%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU575^1 : 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 : n003.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:35:22 AM UTC 2026
% Result : Theorem 2.43s 1.67s
% Output : Refutation 2.43s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 6
% Syntax : Number of formulae : 38 ( 11 unt; 0 typ; 0 def)
% Number of atoms : 1694 ( 469 equ; 0 cnn)
% Maximal formula atoms : 94 ( 44 avg)
% Number of connectives : 877 ( 49 ~; 35 |; 294 &; 118 @)
% ( 4 <=>; 377 =>; 0 <=; 0 <~>)
% Maximal formula depth : 97 ( 21 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 206 ( 203 usr; 98 con; 0-4 aty)
% Number of variables : 47 ( 0 ^; 35 !; 12 ?; 47 :)
% 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_120,type,
sP1: $i > $i > $o ).
thf(func_def_121,type,
sP2: $i > $o ).
thf(func_def_122,type,
sP3: $i > $i > $o ).
thf(func_def_123,type,
sP4: $i > $i > $o ).
thf(func_def_124,type,
sK5: $i > $o ).
thf(func_def_125,type,
sK6: ( $i > $o ) > $i ).
thf(func_def_135,type,
sK16: $i > $i > $i ).
thf(func_def_138,type,
sK19: $i > $i > $i ).
thf(func_def_143,type,
sK24: $i > $o ).
thf(func_def_145,type,
sK26: ( $i > $o ) > $i > $i ).
thf(func_def_147,type,
sK28: $i > $o ).
thf(func_def_149,type,
sK30: ( $i > $o ) > $i > $i ).
thf(func_def_155,type,
sK36: $i > $o ).
thf(func_def_157,type,
sK38: ( $i > $o ) > $i > $i ).
thf(func_def_159,type,
sK40: $i > $i > $o ).
thf(func_def_160,type,
sK41: $i > $i ).
thf(func_def_161,type,
sK42: $i > $i ).
thf(func_def_162,type,
sK43: ( $i > $i > $o ) > $i > $i ).
thf(func_def_163,type,
sK44: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_164,type,
sK45: ( $i > $i > $o ) > $i > $i ).
thf(func_def_166,type,
sK47: $i > $o ).
thf(func_def_169,type,
sK50: $i > $i ).
thf(func_def_180,type,
sK61: $i > $i > $i ).
thf(func_def_188,type,
sK69: $i > $o ).
thf(func_def_190,type,
sK71: $i > ( $i > $o ) > $i ).
thf(func_def_193,type,
sK74: $i > $i > $i ).
thf(func_def_198,type,
sK79: $i > $o ).
thf(func_def_199,type,
sK80: $i > $o ).
thf(func_def_200,type,
sK81: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_201,type,
sK82: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_205,type,
sK86: $i > $i ).
thf(func_def_208,type,
sK89: $i > $o ).
thf(func_def_219,type,
sK100: $i > $o ).
thf(func_def_223,type,
sK104: $i > $i ).
thf(func_def_225,type,
sK106: $i > $o ).
thf(func_def_230,type,
sK111: $i > ( $i > $o ) > $i ).
thf(func_def_231,type,
sK112: $i > $o ).
thf(func_def_242,type,
sK123: $i > $i ).
thf(func_def_243,type,
sK124: $i > $i ).
thf(func_def_244,type,
sK125: ( $i > $o ) > $i ).
thf(func_def_245,type,
sK126: ( $i > $o ) > $i ).
thf(func_def_246,type,
sK127: $i > $o ).
thf(func_def_252,type,
sK133: $i > $o ).
thf(func_def_256,type,
sK137: $i > $o ).
thf(func_def_260,type,
sK141: ( $i > $o ) > $i ).
thf(func_def_261,type,
sK142: $i > $o ).
thf(func_def_262,type,
sK143: $i > $i ).
thf(func_def_264,type,
sK145: $i > $i ).
thf(func_def_269,type,
sK150: $i > $o ).
thf(func_def_270,type,
sK151: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_271,type,
sK152: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_274,type,
sK155: $i > $o ).
thf(func_def_275,type,
sK156: $i > $o ).
thf(func_def_280,type,
sK161: $i > $i > $i ).
thf(func_def_281,type,
sK162: $i > $i > $i ).
thf(func_def_284,type,
sK165: ( $i > $o ) > $i ).
thf(func_def_285,type,
sK166: $i > $o ).
thf(func_def_286,type,
sK167: $i > $i ).
thf(func_def_287,type,
sK168: $i > $o ).
thf(func_def_292,type,
sK173: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_293,type,
sK174: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_294,type,
sK175: $i > $o ).
thf(func_def_295,type,
sK176: $i > $o ).
thf(func_def_297,type,
sK178: $i > $i ).
thf(func_def_301,type,
sK182: $i > $o ).
thf(func_def_304,type,
sK185: $i > ( $i > $o ) > $i ).
thf(func_def_309,type,
sK190: ( $i > $o ) > $i > $i ).
thf(func_def_311,type,
sK192: $i > $o ).
thf(func_def_325,type,
sK206: $i > $i > $i ).
