%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU717^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n015.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 14.28s 3.27s
% Output : Refutation 14.28s
% Verified :
% SZS Type : Refutation
% Derivation depth : 16
% Number of leaves : 12
% Syntax : Number of formulae : 61 ( 14 unt; 0 typ; 0 def)
% Number of atoms : 4391 (1184 equ; 0 cnn)
% Maximal formula atoms : 239 ( 71 avg)
% Number of connectives : 2546 ( 126 ~; 106 |; 763 &; 569 @)
% ( 10 <=>; 972 =>; 0 <=; 0 <~>)
% Maximal formula depth : 246 ( 32 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 433 ( 430 usr; 245 con; 0-4 aty)
% Number of variables : 147 ( 0 ^; 106 !; 41 ?; 147 :)
% 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_287,type,
sP2: $i > $i > $i > $i > $o ).
thf(func_def_290,type,
sP5: $i > $i > $o ).
thf(func_def_291,type,
sP6: $i > $o ).
thf(func_def_292,type,
sP7: $i > $i > $o ).
thf(func_def_293,type,
sP8: $i > $i > $o ).
thf(func_def_294,type,
sP9: $i > $i > $i > $o ).
thf(func_def_296,type,
sP11: $i > $o > $i > $i > $o ).
thf(func_def_303,type,
sK18: $i > $o ).
thf(func_def_306,type,
sK21: $i > ( $i > $o ) > $i ).
thf(func_def_307,type,
sK22: $i > ( $i > $o ) > $i ).
thf(func_def_323,type,
sK38: ( $i > $o ) > $i ).
thf(func_def_324,type,
sK39: ( $i > $o ) > $i ).
thf(func_def_325,type,
sK40: $i > $o ).
thf(func_def_328,type,
sK43: $i > $i > $i > $i > $i ).
thf(func_def_329,type,
sK44: $i > $i > $i > $i > $i ).
thf(func_def_338,type,
sK53: $i > $o ).
thf(func_def_340,type,
sK55: ( $i > $o ) > $i > $i ).
thf(func_def_341,type,
sK56: ( $i > $o ) > $i > $i ).
thf(func_def_343,type,
sK58: $i > $o ).
thf(func_def_359,type,
sK74: $i > $i > $o ).
thf(func_def_361,type,
sK76: $i > $o ).
thf(func_def_363,type,
sK78: ( $i > $o ) > $i > $i ).
thf(func_def_364,type,
sK79: $i > ( $i > $i ) > $i > $i ).
thf(func_def_366,type,
sK81: $i > $i ).
thf(func_def_369,type,
sK84: $i > $i > $i > $i ).
thf(func_def_379,type,
sK94: $i > $i > $i > $i > $i ).
thf(func_def_380,type,
sK95: $i > $i > $i > $i > $i ).
thf(func_def_381,type,
sK96: $i > $i > $i > $i > $i ).
thf(func_def_382,type,
sK97: $i > $i > $i > $i > $i ).
thf(func_def_388,type,
sK103: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_389,type,
sK104: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_393,type,
sK108: $i > $o ).
thf(func_def_396,type,
sK111: $i > $i ).
thf(func_def_399,type,
sK114: $i > $o ).
thf(func_def_405,type,
sK120: $i > ( $i > $i ) > $i > $i ).
thf(func_def_407,type,
sK122: $i > $i ).
thf(func_def_409,type,
sK124: $i > $i ).
thf(func_def_417,type,
sK132: $i > $o ).
thf(func_def_432,type,
sK147: $i > $o ).
thf(func_def_433,type,
sK148: ( $i > $o ) > $i > $i ).
thf(func_def_441,type,
sK156: $i > $i ).
thf(func_def_473,type,
sK188: ( $i > $o ) > $i > $i ).
thf(func_def_475,type,
sK190: $i > $o ).
thf(func_def_477,type,
sK192: $i > $o ).
thf(func_def_478,type,
sK193: ( $i > $o ) > $i ).
thf(func_def_479,type,
sK194: $i > $i ).
thf(func_def_482,type,
sK197: $i > $i > ( $i > $i ) > $i ).
thf(func_def_497,type,
sK212: $i > $i ).
thf(func_def_500,type,
sK215: $i > $i ).
thf(func_def_504,type,
sK219: $i > $i > ( $i > $i ) > $i ).
thf(func_def_511,type,
sK226: $i > $o ).
thf(func_def_514,type,
sK229: $i > ( $i > $o ) > $i ).
thf(func_def_517,type,
sK232: $i > $i > $i ).
thf(func_def_521,type,
sK236: ( $i > $o ) > $i ).
thf(func_def_522,type,
sK237: $i > $o ).
thf(func_def_523,type,
sK238: $i > $i ).
thf(func_def_524,type,
sK239: ( $i > $o ) > $i > $i ).
thf(func_def_526,type,
sK241: $i > $o ).
thf(func_def_532,type,
sK247: $i > $o ).
thf(func_def_537,type,
sK252: $i > $i > $o ).
thf(func_def_553,type,
sK268: $i > $o ).
thf(func_def_555,type,
sK270: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_556,type,
sK271: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_565,type,
sK280: $i > $i ).
thf(func_def_574,type,
sK289: $i > $i > $i ).
thf(func_def_581,type,
sK296: $i > $i > $i > $i ).
thf(func_def_582,type,
sK297: $i > $i > $i > $i ).
thf(func_def_589,type,
sK304: $i > $o ).
thf(func_def_605,type,
sK320: $i > $i > $i ).
thf(func_def_606,type,
sK321: $i > $i > $i ).
thf(func_def_613,type,
sK328: $i > $o ).
thf(func_def_624,type,
sK339: $i > ( $i > $o ) > $i ).
thf(func_def_625,type,
sK340: $i > $o ).
thf(func_def_657,type,
sK372: $i > $i > $o ).
thf(func_def_668,type,
sK383: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_669,type,
sK384: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_672,type,
sK387: $i > $o ).
thf(func_def_673,type,
sK388: $i > $o ).
thf(func_def_680,type,
sK395: $i > $i > ( $i > $i ) > $i ).
thf(func_def_681,type,
sK396: $i > $i ).
thf(func_def_696,type,
sK411: ( $i > $o ) > $i ).
thf(func_def_697,type,
sK412: $i > $o ).
thf(func_def_698,type,
sK413: $i > $i ).
thf(func_def_699,type,
sK414: $i > $i > $o ).
thf(func_def_701,type,
sK416: $i > $i ).
thf(func_def_702,type,
sK417: $i > $i ).
thf(func_def_703,type,
sK418: $i > ( $i > $i > $o ) > $i ).
thf(func_def_704,type,
sK419: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_705,type,
sK420: $i > ( $i > $i > $o ) > $i ).
thf(func_def_709,type,
sK424: $i > ( $i > $o ) > $i > $i ).
thf(func_def_711,type,
sK426: $i > $o ).
thf(func_def_714,type,
sK429: $i > $o ).
thf(func_def_716,type,
sK431: ( $i > $o ) > $i > $i ).
thf(func_def_723,type,
sK438: $i > $i > $i ).
thf(func_def_740,type,
sK455: $i > $o ).
thf(func_def_748,type,
sK463: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_749,type,
sK464: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_750,type,
sK465: $i > $o ).
thf(func_def_751,type,
sK466: $i > $o ).
thf(func_def_758,type,
sK473: $i > $i > $i > $i > $i ).
thf(func_def_759,type,
sK474: $i > $i > $i ).
thf(func_def_760,type,
sK475: $i > $i > $i ).
thf(func_def_761,type,
sK476: $i > $i > $i ).
thf(func_def_762,type,
sK477: $i > $i > $i ).
thf(func_def_763,type,
sK478: $i > $i > $i ).
thf(func_def_764,type,
sK479: $i > $i > $i > $i ).
thf(func_def_765,type,
sK480: $i > $i ).
thf(func_def_766,type,
sK481: $i > $i ).
thf(func_def_767,type,
sK482: $i > $i ).
thf(func_def_768,type,
sK483: $i > $i ).
thf(func_def_769,type,
sK484: $i > $i > $i > $i ).
thf(func_def_770,type,
sK485: $i > $i > $i > $i ).
thf(func_def_771,type,
sK486: $i > $i > $i ).
thf(func_def_772,type,
sK487: $i > $i > $i ).
thf(func_def_773,type,
sK488: $i > $i > $i ).
thf(func_def_774,type,
sK489: $i > $i ).
thf(func_def_776,type,
sK491: $i > $i ).
thf(func_def_777,type,
sK492: $i > $i ).
thf(func_def_785,type,
sK500: $i > $o ).
thf(func_def_795,type,
sK510: $i > $i > $o ).
thf(func_def_803,type,
sK518: $i > $i > $i > $i ).
thf(func_def_807,type,
sK522: $i > $i > $i > $i ).
thf(func_def_808,type,
sK523: $i > ( $i > $o ) > $i ).
thf(func_def_809,type,
sK524: $i > $o ).
thf(func_def_815,type,
sK530: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_816,type,
sK531: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_817,type,
sK532: $i > $o ).
thf(func_def_818,type,
sK533: $i > $o ).
thf(func_def_826,type,
sK541: $i > $i > $o ).
thf(func_def_846,type,
sK561: $i > $i > $i ).
thf(func_def_852,type,
sK567: $i > $i ).
thf(func_def_853,type,
sK568: $i > $i ).
thf(func_def_860,type,
sK575: $i > ( $i > $o ) > $i ).
thf(func_def_861,type,
sK576: $i > $o ).
thf(func_def_878,type,
sK593: $i > $i > $i > $i > $i ).
thf(func_def_883,type,
sK598: $i > $i > $i ).
thf(func_def_887,type,
sK602: $i > $i > $o ).
thf(func_def_898,type,
sK613: $i > $i > $i ).
thf(func_def_926,type,
sK641: $i > $o ).
thf(func_def_932,type,
sK647: $i > $o ).
thf(func_def_939,type,
sK654: $i > $o ).
thf(func_def_942,type,
sK657: $i > $i > $o ).
thf(func_def_951,type,
sK666: $i > $o ).
thf(func_def_954,type,
sK669: $i > $i > $i ).
thf(func_def_982,type,
sK697: $i > $i > $i > $i > $i ).
thf(func_def_983,type,
sK698: $i > $i > $i ).
thf(func_def_991,type,
sK706: $i > $o ).
thf(func_def_999,type,
sK714: $o > $i > $i > $i ).
thf(func_def_1017,type,
sK732: $i > $i ).
thf(func_def_1020,type,
sK735: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1022,type,
ph737:
!>[X0: $tType] : X0 ).
thf(f4679,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f4678,f2642]) ).
thf(f2642,plain,
subsetI1 = $true,
inference(cnf_transformation,[status(thm)],[f1550]) ).
thf(f1550,plain,
( ( subPowSU = $true )
& ( setunionsingleton2 = $true )
& ( setadjoinSub2 = $true )
& ( exuE2 = $true )
& ( ksndpairEq = $true )
& ( setadjoin__Cong = $true )
& ( cartprodfstin = $true )
& ( notinsingleton = $true )
& ( setextT = $true )
& ( quantDeMorgan1 = $true )
& ( ( in @ sK345 @ ( powerset @ sK347 ) )
!= $true )
& ! [X3: $i] :
( ( $true
= ( in @ X3 @ sK347 ) )
| ( $true
!= ( in @ X3 @ sK346 ) )
| ( $true
!= ( in @ X3 @ sK345 ) ) )
& ( $true
= ( in @ sK347 @ ( powerset @ sK346 ) ) )
& ( $true
= ( in @ sK345 @ ( powerset @ sK346 ) ) )
& ( bs114d = $true )
& ( exuEu = $true )
& ( emptysetE = $true )
& ( subbreln = $true )
& ( funcGraphProp3 = $true )
& ( iffalseProp2 = $true )
& ( symdiffIneg1 = $true )
& ( subset2powerset = $true )
& ( funcGraphProp1 = $true )
& ( singletoninpowunion = $true )
& ( subsetTI = $true )
& ( setminusIRneg = $true )
& ( singletonsswitch = $true )
& ( quantDeMorgan4 = $true )
& ( lamProp = $true )
& ( setoftrueEq = $true )
& ( binunionE = $true )
& ( setadjoinAx = $true )
& ( setminusEL = $true )
& ( setminusLsub = $true )
& ( sepSubset = $true )
& ( binintersectSubset4 = $true )
& ( subsetRefl = $true )
& ( emptysetAx = $true )
& ( powersetsubset = $true )
& ( wellorderingAx = $true )
& ( quantDeMorgan3 = $true )
& ( dpsetconstrEL2 = $true )
& ( beta2 = $true )
& ( in__Cong = $true )
& ( exuI3 = $true )
& ( setukpairIL = $true )
& ( binintersectSubset3 = $true )
& ( ap2p = $true )
& ( cartprodpairsurjEq = $true )
& ( funcextLem = $true )
& ( ex1E2 = $true )
& ( binintersectLsub = $true )
& ( emptyInPowerset = $true )
& ( quantDeMorgan2 = $true )
& ( dsetconstr__Cong = $true )
& ( setukpairIR = $true )
& ( omegaIndAx = $true )
& ( setadjoinIL = $true )
& ( exuE1 = $true )
& ( ubforcartprodlem1 = $true )
& ( setminusSubset1 = $true )
& ( emptysetimpfalse = $true )
& ( singletonsuniq = $true )
& ( powersetI1 = $true )
& ( notequalI2 = $true )
& ( descr__Cong = $true )
& ( dsetconstrEL = $true )
& ( brelnall2 = $true )
& ( ksndsingleton = $true )
& ( funcImageSingleton = $true )
& ( ex1I2 = $true )
& ( kpairiskpair = $true )
& ( binintersectSubset1 = $true )
& ( dpsetconstrERa = $true )
& ( binintersectSubset2 = $true )
& ( setukpairinjL = $true )
& ( singletoninpowerset = $true )
& ( binintersectT_lem = $true )
& ( inCongP = $true )
& ( binintersectER = $true )
& ( sepInPowerset = $true )
& ( eqinunit = $true )
& ( emptyI = $true )
& ( ubforcartprodlem3 = $true )
& ( replAx = $true )
& ( theeq = $true )
& ( binintersectSubset5 = $true )
& ( upairsetIR = $true )
& ( dpsetconstrER = $true )
& ( dpsetconstrSub = $true )
& ( symdiffIneg2 = $true )
& ( setunion__Cong = $true )
& ( notequalI1 = $true )
& ( upairequniteq = $true )
& ( eqimpsubset1 = $true )
& ( kfstpairEq = $true )
& ( apProp = $true )
& ( funcinfuncset = $true )
& ( binunionIR = $true )
& ( singletonsubset = $true )
& ( cartprodsndpairEq = $true )
& ( funcext = $true )
& ( ex1I = $true )
& ( notinemptyset = $true )
& ( noeltsimpempty = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( iftrueorfalse = $true )
& ( dpsetconstrI = $true )
& ( setukpairinjR1 = $true )
& ( binunionIL = $true )
& ( upairsetIL = $true )
& ( binintersectEL = $true )
& ( binunionEcases = $true )
& ( eta1 = $true )
& ( complementT_lem = $true )
& ( setunionsingleton1 = $true )
& ( beta1 = $true )
& ( iftrueProp1 = $true )
& ( powersetE1 = $true )
& ( binintersectRsub = $true )
& ( powerset__Cong = $true )
& ( symdiffI1 = $true )
& ( setukpairinjR = $true )
& ( prop2set2propI = $true )
& ( setext = $true )
& ( dsetconstrI = $true )
& ( setukpairinjR12 = $true )
& ( setminusILneg = $true )
& ( cartprodpairin = $true )
& ( exuE3e = $true )
& ( subsetE2 = $true )
& ( setadjoinOr = $true )
& ( powersetAx = $true )
& ( nonemptyI1 = $true )
& ( emptysetsubset = $true )
& ( funcGraphProp2 = $true )
& ( omegaSAx = $true )
& ( setukpairinjL1 = $true )
& ( setminusER = $true )
& ( cartprodpairmemEL = $true )
& ( powersetI = $true )
& ( setukpairinjR2 = $true )
& ( emptyinunitempty = $true )
& ( notsubsetI = $true )
& ( inPowerset = $true )
& ( notdallE = $true )
& ( iftrue = $true )
& ( upairset2E = $true )
& ( setextAx = $true )
& ( app = $true )
& ( infuncsetfunc = $true )
& ( ap2apEq2 = $true )
& ( eta2 = $true )
& ( subsetemptysetimpeq = $true )
& ( lam2p = $true )
& ( emptyset__Cong = $true )
& ( symdiffI2 = $true )
& ( disjointsetsI1 = $true )
& ( subsetI2 = $true )
& ( vacuousDall = $true )
& ( emptyinPowerset = $true )
& ( symdiffE = $true )
& ( omega__Cong = $true )
& ( setminusSubset2 = $true )
& ( lam2lamEq = $true )
& ( powersetE = $true )
& ( dpsetconstrEL1 = $true )
& ( uniqinunit = $true )
& ( funcGraphProp4 = $true )
& ( foundationAx = $true )
& ( setunionE = $true )
& ( setextsub = $true )
& ( subsetI1 = $true )
& ( ifp = $true )
& ( upairsubunion = $true )
& ( exu__Cong = $true )
& ( setadjoinIR = $true )
& ( theprop = $true )
& ( eqbreln = $true )
& ( setminusELneg = $true )
& ( binunionT_lem = $true )
& ( ubforcartprodlem2 = $true )
& ( ap2apEq1 = $true )
& ( nonemptyI = $true )
& ( exuI1 = $true )
& ( ifSingleton = $true )
& ( singletonprop = $true )
& ( exuI2 = $true )
& ( iftrueProp2 = $true )
& ( binintersectI = $true )
& ( cartprodmempaircEq = $true )
& ( upairinpowunion = $true )
& ( setminusERneg = $true )
& ( nonemptyImpWitness = $true )
& ( emptyE1 = $true )
& ( eqimpsubset2 = $true )
& ( upairset2IR = $true )
& ( exuE3u = $true )
& ( setminusT_lem = $true )
& ( setOfPairsIsBReln = $true )
& ( brelnall1 = $true )
& ( prop2setI = $true )
& ( setbeta = $true )
& ( dsetconstrER = $true )
& ( cartprodmempair1 = $true )
& ( setukpairinjR11 = $true )
& ( setunionE2 = $true )
& ( kfstsingleton = $true )
& ( descrp = $true )
& ( powersetT_lem = $true )
& ( secondinupair = $true )
& ( iffalse = $true )
& ( nonemptyE1 = $true )
& ( lamp = $true )
& ( upairsetE = $true )
& ( funcext2 = $true )
& ( kpairp = $true )
& ( prop2setE = $true )
& ( setunionI = $true )
& ( ex1E1 = $true )
& ( cartprodmempair = $true )
& ( kpairsurjEq = $true )
& ( omega0Ax = $true )
& ( setunionsingleton = $true )
& ( setunionAx = $true )
& ( cartprodsndin = $true )
& ( cartprodpairmemER = $true )
& ( iffalseProp1 = $true )
& ( binunionRsub = $true )
& ( setminusI = $true )
& ( cartprodfstpairEq = $true )
& ( binunionLsub = $true )
& ( subsetE = $true )
& ( setukpairinjL2 = $true )
& ( notdexE = $true )
& ( setadjoinE = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK345,sK346,sK347])],[f786,f1549,f1548]) ).
thf(f1548,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( $true
!= ( in @ X0 @ ( powerset @ X2 ) ) )
& ! [X3: $i] :
( ( $true
= ( in @ X3 @ X2 ) )
| ( $true
!= ( in @ X3 @ X1 ) )
| ( $true
!= ( in @ X3 @ X0 ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ X1 ) ) ) )
& ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) ) )
=> ( ? [X2: $i] :
( ( $true
!= ( in @ sK345 @ ( powerset @ X2 ) ) )
& ! [X3: $i] :
( ( $true
= ( in @ X3 @ X2 ) )
| ( $true
!= ( in @ X3 @ sK346 ) )
| ( $true
!= ( in @ X3 @ sK345 ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ sK346 ) ) ) )
& ( $true
= ( in @ sK345 @ ( powerset @ sK346 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1549,plain,
( ? [X2: $i] :
( ( $true
!= ( in @ sK345 @ ( powerset @ X2 ) ) )
& ! [X3: $i] :
( ( $true
= ( in @ X3 @ X2 ) )
| ( $true
!= ( in @ X3 @ sK346 ) )
| ( $true
!= ( in @ X3 @ sK345 ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ sK346 ) ) ) )
=> ( ( ( in @ sK345 @ ( powerset @ sK347 ) )
!= $true )
& ! [X3: $i] :
( ( $true
= ( in @ X3 @ sK347 ) )
| ( $true
!= ( in @ X3 @ sK346 ) )
| ( $true
!= ( in @ X3 @ sK345 ) ) )
& ( $true
= ( in @ sK347 @ ( powerset @ sK346 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f786,plain,
( ( subPowSU = $true )
& ( setunionsingleton2 = $true )
& ( setadjoinSub2 = $true )
& ( exuE2 = $true )
& ( ksndpairEq = $true )
& ( setadjoin__Cong = $true )
& ( cartprodfstin = $true )
& ( notinsingleton = $true )
& ( setextT = $true )
& ( quantDeMorgan1 = $true )
& ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( $true
!= ( in @ X0 @ ( powerset @ X2 ) ) )
& ! [X3: $i] :
( ( $true
= ( in @ X3 @ X2 ) )
| ( $true
!= ( in @ X3 @ X1 ) )
| ( $true
!= ( in @ X3 @ X0 ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ X1 ) ) ) )
& ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) ) )
& ( bs114d = $true )
& ( exuEu = $true )
& ( emptysetE = $true )
& ( subbreln = $true )
& ( funcGraphProp3 = $true )
& ( iffalseProp2 = $true )
& ( symdiffIneg1 = $true )
& ( subset2powerset = $true )
& ( funcGraphProp1 = $true )
& ( singletoninpowunion = $true )
& ( subsetTI = $true )
& ( setminusIRneg = $true )
& ( singletonsswitch = $true )
& ( quantDeMorgan4 = $true )
& ( lamProp = $true )
& ( setoftrueEq = $true )
& ( binunionE = $true )
& ( setadjoinAx = $true )
& ( setminusEL = $true )
& ( setminusLsub = $true )
& ( sepSubset = $true )
& ( binintersectSubset4 = $true )
& ( subsetRefl = $true )
& ( emptysetAx = $true )
& ( powersetsubset = $true )
& ( wellorderingAx = $true )
& ( quantDeMorgan3 = $true )
& ( dpsetconstrEL2 = $true )
& ( beta2 = $true )
& ( in__Cong = $true )
& ( exuI3 = $true )
& ( setukpairIL = $true )
& ( binintersectSubset3 = $true )
& ( ap2p = $true )
& ( cartprodpairsurjEq = $true )
& ( funcextLem = $true )
& ( ex1E2 = $true )
& ( binintersectLsub = $true )
& ( emptyInPowerset = $true )
& ( quantDeMorgan2 = $true )
& ( dsetconstr__Cong = $true )
& ( setukpairIR = $true )
& ( omegaIndAx = $true )
& ( setadjoinIL = $true )
& ( exuE1 = $true )
& ( ubforcartprodlem1 = $true )
& ( setminusSubset1 = $true )
& ( emptysetimpfalse = $true )
& ( singletonsuniq = $true )
& ( powersetI1 = $true )
& ( notequalI2 = $true )
& ( descr__Cong = $true )
& ( dsetconstrEL = $true )
& ( brelnall2 = $true )
& ( ksndsingleton = $true )
& ( funcImageSingleton = $true )
& ( ex1I2 = $true )
& ( kpairiskpair = $true )
& ( binintersectSubset1 = $true )
& ( dpsetconstrERa = $true )
& ( binintersectSubset2 = $true )
& ( setukpairinjL = $true )
& ( singletoninpowerset = $true )
& ( binintersectT_lem = $true )
& ( inCongP = $true )
& ( binintersectER = $true )
& ( sepInPowerset = $true )
& ( eqinunit = $true )
& ( emptyI = $true )
& ( ubforcartprodlem3 = $true )
& ( replAx = $true )
& ( theeq = $true )
& ( binintersectSubset5 = $true )
& ( upairsetIR = $true )
& ( dpsetconstrER = $true )
& ( dpsetconstrSub = $true )
& ( symdiffIneg2 = $true )
& ( setunion__Cong = $true )
& ( notequalI1 = $true )
& ( upairequniteq = $true )
& ( eqimpsubset1 = $true )
& ( kfstpairEq = $true )
& ( apProp = $true )
& ( funcinfuncset = $true )
& ( binunionIR = $true )
& ( singletonsubset = $true )
& ( cartprodsndpairEq = $true )
& ( funcext = $true )
& ( ex1I = $true )
& ( notinemptyset = $true )
& ( noeltsimpempty = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( iftrueorfalse = $true )
& ( dpsetconstrI = $true )
& ( setukpairinjR1 = $true )
& ( binunionIL = $true )
& ( upairsetIL = $true )
& ( binintersectEL = $true )
& ( binunionEcases = $true )
& ( eta1 = $true )
& ( complementT_lem = $true )
& ( setunionsingleton1 = $true )
& ( beta1 = $true )
& ( iftrueProp1 = $true )
& ( powersetE1 = $true )
& ( binintersectRsub = $true )
& ( powerset__Cong = $true )
& ( symdiffI1 = $true )
& ( setukpairinjR = $true )
& ( prop2set2propI = $true )
& ( setext = $true )
& ( dsetconstrI = $true )
& ( setukpairinjR12 = $true )
& ( setminusILneg = $true )
& ( cartprodpairin = $true )
& ( exuE3e = $true )
& ( subsetE2 = $true )
& ( setadjoinOr = $true )
& ( powersetAx = $true )
& ( nonemptyI1 = $true )
& ( emptysetsubset = $true )
& ( funcGraphProp2 = $true )
& ( omegaSAx = $true )
& ( setukpairinjL1 = $true )
& ( setminusER = $true )
& ( cartprodpairmemEL = $true )
& ( powersetI = $true )
& ( setukpairinjR2 = $true )
& ( emptyinunitempty = $true )
& ( notsubsetI = $true )
& ( inPowerset = $true )
& ( notdallE = $true )
& ( iftrue = $true )
& ( upairset2E = $true )
& ( setextAx = $true )
& ( app = $true )
& ( infuncsetfunc = $true )
& ( ap2apEq2 = $true )
& ( eta2 = $true )
& ( subsetemptysetimpeq = $true )
& ( lam2p = $true )
& ( emptyset__Cong = $true )
& ( symdiffI2 = $true )
& ( disjointsetsI1 = $true )
& ( subsetI2 = $true )
& ( vacuousDall = $true )
& ( emptyinPowerset = $true )
& ( symdiffE = $true )
& ( omega__Cong = $true )
& ( setminusSubset2 = $true )
& ( lam2lamEq = $true )
& ( powersetE = $true )
& ( dpsetconstrEL1 = $true )
& ( uniqinunit = $true )
& ( funcGraphProp4 = $true )
& ( foundationAx = $true )
& ( setunionE = $true )
& ( setextsub = $true )
& ( subsetI1 = $true )
& ( ifp = $true )
& ( upairsubunion = $true )
& ( exu__Cong = $true )
& ( setadjoinIR = $true )
& ( theprop = $true )
& ( eqbreln = $true )
& ( setminusELneg = $true )
& ( binunionT_lem = $true )
& ( ubforcartprodlem2 = $true )
& ( ap2apEq1 = $true )
& ( nonemptyI = $true )
& ( exuI1 = $true )
& ( ifSingleton = $true )
& ( singletonprop = $true )
& ( exuI2 = $true )
& ( iftrueProp2 = $true )
& ( binintersectI = $true )
& ( cartprodmempaircEq = $true )
& ( upairinpowunion = $true )
& ( setminusERneg = $true )
& ( nonemptyImpWitness = $true )
& ( emptyE1 = $true )
& ( eqimpsubset2 = $true )
& ( upairset2IR = $true )
& ( exuE3u = $true )
& ( setminusT_lem = $true )
& ( setOfPairsIsBReln = $true )
& ( brelnall1 = $true )
& ( prop2setI = $true )
& ( setbeta = $true )
& ( dsetconstrER = $true )
& ( cartprodmempair1 = $true )
& ( setukpairinjR11 = $true )
& ( setunionE2 = $true )
& ( kfstsingleton = $true )
& ( descrp = $true )
& ( powersetT_lem = $true )
& ( secondinupair = $true )
& ( iffalse = $true )
& ( nonemptyE1 = $true )
& ( lamp = $true )
& ( upairsetE = $true )
& ( funcext2 = $true )
& ( kpairp = $true )
& ( prop2setE = $true )
& ( setunionI = $true )
& ( ex1E1 = $true )
& ( cartprodmempair = $true )
& ( kpairsurjEq = $true )
& ( omega0Ax = $true )
& ( setunionsingleton = $true )
& ( setunionAx = $true )
& ( cartprodsndin = $true )
& ( cartprodpairmemER = $true )
& ( iffalseProp1 = $true )
& ( binunionRsub = $true )
& ( setminusI = $true )
& ( cartprodfstpairEq = $true )
& ( binunionLsub = $true )
& ( subsetE = $true )
& ( setukpairinjL2 = $true )
& ( notdexE = $true )
& ( setadjoinE = $true ) ),
inference(flattening,[status(thm)],[f785]) ).
thf(f785,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( $true
!= ( in @ X0 @ ( powerset @ X2 ) ) )
& ! [X3: $i] :
( ( $true
= ( in @ X3 @ X2 ) )
| ( $true
!= ( in @ X3 @ X0 ) )
| ( $true
!= ( in @ X3 @ X1 ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ X1 ) ) ) )
& ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) ) )
& ( subsetTI = $true )
& ( 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)],[f450]) ).
thf(f450,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 )
=> ( ( subsetTI = $true )
=> ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) )
=> ! [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X3: $i] :
( ( $true
= ( in @ X3 @ X1 ) )
=> ( ( $true
= ( in @ X3 @ X0 ) )
=> ( $true
= ( in @ X3 @ X2 ) ) ) )
=> ( $true
= ( in @ X0 @ ( powerset @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f449]) ).
thf(f449,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
=> ( subsetTI
=> ! [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 ) ) )
=> ( in @ X0 @ ( powerset @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f239]) ).
thf(f239,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
=> ( subsetTI
=> ! [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 ) ) )
=> ( in @ X11 @ ( powerset @ X16 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f238]) ).
thf(f238,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
=> ( subsetTI
=> ! [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 ) ) )
=> ( in @ X11 @ ( powerset @ X16 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.rvsH9IYwBY/DTF2THF_12192.p',powersetTI1) ).
thf(f4678,plain,
subsetI1 != $true,
inference(subsumption_resolution,[status(thm)],[f4673,f3598]) ).
thf(f3598,plain,
( $true
!= ( subset @ sK345 @ sK347 ) ),
inference(subsumption_resolution,[status(thm)],[f3592,f2752]) ).
thf(f2752,plain,
powersetI1 = $true,
inference(cnf_transformation,[status(thm)],[f1550]) ).
thf(f3592,plain,
( ( powersetI1 != $true )
| ( $true
!= ( subset @ sK345 @ sK347 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3585]) ).
thf(f3585,plain,
( ( powersetI1 != $true )
| ( $true != $true )
| ( $true
!= ( subset @ sK345 @ sK347 ) ) ),
inference(superposition,[status(thm)],[f2805,f3017]) ).
thf(f3017,plain,
! [X2: $i,X3: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X3 ) ) )
| ( powersetI1 != $true )
| ( $true
!= ( subset @ X2 @ X3 ) ) ),
inference(cnf_transformation,[status(thm)],[f1786]) ).
thf(f1786,plain,
( ( ( powersetI1 = $true )
| ( ( $true
!= ( in @ sK502 @ ( powerset @ sK503 ) ) )
& ( $true
= ( subset @ sK502 @ sK503 ) ) ) )
& ( ! [X2: $i,X3: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X3 ) ) )
| ( $true
!= ( subset @ X2 @ X3 ) ) )
| ( powersetI1 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK502,sK503])],[f1784,f1785]) ).
thf(f1785,plain,
( ? [X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ ( powerset @ X1 ) ) )
& ( ( subset @ X0 @ X1 )
= $true ) )
=> ( ( $true
!= ( in @ sK502 @ ( powerset @ sK503 ) ) )
& ( $true
= ( subset @ sK502 @ sK503 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1784,plain,
( ( ( powersetI1 = $true )
| ? [X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ ( powerset @ X1 ) ) )
& ( ( subset @ X0 @ X1 )
= $true ) ) )
& ( ! [X2: $i,X3: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X3 ) ) )
| ( $true
!= ( subset @ X2 @ X3 ) ) )
| ( powersetI1 != $true ) ) ),
inference(rectify,[status(thm)],[f1783]) ).
thf(f1783,plain,
( ( ( powersetI1 = $true )
| ? [X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ ( powerset @ X1 ) ) )
& ( ( subset @ X0 @ X1 )
= $true ) ) )
& ( ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) )
| ( ( subset @ X0 @ X1 )
!= $true ) )
| ( powersetI1 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f770]) ).
thf(f770,plain,
( ( powersetI1 = $true )
<=> ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) )
| ( ( subset @ X0 @ X1 )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f629]) ).
thf(f629,plain,
( ! [X0: $i,X1: $i] :
( ( ( subset @ X0 @ X1 )
= $true )
=> ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) ) )
<=> ( powersetI1 = $true ) ),
inference(fool_elimination,[status(thm)],[f628]) ).
thf(f628,plain,
( powersetI1
= ( ! [X0: $i,X1: $i] :
( ( subset @ X0 @ X1 )
=> ( in @ X0 @ ( powerset @ X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f95]) ).
thf(f95,axiom,
( powersetI1
= ( ! [X4: $i,X3: $i] :
( ( subset @ X4 @ X3 )
=> ( in @ X4 @ ( powerset @ X3 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.rvsH9IYwBY/DTF2THF_12192.p',powersetI1) ).
thf(f2805,plain,
( ( in @ sK345 @ ( powerset @ sK347 ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f1550]) ).
thf(f4673,plain,
( ( $true
= ( subset @ sK345 @ sK347 ) )
| ( subsetI1 != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4672]) ).
thf(f4672,plain,
( ( $true != $true )
| ( $true
= ( subset @ sK345 @ sK347 ) )
| ( subsetI1 != $true ) ),
inference(duplicate_literal_removal,[status(thm)],[f4666]) ).
thf(f4666,plain,
( ( $true
= ( subset @ sK345 @ sK347 ) )
| ( $true != $true )
| ( $true
= ( subset @ sK345 @ sK347 ) )
| ( subsetI1 != $true ) ),
inference(superposition,[status(thm)],[f3706,f2441]) ).
thf(f2441,plain,
! [X0: $i,X1: $i] :
( ( $true
= ( in @ ( sK232 @ X1 @ X0 ) @ X1 ) )
| ( $true
= ( subset @ X1 @ X0 ) )
| ( subsetI1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f1382]) ).
thf(f1382,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( $true
= ( in @ ( sK232 @ X1 @ X0 ) @ X1 ) )
& ( $true
!= ( in @ ( sK232 @ X1 @ X0 ) @ X0 ) ) )
| ( $true
= ( subset @ X1 @ X0 ) ) )
| ( subsetI1 != $true ) )
& ( ( subsetI1 = $true )
| ( ! [X5: $i] :
( ( ( in @ X5 @ sK234 )
!= $true )
| ( $true
= ( in @ X5 @ sK233 ) ) )
& ( ( subset @ sK234 @ sK233 )
!= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK232,sK233,sK234])],[f1379,f1381,f1380]) ).
thf(f1380,plain,
! [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( $true
= ( in @ X2 @ X1 ) )
& ( $true
!= ( in @ X2 @ X0 ) ) )
=> ( ( $true
= ( in @ ( sK232 @ X1 @ X0 ) @ X1 ) )
& ( $true
!= ( in @ ( sK232 @ X1 @ X0 ) @ X0 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1381,plain,
( ? [X3: $i,X4: $i] :
( ! [X5: $i] :
( ( ( in @ X5 @ X4 )
!= $true )
| ( ( in @ X5 @ X3 )
= $true ) )
& ( ( subset @ X4 @ X3 )
!= $true ) )
=> ( ! [X5: $i] :
( ( ( in @ X5 @ sK234 )
!= $true )
| ( $true
= ( in @ X5 @ sK233 ) ) )
& ( ( subset @ sK234 @ sK233 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1379,plain,
( ( ! [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( $true
= ( in @ X2 @ X1 ) )
& ( $true
!= ( in @ X2 @ X0 ) ) )
| ( $true
= ( subset @ X1 @ X0 ) ) )
| ( subsetI1 != $true ) )
& ( ( subsetI1 = $true )
| ? [X3: $i,X4: $i] :
( ! [X5: $i] :
( ( ( in @ X5 @ X4 )
!= $true )
| ( ( in @ X5 @ X3 )
= $true ) )
& ( ( subset @ X4 @ X3 )
!= $true ) ) ) ),
inference(rectify,[status(thm)],[f1378]) ).
thf(f1378,plain,
( ( ! [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( $true
= ( in @ X2 @ X1 ) )
& ( $true
!= ( in @ X2 @ X0 ) ) )
| ( $true
= ( subset @ X1 @ X0 ) ) )
| ( subsetI1 != $true ) )
& ( ( subsetI1 = $true )
| ? [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( $true
!= ( in @ X2 @ X1 ) )
| ( $true
= ( in @ X2 @ X0 ) ) )
& ( $true
!= ( subset @ X1 @ X0 ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f912]) ).
thf(f912,plain,
( ! [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( $true
= ( in @ X2 @ X1 ) )
& ( $true
!= ( in @ X2 @ X0 ) ) )
| ( $true
= ( subset @ X1 @ X0 ) ) )
<=> ( subsetI1 = $true ) ),
inference(ennf_transformation,[status(thm)],[f615]) ).
thf(f615,plain,
( ( subsetI1 = $true )
<=> ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( $true
= ( in @ X2 @ X1 ) )
=> ( $true
= ( in @ X2 @ X0 ) ) )
=> ( $true
= ( subset @ X1 @ X0 ) ) ) ),
inference(fool_elimination,[status(thm)],[f614]) ).
thf(f614,plain,
! [X0: $i,X1: $i] :
( ( ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ X0 ) )
=> ( subset @ X1 @ X0 ) )
= subsetI1 ),
inference(rectify,[status(thm)],[f78]) ).
thf(f78,axiom,
! [X4: $i,X3: $i] :
( ( ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( in @ X1 @ X4 ) )
=> ( subset @ X3 @ X4 ) )
= subsetI1 ),
file('/export/starexec/sandbox/tmp/tmp.rvsH9IYwBY/DTF2THF_12192.p',subsetI1) ).
thf(f3706,plain,
! [X0: $i] :
( ( $true
!= ( in @ ( sK232 @ X0 @ sK347 ) @ sK345 ) )
| ( $true
= ( subset @ X0 @ sK347 ) ) ),
inference(subsumption_resolution,[status(thm)],[f3705,f3509]) ).
thf(f3509,plain,
! [X0: $i] :
( ( $true
= ( in @ X0 @ sK346 ) )
| ( $true
!= ( in @ X0 @ sK345 ) ) ),
inference(subsumption_resolution,[status(thm)],[f3508,f2682]) ).
thf(f2682,plain,
powersetAx = $true,
inference(cnf_transformation,[status(thm)],[f1550]) ).
thf(f3508,plain,
! [X0: $i] :
( ( $true
= ( in @ X0 @ sK346 ) )
| ( $true
!= ( in @ X0 @ sK345 ) )
| ( powersetAx != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3499]) ).
thf(f3499,plain,
! [X0: $i] :
( ( $true
= ( in @ X0 @ sK346 ) )
| ( powersetAx != $true )
| ( $true != $true )
| ( $true
!= ( in @ X0 @ sK345 ) ) ),
inference(superposition,[status(thm)],[f3099,f2802]) ).
thf(f2802,plain,
( $true
= ( in @ sK345 @ ( powerset @ sK346 ) ) ),
inference(cnf_transformation,[status(thm)],[f1550]) ).
thf(f3099,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) )
| ( $true
= ( in @ X2 @ X0 ) )
| ( powersetAx != $true )
| ( $true
!= ( in @ X2 @ X1 ) ) ),
inference(cnf_transformation,[status(thm)],[f1877]) ).
thf(f1877,plain,
( ( ! [X0: $i,X1: $i] :
( ( ! [X2: $i] :
( ( $true
!= ( in @ X2 @ X1 ) )
| ( $true
= ( in @ X2 @ X0 ) ) )
| ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) ) )
& ( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
| ( ( $true
= ( in @ ( sK561 @ X1 @ X0 ) @ X1 ) )
& ( ( in @ ( sK561 @ X1 @ X0 ) @ X0 )
!= $true ) ) ) )
| ( powersetAx != $true ) )
& ( ( powersetAx = $true )
| ( ( ( $true
!= ( in @ sK563 @ ( powerset @ sK562 ) ) )
| ( ( $true
= ( in @ sK564 @ sK563 ) )
& ( ( in @ sK564 @ sK562 )
!= $true ) ) )
& ( ( $true
= ( in @ sK563 @ ( powerset @ sK562 ) ) )
| ! [X7: $i] :
( ( ( in @ X7 @ sK563 )
!= $true )
| ( $true
= ( in @ X7 @ sK562 ) ) ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK561,sK562,sK563,sK564])],[f1873,f1876,f1875,f1874]) ).
thf(f1874,plain,
! [X0: $i,X1: $i] :
( ? [X3: $i] :
( ( $true
= ( in @ X3 @ X1 ) )
& ( $true
!= ( in @ X3 @ X0 ) ) )
=> ( ( $true
= ( in @ ( sK561 @ X1 @ X0 ) @ X1 ) )
& ( ( in @ ( sK561 @ X1 @ X0 ) @ X0 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1875,plain,
( ? [X4: $i,X5: $i] :
( ( ( $true
!= ( in @ X5 @ ( powerset @ X4 ) ) )
| ? [X6: $i] :
( ( $true
= ( in @ X6 @ X5 ) )
& ( ( in @ X6 @ X4 )
!= $true ) ) )
& ( ( $true
= ( in @ X5 @ ( powerset @ X4 ) ) )
| ! [X7: $i] :
( ( $true
!= ( in @ X7 @ X5 ) )
| ( ( in @ X7 @ X4 )
= $true ) ) ) )
=> ( ( ( $true
!= ( in @ sK563 @ ( powerset @ sK562 ) ) )
| ? [X6: $i] :
( ( $true
= ( in @ X6 @ sK563 ) )
& ( ( in @ X6 @ sK562 )
!= $true ) ) )
& ( ( $true
= ( in @ sK563 @ ( powerset @ sK562 ) ) )
| ! [X7: $i] :
( ( ( in @ X7 @ sK563 )
!= $true )
| ( $true
= ( in @ X7 @ sK562 ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1876,plain,
( ? [X6: $i] :
( ( $true
= ( in @ X6 @ sK563 ) )
& ( ( in @ X6 @ sK562 )
!= $true ) )
=> ( ( $true
= ( in @ sK564 @ sK563 ) )
& ( ( in @ sK564 @ sK562 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1873,plain,
( ( ! [X0: $i,X1: $i] :
( ( ! [X2: $i] :
( ( $true
!= ( in @ X2 @ X1 ) )
| ( $true
= ( in @ X2 @ X0 ) ) )
| ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) ) )
& ( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
| ? [X3: $i] :
( ( $true
= ( in @ X3 @ X1 ) )
& ( $true
!= ( in @ X3 @ X0 ) ) ) ) )
| ( powersetAx != $true ) )
& ( ( powersetAx = $true )
| ? [X4: $i,X5: $i] :
( ( ( $true
!= ( in @ X5 @ ( powerset @ X4 ) ) )
| ? [X6: $i] :
( ( $true
= ( in @ X6 @ X5 ) )
& ( ( in @ X6 @ X4 )
!= $true ) ) )
& ( ( $true
= ( in @ X5 @ ( powerset @ X4 ) ) )
| ! [X7: $i] :
( ( $true
!= ( in @ X7 @ X5 ) )
| ( ( in @ X7 @ X4 )
= $true ) ) ) ) ) ),
inference(rectify,[status(thm)],[f1872]) ).
thf(f1872,plain,
( ( ! [X0: $i,X1: $i] :
( ( ! [X2: $i] :
( ( $true
!= ( in @ X2 @ X1 ) )
| ( $true
= ( in @ X2 @ X0 ) ) )
| ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) ) )
& ( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
| ? [X2: $i] :
( ( $true
= ( in @ X2 @ X1 ) )
& ( $true
!= ( in @ X2 @ X0 ) ) ) ) )
| ( powersetAx != $true ) )
& ( ( powersetAx = $true )
| ? [X0: $i,X1: $i] :
( ( ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) )
| ? [X2: $i] :
( ( $true
= ( in @ X2 @ X1 ) )
& ( $true
!= ( in @ X2 @ X0 ) ) ) )
& ( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
| ! [X2: $i] :
( ( $true
!= ( in @ X2 @ X1 ) )
| ( $true
= ( in @ X2 @ X0 ) ) ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f951]) ).
thf(f951,plain,
( ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( $true
!= ( in @ X2 @ X1 ) )
| ( $true
= ( in @ X2 @ X0 ) ) )
<=> ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) ) )
<=> ( powersetAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f410]) ).
thf(f410,plain,
( ( powersetAx = $true )
<=> ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
<=> ! [X2: $i] :
( ( $true
= ( in @ X2 @ X1 ) )
=> ( $true
= ( in @ X2 @ X0 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f409]) ).
thf(f409,plain,
( powersetAx
= ( ! [X0: $i,X1: $i] :
( ( in @ X1 @ ( powerset @ X0 ) )
<=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ X0 ) ) ) ) ),
inference(rectify,[status(thm)],[f5]) ).
thf(f5,axiom,
( powersetAx
= ( ! [X3: $i,X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
<=> ! [X1: $i] :
( ( in @ X1 @ X4 )
=> ( in @ X1 @ X3 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.rvsH9IYwBY/DTF2THF_12192.p',powersetAx) ).
thf(f3705,plain,
! [X0: $i] :
( ( $true
= ( subset @ X0 @ sK347 ) )
| ( $true
!= ( in @ ( sK232 @ X0 @ sK347 ) @ sK346 ) )
| ( $true
!= ( in @ ( sK232 @ X0 @ sK347 ) @ sK345 ) ) ),
inference(subsumption_resolution,[status(thm)],[f3701,f2642]) ).
thf(f3701,plain,
! [X0: $i] :
( ( $true
= ( subset @ X0 @ sK347 ) )
| ( subsetI1 != $true )
| ( $true
!= ( in @ ( sK232 @ X0 @ sK347 ) @ sK346 ) )
| ( $true
!= ( in @ ( sK232 @ X0 @ sK347 ) @ sK345 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3664]) ).
thf(f3664,plain,
! [X0: $i] :
( ( $true != $true )
| ( subsetI1 != $true )
| ( $true
= ( subset @ X0 @ sK347 ) )
| ( $true
!= ( in @ ( sK232 @ X0 @ sK347 ) @ sK346 ) )
| ( $true
!= ( in @ ( sK232 @ X0 @ sK347 ) @ sK345 ) ) ),
inference(superposition,[status(thm)],[f2440,f2804]) ).
thf(f2804,plain,
! [X3: $i] :
( ( $true
= ( in @ X3 @ sK347 ) )
| ( $true
!= ( in @ X3 @ sK346 ) )
| ( $true
!= ( in @ X3 @ sK345 ) ) ),
inference(cnf_transformation,[status(thm)],[f1550]) ).
thf(f2440,plain,
! [X0: $i,X1: $i] :
( ( $true
!= ( in @ ( sK232 @ X1 @ X0 ) @ X0 ) )
| ( $true
= ( subset @ X1 @ X0 ) )
| ( subsetI1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f1382]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU717^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n015.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 12:32:01 EST 2026
% 0.16/0.33 % CPUTime :
% 0.16/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.45 ---- Original DTF file ---
% 0.19/0.45 thf(spec,logic,$$dhol).
