, type, sK588: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.73/1.93 thf(func_def_873, type, sK589: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.73/1.93 thf(func_def_876, type, sK592: $i > $o).
% 2.73/1.93 thf(func_def_877, type, sK593: $i > $o).
% 2.73/1.93 thf(func_def_905, type, sK621: $i > $i > $i > $i > $i).
% 2.73/1.93 thf(func_def_906, type, sK622: $i > $i > $i > $i > $i).
% 2.73/1.93 thf(func_def_907, type, sK623: $i > $i > $i > $i > $i).
% 2.73/1.93 thf(func_def_908, type, sK624: $i > $i > $i > $i > $i).
% 2.73/1.93 thf(func_def_917, type, sK633: $i > $o).
% 2.73/1.93 thf(func_def_918, type, sK634: $i > $o).
% 2.73/1.93 thf(func_def_924, type, sK640: $i > $o).
% 2.73/1.93 thf(func_def_934, type, sK650: $i > $o).
% 2.73/1.93 thf(func_def_939, type, sK655: $i > $i).
% 2.73/1.93 thf(func_def_940, type, sK656: ($i > $i) > $i > $i > $i).
% 2.73/1.93 thf(func_def_945, type, sK661: $i > $o).
% 2.73/1.93 thf(func_def_954, type, sK670: $i > $i > $i).
% 2.73/1.93 thf(func_def_961, type, sK677: $i > $i > $i > $i > $i).
% 2.73/1.93 thf(func_def_962, type, sK678: $i > $i > $i).
% 2.73/1.93 thf(func_def_963, type, sK679: $i > $i > $i).
% 2.73/1.93 thf(func_def_969, type, sK685: ($i > $i) > $i > $i > $i).
% 2.73/1.93 thf(func_def_972, type, sK688: $i > $i).
% 2.73/1.93 thf(func_def_975, type, sK691: $i > $i > $i).
% 2.73/1.93 thf(func_def_997, type, sK713: $i > $i > $o).
% 2.73/1.93 thf(func_def_1007, type, sK723: ($i > $o) > $i > $i).
% 2.73/1.93 thf(func_def_1009, type, sK725: $i > $o).
% 2.73/1.93 thf(func_def_1017, type, ph733: !>[X0: $tType]:(X0)).
% 2.73/1.93 thf(f7576,plain,(
% 2.73/1.93 $false),
% 2.73/1.93 inference(avatar_sat_refutation,[status(thm)],[f6603,f7545,f7558])).
% 2.73/1.93 thf(f7558,plain,(
% 2.73/1.93 spl732_70 | ~spl732_72),
% 2.73/1.93 inference(avatar_split_clause,[status(thm)],[f7557,f4730,f4678])).
% 2.73/1.93 thf(f4678,definition,(
% 2.73/1.93 spl732_70 <=> ($true = (subset @ sK310 @ sK312))),
% 2.73/1.93 introduced(definition,[new_symbols(naming,[spl732_70])],[avatar_definition])).
% 2.73/1.93 thf(f4730,definition,(
% 2.73/1.93 spl732_72 <=> ((in @ sK310 @ (powerset @ sK312)) = $true)),
% 2.73/1.93 introduced(definition,[new_symbols(naming,[spl732_72])],[avatar_definition])).
% 2.73/1.93 thf(f7557,plain,(
% 2.73/1.93 ($true = (subset @ sK310 @ sK312)) | ~spl732_72),
% 2.73/1.93 inference(trivial_inequality_removal,[status(thm)],[f7556])).
% 2.73/1.93 thf(f7556,plain,(
% 2.73/1.93 ($true = (subset @ sK310 @ sK312)) | ($true != $true) | ~spl732_72),
% 2.73/1.93 inference(forward_demodulation,[status(thm)],[f7555,f2567])).
% 2.73/1.93 thf(f2567,plain,(
% 2.73/1.93 (powersetE1 = $true)),
% 2.73/1.93 inference(cnf_transformation,[status(thm)],[f1496])).
% 2.73/1.93 thf(f1496,plain,(
% 2.73/1.93 (exuE3e = $true) & (setextsub = $true) & (setminusIRneg = $true) & (binintersectSubset3 = $true) & (setadjoinE = $true) & (cartprodmempair1 = $true) & (setminusEL = $true) & (emptysetimpfalse = $true) & (emptyE1 = $true) & (kpairsurjEq = $true) & (cartprodpairmemER = $true) & (ex1I2 = $true) & (setminusILneg = $true) & (emptyI = $true) & (setunionAx = $true) & (setukpairIR = $true) & (emptysetE = $true) & (ubforcartprodlem3 = $true) & (setminusELneg = $true) & (setukpairinjL1 = $true) & (cartprodmempair = $true) & (binintersectSubset4 = $true) & (omegaIndAx = $true) & (complementT_lem = $true) & (dsetconstrI = $true) & (subPowSU = $true) & (setunionE = $true) & (exu__Cong = $true) & (singletonsswitch = $true) & (singletoninpowunion = $true) & (funcextLem = $true) & (binunionIR = $true) & (upairsetIR = $true) & (setadjoinIR = $true) & (setminusI = $true) & (setukpairIL = $true) & (dpsetconstrSub = $true) & (((in @ sK310 @ (powerset @ sK311)) = $true) & (($true = (in @ sK312 @ (powerset @ sK311))) & ! [X3 : $i] : (($true != (in @ X3 @ sK310)) | ($true = (in @ X3 @ sK312)) | ($true != (in @ X3 @ sK311))) & ($true != (subset @ sK310 @ sK312)))) & (secondinupair = $true) & (nonemptyI1 = $true) & (singletonprop = $true) & (exuEu = $true) & (cartprodfstpairEq = $true) & (binintersectEL = $true) & (upairsetE = $true) & (setukpairinjL2 = $true) & (iftrueProp1 = $true) & (iftrueProp2 = $true) & (cartprodpairsurjEq = $true) & (beta1 = $true) & (apProp = $true) & (subsetI1 = $true) & (subsetE = $true) & (dpsetconstrEL1 = $true) & (subsetRefl = $true) & (setadjoinSub2 = $true) & (ex1E1 = $true) & (exuE1 = $true) & (theeq = $true) & (disjointsetsI1 = $true) & (binintersectI = $true) & (ubforcartprodlem1 = $true) & (funcGraphProp4 = $true) & (setadjoin__Cong = $true) & (ap2p = $true) & (brelnall1 = $true) & (symdiffIneg1 = $true) & (funcGraphProp2 = $true) & (binintersectLsub = $true) & (cartprodsndin = $true) & (nonemptyE1 = $true) & (notsubsetI = $true) & (setminusSubset2 = $true) & (ifp = $true) & (in__Cong = $true) & (prop2setE = $true) & (notdexE = $true) & (iffalse = $true) & (funcGraphProp3 = $true) & (setextAx = $true) & (setadjoinOr = $true) & (eqbreln = $true) & (powersetAx = $true) & (sepSubset = $true) & (vacuousDall = $true) & (emptysetAx = $true) & (symdiffIneg2 = $true) & (prop2set2propI = $true) & (iftrueorfalse = $true) & (dsetconstrEL = $true) & (brelnall2 = $true) & (exuE2 = $true) & (upairset2E = $true) & (exuE3u = $true) & (subset2powerset = $true) & (infuncsetfunc = $true) & (omega0Ax = $true) & (exuI3 = $true) & (emptyset__Cong = $true) & (setminusSubset1 = $true) & (powersetI = $true) & (funcImageSingleton = $true) & (quantDeMorgan2 = $true) & (powersetE = $true) & (lamProp = $true) & (binintersectSubset1 = $true) & (setukpairinjR = $true) & (omega__Cong = $true) & (binunionRsub = $true) & (subsetTrans = $true) & (eqimpsubset1 = $true) & (symdiffI1 = $true) & (kfstsingleton = $true) & (setukpairinjR11 = $true) & (binintersectT_lem = $true) & (binintersectSubset5 = $true) & (upairequniteq = $true) & (noeltsimpempty = $true) & (ifSingleton = $true) & (subsetI2 = $true) & (inCongP = $true) & (emptysetsubset = $true) & (setextT = $true) & (cartprodsndpairEq = $true) & (ap2apEq2 = $true) & (ex1I = $true) & (sepInPowerset = $true) & (setoftrueEq = $true) & (setadjoinIL = $true) & (symdiffE = $true) & (notequalI2 = $true) & (binunionEcases = $true) & (setadjoinAx = $true) & (ap2apEq1 = $true) & (setunionsingleton = $true) & (emptyinunitempty = $true) & (ksndsingleton = $true) & (eta1 = $true) & (cartprodmempaircEq = $true) & (descrp = $true) & (cartprodpairin = $true) & (binintersectER = $true) & (ex1E2 = $true) & (symdiffI2 = $true) & (setukpairinjL = $true) & (prop2setI = $true) & (powerset__Cong = $true) & (inPowerset = $true) & (quantDeMorgan4 = $true) & (notequalI1 = $true) & (upairsetIL = $true) & (ksndpairEq = $true) & (setunion__Cong = $true) & (setunionE2 = $true) & (exuI1 = $true) & (iftrue = $true) & (subbreln = $true) & (setext = $true) & (notinsingleton = $true) & (setminusERneg = $true) & (binintersectSubset2 = $true) & (binunionIL = $true) & (setminusER = $true) & (eqimpsubset2 = $true) & (funcGraphProp1 = $true) & (dpsetconstrER = $true) & (setminusLsub = $true) & (nonemptyImpWitness = $true) & (setunionI = $true) & (replAx = $true) & (app = $true) & (singletonsuniq = $true) & (omegaSAx = $true) & (powersetsubset = $true) & (exuI2 = $true) & (powersetI1 = $true) & (dsetconstrER = $true) & (foundationAx = $true) & (kpairiskpair = $true) & (quantDeMorgan3 = $true) & (nonemptyI = $true) & (wellorderingAx = $true) & (singletonsubset = $true) & (cartprodpairmemEL = $true) & (setadjoinSub = $true) & (setbeta = $true) & (bs114d = $true) & (theprop = $true) & (setukpairinjR1 = $true) & (emptyinPowerset = $true) & (emptyInPowerset = $true) & (subsetemptysetimpeq = $true) & (subsetE2 = $true) & (lam2lamEq = $true) & (funcinfuncset = $true) & (iffalseProp2 = $true) & (funcext = $true) & (upairset2IR = $true) & (lamp = $true) & (dpsetconstrERa = $true) & (beta2 = $true) & (binunionE = $true) & (iffalseProp1 = $true) & (binunionT_lem = $true) & (notinemptyset = $true) & (powersetE1 = $true) & (kpairp = $true) & (dpsetconstrI = $true) & (uniqinunit = $true) & (binunionLsub = $true) & (ubforcartprodlem2 = $true) & (descr__Cong = $true) & (dsetconstr__Cong = $true) & (binintersectRsub = $true) & (setukpairinjR2 = $true) & (eta2 = $true) & (singletoninpowerset = $true) & (setukpairinjR12 = $true) & (funcext2 = $true) & (eqinunit = $true) & (upairinpowunion = $true) & (dpsetconstrEL2 = $true) & (cartprodfstin = $true) & (setunionsingleton1 = $true) & (upairsubunion = $true) & (setminusT_lem = $true) & (kfstpairEq = $true) & (setOfPairsIsBReln = $true) & (powersetT_lem = $true) & (lam2p = $true) & (quantDeMorgan1 = $true) & (setunionsingleton2 = $true) & (notdallE = $true)),
% 2.73/1.93 inference(skolemisation,[status(esa),new_symbols(skolem,[sK310,sK311,sK312])],[f755,f1495,f1494])).
% 2.73/1.93 thf(f1494,plain,(
% 2.73/1.93 ? [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (powerset @ X1))) & ? [X2 : $i] : (((in @ X2 @ (powerset @ X1)) = $true) & ! [X3 : $i] : (((in @ X3 @ X0) != $true) | ((in @ X3 @ X2) = $true) | ($true != (in @ X3 @ X1))) & ((subset @ X0 @ X2) != $true))) => (((in @ sK310 @ (powerset @ sK311)) = $true) & ? [X2 : $i] : (((in @ X2 @ (powerset @ sK311)) = $true) & ! [X3 : $i] : (($true != (in @ X3 @ sK310)) | ((in @ X3 @ X2) = $true) | ($true != (in @ X3 @ sK311))) & ((subset @ sK310 @ X2) != $true)))),
% 2.73/1.93 introduced(definition,[],[choice_axiom])).
% 2.73/1.93 thf(f1495,plain,(
% 2.73/1.93 ? [X2 : $i] : (((in @ X2 @ (powerset @ sK311)) = $true) & ! [X3 : $i] : (($true != (in @ X3 @ sK310)) | ((in @ X3 @ X2) = $true) | ($true != (in @ X3 @ sK311))) & ((subset @ sK310 @ X2) != $true)) => (($true = (in @ sK312 @ (powerset @ sK311))) & ! [X3 : $i] : (($true != (in @ X3 @ sK310)) | ($true = (in @ X3 @ sK312)) | ($true != (in @ X3 @ sK311))) & ($true != (subset @ sK310 @ sK312)))),
% 2.73/1.93 introduced(definition,[],[choice_axiom])).
% 2.73/1.93 thf(f755,plain,(
% 2.73/1.93 (exuE3e = $true) & (setextsub = $true) & (setminusIRneg = $true) & (binintersectSubset3 = $true) & (setadjoinE = $true) & (cartprodmempair1 = $true) & (setminusEL = $true) & (emptysetimpfalse = $true) & (emptyE1 = $true) & (kpairsurjEq = $true) & (cartprodpairmemER = $true) & (ex1I2 = $true) & (setminusILneg = $true) & (emptyI = $true) & (setunionAx = $true) & (setukpairIR = $true) & (emptysetE = $true) & (ubforcartprodlem3 = $true) & (setminusELneg = $true) & (setukpairinjL1 = $true) & (cartprodmempair = $true) & (binintersectSubset4 = $true) & (omegaIndAx = $true) & (complementT_lem = $true) & (dsetconstrI = $true) & (subPowSU = $true) & (setunionE = $true) & (exu__Cong = $true) & (singletonsswitch = $true) & (singletoninpowunion = $true) & (funcextLem = $true) & (binunionIR = $true) & (upairsetIR = $true) & (setadjoinIR = $true) & (setminusI = $true) & (setukpairIL = $true) & (dpsetconstrSub = $true) & ? [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (powerset @ X1))) & ? [X2 : $i] : (((in @ X2 @ (powerset @ X1)) = $true) & ! [X3 : $i] : (((in @ X3 @ X0) != $true) | ((in @ X3 @ X2) = $true) | ($true != (in @ X3 @ X1))) & ((subset @ X0 @ X2) != $true))) & (secondinupair = $true) & (nonemptyI1 = $true) & (singletonprop = $true) & (exuEu = $true) & (cartprodfstpairEq = $true) & (binintersectEL = $true) & (upairsetE = $true) & (setukpairinjL2 = $true) & (iftrueProp1 = $true) & (iftrueProp2 = $true) & (cartprodpairsurjEq = $true) & (beta1 = $true) & (apProp = $true) & (subsetI1 = $true) & (subsetE = $true) & (dpsetconstrEL1 = $true) & (subsetRefl = $true) & (setadjoinSub2 = $true) & (ex1E1 = $true) & (exuE1 = $true) & (theeq = $true) & (disjointsetsI1 = $true) & (binintersectI = $true) & (ubforcartprodlem1 = $true) & (funcGraphProp4 = $true) & (setadjoin__Cong = $true) & (ap2p = $true) & (brelnall1 = $true) & (symdiffIneg1 = $true) & (funcGraphProp2 = $true) & (binintersectLsub = $true) & (cartprodsndin = $true) & (nonemptyE1 = $true) & (notsubsetI = $true) & (setminusSubset2 = $true) & (ifp = $true) & (in__Cong = $true) & (prop2setE = $true) & (notdexE = $true) & (iffalse = $true) & (funcGraphProp3 = $true) & (setextAx = $true) & (setadjoinOr = $true) & (eqbreln = $true) & (powersetAx = $true) & (sepSubset = $true) & (vacuousDall = $true) & (emptysetAx = $true) & (symdiffIneg2 = $true) & (prop2set2propI = $true) & (iftrueorfalse = $true) & (dsetconstrEL = $true) & (brelnall2 = $true) & (exuE2 = $true) & (upairset2E = $true) & (exuE3u = $true) & (subset2powerset = $true) & (infuncsetfunc = $true) & (omega0Ax = $true) & (exuI3 = $true) & (emptyset__Cong = $true) & (setminusSubset1 = $true) & (powersetI = $true) & (funcImageSingleton = $true) & (quantDeMorgan2 = $true) & (powersetE = $true) & (lamProp = $true) & (binintersectSubset1 = $true) & (setukpairinjR = $true) & (omega__Cong = $true) & (binunionRsub = $true) & (subsetTrans = $true) & (eqimpsubset1 = $true) & (symdiffI1 = $true) & (kfstsingleton = $true) & (setukpairinjR11 = $true) & (binintersectT_lem = $true) & (binintersectSubset5 = $true) & (upairequniteq = $true) & (noeltsimpempty = $true) & (ifSingleton = $true) & (subsetI2 = $true) & (inCongP = $true) & (emptysetsubset = $true) & (setextT = $true) & (cartprodsndpairEq = $true) & (ap2apEq2 = $true) & (ex1I = $true) & (sepInPowerset = $true) & (setoftrueEq = $true) & (setadjoinIL = $true) & (symdiffE = $true) & (notequalI2 = $true) & (binunionEcases = $true) & (setadjoinAx = $true) & (ap2apEq1 = $true) & (setunionsingleton = $true) & (emptyinunitempty = $true) & (ksndsingleton = $true) & (eta1 = $true) & (cartprodmempaircEq = $true) & (descrp = $true) & (cartprodpairin = $true) & (binintersectER = $true) & (ex1E2 = $true) & (symdiffI2 = $true) & (setukpairinjL = $true) & (prop2setI = $true) & (powerset__Cong = $true) & (inPowerset = $true) & (quantDeMorgan4 = $true) & (notequalI1 = $true) & (upairsetIL = $true) & (ksndpairEq = $true) & (setunion__Cong = $true) & (setunionE2 = $true) & (exuI1 = $true) & (iftrue = $true) & (subbreln = $true) & (setext = $true) & (notinsingleton = $true) & (setminusERneg = $true) & (binintersectSubset2 = $true) & (binunionIL = $true) & (setminusER = $true) & (eqimpsubset2 = $true) & (funcGraphProp1 = $true) & (dpsetconstrER = $true) & (setminusLsub = $true) & (nonemptyImpWitness = $true) & (setunionI = $true) & (replAx = $true) & (app = $true) & (singletonsuniq = $true) & (omegaSAx = $true) & (powersetsubset = $true) & (exuI2 = $true) & (powersetI1 = $true) & (dsetconstrER = $true) & (foundationAx = $true) & (kpairiskpair = $true) & (quantDeMorgan3 = $true) & (nonemptyI = $true) & (wellorderingAx = $true) & (singletonsubset = $true) & (cartprodpairmemEL = $true) & (setadjoinSub = $true) & (setbeta = $true) & (bs114d = $true) & (theprop = $true) & (setukpairinjR1 = $true) & (emptyinPowerset = $true) & (emptyInPowerset = $true) & (subsetemptysetimpeq = $true) & (subsetE2 = $true) & (lam2lamEq = $true) & (funcinfuncset = $true) & (iffalseProp2 = $true) & (funcext = $true) & (upairset2IR = $true) & (lamp = $true) & (dpsetconstrERa = $true) & (beta2 = $true) & (binunionE = $true) & (iffalseProp1 = $true) & (binunionT_lem = $true) & (notinemptyset = $true) & (powersetE1 = $true) & (kpairp = $true) & (dpsetconstrI = $true) & (uniqinunit = $true) & (binunionLsub = $true) & (ubforcartprodlem2 = $true) & (descr__Cong = $true) & (dsetconstr__Cong = $true) & (binintersectRsub = $true) & (setukpairinjR2 = $true) & (eta2 = $true) & (singletoninpowerset = $true) & (setukpairinjR12 = $true) & (funcext2 = $true) & (eqinunit = $true) & (upairinpowunion = $true) & (dpsetconstrEL2 = $true) & (cartprodfstin = $true) & (setunionsingleton1 = $true) & (upairsubunion = $true) & (setminusT_lem = $true) & (kfstpairEq = $true) & (setOfPairsIsBReln = $true) & (powersetT_lem = $true) & (lam2p = $true) & (quantDeMorgan1 = $true) & (setunionsingleton2 = $true) & (notdallE = $true)),
% 2.73/1.93 inference(flattening,[status(thm)],[f754])).
% 2.73/1.93 thf(f754,plain,(
% 2.73/1.93 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (? [X2 : $i] : ((((subset @ X0 @ X2) != $true) & ! [X3 : $i] : ((((in @ X3 @ X2) = $true) | ((in @ X3 @ X0) != $true)) | ($true != (in @ X3 @ X1)))) & ((in @ X2 @ (powerset @ X1)) = $true)) & ($true = (in @ X0 @ (powerset @ X1)))) & (setextT = $true)) & (complementT_lem = $true)) & (setminusT_lem = $true)) & (powersetT_lem = $true)) & (binunionT_lem = $true)) & (binintersectT_lem = $true)) & (iftrueorfalse = $true)) & (iffalse = $true)) & (iftrue = $true)) & (theeq = $true)) & (ifp = $true)) & (ifSingleton = $true)) & (iftrueProp2 = $true)) & (iftrueProp1 = $true)) & (iffalseProp2 = $true)) & (iffalseProp1 = $true)) & (eta2 = $true)) & (beta2 = $true)) & (lam2lamEq = $true)) & (eta1 = $true)) & (beta1 = $true)) & (ap2apEq2 = $true)) & (ap2apEq1 = $true)) & (funcext2 = $true)) & (funcext = $true)) & (eqbreln = $true)) & (subbreln = $true)) & (funcGraphProp4 = $true)) & (funcextLem = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp3 = $true)) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (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)),
% 2.73/1.93 inference(ennf_transformation,[status(thm)],[f650])).
% 2.73/1.93 thf(f650,plain,(
% 2.73/1.93 ~((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) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ((funcGraphProp3 = $true) => ((funcGraphProp2 = $true) => ((funcextLem = $true) => ((funcGraphProp4 = $true) => ((subbreln = $true) => ((eqbreln = $true) => ((funcext = $true) => ((funcext2 = $true) => ((ap2apEq1 = $true) => ((ap2apEq2 = $true) => ((beta1 = $true) => ((eta1 = $true) => ((lam2lamEq = $true) => ((beta2 = $true) => ((eta2 = $true) => ((iffalseProp1 = $true) => ((iffalseProp2 = $true) => ((iftrueProp1 = $true) => ((iftrueProp2 = $true) => ((ifSingleton = $true) => ((ifp = $true) => ((theeq = $true) => ((iftrue = $true) => ((iffalse = $true) => ((iftrueorfalse = $true) => ((binintersectT_lem = $true) => ((binunionT_lem = $true) => ((powersetT_lem = $true) => ((setminusT_lem = $true) => ((complementT_lem = $true) => ((setextT = $true) => ! [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (powerset @ X1))) => ! [X2 : $i] : (((in @ X2 @ (powerset @ X1)) = $true) => (! [X3 : $i] : (($true = (in @ X3 @ X1)) => (((in @ X3 @ X0) = $true) => ((in @ X3 @ X2) = $true))) => ((subset @ X0 @ X2) = $true))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.73/1.93 inference(fool_elimination,[status(thm)],[f649])).