thf(func_def_329,type,
sK210: $i > $i > $i ).
thf(func_def_330,type,
sK211: $i > $i > $i ).
thf(func_def_331,type,
sK212: $i > $i > $i ).
thf(func_def_332,type,
sK213: $i > $i > $i ).
thf(func_def_333,type,
sK214: $i > $i > $i > $i ).
thf(func_def_334,type,
sK215: $i > $i ).
thf(func_def_335,type,
sK216: $i > $i ).
thf(func_def_336,type,
sK217: $i > $i ).
thf(func_def_337,type,
sK218: $i > $i ).
thf(func_def_338,type,
sK219: $i > $i > $i ).
thf(func_def_339,type,
sK220: $i > $i > $i > $i ).
thf(func_def_340,type,
sK221: $i > $i > $i > $i ).
thf(func_def_341,type,
sK222: $i > $i > $i ).
thf(func_def_342,type,
sK223: $i > $i > $i ).
thf(func_def_343,type,
sK224: $i > $i ).
thf(func_def_344,type,
sK225: $i > $i > $i ).
thf(func_def_346,type,
sK227: $i > $i ).
thf(func_def_347,type,
sK228: $i > $i ).
thf(func_def_353,type,
sK234: $i > $i > $i ).
thf(func_def_354,type,
sK235: $i > ( $i > $o ) > $i ).
thf(func_def_355,type,
sK236: $i > $o ).
thf(func_def_361,type,
sK242: $i > $o ).
thf(func_def_365,type,
sK246: $o > $i > $i > $i ).
thf(func_def_369,type,
sK250: $i > $o ).
thf(func_def_378,type,
ph259:
!>[X0: $tType] : X0 ).
thf(f2333,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f2332,f1723]) ).
thf(f1723,plain,
! [X0: $i] :
( ( subset @ emptyset @ X0 )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f1423]) ).
thf(f1423,plain,
! [X0: $i] :
( ( $true != $true )
| ( ( subset @ emptyset @ X0 )
= $true ) ),
inference(definition_unfolding,[status(thm)],[f1057,f909]) ).
thf(f909,plain,
emptysetsubset = $true,
inference(cnf_transformation,[status(thm)],[f415]) ).
thf(f415,plain,
( ( subsetI1 = $true )
& ( emptyI = $true )
& ( descr__Cong = $true )
& ( notdallE = $true )
& ( emptyinPowerset = $true )
& ( nonemptyI = $true )
& ( upairsetE = $true )
& ( exuE1 = $true )
& ( powersetI = $true )
& ( exuI3 = $true )
& ( descrp = $true )
& ( notsubsetI = $true )
& ( vacuousDall = $true )
& ( subPowSU = $true )
& ( nonemptyE1 = $true )
& ( powersetAx = $true )
& ( eqimpsubset1 = $true )
& ( subset2powerset = $true )
& ( dsetconstrEL = $true )
& ( omegaSAx = $true )
& ( notinsingleton = $true )
& ( eqinunit = $true )
& ( uniqinunit = $true )
& ( exuE3e = $true )
& ( emptysetAx = $true )
& ( dsetconstrI = $true )
& ( quantDeMorgan3 = $true )
& ( notdexE = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( in__Cong = $true )
& ( setadjoinAx = $true )
& ( prop2setE = $true )
& ( setadjoinOr = $true )
& ( quantDeMorgan4 = $true )
& ( emptysetsubset = $true )
& ( setoftrueEq = $true )
& ( prop2set2propI = $true )
& ( subsetE = $true )
& ( nonemptyImpWitness = $true )
& ( setadjoin__Cong = $true )
& ( setadjoinIR = $true )
& ( setextsub = $true )
& ( emptysetimpfalse = $true )
& ( emptyE1 = $true )
& ( exuI2 = $true )
& ( subsetTrans = $true )
& ( singletonsswitch = $true )
& ( upairsetIR = $true )
& ( exuE2 = $true )
& ( powerset__Cong = $true )
& ( emptysetE = $true )
& ( setunionI = $true )
& ( setadjoinE = $true )
& ( eqimpsubset2 = $true )
& ( subsetRefl = $true )
& ( notinemptyset = $true )
& ( dsetconstrER = $true )
& ( wellorderingAx = $true )
& ( setunionAx = $true )
& ( notequalI1 = $true )
& ( exuI1 = $true )
& ( omega__Cong = $true )
& ( setadjoinSub = $true )
& ( quantDeMorgan1 = $true )
& ( setunion__Cong = $true )
& ( foundationAx = $true )
& ( inCongP = $true )
& ( setunionE = $true )
& ( quantDeMorgan2 = $true )
& ( exu__Cong = $true )
& ( setadjoinSub2 = $true )
& ( notequalI2 = $true )
& ( emptyinunitempty = $true )
& ( upairsetIL = $true )
& ( setbeta = $true )
& ( omega0Ax = $true )
& ( emptyset__Cong = $true )
& ( setadjoinIL = $true )
& ( noeltsimpempty = $true )
& ( setextAx = $true )
& ( exuE3u = $true )
& ( subsetI2 = $true )
& ( dsetconstr__Cong = $true )
& ( subsetE2 = $true )
& ( exuEu = $true )
& ( omegaIndAx = $true )
& ( replAx = $true )
& ( nonemptyI1 = $true )
& ( setext = $true )
& ( ( subset @ sK7 @ emptyset )
= $true )
& ( emptyset != sK7 )
& ( prop2setI = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK7])],[f359,f414]) ).