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 % File : SEU717^1 : TPTP v9.2.1. Released v3.7.0.
% 0.19/0.45 % Domain : Set Theory
% 0.19/0.45 % Problem : Typed Set Theory - Laws for Typed Sets
% 0.19/0.45 % Version : Especial.
% 0.19/0.45 % English : (! A:i.! X:i.in X (powerset A) -> (! Y:i.in Y (powerset A) ->
% 0.19/0.45 % (! x:i.in x A -> in x X -> in x Y) -> in X (powerset Y)))
% 0.19/0.45
% 0.19/0.45 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.19/0.45 % Source : [Bro08]
% 0.19/0.45 % Names : ZFC219g [Bro08]
% 0.19/0.45
% 0.19/0.45 % Status : Theorem
% 0.19/0.45 % Rating : 0.33 v9.1.0, 0.25 v9.0.0, 0.30 v8.2.0, 0.38 v8.1.0, 0.18 v7.5.0, 0.14 v7.4.0, 0.44 v7.2.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.43 v6.0.0, 0.57 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 1.00 v3.7.0
% 0.19/0.45 % Syntax : Number of formulae : 509 ( 237 unt; 271 typ; 237 def)
% 0.19/0.45 % Number of atoms : 1339 ( 362 equ; 0 cnn)
% 0.19/0.45 % Maximal formula atoms : 239 ( 5 avg)
% 0.19/0.45 % Number of connectives : 2609 ( 55 ~; 12 |; 44 &;1847 @)
% 0.19/0.45 % ( 14 <=>; 637 =>; 0 <=; 0 <~>)
% 0.19/0.45 % Maximal formula depth : 245 ( 2 avg)
% 0.19/0.45 % Number of types : 2 ( 0 usr)
% 0.19/0.45 % Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% 0.19/0.45 % Number of symbols : 274 ( 271 usr; 237 con; 0-4 aty)
% 0.19/0.45 % Number of variables : 798 ( 74 ^; 689 !; 35 ?; 798 :)
% 0.19/0.45 % SPC : TH0_THM_EQU_NAR
% 0.19/0.45
% 0.19/0.45 % Comments : http://mathgate.info/detsetitem.php?id=275
% 0.19/0.45 %------------------------------------------------------------------------------
% 0.19/0.45 thf(in_type,type,
% 0.19/0.45 in: $i > $i > $o ).
% 0.19/0.45
% 0.19/0.45 thf(exu_type,type,
% 0.19/0.45 exu: ( $i > $o ) > $o ).
% 0.19/0.45
% 0.19/0.45 thf(exu,definition,
% 0.19/0.45 ( exu
% 0.19/0.45 = ( ^ [Xphi: $i > $o] :
% 0.19/0.45 ? [Xx: $i] :
% 0.19/0.45 ( ( Xphi @ Xx )
% 0.19/0.45 & ! [Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xy )
% 0.19/0.45 => ( Xx = Xy ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setextAx_type,type,
% 0.19/0.45 setextAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setextAx,definition,
% 0.19/0.45 ( setextAx
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 <=> ( in @ Xx @ B ) )
% 0.19/0.45 => ( A = B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyset_type,type,
% 0.19/0.45 emptyset: $i ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetAx_type,type,
% 0.19/0.45 emptysetAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetAx,definition,
% 0.19/0.45 ( emptysetAx
% 0.19/0.45 = ( ! [Xx: $i] :
% 0.19/0.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoin_type,type,
% 0.19/0.45 setadjoin: $i > $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinAx_type,type,
% 0.19/0.45 setadjoinAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinAx,definition,
% 0.19/0.45 ( setadjoinAx
% 0.19/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.45 <=> ( ( Xy = Xx )
% 0.19/0.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(powerset_type,type,
% 0.19/0.45 powerset: $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(powersetAx_type,type,
% 0.19/0.45 powersetAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(powersetAx,definition,
% 0.19/0.45 ( powersetAx
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.45 <=> ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setunion_type,type,
% 0.19/0.45 setunion: $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(setunionAx_type,type,
% 0.19/0.45 setunionAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setunionAx,definition,
% 0.19/0.45 ( setunionAx
% 0.19/0.45 = ( ! [A: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.45 <=> ? [B: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 & ( in @ B @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(omega_type,type,
% 0.19/0.45 omega: $i ).
% 0.19/0.45
% 0.19/0.45 thf(omega0Ax_type,type,
% 0.19/0.45 omega0Ax: $o ).
% 0.19/0.45
% 0.19/0.45 thf(omega0Ax,definition,
% 0.19/0.45 ( omega0Ax
% 0.19/0.45 = ( in @ emptyset @ omega ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(omegaSAx_type,type,
% 0.19/0.45 omegaSAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(omegaSAx,definition,
% 0.19/0.45 ( omegaSAx
% 0.19/0.45 = ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ omega )
% 0.19/0.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(omegaIndAx_type,type,
% 0.19/0.45 omegaIndAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(omegaIndAx,definition,
% 0.19/0.45 ( omegaIndAx
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ( ( in @ emptyset @ A )
% 0.19/0.45 & ! [Xx: $i] :
% 0.19/0.45 ( ( ( in @ Xx @ omega )
% 0.19/0.45 & ( in @ Xx @ A ) )
% 0.19/0.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.19/0.45 => ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ omega )
% 0.19/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(replAx_type,type,
% 0.19/0.45 replAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(replAx,definition,
% 0.19/0.45 ( replAx
% 0.19/0.45 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( exu
% 0.19/0.45 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.19/0.45 => ? [B: $i] :
% 0.19/0.45 ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 <=> ? [Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ A )
% 0.19/0.45 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(foundationAx_type,type,
% 0.19/0.45 foundationAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(foundationAx,definition,
% 0.19/0.45 ( foundationAx
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.45 => ? [B: $i] :
% 0.19/0.45 ( ( in @ B @ A )
% 0.19/0.45 & ~ ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(wellorderingAx_type,type,
% 0.19/0.45 wellorderingAx: $o ).
% 0.19/0.45
% 0.19/0.45 thf(wellorderingAx,definition,
% 0.19/0.45 ( wellorderingAx
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ? [B: $i] :
% 0.19/0.45 ( ! [C: $i] :
% 0.19/0.45 ( ( in @ C @ B )
% 0.19/0.45 => ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ C )
% 0.19/0.45 => ( in @ Xx @ A ) ) )
% 0.19/0.45 & ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( ( in @ Xx @ A )
% 0.19/0.45 & ( in @ Xy @ A ) )
% 0.19/0.45 => ( ! [C: $i] :
% 0.19/0.45 ( ( in @ C @ B )
% 0.19/0.45 => ( ( in @ Xx @ C )
% 0.19/0.45 <=> ( in @ Xy @ C ) ) )
% 0.19/0.45 => ( Xx = Xy ) ) )
% 0.19/0.45 & ! [C: $i,D: $i] :
% 0.19/0.45 ( ( ( in @ C @ B )
% 0.19/0.45 & ( in @ D @ B ) )
% 0.19/0.45 => ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ C )
% 0.19/0.45 => ( in @ Xx @ D ) )
% 0.19/0.45 | ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ D )
% 0.19/0.45 => ( in @ Xx @ C ) ) ) )
% 0.19/0.45 & ! [C: $i] :
% 0.19/0.45 ( ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ C )
% 0.19/0.45 => ( in @ Xx @ A ) )
% 0.19/0.45 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.19/0.45 => ? [D: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ D @ B )
% 0.19/0.45 & ( in @ Xx @ C )
% 0.19/0.45 & ~ ? [Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ D )
% 0.19/0.45 & ( in @ Xy @ C ) )
% 0.19/0.45 & ! [E: $i] :
% 0.19/0.45 ( ( in @ E @ B )
% 0.19/0.45 => ( ! [Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ E )
% 0.19/0.45 => ( in @ Xy @ D ) )
% 0.19/0.45 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(descr_type,type,
% 0.19/0.45 descr: ( $i > $o ) > $i ).
% 0.19/0.45
% 0.19/0.45 thf(descrp_type,type,
% 0.19/0.45 descrp: $o ).
% 0.19/0.45
% 0.19/0.45 thf(descrp,definition,
% 0.19/0.45 ( descrp
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 => ( Xphi
% 0.19/0.45 @ ( descr
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstr_type,type,
% 0.19/0.45 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrI_type,type,
% 0.19/0.45 dsetconstrI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrI,definition,
% 0.19/0.45 ( dsetconstrI
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ( Xphi @ Xx )
% 0.19/0.45 => ( in @ Xx
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrEL_type,type,
% 0.19/0.45 dsetconstrEL: $o ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrEL,definition,
% 0.19/0.45 ( dsetconstrEL
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrER_type,type,
% 0.19/0.45 dsetconstrER: $o ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstrER,definition,
% 0.19/0.45 ( dsetconstrER
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.45 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuE1_type,type,
% 0.19/0.45 exuE1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuE1,definition,
% 0.19/0.45 ( exuE1
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 => ? [Xx: $i] :
% 0.19/0.45 ( ( Xphi @ Xx )
% 0.19/0.45 & ! [Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xy )
% 0.19/0.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(prop2set_type,type,
% 0.19/0.45 prop2set: $o > $i ).
% 0.19/0.45
% 0.19/0.45 thf(prop2setE_type,type,
% 0.19/0.45 prop2setE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(prop2setE,definition,
% 0.19/0.45 ( prop2setE
% 0.19/0.45 = ( ! [Xphi: $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.19/0.45 => Xphi ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetE_type,type,
% 0.19/0.45 emptysetE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetE,definition,
% 0.19/0.45 ( emptysetE
% 0.19/0.45 = ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ emptyset )
% 0.19/0.45 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetimpfalse_type,type,
% 0.19/0.45 emptysetimpfalse: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetimpfalse,definition,
% 0.19/0.45 ( emptysetimpfalse
% 0.19/0.45 = ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ emptyset )
% 0.19/0.45 => $false ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(notinemptyset_type,type,
% 0.19/0.45 notinemptyset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(notinemptyset,definition,
% 0.19/0.45 ( notinemptyset
% 0.19/0.45 = ( ! [Xx: $i] :
% 0.19/0.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuE3e_type,type,
% 0.19/0.45 exuE3e: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuE3e,definition,
% 0.19/0.45 ( exuE3e
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setext_type,type,
% 0.19/0.45 setext: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setext,definition,
% 0.19/0.45 ( setext
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( in @ Xx @ B ) )
% 0.19/0.45 => ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ A ) )
% 0.19/0.45 => ( A = B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyI_type,type,
% 0.19/0.45 emptyI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptyI,definition,
% 0.19/0.45 ( emptyI
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ~ ( in @ Xx @ A )
% 0.19/0.45 => ( A = emptyset ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(noeltsimpempty_type,type,
% 0.19/0.45 noeltsimpempty: $o ).
% 0.19/0.45
% 0.19/0.45 thf(noeltsimpempty,definition,
% 0.19/0.45 ( noeltsimpempty
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ~ ( in @ Xx @ A )
% 0.19/0.45 => ( A = emptyset ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setbeta_type,type,
% 0.19/0.45 setbeta: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setbeta,definition,
% 0.19/0.45 ( setbeta
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ( in @ Xx
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.45 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(nonempty_type,type,
% 0.19/0.45 nonempty: $i > $o ).
% 0.19/0.45
% 0.19/0.45 thf(nonempty,definition,
% 0.19/0.45 ( nonempty
% 0.19/0.45 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyE1_type,type,
% 0.19/0.45 nonemptyE1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyE1,definition,
% 0.19/0.45 ( nonemptyE1
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ( nonempty @ A )
% 0.19/0.45 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyI_type,type,
% 0.19/0.45 nonemptyI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyI,definition,
% 0.19/0.45 ( nonemptyI
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ( Xphi @ Xx )
% 0.19/0.45 => ( nonempty
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyI1_type,type,
% 0.19/0.45 nonemptyI1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyI1,definition,
% 0.19/0.45 ( nonemptyI1
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.45 => ( nonempty @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinIL_type,type,
% 0.19/0.45 setadjoinIL: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinIL,definition,
% 0.19/0.45 ( setadjoinIL
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyinunitempty_type,type,
% 0.19/0.45 emptyinunitempty: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptyinunitempty,definition,
% 0.19/0.45 ( emptyinunitempty
% 0.19/0.45 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinIR_type,type,
% 0.19/0.45 setadjoinIR: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinIR,definition,
% 0.19/0.45 ( setadjoinIR
% 0.19/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ A )
% 0.19/0.45 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinE_type,type,
% 0.19/0.45 setadjoinE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinE,definition,
% 0.19/0.45 ( setadjoinE
% 0.19/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.45 => ! [Xphi: $o] :
% 0.19/0.45 ( ( ( Xy = Xx )
% 0.19/0.45 => Xphi )
% 0.19/0.45 => ( ( ( in @ Xy @ A )
% 0.19/0.45 => Xphi )
% 0.19/0.45 => Xphi ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinOr_type,type,
% 0.19/0.45 setadjoinOr: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinOr,definition,
% 0.19/0.45 ( setadjoinOr
% 0.19/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.45 => ( ( Xy = Xx )
% 0.19/0.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setoftrueEq_type,type,
% 0.19/0.45 setoftrueEq: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setoftrueEq,definition,
% 0.19/0.45 ( setoftrueEq
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xx: $i] : $true )
% 0.19/0.45 = A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(powersetI_type,type,
% 0.19/0.45 powersetI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(powersetI,definition,
% 0.19/0.45 ( powersetI
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ A ) )
% 0.19/0.45 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyinPowerset_type,type,
% 0.19/0.45 emptyinPowerset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptyinPowerset,definition,
% 0.19/0.45 ( emptyinPowerset
% 0.19/0.45 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyInPowerset_type,type,
% 0.19/0.45 emptyInPowerset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptyInPowerset,definition,
% 0.19/0.45 ( emptyInPowerset
% 0.19/0.45 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(powersetE_type,type,
% 0.19/0.45 powersetE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(powersetE,definition,
% 0.19/0.45 ( powersetE
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.45 => ( ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setunionI_type,type,
% 0.19/0.45 setunionI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setunionI,definition,
% 0.19/0.45 ( setunionI
% 0.19/0.45 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ( ( in @ B @ A )
% 0.19/0.45 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setunionE_type,type,
% 0.19/0.45 setunionE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setunionE,definition,
% 0.19/0.45 ( setunionE
% 0.19/0.45 = ( ! [A: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.45 => ! [Xphi: $o] :
% 0.19/0.45 ( ! [B: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ( ( in @ B @ A )
% 0.19/0.45 => Xphi ) )
% 0.19/0.45 => Xphi ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(subPowSU_type,type,
% 0.19/0.45 subPowSU: $o ).
% 0.19/0.45
% 0.19/0.45 thf(subPowSU,definition,
% 0.19/0.45 ( subPowSU
% 0.19/0.45 = ( ! [A: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuE2_type,type,
% 0.19/0.45 exuE2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuE2,definition,
% 0.19/0.45 ( exuE2
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 => ? [Xx: $i] :
% 0.19/0.45 ! [Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xy )
% 0.19/0.45 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyImpWitness_type,type,
% 0.19/0.45 nonemptyImpWitness: $o ).
% 0.19/0.45
% 0.19/0.45 thf(nonemptyImpWitness,definition,
% 0.19/0.45 ( nonemptyImpWitness
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ( nonempty @ A )
% 0.19/0.45 => ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & $true ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(uniqinunit_type,type,
% 0.19/0.45 uniqinunit: $o ).
% 0.19/0.45
% 0.19/0.45 thf(uniqinunit,definition,
% 0.19/0.45 ( uniqinunit
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.45 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(notinsingleton_type,type,
% 0.19/0.45 notinsingleton: $o ).
% 0.19/0.45
% 0.19/0.45 thf(notinsingleton,definition,
% 0.19/0.45 ( notinsingleton
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xx != Xy )
% 0.19/0.45 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(eqinunit_type,type,
% 0.19/0.45 eqinunit: $o ).
% 0.19/0.45
% 0.19/0.45 thf(eqinunit,definition,
% 0.19/0.45 ( eqinunit
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xx = Xy )
% 0.19/0.45 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(singletonsswitch_type,type,
% 0.19/0.45 singletonsswitch: $o ).
% 0.19/0.45
% 0.19/0.45 thf(singletonsswitch,definition,
% 0.19/0.45 ( singletonsswitch
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.45 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(upairsetE_type,type,
% 0.19/0.45 upairsetE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(upairsetE,definition,
% 0.19/0.45 ( upairsetE
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.45 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.19/0.45 => ( ( Xz = Xx )
% 0.19/0.45 | ( Xz = Xy ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(upairsetIL_type,type,
% 0.19/0.45 upairsetIL: $o ).
% 0.19/0.45
% 0.19/0.45 thf(upairsetIL,definition,
% 0.19/0.45 ( upairsetIL
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(upairsetIR_type,type,
% 0.19/0.45 upairsetIR: $o ).
% 0.19/0.45
% 0.19/0.45 thf(upairsetIR,definition,
% 0.19/0.45 ( upairsetIR
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyE1_type,type,
% 0.19/0.45 emptyE1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptyE1,definition,
% 0.19/0.45 ( emptyE1
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ( Xphi @ Xx ) )
% 0.19/0.45 => ( ( ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 = emptyset )
% 0.19/0.45 => $false ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(vacuousDall_type,type,
% 0.19/0.45 vacuousDall: $o ).
% 0.19/0.45
% 0.19/0.45 thf(vacuousDall,definition,
% 0.19/0.45 ( vacuousDall
% 0.19/0.45 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ emptyset )
% 0.19/0.45 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan1_type,type,
% 0.19/0.45 quantDeMorgan1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan1,definition,
% 0.19/0.45 ( quantDeMorgan1
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ~ ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( Xphi @ Xx ) )
% 0.19/0.45 => ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan2_type,type,
% 0.19/0.45 quantDeMorgan2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan2,definition,
% 0.19/0.45 ( quantDeMorgan2
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ~ ( Xphi @ Xx ) )
% 0.19/0.45 => ~ ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan3_type,type,
% 0.19/0.45 quantDeMorgan3: $o ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan3,definition,
% 0.19/0.45 ( quantDeMorgan3
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ~ ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ( Xphi @ Xx ) )
% 0.19/0.45 => ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan4_type,type,
% 0.19/0.45 quantDeMorgan4: $o ).
% 0.19/0.45
% 0.19/0.45 thf(quantDeMorgan4,definition,
% 0.19/0.45 ( quantDeMorgan4
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ~ ( Xphi @ Xx ) )
% 0.19/0.45 => ~ ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(prop2setI_type,type,
% 0.19/0.45 prop2setI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(prop2setI,definition,
% 0.19/0.45 ( prop2setI
% 0.19/0.45 = ( ! [Xphi: $o] :
% 0.19/0.45 ( Xphi
% 0.19/0.45 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(set2prop_type,type,
% 0.19/0.45 set2prop: $i > $o ).
% 0.19/0.45
% 0.19/0.45 thf(prop2set2propI_type,type,
% 0.19/0.45 prop2set2propI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(prop2set2propI,definition,
% 0.19/0.45 ( prop2set2propI
% 0.19/0.45 = ( ! [Xphi: $o] :
% 0.19/0.45 ( Xphi
% 0.19/0.45 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(notdexE_type,type,
% 0.19/0.45 notdexE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(notdexE,definition,
% 0.19/0.45 ( notdexE
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ~ ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ( Xphi @ Xx ) )
% 0.19/0.45 => ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(notdallE_type,type,
% 0.19/0.45 notdallE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(notdallE,definition,
% 0.19/0.45 ( notdallE
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( ~ ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( Xphi @ Xx ) )
% 0.19/0.45 => ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuI1_type,type,
% 0.19/0.45 exuI1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuI1,definition,
% 0.19/0.45 ( exuI1
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ? [Xx: $i] :
% 0.19/0.45 ( ( Xphi @ Xx )
% 0.19/0.45 & ! [Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xy )
% 0.19/0.45 => ( Xx = Xy ) ) )
% 0.19/0.45 => ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuI3_type,type,
% 0.19/0.45 exuI3: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuI3,definition,
% 0.19/0.45 ( exuI3
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.19/0.45 => ( ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xx )
% 0.19/0.45 => ( ( Xphi @ Xy )
% 0.19/0.45 => ( Xx = Xy ) ) )
% 0.19/0.45 => ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuI2_type,type,
% 0.19/0.45 exuI2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuI2,definition,
% 0.19/0.45 ( exuI2
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ? [Xx: $i] :
% 0.19/0.45 ! [Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xy )
% 0.19/0.45 <=> ( Xy = Xx ) )
% 0.19/0.45 => ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(inCongP_type,type,
% 0.19/0.45 inCongP: $o ).
% 0.19/0.45
% 0.19/0.45 thf(inCongP,definition,
% 0.19/0.45 ( inCongP
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( A = B )
% 0.19/0.45 => ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xx = Xy )
% 0.19/0.45 => ( ( in @ Xx @ A )
% 0.19/0.45 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(in__Cong_type,type,
% 0.19/0.45 in__Cong: $o ).
% 0.19/0.45
% 0.19/0.45 thf(in__Cong,definition,
% 0.19/0.45 ( in__Cong
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( A = B )
% 0.19/0.45 => ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xx = Xy )
% 0.19/0.45 => ( ( in @ Xx @ A )
% 0.19/0.45 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuE3u_type,type,
% 0.19/0.45 exuE3u: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuE3u,definition,
% 0.19/0.45 ( exuE3u
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 => ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xx )
% 0.19/0.45 => ( ( Xphi @ Xy )
% 0.19/0.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exu__Cong_type,type,
% 0.19/0.45 exu__Cong: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exu__Cong,definition,
% 0.19/0.45 ( exu__Cong
% 0.19/0.45 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.45 ( ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xx = Xy )
% 0.19/0.45 => ( ( Xphi @ Xx )
% 0.19/0.45 <=> ( Xpsi @ Xy ) ) )
% 0.19/0.45 => ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 <=> ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptyset__Cong_type,type,
% 0.19/0.45 emptyset__Cong: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptyset__Cong,definition,
% 0.19/0.45 ( emptyset__Cong
% 0.19/0.45 = ( emptyset = emptyset ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoin__Cong_type,type,
% 0.19/0.45 setadjoin__Cong: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoin__Cong,definition,
% 0.19/0.45 ( setadjoin__Cong
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xx = Xy )
% 0.19/0.45 => ! [Xz: $i,Xu: $i] :
% 0.19/0.45 ( ( Xz = Xu )
% 0.19/0.45 => ( ( setadjoin @ Xx @ Xz )
% 0.19/0.45 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(powerset__Cong_type,type,
% 0.19/0.45 powerset__Cong: $o ).
% 0.19/0.45
% 0.19/0.45 thf(powerset__Cong,definition,
% 0.19/0.45 ( powerset__Cong
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( A = B )
% 0.19/0.45 => ( ( powerset @ A )
% 0.19/0.45 = ( powerset @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setunion__Cong_type,type,
% 0.19/0.45 setunion__Cong: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setunion__Cong,definition,
% 0.19/0.45 ( setunion__Cong
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( A = B )
% 0.19/0.45 => ( ( setunion @ A )
% 0.19/0.45 = ( setunion @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(omega__Cong_type,type,
% 0.19/0.45 omega__Cong: $o ).
% 0.19/0.45
% 0.19/0.45 thf(omega__Cong,definition,
% 0.19/0.45 ( omega__Cong
% 0.19/0.45 = ( omega = omega ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(exuEu_type,type,
% 0.19/0.45 exuEu: $o ).
% 0.19/0.45
% 0.19/0.45 thf(exuEu,definition,
% 0.19/0.45 ( exuEu
% 0.19/0.45 = ( ! [Xphi: $i > $o] :
% 0.19/0.45 ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 => ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xphi @ Xx )
% 0.19/0.45 => ( ( Xphi @ Xy )
% 0.19/0.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(descr__Cong_type,type,
% 0.19/0.45 descr__Cong: $o ).
% 0.19/0.45
% 0.19/0.45 thf(descr__Cong,definition,
% 0.19/0.45 ( descr__Cong
% 0.19/0.45 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.45 ( ! [Xx: $i,Xy: $i] :
% 0.19/0.45 ( ( Xx = Xy )
% 0.19/0.45 => ( ( Xphi @ Xx )
% 0.19/0.45 <=> ( Xpsi @ Xy ) ) )
% 0.19/0.45 => ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 => ( ( exu
% 0.19/0.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.19/0.45 => ( ( descr
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 = ( descr
% 0.19/0.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstr__Cong_type,type,
% 0.19/0.45 dsetconstr__Cong: $o ).
% 0.19/0.45
% 0.19/0.45 thf(dsetconstr__Cong,definition,
% 0.19/0.45 ( dsetconstr__Cong
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( A = B )
% 0.19/0.45 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ! [Xy: $i] :
% 0.19/0.45 ( ( in @ Xy @ B )
% 0.19/0.45 => ( ( Xx = Xy )
% 0.19/0.45 => ( ( Xphi @ Xx )
% 0.19/0.45 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.19/0.45 => ( ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 = ( dsetconstr @ B
% 0.19/0.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(subset_type,type,
% 0.19/0.45 subset: $i > $i > $o ).
% 0.19/0.45
% 0.19/0.45 thf(disjoint_type,type,
% 0.19/0.45 disjoint: $i > $i > $o ).
% 0.19/0.45
% 0.19/0.45 thf(setsmeet_type,type,
% 0.19/0.45 setsmeet: $i > $i > $o ).
% 0.19/0.45
% 0.19/0.45 thf(subsetI1_type,type,
% 0.19/0.45 subsetI1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(subsetI1,definition,
% 0.19/0.45 ( subsetI1
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( in @ Xx @ B ) )
% 0.19/0.45 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(eqimpsubset2_type,type,
% 0.19/0.45 eqimpsubset2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(eqimpsubset2,definition,
% 0.19/0.45 ( eqimpsubset2
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( A = B )
% 0.19/0.45 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(eqimpsubset1_type,type,
% 0.19/0.45 eqimpsubset1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(eqimpsubset1,definition,
% 0.19/0.45 ( eqimpsubset1
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( A = B )
% 0.19/0.45 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(subsetI2_type,type,
% 0.19/0.45 subsetI2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(subsetI2,definition,
% 0.19/0.45 ( subsetI2
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ! [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( in @ Xx @ B ) )
% 0.19/0.45 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetsubset_type,type,
% 0.19/0.45 emptysetsubset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(emptysetsubset,definition,
% 0.19/0.45 ( emptysetsubset
% 0.19/0.45 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(subsetE_type,type,
% 0.19/0.45 subsetE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(subsetE,definition,
% 0.19/0.45 ( subsetE
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( subset @ A @ B )
% 0.19/0.45 => ( ( in @ Xx @ A )
% 0.19/0.45 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(subsetE2_type,type,
% 0.19/0.45 subsetE2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(subsetE2,definition,
% 0.19/0.45 ( subsetE2
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( subset @ A @ B )
% 0.19/0.45 => ( ~ ( in @ Xx @ B )
% 0.19/0.45 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(notsubsetI_type,type,
% 0.19/0.45 notsubsetI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(notsubsetI,definition,
% 0.19/0.45 ( notsubsetI
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ~ ( in @ Xx @ B )
% 0.19/0.45 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(notequalI1_type,type,
% 0.19/0.45 notequalI1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(notequalI1,definition,
% 0.19/0.45 ( notequalI1
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ~ ( subset @ A @ B )
% 0.19/0.45 => ( A != B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(notequalI2_type,type,
% 0.19/0.45 notequalI2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(notequalI2,definition,
% 0.19/0.45 ( notequalI2
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ~ ( in @ Xx @ B )
% 0.19/0.45 => ( A != B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(subsetRefl_type,type,
% 0.19/0.45 subsetRefl: $o ).
% 0.19/0.45
% 0.19/0.45 thf(subsetRefl,definition,
% 0.19/0.45 ( subsetRefl
% 0.19/0.45 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(subsetTrans_type,type,
% 0.19/0.45 subsetTrans: $o ).
% 0.19/0.45
% 0.19/0.45 thf(subsetTrans,definition,
% 0.19/0.45 ( subsetTrans
% 0.19/0.45 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.45 ( ( subset @ A @ B )
% 0.19/0.45 => ( ( subset @ B @ C )
% 0.19/0.45 => ( subset @ A @ C ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinSub_type,type,
% 0.19/0.45 setadjoinSub: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinSub,definition,
% 0.19/0.45 ( setadjoinSub
% 0.19/0.45 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinSub2_type,type,
% 0.19/0.45 setadjoinSub2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setadjoinSub2,definition,
% 0.19/0.45 ( setadjoinSub2
% 0.19/0.45 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.19/0.45 ( ( subset @ A @ B )
% 0.19/0.45 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(subset2powerset_type,type,
% 0.19/0.45 subset2powerset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(subset2powerset,definition,
% 0.19/0.45 ( subset2powerset
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( subset @ A @ B )
% 0.19/0.45 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setextsub_type,type,
% 0.19/0.45 setextsub: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setextsub,definition,
% 0.19/0.45 ( setextsub
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( subset @ A @ B )
% 0.19/0.45 => ( ( subset @ B @ A )
% 0.19/0.45 => ( A = B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(subsetemptysetimpeq_type,type,
% 0.19/0.45 subsetemptysetimpeq: $o ).
% 0.19/0.45
% 0.19/0.45 thf(subsetemptysetimpeq,definition,
% 0.19/0.45 ( subsetemptysetimpeq
% 0.19/0.45 = ( ! [A: $i] :
% 0.19/0.45 ( ( subset @ A @ emptyset )
% 0.19/0.45 => ( A = emptyset ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(powersetI1_type,type,
% 0.19/0.45 powersetI1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(powersetI1,definition,
% 0.19/0.45 ( powersetI1
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( subset @ B @ A )
% 0.19/0.45 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(powersetE1_type,type,
% 0.19/0.45 powersetE1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(powersetE1,definition,
% 0.19/0.45 ( powersetE1
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.45 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(inPowerset_type,type,
% 0.19/0.45 inPowerset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(inPowerset,definition,
% 0.19/0.45 ( inPowerset
% 0.19/0.45 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(powersetsubset_type,type,
% 0.19/0.45 powersetsubset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(powersetsubset,definition,
% 0.19/0.45 ( powersetsubset
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( subset @ A @ B )
% 0.19/0.45 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sepInPowerset_type,type,
% 0.19/0.45 sepInPowerset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(sepInPowerset,definition,
% 0.19/0.45 ( sepInPowerset
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( in
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 @ ( powerset @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(sepSubset_type,type,
% 0.19/0.45 sepSubset: $o ).
% 0.19/0.45
% 0.19/0.45 thf(sepSubset,definition,
% 0.19/0.45 ( sepSubset
% 0.19/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.45 ( subset
% 0.19/0.45 @ ( dsetconstr @ A
% 0.19/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.45 @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binunion_type,type,
% 0.19/0.45 binunion: $i > $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(binunionIL_type,type,
% 0.19/0.45 binunionIL: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binunionIL,definition,
% 0.19/0.45 ( binunionIL
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(upairset2IR_type,type,
% 0.19/0.45 upairset2IR: $o ).
% 0.19/0.45
% 0.19/0.45 thf(upairset2IR,definition,
% 0.19/0.45 ( upairset2IR
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binunionIR_type,type,
% 0.19/0.45 binunionIR: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binunionIR,definition,
% 0.19/0.45 ( binunionIR
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binunionEcases_type,type,
% 0.19/0.45 binunionEcases: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binunionEcases,definition,
% 0.19/0.45 ( binunionEcases
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.19/0.45 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.19/0.45 => ( ( ( in @ Xx @ A )
% 0.19/0.45 => Xphi )
% 0.19/0.45 => ( ( ( in @ Xx @ B )
% 0.19/0.45 => Xphi )
% 0.19/0.45 => Xphi ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binunionE_type,type,
% 0.19/0.45 binunionE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binunionE,definition,
% 0.19/0.45 ( binunionE
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.19/0.45 => ( ( in @ Xx @ A )
% 0.19/0.45 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binunionLsub_type,type,
% 0.19/0.45 binunionLsub: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binunionLsub,definition,
% 0.19/0.45 ( binunionLsub
% 0.19/0.45 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binunionRsub_type,type,
% 0.19/0.45 binunionRsub: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binunionRsub,definition,
% 0.19/0.45 ( binunionRsub
% 0.19/0.45 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersect_type,type,
% 0.19/0.45 binintersect: $i > $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectI_type,type,
% 0.19/0.45 binintersectI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectI,definition,
% 0.19/0.45 ( binintersectI
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset5_type,type,
% 0.19/0.45 binintersectSubset5: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset5,definition,
% 0.19/0.45 ( binintersectSubset5
% 0.19/0.45 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.45 ( ( subset @ C @ A )
% 0.19/0.45 => ( ( subset @ C @ B )
% 0.19/0.45 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectEL_type,type,
% 0.19/0.45 binintersectEL: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectEL,definition,
% 0.19/0.45 ( binintersectEL
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.19/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectLsub_type,type,
% 0.19/0.45 binintersectLsub: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectLsub,definition,
% 0.19/0.45 ( binintersectLsub
% 0.19/0.45 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset2_type,type,
% 0.19/0.45 binintersectSubset2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset2,definition,
% 0.19/0.45 ( binintersectSubset2
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( subset @ A @ B )
% 0.19/0.45 => ( ( binintersect @ A @ B )
% 0.19/0.45 = A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset3_type,type,
% 0.19/0.45 binintersectSubset3: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset3,definition,
% 0.19/0.45 ( binintersectSubset3
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( ( binintersect @ A @ B )
% 0.19/0.45 = B )
% 0.19/0.45 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectER_type,type,
% 0.19/0.45 binintersectER: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectER,definition,
% 0.19/0.45 ( binintersectER
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.19/0.45 => ( in @ Xx @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(disjointsetsI1_type,type,
% 0.19/0.45 disjointsetsI1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(disjointsetsI1,definition,
% 0.19/0.45 ( disjointsetsI1
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ~ ? [Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 & ( in @ Xx @ B ) )
% 0.19/0.45 => ( ( binintersect @ A @ B )
% 0.19/0.45 = emptyset ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectRsub_type,type,
% 0.19/0.45 binintersectRsub: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectRsub,definition,
% 0.19/0.45 ( binintersectRsub
% 0.19/0.45 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset4_type,type,
% 0.19/0.45 binintersectSubset4: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset4,definition,
% 0.19/0.45 ( binintersectSubset4
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( subset @ B @ A )
% 0.19/0.45 => ( ( binintersect @ A @ B )
% 0.19/0.45 = B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset1_type,type,
% 0.19/0.45 binintersectSubset1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(binintersectSubset1,definition,
% 0.19/0.45 ( binintersectSubset1
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( ( binintersect @ A @ B )
% 0.19/0.45 = A )
% 0.19/0.45 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(bs114d_type,type,
% 0.19/0.45 bs114d: $o ).
% 0.19/0.45
% 0.19/0.45 thf(bs114d,definition,
% 0.19/0.45 ( bs114d
% 0.19/0.45 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.45 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.19/0.45 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(regular_type,type,
% 0.19/0.45 regular: $i > $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminus_type,type,
% 0.19/0.45 setminus: $i > $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(setminusI_type,type,
% 0.19/0.45 setminusI: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusI,definition,
% 0.19/0.45 ( setminusI
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ~ ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusEL_type,type,
% 0.19/0.45 setminusEL: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusEL,definition,
% 0.19/0.45 ( setminusEL
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusER_type,type,
% 0.19/0.45 setminusER: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusER,definition,
% 0.19/0.45 ( setminusER
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.45 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusSubset2_type,type,
% 0.19/0.45 setminusSubset2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusSubset2,definition,
% 0.19/0.45 ( setminusSubset2
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( subset @ A @ B )
% 0.19/0.45 => ( ( setminus @ A @ B )
% 0.19/0.45 = emptyset ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusERneg_type,type,
% 0.19/0.45 setminusERneg: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusERneg,definition,
% 0.19/0.45 ( setminusERneg
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.45 => ( ( in @ Xx @ A )
% 0.19/0.45 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusELneg_type,type,
% 0.19/0.45 setminusELneg: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusELneg,definition,
% 0.19/0.45 ( setminusELneg
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.45 => ( ~ ( in @ Xx @ B )
% 0.19/0.45 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusILneg_type,type,
% 0.19/0.45 setminusILneg: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusILneg,definition,
% 0.19/0.45 ( setminusILneg
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ~ ( in @ Xx @ A )
% 0.19/0.45 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusIRneg_type,type,
% 0.19/0.45 setminusIRneg: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusIRneg,definition,
% 0.19/0.45 ( setminusIRneg
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ B )
% 0.19/0.45 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusLsub_type,type,
% 0.19/0.45 setminusLsub: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusLsub,definition,
% 0.19/0.45 ( setminusLsub
% 0.19/0.45 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setminusSubset1_type,type,
% 0.19/0.45 setminusSubset1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setminusSubset1,definition,
% 0.19/0.45 ( setminusSubset1
% 0.19/0.45 = ( ! [A: $i,B: $i] :
% 0.19/0.45 ( ( ( setminus @ A @ B )
% 0.19/0.45 = emptyset )
% 0.19/0.45 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(symdiff_type,type,
% 0.19/0.45 symdiff: $i > $i > $i ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffE_type,type,
% 0.19/0.45 symdiffE: $o ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffE,definition,
% 0.19/0.45 ( symdiffE
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.19/0.45 => ! [Xphi: $o] :
% 0.19/0.45 ( ( ( in @ Xx @ A )
% 0.19/0.45 => ( ~ ( in @ Xx @ B )
% 0.19/0.45 => Xphi ) )
% 0.19/0.45 => ( ( ~ ( in @ Xx @ A )
% 0.19/0.45 => ( ( in @ Xx @ B )
% 0.19/0.45 => Xphi ) )
% 0.19/0.45 => Xphi ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffI1_type,type,
% 0.19/0.45 symdiffI1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffI1,definition,
% 0.19/0.45 ( symdiffI1
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ~ ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffI2_type,type,
% 0.19/0.45 symdiffI2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffI2,definition,
% 0.19/0.45 ( symdiffI2
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ~ ( in @ Xx @ A )
% 0.19/0.45 => ( ( in @ Xx @ B )
% 0.19/0.45 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffIneg1_type,type,
% 0.19/0.45 symdiffIneg1: $o ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffIneg1,definition,
% 0.19/0.45 ( symdiffIneg1
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ( in @ Xx @ A )
% 0.19/0.45 => ( ( in @ Xx @ B )
% 0.19/0.45 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffIneg2_type,type,
% 0.19/0.45 symdiffIneg2: $o ).
% 0.19/0.45
% 0.19/0.45 thf(symdiffIneg2,definition,
% 0.19/0.45 ( symdiffIneg2
% 0.19/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.45 ( ~ ( in @ Xx @ A )
% 0.19/0.45 => ( ~ ( in @ Xx @ B )
% 0.19/0.45 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(iskpair_type,type,
% 0.19/0.45 iskpair: $i > $o ).
% 0.19/0.45
% 0.19/0.45 thf(secondinupair_type,type,
% 0.19/0.45 secondinupair: $o ).
% 0.19/0.45
% 0.19/0.45 thf(secondinupair,definition,
% 0.19/0.45 ( secondinupair
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setukpairIL_type,type,
% 0.19/0.45 setukpairIL: $o ).
% 0.19/0.45
% 0.19/0.45 thf(setukpairIL,definition,
% 0.19/0.45 ( setukpairIL
% 0.19/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 thf(setukpairIR_type,type,
% 0.19/0.46 setukpairIR: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairIR,definition,
% 0.19/0.46 ( setukpairIR
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(kpairiskpair_type,type,
% 0.19/0.46 kpairiskpair: $o ).
% 0.19/0.46
% 0.19/0.46 thf(kpairiskpair,definition,
% 0.19/0.46 ( kpairiskpair
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(kpair_type,type,
% 0.19/0.46 kpair: $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(kpairp_type,type,
% 0.19/0.46 kpairp: $o ).
% 0.19/0.46
% 0.19/0.46 thf(kpairp,definition,
% 0.19/0.46 ( kpairp
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprod_type,type,
% 0.19/0.46 cartprod: $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(singletonsubset_type,type,
% 0.19/0.46 singletonsubset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(singletonsubset,definition,
% 0.19/0.46 ( singletonsubset
% 0.19/0.46 = ( ! [A: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(singletoninpowerset_type,type,
% 0.19/0.46 singletoninpowerset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(singletoninpowerset,definition,
% 0.19/0.46 ( singletoninpowerset
% 0.19/0.46 = ( ! [A: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(singletoninpowunion_type,type,
% 0.19/0.46 singletoninpowunion: $o ).
% 0.19/0.46
% 0.19/0.46 thf(singletoninpowunion,definition,
% 0.19/0.46 ( singletoninpowunion
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(upairset2E_type,type,
% 0.19/0.46 upairset2E: $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairset2E,definition,
% 0.19/0.46 ( upairset2E
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.19/0.46 => ( ( Xz = Xx )
% 0.19/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(upairsubunion_type,type,
% 0.19/0.46 upairsubunion: $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairsubunion,definition,
% 0.19/0.46 ( upairsubunion
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(upairinpowunion_type,type,
% 0.19/0.46 upairinpowunion: $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairinpowunion,definition,
% 0.19/0.46 ( upairinpowunion
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ubforcartprodlem1_type,type,
% 0.19/0.46 ubforcartprodlem1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ubforcartprodlem1,definition,
% 0.19/0.46 ( ubforcartprodlem1
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ubforcartprodlem2_type,type,
% 0.19/0.46 ubforcartprodlem2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ubforcartprodlem2,definition,
% 0.19/0.46 ( ubforcartprodlem2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ubforcartprodlem3_type,type,
% 0.19/0.46 ubforcartprodlem3: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ubforcartprodlem3,definition,
% 0.19/0.46 ( ubforcartprodlem3
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodpairin_type,type,
% 0.19/0.46 cartprodpairin: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodpairin,definition,
% 0.19/0.46 ( cartprodpairin
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodmempair1_type,type,
% 0.19/0.46 cartprodmempair1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodmempair1,definition,
% 0.19/0.46 ( cartprodmempair1
% 0.19/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.46 => ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ? [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 & ( Xu
% 0.19/0.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodmempair_type,type,
% 0.19/0.46 cartprodmempair: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodmempair,definition,
% 0.19/0.46 ( cartprodmempair
% 0.19/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.46 => ( iskpair @ Xu ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunionE2_type,type,
% 0.19/0.46 setunionE2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunionE2,definition,
% 0.19/0.46 ( setunionE2
% 0.19/0.46 = ( ! [A: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.46 => ? [X: $i] :
% 0.19/0.46 ( ( in @ X @ A )
% 0.19/0.46 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunionsingleton1_type,type,
% 0.19/0.46 setunionsingleton1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunionsingleton1,definition,
% 0.19/0.46 ( setunionsingleton1
% 0.19/0.46 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunionsingleton2_type,type,
% 0.19/0.46 setunionsingleton2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunionsingleton2,definition,
% 0.19/0.46 ( setunionsingleton2
% 0.19/0.46 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunionsingleton_type,type,
% 0.19/0.46 setunionsingleton: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunionsingleton,definition,
% 0.19/0.46 ( setunionsingleton
% 0.19/0.46 = ( ! [Xx: $i] :
% 0.19/0.46 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.19/0.46 = Xx ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(singleton_type,type,
% 0.19/0.46 singleton: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(singletonprop_type,type,
% 0.19/0.46 singletonprop: $o ).