% 2.73/1.93 thf(f649,plain,(
% 2.73/1.93 ~(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 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => ! [X0 : $i,X1 : $i] : ((in @ X0 @ (powerset @ X1)) => ! [X2 : $i] : ((in @ X2 @ (powerset @ X1)) => (! [X3 : $i] : ((in @ X3 @ X1) => ((in @ X3 @ X0) => (in @ X3 @ X2))) => (subset @ X0 @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.73/1.93 inference(rectify,[status(thm)],[f238])).
% 2.73/1.93 thf(f238,negated_conjecture,(
% 2.73/1.93 ~(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 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => ! [X11 : $i,X3 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => (! [X1 : $i] : ((in @ X1 @ X3) => ((in @ X1 @ X11) => (in @ X1 @ X16))) => (subset @ X11 @ X16))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.73/1.93 inference(negated_conjecture,[status(cth)],[f237])).
% 2.73/1.93 thf(f237,conjecture,(
% 2.73/1.93 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 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => ! [X11 : $i,X3 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => (! [X1 : $i] : ((in @ X1 @ X3) => ((in @ X1 @ X11) => (in @ X1 @ X16))) => (subset @ X11 @ X16)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.73/1.93 file('/export/starexec/sandbox2/tmp/tmp.DvUejQI5fj/DTF2THF_12059.p',subsetTI)).
% 2.73/1.93 thf(f7555,plain,(
% 2.73/1.93 ($true = (subset @ sK310 @ sK312)) | (powersetE1 != $true) | ~spl732_72),
% 2.73/1.93 inference(trivial_inequality_removal,[status(thm)],[f7548])).
% 2.73/1.93 thf(f7548,plain,(
% 2.73/1.93 ($true = (subset @ sK310 @ sK312)) | (powersetE1 != $true) | ($true != $true) | ~spl732_72),
% 2.73/1.93 inference(superposition,[status(thm)],[f3032,f4732])).
% 2.73/1.93 thf(f4732,plain,(
% 2.73/1.93 ((in @ sK310 @ (powerset @ sK312)) = $true) | ~spl732_72),
% 2.73/1.93 inference(avatar_component_clause,[status(thm)],[f4730])).
% 2.73/1.93 thf(f3032,plain,(
% 2.73/1.93 ( ! [X0 : $i,X1 : $i] : (($true != (in @ X0 @ (powerset @ X1))) | ((subset @ X0 @ X1) = $true) | (powersetE1 != $true)) )),
% 2.73/1.93 inference(cnf_transformation,[status(thm)],[f1807])).
% 2.73/1.93 thf(f1807,plain,(
% 2.73/1.93 (! [X0 : $i,X1 : $i] : (($true != (in @ X0 @ (powerset @ X1))) | ((subset @ X0 @ X1) = $true)) | (powersetE1 != $true)) & ((powersetE1 = $true) | (($true = (in @ sK519 @ (powerset @ sK520))) & ((subset @ sK519 @ sK520) != $true)))),
% 2.73/1.93 inference(skolemisation,[status(esa),new_symbols(skolem,[sK519,sK520])],[f1805,f1806])).
% 2.73/1.93 thf(f1806,plain,(
% 2.73/1.93 ? [X2 : $i,X3 : $i] : (((in @ X2 @ (powerset @ X3)) = $true) & ((subset @ X2 @ X3) != $true)) => (($true = (in @ sK519 @ (powerset @ sK520))) & ((subset @ sK519 @ sK520) != $true))),
% 2.73/1.93 introduced(definition,[],[choice_axiom])).
% 2.73/1.93 thf(f1805,plain,(
% 2.73/1.93 (! [X0 : $i,X1 : $i] : (($true != (in @ X0 @ (powerset @ X1))) | ((subset @ X0 @ X1) = $true)) | (powersetE1 != $true)) & ((powersetE1 = $true) | ? [X2 : $i,X3 : $i] : (((in @ X2 @ (powerset @ X3)) = $true) & ((subset @ X2 @ X3) != $true)))),
% 2.73/1.93 inference(rectify,[status(thm)],[f1804])).
% 2.73/1.93 thf(f1804,plain,(
% 2.73/1.93 (! [X0 : $i,X1 : $i] : (($true != (in @ X0 @ (powerset @ X1))) | ((subset @ X0 @ X1) = $true)) | (powersetE1 != $true)) & ((powersetE1 = $true) | ? [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (powerset @ X1))) & ((subset @ X0 @ X1) != $true)))),
% 2.73/1.93 inference(nnf_transformation,[status(thm)],[f962])).
% 2.73/1.93 thf(f962,plain,(
% 2.73/1.93 ! [X0 : $i,X1 : $i] : (($true != (in @ X0 @ (powerset @ X1))) | ((subset @ X0 @ X1) = $true)) <=> (powersetE1 = $true)),
% 2.73/1.93 inference(ennf_transformation,[status(thm)],[f520])).
% 2.73/1.93 thf(f520,plain,(
% 2.73/1.93 ! [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (powerset @ X1))) => ((subset @ X0 @ X1) = $true)) <=> (powersetE1 = $true)),
% 2.73/1.93 inference(fool_elimination,[status(thm)],[f519])).
% 2.73/1.93 thf(f519,plain,(
% 2.73/1.93 (! [X0 : $i,X1 : $i] : ((in @ X0 @ (powerset @ X1)) => (subset @ X0 @ X1)) = powersetE1)),
% 2.73/1.93 inference(rectify,[status(thm)],[f96])).
% 2.73/1.93 thf(f96,axiom,(
% 2.73/1.93 (! [X4 : $i,X3 : $i] : ((in @ X4 @ (powerset @ X3)) => (subset @ X4 @ X3)) = powersetE1)),
% 2.73/1.93 file('/export/starexec/sandbox2/tmp/tmp.DvUejQI5fj/DTF2THF_12059.p',powersetE1)).
% 2.73/1.93 thf(f7545,plain,(
% 2.73/1.93 spl732_72),
% 2.73/1.93 inference(avatar_split_clause,[status(thm)],[f7544,f4730])).
% 2.73/1.93 thf(f7544,plain,(
% 2.73/1.93 ((in @ sK310 @ (powerset @ sK312)) = $true)),
% 2.73/1.93 inference(trivial_inequality_removal,[status(thm)],[f7543])).
% 2.73/1.93 thf(f7543,plain,(
% 2.73/1.93 ($true != $true) | ((in @ sK310 @ (powerset @ sK312)) = $true)),
% 2.73/1.93 inference(forward_demodulation,[status(thm)],[f7532,f2672])).
% 2.73/1.93 thf(f2672,plain,(
% 2.73/1.93 (powersetI = $true)),
% 2.73/1.93 inference(cnf_transformation,[status(thm)],[f1496])).
% 2.73/1.93 thf(f7532,plain,(
% 2.73/1.93 ((in @ sK310 @ (powerset @ sK312)) = $true) | (powersetI != $true)),
% 2.73/1.93 inference(trivial_inequality_removal,[status(thm)],[f7531])).
% 2.73/1.93 thf(f7531,plain,(
% 2.73/1.93 ($true != $true) | ((in @ sK310 @ (powerset @ sK312)) = $true) | (powersetI != $true)),
% 2.73/1.93 inference(duplicate_literal_removal,[status(thm)],[f7524])).
% 2.73/1.93 thf(f7524,plain,(
% 2.73/1.93 (powersetI != $true) | ($true != $true) | ((in @ sK310 @ (powerset @ sK312)) = $true) | ((in @ sK310 @ (powerset @ sK312)) = $true)),
% 2.73/1.93 inference(superposition,[status(thm)],[f4761,f2256])).
% 2.73/1.93 thf(f2256,plain,(
% 2.73/1.93 ( ! [X0 : $i,X1 : $i] : (((in @ (sK99 @ X1 @ X0) @ X1) = $true) | (powersetI != $true) | ((in @ X1 @ (powerset @ X0)) = $true)) )),
% 2.73/1.93 inference(cnf_transformation,[status(thm)],[f1162])).