thf(f414,plain,
( ? [X0: $i] :
( ( ( subset @ X0 @ emptyset )
= $true )
& ( emptyset != X0 ) )
=> ( ( ( subset @ sK7 @ emptyset )
= $true )
& ( emptyset != sK7 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f359,plain,
( ( subsetI1 = $true )
& ( emptyI = $true )
& ( descr__Cong = $true )
& ( notdallE = $true )
& ( emptyinPowerset = $true )
& ( nonemptyI = $true )
& ( upairsetE = $true )
& ( exuE1 = $true )
& ( powersetI = $true )
& ( exuI3 = $true )
& ( descrp = $true )
& ( notsubsetI = $true )
& ( vacuousDall = $true )
& ( subPowSU = $true )
& ( nonemptyE1 = $true )
& ( powersetAx = $true )
& ( eqimpsubset1 = $true )
& ( subset2powerset = $true )
& ( dsetconstrEL = $true )
& ( omegaSAx = $true )
& ( notinsingleton = $true )
& ( eqinunit = $true )
& ( uniqinunit = $true )
& ( exuE3e = $true )
& ( emptysetAx = $true )
& ( dsetconstrI = $true )
& ( quantDeMorgan3 = $true )
& ( notdexE = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( in__Cong = $true )
& ( setadjoinAx = $true )
& ( prop2setE = $true )
& ( setadjoinOr = $true )
& ( quantDeMorgan4 = $true )
& ( emptysetsubset = $true )
& ( setoftrueEq = $true )
& ( prop2set2propI = $true )
& ( subsetE = $true )
& ( nonemptyImpWitness = $true )
& ( setadjoin__Cong = $true )
& ( setadjoinIR = $true )
& ( setextsub = $true )
& ( emptysetimpfalse = $true )
& ( emptyE1 = $true )
& ( exuI2 = $true )
& ( subsetTrans = $true )
& ( singletonsswitch = $true )
& ( upairsetIR = $true )
& ( exuE2 = $true )
& ( powerset__Cong = $true )
& ( emptysetE = $true )
& ( setunionI = $true )
& ( setadjoinE = $true )
& ( eqimpsubset2 = $true )
& ( subsetRefl = $true )
& ( notinemptyset = $true )
& ( dsetconstrER = $true )
& ( wellorderingAx = $true )
& ( setunionAx = $true )
& ( notequalI1 = $true )
& ( exuI1 = $true )
& ( omega__Cong = $true )
& ( setadjoinSub = $true )
& ( quantDeMorgan1 = $true )
& ( setunion__Cong = $true )
& ( foundationAx = $true )
& ( inCongP = $true )
& ( setunionE = $true )
& ( quantDeMorgan2 = $true )
& ( exu__Cong = $true )
& ( setadjoinSub2 = $true )
& ( notequalI2 = $true )
& ( emptyinunitempty = $true )
& ( upairsetIL = $true )
& ( setbeta = $true )
& ( omega0Ax = $true )
& ( emptyset__Cong = $true )
& ( setadjoinIL = $true )
& ( noeltsimpempty = $true )
& ( setextAx = $true )
& ( exuE3u = $true )
& ( subsetI2 = $true )
& ( dsetconstr__Cong = $true )
& ( subsetE2 = $true )
& ( exuEu = $true )
& ( omegaIndAx = $true )
& ( replAx = $true )
& ( nonemptyI1 = $true )
& ( setext = $true )
& ? [X0: $i] :
( ( ( subset @ X0 @ emptyset )
= $true )
& ( emptyset != X0 ) )
& ( prop2setI = $true ) ),
inference(flattening,[status(thm)],[f358]) ).
thf(f358,plain,
( ? [X0: $i] :
( ( ( subset @ X0 @ emptyset )
= $true )
& ( emptyset != X0 ) )
& ( 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)],[f134]) ).
thf(f134,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 )
=> ! [X0: $i] :
( ( ( subset @ X0 @ emptyset )
= $true )
=> ( emptyset = X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f133]) ).
thf(f133,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
=> ! [X0: $i] :
( ( subset @ X0 @ emptyset )
=> ( emptyset = X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f95]) ).
thf(f95,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
=> ! [X3: $i] :
( ( subset @ X3 @ emptyset )
=> ( emptyset = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f94]) ).
thf(f94,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
=> ! [X3: $i] :
( ( subset @ X3 @ emptyset )
=> ( emptyset = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.x7ZFQe8mUJ/DTF2THF_11764.p',subsetemptysetimpeq) ).
thf(f1057,plain,
! [X0: $i] :
( ( ( subset @ emptyset @ X0 )
= $true )
| ( emptysetsubset != $true ) ),
inference(cnf_transformation,[status(thm)],[f542]) ).