% 0.19/0.46
% 0.19/0.46 thf(singletonprop,definition,
% 0.19/0.46 ( singletonprop
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 => ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) )
% 0.19/0.46 => ( ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ( Xphi @ Xx ) )
% 0.19/0.46 => ( singleton
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ex1_type,type,
% 0.19/0.46 ex1: $i > ( $i > $o ) > $o ).
% 0.19/0.46
% 0.19/0.46 thf(ex1,definition,
% 0.19/0.46 ( ex1
% 0.19/0.46 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( singleton
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ex1E1_type,type,
% 0.19/0.46 ex1E1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ex1E1,definition,
% 0.19/0.46 ( ex1E1
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( ( ex1 @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ex1I_type,type,
% 0.19/0.46 ex1I: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ex1I,definition,
% 0.19/0.46 ( ex1I
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 => ( ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xy = Xx ) ) )
% 0.19/0.46 => ( ex1 @ A
% 0.19/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ex1I2_type,type,
% 0.19/0.46 ex1I2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ex1I2,definition,
% 0.19/0.46 ( ex1I2
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 => ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) )
% 0.19/0.46 => ( ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ( Xphi @ Xx ) )
% 0.19/0.46 => ( ex1 @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(singletonsuniq_type,type,
% 0.19/0.46 singletonsuniq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(singletonsuniq,definition,
% 0.19/0.46 ( singletonsuniq
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( ( setadjoin @ Xx @ emptyset )
% 0.19/0.46 = ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(atmost1p_type,type,
% 0.19/0.46 atmost1p: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(atleast2p_type,type,
% 0.19/0.46 atleast2p: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(atmost2p_type,type,
% 0.19/0.46 atmost2p: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairsetp_type,type,
% 0.19/0.46 upairsetp: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjL1_type,type,
% 0.19/0.46 setukpairinjL1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjL1,definition,
% 0.19/0.46 ( setukpairinjL1
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.46 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.19/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(kfstsingleton_type,type,
% 0.19/0.46 kfstsingleton: $o ).
% 0.19/0.46
% 0.19/0.46 thf(kfstsingleton,definition,
% 0.19/0.46 ( kfstsingleton
% 0.19/0.46 = ( ! [Xu: $i] :
% 0.19/0.46 ( ( iskpair @ Xu )
% 0.19/0.46 => ( singleton
% 0.19/0.46 @ ( dsetconstr @ ( setunion @ Xu )
% 0.19/0.46 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(theprop_type,type,
% 0.19/0.46 theprop: $o ).
% 0.19/0.46
% 0.19/0.46 thf(theprop,definition,
% 0.19/0.46 ( theprop
% 0.19/0.46 = ( ! [X: $i] :
% 0.19/0.46 ( ( singleton @ X )
% 0.19/0.46 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(kfst_type,type,
% 0.19/0.46 kfst: $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(kfstpairEq_type,type,
% 0.19/0.46 kfstpairEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(kfstpairEq,definition,
% 0.19/0.46 ( kfstpairEq
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.19/0.46 = Xx ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodfstin_type,type,
% 0.19/0.46 cartprodfstin: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodfstin,definition,
% 0.19/0.46 ( cartprodfstin
% 0.19/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.46 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjL2_type,type,
% 0.19/0.46 setukpairinjL2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjL2,definition,
% 0.19/0.46 ( setukpairinjL2
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.19/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.19/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjL_type,type,
% 0.19/0.46 setukpairinjL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjL,definition,
% 0.19/0.46 ( setukpairinjL
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.46 ( ( ( kpair @ Xx @ Xy )
% 0.19/0.46 = ( kpair @ Xz @ Xu ) )
% 0.19/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR11_type,type,
% 0.19/0.46 setukpairinjR11: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR11,definition,
% 0.19/0.46 ( setukpairinjR11
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.46 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR12_type,type,
% 0.19/0.46 setukpairinjR12: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR12,definition,
% 0.19/0.46 ( setukpairinjR12
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.19/0.46 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR1_type,type,
% 0.19/0.46 setukpairinjR1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR1,definition,
% 0.19/0.46 ( setukpairinjR1
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.19/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.19/0.46 => ( ( Xz = Xu )
% 0.19/0.46 => ( Xy = Xu ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(upairequniteq_type,type,
% 0.19/0.46 upairequniteq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairequniteq,definition,
% 0.19/0.46 ( upairequniteq
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.46 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.46 = ( setadjoin @ Xz @ emptyset ) )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR2_type,type,
% 0.19/0.46 setukpairinjR2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR2,definition,
% 0.19/0.46 ( setukpairinjR2
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.19/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.19/0.46 => ( Xy = Xu ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR_type,type,
% 0.19/0.46 setukpairinjR: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR,definition,
% 0.19/0.46 ( setukpairinjR
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.46 ( ( ( kpair @ Xx @ Xy )
% 0.19/0.46 = ( kpair @ Xz @ Xu ) )
% 0.19/0.46 => ( Xy = Xu ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ksndsingleton_type,type,
% 0.19/0.46 ksndsingleton: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ksndsingleton,definition,
% 0.19/0.46 ( ksndsingleton
% 0.19/0.46 = ( ! [Xu: $i] :
% 0.19/0.46 ( ( iskpair @ Xu )
% 0.19/0.46 => ( singleton
% 0.19/0.46 @ ( dsetconstr @ ( setunion @ Xu )
% 0.19/0.46 @ ^ [Xx: $i] :
% 0.19/0.46 ( Xu
% 0.19/0.46 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ksnd_type,type,
% 0.19/0.46 ksnd: $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(ksndpairEq_type,type,
% 0.19/0.46 ksndpairEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ksndpairEq,definition,
% 0.19/0.46 ( ksndpairEq
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.19/0.46 = Xy ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(kpairsurjEq_type,type,
% 0.19/0.46 kpairsurjEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(kpairsurjEq,definition,
% 0.19/0.46 ( kpairsurjEq
% 0.19/0.46 = ( ! [Xu: $i] :
% 0.19/0.46 ( ( iskpair @ Xu )
% 0.19/0.46 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.19/0.46 = Xu ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodsndin_type,type,
% 0.19/0.46 cartprodsndin: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodsndin,definition,
% 0.19/0.46 ( cartprodsndin
% 0.19/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.46 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodpairmemEL_type,type,
% 0.19/0.46 cartprodpairmemEL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodpairmemEL,definition,
% 0.19/0.46 ( cartprodpairmemEL
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodpairmemER_type,type,
% 0.19/0.46 cartprodpairmemER: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodpairmemER,definition,
% 0.19/0.46 ( cartprodpairmemER
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.19/0.46 => ( in @ Xy @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodmempaircEq_type,type,
% 0.19/0.46 cartprodmempaircEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodmempaircEq,definition,
% 0.19/0.46 ( cartprodmempaircEq
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( kpair @ Xx @ Xy )
% 0.19/0.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodfstpairEq_type,type,
% 0.19/0.46 cartprodfstpairEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodfstpairEq,definition,
% 0.19/0.46 ( cartprodfstpairEq
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.19/0.46 = Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodsndpairEq_type,type,
% 0.19/0.46 cartprodsndpairEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodsndpairEq,definition,
% 0.19/0.46 ( cartprodsndpairEq
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.19/0.46 = Xy ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodpairsurjEq_type,type,
% 0.19/0.46 cartprodpairsurjEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodpairsurjEq,definition,
% 0.19/0.46 ( cartprodpairsurjEq
% 0.19/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.46 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.19/0.46 = Xu ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(breln_type,type,
% 0.19/0.46 breln: $i > $i > $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(breln,definition,
% 0.19/0.46 ( breln
% 0.19/0.46 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstr_type,type,
% 0.19/0.46 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrI_type,type,
% 0.19/0.46 dpsetconstrI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrI,definition,
% 0.19/0.46 ( dpsetconstrI
% 0.19/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( Xphi @ Xx @ Xy )
% 0.19/0.46 => ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.46 @ ( dpsetconstr @ A @ B
% 0.19/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrSub_type,type,
% 0.19/0.46 dpsetconstrSub: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrSub,definition,
% 0.19/0.46 ( dpsetconstrSub
% 0.19/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.19/0.46 ( subset
% 0.19/0.46 @ ( dpsetconstr @ A @ B
% 0.19/0.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.19/0.46 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setOfPairsIsBReln_type,type,
% 0.19/0.46 setOfPairsIsBReln: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setOfPairsIsBReln,definition,
% 0.19/0.46 ( setOfPairsIsBReln
% 0.19/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.19/0.46 ( breln @ A @ B
% 0.19/0.46 @ ( dpsetconstr @ A @ B
% 0.19/0.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrERa_type,type,
% 0.19/0.46 dpsetconstrERa: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrERa,definition,
% 0.19/0.46 ( dpsetconstrERa
% 0.19/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.46 @ ( dpsetconstr @ A @ B
% 0.19/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.19/0.46 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrEL1_type,type,
% 0.19/0.46 dpsetconstrEL1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrEL1,definition,
% 0.19/0.46 ( dpsetconstrEL1
% 0.19/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.46 @ ( dpsetconstr @ A @ B
% 0.19/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrEL2_type,type,
% 0.19/0.46 dpsetconstrEL2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrEL2,definition,
% 0.19/0.46 ( dpsetconstrEL2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.46 @ ( dpsetconstr @ A @ B
% 0.19/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.19/0.46 => ( in @ Xy @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrER_type,type,
% 0.19/0.46 dpsetconstrER: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrER,definition,
% 0.19/0.46 ( dpsetconstrER
% 0.19/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.46 @ ( dpsetconstr @ A @ B
% 0.19/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.19/0.46 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(func_type,type,
% 0.19/0.46 func: $i > $i > $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(funcSet_type,type,
% 0.19/0.46 funcSet: $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(funcImageSingleton_type,type,
% 0.19/0.46 funcImageSingleton: $o ).
% 0.19/0.46
% 0.19/0.46 thf(funcImageSingleton,definition,
% 0.19/0.46 ( funcImageSingleton
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xf )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( singleton
% 0.19/0.46 @ ( dsetconstr @ B
% 0.19/0.46 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(apProp_type,type,
% 0.19/0.46 apProp: $o ).
% 0.19/0.46
% 0.19/0.46 thf(apProp,definition,
% 0.19/0.46 ( apProp
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xf )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in
% 0.19/0.46 @ ( setunion
% 0.19/0.46 @ ( dsetconstr @ B
% 0.19/0.46 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.19/0.46 @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ap_type,type,
% 0.19/0.46 ap: $i > $i > $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(app_type,type,
% 0.19/0.46 app: $o ).
% 0.19/0.46
% 0.19/0.46 thf(app,definition,
% 0.19/0.46 ( app
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xf )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(infuncsetfunc_type,type,
% 0.19/0.46 infuncsetfunc: $o ).
% 0.19/0.46
% 0.19/0.46 thf(infuncsetfunc,definition,
% 0.19/0.46 ( infuncsetfunc
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.46 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ap2p_type,type,
% 0.19/0.46 ap2p: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ap2p,definition,
% 0.19/0.46 ( ap2p
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(funcinfuncset_type,type,
% 0.19/0.46 funcinfuncset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(funcinfuncset,definition,
% 0.19/0.46 ( funcinfuncset
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xf )
% 0.19/0.46 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(lamProp_type,type,
% 0.19/0.46 lamProp: $o ).
% 0.19/0.46
% 0.19/0.46 thf(lamProp,definition,
% 0.19/0.46 ( lamProp
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.46 => ( func @ A @ B
% 0.19/0.46 @ ( dpsetconstr @ A @ B
% 0.19/0.46 @ ^ [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xf @ Xx )
% 0.19/0.46 = Xy ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(lam_type,type,
% 0.19/0.46 lam: $i > $i > ( $i > $i ) > $i ).
% 0.19/0.46
% 0.19/0.46 thf(lamp_type,type,
% 0.19/0.46 lamp: $o ).
% 0.19/0.46
% 0.19/0.46 thf(lamp,definition,
% 0.19/0.46 ( lamp
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.46 => ( func @ A @ B
% 0.19/0.46 @ ( lam @ A @ B
% 0.19/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(lam2p_type,type,
% 0.19/0.46 lam2p: $o ).
% 0.19/0.46
% 0.19/0.46 thf(lam2p,definition,
% 0.19/0.46 ( lam2p
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.46 => ( in
% 0.19/0.46 @ ( lam @ A @ B
% 0.19/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.19/0.46 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(brelnall1_type,type,
% 0.19/0.46 brelnall1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(brelnall1,definition,
% 0.19/0.46 ( brelnall1
% 0.19/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.19/0.46 ( ( breln @ A @ B @ R )
% 0.19/0.46 => ! [Xphi: $i > $o] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ R )
% 0.19/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(brelnall2_type,type,
% 0.19/0.46 brelnall2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(brelnall2,definition,
% 0.19/0.46 ( brelnall2
% 0.19/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.19/0.46 ( ( breln @ A @ B @ R )
% 0.19/0.46 => ! [Xphi: $i > $o] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ R )
% 0.19/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ex1E2_type,type,
% 0.19/0.46 ex1E2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ex1E2,definition,
% 0.19/0.46 ( ex1E2
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( ( ex1 @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 => ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(funcGraphProp1_type,type,
% 0.19/0.46 funcGraphProp1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(funcGraphProp1,definition,
% 0.19/0.46 ( funcGraphProp1
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xf )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(funcGraphProp3_type,type,
% 0.19/0.46 funcGraphProp3: $o ).
% 0.19/0.46
% 0.19/0.46 thf(funcGraphProp3,definition,
% 0.19/0.46 ( funcGraphProp3
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(funcGraphProp2_type,type,
% 0.19/0.46 funcGraphProp2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(funcGraphProp2,definition,
% 0.19/0.46 ( funcGraphProp2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xf )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.19/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.46 = Xy ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(funcextLem_type,type,
% 0.19/0.46 funcextLem: $o ).
% 0.19/0.46
% 0.19/0.46 thf(funcextLem,definition,
% 0.19/0.46 ( funcextLem
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xf )
% 0.19/0.46 => ! [Xg: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xg )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.19/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(funcGraphProp4_type,type,
% 0.19/0.46 funcGraphProp4: $o ).
% 0.19/0.46
% 0.19/0.46 thf(funcGraphProp4,definition,
% 0.19/0.46 ( funcGraphProp4
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.19/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.46 = Xy ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subbreln_type,type,
% 0.19/0.46 subbreln: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subbreln,definition,
% 0.19/0.46 ( subbreln
% 0.19/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.19/0.46 ( ( breln @ A @ B @ R )
% 0.19/0.46 => ! [S: $i] :
% 0.19/0.46 ( ( breln @ A @ B @ S )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.19/0.46 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(eqbreln_type,type,
% 0.19/0.46 eqbreln: $o ).
% 0.19/0.46
% 0.19/0.46 thf(eqbreln,definition,
% 0.19/0.46 ( eqbreln
% 0.19/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.19/0.46 ( ( breln @ A @ B @ R )
% 0.19/0.46 => ! [S: $i] :
% 0.19/0.46 ( ( breln @ A @ B @ S )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.19/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.19/0.46 => ( R = S ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(funcext_type,type,
% 0.19/0.46 funcext: $o ).
% 0.19/0.46
% 0.19/0.46 thf(funcext,definition,
% 0.19/0.46 ( funcext
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xf )
% 0.19/0.46 => ! [Xg: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xg )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.19/0.46 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(funcext2_type,type,
% 0.19/0.46 funcext2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(funcext2,definition,
% 0.19/0.46 ( funcext2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.46 => ! [Xg: $i] :
% 0.19/0.46 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.19/0.46 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ap2apEq1_type,type,
% 0.19/0.46 ap2apEq1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ap2apEq1,definition,
% 0.19/0.46 ( ap2apEq1
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.46 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ap2apEq2_type,type,
% 0.19/0.46 ap2apEq2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ap2apEq2,definition,
% 0.19/0.46 ( ap2apEq2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xf )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.46 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(beta1_type,type,
% 0.19/0.46 beta1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(beta1,definition,
% 0.19/0.46 ( beta1
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( ap @ A @ B
% 0.19/0.46 @ ( lam @ A @ B
% 0.19/0.46 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.19/0.46 @ Xx )
% 0.19/0.46 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(eta1_type,type,
% 0.19/0.46 eta1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(eta1,definition,
% 0.19/0.46 ( eta1
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xf )
% 0.19/0.46 => ( ( lam @ A @ B
% 0.19/0.46 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.19/0.46 = Xf ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(lam2lamEq_type,type,
% 0.19/0.46 lam2lamEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(lam2lamEq,definition,
% 0.19/0.46 ( lam2lamEq
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.46 => ( ( lam @ A @ B
% 0.19/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.19/0.46 = ( lam @ A @ B
% 0.19/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(beta2_type,type,
% 0.19/0.46 beta2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(beta2,definition,
% 0.19/0.46 ( beta2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( ap @ A @ B
% 0.19/0.46 @ ( lam @ A @ B
% 0.19/0.46 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.19/0.46 @ Xx )
% 0.19/0.46 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(eta2_type,type,
% 0.19/0.46 eta2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(eta2,definition,
% 0.19/0.46 ( eta2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.46 => ( ( lam @ A @ B
% 0.19/0.46 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.19/0.46 = Xf ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(iffalseProp1_type,type,
% 0.19/0.46 iffalseProp1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(iffalseProp1,definition,
% 0.19/0.46 ( iffalseProp1
% 0.19/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( ~ Xphi
% 0.19/0.46 => ( in @ Xy
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xz: $i] :
% 0.19/0.46 ( ( Xphi
% 0.19/0.46 & ( Xz = Xx ) )
% 0.19/0.46 | ( ~ Xphi
% 0.19/0.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(iffalseProp2_type,type,
% 0.19/0.46 iffalseProp2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(iffalseProp2,definition,
% 0.19/0.46 ( iffalseProp2
% 0.19/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( ~ Xphi
% 0.19/0.46 => ( ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xz: $i] :
% 0.19/0.46 ( ( Xphi
% 0.19/0.46 & ( Xz = Xx ) )
% 0.19/0.46 | ( ~ Xphi
% 0.19/0.46 & ( Xz = Xy ) ) ) )
% 0.19/0.46 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(iftrueProp1_type,type,
% 0.19/0.46 iftrueProp1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(iftrueProp1,definition,
% 0.19/0.46 ( iftrueProp1
% 0.19/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( Xphi
% 0.19/0.46 => ( in @ Xx
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xz: $i] :
% 0.19/0.46 ( ( Xphi
% 0.19/0.46 & ( Xz = Xx ) )
% 0.19/0.46 | ( ~ Xphi
% 0.19/0.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(iftrueProp2_type,type,
% 0.19/0.46 iftrueProp2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(iftrueProp2,definition,
% 0.19/0.46 ( iftrueProp2
% 0.19/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( Xphi
% 0.19/0.46 => ( ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xz: $i] :
% 0.19/0.46 ( ( Xphi
% 0.19/0.46 & ( Xz = Xx ) )
% 0.19/0.46 | ( ~ Xphi
% 0.19/0.46 & ( Xz = Xy ) ) ) )
% 0.19/0.46 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ifSingleton_type,type,
% 0.19/0.46 ifSingleton: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ifSingleton,definition,
% 0.19/0.46 ( ifSingleton
% 0.19/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( singleton
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xz: $i] :
% 0.19/0.46 ( ( Xphi
% 0.19/0.46 & ( Xz = Xx ) )
% 0.19/0.46 | ( ~ Xphi
% 0.19/0.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(if_type,type,
% 0.19/0.46 if: $i > $o > $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(ifp_type,type,
% 0.19/0.46 ifp: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ifp,definition,
% 0.19/0.46 ( ifp
% 0.19/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(theeq_type,type,
% 0.19/0.46 theeq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(theeq,definition,
% 0.19/0.46 ( theeq
% 0.19/0.46 = ( ! [X: $i] :
% 0.19/0.46 ( ( singleton @ X )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ X )
% 0.19/0.46 => ( ( setunion @ X )
% 0.19/0.46 = Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(iftrue_type,type,
% 0.19/0.46 iftrue: $o ).
% 0.19/0.46
% 0.19/0.46 thf(iftrue,definition,
% 0.19/0.46 ( iftrue
% 0.19/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( Xphi
% 0.19/0.46 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.19/0.46 = Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(iffalse_type,type,
% 0.19/0.46 iffalse: $o ).
% 0.19/0.46
% 0.19/0.46 thf(iffalse,definition,
% 0.19/0.46 ( iffalse
% 0.19/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( ~ Xphi
% 0.19/0.46 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.19/0.46 = Xy ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(iftrueorfalse_type,type,
% 0.19/0.46 iftrueorfalse: $o ).
% 0.19/0.46
% 0.19/0.46 thf(iftrueorfalse,definition,
% 0.19/0.46 ( iftrueorfalse
% 0.19/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectT_lem_type,type,
% 0.19/0.46 binintersectT_lem: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectT_lem,definition,
% 0.19/0.46 ( binintersectT_lem
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Y: $i] :
% 0.19/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.46 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binunionT_lem_type,type,
% 0.19/0.46 binunionT_lem: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binunionT_lem,definition,
% 0.19/0.46 ( binunionT_lem
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Y: $i] :
% 0.19/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.46 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powersetT_lem_type,type,
% 0.19/0.46 powersetT_lem: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powersetT_lem,definition,
% 0.19/0.46 ( powersetT_lem
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusT_lem_type,type,
% 0.19/0.46 setminusT_lem: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusT_lem,definition,
% 0.19/0.46 ( setminusT_lem
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Y: $i] :
% 0.19/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.46 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(complementT_lem_type,type,
% 0.19/0.46 complementT_lem: $o ).
% 0.19/0.46
% 0.19/0.46 thf(complementT_lem,definition,
% 0.19/0.46 ( complementT_lem
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setextT_type,type,
% 0.19/0.46 setextT: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setextT,definition,
% 0.19/0.46 ( setextT
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Y: $i] :
% 0.19/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx @ X )
% 0.19/0.46 => ( in @ Xx @ Y ) ) )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx @ Y )
% 0.19/0.46 => ( in @ Xx @ X ) ) )
% 0.19/0.46 => ( X = Y ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subsetTI_type,type,
% 0.19/0.46 subsetTI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetTI,definition,
% 0.19/0.46 ( subsetTI
% 0.19/0.46 = ( ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Y: $i] :
% 0.19/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx @ X )
% 0.19/0.46 => ( in @ Xx @ Y ) ) )
% 0.19/0.46 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powersetTI1,conjecture,
% 0.19/0.46 ( setextAx
% 0.19/0.46 => ( emptysetAx
% 0.19/0.46 => ( setadjoinAx
% 0.19/0.46 => ( powersetAx
% 0.19/0.46 => ( setunionAx
% 0.19/0.46 => ( omega0Ax
% 0.19/0.46 => ( omegaSAx
% 0.19/0.46 => ( omegaIndAx
% 0.19/0.46 => ( replAx
% 0.19/0.46 => ( foundationAx
% 0.19/0.46 => ( wellorderingAx
% 0.19/0.46 => ( descrp
% 0.19/0.46 => ( dsetconstrI
% 0.19/0.46 => ( dsetconstrEL
% 0.19/0.46 => ( dsetconstrER
% 0.19/0.46 => ( exuE1
% 0.19/0.46 => ( prop2setE
% 0.19/0.46 => ( emptysetE
% 0.19/0.46 => ( emptysetimpfalse
% 0.19/0.46 => ( notinemptyset
% 0.19/0.46 => ( exuE3e
% 0.19/0.46 => ( setext
% 0.19/0.46 => ( emptyI
% 0.19/0.46 => ( noeltsimpempty
% 0.19/0.46 => ( setbeta
% 0.19/0.46 => ( nonemptyE1
% 0.19/0.46 => ( nonemptyI
% 0.19/0.46 => ( nonemptyI1
% 0.19/0.46 => ( setadjoinIL
% 0.19/0.46 => ( emptyinunitempty
% 0.19/0.46 => ( setadjoinIR
% 0.19/0.46 => ( setadjoinE
% 0.19/0.46 => ( setadjoinOr
% 0.19/0.46 => ( setoftrueEq
% 0.19/0.46 => ( powersetI
% 0.19/0.46 => ( emptyinPowerset
% 0.19/0.46 => ( emptyInPowerset
% 0.19/0.46 => ( powersetE
% 0.19/0.46 => ( setunionI
% 0.19/0.46 => ( setunionE
% 0.19/0.46 => ( subPowSU
% 0.19/0.46 => ( exuE2
% 0.19/0.46 => ( nonemptyImpWitness
% 0.19/0.46 => ( uniqinunit
% 0.19/0.46 => ( notinsingleton
% 0.19/0.46 => ( eqinunit
% 0.19/0.46 => ( singletonsswitch
% 0.19/0.46 => ( upairsetE
% 0.19/0.46 => ( upairsetIL
% 0.19/0.46 => ( upairsetIR
% 0.19/0.46 => ( emptyE1
% 0.19/0.46 => ( vacuousDall
% 0.19/0.46 => ( quantDeMorgan1
% 0.19/0.46 => ( quantDeMorgan2
% 0.19/0.46 => ( quantDeMorgan3
% 0.19/0.46 => ( quantDeMorgan4
% 0.19/0.46 => ( prop2setI
% 0.19/0.46 => ( prop2set2propI
% 0.19/0.46 => ( notdexE
% 0.19/0.46 => ( notdallE
% 0.19/0.46 => ( exuI1
% 0.19/0.46 => ( exuI3
% 0.19/0.46 => ( exuI2
% 0.19/0.46 => ( inCongP
% 0.19/0.46 => ( in__Cong
% 0.19/0.46 => ( exuE3u
% 0.19/0.46 => ( exu__Cong
% 0.19/0.46 => ( emptyset__Cong
% 0.19/0.46 => ( setadjoin__Cong
% 0.19/0.46 => ( powerset__Cong
% 0.19/0.46 => ( setunion__Cong
% 0.19/0.46 => ( omega__Cong
% 0.19/0.46 => ( exuEu
% 0.19/0.46 => ( descr__Cong
% 0.19/0.46 => ( dsetconstr__Cong
% 0.19/0.46 => ( subsetI1
% 0.19/0.46 => ( eqimpsubset2
% 0.19/0.46 => ( eqimpsubset1
% 0.19/0.46 => ( subsetI2
% 0.19/0.46 => ( emptysetsubset
% 0.19/0.46 => ( subsetE
% 0.19/0.46 => ( subsetE2
% 0.19/0.46 => ( notsubsetI
% 0.19/0.46 => ( notequalI1
% 0.19/0.46 => ( notequalI2
% 0.19/0.46 => ( subsetRefl
% 0.19/0.46 => ( subsetTrans
% 0.19/0.46 => ( setadjoinSub
% 0.19/0.46 => ( setadjoinSub2
% 0.19/0.46 => ( subset2powerset
% 0.19/0.46 => ( setextsub
% 0.19/0.46 => ( subsetemptysetimpeq
% 0.19/0.46 => ( powersetI1
% 0.19/0.46 => ( powersetE1
% 0.19/0.46 => ( inPowerset
% 0.19/0.46 => ( powersetsubset
% 0.19/0.46 => ( sepInPowerset
% 0.19/0.46 => ( sepSubset
% 0.19/0.46 => ( binunionIL
% 0.19/0.46 => ( upairset2IR
% 0.19/0.46 => ( binunionIR
% 0.19/0.46 => ( binunionEcases
% 0.19/0.46 => ( binunionE
% 0.19/0.46 => ( binunionLsub
% 0.19/0.46 => ( binunionRsub
% 0.19/0.46 => ( binintersectI
% 0.19/0.46 => ( binintersectSubset5
% 0.19/0.46 => ( binintersectEL
% 0.19/0.46 => ( binintersectLsub
% 0.19/0.46 => ( binintersectSubset2
% 0.19/0.46 => ( binintersectSubset3
% 0.19/0.46 => ( binintersectER
% 0.19/0.46 => ( disjointsetsI1
% 0.19/0.46 => ( binintersectRsub
% 0.19/0.46 => ( binintersectSubset4
% 0.19/0.46 => ( binintersectSubset1
% 0.19/0.46 => ( bs114d
% 0.19/0.46 => ( setminusI
% 0.19/0.46 => ( setminusEL
% 0.19/0.46 => ( setminusER
% 0.19/0.46 => ( setminusSubset2
% 0.19/0.46 => ( setminusERneg
% 0.19/0.46 => ( setminusELneg
% 0.19/0.46 => ( setminusILneg
% 0.19/0.46 => ( setminusIRneg
% 0.19/0.46 => ( setminusLsub
% 0.19/0.46 => ( setminusSubset1
% 0.19/0.46 => ( symdiffE
% 0.19/0.46 => ( symdiffI1
% 0.19/0.46 => ( symdiffI2
% 0.19/0.46 => ( symdiffIneg1
% 0.19/0.46 => ( symdiffIneg2
% 0.19/0.46 => ( secondinupair
% 0.19/0.46 => ( setukpairIL
% 0.19/0.46 => ( setukpairIR
% 0.19/0.46 => ( kpairiskpair
% 0.19/0.46 => ( kpairp
% 0.19/0.46 => ( singletonsubset
% 0.19/0.46 => ( singletoninpowerset
% 0.19/0.46 => ( singletoninpowunion
% 0.19/0.46 => ( upairset2E
% 0.19/0.46 => ( upairsubunion
% 0.19/0.46 => ( upairinpowunion
% 0.19/0.46 => ( ubforcartprodlem1
% 0.19/0.46 => ( ubforcartprodlem2
% 0.19/0.46 => ( ubforcartprodlem3
% 0.19/0.46 => ( cartprodpairin
% 0.19/0.46 => ( cartprodmempair1
% 0.19/0.46 => ( cartprodmempair
% 0.19/0.46 => ( setunionE2
% 0.19/0.46 => ( setunionsingleton1
% 0.19/0.46 => ( setunionsingleton2
% 0.19/0.46 => ( setunionsingleton
% 0.19/0.46 => ( singletonprop
% 0.19/0.46 => ( ex1E1
% 0.19/0.46 => ( ex1I
% 0.19/0.46 => ( ex1I2
% 0.19/0.46 => ( singletonsuniq
% 0.19/0.46 => ( setukpairinjL1
% 0.19/0.46 => ( kfstsingleton
% 0.19/0.46 => ( theprop
% 0.19/0.46 => ( kfstpairEq
% 0.19/0.46 => ( cartprodfstin
% 0.19/0.46 => ( setukpairinjL2
% 0.19/0.46 => ( setukpairinjL
% 0.19/0.46 => ( setukpairinjR11
% 0.19/0.46 => ( setukpairinjR12
% 0.19/0.46 => ( setukpairinjR1
% 0.19/0.46 => ( upairequniteq
% 0.19/0.46 => ( setukpairinjR2
% 0.19/0.46 => ( setukpairinjR
% 0.19/0.46 => ( ksndsingleton
% 0.19/0.46 => ( ksndpairEq
% 0.19/0.46 => ( kpairsurjEq
% 0.19/0.46 => ( cartprodsndin
% 0.19/0.46 => ( cartprodpairmemEL
% 0.19/0.46 => ( cartprodpairmemER
% 0.19/0.46 => ( cartprodmempaircEq
% 0.19/0.46 => ( cartprodfstpairEq
% 0.19/0.46 => ( cartprodsndpairEq
% 0.19/0.46 => ( cartprodpairsurjEq
% 0.19/0.46 => ( dpsetconstrI
% 0.19/0.46 => ( dpsetconstrSub
% 0.19/0.46 => ( setOfPairsIsBReln
% 0.19/0.46 => ( dpsetconstrERa
% 0.19/0.46 => ( dpsetconstrEL1
% 0.19/0.46 => ( dpsetconstrEL2
% 0.19/0.46 => ( dpsetconstrER
% 0.19/0.46 => ( funcImageSingleton
% 0.19/0.46 => ( apProp
% 0.19/0.46 => ( app
% 0.19/0.46 => ( infuncsetfunc
% 0.19/0.46 => ( ap2p
% 0.19/0.46 => ( funcinfuncset
% 0.19/0.46 => ( lamProp
% 0.19/0.46 => ( lamp
% 0.19/0.46 => ( lam2p
% 0.19/0.46 => ( brelnall1
% 0.19/0.46 => ( brelnall2
% 0.19/0.46 => ( ex1E2
% 0.19/0.46 => ( funcGraphProp1
% 0.19/0.46 => ( funcGraphProp3
% 0.19/0.46 => ( funcGraphProp2
% 0.19/0.46 => ( funcextLem
% 0.19/0.46 => ( funcGraphProp4
% 0.19/0.46 => ( subbreln
% 0.19/0.46 => ( eqbreln
% 0.19/0.46 => ( funcext
% 0.19/0.46 => ( funcext2
% 0.19/0.46 => ( ap2apEq1
% 0.19/0.46 => ( ap2apEq2
% 0.19/0.46 => ( beta1
% 0.19/0.46 => ( eta1
% 0.19/0.46 => ( lam2lamEq
% 0.19/0.46 => ( beta2
% 0.19/0.46 => ( eta2
% 0.19/0.46 => ( iffalseProp1
% 0.19/0.46 => ( iffalseProp2
% 0.19/0.46 => ( iftrueProp1
% 0.19/0.46 => ( iftrueProp2
% 0.19/0.46 => ( ifSingleton
% 0.19/0.46 => ( ifp
% 0.19/0.46 => ( theeq
% 0.19/0.46 => ( iftrue
% 0.19/0.46 => ( iffalse
% 0.19/0.46 => ( iftrueorfalse
% 0.19/0.46 => ( binintersectT_lem
% 0.19/0.46 => ( binunionT_lem
% 0.19/0.46 => ( powersetT_lem
% 0.19/0.46 => ( setminusT_lem
% 0.19/0.46 => ( complementT_lem
% 0.19/0.46 => ( setextT
% 0.19/0.46 => ( subsetTI
% 0.19/0.46 => ! [A: $i,X: $i] :
% 0.19/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.46 => ! [Y: $i] :
% 0.19/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx @ X )
% 0.19/0.46 => ( in @ Xx @ Y ) ) )
% 0.19/0.46 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 ------------------------
% 0.30/1.50 ---- Embedded in THF ---
% 0.30/1.50 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.30/1.50 %%% Generated on Tue Feb 24 12:32:02 EST 2026
% 0.30/1.50 %%% using '$$dhol' embedding, version 1.3.0.
% 0.30/1.50 %%% Logic specification used:
% 0.30/1.50 %%% thf(spec, logic, $$dhol).
% 0.30/1.50
% 0.30/1.50 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.rvsH9IYwBY/DTF2DTF12192.p
% See solution above
% 0.30/1.51 ------------------------
% 0.30/1.54 ---- Cleaned THF ---
% 0.30/1.54 thf(in_type,type,
% 0.30/1.54 in: $i > $i > $o ).
% 0.30/1.54
% 0.30/1.54 thf(exu_type,type,
% 0.30/1.54 exu: ( $i > $o ) > $o ).
% 0.30/1.54
% 0.30/1.54 thf(exu,definition,
% 0.30/1.54 ( exu
% 0.30/1.54 = ( ^ [Xphi: $i > $o] :
% 0.30/1.54 ? [Xx: $i] :
% 0.30/1.54 ( ( Xphi @ Xx )
% 0.30/1.54 & ! [Xy: $i] :
% 0.30/1.54 ( ( Xphi @ Xy )
% 0.30/1.54 => ( Xx = Xy ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setextAx_type,type,
% 0.30/1.54 setextAx: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setextAx,definition,
% 0.30/1.54 ( setextAx
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 <=> ( in @ Xx @ B ) )
% 0.30/1.54 => ( A = B ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(emptyset_type,type,
% 0.30/1.54 emptyset: $i ).
% 0.30/1.54
% 0.30/1.54 thf(emptysetAx_type,type,
% 0.30/1.54 emptysetAx: $o ).
% 0.30/1.54
% 0.30/1.54 thf(emptysetAx,definition,
% 0.30/1.54 ( emptysetAx
% 0.30/1.54 = ( ! [Xx: $i] :
% 0.30/1.54 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setadjoin_type,type,
% 0.30/1.54 setadjoin: $i > $i > $i ).
% 0.30/1.54
% 0.30/1.54 thf(setadjoinAx_type,type,
% 0.30/1.54 setadjoinAx: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setadjoinAx,definition,
% 0.30/1.54 ( setadjoinAx
% 0.30/1.54 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/1.54 <=> ( ( Xy = Xx )
% 0.30/1.54 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(powerset_type,type,
% 0.30/1.54 powerset: $i > $i ).
% 0.30/1.54
% 0.30/1.54 thf(powersetAx_type,type,
% 0.30/1.54 powersetAx: $o ).
% 0.30/1.54
% 0.30/1.54 thf(powersetAx,definition,
% 0.30/1.54 ( powersetAx
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ( in @ B @ ( powerset @ A ) )
% 0.30/1.54 <=> ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ B )
% 0.30/1.54 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setunion_type,type,
% 0.30/1.54 setunion: $i > $i ).
% 0.30/1.54
% 0.30/1.54 thf(setunionAx_type,type,
% 0.30/1.54 setunionAx: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setunionAx,definition,
% 0.30/1.54 ( setunionAx
% 0.30/1.54 = ( ! [A: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/1.54 <=> ? [B: $i] :
% 0.30/1.54 ( ( in @ Xx @ B )
% 0.30/1.54 & ( in @ B @ A ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(omega_type,type,
% 0.30/1.54 omega: $i ).
% 0.30/1.54
% 0.30/1.54 thf(omega0Ax_type,type,
% 0.30/1.54 omega0Ax: $o ).
% 0.30/1.54
% 0.30/1.54 thf(omega0Ax,definition,
% 0.30/1.54 ( omega0Ax
% 0.30/1.54 = ( in @ emptyset @ omega ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(omegaSAx_type,type,
% 0.30/1.54 omegaSAx: $o ).
% 0.30/1.54
% 0.30/1.54 thf(omegaSAx,definition,
% 0.30/1.54 ( omegaSAx
% 0.30/1.54 = ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ omega )
% 0.30/1.54 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(omegaIndAx_type,type,
% 0.30/1.54 omegaIndAx: $o ).
% 0.30/1.54
% 0.30/1.54 thf(omegaIndAx,definition,
% 0.30/1.54 ( omegaIndAx
% 0.30/1.54 = ( ! [A: $i] :
% 0.30/1.54 ( ( ( in @ emptyset @ A )
% 0.30/1.54 & ! [Xx: $i] :
% 0.30/1.54 ( ( ( in @ Xx @ omega )
% 0.30/1.54 & ( in @ Xx @ A ) )
% 0.30/1.54 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.30/1.54 => ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ omega )
% 0.30/1.54 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(replAx_type,type,
% 0.30/1.54 replAx: $o ).
% 0.30/1.54
% 0.30/1.54 thf(replAx,definition,
% 0.30/1.54 ( replAx
% 0.30/1.54 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.30/1.54 ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( exu
% 0.30/1.54 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.30/1.54 => ? [B: $i] :
% 0.30/1.54 ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ B )
% 0.30/1.54 <=> ? [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ A )
% 0.30/1.54 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(foundationAx_type,type,
% 0.30/1.54 foundationAx: $o ).
% 0.30/1.54
% 0.30/1.54 thf(foundationAx,definition,
% 0.30/1.54 ( foundationAx
% 0.30/1.54 = ( ! [A: $i] :
% 0.30/1.54 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/1.54 => ? [B: $i] :
% 0.30/1.54 ( ( in @ B @ A )
% 0.30/1.54 & ~ ? [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ B )
% 0.30/1.54 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(wellorderingAx_type,type,
% 0.30/1.54 wellorderingAx: $o ).
% 0.30/1.54
% 0.30/1.54 thf(wellorderingAx,definition,
% 0.30/1.54 ( wellorderingAx
% 0.30/1.54 = ( ! [A: $i] :
% 0.30/1.54 ? [B: $i] :
% 0.30/1.54 ( ! [C: $i] :
% 0.30/1.54 ( ( in @ C @ B )
% 0.30/1.54 => ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ C )
% 0.30/1.54 => ( in @ Xx @ A ) ) )
% 0.30/1.54 & ! [Xx: $i,Xy: $i] :
% 0.30/1.54 ( ( ( in @ Xx @ A )
% 0.30/1.54 & ( in @ Xy @ A ) )
% 0.30/1.54 => ( ! [C: $i] :
% 0.30/1.54 ( ( in @ C @ B )
% 0.30/1.54 => ( ( in @ Xx @ C )
% 0.30/1.54 <=> ( in @ Xy @ C ) ) )
% 0.30/1.54 => ( Xx = Xy ) ) )
% 0.30/1.54 & ! [C: $i,D: $i] :
% 0.30/1.54 ( ( ( in @ C @ B )
% 0.30/1.54 & ( in @ D @ B ) )
% 0.30/1.54 => ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ C )
% 0.30/1.54 => ( in @ Xx @ D ) )
% 0.30/1.54 | ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ D )
% 0.30/1.54 => ( in @ Xx @ C ) ) ) )
% 0.30/1.54 & ! [C: $i] :
% 0.30/1.54 ( ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ C )
% 0.30/1.54 => ( in @ Xx @ A ) )
% 0.30/1.54 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.30/1.54 => ? [D: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ D @ B )
% 0.30/1.54 & ( in @ Xx @ C )
% 0.30/1.54 & ~ ? [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ D )
% 0.30/1.54 & ( in @ Xy @ C ) )
% 0.30/1.54 & ! [E: $i] :
% 0.30/1.54 ( ( in @ E @ B )
% 0.30/1.54 => ( ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ E )
% 0.30/1.54 => ( in @ Xy @ D ) )
% 0.30/1.54 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(descr_type,type,
% 0.30/1.54 descr: ( $i > $o ) > $i ).
% 0.30/1.54
% 0.30/1.54 thf(descrp_type,type,
% 0.30/1.54 descrp: $o ).
% 0.30/1.54
% 0.30/1.54 thf(descrp,definition,
% 0.30/1.54 ( descrp
% 0.30/1.54 = ( ! [Xphi: $i > $o] :
% 0.30/1.54 ( ( exu
% 0.30/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.54 => ( Xphi
% 0.30/1.54 @ ( descr
% 0.30/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(dsetconstr_type,type,
% 0.30/1.54 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.30/1.54
% 0.30/1.54 thf(dsetconstrI_type,type,
% 0.30/1.54 dsetconstrI: $o ).
% 0.30/1.54
% 0.30/1.54 thf(dsetconstrI,definition,
% 0.30/1.54 ( dsetconstrI
% 0.30/1.54 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( ( Xphi @ Xx )
% 0.30/1.54 => ( in @ Xx
% 0.30/1.54 @ ( dsetconstr @ A
% 0.30/1.54 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(dsetconstrEL_type,type,
% 0.30/1.54 dsetconstrEL: $o ).
% 0.30/1.54
% 0.30/1.54 thf(dsetconstrEL,definition,
% 0.30/1.54 ( dsetconstrEL
% 0.30/1.54 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx
% 0.30/1.54 @ ( dsetconstr @ A
% 0.30/1.54 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/1.54 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(dsetconstrER_type,type,
% 0.30/1.54 dsetconstrER: $o ).
% 0.30/1.54
% 0.30/1.54 thf(dsetconstrER,definition,
% 0.30/1.54 ( dsetconstrER
% 0.30/1.54 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx
% 0.30/1.54 @ ( dsetconstr @ A
% 0.30/1.54 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/1.54 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(exuE1_type,type,
% 0.30/1.54 exuE1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(exuE1,definition,
% 0.30/1.54 ( exuE1
% 0.30/1.54 = ( ! [Xphi: $i > $o] :
% 0.30/1.54 ( ( exu
% 0.30/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.54 => ? [Xx: $i] :
% 0.30/1.54 ( ( Xphi @ Xx )
% 0.30/1.54 & ! [Xy: $i] :
% 0.30/1.54 ( ( Xphi @ Xy )
% 0.30/1.54 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(prop2set_type,type,
% 0.30/1.54 prop2set: $o > $i ).
% 0.30/1.54
% 0.30/1.54 thf(prop2setE_type,type,
% 0.30/1.54 prop2setE: $o ).
% 0.30/1.54
% 0.30/1.54 thf(prop2setE,definition,
% 0.30/1.54 ( prop2setE
% 0.30/1.54 = ( ! [Xphi: $o,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.30/1.54 => Xphi ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(emptysetE_type,type,
% 0.30/1.54 emptysetE: $o ).
% 0.30/1.54
% 0.30/1.54 thf(emptysetE,definition,
% 0.30/1.54 ( emptysetE
% 0.30/1.54 = ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ emptyset )
% 0.30/1.54 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(emptysetimpfalse_type,type,
% 0.30/1.54 emptysetimpfalse: $o ).