% 2.73/1.93 thf(f1162,plain,(
% 2.73/1.93 (! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) | (((in @ (sK99 @ X1 @ X0) @ X1) = $true) & ((in @ (sK99 @ X1 @ X0) @ X0) != $true))) | (powersetI != $true)) & ((powersetI = $true) | (((in @ sK101 @ (powerset @ sK100)) != $true) & ! [X5 : $i] : (((in @ X5 @ sK101) != $true) | ($true = (in @ X5 @ sK100)))))),
% 2.73/1.93 inference(skolemisation,[status(esa),new_symbols(skolem,[sK99,sK100,sK101])],[f1159,f1161,f1160])).
% 2.73/1.93 thf(f1160,plain,(
% 2.73/1.93 ! [X0 : $i,X1 : $i] : (? [X2 : $i] : (((in @ X2 @ X1) = $true) & ((in @ X2 @ X0) != $true)) => (((in @ (sK99 @ X1 @ X0) @ X1) = $true) & ((in @ (sK99 @ X1 @ X0) @ X0) != $true)))),
% 2.73/1.93 introduced(definition,[],[choice_axiom])).
% 2.73/1.93 thf(f1161,plain,(
% 2.73/1.93 ? [X3 : $i,X4 : $i] : (((in @ X4 @ (powerset @ X3)) != $true) & ! [X5 : $i] : (((in @ X5 @ X4) != $true) | ((in @ X5 @ X3) = $true))) => (((in @ sK101 @ (powerset @ sK100)) != $true) & ! [X5 : $i] : (((in @ X5 @ sK101) != $true) | ($true = (in @ X5 @ sK100))))),
% 2.73/1.93 introduced(definition,[],[choice_axiom])).
% 2.73/1.93 thf(f1159,plain,(
% 2.73/1.93 (! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) | ? [X2 : $i] : (((in @ X2 @ X1) = $true) & ((in @ X2 @ X0) != $true))) | (powersetI != $true)) & ((powersetI = $true) | ? [X3 : $i,X4 : $i] : (((in @ X4 @ (powerset @ X3)) != $true) & ! [X5 : $i] : (((in @ X5 @ X4) != $true) | ((in @ X5 @ X3) = $true))))),
% 2.73/1.93 inference(rectify,[status(thm)],[f1158])).
% 2.73/1.93 thf(f1158,plain,(
% 2.73/1.93 (! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) | ? [X2 : $i] : (((in @ X2 @ X1) = $true) & ((in @ X2 @ X0) != $true))) | (powersetI != $true)) & ((powersetI = $true) | ? [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) != $true) & ! [X2 : $i] : (((in @ X2 @ X1) != $true) | ((in @ X2 @ X0) = $true))))),
% 2.73/1.93 inference(nnf_transformation,[status(thm)],[f767])).
% 2.73/1.93 thf(f767,plain,(
% 2.73/1.93 ! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) | ? [X2 : $i] : (((in @ X2 @ X1) = $true) & ((in @ X2 @ X0) != $true))) <=> (powersetI = $true)),
% 2.73/1.93 inference(ennf_transformation,[status(thm)],[f310])).
% 2.73/1.93 thf(f310,plain,(
% 2.73/1.93 (powersetI = $true) <=> ! [X0 : $i,X1 : $i] : (! [X2 : $i] : (((in @ X2 @ X1) = $true) => ((in @ X2 @ X0) = $true)) => ((in @ X1 @ (powerset @ X0)) = $true))),
% 2.73/1.93 inference(fool_elimination,[status(thm)],[f309])).
% 2.73/1.93 thf(f309,plain,(
% 2.73/1.93 (powersetI = ! [X0 : $i,X1 : $i] : (! [X2 : $i] : ((in @ X2 @ X1) => (in @ X2 @ X0)) => (in @ X1 @ (powerset @ X0))))),
% 2.73/1.93 inference(rectify,[status(thm)],[f37])).
% 2.73/1.93 thf(f37,axiom,(
% 2.73/1.93 (powersetI = ! [X3 : $i,X4 : $i] : (! [X1 : $i] : ((in @ X1 @ X4) => (in @ X1 @ X3)) => (in @ X4 @ (powerset @ X3))))),
% 2.73/1.93 file('/export/starexec/sandbox2/tmp/tmp.DvUejQI5fj/DTF2THF_12059.p',powersetI)).
% 2.73/1.93 thf(f4761,plain,(
% 2.73/1.93 ( ! [X0 : $i] : (($true != (in @ (sK99 @ X0 @ sK312) @ sK310)) | ((in @ X0 @ (powerset @ sK312)) = $true)) )),
% 2.73/1.93 inference(trivial_inequality_removal,[status(thm)],[f4760])).
% 2.73/1.93 thf(f4760,plain,(
% 2.73/1.93 ( ! [X0 : $i] : (($true != $true) | ($true != (in @ (sK99 @ X0 @ sK312) @ sK310)) | ((in @ X0 @ (powerset @ sK312)) = $true)) )),
% 2.73/1.93 inference(duplicate_literal_removal,[status(thm)],[f4743])).
% 2.73/1.93 thf(f4743,plain,(
% 2.73/1.93 ( ! [X0 : $i] : (($true != (in @ (sK99 @ X0 @ sK312) @ sK310)) | ((in @ X0 @ (powerset @ sK312)) = $true) | ($true != $true) | ($true != (in @ (sK99 @ X0 @ sK312) @ sK310))) )),
% 2.73/1.93 inference(superposition,[status(thm)],[f3629,f3540])).
% 2.73/1.93 thf(f3540,plain,(
% 2.73/1.93 ( ! [X0 : $i] : (($true = (in @ X0 @ sK311)) | ($true != (in @ X0 @ sK310))) )),
% 2.73/1.93 inference(trivial_inequality_removal,[status(thm)],[f3539])).
% 2.73/1.93 thf(f3539,plain,(
% 2.73/1.93 ( ! [X0 : $i] : (($true != (in @ X0 @ sK310)) | ($true = (in @ X0 @ sK311)) | ($true != $true)) )),
% 2.73/1.93 inference(forward_demodulation,[status(thm)],[f3532,f2690])).
% 2.73/1.93 thf(f2690,plain,(
% 2.73/1.93 (powersetAx = $true)),
% 2.73/1.93 inference(cnf_transformation,[status(thm)],[f1496])).
% 2.73/1.93 thf(f3532,plain,(
% 2.73/1.93 ( ! [X0 : $i] : (($true != (in @ X0 @ sK310)) | (powersetAx != $true) | ($true = (in @ X0 @ sK311))) )),
% 2.73/1.93 inference(trivial_inequality_removal,[status(thm)],[f3526])).
% 2.73/1.93 thf(f3526,plain,(
% 2.73/1.93 ( ! [X0 : $i] : (($true != $true) | (powersetAx != $true) | ($true = (in @ X0 @ sK311)) | ($true != (in @ X0 @ sK310))) )),
% 2.73/1.93 inference(superposition,[status(thm)],[f2891,f2738])).
% 2.73/1.93 thf(f2738,plain,(
% 2.73/1.93 ((in @ sK310 @ (powerset @ sK311)) = $true)),
% 2.73/1.93 inference(cnf_transformation,[status(thm)],[f1496])).
% 2.73/1.93 thf(f2891,plain,(
% 2.73/1.93 ( ! [X7 : $i,X4 : $i,X5 : $i] : (((in @ X5 @ (powerset @ X4)) != $true) | ((in @ X7 @ X4) = $true) | (powersetAx != $true) | ($true != (in @ X7 @ X5))) )),
% 2.73/1.93 inference(cnf_transformation,[status(thm)],[f1635])).