thf(f542,plain,
( ( ! [X0: $i] :
( ( subset @ emptyset @ X0 )
= $true )
| ( emptysetsubset != $true ) )
& ( ( emptysetsubset = $true )
| ( ( subset @ emptyset @ sK85 )
!= $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK85])],[f540,f541]) ).
thf(f541,plain,
( ? [X1: $i] :
( ( subset @ emptyset @ X1 )
!= $true )
=> ( ( subset @ emptyset @ sK85 )
!= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f540,plain,
( ( ! [X0: $i] :
( ( subset @ emptyset @ X0 )
= $true )
| ( emptysetsubset != $true ) )
& ( ( emptysetsubset = $true )
| ? [X1: $i] :
( ( subset @ emptyset @ X1 )
!= $true ) ) ),
inference(rectify,[status(thm)],[f539]) ).
thf(f539,plain,
( ( ! [X0: $i] :
( ( subset @ emptyset @ X0 )
= $true )
| ( emptysetsubset != $true ) )
& ( ( emptysetsubset = $true )
| ? [X0: $i] :
( ( subset @ emptyset @ X0 )
!= $true ) ) ),
inference(nnf_transformation,[status(thm)],[f268]) ).
thf(f268,plain,
( ! [X0: $i] :
( ( subset @ emptyset @ X0 )
= $true )
<=> ( emptysetsubset = $true ) ),
inference(fool_elimination,[status(thm)],[f267]) ).
thf(f267,plain,
! [X0: $i] :
( ( subset @ emptyset @ X0 )
= emptysetsubset ),
inference(rectify,[status(thm)],[f82]) ).
thf(f82,axiom,
! [X3: $i] :
( ( subset @ emptyset @ X3 )
= emptysetsubset ),
file('/export/starexec/sandbox/tmp/tmp.x7ZFQe8mUJ/DTF2THF_11764.p',emptysetsubset) ).
thf(f2332,plain,
( ( subset @ emptyset @ sK7 )
!= $true ),
inference(subsumption_resolution,[status(thm)],[f2329,f853]) ).
thf(f853,plain,
emptyset != sK7,
inference(cnf_transformation,[status(thm)],[f415]) ).
thf(f2329,plain,
( ( emptyset = sK7 )
| ( ( subset @ emptyset @ sK7 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f2309]) ).
thf(f2309,plain,
( ( emptyset = sK7 )
| ( $true != $true )
| ( ( subset @ emptyset @ sK7 )
!= $true ) ),
inference(superposition,[status(thm)],[f1840,f854]) ).
thf(f854,plain,
( ( subset @ sK7 @ emptyset )
= $true ),
inference(cnf_transformation,[status(thm)],[f415]) ).
thf(f1840,plain,
! [X2: $i,X3: $i] :
( ( ( subset @ X3 @ X2 )
!= $true )
| ( ( subset @ X2 @ X3 )
!= $true )
| ( X2 = X3 ) ),
inference(trivial_inequality_removal,[status(thm)],[f1396]) ).
thf(f1396,plain,
! [X2: $i,X3: $i] :
( ( ( subset @ X3 @ X2 )
!= $true )
| ( $true != $true )
| ( ( subset @ X2 @ X3 )
!= $true )
| ( X2 = X3 ) ),
inference(definition_unfolding,[status(thm)],[f1026,f902]) ).
thf(f902,plain,
setextsub = $true,
inference(cnf_transformation,[status(thm)],[f415]) ).
thf(f1026,plain,
! [X2: $i,X3: $i] :
( ( X2 = X3 )
| ( ( subset @ X3 @ X2 )
!= $true )
| ( ( subset @ X2 @ X3 )
!= $true )
| ( setextsub != $true ) ),
inference(cnf_transformation,[status(thm)],[f508]) ).
thf(f508,plain,
( ( ( setextsub = $true )
| ( ( sK67 != sK68 )
& ( ( subset @ sK68 @ sK67 )
= $true )
& ( ( subset @ sK67 @ sK68 )
= $true ) ) )
& ( ! [X2: $i,X3: $i] :
( ( X2 = X3 )
| ( ( subset @ X3 @ X2 )
!= $true )
| ( ( subset @ X2 @ X3 )
!= $true ) )
| ( setextsub != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK67,sK68])],[f506,f507]) ).
thf(f507,plain,
( ? [X0: $i,X1: $i] :
( ( X0 != X1 )
& ( $true
= ( subset @ X1 @ X0 ) )
& ( ( subset @ X0 @ X1 )
= $true ) )
=> ( ( sK67 != sK68 )
& ( ( subset @ sK68 @ sK67 )
= $true )
& ( ( subset @ sK67 @ sK68 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f506,plain,
( ( ( setextsub = $true )
| ? [X0: $i,X1: $i] :
( ( X0 != X1 )
& ( $true
= ( subset @ X1 @ X0 ) )
& ( ( subset @ X0 @ X1 )
= $true ) ) )
& ( ! [X2: $i,X3: $i] :
( ( X2 = X3 )
| ( ( subset @ X3 @ X2 )
!= $true )
| ( ( subset @ X2 @ X3 )
!= $true ) )
| ( setextsub != $true ) ) ),
inference(rectify,[status(thm)],[f505]) ).