% 0.30/1.54
% 0.30/1.54 thf(emptysetimpfalse,definition,
% 0.30/1.54 ( emptysetimpfalse
% 0.30/1.54 = ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ emptyset )
% 0.30/1.54 => $false ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(notinemptyset_type,type,
% 0.30/1.54 notinemptyset: $o ).
% 0.30/1.54
% 0.30/1.54 thf(notinemptyset,definition,
% 0.30/1.54 ( notinemptyset
% 0.30/1.54 = ( ! [Xx: $i] :
% 0.30/1.54 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(exuE3e_type,type,
% 0.30/1.54 exuE3e: $o ).
% 0.30/1.54
% 0.30/1.54 thf(exuE3e,definition,
% 0.30/1.54 ( exuE3e
% 0.30/1.54 = ( ! [Xphi: $i > $o] :
% 0.30/1.54 ( ( exu
% 0.30/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.54 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setext_type,type,
% 0.30/1.54 setext: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setext,definition,
% 0.30/1.54 ( setext
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( in @ Xx @ B ) )
% 0.30/1.54 => ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ B )
% 0.30/1.54 => ( in @ Xx @ A ) )
% 0.30/1.54 => ( A = B ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(emptyI_type,type,
% 0.30/1.54 emptyI: $o ).
% 0.30/1.54
% 0.30/1.54 thf(emptyI,definition,
% 0.30/1.54 ( emptyI
% 0.30/1.54 = ( ! [A: $i] :
% 0.30/1.54 ( ! [Xx: $i] :
% 0.30/1.54 ~ ( in @ Xx @ A )
% 0.30/1.54 => ( A = emptyset ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(noeltsimpempty_type,type,
% 0.30/1.54 noeltsimpempty: $o ).
% 0.30/1.54
% 0.30/1.54 thf(noeltsimpempty,definition,
% 0.30/1.54 ( noeltsimpempty
% 0.30/1.54 = ( ! [A: $i] :
% 0.30/1.54 ( ! [Xx: $i] :
% 0.30/1.54 ~ ( in @ Xx @ A )
% 0.30/1.54 => ( A = emptyset ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setbeta_type,type,
% 0.30/1.54 setbeta: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setbeta,definition,
% 0.30/1.54 ( setbeta
% 0.30/1.54 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( ( in @ Xx
% 0.30/1.54 @ ( dsetconstr @ A
% 0.30/1.54 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/1.54 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(nonempty_type,type,
% 0.30/1.54 nonempty: $i > $o ).
% 0.30/1.54
% 0.30/1.54 thf(nonempty,definition,
% 0.30/1.54 ( nonempty
% 0.30/1.54 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(nonemptyE1_type,type,
% 0.30/1.54 nonemptyE1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(nonemptyE1,definition,
% 0.30/1.54 ( nonemptyE1
% 0.30/1.54 = ( ! [A: $i] :
% 0.30/1.54 ( ( nonempty @ A )
% 0.30/1.54 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(nonemptyI_type,type,
% 0.30/1.54 nonemptyI: $o ).
% 0.30/1.54
% 0.30/1.54 thf(nonemptyI,definition,
% 0.30/1.54 ( nonemptyI
% 0.30/1.54 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( ( Xphi @ Xx )
% 0.30/1.54 => ( nonempty
% 0.30/1.54 @ ( dsetconstr @ A
% 0.30/1.54 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(nonemptyI1_type,type,
% 0.30/1.54 nonemptyI1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(nonemptyI1,definition,
% 0.30/1.54 ( nonemptyI1
% 0.30/1.54 = ( ! [A: $i] :
% 0.30/1.54 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/1.54 => ( nonempty @ A ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setadjoinIL_type,type,
% 0.30/1.54 setadjoinIL: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setadjoinIL,definition,
% 0.30/1.54 ( setadjoinIL
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(emptyinunitempty_type,type,
% 0.30/1.54 emptyinunitempty: $o ).
% 0.30/1.54
% 0.30/1.54 thf(emptyinunitempty,definition,
% 0.30/1.54 ( emptyinunitempty
% 0.30/1.54 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setadjoinIR_type,type,
% 0.30/1.54 setadjoinIR: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setadjoinIR,definition,
% 0.30/1.54 ( setadjoinIR
% 0.30/1.54 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ A )
% 0.30/1.54 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setadjoinE_type,type,
% 0.30/1.54 setadjoinE: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setadjoinE,definition,
% 0.30/1.54 ( setadjoinE
% 0.30/1.54 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/1.54 => ! [Xphi: $o] :
% 0.30/1.54 ( ( ( Xy = Xx )
% 0.30/1.54 => Xphi )
% 0.30/1.54 => ( ( ( in @ Xy @ A )
% 0.30/1.54 => Xphi )
% 0.30/1.54 => Xphi ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setadjoinOr_type,type,
% 0.30/1.54 setadjoinOr: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setadjoinOr,definition,
% 0.30/1.54 ( setadjoinOr
% 0.30/1.54 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/1.54 => ( ( Xy = Xx )
% 0.30/1.54 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setoftrueEq_type,type,
% 0.30/1.54 setoftrueEq: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setoftrueEq,definition,
% 0.30/1.54 ( setoftrueEq
% 0.30/1.54 = ( ! [A: $i] :
% 0.30/1.54 ( ( dsetconstr @ A
% 0.30/1.54 @ ^ [Xx: $i] : $true )
% 0.30/1.54 = A ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(powersetI_type,type,
% 0.30/1.54 powersetI: $o ).
% 0.30/1.54
% 0.30/1.54 thf(powersetI,definition,
% 0.30/1.54 ( powersetI
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ B )
% 0.30/1.54 => ( in @ Xx @ A ) )
% 0.30/1.54 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(emptyinPowerset_type,type,
% 0.30/1.54 emptyinPowerset: $o ).
% 0.30/1.54
% 0.30/1.54 thf(emptyinPowerset,definition,
% 0.30/1.54 ( emptyinPowerset
% 0.30/1.54 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(emptyInPowerset_type,type,
% 0.30/1.54 emptyInPowerset: $o ).
% 0.30/1.54
% 0.30/1.54 thf(emptyInPowerset,definition,
% 0.30/1.54 ( emptyInPowerset
% 0.30/1.54 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(powersetE_type,type,
% 0.30/1.54 powersetE: $o ).
% 0.30/1.54
% 0.30/1.54 thf(powersetE,definition,
% 0.30/1.54 ( powersetE
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ B @ ( powerset @ A ) )
% 0.30/1.54 => ( ( in @ Xx @ B )
% 0.30/1.54 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setunionI_type,type,
% 0.30/1.54 setunionI: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setunionI,definition,
% 0.30/1.54 ( setunionI
% 0.30/1.54 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/1.54 ( ( in @ Xx @ B )
% 0.30/1.54 => ( ( in @ B @ A )
% 0.30/1.54 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setunionE_type,type,
% 0.30/1.54 setunionE: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setunionE,definition,
% 0.30/1.54 ( setunionE
% 0.30/1.54 = ( ! [A: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/1.54 => ! [Xphi: $o] :
% 0.30/1.54 ( ! [B: $i] :
% 0.30/1.54 ( ( in @ Xx @ B )
% 0.30/1.54 => ( ( in @ B @ A )
% 0.30/1.54 => Xphi ) )
% 0.30/1.54 => Xphi ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(subPowSU_type,type,
% 0.30/1.54 subPowSU: $o ).
% 0.30/1.54
% 0.30/1.54 thf(subPowSU,definition,
% 0.30/1.54 ( subPowSU
% 0.30/1.54 = ( ! [A: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(exuE2_type,type,
% 0.30/1.54 exuE2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(exuE2,definition,
% 0.30/1.54 ( exuE2
% 0.30/1.54 = ( ! [Xphi: $i > $o] :
% 0.30/1.54 ( ( exu
% 0.30/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.54 => ? [Xx: $i] :
% 0.30/1.54 ! [Xy: $i] :
% 0.30/1.54 ( ( Xphi @ Xy )
% 0.30/1.54 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(nonemptyImpWitness_type,type,
% 0.30/1.54 nonemptyImpWitness: $o ).
% 0.30/1.54
% 0.30/1.54 thf(nonemptyImpWitness,definition,
% 0.30/1.54 ( nonemptyImpWitness
% 0.30/1.54 = ( ! [A: $i] :
% 0.30/1.54 ( ( nonempty @ A )
% 0.30/1.54 => ? [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 & $true ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(uniqinunit_type,type,
% 0.30/1.54 uniqinunit: $o ).
% 0.30/1.54
% 0.30/1.54 thf(uniqinunit,definition,
% 0.30/1.54 ( uniqinunit
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i] :
% 0.30/1.54 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/1.54 => ( Xx = Xy ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(notinsingleton_type,type,
% 0.30/1.54 notinsingleton: $o ).
% 0.30/1.54
% 0.30/1.54 thf(notinsingleton,definition,
% 0.30/1.54 ( notinsingleton
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i] :
% 0.30/1.54 ( ( Xx != Xy )
% 0.30/1.54 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(eqinunit_type,type,
% 0.30/1.54 eqinunit: $o ).
% 0.30/1.54
% 0.30/1.54 thf(eqinunit,definition,
% 0.30/1.54 ( eqinunit
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i] :
% 0.30/1.54 ( ( Xx = Xy )
% 0.30/1.54 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(singletonsswitch_type,type,
% 0.30/1.54 singletonsswitch: $o ).
% 0.30/1.54
% 0.30/1.54 thf(singletonsswitch,definition,
% 0.30/1.54 ( singletonsswitch
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i] :
% 0.30/1.54 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/1.54 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(upairsetE_type,type,
% 0.30/1.54 upairsetE: $o ).
% 0.30/1.54
% 0.30/1.54 thf(upairsetE,definition,
% 0.30/1.54 ( upairsetE
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/1.54 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.30/1.54 => ( ( Xz = Xx )
% 0.30/1.54 | ( Xz = Xy ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(upairsetIL_type,type,
% 0.30/1.54 upairsetIL: $o ).
% 0.30/1.54
% 0.30/1.54 thf(upairsetIL,definition,
% 0.30/1.54 ( upairsetIL
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(upairsetIR_type,type,
% 0.30/1.54 upairsetIR: $o ).
% 0.30/1.54
% 0.30/1.54 thf(upairsetIR,definition,
% 0.30/1.54 ( upairsetIR
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(emptyE1_type,type,
% 0.30/1.54 emptyE1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(emptyE1,definition,
% 0.30/1.54 ( emptyE1
% 0.30/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.54 ( ? [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 & ( Xphi @ Xx ) )
% 0.30/1.54 => ( ( ( dsetconstr @ A
% 0.30/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.54 = emptyset )
% 0.30/1.54 => $false ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(vacuousDall_type,type,
% 0.30/1.54 vacuousDall: $o ).
% 0.30/1.54
% 0.30/1.54 thf(vacuousDall,definition,
% 0.30/1.54 ( vacuousDall
% 0.30/1.54 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ emptyset )
% 0.30/1.54 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(quantDeMorgan1_type,type,
% 0.30/1.54 quantDeMorgan1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(quantDeMorgan1,definition,
% 0.30/1.54 ( quantDeMorgan1
% 0.30/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.54 ( ~ ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( Xphi @ Xx ) )
% 0.30/1.54 => ? [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(quantDeMorgan2_type,type,
% 0.30/1.54 quantDeMorgan2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(quantDeMorgan2,definition,
% 0.30/1.54 ( quantDeMorgan2
% 0.30/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.54 ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ~ ( Xphi @ Xx ) )
% 0.30/1.54 => ~ ? [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(quantDeMorgan3_type,type,
% 0.30/1.54 quantDeMorgan3: $o ).
% 0.30/1.54
% 0.30/1.54 thf(quantDeMorgan3,definition,
% 0.30/1.54 ( quantDeMorgan3
% 0.30/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.54 ( ~ ? [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 & ( Xphi @ Xx ) )
% 0.30/1.54 => ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(quantDeMorgan4_type,type,
% 0.30/1.54 quantDeMorgan4: $o ).
% 0.30/1.54
% 0.30/1.54 thf(quantDeMorgan4,definition,
% 0.30/1.54 ( quantDeMorgan4
% 0.30/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.54 ( ? [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 & ~ ( Xphi @ Xx ) )
% 0.30/1.54 => ~ ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(prop2setI_type,type,
% 0.30/1.54 prop2setI: $o ).
% 0.30/1.54
% 0.30/1.54 thf(prop2setI,definition,
% 0.30/1.54 ( prop2setI
% 0.30/1.54 = ( ! [Xphi: $o] :
% 0.30/1.54 ( Xphi
% 0.30/1.54 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(set2prop_type,type,
% 0.30/1.54 set2prop: $i > $o ).
% 0.30/1.54
% 0.30/1.54 thf(prop2set2propI_type,type,
% 0.30/1.54 prop2set2propI: $o ).
% 0.30/1.54
% 0.30/1.54 thf(prop2set2propI,definition,
% 0.30/1.54 ( prop2set2propI
% 0.30/1.54 = ( ! [Xphi: $o] :
% 0.30/1.54 ( Xphi
% 0.30/1.54 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(notdexE_type,type,
% 0.30/1.54 notdexE: $o ).
% 0.30/1.54
% 0.30/1.54 thf(notdexE,definition,
% 0.30/1.54 ( notdexE
% 0.30/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.54 ( ~ ? [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 & ( Xphi @ Xx ) )
% 0.30/1.54 => ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(notdallE_type,type,
% 0.30/1.54 notdallE: $o ).
% 0.30/1.54
% 0.30/1.54 thf(notdallE,definition,
% 0.30/1.54 ( notdallE
% 0.30/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.54 ( ~ ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( Xphi @ Xx ) )
% 0.30/1.54 => ? [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(exuI1_type,type,
% 0.30/1.54 exuI1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(exuI1,definition,
% 0.30/1.54 ( exuI1
% 0.30/1.54 = ( ! [Xphi: $i > $o] :
% 0.30/1.54 ( ? [Xx: $i] :
% 0.30/1.54 ( ( Xphi @ Xx )
% 0.30/1.54 & ! [Xy: $i] :
% 0.30/1.54 ( ( Xphi @ Xy )
% 0.30/1.54 => ( Xx = Xy ) ) )
% 0.30/1.54 => ( exu
% 0.30/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(exuI3_type,type,
% 0.30/1.54 exuI3: $o ).
% 0.30/1.54
% 0.30/1.54 thf(exuI3,definition,
% 0.30/1.54 ( exuI3
% 0.30/1.54 = ( ! [Xphi: $i > $o] :
% 0.30/1.54 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.30/1.54 => ( ! [Xx: $i,Xy: $i] :
% 0.30/1.54 ( ( Xphi @ Xx )
% 0.30/1.54 => ( ( Xphi @ Xy )
% 0.30/1.54 => ( Xx = Xy ) ) )
% 0.30/1.54 => ( exu
% 0.30/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(exuI2_type,type,
% 0.30/1.54 exuI2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(exuI2,definition,
% 0.30/1.54 ( exuI2
% 0.30/1.54 = ( ! [Xphi: $i > $o] :
% 0.30/1.54 ( ? [Xx: $i] :
% 0.30/1.54 ! [Xy: $i] :
% 0.30/1.54 ( ( Xphi @ Xy )
% 0.30/1.54 <=> ( Xy = Xx ) )
% 0.30/1.54 => ( exu
% 0.30/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(inCongP_type,type,
% 0.30/1.54 inCongP: $o ).
% 0.30/1.54
% 0.30/1.54 thf(inCongP,definition,
% 0.30/1.54 ( inCongP
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ( A = B )
% 0.30/1.54 => ! [Xx: $i,Xy: $i] :
% 0.30/1.54 ( ( Xx = Xy )
% 0.30/1.54 => ( ( in @ Xx @ A )
% 0.30/1.54 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(in__Cong_type,type,
% 0.30/1.54 in__Cong: $o ).
% 0.30/1.54
% 0.30/1.54 thf(in__Cong,definition,
% 0.30/1.54 ( in__Cong
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ( A = B )
% 0.30/1.54 => ! [Xx: $i,Xy: $i] :
% 0.30/1.54 ( ( Xx = Xy )
% 0.30/1.54 => ( ( in @ Xx @ A )
% 0.30/1.54 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(exuE3u_type,type,
% 0.30/1.54 exuE3u: $o ).
% 0.30/1.54
% 0.30/1.54 thf(exuE3u,definition,
% 0.30/1.54 ( exuE3u
% 0.30/1.54 = ( ! [Xphi: $i > $o] :
% 0.30/1.54 ( ( exu
% 0.30/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.54 => ! [Xx: $i,Xy: $i] :
% 0.30/1.54 ( ( Xphi @ Xx )
% 0.30/1.54 => ( ( Xphi @ Xy )
% 0.30/1.54 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(exu__Cong_type,type,
% 0.30/1.54 exu__Cong: $o ).
% 0.30/1.54
% 0.30/1.54 thf(exu__Cong,definition,
% 0.30/1.54 ( exu__Cong
% 0.30/1.54 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/1.54 ( ! [Xx: $i,Xy: $i] :
% 0.30/1.54 ( ( Xx = Xy )
% 0.30/1.54 => ( ( Xphi @ Xx )
% 0.30/1.54 <=> ( Xpsi @ Xy ) ) )
% 0.30/1.54 => ( ( exu
% 0.30/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.54 <=> ( exu
% 0.30/1.54 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(emptyset__Cong_type,type,
% 0.30/1.54 emptyset__Cong: $o ).
% 0.30/1.54
% 0.30/1.54 thf(emptyset__Cong,definition,
% 0.30/1.54 ( emptyset__Cong
% 0.30/1.54 = ( emptyset = emptyset ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setadjoin__Cong_type,type,
% 0.30/1.54 setadjoin__Cong: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setadjoin__Cong,definition,
% 0.30/1.54 ( setadjoin__Cong
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i] :
% 0.30/1.54 ( ( Xx = Xy )
% 0.30/1.54 => ! [Xz: $i,Xu: $i] :
% 0.30/1.54 ( ( Xz = Xu )
% 0.30/1.54 => ( ( setadjoin @ Xx @ Xz )
% 0.30/1.54 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(powerset__Cong_type,type,
% 0.30/1.54 powerset__Cong: $o ).
% 0.30/1.54
% 0.30/1.54 thf(powerset__Cong,definition,
% 0.30/1.54 ( powerset__Cong
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ( A = B )
% 0.30/1.54 => ( ( powerset @ A )
% 0.30/1.54 = ( powerset @ B ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setunion__Cong_type,type,
% 0.30/1.54 setunion__Cong: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setunion__Cong,definition,
% 0.30/1.54 ( setunion__Cong
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ( A = B )
% 0.30/1.54 => ( ( setunion @ A )
% 0.30/1.54 = ( setunion @ B ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(omega__Cong_type,type,
% 0.30/1.54 omega__Cong: $o ).
% 0.30/1.54
% 0.30/1.54 thf(omega__Cong,definition,
% 0.30/1.54 ( omega__Cong
% 0.30/1.54 = ( omega = omega ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(exuEu_type,type,
% 0.30/1.54 exuEu: $o ).
% 0.30/1.54
% 0.30/1.54 thf(exuEu,definition,
% 0.30/1.54 ( exuEu
% 0.30/1.54 = ( ! [Xphi: $i > $o] :
% 0.30/1.54 ( ( exu
% 0.30/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.54 => ! [Xx: $i,Xy: $i] :
% 0.30/1.54 ( ( Xphi @ Xx )
% 0.30/1.54 => ( ( Xphi @ Xy )
% 0.30/1.54 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(descr__Cong_type,type,
% 0.30/1.54 descr__Cong: $o ).
% 0.30/1.54
% 0.30/1.54 thf(descr__Cong,definition,
% 0.30/1.54 ( descr__Cong
% 0.30/1.54 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/1.54 ( ! [Xx: $i,Xy: $i] :
% 0.30/1.54 ( ( Xx = Xy )
% 0.30/1.54 => ( ( Xphi @ Xx )
% 0.30/1.54 <=> ( Xpsi @ Xy ) ) )
% 0.30/1.54 => ( ( exu
% 0.30/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.54 => ( ( exu
% 0.30/1.54 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.30/1.54 => ( ( descr
% 0.30/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.54 = ( descr
% 0.30/1.54 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(dsetconstr__Cong_type,type,
% 0.30/1.54 dsetconstr__Cong: $o ).
% 0.30/1.54
% 0.30/1.54 thf(dsetconstr__Cong,definition,
% 0.30/1.54 ( dsetconstr__Cong
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ( A = B )
% 0.30/1.54 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/1.54 ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ B )
% 0.30/1.54 => ( ( Xx = Xy )
% 0.30/1.54 => ( ( Xphi @ Xx )
% 0.30/1.54 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.30/1.54 => ( ( dsetconstr @ A
% 0.30/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.54 = ( dsetconstr @ B
% 0.30/1.54 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(subset_type,type,
% 0.30/1.54 subset: $i > $i > $o ).
% 0.30/1.54
% 0.30/1.54 thf(disjoint_type,type,
% 0.30/1.54 disjoint: $i > $i > $o ).
% 0.30/1.54
% 0.30/1.54 thf(setsmeet_type,type,
% 0.30/1.54 setsmeet: $i > $i > $o ).
% 0.30/1.54
% 0.30/1.54 thf(subsetI1_type,type,
% 0.30/1.54 subsetI1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(subsetI1,definition,
% 0.30/1.54 ( subsetI1
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( in @ Xx @ B ) )
% 0.30/1.54 => ( subset @ A @ B ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(eqimpsubset2_type,type,
% 0.30/1.54 eqimpsubset2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(eqimpsubset2,definition,
% 0.30/1.54 ( eqimpsubset2
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ( A = B )
% 0.30/1.54 => ( subset @ B @ A ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(eqimpsubset1_type,type,
% 0.30/1.54 eqimpsubset1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(eqimpsubset1,definition,
% 0.30/1.54 ( eqimpsubset1
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ( A = B )
% 0.30/1.54 => ( subset @ A @ B ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(subsetI2_type,type,
% 0.30/1.54 subsetI2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(subsetI2,definition,
% 0.30/1.54 ( subsetI2
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( in @ Xx @ B ) )
% 0.30/1.54 => ( subset @ A @ B ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(emptysetsubset_type,type,
% 0.30/1.54 emptysetsubset: $o ).
% 0.30/1.54
% 0.30/1.54 thf(emptysetsubset,definition,
% 0.30/1.54 ( emptysetsubset
% 0.30/1.54 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(subsetE_type,type,
% 0.30/1.54 subsetE: $o ).
% 0.30/1.54
% 0.30/1.54 thf(subsetE,definition,
% 0.30/1.54 ( subsetE
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( subset @ A @ B )
% 0.30/1.54 => ( ( in @ Xx @ A )
% 0.30/1.54 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(subsetE2_type,type,
% 0.30/1.54 subsetE2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(subsetE2,definition,
% 0.30/1.54 ( subsetE2
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( subset @ A @ B )
% 0.30/1.54 => ( ~ ( in @ Xx @ B )
% 0.30/1.54 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(notsubsetI_type,type,
% 0.30/1.54 notsubsetI: $o ).
% 0.30/1.54
% 0.30/1.54 thf(notsubsetI,definition,
% 0.30/1.54 ( notsubsetI
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( ~ ( in @ Xx @ B )
% 0.30/1.54 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(notequalI1_type,type,
% 0.30/1.54 notequalI1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(notequalI1,definition,
% 0.30/1.54 ( notequalI1
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ~ ( subset @ A @ B )
% 0.30/1.54 => ( A != B ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(notequalI2_type,type,
% 0.30/1.54 notequalI2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(notequalI2,definition,
% 0.30/1.54 ( notequalI2
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( ~ ( in @ Xx @ B )
% 0.30/1.54 => ( A != B ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(subsetRefl_type,type,
% 0.30/1.54 subsetRefl: $o ).
% 0.30/1.54
% 0.30/1.54 thf(subsetRefl,definition,
% 0.30/1.54 ( subsetRefl
% 0.30/1.54 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(subsetTrans_type,type,
% 0.30/1.54 subsetTrans: $o ).
% 0.30/1.54
% 0.30/1.54 thf(subsetTrans,definition,
% 0.30/1.54 ( subsetTrans
% 0.30/1.54 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/1.54 ( ( subset @ A @ B )
% 0.30/1.54 => ( ( subset @ B @ C )
% 0.30/1.54 => ( subset @ A @ C ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setadjoinSub_type,type,
% 0.30/1.54 setadjoinSub: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setadjoinSub,definition,
% 0.30/1.54 ( setadjoinSub
% 0.30/1.54 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setadjoinSub2_type,type,
% 0.30/1.54 setadjoinSub2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setadjoinSub2,definition,
% 0.30/1.54 ( setadjoinSub2
% 0.30/1.54 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/1.54 ( ( subset @ A @ B )
% 0.30/1.54 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(subset2powerset_type,type,
% 0.30/1.54 subset2powerset: $o ).
% 0.30/1.54
% 0.30/1.54 thf(subset2powerset,definition,
% 0.30/1.54 ( subset2powerset
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ( subset @ A @ B )
% 0.30/1.54 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setextsub_type,type,
% 0.30/1.54 setextsub: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setextsub,definition,
% 0.30/1.54 ( setextsub
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ( subset @ A @ B )
% 0.30/1.54 => ( ( subset @ B @ A )
% 0.30/1.54 => ( A = B ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(subsetemptysetimpeq_type,type,
% 0.30/1.54 subsetemptysetimpeq: $o ).
% 0.30/1.54
% 0.30/1.54 thf(subsetemptysetimpeq,definition,
% 0.30/1.54 ( subsetemptysetimpeq
% 0.30/1.54 = ( ! [A: $i] :
% 0.30/1.54 ( ( subset @ A @ emptyset )
% 0.30/1.54 => ( A = emptyset ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(powersetI1_type,type,
% 0.30/1.54 powersetI1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(powersetI1,definition,
% 0.30/1.54 ( powersetI1
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ( subset @ B @ A )
% 0.30/1.54 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(powersetE1_type,type,
% 0.30/1.54 powersetE1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(powersetE1,definition,
% 0.30/1.54 ( powersetE1
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ( in @ B @ ( powerset @ A ) )
% 0.30/1.54 => ( subset @ B @ A ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(inPowerset_type,type,
% 0.30/1.54 inPowerset: $o ).
% 0.30/1.54
% 0.30/1.54 thf(inPowerset,definition,
% 0.30/1.54 ( inPowerset
% 0.30/1.54 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(powersetsubset_type,type,
% 0.30/1.54 powersetsubset: $o ).
% 0.30/1.54
% 0.30/1.54 thf(powersetsubset,definition,
% 0.30/1.54 ( powersetsubset
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ( subset @ A @ B )
% 0.30/1.54 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(sepInPowerset_type,type,
% 0.30/1.54 sepInPowerset: $o ).
% 0.30/1.54
% 0.30/1.54 thf(sepInPowerset,definition,
% 0.30/1.54 ( sepInPowerset
% 0.30/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.54 ( in
% 0.30/1.54 @ ( dsetconstr @ A
% 0.30/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.54 @ ( powerset @ A ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(sepSubset_type,type,
% 0.30/1.54 sepSubset: $o ).
% 0.30/1.54
% 0.30/1.54 thf(sepSubset,definition,
% 0.30/1.54 ( sepSubset
% 0.30/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.54 ( subset
% 0.30/1.54 @ ( dsetconstr @ A
% 0.30/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.54 @ A ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(binunion_type,type,
% 0.30/1.54 binunion: $i > $i > $i ).
% 0.30/1.54
% 0.30/1.54 thf(binunionIL_type,type,
% 0.30/1.54 binunionIL: $o ).
% 0.30/1.54
% 0.30/1.54 thf(binunionIL,definition,
% 0.30/1.54 ( binunionIL
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(upairset2IR_type,type,
% 0.30/1.54 upairset2IR: $o ).
% 0.30/1.54
% 0.30/1.54 thf(upairset2IR,definition,
% 0.30/1.54 ( upairset2IR
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(binunionIR_type,type,
% 0.30/1.54 binunionIR: $o ).
% 0.30/1.54
% 0.30/1.54 thf(binunionIR,definition,
% 0.30/1.54 ( binunionIR
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ B )
% 0.30/1.54 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(binunionEcases_type,type,
% 0.30/1.54 binunionEcases: $o ).
% 0.30/1.54
% 0.30/1.54 thf(binunionEcases,definition,
% 0.30/1.54 ( binunionEcases
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.30/1.54 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.30/1.54 => ( ( ( in @ Xx @ A )
% 0.30/1.54 => Xphi )
% 0.30/1.54 => ( ( ( in @ Xx @ B )
% 0.30/1.54 => Xphi )
% 0.30/1.54 => Xphi ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(binunionE_type,type,
% 0.30/1.54 binunionE: $o ).
% 0.30/1.54
% 0.30/1.54 thf(binunionE,definition,
% 0.30/1.54 ( binunionE
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.30/1.54 => ( ( in @ Xx @ A )
% 0.30/1.54 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(binunionLsub_type,type,
% 0.30/1.54 binunionLsub: $o ).
% 0.30/1.54
% 0.30/1.54 thf(binunionLsub,definition,
% 0.30/1.54 ( binunionLsub
% 0.30/1.54 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(binunionRsub_type,type,
% 0.30/1.54 binunionRsub: $o ).
% 0.30/1.54
% 0.30/1.54 thf(binunionRsub,definition,
% 0.30/1.54 ( binunionRsub
% 0.30/1.54 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(binintersect_type,type,
% 0.30/1.54 binintersect: $i > $i > $i ).
% 0.30/1.54
% 0.30/1.54 thf(binintersectI_type,type,
% 0.30/1.54 binintersectI: $o ).
% 0.30/1.54
% 0.30/1.54 thf(binintersectI,definition,
% 0.30/1.54 ( binintersectI
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( ( in @ Xx @ B )
% 0.30/1.54 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(binintersectSubset5_type,type,
% 0.30/1.54 binintersectSubset5: $o ).
% 0.30/1.54
% 0.30/1.54 thf(binintersectSubset5,definition,
% 0.30/1.54 ( binintersectSubset5
% 0.30/1.54 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/1.54 ( ( subset @ C @ A )
% 0.30/1.54 => ( ( subset @ C @ B )
% 0.30/1.54 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(binintersectEL_type,type,
% 0.30/1.54 binintersectEL: $o ).
% 0.30/1.54
% 0.30/1.54 thf(binintersectEL,definition,
% 0.30/1.54 ( binintersectEL
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.30/1.54 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(binintersectLsub_type,type,
% 0.30/1.54 binintersectLsub: $o ).
% 0.30/1.54
% 0.30/1.54 thf(binintersectLsub,definition,
% 0.30/1.54 ( binintersectLsub
% 0.30/1.54 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(binintersectSubset2_type,type,
% 0.30/1.54 binintersectSubset2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(binintersectSubset2,definition,
% 0.30/1.54 ( binintersectSubset2
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ( subset @ A @ B )
% 0.30/1.54 => ( ( binintersect @ A @ B )
% 0.30/1.54 = A ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(binintersectSubset3_type,type,
% 0.30/1.54 binintersectSubset3: $o ).
% 0.30/1.54
% 0.30/1.54 thf(binintersectSubset3,definition,
% 0.30/1.54 ( binintersectSubset3
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ( ( binintersect @ A @ B )
% 0.30/1.54 = B )
% 0.30/1.54 => ( subset @ B @ A ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(binintersectER_type,type,
% 0.30/1.54 binintersectER: $o ).
% 0.30/1.54
% 0.30/1.54 thf(binintersectER,definition,
% 0.30/1.54 ( binintersectER
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.30/1.54 => ( in @ Xx @ B ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(disjointsetsI1_type,type,
% 0.30/1.54 disjointsetsI1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(disjointsetsI1,definition,
% 0.30/1.54 ( disjointsetsI1
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ~ ? [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 & ( in @ Xx @ B ) )
% 0.30/1.54 => ( ( binintersect @ A @ B )
% 0.30/1.54 = emptyset ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(binintersectRsub_type,type,
% 0.30/1.54 binintersectRsub: $o ).
% 0.30/1.54
% 0.30/1.54 thf(binintersectRsub,definition,
% 0.30/1.54 ( binintersectRsub
% 0.30/1.54 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(binintersectSubset4_type,type,
% 0.30/1.54 binintersectSubset4: $o ).
% 0.30/1.54
% 0.30/1.54 thf(binintersectSubset4,definition,
% 0.30/1.54 ( binintersectSubset4
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ( subset @ B @ A )
% 0.30/1.54 => ( ( binintersect @ A @ B )
% 0.30/1.54 = B ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(binintersectSubset1_type,type,
% 0.30/1.54 binintersectSubset1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(binintersectSubset1,definition,
% 0.30/1.54 ( binintersectSubset1
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ( ( binintersect @ A @ B )
% 0.30/1.54 = A )
% 0.30/1.54 => ( subset @ A @ B ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(bs114d_type,type,
% 0.30/1.54 bs114d: $o ).
% 0.30/1.54
% 0.30/1.54 thf(bs114d,definition,
% 0.30/1.54 ( bs114d
% 0.30/1.54 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/1.54 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.30/1.54 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(regular_type,type,
% 0.30/1.54 regular: $i > $o ).
% 0.30/1.54
% 0.30/1.54 thf(setminus_type,type,
% 0.30/1.54 setminus: $i > $i > $i ).
% 0.30/1.54
% 0.30/1.54 thf(setminusI_type,type,
% 0.30/1.54 setminusI: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setminusI,definition,
% 0.30/1.54 ( setminusI
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( ~ ( in @ Xx @ B )
% 0.30/1.54 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setminusEL_type,type,
% 0.30/1.54 setminusEL: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setminusEL,definition,
% 0.30/1.54 ( setminusEL
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/1.54 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setminusER_type,type,
% 0.30/1.54 setminusER: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setminusER,definition,
% 0.30/1.54 ( setminusER
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/1.54 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setminusSubset2_type,type,
% 0.30/1.54 setminusSubset2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setminusSubset2,definition,
% 0.30/1.54 ( setminusSubset2
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ( subset @ A @ B )
% 0.30/1.54 => ( ( setminus @ A @ B )
% 0.30/1.54 = emptyset ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setminusERneg_type,type,
% 0.30/1.54 setminusERneg: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setminusERneg,definition,
% 0.30/1.54 ( setminusERneg
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/1.54 => ( ( in @ Xx @ A )
% 0.30/1.54 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setminusELneg_type,type,
% 0.30/1.54 setminusELneg: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setminusELneg,definition,
% 0.30/1.54 ( setminusELneg
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/1.54 => ( ~ ( in @ Xx @ B )
% 0.30/1.54 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setminusILneg_type,type,
% 0.30/1.54 setminusILneg: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setminusILneg,definition,
% 0.30/1.54 ( setminusILneg
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ~ ( in @ Xx @ A )
% 0.30/1.54 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setminusIRneg_type,type,
% 0.30/1.54 setminusIRneg: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setminusIRneg,definition,
% 0.30/1.54 ( setminusIRneg
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ B )
% 0.30/1.54 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setminusLsub_type,type,
% 0.30/1.54 setminusLsub: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setminusLsub,definition,
% 0.30/1.54 ( setminusLsub
% 0.30/1.54 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setminusSubset1_type,type,
% 0.30/1.54 setminusSubset1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setminusSubset1,definition,
% 0.30/1.54 ( setminusSubset1
% 0.30/1.54 = ( ! [A: $i,B: $i] :
% 0.30/1.54 ( ( ( setminus @ A @ B )
% 0.30/1.54 = emptyset )
% 0.30/1.54 => ( subset @ A @ B ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(symdiff_type,type,
% 0.30/1.54 symdiff: $i > $i > $i ).
% 0.30/1.54
% 0.30/1.54 thf(symdiffE_type,type,
% 0.30/1.54 symdiffE: $o ).
% 0.30/1.54
% 0.30/1.54 thf(symdiffE,definition,
% 0.30/1.54 ( symdiffE
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.30/1.54 => ! [Xphi: $o] :
% 0.30/1.54 ( ( ( in @ Xx @ A )
% 0.30/1.54 => ( ~ ( in @ Xx @ B )
% 0.30/1.54 => Xphi ) )
% 0.30/1.54 => ( ( ~ ( in @ Xx @ A )
% 0.30/1.54 => ( ( in @ Xx @ B )
% 0.30/1.54 => Xphi ) )
% 0.30/1.54 => Xphi ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(symdiffI1_type,type,
% 0.30/1.54 symdiffI1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(symdiffI1,definition,
% 0.30/1.54 ( symdiffI1
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( ~ ( in @ Xx @ B )
% 0.30/1.54 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(symdiffI2_type,type,
% 0.30/1.54 symdiffI2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(symdiffI2,definition,
% 0.30/1.54 ( symdiffI2
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ~ ( in @ Xx @ A )
% 0.30/1.54 => ( ( in @ Xx @ B )
% 0.30/1.54 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(symdiffIneg1_type,type,
% 0.30/1.54 symdiffIneg1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(symdiffIneg1,definition,
% 0.30/1.54 ( symdiffIneg1
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( ( in @ Xx @ B )
% 0.30/1.54 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(symdiffIneg2_type,type,
% 0.30/1.54 symdiffIneg2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(symdiffIneg2,definition,
% 0.30/1.54 ( symdiffIneg2
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ~ ( in @ Xx @ A )
% 0.30/1.54 => ( ~ ( in @ Xx @ B )
% 0.30/1.54 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(iskpair_type,type,
% 0.30/1.54 iskpair: $i > $o ).
% 0.30/1.54
% 0.30/1.54 thf(secondinupair_type,type,
% 0.30/1.54 secondinupair: $o ).
% 0.30/1.54
% 0.30/1.54 thf(secondinupair,definition,
% 0.30/1.54 ( secondinupair
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setukpairIL_type,type,
% 0.30/1.54 setukpairIL: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setukpairIL,definition,
% 0.30/1.54 ( setukpairIL
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setukpairIR_type,type,
% 0.30/1.54 setukpairIR: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setukpairIR,definition,
% 0.30/1.54 ( setukpairIR
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(kpairiskpair_type,type,
% 0.30/1.54 kpairiskpair: $o ).
% 0.30/1.54
% 0.30/1.54 thf(kpairiskpair,definition,
% 0.30/1.54 ( kpairiskpair
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(kpair_type,type,
% 0.30/1.54 kpair: $i > $i > $i ).
% 0.30/1.54
% 0.30/1.54 thf(kpairp_type,type,
% 0.30/1.54 kpairp: $o ).
% 0.30/1.54
% 0.30/1.54 thf(kpairp,definition,
% 0.30/1.54 ( kpairp
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(cartprod_type,type,
% 0.30/1.54 cartprod: $i > $i > $i ).
% 0.30/1.54
% 0.30/1.54 thf(singletonsubset_type,type,
% 0.30/1.54 singletonsubset: $o ).
% 0.30/1.54
% 0.30/1.54 thf(singletonsubset,definition,
% 0.30/1.54 ( singletonsubset
% 0.30/1.54 = ( ! [A: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(singletoninpowerset_type,type,
% 0.30/1.54 singletoninpowerset: $o ).
% 0.30/1.54
% 0.30/1.54 thf(singletoninpowerset,definition,
% 0.30/1.54 ( singletoninpowerset
% 0.30/1.54 = ( ! [A: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(singletoninpowunion_type,type,
% 0.30/1.54 singletoninpowunion: $o ).
% 0.30/1.54
% 0.30/1.54 thf(singletoninpowunion,definition,
% 0.30/1.54 ( singletoninpowunion
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(upairset2E_type,type,
% 0.30/1.54 upairset2E: $o ).
% 0.30/1.54
% 0.30/1.54 thf(upairset2E,definition,
% 0.30/1.54 ( upairset2E
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/1.54 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.30/1.54 => ( ( Xz = Xx )
% 0.30/1.54 | ( Xz = Xy ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(upairsubunion_type,type,
% 0.30/1.54 upairsubunion: $o ).
% 0.30/1.54
% 0.30/1.54 thf(upairsubunion,definition,
% 0.30/1.54 ( upairsubunion
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ B )
% 0.30/1.54 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(upairinpowunion_type,type,
% 0.30/1.54 upairinpowunion: $o ).
% 0.30/1.54
% 0.30/1.54 thf(upairinpowunion,definition,
% 0.30/1.54 ( upairinpowunion
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ B )
% 0.30/1.54 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(ubforcartprodlem1_type,type,
% 0.30/1.54 ubforcartprodlem1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(ubforcartprodlem1,definition,
% 0.30/1.54 ( ubforcartprodlem1
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ B )
% 0.30/1.54 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(ubforcartprodlem2_type,type,
% 0.30/1.54 ubforcartprodlem2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(ubforcartprodlem2,definition,
% 0.30/1.54 ( ubforcartprodlem2
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ B )
% 0.30/1.54 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(ubforcartprodlem3_type,type,
% 0.30/1.54 ubforcartprodlem3: $o ).
% 0.30/1.54
% 0.30/1.54 thf(ubforcartprodlem3,definition,
% 0.30/1.54 ( ubforcartprodlem3
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ B )
% 0.30/1.54 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(cartprodpairin_type,type,
% 0.30/1.54 cartprodpairin: $o ).
% 0.30/1.54
% 0.30/1.54 thf(cartprodpairin,definition,
% 0.30/1.54 ( cartprodpairin
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ B )
% 0.30/1.54 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(cartprodmempair1_type,type,
% 0.30/1.54 cartprodmempair1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(cartprodmempair1,definition,
% 0.30/1.54 ( cartprodmempair1
% 0.30/1.54 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/1.54 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/1.54 => ? [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 & ? [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ B )
% 0.30/1.54 & ( Xu
% 0.30/1.54 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(cartprodmempair_type,type,
% 0.30/1.54 cartprodmempair: $o ).
% 0.30/1.54
% 0.30/1.54 thf(cartprodmempair,definition,
% 0.30/1.54 ( cartprodmempair
% 0.30/1.54 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/1.54 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/1.54 => ( iskpair @ Xu ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setunionE2_type,type,
% 0.30/1.54 setunionE2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setunionE2,definition,
% 0.30/1.54 ( setunionE2
% 0.30/1.54 = ( ! [A: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/1.54 => ? [X: $i] :
% 0.30/1.54 ( ( in @ X @ A )
% 0.30/1.54 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setunionsingleton1_type,type,
% 0.30/1.54 setunionsingleton1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setunionsingleton1,definition,
% 0.30/1.54 ( setunionsingleton1
% 0.30/1.54 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setunionsingleton2_type,type,
% 0.30/1.54 setunionsingleton2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setunionsingleton2,definition,
% 0.30/1.54 ( setunionsingleton2
% 0.30/1.54 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setunionsingleton_type,type,
% 0.30/1.54 setunionsingleton: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setunionsingleton,definition,
% 0.30/1.54 ( setunionsingleton
% 0.30/1.54 = ( ! [Xx: $i] :
% 0.30/1.54 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.30/1.54 = Xx ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(singleton_type,type,
% 0.30/1.54 singleton: $i > $o ).
% 0.30/1.54
% 0.30/1.54 thf(singletonprop_type,type,
% 0.30/1.54 singletonprop: $o ).
% 0.30/1.54
% 0.30/1.54 thf(singletonprop,definition,
% 0.30/1.54 ( singletonprop
% 0.30/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.54 ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ A )
% 0.30/1.54 => ( ( Xphi @ Xx )
% 0.30/1.54 => ( ( Xphi @ Xy )
% 0.30/1.54 => ( Xx = Xy ) ) ) ) )
% 0.30/1.54 => ( ? [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 & ( Xphi @ Xx ) )
% 0.30/1.54 => ( singleton
% 0.30/1.54 @ ( dsetconstr @ A
% 0.30/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(ex1_type,type,
% 0.30/1.54 ex1: $i > ( $i > $o ) > $o ).
% 0.30/1.54
% 0.30/1.54 thf(ex1,definition,
% 0.30/1.54 ( ex1
% 0.30/1.54 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.30/1.54 ( singleton
% 0.30/1.54 @ ( dsetconstr @ A
% 0.30/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(ex1E1_type,type,
% 0.30/1.54 ex1E1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(ex1E1,definition,
% 0.30/1.54 ( ex1E1
% 0.30/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.54 ( ( ex1 @ A
% 0.30/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.54 => ? [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(ex1I_type,type,
% 0.30/1.54 ex1I: $o ).
% 0.30/1.54
% 0.30/1.54 thf(ex1I,definition,
% 0.30/1.54 ( ex1I
% 0.30/1.54 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( ( Xphi @ Xx )
% 0.30/1.54 => ( ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ A )
% 0.30/1.54 => ( ( Xphi @ Xy )
% 0.30/1.54 => ( Xy = Xx ) ) )
% 0.30/1.54 => ( ex1 @ A
% 0.30/1.54 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(ex1I2_type,type,
% 0.30/1.54 ex1I2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(ex1I2,definition,
% 0.30/1.54 ( ex1I2
% 0.30/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.54 ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ A )
% 0.30/1.54 => ( ( Xphi @ Xx )
% 0.30/1.54 => ( ( Xphi @ Xy )
% 0.30/1.54 => ( Xx = Xy ) ) ) ) )
% 0.30/1.54 => ( ? [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 & ( Xphi @ Xx ) )
% 0.30/1.54 => ( ex1 @ A
% 0.30/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(singletonsuniq_type,type,
% 0.30/1.54 singletonsuniq: $o ).