% 2.73/1.93 thf(f1635,plain,(
% 2.73/1.93 ((powersetAx = $true) | (((((in @ sK406 @ sK404) != $true) & ((in @ sK406 @ sK405) = $true)) | ($true != (in @ sK405 @ (powerset @ sK404)))) & (! [X3 : $i] : (($true = (in @ X3 @ sK404)) | ((in @ X3 @ sK405) != $true)) | ($true = (in @ sK405 @ (powerset @ sK404)))))) & (! [X4 : $i,X5 : $i] : ((((in @ X5 @ (powerset @ X4)) = $true) | (($true != (in @ (sK407 @ X5 @ X4) @ X4)) & ((in @ (sK407 @ X5 @ X4) @ X5) = $true))) & (! [X7 : $i] : (((in @ X7 @ X4) = $true) | ($true != (in @ X7 @ X5))) | ((in @ X5 @ (powerset @ X4)) != $true))) | (powersetAx != $true))),
% 2.73/1.93 inference(skolemisation,[status(esa),new_symbols(skolem,[sK404,sK405,sK406,sK407])],[f1631,f1634,f1633,f1632])).
% 2.73/1.93 thf(f1632,plain,(
% 2.73/1.93 ? [X0 : $i,X1 : $i] : ((? [X2 : $i] : (((in @ X2 @ X0) != $true) & ((in @ X2 @ X1) = $true)) | ((in @ X1 @ (powerset @ X0)) != $true)) & (! [X3 : $i] : (((in @ X3 @ X0) = $true) | ($true != (in @ X3 @ X1))) | ((in @ X1 @ (powerset @ X0)) = $true))) => ((? [X2 : $i] : (((in @ X2 @ sK404) != $true) & ($true = (in @ X2 @ sK405))) | ($true != (in @ sK405 @ (powerset @ sK404)))) & (! [X3 : $i] : (($true = (in @ X3 @ sK404)) | ((in @ X3 @ sK405) != $true)) | ($true = (in @ sK405 @ (powerset @ sK404)))))),
% 2.73/1.93 introduced(definition,[],[choice_axiom])).
% 2.73/1.93 thf(f1633,plain,(
% 2.73/1.93 ? [X2 : $i] : (((in @ X2 @ sK404) != $true) & ($true = (in @ X2 @ sK405))) => (((in @ sK406 @ sK404) != $true) & ((in @ sK406 @ sK405) = $true))),
% 2.73/1.93 introduced(definition,[],[choice_axiom])).
% 2.73/1.93 thf(f1634,plain,(
% 2.73/1.93 ! [X4 : $i,X5 : $i] : (? [X6 : $i] : (((in @ X6 @ X4) != $true) & ((in @ X6 @ X5) = $true)) => (($true != (in @ (sK407 @ X5 @ X4) @ X4)) & ((in @ (sK407 @ X5 @ X4) @ X5) = $true)))),
% 2.73/1.93 introduced(definition,[],[choice_axiom])).
% 2.73/1.93 thf(f1631,plain,(
% 2.73/1.93 ((powersetAx = $true) | ? [X0 : $i,X1 : $i] : ((? [X2 : $i] : (((in @ X2 @ X0) != $true) & ((in @ X2 @ X1) = $true)) | ((in @ X1 @ (powerset @ X0)) != $true)) & (! [X3 : $i] : (((in @ X3 @ X0) = $true) | ($true != (in @ X3 @ X1))) | ((in @ X1 @ (powerset @ X0)) = $true)))) & (! [X4 : $i,X5 : $i] : ((((in @ X5 @ (powerset @ X4)) = $true) | ? [X6 : $i] : (((in @ X6 @ X4) != $true) & ((in @ X6 @ X5) = $true))) & (! [X7 : $i] : (((in @ X7 @ X4) = $true) | ($true != (in @ X7 @ X5))) | ((in @ X5 @ (powerset @ X4)) != $true))) | (powersetAx != $true))),
% 2.73/1.93 inference(rectify,[status(thm)],[f1630])).
% 2.73/1.93 thf(f1630,plain,(
% 2.73/1.93 ((powersetAx = $true) | ? [X0 : $i,X1 : $i] : ((? [X2 : $i] : (((in @ X2 @ X0) != $true) & ((in @ X2 @ X1) = $true)) | ((in @ X1 @ (powerset @ X0)) != $true)) & (! [X2 : $i] : (((in @ X2 @ X0) = $true) | ((in @ X2 @ X1) != $true)) | ((in @ X1 @ (powerset @ X0)) = $true)))) & (! [X0 : $i,X1 : $i] : ((((in @ X1 @ (powerset @ X0)) = $true) | ? [X2 : $i] : (((in @ X2 @ X0) != $true) & ((in @ X2 @ X1) = $true))) & (! [X2 : $i] : (((in @ X2 @ X0) = $true) | ((in @ X2 @ X1) != $true)) | ((in @ X1 @ (powerset @ X0)) != $true))) | (powersetAx != $true))),
% 2.73/1.93 inference(nnf_transformation,[status(thm)],[f864])).
% 2.73/1.93 thf(f864,plain,(
% 2.73/1.93 (powersetAx = $true) <=> ! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) <=> ! [X2 : $i] : (((in @ X2 @ X0) = $true) | ((in @ X2 @ X1) != $true)))),
% 2.73/1.93 inference(ennf_transformation,[status(thm)],[f604])).
% 2.73/1.93 thf(f604,plain,(
% 2.73/1.93 ! [X0 : $i,X1 : $i] : (! [X2 : $i] : (((in @ X2 @ X1) = $true) => ((in @ X2 @ X0) = $true)) <=> ((in @ X1 @ (powerset @ X0)) = $true)) <=> (powersetAx = $true)),
% 2.73/1.93 inference(fool_elimination,[status(thm)],[f603])).
% 2.73/1.93 thf(f603,plain,(
% 2.73/1.93 (! [X0 : $i,X1 : $i] : ((in @ X1 @ (powerset @ X0)) <=> ! [X2 : $i] : ((in @ X2 @ X1) => (in @ X2 @ X0))) = powersetAx)),
% 2.73/1.93 inference(rectify,[status(thm)],[f5])).
% 2.73/1.93 thf(f5,axiom,(
% 2.73/1.93 (! [X3 : $i,X4 : $i] : ((in @ X4 @ (powerset @ X3)) <=> ! [X1 : $i] : ((in @ X1 @ X4) => (in @ X1 @ X3))) = powersetAx)),
% 2.73/1.93 file('/export/starexec/sandbox2/tmp/tmp.DvUejQI5fj/DTF2THF_12059.p',powersetAx)).
% 2.73/1.93 thf(f3629,plain,(
% 2.73/1.93 ( ! [X0 : $i] : (($true != (in @ (sK99 @ X0 @ sK312) @ sK311)) | ($true != (in @ (sK99 @ X0 @ sK312) @ sK310)) | ((in @ X0 @ (powerset @ sK312)) = $true)) )),
% 2.73/1.93 inference(trivial_inequality_removal,[status(thm)],[f3628])).
% 2.73/1.93 thf(f3628,plain,(
% 2.73/1.93 ( ! [X0 : $i] : (($true != (in @ (sK99 @ X0 @ sK312) @ sK310)) | ((in @ X0 @ (powerset @ sK312)) = $true) | ($true != $true) | ($true != (in @ (sK99 @ X0 @ sK312) @ sK311))) )),
% 2.73/1.93 inference(forward_demodulation,[status(thm)],[f3579,f2672])).
% 2.73/1.93 thf(f3579,plain,(
% 2.73/1.93 ( ! [X0 : $i] : (($true != (in @ (sK99 @ X0 @ sK312) @ sK311)) | ($true != (in @ (sK99 @ X0 @ sK312) @ sK310)) | ((in @ X0 @ (powerset @ sK312)) = $true) | (powersetI != $true)) )),
% 2.73/1.93 inference(trivial_inequality_removal,[status(thm)],[f3564])).