thf(f505,plain,
( ( ( setextsub = $true )
| ? [X0: $i,X1: $i] :
( ( X0 != X1 )
& ( $true
= ( subset @ X1 @ X0 ) )
& ( ( subset @ X0 @ X1 )
= $true ) ) )
& ( ! [X0: $i,X1: $i] :
( ( X0 = X1 )
| ( $true
!= ( subset @ X1 @ X0 ) )
| ( ( subset @ X0 @ X1 )
!= $true ) )
| ( setextsub != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f373]) ).
thf(f373,plain,
( ( setextsub = $true )
<=> ! [X0: $i,X1: $i] :
( ( X0 = X1 )
| ( $true
!= ( subset @ X1 @ X0 ) )
| ( ( subset @ X0 @ X1 )
!= $true ) ) ),
inference(flattening,[status(thm)],[f372]) ).
thf(f372,plain,
( ( setextsub = $true )
<=> ! [X0: $i,X1: $i] :
( ( X0 = X1 )
| ( $true
!= ( subset @ X1 @ X0 ) )
| ( ( subset @ X0 @ X1 )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f192]) ).
thf(f192,plain,
( ( setextsub = $true )
<=> ! [X0: $i,X1: $i] :
( ( ( subset @ X0 @ X1 )
= $true )
=> ( ( $true
= ( subset @ X1 @ X0 ) )
=> ( X0 = X1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f191]) ).
thf(f191,plain,
( setextsub
= ( ! [X0: $i,X1: $i] :
( ( subset @ X0 @ X1 )
=> ( ( subset @ X1 @ X0 )
=> ( X0 = X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f93]) ).
thf(f93,axiom,
( setextsub
= ( ! [X3: $i,X4: $i] :
( ( subset @ X3 @ X4 )
=> ( ( subset @ X4 @ X3 )
=> ( X3 = X4 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.x7ZFQe8mUJ/DTF2THF_11764.p',setextsub) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : SEU575^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.09 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.11/0.29 % Computer : n003.cluster.edu
% 0.11/0.29 % Model : x86_64 x86_64
% 0.11/0.29 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.29 % Memory : 8042.1875MB
% 0.11/0.29 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.29 % CPULimit : 300
% 0.11/0.29 % WCLimit : 300
% 0.11/0.29 % DateTime : Tue Feb 24 12:27:10 EST 2026
% 0.11/0.29 % CPUTime :
% 0.11/0.29 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.18/0.38 ---- Original DTF file ---
% 0.18/0.38 thf(spec,logic,$$dhol).
% 0.18/0.38 %------------------------------------------------------------------------------
% 0.18/0.38 % File : SEU575^1 : TPTP v9.2.1. Released v3.7.0.
% 0.18/0.38 % Domain : Set Theory
% 0.18/0.38 % Problem : Preliminary Notions - Relations on Sets - Subsets
% 0.18/0.38 % Version : Especial.
% 0.18/0.38 % English : (! A:i.subset A emptyset -> A = emptyset)
% 0.18/0.38
% 0.18/0.38 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.18/0.38 % Source : [Bro08]
% 0.18/0.38 % Names : ZFC077g [Bro08]
% 0.18/0.38
% 0.18/0.38 % Status : Theorem
% 0.18/0.38 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.23 v8.1.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.67 v3.7.0
% 0.18/0.38 % Syntax : Number of formulae : 200 ( 93 unt; 106 typ; 93 def)
% 0.18/0.38 % Number of atoms : 508 ( 142 equ; 0 cnn)
% 0.18/0.38 % Maximal formula atoms : 93 ( 5 avg)
% 0.18/0.38 % Number of connectives : 798 ( 29 ~; 5 |; 27 &; 491 @)
% 0.18/0.38 % ( 14 <=>; 232 =>; 0 <=; 0 <~>)
% 0.18/0.38 % Maximal formula depth : 96 ( 2 avg)
% 0.18/0.38 % Number of types : 2 ( 0 usr)
% 0.18/0.38 % Number of type conns : 53 ( 53 >; 0 *; 0 +; 0 <<)
% 0.18/0.38 % Number of symbols : 109 ( 106 usr; 95 con; 0-2 aty)
% 0.18/0.38 % Number of variables : 281 ( 28 ^; 225 !; 28 ?; 281 :)
% 0.18/0.38 % SPC : TH0_THM_EQU_NAR
% 0.18/0.38
% 0.18/0.38 % Comments : http://mathgate.info/detsetitem.php?id=479
% 0.18/0.38 %------------------------------------------------------------------------------
% 0.18/0.38 thf(in_type,type,
% 0.18/0.38 in: $i > $i > $o ).
% 0.18/0.38
% 0.18/0.38 thf(exu_type,type,
% 0.18/0.38 exu: ( $i > $o ) > $o ).