% 0.30/1.54
% 0.30/1.54 thf(singletonsuniq,definition,
% 0.30/1.54 ( singletonsuniq
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i] :
% 0.30/1.54 ( ( ( setadjoin @ Xx @ emptyset )
% 0.30/1.54 = ( setadjoin @ Xy @ emptyset ) )
% 0.30/1.54 => ( Xx = Xy ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(atmost1p_type,type,
% 0.30/1.54 atmost1p: $i > $o ).
% 0.30/1.54
% 0.30/1.54 thf(atleast2p_type,type,
% 0.30/1.54 atleast2p: $i > $o ).
% 0.30/1.54
% 0.30/1.54 thf(atmost2p_type,type,
% 0.30/1.54 atmost2p: $i > $o ).
% 0.30/1.54
% 0.30/1.54 thf(upairsetp_type,type,
% 0.30/1.54 upairsetp: $i > $o ).
% 0.30/1.54
% 0.30/1.54 thf(setukpairinjL1_type,type,
% 0.30/1.54 setukpairinjL1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setukpairinjL1,definition,
% 0.30/1.54 ( setukpairinjL1
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/1.54 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.30/1.54 => ( Xx = Xz ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(kfstsingleton_type,type,
% 0.30/1.54 kfstsingleton: $o ).
% 0.30/1.54
% 0.30/1.54 thf(kfstsingleton,definition,
% 0.30/1.54 ( kfstsingleton
% 0.30/1.54 = ( ! [Xu: $i] :
% 0.30/1.54 ( ( iskpair @ Xu )
% 0.30/1.54 => ( singleton
% 0.30/1.54 @ ( dsetconstr @ ( setunion @ Xu )
% 0.30/1.54 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(theprop_type,type,
% 0.30/1.54 theprop: $o ).
% 0.30/1.54
% 0.30/1.54 thf(theprop,definition,
% 0.30/1.54 ( theprop
% 0.30/1.54 = ( ! [X: $i] :
% 0.30/1.54 ( ( singleton @ X )
% 0.30/1.54 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(kfst_type,type,
% 0.30/1.54 kfst: $i > $i ).
% 0.30/1.54
% 0.30/1.54 thf(kfstpairEq_type,type,
% 0.30/1.54 kfstpairEq: $o ).
% 0.30/1.54
% 0.30/1.54 thf(kfstpairEq,definition,
% 0.30/1.54 ( kfstpairEq
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i] :
% 0.30/1.54 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.30/1.54 = Xx ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(cartprodfstin_type,type,
% 0.30/1.54 cartprodfstin: $o ).
% 0.30/1.54
% 0.30/1.54 thf(cartprodfstin,definition,
% 0.30/1.54 ( cartprodfstin
% 0.30/1.54 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/1.54 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/1.54 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setukpairinjL2_type,type,
% 0.30/1.54 setukpairinjL2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setukpairinjL2,definition,
% 0.30/1.54 ( setukpairinjL2
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/1.54 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/1.54 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/1.54 => ( Xx = Xz ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setukpairinjL_type,type,
% 0.30/1.54 setukpairinjL: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setukpairinjL,definition,
% 0.30/1.54 ( setukpairinjL
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/1.54 ( ( ( kpair @ Xx @ Xy )
% 0.30/1.54 = ( kpair @ Xz @ Xu ) )
% 0.30/1.54 => ( Xx = Xz ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setukpairinjR11_type,type,
% 0.30/1.54 setukpairinjR11: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setukpairinjR11,definition,
% 0.30/1.54 ( setukpairinjR11
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i] :
% 0.30/1.54 ( ( Xx = Xy )
% 0.30/1.54 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/1.54 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setukpairinjR12_type,type,
% 0.30/1.54 setukpairinjR12: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setukpairinjR12,definition,
% 0.30/1.54 ( setukpairinjR12
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i] :
% 0.30/1.54 ( ( Xx = Xy )
% 0.30/1.54 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/1.54 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setukpairinjR1_type,type,
% 0.30/1.54 setukpairinjR1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setukpairinjR1,definition,
% 0.30/1.54 ( setukpairinjR1
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/1.54 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/1.54 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/1.54 => ( ( Xz = Xu )
% 0.30/1.54 => ( Xy = Xu ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(upairequniteq_type,type,
% 0.30/1.54 upairequniteq: $o ).
% 0.30/1.54
% 0.30/1.54 thf(upairequniteq,definition,
% 0.30/1.54 ( upairequniteq
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/1.54 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/1.54 = ( setadjoin @ Xz @ emptyset ) )
% 0.30/1.54 => ( Xx = Xy ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setukpairinjR2_type,type,
% 0.30/1.54 setukpairinjR2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setukpairinjR2,definition,
% 0.30/1.54 ( setukpairinjR2
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/1.54 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/1.54 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/1.54 => ( Xy = Xu ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setukpairinjR_type,type,
% 0.30/1.54 setukpairinjR: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setukpairinjR,definition,
% 0.30/1.54 ( setukpairinjR
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/1.54 ( ( ( kpair @ Xx @ Xy )
% 0.30/1.54 = ( kpair @ Xz @ Xu ) )
% 0.30/1.54 => ( Xy = Xu ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(ksndsingleton_type,type,
% 0.30/1.54 ksndsingleton: $o ).
% 0.30/1.54
% 0.30/1.54 thf(ksndsingleton,definition,
% 0.30/1.54 ( ksndsingleton
% 0.30/1.54 = ( ! [Xu: $i] :
% 0.30/1.54 ( ( iskpair @ Xu )
% 0.30/1.54 => ( singleton
% 0.30/1.54 @ ( dsetconstr @ ( setunion @ Xu )
% 0.30/1.54 @ ^ [Xx: $i] :
% 0.30/1.54 ( Xu
% 0.30/1.54 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(ksnd_type,type,
% 0.30/1.54 ksnd: $i > $i ).
% 0.30/1.54
% 0.30/1.54 thf(ksndpairEq_type,type,
% 0.30/1.54 ksndpairEq: $o ).
% 0.30/1.54
% 0.30/1.54 thf(ksndpairEq,definition,
% 0.30/1.54 ( ksndpairEq
% 0.30/1.54 = ( ! [Xx: $i,Xy: $i] :
% 0.30/1.54 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.30/1.54 = Xy ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(kpairsurjEq_type,type,
% 0.30/1.54 kpairsurjEq: $o ).
% 0.30/1.54
% 0.30/1.54 thf(kpairsurjEq,definition,
% 0.30/1.54 ( kpairsurjEq
% 0.30/1.54 = ( ! [Xu: $i] :
% 0.30/1.54 ( ( iskpair @ Xu )
% 0.30/1.54 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.30/1.54 = Xu ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(cartprodsndin_type,type,
% 0.30/1.54 cartprodsndin: $o ).
% 0.30/1.54
% 0.30/1.54 thf(cartprodsndin,definition,
% 0.30/1.54 ( cartprodsndin
% 0.30/1.54 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/1.54 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/1.54 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(cartprodpairmemEL_type,type,
% 0.30/1.54 cartprodpairmemEL: $o ).
% 0.30/1.54
% 0.30/1.54 thf(cartprodpairmemEL,definition,
% 0.30/1.54 ( cartprodpairmemEL
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.30/1.54 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.30/1.54 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(cartprodpairmemER_type,type,
% 0.30/1.54 cartprodpairmemER: $o ).
% 0.30/1.54
% 0.30/1.54 thf(cartprodpairmemER,definition,
% 0.30/1.54 ( cartprodpairmemER
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.30/1.54 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.30/1.54 => ( in @ Xy @ B ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(cartprodmempaircEq_type,type,
% 0.30/1.54 cartprodmempaircEq: $o ).
% 0.30/1.54
% 0.30/1.54 thf(cartprodmempaircEq,definition,
% 0.30/1.54 ( cartprodmempaircEq
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ B )
% 0.30/1.54 => ( ( kpair @ Xx @ Xy )
% 0.30/1.54 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(cartprodfstpairEq_type,type,
% 0.30/1.54 cartprodfstpairEq: $o ).
% 0.30/1.54
% 0.30/1.54 thf(cartprodfstpairEq,definition,
% 0.30/1.54 ( cartprodfstpairEq
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ B )
% 0.30/1.54 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.30/1.54 = Xx ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(cartprodsndpairEq_type,type,
% 0.30/1.54 cartprodsndpairEq: $o ).
% 0.30/1.54
% 0.30/1.54 thf(cartprodsndpairEq,definition,
% 0.30/1.54 ( cartprodsndpairEq
% 0.30/1.54 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ B )
% 0.30/1.54 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.30/1.54 = Xy ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(cartprodpairsurjEq_type,type,
% 0.30/1.54 cartprodpairsurjEq: $o ).
% 0.30/1.54
% 0.30/1.54 thf(cartprodpairsurjEq,definition,
% 0.30/1.54 ( cartprodpairsurjEq
% 0.30/1.54 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/1.54 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/1.54 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.30/1.54 = Xu ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(breln_type,type,
% 0.30/1.54 breln: $i > $i > $i > $o ).
% 0.30/1.54
% 0.30/1.54 thf(breln,definition,
% 0.30/1.54 ( breln
% 0.30/1.54 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(dpsetconstr_type,type,
% 0.30/1.54 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.30/1.54
% 0.30/1.54 thf(dpsetconstrI_type,type,
% 0.30/1.54 dpsetconstrI: $o ).
% 0.30/1.54
% 0.30/1.54 thf(dpsetconstrI,definition,
% 0.30/1.54 ( dpsetconstrI
% 0.30/1.54 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ B )
% 0.30/1.54 => ( ( Xphi @ Xx @ Xy )
% 0.30/1.54 => ( in @ ( kpair @ Xx @ Xy )
% 0.30/1.54 @ ( dpsetconstr @ A @ B
% 0.30/1.54 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(dpsetconstrSub_type,type,
% 0.30/1.54 dpsetconstrSub: $o ).
% 0.30/1.54
% 0.30/1.54 thf(dpsetconstrSub,definition,
% 0.30/1.54 ( dpsetconstrSub
% 0.30/1.54 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.30/1.54 ( subset
% 0.30/1.54 @ ( dpsetconstr @ A @ B
% 0.30/1.54 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.30/1.54 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setOfPairsIsBReln_type,type,
% 0.30/1.54 setOfPairsIsBReln: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setOfPairsIsBReln,definition,
% 0.30/1.54 ( setOfPairsIsBReln
% 0.30/1.54 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.30/1.54 ( breln @ A @ B
% 0.30/1.54 @ ( dpsetconstr @ A @ B
% 0.30/1.54 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(dpsetconstrERa_type,type,
% 0.30/1.54 dpsetconstrERa: $o ).
% 0.30/1.54
% 0.30/1.54 thf(dpsetconstrERa,definition,
% 0.30/1.54 ( dpsetconstrERa
% 0.30/1.54 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ B )
% 0.30/1.54 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/1.54 @ ( dpsetconstr @ A @ B
% 0.30/1.54 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/1.54 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(dpsetconstrEL1_type,type,
% 0.30/1.54 dpsetconstrEL1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(dpsetconstrEL1,definition,
% 0.30/1.54 ( dpsetconstrEL1
% 0.30/1.54 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.30/1.54 ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/1.54 @ ( dpsetconstr @ A @ B
% 0.30/1.54 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/1.54 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(dpsetconstrEL2_type,type,
% 0.30/1.54 dpsetconstrEL2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(dpsetconstrEL2,definition,
% 0.30/1.54 ( dpsetconstrEL2
% 0.30/1.54 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.30/1.54 ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/1.54 @ ( dpsetconstr @ A @ B
% 0.30/1.54 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/1.54 => ( in @ Xy @ B ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(dpsetconstrER_type,type,
% 0.30/1.54 dpsetconstrER: $o ).
% 0.30/1.54
% 0.30/1.54 thf(dpsetconstrER,definition,
% 0.30/1.54 ( dpsetconstrER
% 0.30/1.54 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.30/1.54 ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/1.54 @ ( dpsetconstr @ A @ B
% 0.30/1.54 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/1.54 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(func_type,type,
% 0.30/1.54 func: $i > $i > $i > $o ).
% 0.30/1.54
% 0.30/1.54 thf(funcSet_type,type,
% 0.30/1.54 funcSet: $i > $i > $i ).
% 0.30/1.54
% 0.30/1.54 thf(funcImageSingleton_type,type,
% 0.30/1.54 funcImageSingleton: $o ).
% 0.30/1.54
% 0.30/1.54 thf(funcImageSingleton,definition,
% 0.30/1.54 ( funcImageSingleton
% 0.30/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/1.54 ( ( func @ A @ B @ Xf )
% 0.30/1.54 => ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( singleton
% 0.30/1.54 @ ( dsetconstr @ B
% 0.30/1.54 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(apProp_type,type,
% 0.30/1.54 apProp: $o ).
% 0.30/1.54
% 0.30/1.54 thf(apProp,definition,
% 0.30/1.54 ( apProp
% 0.30/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/1.54 ( ( func @ A @ B @ Xf )
% 0.30/1.54 => ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( in
% 0.30/1.54 @ ( setunion
% 0.30/1.54 @ ( dsetconstr @ B
% 0.30/1.54 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.30/1.54 @ B ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(ap_type,type,
% 0.30/1.54 ap: $i > $i > $i > $i > $i ).
% 0.30/1.54
% 0.30/1.54 thf(app_type,type,
% 0.30/1.54 app: $o ).
% 0.30/1.54
% 0.30/1.54 thf(app,definition,
% 0.30/1.54 ( app
% 0.30/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/1.54 ( ( func @ A @ B @ Xf )
% 0.30/1.54 => ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(infuncsetfunc_type,type,
% 0.30/1.54 infuncsetfunc: $o ).
% 0.30/1.54
% 0.30/1.54 thf(infuncsetfunc,definition,
% 0.30/1.54 ( infuncsetfunc
% 0.30/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/1.54 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/1.54 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(ap2p_type,type,
% 0.30/1.54 ap2p: $o ).
% 0.30/1.54
% 0.30/1.54 thf(ap2p,definition,
% 0.30/1.54 ( ap2p
% 0.30/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/1.54 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/1.54 => ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(funcinfuncset_type,type,
% 0.30/1.54 funcinfuncset: $o ).
% 0.30/1.54
% 0.30/1.54 thf(funcinfuncset,definition,
% 0.30/1.54 ( funcinfuncset
% 0.30/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/1.54 ( ( func @ A @ B @ Xf )
% 0.30/1.54 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(lamProp_type,type,
% 0.30/1.54 lamProp: $o ).
% 0.30/1.54
% 0.30/1.54 thf(lamProp,definition,
% 0.30/1.54 ( lamProp
% 0.30/1.54 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/1.54 ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/1.54 => ( func @ A @ B
% 0.30/1.54 @ ( dpsetconstr @ A @ B
% 0.30/1.54 @ ^ [Xx: $i,Xy: $i] :
% 0.30/1.54 ( ( Xf @ Xx )
% 0.30/1.54 = Xy ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(lam_type,type,
% 0.30/1.54 lam: $i > $i > ( $i > $i ) > $i ).
% 0.30/1.54
% 0.30/1.54 thf(lamp_type,type,
% 0.30/1.54 lamp: $o ).
% 0.30/1.54
% 0.30/1.54 thf(lamp,definition,
% 0.30/1.54 ( lamp
% 0.30/1.54 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/1.54 ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/1.54 => ( func @ A @ B
% 0.30/1.54 @ ( lam @ A @ B
% 0.30/1.54 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(lam2p_type,type,
% 0.30/1.54 lam2p: $o ).
% 0.30/1.54
% 0.30/1.54 thf(lam2p,definition,
% 0.30/1.54 ( lam2p
% 0.30/1.54 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/1.54 ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/1.54 => ( in
% 0.30/1.54 @ ( lam @ A @ B
% 0.30/1.54 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.30/1.54 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(brelnall1_type,type,
% 0.30/1.54 brelnall1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(brelnall1,definition,
% 0.30/1.54 ( brelnall1
% 0.30/1.54 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/1.54 ( ( breln @ A @ B @ R )
% 0.30/1.54 => ! [Xphi: $i > $o] :
% 0.30/1.54 ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ B )
% 0.30/1.54 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/1.54 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.30/1.54 => ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ R )
% 0.30/1.54 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(brelnall2_type,type,
% 0.30/1.54 brelnall2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(brelnall2,definition,
% 0.30/1.54 ( brelnall2
% 0.30/1.54 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/1.54 ( ( breln @ A @ B @ R )
% 0.30/1.54 => ! [Xphi: $i > $o] :
% 0.30/1.54 ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ B )
% 0.30/1.54 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/1.54 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.30/1.54 => ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ R )
% 0.30/1.54 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(ex1E2_type,type,
% 0.30/1.54 ex1E2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(ex1E2,definition,
% 0.30/1.54 ( ex1E2
% 0.30/1.54 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/1.54 ( ( ex1 @ A
% 0.30/1.54 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/1.54 => ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ A )
% 0.30/1.54 => ( ( Xphi @ Xx )
% 0.30/1.54 => ( ( Xphi @ Xy )
% 0.30/1.54 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(funcGraphProp1_type,type,
% 0.30/1.54 funcGraphProp1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(funcGraphProp1,definition,
% 0.30/1.54 ( funcGraphProp1
% 0.30/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/1.54 ( ( func @ A @ B @ Xf )
% 0.30/1.54 => ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(funcGraphProp3_type,type,
% 0.30/1.54 funcGraphProp3: $o ).
% 0.30/1.54
% 0.30/1.54 thf(funcGraphProp3,definition,
% 0.30/1.54 ( funcGraphProp3
% 0.30/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/1.54 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/1.54 => ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(funcGraphProp2_type,type,
% 0.30/1.54 funcGraphProp2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(funcGraphProp2,definition,
% 0.30/1.54 ( funcGraphProp2
% 0.30/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/1.54 ( ( func @ A @ B @ Xf )
% 0.30/1.54 => ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ B )
% 0.30/1.54 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.30/1.54 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/1.54 = Xy ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(funcextLem_type,type,
% 0.30/1.54 funcextLem: $o ).
% 0.30/1.54
% 0.30/1.54 thf(funcextLem,definition,
% 0.30/1.54 ( funcextLem
% 0.30/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/1.54 ( ( func @ A @ B @ Xf )
% 0.30/1.54 => ! [Xg: $i] :
% 0.30/1.54 ( ( func @ A @ B @ Xg )
% 0.30/1.54 => ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/1.54 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.30/1.54 => ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ B )
% 0.30/1.54 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.30/1.54 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(funcGraphProp4_type,type,
% 0.30/1.54 funcGraphProp4: $o ).
% 0.30/1.54
% 0.30/1.54 thf(funcGraphProp4,definition,
% 0.30/1.54 ( funcGraphProp4
% 0.30/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/1.54 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/1.54 => ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ B )
% 0.30/1.54 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.30/1.54 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/1.54 = Xy ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(subbreln_type,type,
% 0.30/1.54 subbreln: $o ).
% 0.30/1.54
% 0.30/1.54 thf(subbreln,definition,
% 0.30/1.54 ( subbreln
% 0.30/1.54 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/1.54 ( ( breln @ A @ B @ R )
% 0.30/1.54 => ! [S: $i] :
% 0.30/1.54 ( ( breln @ A @ B @ S )
% 0.30/1.54 => ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ B )
% 0.30/1.54 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/1.54 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.30/1.54 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(eqbreln_type,type,
% 0.30/1.54 eqbreln: $o ).
% 0.30/1.54
% 0.30/1.54 thf(eqbreln,definition,
% 0.30/1.54 ( eqbreln
% 0.30/1.54 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/1.54 ( ( breln @ A @ B @ R )
% 0.30/1.54 => ! [S: $i] :
% 0.30/1.54 ( ( breln @ A @ B @ S )
% 0.30/1.54 => ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ B )
% 0.30/1.54 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/1.54 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.30/1.54 => ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ B )
% 0.30/1.54 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.30/1.54 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.30/1.54 => ( R = S ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(funcext_type,type,
% 0.30/1.54 funcext: $o ).
% 0.30/1.54
% 0.30/1.54 thf(funcext,definition,
% 0.30/1.54 ( funcext
% 0.30/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/1.54 ( ( func @ A @ B @ Xf )
% 0.30/1.54 => ! [Xg: $i] :
% 0.30/1.54 ( ( func @ A @ B @ Xg )
% 0.30/1.54 => ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/1.54 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.30/1.54 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(funcext2_type,type,
% 0.30/1.54 funcext2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(funcext2,definition,
% 0.30/1.54 ( funcext2
% 0.30/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/1.54 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/1.54 => ! [Xg: $i] :
% 0.30/1.54 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.30/1.54 => ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/1.54 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.30/1.54 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(ap2apEq1_type,type,
% 0.30/1.54 ap2apEq1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(ap2apEq1,definition,
% 0.30/1.54 ( ap2apEq1
% 0.30/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/1.54 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/1.54 => ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/1.54 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(ap2apEq2_type,type,
% 0.30/1.54 ap2apEq2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(ap2apEq2,definition,
% 0.30/1.54 ( ap2apEq2
% 0.30/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/1.54 ( ( func @ A @ B @ Xf )
% 0.30/1.54 => ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/1.54 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(beta1_type,type,
% 0.30/1.54 beta1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(beta1,definition,
% 0.30/1.54 ( beta1
% 0.30/1.54 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/1.54 ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/1.54 => ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( ( ap @ A @ B
% 0.30/1.54 @ ( lam @ A @ B
% 0.30/1.54 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.30/1.54 @ Xx )
% 0.30/1.54 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(eta1_type,type,
% 0.30/1.54 eta1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(eta1,definition,
% 0.30/1.54 ( eta1
% 0.30/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/1.54 ( ( func @ A @ B @ Xf )
% 0.30/1.54 => ( ( lam @ A @ B
% 0.30/1.54 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.30/1.54 = Xf ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(lam2lamEq_type,type,
% 0.30/1.54 lam2lamEq: $o ).
% 0.30/1.54
% 0.30/1.54 thf(lam2lamEq,definition,
% 0.30/1.54 ( lam2lamEq
% 0.30/1.54 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/1.54 ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/1.54 => ( ( lam @ A @ B
% 0.30/1.54 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.30/1.54 = ( lam @ A @ B
% 0.30/1.54 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(beta2_type,type,
% 0.30/1.54 beta2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(beta2,definition,
% 0.30/1.54 ( beta2
% 0.30/1.54 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/1.54 ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/1.54 => ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( ( ap @ A @ B
% 0.30/1.54 @ ( lam @ A @ B
% 0.30/1.54 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.30/1.54 @ Xx )
% 0.30/1.54 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(eta2_type,type,
% 0.30/1.54 eta2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(eta2,definition,
% 0.30/1.54 ( eta2
% 0.30/1.54 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/1.54 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/1.54 => ( ( lam @ A @ B
% 0.30/1.54 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.30/1.54 = Xf ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(iffalseProp1_type,type,
% 0.30/1.54 iffalseProp1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(iffalseProp1,definition,
% 0.30/1.54 ( iffalseProp1
% 0.30/1.54 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ A )
% 0.30/1.54 => ( ~ Xphi
% 0.30/1.54 => ( in @ Xy
% 0.30/1.54 @ ( dsetconstr @ A
% 0.30/1.54 @ ^ [Xz: $i] :
% 0.30/1.54 ( ( Xphi
% 0.30/1.54 & ( Xz = Xx ) )
% 0.30/1.54 | ( ~ Xphi
% 0.30/1.54 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(iffalseProp2_type,type,
% 0.30/1.54 iffalseProp2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(iffalseProp2,definition,
% 0.30/1.54 ( iffalseProp2
% 0.30/1.54 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ A )
% 0.30/1.54 => ( ~ Xphi
% 0.30/1.54 => ( ( dsetconstr @ A
% 0.30/1.54 @ ^ [Xz: $i] :
% 0.30/1.54 ( ( Xphi
% 0.30/1.54 & ( Xz = Xx ) )
% 0.30/1.54 | ( ~ Xphi
% 0.30/1.54 & ( Xz = Xy ) ) ) )
% 0.30/1.54 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(iftrueProp1_type,type,
% 0.30/1.54 iftrueProp1: $o ).
% 0.30/1.54
% 0.30/1.54 thf(iftrueProp1,definition,
% 0.30/1.54 ( iftrueProp1
% 0.30/1.54 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ A )
% 0.30/1.54 => ( Xphi
% 0.30/1.54 => ( in @ Xx
% 0.30/1.54 @ ( dsetconstr @ A
% 0.30/1.54 @ ^ [Xz: $i] :
% 0.30/1.54 ( ( Xphi
% 0.30/1.54 & ( Xz = Xx ) )
% 0.30/1.54 | ( ~ Xphi
% 0.30/1.54 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(iftrueProp2_type,type,
% 0.30/1.54 iftrueProp2: $o ).
% 0.30/1.54
% 0.30/1.54 thf(iftrueProp2,definition,
% 0.30/1.54 ( iftrueProp2
% 0.30/1.54 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ A )
% 0.30/1.54 => ( Xphi
% 0.30/1.54 => ( ( dsetconstr @ A
% 0.30/1.54 @ ^ [Xz: $i] :
% 0.30/1.54 ( ( Xphi
% 0.30/1.54 & ( Xz = Xx ) )
% 0.30/1.54 | ( ~ Xphi
% 0.30/1.54 & ( Xz = Xy ) ) ) )
% 0.30/1.54 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(ifSingleton_type,type,
% 0.30/1.54 ifSingleton: $o ).
% 0.30/1.54
% 0.30/1.54 thf(ifSingleton,definition,
% 0.30/1.54 ( ifSingleton
% 0.30/1.54 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ A )
% 0.30/1.54 => ( singleton
% 0.30/1.54 @ ( dsetconstr @ A
% 0.30/1.54 @ ^ [Xz: $i] :
% 0.30/1.54 ( ( Xphi
% 0.30/1.54 & ( Xz = Xx ) )
% 0.30/1.54 | ( ~ Xphi
% 0.30/1.54 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(if_type,type,
% 0.30/1.54 if: $i > $o > $i > $i > $i ).
% 0.30/1.54
% 0.30/1.54 thf(ifp_type,type,
% 0.30/1.54 ifp: $o ).
% 0.30/1.54
% 0.30/1.54 thf(ifp,definition,
% 0.30/1.54 ( ifp
% 0.30/1.54 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ A )
% 0.30/1.54 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(theeq_type,type,
% 0.30/1.54 theeq: $o ).
% 0.30/1.54
% 0.30/1.54 thf(theeq,definition,
% 0.30/1.54 ( theeq
% 0.30/1.54 = ( ! [X: $i] :
% 0.30/1.54 ( ( singleton @ X )
% 0.30/1.54 => ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ X )
% 0.30/1.54 => ( ( setunion @ X )
% 0.30/1.54 = Xx ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(iftrue_type,type,
% 0.30/1.54 iftrue: $o ).
% 0.30/1.54
% 0.30/1.54 thf(iftrue,definition,
% 0.30/1.54 ( iftrue
% 0.30/1.54 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ A )
% 0.30/1.54 => ( Xphi
% 0.30/1.54 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.30/1.54 = Xx ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(iffalse_type,type,
% 0.30/1.54 iffalse: $o ).
% 0.30/1.54
% 0.30/1.54 thf(iffalse,definition,
% 0.30/1.54 ( iffalse
% 0.30/1.54 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ A )
% 0.30/1.54 => ( ~ Xphi
% 0.30/1.54 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.30/1.54 = Xy ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(iftrueorfalse_type,type,
% 0.30/1.54 iftrueorfalse: $o ).
% 0.30/1.54
% 0.30/1.54 thf(iftrueorfalse,definition,
% 0.30/1.54 ( iftrueorfalse
% 0.30/1.54 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ! [Xy: $i] :
% 0.30/1.54 ( ( in @ Xy @ A )
% 0.30/1.54 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(binintersectT_lem_type,type,
% 0.30/1.54 binintersectT_lem: $o ).
% 0.30/1.54
% 0.30/1.54 thf(binintersectT_lem,definition,
% 0.30/1.54 ( binintersectT_lem
% 0.30/1.54 = ( ! [A: $i,X: $i] :
% 0.30/1.54 ( ( in @ X @ ( powerset @ A ) )
% 0.30/1.54 => ! [Y: $i] :
% 0.30/1.54 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/1.54 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(binunionT_lem_type,type,
% 0.30/1.54 binunionT_lem: $o ).
% 0.30/1.54
% 0.30/1.54 thf(binunionT_lem,definition,
% 0.30/1.54 ( binunionT_lem
% 0.30/1.54 = ( ! [A: $i,X: $i] :
% 0.30/1.54 ( ( in @ X @ ( powerset @ A ) )
% 0.30/1.54 => ! [Y: $i] :
% 0.30/1.54 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/1.54 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(powersetT_lem_type,type,
% 0.30/1.54 powersetT_lem: $o ).
% 0.30/1.54
% 0.30/1.54 thf(powersetT_lem,definition,
% 0.30/1.54 ( powersetT_lem
% 0.30/1.54 = ( ! [A: $i,X: $i] :
% 0.30/1.54 ( ( in @ X @ ( powerset @ A ) )
% 0.30/1.54 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setminusT_lem_type,type,
% 0.30/1.54 setminusT_lem: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setminusT_lem,definition,
% 0.30/1.54 ( setminusT_lem
% 0.30/1.54 = ( ! [A: $i,X: $i] :
% 0.30/1.54 ( ( in @ X @ ( powerset @ A ) )
% 0.30/1.54 => ! [Y: $i] :
% 0.30/1.54 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/1.54 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(complementT_lem_type,type,
% 0.30/1.54 complementT_lem: $o ).
% 0.30/1.54
% 0.30/1.54 thf(complementT_lem,definition,
% 0.30/1.54 ( complementT_lem
% 0.30/1.54 = ( ! [A: $i,X: $i] :
% 0.30/1.54 ( ( in @ X @ ( powerset @ A ) )
% 0.30/1.54 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(setextT_type,type,
% 0.30/1.54 setextT: $o ).
% 0.30/1.54
% 0.30/1.54 thf(setextT,definition,
% 0.30/1.54 ( setextT
% 0.30/1.54 = ( ! [A: $i,X: $i] :
% 0.30/1.54 ( ( in @ X @ ( powerset @ A ) )
% 0.30/1.54 => ! [Y: $i] :
% 0.30/1.54 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/1.54 => ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( ( in @ Xx @ X )
% 0.30/1.54 => ( in @ Xx @ Y ) ) )
% 0.30/1.54 => ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( ( in @ Xx @ Y )
% 0.30/1.54 => ( in @ Xx @ X ) ) )
% 0.30/1.54 => ( X = Y ) ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(subsetTI_type,type,
% 0.30/1.54 subsetTI: $o ).
% 0.30/1.54
% 0.30/1.54 thf(subsetTI,definition,
% 0.30/1.54 ( subsetTI
% 0.30/1.54 = ( ! [A: $i,X: $i] :
% 0.30/1.54 ( ( in @ X @ ( powerset @ A ) )
% 0.30/1.54 => ! [Y: $i] :
% 0.30/1.54 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/1.54 => ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( ( in @ Xx @ X )
% 0.30/1.54 => ( in @ Xx @ Y ) ) )
% 0.30/1.54 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.30/1.54
% 0.30/1.54 thf(powersetTI1,conjecture,
% 0.30/1.54 ( setextAx
% 0.30/1.54 => ( emptysetAx
% 0.30/1.54 => ( setadjoinAx
% 0.30/1.54 => ( powersetAx
% 0.30/1.54 => ( setunionAx
% 0.30/1.54 => ( omega0Ax
% 0.30/1.54 => ( omegaSAx
% 0.30/1.54 => ( omegaIndAx
% 0.30/1.54 => ( replAx
% 0.30/1.54 => ( foundationAx
% 0.30/1.54 => ( wellorderingAx
% 0.30/1.54 => ( descrp
% 0.30/1.54 => ( dsetconstrI
% 0.30/1.54 => ( dsetconstrEL
% 0.30/1.54 => ( dsetconstrER
% 0.30/1.54 => ( exuE1
% 0.30/1.54 => ( prop2setE
% 0.30/1.54 => ( emptysetE
% 0.30/1.54 => ( emptysetimpfalse
% 0.30/1.54 => ( notinemptyset
% 0.30/1.54 => ( exuE3e
% 0.30/1.54 => ( setext
% 0.30/1.54 => ( emptyI
% 0.30/1.54 => ( noeltsimpempty
% 0.30/1.54 => ( setbeta
% 0.30/1.54 => ( nonemptyE1
% 0.30/1.54 => ( nonemptyI
% 0.30/1.54 => ( nonemptyI1
% 0.30/1.54 => ( setadjoinIL
% 0.30/1.54 => ( emptyinunitempty
% 0.30/1.54 => ( setadjoinIR
% 0.30/1.54 => ( setadjoinE
% 0.30/1.54 => ( setadjoinOr
% 0.30/1.54 => ( setoftrueEq
% 0.30/1.54 => ( powersetI
% 0.30/1.54 => ( emptyinPowerset
% 0.30/1.54 => ( emptyInPowerset
% 0.30/1.54 => ( powersetE
% 0.30/1.54 => ( setunionI
% 0.30/1.54 => ( setunionE
% 0.30/1.54 => ( subPowSU
% 0.30/1.54 => ( exuE2
% 0.30/1.54 => ( nonemptyImpWitness
% 0.30/1.54 => ( uniqinunit
% 0.30/1.54 => ( notinsingleton
% 0.30/1.54 => ( eqinunit
% 0.30/1.54 => ( singletonsswitch
% 0.30/1.54 => ( upairsetE
% 0.30/1.54 => ( upairsetIL
% 0.30/1.54 => ( upairsetIR
% 0.30/1.54 => ( emptyE1
% 0.30/1.54 => ( vacuousDall
% 0.30/1.54 => ( quantDeMorgan1
% 0.30/1.54 => ( quantDeMorgan2
% 0.30/1.54 => ( quantDeMorgan3
% 0.30/1.54 => ( quantDeMorgan4
% 0.30/1.54 => ( prop2setI
% 0.30/1.54 => ( prop2set2propI
% 0.30/1.54 => ( notdexE
% 0.30/1.54 => ( notdallE
% 0.30/1.54 => ( exuI1
% 0.30/1.54 => ( exuI3
% 0.30/1.54 => ( exuI2
% 0.30/1.54 => ( inCongP
% 0.30/1.54 => ( in__Cong
% 0.30/1.54 => ( exuE3u
% 0.30/1.54 => ( exu__Cong
% 0.30/1.54 => ( emptyset__Cong
% 0.30/1.54 => ( setadjoin__Cong
% 0.30/1.54 => ( powerset__Cong
% 0.30/1.54 => ( setunion__Cong
% 0.30/1.54 => ( omega__Cong
% 0.30/1.54 => ( exuEu
% 0.30/1.54 => ( descr__Cong
% 0.30/1.54 => ( dsetconstr__Cong
% 0.30/1.54 => ( subsetI1
% 0.30/1.54 => ( eqimpsubset2
% 0.30/1.54 => ( eqimpsubset1
% 0.30/1.54 => ( subsetI2
% 0.30/1.54 => ( emptysetsubset
% 0.30/1.54 => ( subsetE
% 0.30/1.54 => ( subsetE2
% 0.30/1.54 => ( notsubsetI
% 0.30/1.54 => ( notequalI1
% 0.30/1.54 => ( notequalI2
% 0.30/1.54 => ( subsetRefl
% 0.30/1.54 => ( subsetTrans
% 0.30/1.54 => ( setadjoinSub
% 0.30/1.54 => ( setadjoinSub2
% 0.30/1.54 => ( subset2powerset
% 0.30/1.54 => ( setextsub
% 0.30/1.54 => ( subsetemptysetimpeq
% 0.30/1.54 => ( powersetI1
% 0.30/1.54 => ( powersetE1
% 0.30/1.54 => ( inPowerset
% 0.30/1.54 => ( powersetsubset
% 0.30/1.54 => ( sepInPowerset
% 0.30/1.54 => ( sepSubset
% 0.30/1.54 => ( binunionIL
% 0.30/1.54 => ( upairset2IR
% 0.30/1.54 => ( binunionIR
% 0.30/1.54 => ( binunionEcases
% 0.30/1.54 => ( binunionE
% 0.30/1.54 => ( binunionLsub
% 0.30/1.54 => ( binunionRsub
% 0.30/1.54 => ( binintersectI
% 0.30/1.54 => ( binintersectSubset5
% 0.30/1.54 => ( binintersectEL
% 0.30/1.54 => ( binintersectLsub
% 0.30/1.54 => ( binintersectSubset2
% 0.30/1.54 => ( binintersectSubset3
% 0.30/1.54 => ( binintersectER
% 0.30/1.54 => ( disjointsetsI1
% 0.30/1.54 => ( binintersectRsub
% 0.30/1.54 => ( binintersectSubset4
% 0.30/1.54 => ( binintersectSubset1
% 0.30/1.54 => ( bs114d
% 0.30/1.54 => ( setminusI
% 0.30/1.54 => ( setminusEL
% 0.30/1.54 => ( setminusER
% 0.30/1.54 => ( setminusSubset2
% 0.30/1.54 => ( setminusERneg
% 0.30/1.54 => ( setminusELneg
% 0.30/1.54 => ( setminusILneg
% 0.30/1.54 => ( setminusIRneg
% 0.30/1.54 => ( setminusLsub
% 0.30/1.54 => ( setminusSubset1
% 0.30/1.54 => ( symdiffE
% 0.30/1.54 => ( symdiffI1
% 0.30/1.54 => ( symdiffI2
% 0.30/1.54 => ( symdiffIneg1
% 0.30/1.54 => ( symdiffIneg2
% 0.30/1.54 => ( secondinupair
% 0.30/1.54 => ( setukpairIL
% 0.30/1.54 => ( setukpairIR
% 0.30/1.54 => ( kpairiskpair
% 0.30/1.54 => ( kpairp
% 0.30/1.54 => ( singletonsubset
% 0.30/1.54 => ( singletoninpowerset
% 0.30/1.54 => ( singletoninpowunion
% 0.30/1.54 => ( upairset2E
% 0.30/1.54 => ( upairsubunion
% 0.30/1.54 => ( upairinpowunion
% 0.30/1.54 => ( ubforcartprodlem1
% 0.30/1.54 => ( ubforcartprodlem2
% 0.30/1.54 => ( ubforcartprodlem3
% 0.30/1.54 => ( cartprodpairin
% 0.30/1.54 => ( cartprodmempair1
% 0.30/1.54 => ( cartprodmempair
% 0.30/1.54 => ( setunionE2
% 0.30/1.54 => ( setunionsingleton1
% 0.30/1.54 => ( setunionsingleton2
% 0.30/1.54 => ( setunionsingleton
% 0.30/1.54 => ( singletonprop
% 0.30/1.54 => ( ex1E1
% 0.30/1.54 => ( ex1I
% 0.30/1.54 => ( ex1I2
% 0.30/1.54 => ( singletonsuniq
% 0.30/1.54 => ( setukpairinjL1
% 0.30/1.54 => ( kfstsingleton
% 0.30/1.54 => ( theprop
% 0.30/1.54 => ( kfstpairEq
% 0.30/1.54 => ( cartprodfstin
% 0.30/1.54 => ( setukpairinjL2
% 0.30/1.54 => ( setukpairinjL
% 0.30/1.54 => ( setukpairinjR11
% 0.30/1.54 => ( setukpairinjR12
% 0.30/1.54 => ( setukpairinjR1
% 0.30/1.54 => ( upairequniteq
% 0.30/1.54 => ( setukpairinjR2
% 0.30/1.54 => ( setukpairinjR
% 0.30/1.54 => ( ksndsingleton
% 0.30/1.54 => ( ksndpairEq
% 0.30/1.54 => ( kpairsurjEq
% 0.30/1.54 => ( cartprodsndin
% 0.30/1.54 => ( cartprodpairmemEL
% 0.30/1.54 => ( cartprodpairmemER
% 0.30/1.54 => ( cartprodmempaircEq
% 0.30/1.54 => ( cartprodfstpairEq
% 0.30/1.54 => ( cartprodsndpairEq
% 0.30/1.54 => ( cartprodpairsurjEq
% 0.30/1.54 => ( dpsetconstrI
% 0.30/1.54 => ( dpsetconstrSub
% 0.30/1.54 => ( setOfPairsIsBReln
% 0.30/1.54 => ( dpsetconstrERa
% 0.30/1.54 => ( dpsetconstrEL1
% 0.30/1.54 => ( dpsetconstrEL2
% 0.30/1.54 => ( dpsetconstrER
% 0.30/1.54 => ( funcImageSingleton
% 0.30/1.54 => ( apProp
% 0.30/1.54 => ( app
% 0.30/1.54 => ( infuncsetfunc
% 0.30/1.54 => ( ap2p
% 0.30/1.54 => ( funcinfuncset
% 0.30/1.54 => ( lamProp
% 0.30/1.54 => ( lamp
% 0.30/1.54 => ( lam2p
% 0.30/1.54 => ( brelnall1
% 0.30/1.54 => ( brelnall2
% 0.30/1.54 => ( ex1E2
% 0.30/1.54 => ( funcGraphProp1
% 0.30/1.54 => ( funcGraphProp3
% 0.30/1.54 => ( funcGraphProp2
% 0.30/1.54 => ( funcextLem
% 0.30/1.54 => ( funcGraphProp4
% 0.30/1.54 => ( subbreln
% 0.30/1.54 => ( eqbreln
% 0.30/1.54 => ( funcext
% 0.30/1.54 => ( funcext2
% 0.30/1.54 => ( ap2apEq1
% 0.30/1.54 => ( ap2apEq2
% 0.30/1.54 => ( beta1
% 0.30/1.54 => ( eta1
% 0.30/1.54 => ( lam2lamEq
% 0.30/1.54 => ( beta2
% 0.30/1.54 => ( eta2
% 0.30/1.54 => ( iffalseProp1
% 0.30/1.54 => ( iffalseProp2
% 0.30/1.54 => ( iftrueProp1
% 0.30/1.54 => ( iftrueProp2
% 0.30/1.54 => ( ifSingleton
% 0.30/1.54 => ( ifp
% 0.30/1.54 => ( theeq
% 0.30/1.54 => ( iftrue
% 0.30/1.54 => ( iffalse
% 0.30/1.54 => ( iftrueorfalse
% 0.30/1.54 => ( binintersectT_lem
% 0.30/1.54 => ( binunionT_lem
% 0.30/1.54 => ( powersetT_lem
% 0.30/1.54 => ( setminusT_lem
% 0.30/1.54 => ( complementT_lem
% 0.30/1.54 => ( setextT
% 0.30/1.54 => ( subsetTI
% 0.30/1.54 => ! [A: $i,X: $i] :
% 0.30/1.54 ( ( in @ X @ ( powerset @ A ) )
% 0.30/1.54 => ! [Y: $i] :
% 0.30/1.54 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/1.54 => ( ! [Xx: $i] :
% 0.30/1.54 ( ( in @ Xx @ A )
% 0.30/1.54 => ( ( in @ Xx @ X )
% 0.30/1.54 => ( in @ Xx @ Y ) ) )
% 0.30/1.54 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/1.54 ------------------------
% 0.30/1.56 % (12386)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_12192 for (2999ds/183Mi)
% 0.30/1.56 % (12391)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_12192 for (2999ds/275Mi)
% 0.30/1.56 % (12387)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_12192 for (2999ds/4Mi)
% 0.30/1.57 % (12388)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_12192 for (2999ds/27Mi)
% 0.30/1.57 % (12392)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_12192 for (2999ds/18Mi)
% 0.30/1.57 % (12387)Instruction limit reached!
% 0.30/1.57 % (12387)------------------------------
% 0.30/1.57 % (12387)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.57 % (12387)Termination reason: Unknown
% 0.30/1.57 % (12387)Termination phase: shuffling
% 0.30/1.57
% 0.30/1.57 % (12387)Memory used [KB]: 1535
% 0.30/1.57 % (12387)Time elapsed: 0.003 s
% 0.30/1.57 % (12387)Instructions burned: 6 (million)
% 0.30/1.57 % (12387)------------------------------
% 0.30/1.57 % (12387)------------------------------
% 0.30/1.57 % (12389)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_12192 for (2999ds/2Mi)
% 0.30/1.57 % (12390)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_12192 for (2999ds/2Mi)
% 0.30/1.57 % (12389)Instruction limit reached!
% 0.30/1.57 % (12389)------------------------------
% 0.30/1.57 % (12389)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.57 % (12390)Instruction limit reached!
% 0.30/1.57 % (12390)------------------------------
% 0.30/1.57 % (12390)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.57 % (12390)Termination reason: Unknown
% 0.30/1.57 % (12390)Termination phase: shuffling
% 0.30/1.57
% 0.30/1.57 % (12390)Memory used [KB]: 1407
% 0.30/1.57 % (12390)Time elapsed: 0.003 s
% 0.30/1.57 % (12390)Instructions burned: 3 (million)
% 0.30/1.57 % (12390)------------------------------
% 0.30/1.57 % (12390)------------------------------
% 0.30/1.57 % (12389)Termination reason: Unknown
% 0.30/1.57 % (12389)Termination phase: shuffling
% 0.30/1.57
% 0.30/1.57 % (12389)Memory used [KB]: 1407
% 0.30/1.57 % (12392)Instruction limit reached!