% 2.73/1.93 thf(f3564,plain,(
% 2.73/1.93 ( ! [X0 : $i] : (($true != (in @ (sK99 @ X0 @ sK312) @ sK311)) | ($true != $true) | ($true != (in @ (sK99 @ X0 @ sK312) @ sK310)) | (powersetI != $true) | ((in @ X0 @ (powerset @ sK312)) = $true)) )),
% 2.73/1.93 inference(superposition,[status(thm)],[f2255,f2736])).
% 2.73/1.93 thf(f2736,plain,(
% 2.73/1.93 ( ! [X3 : $i] : (($true = (in @ X3 @ sK312)) | ($true != (in @ X3 @ sK310)) | ($true != (in @ X3 @ sK311))) )),
% 2.73/1.93 inference(cnf_transformation,[status(thm)],[f1496])).
% 2.73/1.93 thf(f2255,plain,(
% 2.73/1.93 ( ! [X0 : $i,X1 : $i] : (((in @ (sK99 @ X1 @ X0) @ X0) != $true) | (powersetI != $true) | ((in @ X1 @ (powerset @ X0)) = $true)) )),
% 2.73/1.93 inference(cnf_transformation,[status(thm)],[f1162])).
% 2.73/1.93 thf(f6603,plain,(
% 2.73/1.93 ~spl732_70),
% 2.73/1.93 inference(avatar_split_clause,[status(thm)],[f6602,f4678])).
% 2.73/1.93 thf(f6602,plain,(
% 2.73/1.93 ($true != (subset @ sK310 @ sK312))),
% 2.73/1.93 inference(trivial_inequality_removal,[status(thm)],[f6601])).
% 2.73/1.93 thf(f6601,plain,(
% 2.73/1.93 ($true != (subset @ sK310 @ sK312)) | ($true != $true)),
% 2.73/1.93 inference(forward_demodulation,[status(thm)],[f6588,f2638])).
% 2.73/1.93 thf(f2638,plain,(
% 2.73/1.93 (setunionsingleton = $true)),
% 2.73/1.93 inference(cnf_transformation,[status(thm)],[f1496])).
% 2.73/1.93 thf(f6588,plain,(
% 2.73/1.93 ($true != (subset @ sK310 @ sK312)) | (setunionsingleton != $true)),
% 2.73/1.93 inference(superposition,[status(thm)],[f3846,f2880])).
% 2.73/1.93 thf(f2880,plain,(
% 2.73/1.93 ( ! [X0 : $i] : (((setunion @ (setadjoin @ X0 @ emptyset)) = X0) | (setunionsingleton != $true)) )),
% 2.73/1.93 inference(cnf_transformation,[status(thm)],[f1617])).
% 2.73/1.93 thf(f1617,plain,(
% 2.73/1.93 (! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0) | (setunionsingleton != $true)) & ((setunionsingleton = $true) | (sK395 != (setunion @ (setadjoin @ sK395 @ emptyset))))),
% 2.73/1.93 inference(skolemisation,[status(esa),new_symbols(skolem,[sK395])],[f1615,f1616])).
% 2.73/1.93 thf(f1616,plain,(
% 2.73/1.93 ? [X1 : $i] : ((setunion @ (setadjoin @ X1 @ emptyset)) != X1) => (sK395 != (setunion @ (setadjoin @ sK395 @ emptyset)))),
% 2.73/1.93 introduced(definition,[],[choice_axiom])).
% 2.73/1.93 thf(f1615,plain,(
% 2.73/1.93 (! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0) | (setunionsingleton != $true)) & ((setunionsingleton = $true) | ? [X1 : $i] : ((setunion @ (setadjoin @ X1 @ emptyset)) != X1))),
% 2.73/1.93 inference(rectify,[status(thm)],[f1614])).
% 2.73/1.93 thf(f1614,plain,(
% 2.73/1.93 (! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0) | (setunionsingleton != $true)) & ((setunionsingleton = $true) | ? [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) != X0))),
% 2.73/1.93 inference(nnf_transformation,[status(thm)],[f246])).
% 2.73/1.93 thf(f246,plain,(
% 2.73/1.93 ! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0) <=> (setunionsingleton = $true)),
% 2.73/1.93 inference(fool_elimination,[status(thm)],[f245])).
% 2.73/1.93 thf(f245,plain,(
% 2.73/1.93 (setunionsingleton = ! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0))),
% 2.73/1.93 inference(rectify,[status(thm)],[f155])).
% 2.73/1.93 thf(f155,axiom,(
% 2.73/1.93 (setunionsingleton = ! [X1 : $i] : ((setunion @ (setadjoin @ X1 @ emptyset)) = X1))),
% 2.73/1.93 file('/export/starexec/sandbox2/tmp/tmp.DvUejQI5fj/DTF2THF_12059.p',setunionsingleton)).
% 2.73/1.93 thf(f3846,plain,(
% 2.73/1.93 ((subset @ (setunion @ (setadjoin @ sK310 @ emptyset)) @ sK312) != $true)),
% 2.73/1.93 inference(trivial_inequality_removal,[status(thm)],[f3845])).
% 2.73/1.93 thf(f3845,plain,(
% 2.73/1.93 ($true != $true) | ((subset @ (setunion @ (setadjoin @ sK310 @ emptyset)) @ sK312) != $true)),
% 2.73/1.93 inference(forward_demodulation,[status(thm)],[f3792,f2541])).
% 2.73/1.93 thf(f2541,plain,(
% 2.73/1.93 (setunionsingleton2 = $true)),
% 2.73/1.93 inference(cnf_transformation,[status(thm)],[f1496])).
% 2.73/1.93 thf(f3792,plain,(
% 2.73/1.93 (setunionsingleton2 != $true) | ((subset @ (setunion @ (setadjoin @ sK310 @ emptyset)) @ sK312) != $true)),
% 2.73/1.93 inference(trivial_inequality_removal,[status(thm)],[f3756])).
% 2.73/1.93 thf(f3756,plain,(
% 2.73/1.93 ($true != $true) | ((subset @ (setunion @ (setadjoin @ sK310 @ emptyset)) @ sK312) != $true) | (setunionsingleton2 != $true)),
% 2.73/1.93 inference(superposition,[status(thm)],[f3493,f3088])).
% 2.73/1.93 thf(f3088,plain,(
% 2.73/1.93 ( ! [X0 : $i] : (((subset @ X0 @ (setunion @ (setadjoin @ X0 @ emptyset))) = $true) | (setunionsingleton2 != $true)) )),
% 2.73/1.93 inference(cnf_transformation,[status(thm)],[f1870])).
% 2.73/1.93 thf(f1870,plain,(
% 2.73/1.93 (! [X0 : $i] : ((subset @ X0 @ (setunion @ (setadjoin @ X0 @ emptyset))) = $true) | (setunionsingleton2 != $true)) & ((setunionsingleton2 = $true) | ($true != (subset @ sK560 @ (setunion @ (setadjoin @ sK560 @ emptyset)))))),
% 2.73/1.93 inference(skolemisation,[status(esa),new_symbols(skolem,[sK560])],[f1868,f1869])).
% 2.73/1.93 thf(f1869,plain,(
% 2.73/1.93 ? [X1 : $i] : ((subset @ X1 @ (setunion @ (setadjoin @ X1 @ emptyset))) != $true) => ($true != (subset @ sK560 @ (setunion @ (setadjoin @ sK560 @ emptyset))))),
% 2.73/1.93 introduced(definition,[],[choice_axiom])).
% 2.73/1.93 thf(f1868,plain,(
% 2.73/1.93 (! [X0 : $i] : ((subset @ X0 @ (setunion @ (setadjoin @ X0 @ emptyset))) = $true) | (setunionsingleton2 != $true)) & ((setunionsingleton2 = $true) | ? [X1 : $i] : ((subset @ X1 @ (setunion @ (setadjoin @ X1 @ emptyset))) != $true))),
% 2.73/1.93 inference(rectify,[status(thm)],[f1867])).