% 0.18/0.38
% 0.18/0.38 thf(exu,definition,
% 0.18/0.38 ( exu
% 0.18/0.38 = ( ^ [Xphi: $i > $o] :
% 0.18/0.38 ? [Xx: $i] :
% 0.18/0.38 ( ( Xphi @ Xx )
% 0.18/0.38 & ! [Xy: $i] :
% 0.18/0.38 ( ( Xphi @ Xy )
% 0.18/0.38 => ( Xx = Xy ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(setextAx_type,type,
% 0.18/0.38 setextAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(setextAx,definition,
% 0.18/0.38 ( setextAx
% 0.18/0.38 = ( ! [A: $i,B: $i] :
% 0.18/0.38 ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 <=> ( in @ Xx @ B ) )
% 0.18/0.38 => ( A = B ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(emptyset_type,type,
% 0.18/0.38 emptyset: $i ).
% 0.18/0.38
% 0.18/0.38 thf(emptysetAx_type,type,
% 0.18/0.38 emptysetAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(emptysetAx,definition,
% 0.18/0.38 ( emptysetAx
% 0.18/0.38 = ( ! [Xx: $i] :
% 0.18/0.38 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(setadjoin_type,type,
% 0.18/0.38 setadjoin: $i > $i > $i ).
% 0.18/0.38
% 0.18/0.38 thf(setadjoinAx_type,type,
% 0.18/0.38 setadjoinAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(setadjoinAx,definition,
% 0.18/0.38 ( setadjoinAx
% 0.18/0.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.18/0.38 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.18/0.38 <=> ( ( Xy = Xx )
% 0.18/0.38 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(powerset_type,type,
% 0.18/0.38 powerset: $i > $i ).
% 0.18/0.38
% 0.18/0.38 thf(powersetAx_type,type,
% 0.18/0.38 powersetAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(powersetAx,definition,
% 0.18/0.38 ( powersetAx
% 0.18/0.38 = ( ! [A: $i,B: $i] :
% 0.18/0.38 ( ( in @ B @ ( powerset @ A ) )
% 0.18/0.38 <=> ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ B )
% 0.18/0.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(setunion_type,type,
% 0.18/0.38 setunion: $i > $i ).
% 0.18/0.38
% 0.18/0.38 thf(setunionAx_type,type,
% 0.18/0.38 setunionAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(setunionAx,definition,
% 0.18/0.38 ( setunionAx
% 0.18/0.38 = ( ! [A: $i,Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ ( setunion @ A ) )
% 0.18/0.38 <=> ? [B: $i] :
% 0.18/0.38 ( ( in @ Xx @ B )
% 0.18/0.38 & ( in @ B @ A ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(omega_type,type,
% 0.18/0.38 omega: $i ).
% 0.18/0.38
% 0.18/0.38 thf(omega0Ax_type,type,
% 0.18/0.38 omega0Ax: $o ).
% 0.18/0.38
% 0.18/0.38 thf(omega0Ax,definition,
% 0.18/0.38 ( omega0Ax
% 0.18/0.38 = ( in @ emptyset @ omega ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(omegaSAx_type,type,
% 0.18/0.38 omegaSAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(omegaSAx,definition,
% 0.18/0.38 ( omegaSAx
% 0.18/0.38 = ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ omega )
% 0.18/0.38 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(omegaIndAx_type,type,
% 0.18/0.38 omegaIndAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(omegaIndAx,definition,
% 0.18/0.38 ( omegaIndAx
% 0.18/0.38 = ( ! [A: $i] :
% 0.18/0.38 ( ( ( in @ emptyset @ A )
% 0.18/0.38 & ! [Xx: $i] :
% 0.18/0.38 ( ( ( in @ Xx @ omega )
% 0.18/0.38 & ( in @ Xx @ A ) )
% 0.18/0.38 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.18/0.38 => ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ omega )
% 0.18/0.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(replAx_type,type,
% 0.18/0.38 replAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(replAx,definition,
% 0.18/0.38 ( replAx
% 0.18/0.38 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( exu
% 0.18/0.39 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.18/0.39 => ? [B: $i] :
% 0.18/0.39 ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ B )
% 0.18/0.39 <=> ? [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ A )
% 0.18/0.39 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(foundationAx_type,type,
% 0.18/0.39 foundationAx: $o ).
% 0.18/0.39
% 0.18/0.39 thf(foundationAx,definition,
% 0.18/0.39 ( foundationAx
% 0.18/0.39 = ( ! [A: $i] :
% 0.18/0.39 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.18/0.39 => ? [B: $i] :
% 0.18/0.39 ( ( in @ B @ A )
% 0.18/0.39 & ~ ? [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ B )
% 0.18/0.39 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(wellorderingAx_type,type,
% 0.18/0.39 wellorderingAx: $o ).