% 0.30/1.57 % (12392)------------------------------
% 0.30/1.57 % (12392)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.57 % (12389)Time elapsed: 0.004 s
% 0.30/1.57 % (12389)Instructions burned: 3 (million)
% 0.30/1.57 % (12389)------------------------------
% 0.30/1.57 % (12389)------------------------------
% 0.30/1.57 % (12392)Termination reason: Unknown
% 0.30/1.57 % (12392)Termination phase: shuffling
% 0.30/1.57
% 0.30/1.57 % (12392)Memory used [KB]: 1791
% 0.30/1.57 % (12392)Time elapsed: 0.010 s
% 0.30/1.57 % (12392)Instructions burned: 18 (million)
% 0.30/1.57 % (12392)------------------------------
% 0.30/1.57 % (12392)------------------------------
% 0.30/1.58 % (12388)Instruction limit reached!
% 0.30/1.58 % (12388)------------------------------
% 0.30/1.58 % (12388)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.58 % (12388)Termination reason: Unknown
% 0.30/1.58 % (12388)Termination phase: shuffling
% 0.30/1.58
% 0.30/1.58 % (12388)Memory used [KB]: 1918
% 0.30/1.58 % (12388)Time elapsed: 0.012 s
% 0.30/1.58 % (12388)Instructions burned: 27 (million)
% 0.30/1.58 % (12388)------------------------------
% 0.30/1.58 % (12388)------------------------------
% 0.30/1.58 % (12393)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_12192 for (2999ds/3Mi)
% 0.30/1.58 % (12393)Instruction limit reached!
% 0.30/1.58 % (12393)------------------------------
% 0.30/1.58 % (12393)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.58 % (12393)Termination reason: Unknown
% 0.30/1.58 % (12393)Termination phase: shuffling
% 0.30/1.58
% 0.30/1.58 % (12393)Memory used [KB]: 1407
% 0.30/1.58 % (12393)Time elapsed: 0.002 s
% 0.30/1.58 % (12393)Instructions burned: 3 (million)
% 0.30/1.58 % (12393)------------------------------
% 0.30/1.58 % (12393)------------------------------
% 0.30/1.59 % (12397)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_12192 for (2999ds/1041Mi)
% 0.30/1.59 % (12395)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_12192 for (2999ds/15Mi)
% 0.30/1.59 % (12394)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_12192 for (2999ds/37Mi)
% 0.30/1.60 % (12398)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_12192 for (2999ds/7Mi)
% 3.08/1.60 % (12396)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_12192 for (2999ds/3Mi)
% 3.08/1.60 % (12395)Instruction limit reached!
% 3.08/1.60 % (12395)------------------------------
% 3.08/1.60 % (12395)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.60 % (12395)Termination reason: Unknown
% 3.08/1.60 % (12395)Termination phase: shuffling
% 3.08/1.60
% 3.08/1.60 % (12395)Memory used [KB]: 1663
% 3.08/1.60 % (12395)Time elapsed: 0.009 s
% 3.08/1.60 % (12395)Instructions burned: 16 (million)
% 3.08/1.60 % (12395)------------------------------
% 3.08/1.60 % (12395)------------------------------
% 3.08/1.60 % (12398)Instruction limit reached!
% 3.08/1.60 % (12398)------------------------------
% 3.08/1.60 % (12398)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.60 % (12398)Termination reason: Unknown
% 3.08/1.60 % (12398)Termination phase: shuffling
% 3.08/1.60
% 3.08/1.60 % (12398)Memory used [KB]: 1535
% 3.08/1.60 % (12396)Instruction limit reached!
% 3.08/1.60 % (12396)------------------------------
% 3.08/1.60 % (12396)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.60 % (12396)Termination reason: Unknown
% 3.08/1.60 % (12396)Termination phase: shuffling
% 3.08/1.60
% 3.08/1.60 % (12396)Memory used [KB]: 1407
% 3.08/1.60 % (12396)Time elapsed: 0.005 s
% 3.08/1.60 % (12396)Instructions burned: 3 (million)
% 3.08/1.60 % (12396)------------------------------
% 3.08/1.60 % (12396)------------------------------
% 3.08/1.60 % (12398)Time elapsed: 0.005 s
% 3.08/1.60 % (12398)Instructions burned: 7 (million)
% 3.08/1.60 % (12398)------------------------------
% 3.08/1.60 % (12398)------------------------------
% 3.08/1.61 % (12394)Instruction limit reached!
% 3.08/1.61 % (12394)------------------------------
% 3.08/1.61 % (12394)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.61 % (12394)Termination reason: Unknown
% 3.08/1.61 % (12394)Termination phase: shuffling
% 3.08/1.61
% 3.08/1.61 % (12394)Memory used [KB]: 2174
% 3.08/1.61 % (12394)Time elapsed: 0.021 s
% 3.08/1.61 % (12394)Instructions burned: 37 (million)
% 3.08/1.61 % (12394)------------------------------
% 3.08/1.61 % (12394)------------------------------
% 3.08/1.61 % (12386)Instruction limit reached!
% 3.08/1.61 % (12386)------------------------------
% 3.08/1.61 % (12386)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.61 % (12386)Termination reason: Unknown
% 3.08/1.61 % (12386)Termination phase: Saturation
% 3.08/1.61
% 3.08/1.61 % (12386)Memory used [KB]: 7547
% 3.08/1.61 % (12386)Time elapsed: 0.050 s
% 3.08/1.61 % (12386)Instructions burned: 183 (million)
% 3.08/1.61 % (12386)------------------------------
% 3.08/1.61 % (12386)------------------------------
% 3.08/1.61 % (12399)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_12192 for (2999ds/16Mi)
% 3.08/1.62 % (12401)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_12192 for (2999ds/3Mi)
% 3.08/1.62 % (12400)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_12192 for (2999ds/3Mi)
% 3.08/1.62 % (12400)Instruction limit reached!
% 3.08/1.62 % (12400)------------------------------
% 3.08/1.62 % (12400)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.62 % (12400)Termination reason: Unknown
% 3.08/1.62 % (12400)Termination phase: shuffling
% 3.08/1.62
% 3.08/1.62 % (12400)Memory used [KB]: 1407
% 3.08/1.62 % (12400)Time elapsed: 0.004 s
% 3.08/1.62 % (12400)Instructions burned: 3 (million)
% 3.08/1.62 % (12400)------------------------------
% 3.08/1.62 % (12400)------------------------------
% 3.08/1.62 % (12401)Instruction limit reached!
% 3.08/1.62 % (12401)------------------------------
% 3.08/1.62 % (12401)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.62 % (12401)Termination reason: Unknown
% 3.08/1.62 % (12401)Termination phase: shuffling
% 3.08/1.62
% 3.08/1.62 % (12401)Memory used [KB]: 1407
% 3.08/1.62 % (12401)Time elapsed: 0.004 s
% 3.08/1.62 % (12401)Instructions burned: 4 (million)
% 3.08/1.62 % (12401)------------------------------
% 3.08/1.62 % (12401)------------------------------
% 3.08/1.62 % (12399)Instruction limit reached!
% 3.08/1.62 % (12399)------------------------------
% 3.08/1.62 % (12399)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.62 % (12399)Termination reason: Unknown
% 3.08/1.62 % (12399)Termination phase: shuffling
% 3.08/1.62
% 3.08/1.62 % (12399)Memory used [KB]: 1663
% 3.08/1.62 % (12399)Time elapsed: 0.009 s
% 3.08/1.62 % (12399)Instructions burned: 16 (million)
% 3.08/1.62 % (12399)------------------------------
% 3.08/1.62 % (12399)------------------------------
% 3.08/1.62 % (12403)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_12192 for (2999ds/3Mi)
% 3.08/1.62 % (12403)Instruction limit reached!
% 3.08/1.62 % (12403)------------------------------
% 3.08/1.62 % (12403)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.62 % (12403)Termination reason: Unknown
% 3.08/1.62 % (12403)Termination phase: shuffling
% 3.08/1.62
% 3.08/1.62 % (12403)Memory used [KB]: 1535
% 3.08/1.62 % (12403)Time elapsed: 0.003 s
% 3.08/1.62 % (12403)Instructions burned: 6 (million)
% 3.08/1.62 % (12403)------------------------------
% 3.08/1.62 % (12403)------------------------------
% 3.08/1.63 % (12402)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_12192 for (2999ds/7Mi)
% 3.08/1.63 % (12405)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_12192 for (2999ds/18Mi)
% 3.08/1.63 % (12402)Instruction limit reached!
% 3.08/1.63 % (12402)------------------------------
% 3.08/1.63 % (12402)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.63 % (12402)Termination reason: Unknown
% 3.08/1.63 % (12402)Termination phase: shuffling
% 3.08/1.63
% 3.08/1.63 % (12402)Memory used [KB]: 1535
% 3.08/1.63 % (12402)Time elapsed: 0.006 s
% 3.08/1.63 % (12402)Instructions burned: 8 (million)
% 3.08/1.63 % (12402)------------------------------
% 3.08/1.63 % (12402)------------------------------
% 3.08/1.63 % (12407)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_12192 for (2999ds/6Mi)
% 3.08/1.64 % (12407)Instruction limit reached!
% 3.08/1.64 % (12407)------------------------------
% 3.08/1.64 % (12407)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.64 % (12407)Termination reason: Unknown
% 3.08/1.64 % (12407)Termination phase: shuffling
% 3.08/1.64
% 3.08/1.64 % (12407)Memory used [KB]: 1535
% 3.08/1.64 % (12407)Time elapsed: 0.003 s
% 3.08/1.64 % (12407)Instructions burned: 7 (million)
% 3.08/1.64 % (12407)------------------------------
% 3.08/1.64 % (12407)------------------------------
% 3.08/1.64 % (12406)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_12192 for (2999ds/710Mi)
% 3.08/1.64 % (12405)Instruction limit reached!
% 3.08/1.64 % (12405)------------------------------
% 3.08/1.64 % (12405)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.64 % (12405)Termination reason: Unknown
% 3.08/1.64 % (12405)Termination phase: shuffling
% 3.08/1.64
% 3.08/1.64 % (12405)Memory used [KB]: 1791
% 3.08/1.64 % (12405)Time elapsed: 0.009 s
% 3.08/1.64 % (12405)Instructions burned: 19 (million)
% 3.08/1.64 % (12405)------------------------------
% 3.08/1.64 % (12405)------------------------------
% 3.08/1.64 % (12404)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_12192 for (2999ds/4Mi)
% 3.08/1.64 % (12404)Instruction limit reached!
% 3.08/1.64 % (12404)------------------------------
% 3.08/1.64 % (12404)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.64 % (12404)Termination reason: Unknown
% 3.08/1.64 % (12404)Termination phase: shuffling
% 3.08/1.64
% 3.08/1.64 % (12404)Memory used [KB]: 1407
% 3.08/1.64 % (12404)Time elapsed: 0.006 s
% 3.08/1.64 % (12404)Instructions burned: 4 (million)
% 3.08/1.64 % (12404)------------------------------
% 3.08/1.64 % (12404)------------------------------
% 3.08/1.65 % (12391)Instruction limit reached!
% 3.08/1.65 % (12391)------------------------------
% 3.08/1.65 % (12391)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.65 % (12391)Termination reason: Unknown
% 3.08/1.65 % (12391)Termination phase: Saturation
% 3.08/1.65
% 3.08/1.65 % (12391)Memory used [KB]: 9210
% 3.08/1.65 % (12391)Time elapsed: 0.082 s
% 3.08/1.65 % (12391)Instructions burned: 275 (million)
% 3.08/1.65 % (12391)------------------------------
% 3.08/1.65 % (12391)------------------------------
% 3.08/1.65 % (12409)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_12192 for (2999ds/21Mi)
% 3.08/1.65 % (12410)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_12192 for (2999ds/5Mi)
% 3.08/1.65 % (12410)Instruction limit reached!
% 3.08/1.65 % (12410)------------------------------
% 3.08/1.65 % (12410)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.65 % (12410)Termination reason: Unknown
% 3.08/1.65 % (12410)Termination phase: shuffling
% 3.08/1.65
% 3.08/1.65 % (12410)Memory used [KB]: 1535
% 3.08/1.65 % (12410)Time elapsed: 0.003 s
% 3.08/1.65 % (12410)Instructions burned: 6 (million)
% 3.08/1.65 % (12410)------------------------------
% 3.08/1.65 % (12410)------------------------------
% 3.08/1.65 % (12408)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_12192 for (2999ds/902Mi)
% 3.08/1.65 % (12409)Instruction limit reached!
% 3.08/1.65 % (12409)------------------------------
% 3.08/1.65 % (12409)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.65 % (12409)Termination reason: Unknown
% 3.08/1.65 % (12409)Termination phase: shuffling
% 3.08/1.65
% 3.08/1.65 % (12409)Memory used [KB]: 1791
% 3.08/1.65 % (12409)Time elapsed: 0.007 s
% 3.08/1.65 % (12409)Instructions burned: 21 (million)
% 3.08/1.65 % (12409)------------------------------
% 3.08/1.65 % (12409)------------------------------
% 3.08/1.65 % (12411)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_12192 for (2999ds/6Mi)
% 3.08/1.65 % (12411)Instruction limit reached!
% 3.08/1.65 % (12411)------------------------------
% 3.08/1.65 % (12411)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.65 % (12411)Termination reason: Unknown
% 3.08/1.65 % (12411)Termination phase: shuffling
% 3.08/1.65
% 3.08/1.65 % (12411)Memory used [KB]: 1535
% 3.08/1.65 % (12411)Time elapsed: 0.003 s
% 3.08/1.65 % (12411)Instructions burned: 6 (million)
% 3.08/1.65 % (12411)------------------------------
% 3.08/1.65 % (12411)------------------------------
% 3.08/1.66 % (12412)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_12192 for (2999ds/377Mi)
% 3.08/1.66 % (12414)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_12192 for (2999ds/19Mi)
% 3.08/1.66 % (12415)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_12192 for (2999ds/879Mi)
% 3.08/1.67 % (12414)Instruction limit reached!
% 3.08/1.67 % (12414)------------------------------
% 3.08/1.67 % (12414)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.67 % (12414)Termination reason: Unknown
% 3.08/1.67 % (12414)Termination phase: shuffling
% 3.08/1.67
% 3.08/1.67 % (12414)Memory used [KB]: 1791
% 3.08/1.67 % (12414)Time elapsed: 0.007 s
% 3.08/1.67 % (12414)Instructions burned: 22 (million)
% 3.08/1.67 % (12414)------------------------------
% 3.08/1.67 % (12414)------------------------------
% 3.08/1.67 % (12413)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_12192 for (2999ds/779Mi)
% 3.08/1.68 % (12416)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_12192 for (2998ds/17Mi)
% 3.08/1.68 % (12416)Instruction limit reached!
% 3.08/1.68 % (12416)------------------------------
% 3.08/1.68 % (12416)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.68 % (12416)Termination reason: Unknown
% 3.08/1.68 % (12416)Termination phase: shuffling
% 3.08/1.68
% 3.08/1.68 % (12416)Memory used [KB]: 1791
% 3.08/1.68 % (12416)Time elapsed: 0.007 s
% 3.08/1.68 % (12416)Instructions burned: 20 (million)
% 3.08/1.68 % (12416)------------------------------
% 3.08/1.68 % (12416)------------------------------
% 3.08/1.69 % (12422)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_12192 for (2998ds/3Mi)
% 3.08/1.69 % (12422)Instruction limit reached!
% 3.08/1.69 % (12422)------------------------------
% 3.08/1.69 % (12422)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.69 % (12422)Termination reason: Unknown
% 3.08/1.69 % (12422)Termination phase: shuffling
% 3.08/1.69
% 3.08/1.69 % (12422)Memory used [KB]: 1407
% 3.08/1.69 % (12422)Time elapsed: 0.002 s
% 3.08/1.69 % (12422)Instructions burned: 3 (million)
% 3.08/1.69 % (12422)------------------------------
% 3.08/1.69 % (12422)------------------------------
% 3.08/1.70 % (12423)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_12192 for (2998ds/30Mi)
% 3.08/1.71 % (12423)Instruction limit reached!
% 3.08/1.71 % (12423)------------------------------
% 3.08/1.71 % (12423)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.71 % (12423)Termination reason: Unknown
% 3.08/1.71 % (12423)Termination phase: shuffling
% 3.08/1.71
% 3.08/1.71 % (12423)Memory used [KB]: 2046
% 3.08/1.71 % (12423)Time elapsed: 0.011 s
% 3.08/1.71 % (12423)Instructions burned: 31 (million)
% 3.08/1.71 % (12423)------------------------------
% 3.08/1.71 % (12423)------------------------------
% 3.08/1.72 % (12425)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_12192 for (2998ds/127Mi)
% 3.08/1.77 % (12425)Instruction limit reached!
% 3.08/1.77 % (12425)------------------------------
% 3.08/1.77 % (12425)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.77 % (12425)Termination reason: Unknown
% 3.08/1.77 % (12425)Termination phase: Property scanning
% 3.08/1.77
% 3.08/1.77 % (12425)Memory used [KB]: 2558
% 3.08/1.77 % (12425)Time elapsed: 0.044 s
% 3.08/1.77 % (12425)Instructions burned: 130 (million)
% 3.08/1.77 % (12425)------------------------------
% 3.08/1.77 % (12425)------------------------------
% 3.08/1.78 % (12440)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_12192 for (2997ds/100Mi)
% 3.08/1.78 % (12412)Instruction limit reached!
% 3.08/1.78 % (12412)------------------------------
% 3.08/1.78 % (12412)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.78 % (12412)Termination reason: Unknown
% 3.08/1.78 % (12412)Termination phase: Saturation
% 3.08/1.78
% 3.08/1.78 % (12412)Memory used [KB]: 8699
% 3.08/1.78 % (12412)Time elapsed: 0.124 s
% 3.08/1.78 % (12412)Instructions burned: 378 (million)
% 3.08/1.78 % (12412)------------------------------
% 3.08/1.78 % (12412)------------------------------
% 3.08/1.79 % (12443)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_12192 for (2997ds/3Mi)
% 3.08/1.79 % (12443)Instruction limit reached!
% 3.08/1.79 % (12443)------------------------------
% 3.08/1.79 % (12443)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.79 % (12443)Termination reason: Unknown
% 3.08/1.79 % (12443)Termination phase: shuffling
% 3.08/1.79
% 3.08/1.79 % (12443)Memory used [KB]: 1407
% 3.08/1.79 % (12443)Time elapsed: 0.003 s
% 3.08/1.79 % (12443)Instructions burned: 4 (million)
% 3.08/1.79 % (12443)------------------------------
% 3.08/1.79 % (12443)------------------------------
% 3.08/1.81 % (12448)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_12192 for (2997ds/20Mi)
% 3.08/1.81 % (12440)Instruction limit reached!
% 3.08/1.81 % (12440)------------------------------
% 3.08/1.81 % (12440)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.81 % (12440)Termination reason: Unknown
% 3.08/1.81 % (12440)Termination phase: Property scanning
% 3.08/1.81
% 3.08/1.81 % (12440)Memory used [KB]: 2558
% 3.08/1.81 % (12440)Time elapsed: 0.037 s
% 3.08/1.81 % (12440)Instructions burned: 100 (million)
% 3.08/1.81 % (12440)------------------------------
% 3.08/1.81 % (12440)------------------------------
% 3.08/1.82 % (12448)Instruction limit reached!
% 3.08/1.82 % (12448)------------------------------
% 3.08/1.82 % (12448)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.82 % (12448)Termination reason: Unknown
% 3.08/1.82 % (12448)Termination phase: shuffling
% 3.08/1.82
% 3.08/1.82 % (12448)Memory used [KB]: 1791
% 3.08/1.82 % (12448)Time elapsed: 0.011 s
% 3.08/1.82 % (12448)Instructions burned: 20 (million)
% 3.08/1.82 % (12448)------------------------------
% 3.08/1.82 % (12448)------------------------------
% 3.08/1.83 % (12453)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_12192 for (2997ds/86Mi)
% 3.08/1.83 % (12455)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_12192 for (2997ds/107Mi)
% 3.08/1.87 % (12453)Instruction limit reached!
% 3.08/1.87 % (12453)------------------------------
% 3.08/1.87 % (12453)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.87 % (12453)Termination reason: Unknown
% 3.08/1.87 % (12453)Termination phase: Property scanning
% 3.08/1.87
% 3.08/1.87 % (12453)Memory used [KB]: 2430
% 3.08/1.87 % (12453)Time elapsed: 0.040 s
% 3.08/1.87 % (12453)Instructions burned: 86 (million)
% 3.08/1.87 % (12453)------------------------------
% 3.08/1.87 % (12453)------------------------------
% 3.08/1.88 % (12455)Instruction limit reached!
% 3.08/1.88 % (12455)------------------------------
% 3.08/1.88 % (12455)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.88 % (12455)Termination reason: Unknown
% 3.08/1.88 % (12455)Termination phase: Property scanning
% 3.08/1.88
% 3.08/1.88 % (12455)Memory used [KB]: 2430
% 3.08/1.88 % (12455)Time elapsed: 0.048 s
% 3.08/1.88 % (12455)Instructions burned: 108 (million)
% 3.08/1.88 % (12455)------------------------------
% 3.08/1.88 % (12455)------------------------------
% 3.08/1.88 % (12461)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_12192 for (2996ds/39Mi)
% 3.08/1.90 % (12464)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_12192 for (2996ds/448Mi)
% 3.08/1.90 % (12461)Instruction limit reached!
% 3.08/1.90 % (12461)------------------------------
% 3.08/1.90 % (12461)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.90 % (12461)Termination reason: Unknown
% 3.08/1.90 % (12461)Termination phase: shuffling
% 3.08/1.90
% 3.08/1.90 % (12461)Memory used [KB]: 2174
% 3.08/1.90 % (12461)Time elapsed: 0.019 s
% 3.08/1.90 % (12461)Instructions burned: 39 (million)
% 3.08/1.90 % (12461)------------------------------
% 3.08/1.90 % (12461)------------------------------
% 3.08/1.91 % (12397)Instruction limit reached!
% 3.08/1.91 % (12397)------------------------------
% 3.08/1.91 % (12397)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.91 % (12397)Termination reason: Unknown
% 3.08/1.91 % (12397)Termination phase: Saturation
% 3.08/1.91
% 3.08/1.91 % (12397)Memory used [KB]: 13560
% 3.08/1.91 % (12397)Time elapsed: 0.322 s
% 3.08/1.91 % (12397)Instructions burned: 1042 (million)
% 3.08/1.91 % (12397)------------------------------
% 3.08/1.91 % (12397)------------------------------
% 3.08/1.92 % (12406)Instruction limit reached!
% 3.08/1.92 % (12406)------------------------------
% 3.08/1.92 % (12406)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.92 % (12406)Termination reason: Unknown
% 3.08/1.92 % (12406)Termination phase: Saturation
% 3.08/1.92
% 3.08/1.92 % (12406)Memory used [KB]: 10234
% 3.08/1.92 % (12406)Time elapsed: 0.282 s
% 3.08/1.92 % (12406)Instructions burned: 710 (million)
% 3.08/1.92 % (12406)------------------------------
% 3.08/1.92 % (12406)------------------------------
% 3.08/1.92 % (12465)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_12192 for (2996ds/46Mi)
% 3.08/1.93 % (12467)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_12192 for (2996ds/98Mi)
% 3.08/1.93 % (12468)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_12192 for (2996ds/507Mi)
% 3.08/1.94 % (12465)Instruction limit reached!
% 3.08/1.94 % (12465)------------------------------
% 3.08/1.94 % (12465)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.94 % (12465)Termination reason: Unknown
% 3.08/1.94 % (12465)Termination phase: Property scanning
% 3.08/1.94
% 3.08/1.94 % (12465)Memory used [KB]: 2302
% 3.08/1.94 % (12465)Time elapsed: 0.023 s
% 3.08/1.94 % (12465)Instructions burned: 46 (million)
% 3.08/1.94 % (12465)------------------------------
% 3.08/1.94 % (12465)------------------------------
% 3.08/1.96 % (12471)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_12192 for (2996ds/149Mi)
% 3.08/1.96 % (12413)Instruction limit reached!
% 3.08/1.96 % (12413)------------------------------
% 3.08/1.96 % (12413)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.96 % (12413)Termination reason: Unknown
% 3.08/1.96 % (12413)Termination phase: Saturation
% 3.08/1.96
% 3.08/1.96 % (12413)Memory used [KB]: 7419
% 3.08/1.96 % (12413)Time elapsed: 0.293 s
% 3.08/1.96 % (12413)Instructions burned: 779 (million)
% 3.08/1.96 % (12413)------------------------------
% 3.08/1.96 % (12413)------------------------------
% 3.08/1.97 % (12467)Instruction limit reached!
% 3.08/1.97 % (12467)------------------------------
% 3.08/1.97 % (12467)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.97 % (12467)Termination reason: Unknown
% 3.08/1.97 % (12467)Termination phase: Function definition elimination
% 3.08/1.97
% 3.08/1.97 % (12467)Memory used [KB]: 2430
% 3.08/1.97 % (12467)Time elapsed: 0.045 s
% 3.08/1.97 % (12467)Instructions burned: 99 (million)
% 3.08/1.97 % (12467)------------------------------
% 3.08/1.97 % (12467)------------------------------
% 3.08/1.97 % (12472)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_12192 for (2995ds/39Mi)
% 3.08/1.98 % (12472)Instruction limit reached!
% 3.08/1.98 % (12472)------------------------------
% 3.08/1.98 % (12472)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.98 % (12472)Termination reason: Unknown
% 3.08/1.98 % (12472)Termination phase: shuffling
% 3.08/1.98
% 3.08/1.98 % (12472)Memory used [KB]: 2174
% 3.08/1.98 % (12472)Time elapsed: 0.012 s
% 3.08/1.98 % (12472)Instructions burned: 39 (million)
% 3.08/1.98 % (12472)------------------------------
% 3.08/1.98 % (12472)------------------------------
% 3.08/1.99 % (12473)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_12192 for (2995ds/6Mi)
% 3.08/1.99 % (12473)Instruction limit reached!
% 3.08/1.99 % (12473)------------------------------
% 3.08/1.99 % (12473)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/1.99 % (12473)Termination reason: Unknown
% 3.08/1.99 % (12473)Termination phase: shuffling
% 3.08/1.99
% 3.08/1.99 % (12473)Memory used [KB]: 1535
% 3.08/1.99 % (12473)Time elapsed: 0.005 s
% 3.08/1.99 % (12473)Instructions burned: 7 (million)
% 3.08/1.99 % (12473)------------------------------
% 3.08/1.99 % (12473)------------------------------
% 3.08/1.99 % (12474)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_12192 for (2995ds/1087Mi)
% 3.08/2.00 % (12415)Instruction limit reached!
% 3.08/2.00 % (12415)------------------------------
% 3.08/2.00 % (12415)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/2.00 % (12415)Termination reason: Unknown
% 3.08/2.00 % (12415)Termination phase: Saturation
% 3.08/2.00
% 3.08/2.00 % (12415)Memory used [KB]: 9594
% 3.08/2.00 % (12415)Time elapsed: 0.341 s
% 3.08/2.00 % (12415)Instructions burned: 879 (million)
% 3.08/2.00 % (12415)------------------------------
% 3.08/2.00 % (12415)------------------------------
% 3.08/2.01 % (12475)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_12192 for (2995ds/47Mi)
% 3.08/2.02 % (12476)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_12192 for (2995ds/44Mi)
% 3.08/2.02 % (12471)Instruction limit reached!
% 3.08/2.02 % (12471)------------------------------
% 3.08/2.02 % (12471)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.08/2.02 % (12471)Termination reason: Unknown
% 3.08/2.02 % (12471)Termination phase: Property scanning
% 3.08/2.02
% 3.08/2.02 % (12471)Memory used [KB]: 2942
% 3.08/2.02 % (12471)Time elapsed: 0.068 s
% 3.08/2.02 % (12471)Instructions burned: 151 (million)
% 3.08/2.02 % (12471)------------------------------
% 3.08/2.02 % (12471)------------------------------
% 6.20/2.03 % (12475)Instruction limit reached!
% 6.20/2.03 % (12475)------------------------------
% 6.20/2.03 % (12475)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.20/2.03 % (12475)Termination reason: Unknown
% 6.20/2.03 % (12475)Termination phase: Property scanning
% 6.20/2.03
% 6.20/2.03 % (12475)Memory used [KB]: 2302
% 6.20/2.03 % (12475)Time elapsed: 0.023 s
% 6.20/2.03 % (12475)Instructions burned: 47 (million)
% 6.20/2.03 % (12475)------------------------------
% 6.20/2.03 % (12475)------------------------------
% 6.20/2.04 % (12479)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_12192 for (2995ds/432Mi)
% 6.20/2.04 % (12476)Instruction limit reached!
% 6.20/2.04 % (12476)------------------------------
% 6.20/2.04 % (12476)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.20/2.04 % (12476)Termination reason: Unknown
% 6.20/2.04 % (12476)Termination phase: Property scanning
% 6.20/2.04
% 6.20/2.04 % (12476)Memory used [KB]: 2302
% 6.20/2.04 % (12476)Time elapsed: 0.021 s
% 6.20/2.04 % (12476)Instructions burned: 44 (million)
% 6.20/2.04 % (12476)------------------------------
% 6.20/2.04 % (12476)------------------------------
% 6.20/2.04 % (12408)Instruction limit reached!
% 6.20/2.04 % (12408)------------------------------
% 6.20/2.04 % (12408)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.20/2.04 % (12408)Termination reason: Unknown
% 6.20/2.04 % (12408)Termination phase: Saturation
% 6.20/2.04
% 6.20/2.04 % (12408)Memory used [KB]: 12025
% 6.20/2.04 % (12408)Time elapsed: 0.396 s
% 6.20/2.04 % (12408)Instructions burned: 902 (million)
% 6.20/2.04 % (12408)------------------------------
% 6.20/2.04 % (12408)------------------------------
% 6.20/2.05 % (12480)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_12192 for (2995ds/62Mi)
% 6.20/2.05 % (12482)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_12192 for (2995ds/31Mi)
% 6.20/2.06 % (12482)Instruction limit reached!
% 6.20/2.06 % (12482)------------------------------
% 6.20/2.06 % (12482)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.20/2.06 % (12482)Termination reason: Unknown
% 6.20/2.06 % (12482)Termination phase: shuffling
% 6.20/2.06
% 6.20/2.06 % (12482)Memory used [KB]: 2046
% 6.20/2.06 % (12482)Time elapsed: 0.011 s
% 6.20/2.06 % (12482)Instructions burned: 31 (million)
% 6.20/2.06 % (12482)------------------------------
% 6.20/2.06 % (12482)------------------------------
% 6.20/2.06 % (12464)Instruction limit reached!
% 6.20/2.06 % (12464)------------------------------
% 6.20/2.06 % (12464)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.20/2.06 % (12464)Termination reason: Unknown
% 6.20/2.06 % (12464)Termination phase: Saturation
% 6.20/2.06
% 6.20/2.06 % (12464)Memory used [KB]: 7291
% 6.20/2.06 % (12464)Time elapsed: 0.157 s
% 6.20/2.06 % (12464)Instructions burned: 449 (million)
% 6.20/2.06 % (12464)------------------------------
% 6.20/2.06 % (12464)------------------------------
% 6.20/2.06 % (12483)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_12192 for (2995ds/96Mi)
% 6.20/2.07 % (12485)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_12192 for (2994ds/71Mi)
% 6.20/2.07 % (12480)Instruction limit reached!
% 6.20/2.07 % (12480)------------------------------
% 6.20/2.07 % (12480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.20/2.07 % (12480)Termination reason: Unknown
% 6.20/2.07 % (12480)Termination phase: Property scanning
% 6.20/2.07
% 6.20/2.07 % (12480)Memory used [KB]: 2302
% 6.20/2.07 % (12480)Time elapsed: 0.029 s
% 6.20/2.07 % (12480)Instructions burned: 62 (million)
% 6.20/2.07 % (12480)------------------------------
% 6.20/2.07 % (12480)------------------------------
% 6.20/2.07 % (12484)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_12192 for (2994ds/453Mi)
% 6.20/2.09 % (12488)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_12192 for (2994ds/83Mi)
% 6.20/2.09 % (12485)Instruction limit reached!
% 6.20/2.09 % (12485)------------------------------
% 6.20/2.09 % (12485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.20/2.09 % (12485)Termination reason: Unknown
% 6.20/2.09 % (12485)Termination phase: Property scanning
% 6.20/2.09
% 6.20/2.09 % (12485)Memory used [KB]: 2430
% 6.20/2.09 % (12485)Time elapsed: 0.021 s
% 6.20/2.09 % (12485)Instructions burned: 73 (million)
% 6.20/2.09 % (12485)------------------------------
% 6.20/2.09 % (12485)------------------------------
% 6.20/2.10 % (12483)Instruction limit reached!
% 6.20/2.10 % (12483)------------------------------
% 6.20/2.10 % (12483)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.20/2.10 % (12483)Termination reason: Unknown
% 6.20/2.10 % (12483)Termination phase: Function definition elimination
% 6.20/2.10
% 6.20/2.10 % (12483)Memory used [KB]: 2430
% 6.20/2.10 % (12483)Time elapsed: 0.044 s
% 6.20/2.10 % (12483)Instructions burned: 98 (million)
% 6.20/2.10 % (12483)------------------------------
% 6.20/2.10 % (12483)------------------------------
% 6.20/2.11 % (12489)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_12192 for (2994ds/11Mi)
% 6.20/2.11 % (12489)Instruction limit reached!
% 6.20/2.11 % (12489)------------------------------
% 6.20/2.11 % (12489)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.20/2.11 % (12489)Termination reason: Unknown
% 6.20/2.11 % (12489)Termination phase: shuffling
% 6.20/2.11
% 6.20/2.11 % (12489)Memory used [KB]: 1663
% 6.20/2.11 % (12489)Time elapsed: 0.005 s
% 6.20/2.11 % (12489)Instructions burned: 13 (million)
% 6.20/2.11 % (12489)------------------------------
% 6.20/2.11 % (12489)------------------------------
% 6.20/2.11 % (12491)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_gen_be_off:fsr=off:spb=non_intro:i=41:si=on:rtra=on_0 on DTF2THF_12192 for (2994ds/41Mi)
% 6.20/2.12 % (12488)Instruction limit reached!
% 6.20/2.12 % (12488)------------------------------
% 6.20/2.12 % (12488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.20/2.12 % (12488)Termination reason: Unknown
% 6.20/2.12 % (12488)Termination phase: SInE selection
% 6.20/2.12
% 6.20/2.12 % (12488)Memory used [KB]: 2302
% 6.20/2.12 % (12488)Time elapsed: 0.028 s
% 6.20/2.12 % (12488)Instructions burned: 85 (million)
% 6.20/2.12 % (12488)------------------------------
% 6.20/2.12 % (12488)------------------------------
% 6.20/2.12 % (12492)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_12192 for (2994ds/6Mi)
% 6.20/2.12 % (12492)Instruction limit reached!
% 6.20/2.12 % (12492)------------------------------
% 6.20/2.12 % (12492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.20/2.12 % (12492)Termination reason: Unknown
% 6.20/2.12 % (12492)Termination phase: shuffling
% 6.20/2.12
% 6.20/2.12 % (12492)Memory used [KB]: 1535
% 6.20/2.12 % (12492)Time elapsed: 0.003 s
% 6.20/2.12 % (12492)Instructions burned: 6 (million)
% 6.20/2.12 % (12492)------------------------------
% 6.20/2.12 % (12492)------------------------------
% 6.20/2.12 % (12491)Instruction limit reached!
% 6.20/2.12 % (12491)------------------------------
% 6.20/2.12 % (12491)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.20/2.12 % (12491)Termination reason: Unknown
% 6.20/2.12 % (12491)Termination phase: Property scanning
% 6.20/2.12
% 6.20/2.12 % (12491)Memory used [KB]: 2302
% 6.20/2.12 % (12491)Time elapsed: 0.013 s
% 6.20/2.12 % (12491)Instructions burned: 43 (million)
% 6.20/2.12 % (12491)------------------------------
% 6.20/2.12 % (12491)------------------------------
% 6.86/2.13 % (12493)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_12192 for (2994ds/150Mi)
% 6.86/2.13 % (12494)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_12192 for (2994ds/343Mi)
% 6.86/2.13 % (12495)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=109:si=on:rtra=on_0 on DTF2THF_12192 for (2994ds/109Mi)
% 6.86/2.15 % (12468)Instruction limit reached!
% 6.86/2.15 % (12468)------------------------------
% 6.86/2.15 % (12468)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.86/2.15 % (12468)Termination reason: Unknown
% 6.86/2.15 % (12468)Termination phase: Saturation
% 6.86/2.15
% 6.86/2.15 % (12468)Memory used [KB]: 7931
% 6.86/2.15 % (12468)Time elapsed: 0.238 s
% 6.86/2.15 % (12468)Instructions burned: 509 (million)
% 6.86/2.15 % (12468)------------------------------
% 6.86/2.15 % (12468)------------------------------
% 6.86/2.16 % (12496)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=253:si=on:rtra=on_0 on DTF2THF_12192 for (2994ds/253Mi)
% 6.86/2.16 % (12495)Instruction limit reached!
% 6.86/2.16 % (12495)------------------------------
% 6.86/2.16 % (12495)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.86/2.16 % (12495)Termination reason: Unknown
% 6.86/2.16 % (12495)Termination phase: Function definition elimination
% 6.86/2.16
% 6.86/2.16 % (12495)Memory used [KB]: 2430
% 6.86/2.16 % (12495)Time elapsed: 0.030 s
% 6.86/2.16 % (12495)Instructions burned: 112 (million)
% 6.86/2.16 % (12495)------------------------------
% 6.86/2.16 % (12495)------------------------------
% 6.86/2.17 % (12493)Instruction limit reached!
% 6.86/2.17 % (12493)------------------------------
% 6.86/2.17 % (12493)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.86/2.17 % (12493)Termination reason: Unknown
% 6.86/2.17 % (12493)Termination phase: Property scanning
% 6.86/2.17
% 6.86/2.17 % (12493)Memory used [KB]: 4093
% 6.86/2.17 % (12493)Time elapsed: 0.044 s
% 6.86/2.17 % (12493)Instructions burned: 152 (million)
% 6.86/2.17 % (12493)------------------------------
% 6.86/2.17 % (12493)------------------------------
% 6.86/2.17 % (12498)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_12192 for (2993ds/140Mi)
% 6.86/2.18 % (12479)Instruction limit reached!
% 6.86/2.18 % (12479)------------------------------
% 6.86/2.18 % (12479)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.86/2.18 % (12479)Termination reason: Unknown
% 6.86/2.18 % (12479)Termination phase: Saturation
% 6.86/2.18
% 6.86/2.18 % (12479)Memory used [KB]: 2814
% 6.86/2.18 % (12479)Time elapsed: 0.139 s
% 6.86/2.18 % (12479)Instructions burned: 433 (million)
% 6.86/2.18 % (12479)------------------------------
% 6.86/2.18 % (12479)------------------------------
% 6.86/2.18 % (12501)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=614:si=on:rtra=on_0 on DTF2THF_12192 for (2993ds/614Mi)
% 6.86/2.19 % (12504)lrs+1002_1:1_e2e=on:fde=unused:fe=axiom:prag=on:sd=1:sos=on:ss=axioms:st=2.0:i=202:si=on:rtra=on_0 on DTF2THF_12192 for (2993ds/202Mi)
% 7.20/2.20 % (12484)Instruction limit reached!
% 7.20/2.20 % (12484)------------------------------
% 7.20/2.20 % (12484)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.20/2.20 % (12484)Termination reason: Unknown
% 7.20/2.20 % (12484)Termination phase: Saturation
% 7.20/2.20
% 7.20/2.20 % (12484)Memory used [KB]: 7419
% 7.20/2.20 % (12484)Time elapsed: 0.128 s
% 7.20/2.20 % (12484)Instructions burned: 454 (million)
% 7.20/2.20 % (12484)------------------------------
% 7.20/2.20 % (12484)------------------------------
% 7.20/2.21 % (12498)Instruction limit reached!
% 7.20/2.21 % (12498)------------------------------
% 7.20/2.21 % (12498)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.20/2.21 % (12498)Termination reason: Unknown
% 7.20/2.21 % (12498)Termination phase: Saturation
% 7.20/2.21
% 7.20/2.21 % (12498)Memory used [KB]: 7036
% 7.20/2.21 % (12498)Time elapsed: 0.038 s
% 7.20/2.21 % (12498)Instructions burned: 142 (million)
% 7.20/2.21 % (12498)------------------------------
% 7.20/2.21 % (12498)------------------------------
% 7.20/2.21 % (12516)dis+1002_1:1_au=on:cs=on:ins=3:kws=inv_frequency:nwc=10.0:ss=axioms:st=3.0:i=339:si=on:rtra=on_0 on DTF2THF_12192 for (2993ds/339Mi)
% 7.20/2.22 % (12521)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=283:si=on:rtra=on_0 on DTF2THF_12192 for (2993ds/283Mi)
% 7.20/2.22 % (12494)Instruction limit reached!
% 7.20/2.22 % (12494)------------------------------
% 7.20/2.22 % (12494)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.20/2.22 % (12494)Termination reason: Unknown
% 7.20/2.22 % (12494)Termination phase: Saturation
% 7.20/2.22
% 7.20/2.22 % (12494)Memory used [KB]: 8443
% 7.20/2.22 % (12494)Time elapsed: 0.095 s
% 7.20/2.22 % (12494)Instructions burned: 345 (million)
% 7.20/2.22 % (12494)------------------------------
% 7.20/2.22 % (12494)------------------------------
% 7.20/2.23 % (12496)Instruction limit reached!
% 7.20/2.23 % (12496)------------------------------
% 7.20/2.23 % (12496)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.20/2.23 % (12496)Termination reason: Unknown
% 7.20/2.23 % (12496)Termination phase: Saturation
% 7.20/2.23
% 7.20/2.23 % (12496)Memory used [KB]: 9466
% 7.20/2.23 % (12496)Time elapsed: 0.076 s
% 7.20/2.23 % (12496)Instructions burned: 255 (million)
% 7.20/2.23 % (12496)------------------------------
% 7.20/2.23 % (12496)------------------------------
% 7.20/2.23 % (12527)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_12192 for (2993ds/134Mi)
% 7.20/2.24 % (12531)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_12192 for (2993ds/298Mi)
% 7.20/2.24 % (12504)Instruction limit reached!
% 7.20/2.24 % (12504)------------------------------
% 7.20/2.24 % (12504)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.20/2.24 % (12504)Termination reason: Unknown
% 7.20/2.24 % (12504)Termination phase: Saturation
% 7.20/2.24
% 7.20/2.24 % (12504)Memory used [KB]: 4605
% 7.20/2.24 % (12504)Time elapsed: 0.059 s
% 7.20/2.24 % (12504)Instructions burned: 202 (million)
% 7.20/2.24 % (12504)------------------------------
% 7.20/2.24 % (12504)------------------------------
% 7.20/2.25 % (12539)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_12192 for (2993ds/192Mi)
% 7.20/2.27 % (12527)Instruction limit reached!
% 7.20/2.27 % (12527)------------------------------
% 7.20/2.27 % (12527)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.20/2.27 % (12527)Termination reason: Unknown
% 7.20/2.27 % (12527)Termination phase: Saturation
% 7.20/2.27
% 7.20/2.27 % (12527)Memory used [KB]: 7036
% 7.20/2.27 % (12527)Time elapsed: 0.036 s
% 7.20/2.27 % (12527)Instructions burned: 136 (million)
% 7.20/2.27 % (12527)------------------------------
% 7.20/2.27 % (12527)------------------------------
% 7.20/2.28 % (12548)lrs+10_1:1_bd=off:hud=1:nm=0:prag=on:sd=2:sgt=32:sos=all:ss=axioms:i=258:si=on:rtra=on_0 on DTF2THF_12192 for (2992ds/258Mi)
% 7.20/2.29 % (12521)Instruction limit reached!
% 7.20/2.29 % (12521)------------------------------
% 7.20/2.29 % (12521)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.20/2.29 % (12521)Termination reason: Unknown
% 7.20/2.29 % (12521)Termination phase: Saturation
% 7.20/2.29
% 7.20/2.29 % (12521)Memory used [KB]: 7291
% 7.20/2.29 % (12521)Time elapsed: 0.073 s
% 7.20/2.29 % (12521)Instructions burned: 285 (million)
% 7.20/2.29 % (12521)------------------------------
% 7.20/2.29 % (12521)------------------------------
% 7.86/2.30 % (12558)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_12192 for (2992ds/90Mi)
% 7.86/2.31 % (12539)Instruction limit reached!
% 7.86/2.31 % (12539)------------------------------
% 7.86/2.31 % (12539)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.86/2.31 % (12539)Termination reason: Unknown
% 7.86/2.31 % (12539)Termination phase: Saturation
% 7.86/2.31
% 7.86/2.31 % (12539)Memory used [KB]: 7547
% 7.86/2.31 % (12539)Time elapsed: 0.053 s
% 7.86/2.31 % (12539)Instructions burned: 192 (million)
% 7.86/2.31 % (12539)------------------------------
% 7.86/2.31 % (12539)------------------------------
% 7.86/2.31 % (12516)Instruction limit reached!