% 2.73/1.93 thf(f1867,plain,(
% 2.73/1.93 (! [X0 : $i] : ((subset @ X0 @ (setunion @ (setadjoin @ X0 @ emptyset))) = $true) | (setunionsingleton2 != $true)) & ((setunionsingleton2 = $true) | ? [X0 : $i] : ((subset @ X0 @ (setunion @ (setadjoin @ X0 @ emptyset))) != $true))),
% 2.73/1.93 inference(nnf_transformation,[status(thm)],[f606])).
% 2.73/1.93 thf(f606,plain,(
% 2.73/1.93 ! [X0 : $i] : ((subset @ X0 @ (setunion @ (setadjoin @ X0 @ emptyset))) = $true) <=> (setunionsingleton2 = $true)),
% 2.73/1.93 inference(fool_elimination,[status(thm)],[f605])).
% 2.73/1.93 thf(f605,plain,(
% 2.73/1.93 (! [X0 : $i] : (subset @ X0 @ (setunion @ (setadjoin @ X0 @ emptyset))) = setunionsingleton2)),
% 2.73/1.93 inference(rectify,[status(thm)],[f154])).
% 2.73/1.93 thf(f154,axiom,(
% 2.73/1.93 (! [X3 : $i] : (subset @ X3 @ (setunion @ (setadjoin @ X3 @ emptyset))) = setunionsingleton2)),
% 2.73/1.93 file('/export/starexec/sandbox2/tmp/tmp.DvUejQI5fj/DTF2THF_12059.p',setunionsingleton2)).
% 2.73/1.93 thf(f3493,plain,(
% 2.73/1.93 ( ! [X0 : $i] : (((subset @ sK310 @ X0) != $true) | ((subset @ X0 @ sK312) != $true)) )),
% 2.73/1.93 inference(trivial_inequality_removal,[status(thm)],[f3492])).
% 2.73/1.93 thf(f3492,plain,(
% 2.73/1.93 ( ! [X0 : $i] : (((subset @ X0 @ sK312) != $true) | ((subset @ sK310 @ X0) != $true) | ($true != $true)) )),
% 2.73/1.93 inference(forward_demodulation,[status(thm)],[f3485,f2663])).
% 2.73/1.93 thf(f2663,plain,(
% 2.73/1.93 (subsetTrans = $true)),
% 2.73/1.93 inference(cnf_transformation,[status(thm)],[f1496])).
% 2.73/1.93 thf(f3485,plain,(
% 2.73/1.93 ( ! [X0 : $i] : (((subset @ sK310 @ X0) != $true) | (subsetTrans != $true) | ((subset @ X0 @ sK312) != $true)) )),
% 2.73/1.93 inference(trivial_inequality_removal,[status(thm)],[f3483])).
% 2.73/1.93 thf(f3483,plain,(
% 2.73/1.93 ( ! [X0 : $i] : (((subset @ X0 @ sK312) != $true) | ($true != $true) | (subsetTrans != $true) | ((subset @ sK310 @ X0) != $true)) )),
% 2.73/1.93 inference(superposition,[status(thm)],[f2735,f3274])).
% 2.73/1.93 thf(f3274,plain,(
% 2.73/1.93 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((subset @ X0 @ X2) = $true) | ((subset @ X0 @ X1) != $true) | (subsetTrans != $true) | ((subset @ X1 @ X2) != $true)) )),
% 2.73/1.93 inference(cnf_transformation,[status(thm)],[f2086])).
% 2.73/1.93 thf(f2086,plain,(
% 2.73/1.93 (! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X0 @ X2) = $true) | ((subset @ X1 @ X2) != $true) | ((subset @ X0 @ X1) != $true)) | (subsetTrans != $true)) & ((subsetTrans = $true) | (($true != (subset @ sK696 @ sK698)) & ($true = (subset @ sK697 @ sK698)) & ($true = (subset @ sK696 @ sK697))))),
% 2.73/1.93 inference(skolemisation,[status(esa),new_symbols(skolem,[sK696,sK697,sK698])],[f2084,f2085])).
% 2.73/1.93 thf(f2085,plain,(
% 2.73/1.93 ? [X3 : $i,X4 : $i,X5 : $i] : (((subset @ X3 @ X5) != $true) & ((subset @ X4 @ X5) = $true) & ((subset @ X3 @ X4) = $true)) => (($true != (subset @ sK696 @ sK698)) & ($true = (subset @ sK697 @ sK698)) & ($true = (subset @ sK696 @ sK697)))),
% 2.73/1.93 introduced(definition,[],[choice_axiom])).
% 2.73/1.93 thf(f2084,plain,(
% 2.73/1.93 (! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X0 @ X2) = $true) | ((subset @ X1 @ X2) != $true) | ((subset @ X0 @ X1) != $true)) | (subsetTrans != $true)) & ((subsetTrans = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (((subset @ X3 @ X5) != $true) & ((subset @ X4 @ X5) = $true) & ((subset @ X3 @ X4) = $true)))),
% 2.73/1.93 inference(rectify,[status(thm)],[f2083])).
% 2.73/1.93 thf(f2083,plain,(
% 2.73/1.93 (! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X0 @ X2) = $true) | ((subset @ X1 @ X2) != $true) | ((subset @ X0 @ X1) != $true)) | (subsetTrans != $true)) & ((subsetTrans = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X0 @ X2) != $true) & ((subset @ X1 @ X2) = $true) & ((subset @ X0 @ X1) = $true)))),
% 2.73/1.93 inference(nnf_transformation,[status(thm)],[f937])).
% 2.73/1.93 thf(f937,plain,(
% 2.73/1.93 ! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X0 @ X2) = $true) | ((subset @ X1 @ X2) != $true) | ((subset @ X0 @ X1) != $true)) <=> (subsetTrans = $true)),
% 2.73/1.93 inference(flattening,[status(thm)],[f936])).
% 2.73/1.93 thf(f936,plain,(
% 2.73/1.93 ! [X0 : $i,X1 : $i,X2 : $i] : ((((subset @ X0 @ X2) = $true) | ((subset @ X1 @ X2) != $true)) | ((subset @ X0 @ X1) != $true)) <=> (subsetTrans = $true)),
% 2.73/1.93 inference(ennf_transformation,[status(thm)],[f322])).
% 2.73/1.93 thf(f322,plain,(
% 2.73/1.93 ! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X0 @ X1) = $true) => (((subset @ X1 @ X2) = $true) => ((subset @ X0 @ X2) = $true))) <=> (subsetTrans = $true)),
% 2.73/1.93 inference(fool_elimination,[status(thm)],[f321])).
% 2.73/1.93 thf(f321,plain,(
% 2.73/1.93 (! [X0 : $i,X1 : $i,X2 : $i] : ((subset @ X0 @ X1) => ((subset @ X1 @ X2) => (subset @ X0 @ X2))) = subsetTrans)),
% 2.73/1.93 inference(rectify,[status(thm)],[f89])).
% 2.73/1.93 thf(f89,axiom,(
% 2.73/1.93 (! [X3 : $i,X4 : $i,X5 : $i] : ((subset @ X3 @ X4) => ((subset @ X4 @ X5) => (subset @ X3 @ X5))) = subsetTrans)),
% 2.73/1.93 file('/export/starexec/sandbox2/tmp/tmp.DvUejQI5fj/DTF2THF_12059.p',subsetTrans)).
% 2.73/1.93 thf(f2735,plain,(
% 2.73/1.93 ($true != (subset @ sK310 @ sK312))),
% 2.73/1.93 inference(cnf_transformation,[status(thm)],[f1496])).
% 2.73/1.93 % SZS output end Proof for DTF2THF_12059
% 2.73/1.93 % (12354)------------------------------
% 2.73/1.93 % (12354)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.73/1.93 % (12354)Termination reason: Refutation
% 2.73/1.93
% 2.73/1.93 % (12354)Memory used [KB]: 11129
% 2.73/1.93 % (12354)Time elapsed: 0.257 s
% 2.73/1.93 % (12354)Instructions burned: 717 (million)
% 2.73/1.93 % (12354)------------------------------
% 2.73/1.93 % (12354)------------------------------
% 2.73/1.93 % (12286)Success in time 0.357 s
%------------------------------------------------------------------------------