% 0.18/0.39
% 0.18/0.39 thf(wellorderingAx,definition,
% 0.18/0.39 ( wellorderingAx
% 0.18/0.39 = ( ! [A: $i] :
% 0.18/0.39 ? [B: $i] :
% 0.18/0.39 ( ! [C: $i] :
% 0.18/0.39 ( ( in @ C @ B )
% 0.18/0.39 => ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ C )
% 0.18/0.39 => ( in @ Xx @ A ) ) )
% 0.18/0.39 & ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( ( in @ Xx @ A )
% 0.18/0.39 & ( in @ Xy @ A ) )
% 0.18/0.39 => ( ! [C: $i] :
% 0.18/0.39 ( ( in @ C @ B )
% 0.18/0.39 => ( ( in @ Xx @ C )
% 0.18/0.39 <=> ( in @ Xy @ C ) ) )
% 0.18/0.39 => ( Xx = Xy ) ) )
% 0.18/0.39 & ! [C: $i,D: $i] :
% 0.18/0.39 ( ( ( in @ C @ B )
% 0.18/0.39 & ( in @ D @ B ) )
% 0.18/0.39 => ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ C )
% 0.18/0.39 => ( in @ Xx @ D ) )
% 0.18/0.39 | ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ D )
% 0.18/0.39 => ( in @ Xx @ C ) ) ) )
% 0.18/0.39 & ! [C: $i] :
% 0.18/0.39 ( ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ C )
% 0.18/0.39 => ( in @ Xx @ A ) )
% 0.18/0.39 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.18/0.39 => ? [D: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ D @ B )
% 0.18/0.39 & ( in @ Xx @ C )
% 0.18/0.39 & ~ ? [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ D )
% 0.18/0.39 & ( in @ Xy @ C ) )
% 0.18/0.39 & ! [E: $i] :
% 0.18/0.39 ( ( in @ E @ B )
% 0.18/0.39 => ( ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ E )
% 0.18/0.39 => ( in @ Xy @ D ) )
% 0.18/0.39 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(descr_type,type,
% 0.18/0.39 descr: ( $i > $o ) > $i ).
% 0.18/0.39
% 0.18/0.39 thf(descrp_type,type,
% 0.18/0.39 descrp: $o ).
% 0.18/0.39
% 0.18/0.39 thf(descrp,definition,
% 0.18/0.39 ( descrp
% 0.18/0.39 = ( ! [Xphi: $i > $o] :
% 0.18/0.39 ( ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 => ( Xphi
% 0.18/0.39 @ ( descr
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(dsetconstr_type,type,
% 0.18/0.39 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.18/0.39
% 0.18/0.39 thf(dsetconstrI_type,type,
% 0.18/0.39 dsetconstrI: $o ).
% 0.18/0.39
% 0.18/0.39 thf(dsetconstrI,definition,
% 0.18/0.39 ( dsetconstrI
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( ( Xphi @ Xx )
% 0.18/0.39 => ( in @ Xx
% 0.18/0.39 @ ( dsetconstr @ A
% 0.18/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(dsetconstrEL_type,type,
% 0.18/0.39 dsetconstrEL: $o ).
% 0.18/0.39
% 0.18/0.39 thf(dsetconstrEL,definition,
% 0.18/0.39 ( dsetconstrEL
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx
% 0.18/0.39 @ ( dsetconstr @ A
% 0.18/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.18/0.39 => ( in @ Xx @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(dsetconstrER_type,type,
% 0.18/0.39 dsetconstrER: $o ).
% 0.18/0.39
% 0.18/0.39 thf(dsetconstrER,definition,
% 0.18/0.39 ( dsetconstrER
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx
% 0.18/0.39 @ ( dsetconstr @ A
% 0.18/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.18/0.39 => ( Xphi @ Xx ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(exuE1_type,type,
% 0.18/0.39 exuE1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(exuE1,definition,
% 0.18/0.39 ( exuE1
% 0.18/0.39 = ( ! [Xphi: $i > $o] :
% 0.18/0.39 ( ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 => ? [Xx: $i] :
% 0.18/0.39 ( ( Xphi @ Xx )
% 0.18/0.39 & ! [Xy: $i] :
% 0.18/0.39 ( ( Xphi @ Xy )
% 0.18/0.39 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(prop2set_type,type,
% 0.18/0.39 prop2set: $o > $i ).
% 0.18/0.39
% 0.18/0.39 thf(prop2setE_type,type,
% 0.18/0.39 prop2setE: $o ).
% 0.18/0.39
% 0.18/0.39 thf(prop2setE,definition,
% 0.18/0.39 ( prop2setE
% 0.18/0.39 = ( ! [Xphi: $o,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.18/0.39 => Xphi ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(emptysetE_type,type,
% 0.18/0.39 emptysetE: $o ).
% 0.18/0.39
% 0.18/0.39 thf(emptysetE,definition,
% 0.18/0.39 ( emptysetE
% 0.18/0.39 = ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ emptyset )
% 0.18/0.39 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(emptysetimpfalse_type,type,
% 0.18/0.39 emptysetimpfalse: $o ).
% 0.18/0.39
% 0.18/0.39 thf(emptysetimpfalse,definition,
% 0.18/0.39 ( emptysetimpfalse
% 0.18/0.39 = ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ emptyset )
% 0.18/0.39 => $false ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(notinemptyset_type,type,
% 0.18/0.39 notinemptyset: $o ).