% 7.86/2.31 % (12516)------------------------------
% 7.86/2.31 % (12516)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.86/2.31 % (12516)Termination reason: Unknown
% 7.86/2.31 % (12516)Termination phase: Saturation
% 7.86/2.31
% 7.86/2.31 % (12516)Memory used [KB]: 7803
% 7.86/2.31 % (12516)Time elapsed: 0.096 s
% 7.86/2.31 % (12516)Instructions burned: 340 (million)
% 7.86/2.31 % (12516)------------------------------
% 7.86/2.31 % (12516)------------------------------
% 7.86/2.31 % (12565)dis+1004_16:1_avsq=on:avsqr=1,16:cnfonf=lazy_gen:cs=on:fe=off:tgt=ground:tnu=1:i=231:si=on:rtra=on_0 on DTF2THF_12192 for (2992ds/231Mi)
% 7.86/2.32 % (12566)lrs+1002_1:1_au=on:s2a=on:sgt=32:slsq=on:slsqc=1:slsql=off:ss=axioms:i=458:si=on:rtra=on_0 on DTF2THF_12192 for (2992ds/458Mi)
% 7.86/2.32 % (12531)Instruction limit reached!
% 7.86/2.32 % (12531)------------------------------
% 7.86/2.32 % (12531)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.86/2.32 % (12531)Termination reason: Unknown
% 7.86/2.32 % (12531)Termination phase: Saturation
% 7.86/2.32
% 7.86/2.32 % (12531)Memory used [KB]: 7291
% 7.86/2.32 % (12531)Time elapsed: 0.076 s
% 7.86/2.32 % (12531)Instructions burned: 299 (million)
% 7.86/2.32 % (12531)------------------------------
% 7.86/2.32 % (12531)------------------------------
% 7.86/2.32 % (12558)Instruction limit reached!
% 7.86/2.32 % (12558)------------------------------
% 7.86/2.32 % (12558)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.86/2.32 % (12558)Termination reason: Unknown
% 7.86/2.32 % (12558)Termination phase: Function definition elimination
% 7.86/2.32
% 7.86/2.32 % (12558)Memory used [KB]: 2430
% 7.86/2.32 % (12558)Time elapsed: 0.025 s
% 7.86/2.32 % (12558)Instructions burned: 93 (million)
% 7.86/2.32 % (12558)------------------------------
% 7.86/2.32 % (12558)------------------------------
% 7.86/2.33 % (12570)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_12192 for (2992ds/31Mi)
% 7.86/2.33 % (12577)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:s2a=on:sp=weighted_frequency:ss=axioms:i=297:si=on:rtra=on_0 on DTF2THF_12192 for (2992ds/297Mi)
% 7.86/2.34 % (12570)Instruction limit reached!
% 7.86/2.34 % (12570)------------------------------
% 7.86/2.34 % (12570)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.86/2.34 % (12570)Termination reason: Unknown
% 7.86/2.34 % (12570)Termination phase: shuffling
% 7.86/2.34
% 7.86/2.34 % (12570)Memory used [KB]: 2046
% 7.86/2.34 % (12570)Time elapsed: 0.010 s
% 7.86/2.34 % (12570)Instructions burned: 33 (million)
% 7.86/2.34 % (12570)------------------------------
% 7.86/2.34 % (12570)------------------------------
% 7.86/2.34 % (12578)dis+2_1:4_amm=off:au=on:cbe=off:cnfonf=lazy_gen:hfaw=0:hfsq=on:hfsql=off:plsq=on:plsqc=1:plsqr=35,4:sac=on:i=987:si=on:rtra=on_0 on DTF2THF_12192 for (2992ds/987Mi)
% 7.86/2.35 % (12548)Instruction limit reached!
% 7.86/2.35 % (12548)------------------------------
% 7.86/2.35 % (12548)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.86/2.35 % (12548)Termination reason: Unknown
% 7.86/2.35 % (12548)Termination phase: Saturation
% 7.86/2.35
% 7.86/2.35 % (12548)Memory used [KB]: 7931
% 7.86/2.35 % (12548)Time elapsed: 0.069 s
% 7.86/2.35 % (12548)Instructions burned: 258 (million)
% 7.86/2.35 % (12548)------------------------------
% 7.86/2.35 % (12548)------------------------------
% 7.86/2.36 % (12584)lrs+10_1:1024_avsq=on:avsqc=4:avsqr=584411,1048576:bd=off:bet=on:br=off:chr=on:fsd=on:fsdmm=1:hud=10:nm=0:ntd=on:nwc=9.0:piset=or:plsq=on:plsqc=2:plsql=on:plsqr=852219,1048576:rawr=on:sp=const_frequency:spb=units:i=182:si=on:rtra=on_0 on DTF2THF_12192 for (2992ds/182Mi)
% 7.86/2.36 % (12501)Instruction limit reached!
% 7.86/2.36 % (12501)------------------------------
% 7.86/2.36 % (12501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.86/2.36 % (12501)Termination reason: Unknown
% 7.86/2.36 % (12501)Termination phase: Saturation
% 7.86/2.36
% 7.86/2.36 % (12501)Memory used [KB]: 9594
% 7.86/2.36 % (12501)Time elapsed: 0.183 s
% 7.86/2.36 % (12501)Instructions burned: 615 (million)
% 7.86/2.36 % (12501)------------------------------
% 7.86/2.36 % (12501)------------------------------
% 7.86/2.37 % (12595)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=182:si=on:rtra=on_0 on DTF2THF_12192 for (2991ds/182Mi)
% 7.86/2.38 % (12565)Instruction limit reached!
% 7.86/2.38 % (12565)------------------------------
% 7.86/2.38 % (12565)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.86/2.38 % (12565)Termination reason: Unknown
% 7.86/2.38 % (12565)Termination phase: Saturation
% 7.86/2.38
% 7.86/2.38 % (12565)Memory used [KB]: 10362
% 7.86/2.38 % (12565)Time elapsed: 0.064 s
% 7.86/2.38 % (12565)Instructions burned: 231 (million)
% 7.86/2.38 % (12565)------------------------------
% 7.86/2.38 % (12565)------------------------------
% 7.86/2.38 % (12474)Instruction limit reached!
% 7.86/2.38 % (12474)------------------------------
% 7.86/2.38 % (12474)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.86/2.38 % (12474)Termination reason: Unknown
% 7.86/2.38 % (12474)Termination phase: Saturation
% 7.86/2.38
% 7.86/2.38 % (12474)Memory used [KB]: 11257
% 7.86/2.38 % (12474)Time elapsed: 0.385 s
% 7.86/2.38 % (12474)Instructions burned: 1088 (million)
% 7.86/2.38 % (12474)------------------------------
% 7.86/2.38 % (12474)------------------------------
% 7.86/2.39 % (12601)dis+10_1:8_cha=on:cnfonf=lazy_pi_sigma_gen:fe=axiom:hud=5:plsq=on:plsqr=32,1:prag=on:sos=all:i=154:si=on:rtra=on_0 on DTF2THF_12192 for (2991ds/154Mi)
% 7.86/2.39 % (12602)dis+1003_1:64_atotf=0.4:au=on:fsr=off:hflw=5:hfsq=on:hfsqc=20:hfsql=off:plsq=on:plsqc=1:plsqr=32,1:i=79:si=on:rtra=on_0 on DTF2THF_12192 for (2991ds/79Mi)
% 8.60/2.41 % (12584)Instruction limit reached!
% 8.60/2.41 % (12584)------------------------------
% 8.60/2.41 % (12584)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.60/2.41 % (12584)Termination reason: Unknown
% 8.60/2.41 % (12584)Termination phase: Saturation
% 8.60/2.41
% 8.60/2.41 % (12584)Memory used [KB]: 7547
% 8.60/2.41 % (12584)Time elapsed: 0.051 s
% 8.60/2.41 % (12584)Instructions burned: 185 (million)
% 8.60/2.41 % (12584)------------------------------
% 8.60/2.41 % (12584)------------------------------
% 8.60/2.41 % (12577)Instruction limit reached!
% 8.60/2.41 % (12577)------------------------------
% 8.60/2.41 % (12577)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.60/2.41 % (12577)Termination reason: Unknown
% 8.60/2.41 % (12577)Termination phase: Saturation
% 8.60/2.41
% 8.60/2.41 % (12577)Memory used [KB]: 7291
% 8.60/2.41 % (12577)Time elapsed: 0.077 s
% 8.60/2.41 % (12577)Instructions burned: 298 (million)
% 8.60/2.41 % (12577)------------------------------
% 8.60/2.41 % (12577)------------------------------
% 8.60/2.41 % (12602)Instruction limit reached!
% 8.60/2.41 % (12602)------------------------------
% 8.60/2.41 % (12602)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.60/2.41 % (12602)Termination reason: Unknown
% 8.60/2.41 % (12602)Termination phase: Property scanning
% 8.60/2.41
% 8.60/2.41 % (12602)Memory used [KB]: 2430
% 8.60/2.41 % (12602)Time elapsed: 0.023 s
% 8.60/2.41 % (12602)Instructions burned: 82 (million)
% 8.60/2.41 % (12602)------------------------------
% 8.60/2.41 % (12602)------------------------------
% 8.60/2.42 % (12619)dis+1010_5:2_anc=all_dependent:awrs=converge:bd=off:bet=on:c=on:cha=on:cnfonf=lazy_not_gen_be_off:fe=axiom:hud=5:kws=precedence:piset=and:prag=on:s2a=on:sac=on:slsq=on:slsqc=1:sp=reverse_arity:ss=axioms:st=1.5:i=216:si=on:rtra=on_0 on DTF2THF_12192 for (2991ds/216Mi)
% 8.60/2.42 % (12623)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_12192 for (2991ds/114Mi)
% 8.60/2.42 % (12627)dis+1002_1:1_acc=on:au=on:bd=off:e2e=on:sgt=8:ss=axioms:i=922:si=on:rtra=on_0 on DTF2THF_12192 for (2991ds/922Mi)
% 8.60/2.42 % (12595)Instruction limit reached!
% 8.60/2.42 % (12595)------------------------------
% 8.60/2.42 % (12595)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.60/2.42 % (12595)Termination reason: Unknown
% 8.60/2.42 % (12595)Termination phase: Saturation
% 8.60/2.42
% 8.60/2.42 % (12595)Memory used [KB]: 8571
% 8.60/2.42 % (12595)Time elapsed: 0.053 s
% 8.60/2.42 % (12595)Instructions burned: 185 (million)
% 8.60/2.42 % (12595)------------------------------
% 8.60/2.42 % (12595)------------------------------
% 8.60/2.43 % (12601)Instruction limit reached!
% 8.60/2.43 % (12601)------------------------------
% 8.60/2.43 % (12601)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.60/2.43 % (12601)Termination reason: Unknown
% 8.60/2.43 % (12601)Termination phase: Saturation
% 8.60/2.43
% 8.60/2.43 % (12601)Memory used [KB]: 7036
% 8.60/2.43 % (12601)Time elapsed: 0.041 s
% 8.60/2.43 % (12601)Instructions burned: 155 (million)
% 8.60/2.43 % (12601)------------------------------
% 8.60/2.43 % (12601)------------------------------
% 8.60/2.43 % (12630)lrs+10_1:128_au=on:bd=off:cnfonf=lazy_not_gen_be_off:nwc=10.0:piset=and:i=182:si=on:rtra=on_0 on DTF2THF_12192 for (2991ds/182Mi)
% 8.60/2.44 % (12631)ott+1010_1:1024_au=on:av=off:c=on:cnfonf=conj_eager:piset=equals:s2a=on:sp=occurrence:spb=units:ss=axioms:i=164:si=on:rtra=on_0 on DTF2THF_12192 for (2991ds/164Mi)
% 8.60/2.45 % (12566)Instruction limit reached!
% 8.60/2.45 % (12566)------------------------------
% 8.60/2.45 % (12566)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.60/2.45 % (12566)Termination reason: Unknown
% 8.60/2.45 % (12566)Termination phase: Saturation
% 8.60/2.45
% 8.60/2.45 % (12566)Memory used [KB]: 9338
% 8.60/2.45 % (12566)Time elapsed: 0.131 s
% 8.60/2.45 % (12566)Instructions burned: 458 (million)
% 8.60/2.45 % (12566)------------------------------
% 8.60/2.45 % (12566)------------------------------
% 8.60/2.45 % (12623)Instruction limit reached!
% 8.60/2.45 % (12623)------------------------------
% 8.60/2.45 % (12623)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.60/2.45 % (12623)Termination reason: Unknown
% 8.60/2.45 % (12623)Termination phase: Property scanning
% 8.60/2.45
% 8.60/2.45 % (12623)Memory used [KB]: 2558
% 8.60/2.45 % (12623)Time elapsed: 0.030 s
% 8.60/2.45 % (12623)Instructions burned: 115 (million)
% 8.60/2.45 % (12623)------------------------------
% 8.60/2.45 % (12623)------------------------------
% 8.60/2.46 % (12644)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=50:si=on:rtra=on_0 on DTF2THF_12192 for (2991ds/50Mi)
% 8.60/2.46 % (12648)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_12192 for (2990ds/368Mi)
% 8.60/2.47 % (12644)Instruction limit reached!
% 8.60/2.47 % (12644)------------------------------
% 8.60/2.47 % (12644)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.60/2.47 % (12644)Termination reason: Unknown
% 8.60/2.47 % (12644)Termination phase: Property scanning
% 8.60/2.47
% 8.60/2.47 % (12644)Memory used [KB]: 2302
% 8.60/2.47 % (12644)Time elapsed: 0.015 s
% 8.60/2.47 % (12644)Instructions burned: 51 (million)
% 8.60/2.47 % (12644)------------------------------
% 8.60/2.47 % (12644)------------------------------
% 8.60/2.47 % (12619)Instruction limit reached!
% 8.60/2.47 % (12619)------------------------------
% 8.60/2.47 % (12619)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.60/2.47 % (12619)Termination reason: Unknown
% 8.60/2.47 % (12619)Termination phase: Saturation
% 8.60/2.47
% 8.60/2.47 % (12619)Memory used [KB]: 7291
% 8.60/2.47 % (12619)Time elapsed: 0.055 s
% 8.60/2.47 % (12619)Instructions burned: 218 (million)
% 8.60/2.47 % (12619)------------------------------
% 8.60/2.47 % (12619)------------------------------
% 8.60/2.48 % (12655)dis+2_1:1_bd=off:e2e=on:hud=1:nm=3:piset=or:prag=on:s2a=on:sp=const_max:i=1021:si=on:rtra=on_0 on DTF2THF_12192 for (2990ds/1021Mi)
% 8.60/2.48 % (12656)dis+10_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=267:si=on:rtra=on_0 on DTF2THF_12192 for (2990ds/267Mi)
% 8.60/2.48 % (12631)Instruction limit reached!
% 8.60/2.48 % (12631)------------------------------
% 8.60/2.48 % (12631)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.60/2.48 % (12631)Termination reason: Unknown
% 8.60/2.48 % (12631)Termination phase: Saturation
% 8.60/2.48
% 8.60/2.48 % (12631)Memory used [KB]: 2558
% 8.60/2.48 % (12631)Time elapsed: 0.043 s
% 8.60/2.48 % (12631)Instructions burned: 166 (million)
% 8.60/2.48 % (12631)------------------------------
% 8.60/2.48 % (12631)------------------------------
% 8.60/2.48 % (12630)Instruction limit reached!
% 8.60/2.48 % (12630)------------------------------
% 8.60/2.48 % (12630)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.60/2.48 % (12630)Termination reason: Unknown
% 8.60/2.48 % (12630)Termination phase: Saturation
% 8.60/2.48
% 8.60/2.48 % (12630)Memory used [KB]: 7291
% 8.60/2.48 % (12630)Time elapsed: 0.047 s
% 8.60/2.48 % (12630)Instructions burned: 182 (million)
% 8.60/2.48 % (12630)------------------------------
% 8.60/2.48 % (12630)------------------------------
% 8.60/2.49 % (12663)dis+21_1:5_au=on:ntd=on:nwc=5.0:s2a=on:ss=axioms:st=5.0:i=204:si=on:rtra=on_0 on DTF2THF_12192 for (2990ds/204Mi)
% 8.60/2.49 % (12664)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_12192 for (2990ds/26Mi)
% 9.27/2.50 % (12664)Instruction limit reached!
% 9.27/2.50 % (12664)------------------------------
% 9.27/2.50 % (12664)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.27/2.50 % (12664)Termination reason: Unknown
% 9.27/2.50 % (12664)Termination phase: shuffling
% 9.27/2.50
% 9.27/2.50 % (12664)Memory used [KB]: 1918
% 9.27/2.50 % (12664)Time elapsed: 0.009 s
% 9.27/2.50 % (12664)Instructions burned: 28 (million)
% 9.27/2.50 % (12664)------------------------------
% 9.27/2.50 % (12664)------------------------------
% 9.27/2.51 % (12669)dis+1002_1:1_bd=preordered:c=on:cbe=off:cnfonf=lazy_not_gen_be_off:hud=10:ixr=off:prag=on:i=53:si=on:rtra=on_0 on DTF2THF_12192 for (2990ds/53Mi)
% 9.27/2.52 % (12669)Instruction limit reached!
% 9.27/2.52 % (12669)------------------------------
% 9.27/2.52 % (12669)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.27/2.52 % (12669)Termination reason: Unknown
% 9.27/2.52 % (12669)Termination phase: Property scanning
% 9.27/2.52
% 9.27/2.52 % (12669)Memory used [KB]: 2302
% 9.27/2.52 % (12669)Time elapsed: 0.015 s
% 9.27/2.52 % (12669)Instructions burned: 53 (million)
% 9.27/2.52 % (12669)------------------------------
% 9.27/2.52 % (12669)------------------------------
% 9.27/2.53 % (12678)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=4436:si=on:rtra=on_0 on DTF2THF_12192 for (2990ds/4436Mi)
% 9.27/2.54 % (12663)Instruction limit reached!
% 9.27/2.54 % (12663)------------------------------
% 9.27/2.54 % (12663)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.27/2.54 % (12663)Termination reason: Unknown
% 9.27/2.54 % (12663)Termination phase: Saturation
% 9.27/2.54
% 9.27/2.54 % (12663)Memory used [KB]: 7291
% 9.27/2.54 % (12663)Time elapsed: 0.055 s
% 9.27/2.54 % (12663)Instructions burned: 205 (million)
% 9.27/2.54 % (12663)------------------------------
% 9.27/2.54 % (12663)------------------------------
% 9.27/2.55 % (12656)Instruction limit reached!
% 9.27/2.55 % (12656)------------------------------
% 9.27/2.55 % (12656)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.27/2.55 % (12656)Termination reason: Unknown
% 9.27/2.55 % (12656)Termination phase: Saturation
% 9.27/2.55
% 9.27/2.55 % (12656)Memory used [KB]: 7291
% 9.27/2.55 % (12656)Time elapsed: 0.069 s
% 9.27/2.55 % (12656)Instructions burned: 269 (million)
% 9.27/2.55 % (12656)------------------------------
% 9.27/2.55 % (12656)------------------------------
% 9.27/2.55 % (12684)dis+1010_1:1_acc=model:avsq=on:ixr=off:prag=on:s2a=on:sac=on:i=1539:si=on:rtra=on_0 on DTF2THF_12192 for (2990ds/1539Mi)
% 9.27/2.56 % (12687)lrs+2_1:1_bd=off:e2e=on:fde=none:kws=inv_precedence:plsq=on:plsqr=32,1:sos=on:i=408:si=on:rtra=on_0 on DTF2THF_12192 for (2990ds/408Mi)
% 9.27/2.56 % (12648)Instruction limit reached!
% 9.27/2.56 % (12648)------------------------------
% 9.27/2.56 % (12648)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.27/2.56 % (12648)Termination reason: Unknown
% 9.27/2.56 % (12648)Termination phase: Saturation
% 9.27/2.56
% 9.27/2.56 % (12648)Memory used [KB]: 8571
% 9.27/2.56 % (12648)Time elapsed: 0.105 s
% 9.27/2.56 % (12648)Instructions burned: 371 (million)
% 9.27/2.56 % (12648)------------------------------
% 9.27/2.56 % (12648)------------------------------
% 9.27/2.57 % (12695)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=5205:si=on:rtra=on_0 on DTF2THF_12192 for (2989ds/5205Mi)
% 9.56/2.60 % (12578)Instruction limit reached!
% 9.56/2.60 % (12578)------------------------------
% 9.56/2.60 % (12578)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.56/2.60 % (12578)Termination reason: Unknown
% 9.56/2.60 % (12578)Termination phase: Saturation
% 9.56/2.60
% 9.56/2.60 % (12578)Memory used [KB]: 13688
% 9.56/2.60 % (12578)Time elapsed: 0.278 s
% 9.56/2.60 % (12578)Instructions burned: 989 (million)
% 9.56/2.60 % (12578)------------------------------
% 9.56/2.60 % (12578)------------------------------
% 9.56/2.61 % (12707)dis+1010_1:4_au=on:av=off:cbe=off:nwc=10.0:s2a=on:s2at=3.0:sd=2:ss=axioms:st=5.0:i=1913:si=on:rtra=on_0 on DTF2THF_12192 for (2989ds/1913Mi)
% 9.56/2.68 % (12687)Instruction limit reached!
% 9.56/2.68 % (12687)------------------------------
% 9.56/2.68 % (12687)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.56/2.68 % (12687)Termination reason: Unknown
% 9.56/2.68 % (12687)Termination phase: Saturation
% 9.56/2.68
% 9.56/2.68 % (12687)Memory used [KB]: 10618
% 9.56/2.68 % (12687)Time elapsed: 0.128 s
% 9.56/2.68 % (12687)Instructions burned: 410 (million)
% 9.56/2.68 % (12687)------------------------------
% 9.56/2.68 % (12687)------------------------------
% 9.56/2.69 % (12732)lrs+2_1:1_bd=off:cbe=off:piset=equals:sd=1:sgt=20:sos=all:ss=axioms:i=4350:si=on:rtra=on_0 on DTF2THF_12192 for (2988ds/4350Mi)
% 9.97/2.71 % (12627)Instruction limit reached!
% 9.97/2.71 % (12627)------------------------------
% 9.97/2.71 % (12627)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.97/2.71 % (12627)Termination reason: Unknown
% 9.97/2.71 % (12627)Termination phase: Saturation
% 9.97/2.71
% 9.97/2.71 % (12627)Memory used [KB]: 10746
% 9.97/2.71 % (12627)Time elapsed: 0.291 s
% 9.97/2.71 % (12627)Instructions burned: 922 (million)
% 9.97/2.71 % (12627)------------------------------
% 9.97/2.71 % (12627)------------------------------
% 9.97/2.72 % (12750)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=68:si=on:rtra=on_0 on DTF2THF_12192 for (2988ds/68Mi)
% 9.97/2.74 % (12750)Instruction limit reached!
% 9.97/2.74 % (12750)------------------------------
% 9.97/2.74 % (12750)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.97/2.74 % (12750)Termination reason: Unknown
% 9.97/2.74 % (12750)Termination phase: SInE selection
% 9.97/2.74
% 9.97/2.74 % (12750)Memory used [KB]: 2302
% 9.97/2.74 % (12750)Time elapsed: 0.019 s
% 9.97/2.74 % (12750)Instructions burned: 68 (million)
% 9.97/2.74 % (12750)------------------------------
% 9.97/2.74 % (12750)------------------------------
% 9.97/2.75 % (12762)dis+10_1:1_au=on:nwc=5.0:s2a=on:s2at=2.0:sac=on:sd=7:ss=axioms:st=2.0:i=358:si=on:rtra=on_0 on DTF2THF_12192 for (2988ds/358Mi)
% 10.30/2.80 % (12655)Instruction limit reached!
% 10.30/2.80 % (12655)------------------------------
% 10.30/2.80 % (12655)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.30/2.80 % (12655)Termination reason: Unknown
% 10.30/2.80 % (12655)Termination phase: Saturation
% 10.30/2.80
% 10.30/2.80 % (12655)Memory used [KB]: 11897
% 10.30/2.80 % (12655)Time elapsed: 0.319 s
% 10.30/2.80 % (12655)Instructions burned: 1021 (million)
% 10.30/2.80 % (12655)------------------------------
% 10.30/2.80 % (12655)------------------------------
% 10.30/2.81 % (12768)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_12192 for (2987ds/31Mi)
% 10.30/2.82 % (12768)Instruction limit reached!
% 10.30/2.82 % (12768)------------------------------
% 10.30/2.82 % (12768)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.30/2.82 % (12768)Termination reason: Unknown
% 10.30/2.82 % (12768)Termination phase: shuffling
% 10.30/2.82
% 10.30/2.82 % (12768)Memory used [KB]: 1918
% 10.30/2.82 % (12768)Time elapsed: 0.010 s
% 10.30/2.82 % (12768)Instructions burned: 31 (million)
% 10.30/2.82 % (12768)------------------------------
% 10.30/2.82 % (12768)------------------------------
% 10.30/2.82 % (12769)lrs+10_1:1_cnfonf=lazy_simp:sgt=5:sos=on:ss=axioms:i=279:si=on:rtra=on_0 on DTF2THF_12192 for (2987ds/279Mi)
% 10.30/2.85 % (12762)Instruction limit reached!
% 10.30/2.85 % (12762)------------------------------
% 10.30/2.85 % (12762)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.30/2.85 % (12762)Termination reason: Unknown
% 10.30/2.85 % (12762)Termination phase: Saturation
% 10.30/2.85
% 10.30/2.85 % (12762)Memory used [KB]: 8443
% 10.30/2.85 % (12762)Time elapsed: 0.100 s
% 10.30/2.85 % (12762)Instructions burned: 359 (million)
% 10.30/2.85 % (12762)------------------------------
% 10.30/2.85 % (12762)------------------------------
% 10.30/2.86 % (12770)lrs+10_8:1_au=on:fde=none:slsq=on:i=258:si=on:rtra=on_0 on DTF2THF_12192 for (2987ds/258Mi)
% 10.30/2.89 % (12769)Instruction limit reached!
% 10.30/2.89 % (12769)------------------------------
% 10.30/2.89 % (12769)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.30/2.89 % (12769)Termination reason: Unknown
% 10.30/2.89 % (12769)Termination phase: Saturation
% 10.30/2.89
% 10.30/2.89 % (12769)Memory used [KB]: 7291
% 10.30/2.89 % (12769)Time elapsed: 0.070 s
% 10.30/2.89 % (12769)Instructions burned: 279 (million)
% 10.30/2.89 % (12769)------------------------------
% 10.30/2.89 % (12769)------------------------------
% 10.83/2.90 % (12771)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_12192 for (2986ds/26Mi)
% 10.83/2.91 % (12771)Instruction limit reached!
% 10.83/2.91 % (12771)------------------------------
% 10.83/2.91 % (12771)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.83/2.91 % (12771)Termination reason: Unknown
% 10.83/2.91 % (12771)Termination phase: shuffling
% 10.83/2.91
% 10.83/2.91 % (12771)Memory used [KB]: 2046
% 10.83/2.91 % (12771)Time elapsed: 0.009 s
% 10.83/2.91 % (12771)Instructions burned: 28 (million)
% 10.83/2.91 % (12771)------------------------------
% 10.83/2.91 % (12771)------------------------------
% 10.83/2.92 % (12772)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_be_gen:ntd=on:pe=on:i=995:si=on:rtra=on_0 on DTF2THF_12192 for (2986ds/995Mi)
% 10.83/2.93 % (12770)Instruction limit reached!
% 10.83/2.93 % (12770)------------------------------
% 10.83/2.93 % (12770)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.83/2.93 % (12770)Termination reason: Unknown
% 10.83/2.93 % (12770)Termination phase: Saturation
% 10.83/2.93
% 10.83/2.93 % (12770)Memory used [KB]: 8315
% 10.83/2.93 % (12770)Time elapsed: 0.070 s
% 10.83/2.93 % (12770)Instructions burned: 260 (million)
% 10.83/2.93 % (12770)------------------------------
% 10.83/2.93 % (12770)------------------------------
% 10.83/2.94 % (12773)dis+10_1:8_amm=sco:cbe=off:hflw=5:hfsq=on:hfsqc=14:hfsql=off:hfsqr=4,1:hud=5:ile=on:lma=on:nm=2:ntd=on:prag=on:sac=on:sp=frequency:i=373:si=on:rtra=on_0 on DTF2THF_12192 for (2986ds/373Mi)
% 13.20/3.03 % (12684)Instruction limit reached!
% 13.20/3.03 % (12684)------------------------------
% 13.20/3.03 % (12684)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.20/3.03 % (12684)Termination reason: Unknown
% 13.20/3.03 % (12684)Termination phase: Saturation
% 13.20/3.03
% 13.20/3.03 % (12684)Memory used [KB]: 10234
% 13.20/3.03 % (12684)Time elapsed: 0.481 s
% 13.20/3.03 % (12684)Instructions burned: 1542 (million)
% 13.20/3.03 % (12684)------------------------------
% 13.20/3.03 % (12684)------------------------------
% 13.20/3.04 % (12774)lrs+1010_1:128_au=on:cnfonf=lazy_pi_sigma_gen:hflw=4:hfsq=on:hfsqc=20:hfsql=off:i=253:si=on:rtra=on_0 on DTF2THF_12192 for (2985ds/253Mi)
% 13.20/3.04 % (12773)Instruction limit reached!
% 13.20/3.04 % (12773)------------------------------
% 13.20/3.04 % (12773)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.20/3.04 % (12773)Termination reason: Unknown
% 13.20/3.04 % (12773)Termination phase: Saturation
% 13.20/3.04
% 13.20/3.04 % (12773)Memory used [KB]: 8571
% 13.20/3.04 % (12773)Time elapsed: 0.107 s
% 13.20/3.04 % (12773)Instructions burned: 375 (million)
% 13.20/3.04 % (12773)------------------------------
% 13.20/3.04 % (12773)------------------------------
% 13.20/3.05 % (12775)lrs+1002_1:14_au=on:bd=off:e2e=on:fde=unused:sac=on:sos=on:i=832:si=on:rtra=on_0 on DTF2THF_12192 for (2984ds/832Mi)
% 13.20/3.11 % (12774)Instruction limit reached!
% 13.20/3.11 % (12774)------------------------------
% 13.20/3.11 % (12774)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.20/3.11 % (12774)Termination reason: Unknown
% 13.20/3.11 % (12774)Termination phase: Saturation
% 13.20/3.11
% 13.20/3.11 % (12774)Memory used [KB]: 7291
% 13.20/3.11 % (12774)Time elapsed: 0.067 s
% 13.20/3.11 % (12774)Instructions burned: 253 (million)
% 13.20/3.11 % (12774)------------------------------
% 13.20/3.11 % (12774)------------------------------
% 13.85/3.12 % (12776)lrs+1010_1:1_chr=on:cnfonf=lazy_not_be_gen:gs=on:hud=5:plsq=on:plsqc=1:plsqr=5,1:prag=on:rawr=on:i=668:si=on:rtra=on_0 on DTF2THF_12192 for (2984ds/668Mi)
% 13.85/3.18 % (12772)Instruction limit reached!
% 13.85/3.18 % (12772)------------------------------
% 13.85/3.18 % (12772)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.85/3.18 % (12772)Termination reason: Unknown
% 13.85/3.18 % (12772)Termination phase: Saturation
% 13.85/3.18
% 13.85/3.18 % (12772)Memory used [KB]: 13048
% 13.85/3.18 % (12772)Time elapsed: 0.258 s
% 13.85/3.18 % (12772)Instructions burned: 997 (million)
% 13.85/3.18 % (12772)------------------------------
% 13.85/3.18 % (12772)------------------------------
% 13.85/3.19 % (12777)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=40:si=on:rtra=on_0 on DTF2THF_12192 for (2983ds/40Mi)
% 14.28/3.20 % (12777)Instruction limit reached!
% 14.28/3.20 % (12777)------------------------------
% 14.28/3.20 % (12777)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.28/3.20 % (12777)Termination reason: Unknown
% 14.28/3.20 % (12777)Termination phase: shuffling
% 14.28/3.20
% 14.28/3.20 % (12777)Memory used [KB]: 2302
% 14.28/3.20 % (12777)Time elapsed: 0.012 s
% 14.28/3.20 % (12777)Instructions burned: 40 (million)
% 14.28/3.20 % (12777)------------------------------
% 14.28/3.20 % (12777)------------------------------
% 14.28/3.20 % (12707)Instruction limit reached!
% 14.28/3.20 % (12707)------------------------------
% 14.28/3.20 % (12707)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.28/3.20 % (12707)Termination reason: Unknown
% 14.28/3.20 % (12707)Termination phase: Saturation
% 14.28/3.20
% 14.28/3.20 % (12707)Memory used [KB]: 5756
% 14.28/3.20 % (12707)Time elapsed: 0.591 s
% 14.28/3.20 % (12707)Instructions burned: 1915 (million)
% 14.28/3.20 % (12707)------------------------------
% 14.28/3.20 % (12707)------------------------------
% 14.28/3.21 % (12778)lrs+1010_1:1_cnfonf=lazy_pi_sigma_gen:fd=off:ntd=on:sd=1:sgt=32:sos=on:ss=axioms:i=294:si=on:rtra=on_0 on DTF2THF_12192 for (2983ds/294Mi)
% 14.28/3.21 % (12779)dis+1010_1:1_avsq=on:fe=off:fsr=off:hud=10:plsq=on:plsqr=32,1:prag=on:i=187:si=on:rtra=on_0 on DTF2THF_12192 for (2983ds/187Mi)
% 14.28/3.26 % (12779)Instruction limit reached!
% 14.28/3.26 % (12779)------------------------------
% 14.28/3.26 % (12779)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.28/3.26 % (12779)Termination reason: Unknown
% 14.28/3.26 % (12779)Termination phase: Saturation
% 14.28/3.26
% 14.28/3.26 % (12779)Memory used [KB]: 7547
% 14.28/3.26 % (12779)Time elapsed: 0.051 s
% 14.28/3.26 % (12779)Instructions burned: 190 (million)
% 14.28/3.26 % (12779)------------------------------
% 14.28/3.26 % (12779)------------------------------
% 14.28/3.27 % (12775)First to succeed.
% 14.28/3.27 % (12780)lrs+1010_2:3_au=on:fsr=off:sd=4:ss=axioms:st=1.5:i=371:si=on:rtra=on_0 on DTF2THF_12192 for (2982ds/371Mi)
% 14.28/3.27 % (12775)Refutation found. Thanks to Tanya!
% 14.28/3.27 % SZS status Theorem for DTF2THF_12192
% 14.28/3.27 % SZS output start Proof for DTF2THF_12192
% 14.28/3.27 thf(func_def_0, type, in: $i > $i > $o).
% 14.28/3.27 thf(func_def_1, type, exu: ($i > $o) > $o).
% 14.28/3.27 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 14.28/3.27 thf(func_def_8, type, powerset: $i > $i).
% 14.28/3.27 thf(func_def_10, type, setunion: $i > $i).
% 14.28/3.27 thf(func_def_19, type, descr: ($i > $o) > $i).
% 14.28/3.27 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 14.28/3.27 thf(func_def_26, type, prop2set: $o > $i).
% 14.28/3.27 thf(func_def_36, type, nonempty: $i > $o).
% 14.28/3.27 thf(func_def_69, type, set2prop: $i > $o).
% 14.28/3.27 thf(func_def_88, type, subset: $i > $i > $o).
% 14.28/3.27 thf(func_def_89, type, disjoint: $i > $i > $o).
% 14.28/3.27 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 14.28/3.27 thf(func_def_114, type, binunion: $i > $i > $i).
% 14.28/3.27 thf(func_def_122, type, binintersect: $i > $i > $i).
% 14.28/3.27 thf(func_def_135, type, regular: $i > $o).
% 14.28/3.27 thf(func_def_136, type, setminus: $i > $i > $i).
% 14.28/3.27 thf(func_def_147, type, symdiff: $i > $i > $i).
% 14.28/3.27 thf(func_def_153, type, iskpair: $i > $o).
% 14.28/3.27 thf(func_def_158, type, kpair: $i > $i > $i).
% 14.28/3.27 thf(func_def_160, type, cartprod: $i > $i > $i).
% 14.28/3.27 thf(func_def_177, type, singleton: $i > $o).
% 14.28/3.27 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 14.28/3.27 thf(func_def_184, type, atmost1p: $i > $o).
% 14.28/3.27 thf(func_def_185, type, atleast2p: $i > $o).
% 14.28/3.27 thf(func_def_186, type, atmost2p: $i > $o).
% 14.28/3.27 thf(func_def_187, type, upairsetp: $i > $o).
% 14.28/3.27 thf(func_def_191, type, kfst: $i > $i).
% 14.28/3.27 thf(func_def_203, type, ksnd: $i > $i).
% 14.28/3.27 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 14.28/3.27 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 14.28/3.27 thf(func_def_222, type, func: $i > $i > $i > $o).
% 14.28/3.27 thf(func_def_223, type, funcSet: $i > $i > $i).
% 14.28/3.27 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 14.28/3.27 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 14.28/3.27 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 14.28/3.27 thf(func_def_287, type, sP2: $i > $i > $i > $i > $o).
% 14.28/3.27 thf(func_def_290, type, sP5: $i > $i > $o).
% 14.28/3.27 thf(func_def_291, type, sP6: $i > $o).
% 14.28/3.27 thf(func_def_292, type, sP7: $i > $i > $o).
% 14.28/3.27 thf(func_def_293, type, sP8: $i > $i > $o).
% 14.28/3.27 thf(func_def_294, type, sP9: $i > $i > $i > $o).
% 14.28/3.27 thf(func_def_296, type, sP11: $i > $o > $i > $i > $o).
% 14.28/3.27 thf(func_def_303, type, sK18: $i > $o).
% 14.28/3.27 thf(func_def_306, type, sK21: $i > ($i > $o) > $i).
% 14.28/3.27 thf(func_def_307, type, sK22: $i > ($i > $o) > $i).
% 14.28/3.27 thf(func_def_323, type, sK38: ($i > $o) > $i).
% 14.28/3.27 thf(func_def_324, type, sK39: ($i > $o) > $i).
% 14.28/3.27 thf(func_def_325, type, sK40: $i > $o).
% 14.28/3.27 thf(func_def_328, type, sK43: $i > $i > $i > $i > $i).
% 14.28/3.27 thf(func_def_329, type, sK44: $i > $i > $i > $i > $i).
% 14.28/3.27 thf(func_def_338, type, sK53: $i > $o).
% 14.28/3.27 thf(func_def_340, type, sK55: ($i > $o) > $i > $i).
% 14.28/3.27 thf(func_def_341, type, sK56: ($i > $o) > $i > $i).
% 14.28/3.27 thf(func_def_343, type, sK58: $i > $o).
% 14.28/3.27 thf(func_def_359, type, sK74: $i > $i > $o).
% 14.28/3.27 thf(func_def_361, type, sK76: $i > $o).
% 14.28/3.27 thf(func_def_363, type, sK78: ($i > $o) > $i > $i).
% 14.28/3.27 thf(func_def_364, type, sK79: $i > ($i > $i) > $i > $i).
% 14.28/3.27 thf(func_def_366, type, sK81: $i > $i).
% 14.28/3.27 thf(func_def_369, type, sK84: $i > $i > $i > $i).
% 14.28/3.27 thf(func_def_379, type, sK94: $i > $i > $i > $i > $i).
% 14.28/3.27 thf(func_def_380, type, sK95: $i > $i > $i > $i > $i).
% 14.28/3.27 thf(func_def_381, type, sK96: $i > $i > $i > $i > $i).
% 14.28/3.27 thf(func_def_382, type, sK97: $i > $i > $i > $i > $i).
% 14.28/3.27 thf(func_def_388, type, sK103: ($i > $o) > $i > $i > $i > $i).
% 14.28/3.27 thf(func_def_389, type, sK104: ($i > $o) > $i > $i > $i > $i).
% 14.28/3.27 thf(func_def_393, type, sK108: $i > $o).
% 14.28/3.27 thf(func_def_396, type, sK111: $i > $i).
% 14.28/3.27 thf(func_def_399, type, sK114: $i > $o).
% 14.28/3.27 thf(func_def_405, type, sK120: $i > ($i > $i) > $i > $i).
% 14.28/3.27 thf(func_def_407, type, sK122: $i > $i).
% 14.28/3.27 thf(func_def_409, type, sK124: $i > $i).
% 14.28/3.27 thf(func_def_417, type, sK132: $i > $o).
% 14.28/3.27 thf(func_def_432, type, sK147: $i > $o).
% 14.28/3.27 thf(func_def_433, type, sK148: ($i > $o) > $i > $i).
% 14.28/3.27 thf(func_def_441, type, sK156: $i > $i).
% 14.28/3.27 thf(func_def_473, type, sK188: ($i > $o) > $i > $i).
% 14.28/3.27 thf(func_def_475, type, sK190: $i > $o).
% 14.28/3.27 thf(func_def_477, type, sK192: $i > $o).
% 14.28/3.27 thf(func_def_478, type, sK193: ($i > $o) > $i).
% 14.28/3.27 thf(func_def_479, type, sK194: $i > $i).
% 14.28/3.27 thf(func_def_482, type, sK197: $i > $i > ($i > $i) > $i).
% 14.28/3.27 thf(func_def_497, type, sK212: $i > $i).
% 14.28/3.27 thf(func_def_500, type, sK215: $i > $i).
% 14.28/3.27 thf(func_def_504, type, sK219: $i > $i > ($i > $i) > $i).
% 14.28/3.27 thf(func_def_511, type, sK226: $i > $o).
% 14.28/3.27 thf(func_def_514, type, sK229: $i > ($i > $o) > $i).
% 14.28/3.27 thf(func_def_517, type, sK232: $i > $i > $i).
% 14.28/3.27 thf(func_def_521, type, sK236: ($i > $o) > $i).
% 14.28/3.27 thf(func_def_522, type, sK237: $i > $o).
% 14.28/3.27 thf(func_def_523, type, sK238: $i > $i).
% 14.28/3.27 thf(func_def_524, type, sK239: ($i > $o) > $i > $i).
% 14.28/3.27 thf(func_def_526, type, sK241: $i > $o).
% 14.28/3.27 thf(func_def_532, type, sK247: $i > $o).
% 14.28/3.27 thf(func_def_537, type, sK252: $i > $i > $o).
% 14.28/3.27 thf(func_def_553, type, sK268: $i > $o).
% 14.28/3.27 thf(func_def_555, type, sK270: ($i > $o) > $i > $i > $i > $i).
% 14.28/3.27 thf(func_def_556, type, sK271: ($i > $o) > $i > $i > $i > $i).
% 14.28/3.27 thf(func_def_565, type, sK280: $i > $i).
% 14.28/3.27 thf(func_def_574, type, sK289: $i > $i > $i).
% 14.28/3.27 thf(func_def_581, type, sK296: $i > $i > $i > $i).
% 14.28/3.27 thf(func_def_582, type, sK297: $i > $i > $i > $i).
% 14.28/3.27 thf(func_def_589, type, sK304: $i > $o).
% 14.28/3.27 thf(func_def_605, type, sK320: $i > $i > $i).
% 14.28/3.27 thf(func_def_606, type, sK321: $i > $i > $i).
% 14.28/3.27 thf(func_def_613, type, sK328: $i > $o).
% 14.28/3.27 thf(func_def_624, type, sK339: $i > ($i > $o) > $i).
% 14.28/3.27 thf(func_def_625, type, sK340: $i > $o).
% 14.28/3.27 thf(func_def_657, type, sK372: $i > $i > $o).
% 14.28/3.27 thf(func_def_668, type, sK383: ($i > $o) > ($i > $o) > $i > $i > $i).
% 14.28/3.27 thf(func_def_669, type, sK384: ($i > $o) > ($i > $o) > $i > $i > $i).
% 14.28/3.27 thf(func_def_672, type, sK387: $i > $o).
% 14.28/3.27 thf(func_def_673, type, sK388: $i > $o).
% 14.28/3.27 thf(func_def_680, type, sK395: $i > $i > ($i > $i) > $i).
% 14.28/3.27 thf(func_def_681, type, sK396: $i > $i).
% 14.28/3.27 thf(func_def_696, type, sK411: ($i > $o) > $i).
% 14.28/3.27 thf(func_def_697, type, sK412: $i > $o).
% 14.28/3.27 thf(func_def_698, type, sK413: $i > $i).
% 14.28/3.27 thf(func_def_699, type, sK414: $i > $i > $o).
% 14.28/3.27 thf(func_def_701, type, sK416: $i > $i).
% 14.28/3.27 thf(func_def_702, type, sK417: $i > $i).
% 14.28/3.27 thf(func_def_703, type, sK418: $i > ($i > $i > $o) > $i).
% 14.28/3.27 thf(func_def_704, type, sK419: $i > $i > ($i > $i > $o) > $i).
% 14.28/3.27 thf(func_def_705, type, sK420: $i > ($i > $i > $o) > $i).
% 14.28/3.27 thf(func_def_709, type, sK424: $i > ($i > $o) > $i > $i).
% 14.28/3.27 thf(func_def_711, type, sK426: $i > $o).
% 14.28/3.27 thf(func_def_714, type, sK429: $i > $o).
% 14.28/3.27 thf(func_def_716, type, sK431: ($i > $o) > $i > $i).
% 14.28/3.27 thf(func_def_723, type, sK438: $i > $i > $i).
% 14.28/3.27 thf(func_def_740, type, sK455: $i > $o).
% 14.28/3.27 thf(func_def_748, type, sK463: ($i > $o) > ($i > $o) > $i).
% 14.28/3.27 thf(func_def_749, type, sK464: ($i > $o) > ($i > $o) > $i).
% 14.28/3.27 thf(func_def_750, type, sK465: $i > $o).
% 14.28/3.27 thf(func_def_751, type, sK466: $i > $o).
% 14.28/3.27 thf(func_def_758, type, sK473: $i > $i > $i > $i > $i).
% 14.28/3.27 thf(func_def_759, type, sK474: $i > $i > $i).
% 14.28/3.27 thf(func_def_760, type, sK475: $i > $i > $i).
% 14.28/3.27 thf(func_def_761, type, sK476: $i > $i > $i).
% 14.28/3.27 thf(func_def_762, type, sK477: $i > $i > $i).
% 14.28/3.27 thf(func_def_763, type, sK478: $i > $i > $i).
% 14.28/3.27 thf(func_def_764, type, sK479: $i > $i > $i > $i).
% 14.28/3.27 thf(func_def_765, type, sK480: $i > $i).
% 14.28/3.27 thf(func_def_766, type, sK481: $i > $i).
% 14.28/3.27 thf(func_def_767, type, sK482: $i > $i).
% 14.28/3.27 thf(func_def_768, type, sK483: $i > $i).
% 14.28/3.27 thf(func_def_769, type, sK484: $i > $i > $i > $i).
% 14.28/3.27 thf(func_def_770, type, sK485: $i > $i > $i > $i).
% 14.28/3.27 thf(func_def_771, type, sK486: $i > $i > $i).
% 14.28/3.27 thf(func_def_772, type, sK487: $i > $i > $i).
% 14.28/3.27 thf(func_def_773, type, sK488: $i > $i > $i).
% 14.28/3.27 thf(func_def_774, type, sK489: $i > $i).
% 14.28/3.27 thf(func_def_776, type, sK491: $i > $i).
% 14.28/3.27 thf(func_def_777, type, sK492: $i > $i).
% 14.28/3.27 thf(func_def_785, type, sK500: $i > $o).
% 14.28/3.27 thf(func_def_795, type, sK510: $i > $i > $o).
% 14.28/3.27 thf(func_def_803, type, sK518: $i > $i > $i > $i).
% 14.28/3.27 thf(func_def_807, type, sK522: $i > $i > $i > $i).
% 14.28/3.27 thf(func_def_808, type, sK523: $i > ($i > $o) > $i).
% 14.28/3.27 thf(func_def_809, type, sK524: $i > $o).
% 14.28/3.27 thf(func_def_815, type, sK530: ($i > $o) > ($i > $o) > $i).
% 14.28/3.27 thf(func_def_816, type, sK531: ($i > $o) > ($i > $o) > $i).
% 14.28/3.27 thf(func_def_817, type, sK532: $i > $o).
% 14.28/3.27 thf(func_def_818, type, sK533: $i > $o).
% 14.28/3.27 thf(func_def_826, type, sK541: $i > $i > $o).
% 14.28/3.27 thf(func_def_846, type, sK561: $i > $i > $i).
% 14.28/3.27 thf(func_def_852, type, sK567: $i > $i).
% 14.28/3.27 thf(func_def_853, type, sK568: $i > $i).
% 14.28/3.27 thf(func_def_860, type, sK575: $i > ($i > $o) > $i).
% 14.28/3.27 thf(func_def_861, type, sK576: $i > $o).
% 14.28/3.27 thf(func_def_878, type, sK593: $i > $i > $i > $i > $i).
% 14.28/3.27 thf(func_def_883, type, sK598: $i > $i > $i).
% 14.28/3.27 thf(func_def_887, type, sK602: $i > $i > $o).
% 14.28/3.27 thf(func_def_898, type, sK613: $i > $i > $i).
% 14.28/3.27 thf(func_def_926, type, sK641: $i > $o).
% 14.28/3.27 thf(func_def_932, type, sK647: $i > $o).
% 14.28/3.27 thf(func_def_939, type, sK654: $i > $o).
% 14.28/3.27 thf(func_def_942, type, sK657: $i > $i > $o).
% 14.28/3.27 thf(func_def_951, type, sK666: $i > $o).
% 14.28/3.27 thf(func_def_954, type, sK669: $i > $i > $i).
% 14.28/3.27 thf(func_def_982, type, sK697: $i > $i > $i > $i > $i).
% 14.28/3.27 thf(func_def_983, type, sK698: $i > $i > $i).
% 14.28/3.27 thf(func_def_991, type, sK706: $i > $o).
% 14.28/3.27 thf(func_def_999, type, sK714: $o > $i > $i > $i).
% 14.28/3.27 thf(func_def_1017, type, sK732: $i > $i).
% 14.28/3.27 thf(func_def_1020, type, sK735: $i > $i > ($i > $i) > $i).
% 14.28/3.27 thf(func_def_1022, type, ph737: !>[X0: $tType]:(X0)).
% 14.28/3.27 thf(f4679,plain,(
% 14.28/3.27 $false),
% 14.28/3.27 inference(subsumption_resolution,[status(thm)],[f4678,f2642])).
% 14.28/3.27 thf(f2642,plain,(
% 14.28/3.27 (subsetI1 = $true)),
% 14.28/3.27 inference(cnf_transformation,[status(thm)],[f1550])).
% 14.28/3.27 thf(f1550,plain,(
% 14.28/3.27 (subPowSU = $true) & (setunionsingleton2 = $true) & (setadjoinSub2 = $true) & (exuE2 = $true) & (ksndpairEq = $true) & (setadjoin__Cong = $true) & (cartprodfstin = $true) & (notinsingleton = $true) & (setextT = $true) & (quantDeMorgan1 = $true) & ((((in @ sK345 @ (powerset @ sK347)) != $true) & ! [X3 : $i] : (($true = (in @ X3 @ sK347)) | ($true != (in @ X3 @ sK346)) | ($true != (in @ X3 @ sK345))) & ($true = (in @ sK347 @ (powerset @ sK346)))) & ($true = (in @ sK345 @ (powerset @ sK346)))) & (bs114d = $true) & (exuEu = $true) & (emptysetE = $true) & (subbreln = $true) & (funcGraphProp3 = $true) & (iffalseProp2 = $true) & (symdiffIneg1 = $true) & (subset2powerset = $true) & (funcGraphProp1 = $true) & (singletoninpowunion = $true) & (subsetTI = $true) & (setminusIRneg = $true) & (singletonsswitch = $true) & (quantDeMorgan4 = $true) & (lamProp = $true) & (setoftrueEq = $true) & (binunionE = $true) & (setadjoinAx = $true) & (setminusEL = $true) & (setminusLsub = $true) & (sepSubset = $true) & (binintersectSubset4 = $true) & (subsetRefl = $true) & (emptysetAx = $true) & (powersetsubset = $true) & (wellorderingAx = $true) & (quantDeMorgan3 = $true) & (dpsetconstrEL2 = $true) & (beta2 = $true) & (in__Cong = $true) & (exuI3 = $true) & (setukpairIL = $true) & (binintersectSubset3 = $true) & (ap2p = $true) & (cartprodpairsurjEq = $true) & (funcextLem = $true) & (ex1E2 = $true) & (binintersectLsub = $true) & (emptyInPowerset = $true) & (quantDeMorgan2 = $true) & (dsetconstr__Cong = $true) & (setukpairIR = $true) & (omegaIndAx = $true) & (setadjoinIL = $true) & (exuE1 = $true) & (ubforcartprodlem1 = $true) & (setminusSubset1 = $true) & (emptysetimpfalse = $true) & (singletonsuniq = $true) & (powersetI1 = $true) & (notequalI2 = $true) & (descr__Cong = $true) & (dsetconstrEL = $true) & (brelnall2 = $true) & (ksndsingleton = $true) & (funcImageSingleton = $true) & (ex1I2 = $true) & (kpairiskpair = $true) & (binintersectSubset1 = $true) & (dpsetconstrERa = $true) & (binintersectSubset2 = $true) & (setukpairinjL = $true) & (singletoninpowerset = $true) & (binintersectT_lem = $true) & (inCongP = $true) & (binintersectER = $true) & (sepInPowerset = $true) & (eqinunit = $true) & (emptyI = $true) & (ubforcartprodlem3 = $true) & (replAx = $true) & (theeq = $true) & (binintersectSubset5 = $true) & (upairsetIR = $true) & (dpsetconstrER = $true) & (dpsetconstrSub = $true) & (symdiffIneg2 = $true) & (setunion__Cong = $true) & (notequalI1 = $true) & (upairequniteq = $true) & (eqimpsubset1 = $true) & (kfstpairEq = $true) & (apProp = $true) & (funcinfuncset = $true) & (binunionIR = $true) & (singletonsubset = $true) & (cartprodsndpairEq = $true) & (funcext = $true) & (ex1I = $true) & (notinemptyset = $true) & (noeltsimpempty = $true) & (setadjoinSub = $true) & (subsetTrans = $true) & (iftrueorfalse = $true) & (dpsetconstrI = $true) & (setukpairinjR1 = $true) & (binunionIL = $true) & (upairsetIL = $true) & (binintersectEL = $true) & (binunionEcases = $true) & (eta1 = $true) & (complementT_lem = $true) & (setunionsingleton1 = $true) & (beta1 = $true) & (iftrueProp1 = $true) & (powersetE1 = $true) & (binintersectRsub = $true) & (powerset__Cong = $true) & (symdiffI1 = $true) & (setukpairinjR = $true) & (prop2set2propI = $true) & (setext = $true) & (dsetconstrI = $true) & (setukpairinjR12 = $true) & (setminusILneg = $true) & (cartprodpairin = $true) & (exuE3e = $true) & (subsetE2 = $true) & (setadjoinOr = $true) & (powersetAx = $true) & (nonemptyI1 = $true) & (emptysetsubset = $true) & (funcGraphProp2 = $true) & (omegaSAx = $true) & (setukpairinjL1 = $true) & (setminusER = $true) & (cartprodpairmemEL = $true) & (powersetI = $true) & (setukpairinjR2 = $true) & (emptyinunitempty = $true) & (notsubsetI = $true) & (inPowerset = $true) & (notdallE = $true) & (iftrue = $true) & (upairset2E = $true) & (setextAx = $true) & (app = $true) & (infuncsetfunc = $true) & (ap2apEq2 = $true) & (eta2 = $true) & (subsetemptysetimpeq = $true) & (lam2p = $true) & (emptyset__Cong = $true) & (symdiffI2 = $true) & (disjointsetsI1 = $true) & (subsetI2 = $true) & (vacuousDall = $true) & (emptyinPowerset = $true) & (symdiffE = $true) & (omega__Cong = $true) & (setminusSubset2 = $true) & (lam2lamEq = $true) & (powersetE = $true) & (dpsetconstrEL1 = $true) & (uniqinunit = $true) & (funcGraphProp4 = $true) & (foundationAx = $true) & (setunionE = $true) & (setextsub = $true) & (subsetI1 = $true) & (ifp = $true) & (upairsubunion = $true) & (exu__Cong = $true) & (setadjoinIR = $true) & (theprop = $true) & (eqbreln = $true) & (setminusELneg = $true) & (binunionT_lem = $true) & (ubforcartprodlem2 = $true) & (ap2apEq1 = $true) & (nonemptyI = $true) & (exuI1 = $true) & (ifSingleton = $true) & (singletonprop = $true) & (exuI2 = $true) & (iftrueProp2 = $true) & (binintersectI = $true) & (cartprodmempaircEq = $true) & (upairinpowunion = $true) & (setminusERneg = $true) & (nonemptyImpWitness = $true) & (emptyE1 = $true) & (eqimpsubset2 = $true) & (upairset2IR = $true) & (exuE3u = $true) & (setminusT_lem = $true) & (setOfPairsIsBReln = $true) & (brelnall1 = $true) & (prop2setI = $true) & (setbeta = $true) & (dsetconstrER = $true) & (cartprodmempair1 = $true) & (setukpairinjR11 = $true) & (setunionE2 = $true) & (kfstsingleton = $true) & (descrp = $true) & (powersetT_lem = $true) & (secondinupair = $true) & (iffalse = $true) & (nonemptyE1 = $true) & (lamp = $true) & (upairsetE = $true) & (funcext2 = $true) & (kpairp = $true) & (prop2setE = $true) & (setunionI = $true) & (ex1E1 = $true) & (cartprodmempair = $true) & (kpairsurjEq = $true) & (omega0Ax = $true) & (setunionsingleton = $true) & (setunionAx = $true) & (cartprodsndin = $true) & (cartprodpairmemER = $true) & (iffalseProp1 = $true) & (binunionRsub = $true) & (setminusI = $true) & (cartprodfstpairEq = $true) & (binunionLsub = $true) & (subsetE = $true) & (setukpairinjL2 = $true) & (notdexE = $true) & (setadjoinE = $true)),
% 14.28/3.27 inference(skolemisation,[status(esa),new_symbols(skolem,[sK345,sK346,sK347])],[f786,f1549,f1548])).
% 14.28/3.27 thf(f1548,plain,(
% 14.28/3.27 ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (($true != (in @ X0 @ (powerset @ X2))) & ! [X3 : $i] : (($true = (in @ X3 @ X2)) | ($true != (in @ X3 @ X1)) | ($true != (in @ X3 @ X0))) & ($true = (in @ X2 @ (powerset @ X1)))) & ($true = (in @ X0 @ (powerset @ X1)))) => (? [X2 : $i] : (($true != (in @ sK345 @ (powerset @ X2))) & ! [X3 : $i] : (($true = (in @ X3 @ X2)) | ($true != (in @ X3 @ sK346)) | ($true != (in @ X3 @ sK345))) & ($true = (in @ X2 @ (powerset @ sK346)))) & ($true = (in @ sK345 @ (powerset @ sK346))))),
% 14.28/3.27 introduced(definition,[],[choice_axiom])).
% 14.28/3.27 thf(f1549,plain,(
% 14.28/3.27 ? [X2 : $i] : (($true != (in @ sK345 @ (powerset @ X2))) & ! [X3 : $i] : (($true = (in @ X3 @ X2)) | ($true != (in @ X3 @ sK346)) | ($true != (in @ X3 @ sK345))) & ($true = (in @ X2 @ (powerset @ sK346)))) => (((in @ sK345 @ (powerset @ sK347)) != $true) & ! [X3 : $i] : (($true = (in @ X3 @ sK347)) | ($true != (in @ X3 @ sK346)) | ($true != (in @ X3 @ sK345))) & ($true = (in @ sK347 @ (powerset @ sK346))))),
% 14.28/3.27 introduced(definition,[],[choice_axiom])).
% 14.28/3.27 thf(f786,plain,(
% 14.28/3.27 (subPowSU = $true) & (setunionsingleton2 = $true) & (setadjoinSub2 = $true) & (exuE2 = $true) & (ksndpairEq = $true) & (setadjoin__Cong = $true) & (cartprodfstin = $true) & (notinsingleton = $true) & (setextT = $true) & (quantDeMorgan1 = $true) & ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (($true != (in @ X0 @ (powerset @ X2))) & ! [X3 : $i] : (($true = (in @ X3 @ X2)) | ($true != (in @ X3 @ X1)) | ($true != (in @ X3 @ X0))) & ($true = (in @ X2 @ (powerset @ X1)))) & ($true = (in @ X0 @ (powerset @ X1)))) & (bs114d = $true) & (exuEu = $true) & (emptysetE = $true) & (subbreln = $true) & (funcGraphProp3 = $true) & (iffalseProp2 = $true) & (symdiffIneg1 = $true) & (subset2powerset = $true) & (funcGraphProp1 = $true) & (singletoninpowunion = $true) & (subsetTI = $true) & (setminusIRneg = $true) & (singletonsswitch = $true) & (quantDeMorgan4 = $true) & (lamProp = $true) & (setoftrueEq = $true) & (binunionE = $true) & (setadjoinAx = $true) & (setminusEL = $true) & (setminusLsub = $true) & (sepSubset = $true) & (binintersectSubset4 = $true) & (subsetRefl = $true) & (emptysetAx = $true) & (powersetsubset = $true) & (wellorderingAx = $true) & (quantDeMorgan3 = $true) & (dpsetconstrEL2 = $true) & (beta2 = $true) & (in__Cong = $true) & (exuI3 = $true) & (setukpairIL = $true) & (binintersectSubset3 = $true) & (ap2p = $true) & (cartprodpairsurjEq = $true) & (funcextLem = $true) & (ex1E2 = $true) & (binintersectLsub = $true) & (emptyInPowerset = $true) & (quantDeMorgan2 = $true) & (dsetconstr__Cong = $true) & (setukpairIR = $true) & (omegaIndAx = $true) & (setadjoinIL = $true) & (exuE1 = $true) & (ubforcartprodlem1 = $true) & (setminusSubset1 = $true) & (emptysetimpfalse = $true) & (singletonsuniq = $true) & (powersetI1 = $true) & (notequalI2 = $true) & (descr__Cong = $true) & (dsetconstrEL = $true) & (brelnall2 = $true) & (ksndsingleton = $true) & (funcImageSingleton = $true) & (ex1I2 = $true) & (kpairiskpair = $true) & (binintersectSubset1 = $true) & (dpsetconstrERa = $true) & (binintersectSubset2 = $true) & (setukpairinjL = $true) & (singletoninpowerset = $true) & (binintersectT_lem = $true) & (inCongP = $true) & (binintersectER = $true) & (sepInPowerset = $true) & (eqinunit = $true) & (emptyI = $true) & (ubforcartprodlem3 = $true) & (replAx = $true) & (theeq = $true) & (binintersectSubset5 = $true) & (upairsetIR = $true) & (dpsetconstrER = $true) & (dpsetconstrSub = $true) & (symdiffIneg2 = $true) & (setunion__Cong = $true) & (notequalI1 = $true) & (upairequniteq = $true) & (eqimpsubset1 = $true) & (kfstpairEq = $true) & (apProp = $true) & (funcinfuncset = $true) & (binunionIR = $true) & (singletonsubset = $true) & (cartprodsndpairEq = $true) & (funcext = $true) & (ex1I = $true) & (notinemptyset = $true) & (noeltsimpempty = $true) & (setadjoinSub = $true) & (subsetTrans = $true) & (iftrueorfalse = $true) & (dpsetconstrI = $true) & (setukpairinjR1 = $true) & (binunionIL = $true) & (upairsetIL = $true) & (binintersectEL = $true) & (binunionEcases = $true) & (eta1 = $true) & (complementT_lem = $true) & (setunionsingleton1 = $true) & (beta1 = $true) & (iftrueProp1 = $true) & (powersetE1 = $true) & (binintersectRsub = $true) & (powerset__Cong = $true) & (symdiffI1 = $true) & (setukpairinjR = $true) & (prop2set2propI = $true) & (setext = $true) & (dsetconstrI = $true) & (setukpairinjR12 = $true) & (setminusILneg = $true) & (cartprodpairin = $true) & (exuE3e = $true) & (subsetE2 = $true) & (setadjoinOr = $true) & (powersetAx = $true) & (nonemptyI1 = $true) & (emptysetsubset = $true) & (funcGraphProp2 = $true) & (omegaSAx = $true) & (setukpairinjL1 = $true) & (setminusER = $true) & (cartprodpairmemEL = $true) & (powersetI = $true) & (setukpairinjR2 = $true) & (emptyinunitempty = $true) & (notsubsetI = $true) & (inPowerset = $true) & (notdallE = $true) & (iftrue = $true) & (upairset2E = $true) & (setextAx = $true) & (app = $true) & (infuncsetfunc = $true) & (ap2apEq2 = $true) & (eta2 = $true) & (subsetemptysetimpeq = $true) & (lam2p = $true) & (emptyset__Cong = $true) & (symdiffI2 = $true) & (disjointsetsI1 = $true) & (subsetI2 = $true) & (vacuousDall = $true) & (emptyinPowerset = $true) & (symdiffE = $true) & (omega__Cong = $true) & (setminusSubset2 = $true) & (lam2lamEq = $true) & (powersetE = $true) & (dpsetconstrEL1 = $true) & (uniqinunit = $true) & (funcGraphProp4 = $true) & (foundationAx = $true) & (setunionE = $true) & (setextsub = $true) & (subsetI1 = $true) & (ifp = $true) & (upairsubunion = $true) & (exu__Cong = $true) & (setadjoinIR = $true) & (theprop = $true) & (eqbreln = $true) & (setminusELneg = $true) & (binunionT_lem = $true) & (ubforcartprodlem2 = $true) & (ap2apEq1 = $true) & (nonemptyI = $true) & (exuI1 = $true) & (ifSingleton = $true) & (singletonprop = $true) & (exuI2 = $true) & (iftrueProp2 = $true) & (binintersectI = $true) & (cartprodmempaircEq = $true) & (upairinpowunion = $true) & (setminusERneg = $true) & (nonemptyImpWitness = $true) & (emptyE1 = $true) & (eqimpsubset2 = $true) & (upairset2IR = $true) & (exuE3u = $true) & (setminusT_lem = $true) & (setOfPairsIsBReln = $true) & (brelnall1 = $true) & (prop2setI = $true) & (setbeta = $true) & (dsetconstrER = $true) & (cartprodmempair1 = $true) & (setukpairinjR11 = $true) & (setunionE2 = $true) & (kfstsingleton = $true) & (descrp = $true) & (powersetT_lem = $true) & (secondinupair = $true) & (iffalse = $true) & (nonemptyE1 = $true) & (lamp = $true) & (upairsetE = $true) & (funcext2 = $true) & (kpairp = $true) & (prop2setE = $true) & (setunionI = $true) & (ex1E1 = $true) & (cartprodmempair = $true) & (kpairsurjEq = $true) & (omega0Ax = $true) & (setunionsingleton = $true) & (setunionAx = $true) & (cartprodsndin = $true) & (cartprodpairmemER = $true) & (iffalseProp1 = $true) & (binunionRsub = $true) & (setminusI = $true) & (cartprodfstpairEq = $true) & (binunionLsub = $true) & (subsetE = $true) & (setukpairinjL2 = $true) & (notdexE = $true) & (setadjoinE = $true)),
% 14.28/3.27 inference(flattening,[status(thm)],[f785])).
% 14.28/3.27 thf(f785,plain,(
% 14.28/3.27 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (? [X2 : $i] : ((($true != (in @ X0 @ (powerset @ X2))) & ! [X3 : $i] : ((($true = (in @ X3 @ X2)) | ($true != (in @ X3 @ X0))) | ($true != (in @ X3 @ X1)))) & ($true = (in @ X2 @ (powerset @ X1)))) & ($true = (in @ X0 @ (powerset @ X1)))) & (subsetTI = $true)) & (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)),
% 14.28/3.28 inference(ennf_transformation,[status(thm)],[f450])).
% 14.28/3.28 thf(f450,plain,(
% 14.28/3.28 ~((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) => ((subsetTI = $true) => ! [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (powerset @ X1))) => ! [X2 : $i] : (($true = (in @ X2 @ (powerset @ X1))) => (! [X3 : $i] : (($true = (in @ X3 @ X1)) => (($true = (in @ X3 @ X0)) => ($true = (in @ X3 @ X2)))) => ($true = (in @ X0 @ (powerset @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 14.28/3.28 inference(fool_elimination,[status(thm)],[f449])).
% 14.28/3.28 thf(f449,plain,(
% 14.28/3.28 ~(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 => (subsetTI => ! [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))) => (in @ X0 @ (powerset @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 14.28/3.28 inference(rectify,[status(thm)],[f239])).
% 14.28/3.28 thf(f239,negated_conjecture,(
% 14.28/3.28 ~(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 => (subsetTI => ! [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))) => (in @ X11 @ (powerset @ X16))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 14.28/3.28 inference(negated_conjecture,[status(cth)],[f238])).
% 14.28/3.28 thf(f238,conjecture,(
% 14.28/3.28 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 => (subsetTI => ! [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))) => (in @ X11 @ (powerset @ X16)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 14.28/3.28 file('/export/starexec/sandbox/tmp/tmp.rvsH9IYwBY/DTF2THF_12192.p',powersetTI1)).
% 14.28/3.28 thf(f4678,plain,(
% 14.28/3.28 (subsetI1 != $true)),
% 14.28/3.28 inference(subsumption_resolution,[status(thm)],[f4673,f3598])).
% 14.28/3.28 thf(f3598,plain,(
% 14.28/3.28 ($true != (subset @ sK345 @ sK347))),
% 14.28/3.28 inference(subsumption_resolution,[status(thm)],[f3592,f2752])).
% 14.28/3.28 thf(f2752,plain,(
% 14.28/3.28 (powersetI1 = $true)),
% 14.28/3.28 inference(cnf_transformation,[status(thm)],[f1550])).
% 14.28/3.28 thf(f3592,plain,(
% 14.28/3.28 (powersetI1 != $true) | ($true != (subset @ sK345 @ sK347))),
% 14.28/3.28 inference(trivial_inequality_removal,[status(thm)],[f3585])).
% 14.28/3.28 thf(f3585,plain,(
% 14.28/3.28 (powersetI1 != $true) | ($true != $true) | ($true != (subset @ sK345 @ sK347))),
% 14.28/3.28 inference(superposition,[status(thm)],[f2805,f3017])).
% 14.28/3.28 thf(f3017,plain,(
% 14.28/3.28 ( ! [X2 : $i,X3 : $i] : (($true = (in @ X2 @ (powerset @ X3))) | (powersetI1 != $true) | ($true != (subset @ X2 @ X3))) )),
% 14.28/3.28 inference(cnf_transformation,[status(thm)],[f1786])).
% 14.28/3.28 thf(f1786,plain,(
% 14.28/3.28 ((powersetI1 = $true) | (($true != (in @ sK502 @ (powerset @ sK503))) & ($true = (subset @ sK502 @ sK503)))) & (! [X2 : $i,X3 : $i] : (($true = (in @ X2 @ (powerset @ X3))) | ($true != (subset @ X2 @ X3))) | (powersetI1 != $true))),
% 14.28/3.28 inference(skolemisation,[status(esa),new_symbols(skolem,[sK502,sK503])],[f1784,f1785])).
% 14.28/3.28 thf(f1785,plain,(
% 14.28/3.28 ? [X0 : $i,X1 : $i] : (($true != (in @ X0 @ (powerset @ X1))) & ((subset @ X0 @ X1) = $true)) => (($true != (in @ sK502 @ (powerset @ sK503))) & ($true = (subset @ sK502 @ sK503)))),
% 14.28/3.28 introduced(definition,[],[choice_axiom])).
% 14.28/3.28 thf(f1784,plain,(
% 14.28/3.28 ((powersetI1 = $true) | ? [X0 : $i,X1 : $i] : (($true != (in @ X0 @ (powerset @ X1))) & ((subset @ X0 @ X1) = $true))) & (! [X2 : $i,X3 : $i] : (($true = (in @ X2 @ (powerset @ X3))) | ($true != (subset @ X2 @ X3))) | (powersetI1 != $true))),
% 14.28/3.28 inference(rectify,[status(thm)],[f1783])).
% 14.28/3.28 thf(f1783,plain,(
% 14.28/3.28 ((powersetI1 = $true) | ? [X0 : $i,X1 : $i] : (($true != (in @ X0 @ (powerset @ X1))) & ((subset @ X0 @ X1) = $true))) & (! [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (powerset @ X1))) | ((subset @ X0 @ X1) != $true)) | (powersetI1 != $true))),
% 14.28/3.28 inference(nnf_transformation,[status(thm)],[f770])).
% 14.28/3.28 thf(f770,plain,(
% 14.28/3.28 (powersetI1 = $true) <=> ! [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (powerset @ X1))) | ((subset @ X0 @ X1) != $true))),
% 14.28/3.28 inference(ennf_transformation,[status(thm)],[f629])).
% 14.28/3.28 thf(f629,plain,(
% 14.28/3.28 ! [X0 : $i,X1 : $i] : (((subset @ X0 @ X1) = $true) => ($true = (in @ X0 @ (powerset @ X1)))) <=> (powersetI1 = $true)),
% 14.28/3.28 inference(fool_elimination,[status(thm)],[f628])).
% 14.28/3.28 thf(f628,plain,(
% 14.28/3.28 (powersetI1 = ! [X0 : $i,X1 : $i] : ((subset @ X0 @ X1) => (in @ X0 @ (powerset @ X1))))),
% 14.28/3.28 inference(rectify,[status(thm)],[f95])).
% 14.28/3.28 thf(f95,axiom,(
% 14.28/3.28 (powersetI1 = ! [X4 : $i,X3 : $i] : ((subset @ X4 @ X3) => (in @ X4 @ (powerset @ X3))))),
% 14.28/3.28 file('/export/starexec/sandbox/tmp/tmp.rvsH9IYwBY/DTF2THF_12192.p',powersetI1)).
% 14.28/3.28 thf(f2805,plain,(
% 14.28/3.28 ((in @ sK345 @ (powerset @ sK347)) != $true)),
% 14.28/3.28 inference(cnf_transformation,[status(thm)],[f1550])).
% 14.28/3.28 thf(f4673,plain,(
% 14.28/3.28 ($true = (subset @ sK345 @ sK347)) | (subsetI1 != $true)),
% 14.28/3.28 inference(trivial_inequality_removal,[status(thm)],[f4672])).
% 14.28/3.28 thf(f4672,plain,(
% 14.28/3.28 ($true != $true) | ($true = (subset @ sK345 @ sK347)) | (subsetI1 != $true)),
% 14.28/3.28 inference(duplicate_literal_removal,[status(thm)],[f4666])).
% 14.28/3.28 thf(f4666,plain,(
% 14.28/3.28 ($true = (subset @ sK345 @ sK347)) | ($true != $true) | ($true = (subset @ sK345 @ sK347)) | (subsetI1 != $true)),
% 14.28/3.28 inference(superposition,[status(thm)],[f3706,f2441])).
% 14.28/3.28 thf(f2441,plain,(
% 14.28/3.28 ( ! [X0 : $i,X1 : $i] : (($true = (in @ (sK232 @ X1 @ X0) @ X1)) | ($true = (subset @ X1 @ X0)) | (subsetI1 != $true)) )),
% 14.28/3.28 inference(cnf_transformation,[status(thm)],[f1382])).
% 14.28/3.28 thf(f1382,plain,(
% 14.28/3.28 (! [X0 : $i,X1 : $i] : ((($true = (in @ (sK232 @ X1 @ X0) @ X1)) & ($true != (in @ (sK232 @ X1 @ X0) @ X0))) | ($true = (subset @ X1 @ X0))) | (subsetI1 != $true)) & ((subsetI1 = $true) | (! [X5 : $i] : (((in @ X5 @ sK234) != $true) | ($true = (in @ X5 @ sK233))) & ((subset @ sK234 @ sK233) != $true)))),
% 14.28/3.28 inference(skolemisation,[status(esa),new_symbols(skolem,[sK232,sK233,sK234])],[f1379,f1381,f1380])).
% 14.28/3.28 thf(f1380,plain,(
% 14.28/3.28 ! [X0 : $i,X1 : $i] : (? [X2 : $i] : (($true = (in @ X2 @ X1)) & ($true != (in @ X2 @ X0))) => (($true = (in @ (sK232 @ X1 @ X0) @ X1)) & ($true != (in @ (sK232 @ X1 @ X0) @ X0))))),
% 14.28/3.28 introduced(definition,[],[choice_axiom])).
% 14.28/3.28 thf(f1381,plain,(
% 14.28/3.28 ? [X3 : $i,X4 : $i] : (! [X5 : $i] : (((in @ X5 @ X4) != $true) | ((in @ X5 @ X3) = $true)) & ((subset @ X4 @ X3) != $true)) => (! [X5 : $i] : (((in @ X5 @ sK234) != $true) | ($true = (in @ X5 @ sK233))) & ((subset @ sK234 @ sK233) != $true))),
% 14.28/3.28 introduced(definition,[],[choice_axiom])).
% 14.28/3.28 thf(f1379,plain,(
% 14.28/3.28 (! [X0 : $i,X1 : $i] : (? [X2 : $i] : (($true = (in @ X2 @ X1)) & ($true != (in @ X2 @ X0))) | ($true = (subset @ X1 @ X0))) | (subsetI1 != $true)) & ((subsetI1 = $true) | ? [X3 : $i,X4 : $i] : (! [X5 : $i] : (((in @ X5 @ X4) != $true) | ((in @ X5 @ X3) = $true)) & ((subset @ X4 @ X3) != $true)))),
% 14.28/3.28 inference(rectify,[status(thm)],[f1378])).
% 14.28/3.28 thf(f1378,plain,(
% 14.28/3.28 (! [X0 : $i,X1 : $i] : (? [X2 : $i] : (($true = (in @ X2 @ X1)) & ($true != (in @ X2 @ X0))) | ($true = (subset @ X1 @ X0))) | (subsetI1 != $true)) & ((subsetI1 = $true) | ? [X0 : $i,X1 : $i] : (! [X2 : $i] : (($true != (in @ X2 @ X1)) | ($true = (in @ X2 @ X0))) & ($true != (subset @ X1 @ X0))))),
% 14.28/3.28 inference(nnf_transformation,[status(thm)],[f912])).
% 14.28/3.28 thf(f912,plain,(
% 14.28/3.28 ! [X0 : $i,X1 : $i] : (? [X2 : $i] : (($true = (in @ X2 @ X1)) & ($true != (in @ X2 @ X0))) | ($true = (subset @ X1 @ X0))) <=> (subsetI1 = $true)),
% 14.28/3.28 inference(ennf_transformation,[status(thm)],[f615])).
% 14.28/3.28 thf(f615,plain,(
% 14.28/3.28 (subsetI1 = $true) <=> ! [X0 : $i,X1 : $i] : (! [X2 : $i] : (($true = (in @ X2 @ X1)) => ($true = (in @ X2 @ X0))) => ($true = (subset @ X1 @ X0)))),
% 14.28/3.28 inference(fool_elimination,[status(thm)],[f614])).
% 14.28/3.28 thf(f614,plain,(
% 14.28/3.28 (! [X0 : $i,X1 : $i] : (! [X2 : $i] : ((in @ X2 @ X1) => (in @ X2 @ X0)) => (subset @ X1 @ X0)) = subsetI1)),
% 14.28/3.28 inference(rectify,[status(thm)],[f78])).
% 14.28/3.28 thf(f78,axiom,(
% 14.28/3.28 (! [X4 : $i,X3 : $i] : (! [X1 : $i] : ((in @ X1 @ X3) => (in @ X1 @ X4)) => (subset @ X3 @ X4)) = subsetI1)),
% 14.28/3.28 file('/export/starexec/sandbox/tmp/tmp.rvsH9IYwBY/DTF2THF_12192.p',subsetI1)).
% 14.28/3.28 thf(f3706,plain,(
% 14.28/3.28 ( ! [X0 : $i] : (($true != (in @ (sK232 @ X0 @ sK347) @ sK345)) | ($true = (subset @ X0 @ sK347))) )),
% 14.28/3.28 inference(subsumption_resolution,[status(thm)],[f3705,f3509])).
% 14.28/3.28 thf(f3509,plain,(
% 14.28/3.28 ( ! [X0 : $i] : (($true = (in @ X0 @ sK346)) | ($true != (in @ X0 @ sK345))) )),
% 14.28/3.28 inference(subsumption_resolution,[status(thm)],[f3508,f2682])).
% 14.28/3.28 thf(f2682,plain,(
% 14.28/3.28 (powersetAx = $true)),
% 14.28/3.28 inference(cnf_transformation,[status(thm)],[f1550])).
% 14.28/3.28 thf(f3508,plain,(
% 14.28/3.28 ( ! [X0 : $i] : (($true = (in @ X0 @ sK346)) | ($true != (in @ X0 @ sK345)) | (powersetAx != $true)) )),
% 14.28/3.28 inference(trivial_inequality_removal,[status(thm)],[f3499])).
% 14.28/3.28 thf(f3499,plain,(
% 14.28/3.28 ( ! [X0 : $i] : (($true = (in @ X0 @ sK346)) | (powersetAx != $true) | ($true != $true) | ($true != (in @ X0 @ sK345))) )),
% 14.28/3.28 inference(superposition,[status(thm)],[f3099,f2802])).
% 14.28/3.28 thf(f2802,plain,(
% 14.28/3.28 ($true = (in @ sK345 @ (powerset @ sK346)))),
% 14.28/3.28 inference(cnf_transformation,[status(thm)],[f1550])).
% 14.28/3.28 thf(f3099,plain,(
% 14.28/3.28 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true != (in @ X1 @ (powerset @ X0))) | ($true = (in @ X2 @ X0)) | (powersetAx != $true) | ($true != (in @ X2 @ X1))) )),
% 14.28/3.28 inference(cnf_transformation,[status(thm)],[f1877])).
% 14.28/3.28 thf(f1877,plain,(
% 14.28/3.28 (! [X0 : $i,X1 : $i] : ((! [X2 : $i] : (($true != (in @ X2 @ X1)) | ($true = (in @ X2 @ X0))) | ($true != (in @ X1 @ (powerset @ X0)))) & (($true = (in @ X1 @ (powerset @ X0))) | (($true = (in @ (sK561 @ X1 @ X0) @ X1)) & ((in @ (sK561 @ X1 @ X0) @ X0) != $true)))) | (powersetAx != $true)) & ((powersetAx = $true) | ((($true != (in @ sK563 @ (powerset @ sK562))) | (($true = (in @ sK564 @ sK563)) & ((in @ sK564 @ sK562) != $true))) & (($true = (in @ sK563 @ (powerset @ sK562))) | ! [X7 : $i] : (((in @ X7 @ sK563) != $true) | ($true = (in @ X7 @ sK562))))))),
% 14.28/3.28 inference(skolemisation,[status(esa),new_symbols(skolem,[sK561,sK562,sK563,sK564])],[f1873,f1876,f1875,f1874])).
% 14.28/3.28 thf(f1874,plain,(
% 14.28/3.28 ! [X0 : $i,X1 : $i] : (? [X3 : $i] : (($true = (in @ X3 @ X1)) & ($true != (in @ X3 @ X0))) => (($true = (in @ (sK561 @ X1 @ X0) @ X1)) & ((in @ (sK561 @ X1 @ X0) @ X0) != $true)))),
% 14.28/3.28 introduced(definition,[],[choice_axiom])).
% 14.28/3.28 thf(f1875,plain,(
% 14.28/3.28 ? [X4 : $i,X5 : $i] : ((($true != (in @ X5 @ (powerset @ X4))) | ? [X6 : $i] : (($true = (in @ X6 @ X5)) & ((in @ X6 @ X4) != $true))) & (($true = (in @ X5 @ (powerset @ X4))) | ! [X7 : $i] : (($true != (in @ X7 @ X5)) | ((in @ X7 @ X4) = $true)))) => ((($true != (in @ sK563 @ (powerset @ sK562))) | ? [X6 : $i] : (($true = (in @ X6 @ sK563)) & ((in @ X6 @ sK562) != $true))) & (($true = (in @ sK563 @ (powerset @ sK562))) | ! [X7 : $i] : (((in @ X7 @ sK563) != $true) | ($true = (in @ X7 @ sK562)))))),
% 14.28/3.28 introduced(definition,[],[choice_axiom])).
% 14.28/3.28 thf(f1876,plain,(
% 14.28/3.28 ? [X6 : $i] : (($true = (in @ X6 @ sK563)) & ((in @ X6 @ sK562) != $true)) => (($true = (in @ sK564 @ sK563)) & ((in @ sK564 @ sK562) != $true))),
% 14.28/3.28 introduced(definition,[],[choice_axiom])).
% 14.28/3.28 thf(f1873,plain,(
% 14.28/3.28 (! [X0 : $i,X1 : $i] : ((! [X2 : $i] : (($true != (in @ X2 @ X1)) | ($true = (in @ X2 @ X0))) | ($true != (in @ X1 @ (powerset @ X0)))) & (($true = (in @ X1 @ (powerset @ X0))) | ? [X3 : $i] : (($true = (in @ X3 @ X1)) & ($true != (in @ X3 @ X0))))) | (powersetAx != $true)) & ((powersetAx = $true) | ? [X4 : $i,X5 : $i] : ((($true != (in @ X5 @ (powerset @ X4))) | ? [X6 : $i] : (($true = (in @ X6 @ X5)) & ((in @ X6 @ X4) != $true))) & (($true = (in @ X5 @ (powerset @ X4))) | ! [X7 : $i] : (($true != (in @ X7 @ X5)) | ((in @ X7 @ X4) = $true)))))),
% 14.28/3.28 inference(rectify,[status(thm)],[f1872])).
% 14.28/3.28 thf(f1872,plain,(
% 14.28/3.28 (! [X0 : $i,X1 : $i] : ((! [X2 : $i] : (($true != (in @ X2 @ X1)) | ($true = (in @ X2 @ X0))) | ($true != (in @ X1 @ (powerset @ X0)))) & (($true = (in @ X1 @ (powerset @ X0))) | ? [X2 : $i] : (($true = (in @ X2 @ X1)) & ($true != (in @ X2 @ X0))))) | (powersetAx != $true)) & ((powersetAx = $true) | ? [X0 : $i,X1 : $i] : ((($true != (in @ X1 @ (powerset @ X0))) | ? [X2 : $i] : (($true = (in @ X2 @ X1)) & ($true != (in @ X2 @ X0)))) & (($true = (in @ X1 @ (powerset @ X0))) | ! [X2 : $i] : (($true != (in @ X2 @ X1)) | ($true = (in @ X2 @ X0))))))),
% 14.28/3.28 inference(nnf_transformation,[status(thm)],[f951])).
% 14.28/3.28 thf(f951,plain,(
% 14.28/3.28 ! [X0 : $i,X1 : $i] : (! [X2 : $i] : (($true != (in @ X2 @ X1)) | ($true = (in @ X2 @ X0))) <=> ($true = (in @ X1 @ (powerset @ X0)))) <=> (powersetAx = $true)),
% 14.28/3.28 inference(ennf_transformation,[status(thm)],[f410])).
% 14.28/3.28 thf(f410,plain,(
% 14.28/3.28 (powersetAx = $true) <=> ! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) <=> ! [X2 : $i] : (($true = (in @ X2 @ X1)) => ($true = (in @ X2 @ X0))))),
% 14.28/3.28 inference(fool_elimination,[status(thm)],[f409])).
% 14.28/3.28 thf(f409,plain,(
% 14.28/3.28 (powersetAx = ! [X0 : $i,X1 : $i] : ((in @ X1 @ (powerset @ X0)) <=> ! [X2 : $i] : ((in @ X2 @ X1) => (in @ X2 @ X0))))),
% 14.28/3.28 inference(rectify,[status(thm)],[f5])).
% 14.28/3.28 thf(f5,axiom,(
% 14.28/3.28 (powersetAx = ! [X3 : $i,X4 : $i] : ((in @ X4 @ (powerset @ X3)) <=> ! [X1 : $i] : ((in @ X1 @ X4) => (in @ X1 @ X3))))),
% 14.28/3.28 file('/export/starexec/sandbox/tmp/tmp.rvsH9IYwBY/DTF2THF_12192.p',powersetAx)).
% 14.28/3.28 thf(f3705,plain,(
% 14.28/3.28 ( ! [X0 : $i] : (($true = (subset @ X0 @ sK347)) | ($true != (in @ (sK232 @ X0 @ sK347) @ sK346)) | ($true != (in @ (sK232 @ X0 @ sK347) @ sK345))) )),
% 14.28/3.28 inference(subsumption_resolution,[status(thm)],[f3701,f2642])).
% 14.28/3.28 thf(f3701,plain,(
% 14.28/3.28 ( ! [X0 : $i] : (($true = (subset @ X0 @ sK347)) | (subsetI1 != $true) | ($true != (in @ (sK232 @ X0 @ sK347) @ sK346)) | ($true != (in @ (sK232 @ X0 @ sK347) @ sK345))) )),
% 14.28/3.28 inference(trivial_inequality_removal,[status(thm)],[f3664])).
% 14.28/3.28 thf(f3664,plain,(
% 14.28/3.28 ( ! [X0 : $i] : (($true != $true) | (subsetI1 != $true) | ($true = (subset @ X0 @ sK347)) | ($true != (in @ (sK232 @ X0 @ sK347) @ sK346)) | ($true != (in @ (sK232 @ X0 @ sK347) @ sK345))) )),
% 14.28/3.28 inference(superposition,[status(thm)],[f2440,f2804])).
% 14.28/3.28 thf(f2804,plain,(
% 14.28/3.28 ( ! [X3 : $i] : (($true = (in @ X3 @ sK347)) | ($true != (in @ X3 @ sK346)) | ($true != (in @ X3 @ sK345))) )),
% 14.28/3.28 inference(cnf_transformation,[status(thm)],[f1550])).
% 14.28/3.28 thf(f2440,plain,(
% 14.28/3.28 ( ! [X0 : $i,X1 : $i] : (($true != (in @ (sK232 @ X1 @ X0) @ X0)) | ($true = (subset @ X1 @ X0)) | (subsetI1 != $true)) )),
% 14.28/3.28 inference(cnf_transformation,[status(thm)],[f1382])).
% 14.28/3.28 % SZS output end Proof for DTF2THF_12192
% 14.28/3.28 % (12775)------------------------------
% 14.28/3.28 % (12775)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.28/3.28 % (12775)Termination reason: Refutation
% 14.28/3.28
% 14.28/3.28 % (12775)Memory used [KB]: 10746
% 14.28/3.28 % (12775)Time elapsed: 0.220 s
% 14.28/3.28 % (12775)Instructions burned: 696 (million)
% 14.28/3.28 % (12775)------------------------------
% 14.28/3.28 % (12775)------------------------------
% 14.28/3.28 % (12385)Success in time 1.729 s
%------------------------------------------------------------------------------