% 0.18/0.39
% 0.18/0.39 thf(notinemptyset,definition,
% 0.18/0.39 ( notinemptyset
% 0.18/0.39 = ( ! [Xx: $i] :
% 0.18/0.39 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(exuE3e_type,type,
% 0.18/0.39 exuE3e: $o ).
% 0.18/0.39
% 0.18/0.39 thf(exuE3e,definition,
% 0.18/0.39 ( exuE3e
% 0.18/0.39 = ( ! [Xphi: $i > $o] :
% 0.18/0.39 ( ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setext_type,type,
% 0.18/0.39 setext: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setext,definition,
% 0.18/0.39 ( setext
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ Xx @ B ) )
% 0.18/0.39 => ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ B )
% 0.18/0.39 => ( in @ Xx @ A ) )
% 0.18/0.39 => ( A = B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(emptyI_type,type,
% 0.18/0.39 emptyI: $o ).
% 0.18/0.39
% 0.18/0.39 thf(emptyI,definition,
% 0.18/0.39 ( emptyI
% 0.18/0.39 = ( ! [A: $i] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ~ ( in @ Xx @ A )
% 0.18/0.39 => ( A = emptyset ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(noeltsimpempty_type,type,
% 0.18/0.39 noeltsimpempty: $o ).
% 0.18/0.39
% 0.18/0.39 thf(noeltsimpempty,definition,
% 0.18/0.39 ( noeltsimpempty
% 0.18/0.39 = ( ! [A: $i] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ~ ( in @ Xx @ A )
% 0.18/0.39 => ( A = emptyset ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setbeta_type,type,
% 0.18/0.39 setbeta: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setbeta,definition,
% 0.18/0.39 ( setbeta
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( ( in @ Xx
% 0.18/0.39 @ ( dsetconstr @ A
% 0.18/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.18/0.39 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(nonempty_type,type,
% 0.18/0.39 nonempty: $i > $o ).
% 0.18/0.39
% 0.18/0.39 thf(nonempty,definition,
% 0.18/0.39 ( nonempty
% 0.18/0.39 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(nonemptyE1_type,type,
% 0.18/0.39 nonemptyE1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(nonemptyE1,definition,
% 0.18/0.39 ( nonemptyE1
% 0.18/0.39 = ( ! [A: $i] :
% 0.18/0.39 ( ( nonempty @ A )
% 0.18/0.39 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(nonemptyI_type,type,
% 0.18/0.39 nonemptyI: $o ).
% 0.18/0.39
% 0.18/0.39 thf(nonemptyI,definition,
% 0.18/0.39 ( nonemptyI
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( ( Xphi @ Xx )
% 0.18/0.39 => ( nonempty
% 0.18/0.39 @ ( dsetconstr @ A
% 0.18/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(nonemptyI1_type,type,
% 0.18/0.39 nonemptyI1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(nonemptyI1,definition,
% 0.18/0.39 ( nonemptyI1
% 0.18/0.39 = ( ! [A: $i] :
% 0.18/0.39 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.18/0.39 => ( nonempty @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinIL_type,type,
% 0.18/0.39 setadjoinIL: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinIL,definition,
% 0.18/0.39 ( setadjoinIL
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(emptyinunitempty_type,type,
% 0.18/0.39 emptyinunitempty: $o ).
% 0.18/0.39
% 0.18/0.39 thf(emptyinunitempty,definition,
% 0.18/0.39 ( emptyinunitempty
% 0.18/0.39 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinIR_type,type,
% 0.18/0.39 setadjoinIR: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinIR,definition,
% 0.18/0.39 ( setadjoinIR
% 0.18/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ A )
% 0.18/0.39 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinE_type,type,
% 0.18/0.39 setadjoinE: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinE,definition,
% 0.18/0.39 ( setadjoinE
% 0.18/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.18/0.39 => ! [Xphi: $o] :
% 0.18/0.39 ( ( ( Xy = Xx )
% 0.18/0.39 => Xphi )
% 0.18/0.39 => ( ( ( in @ Xy @ A )
% 0.18/0.39 => Xphi )
% 0.18/0.39 => Xphi ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinOr_type,type,
% 0.18/0.39 setadjoinOr: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinOr,definition,
% 0.18/0.39 ( setadjoinOr
% 0.18/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.18/0.39 => ( ( Xy = Xx )
% 0.18/0.39 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setoftrueEq_type,type,
% 0.18/0.39 setoftrueEq: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setoftrueEq,definition,
% 0.18/0.39 ( setoftrueEq
% 0.18/0.39 = ( ! [A: $i] :
% 0.18/0.39 ( ( dsetconstr @ A
% 0.18/0.39 @ ^ [Xx: $i] : $true )
% 0.18/0.39 = A ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(powersetI_type,type,
% 0.18/0.39 powersetI: $o ).
% 0.18/0.39
% 0.18/0.39 thf(powersetI,definition,
% 0.18/0.39 ( powersetI
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ B )
% 0.18/0.39 => ( in @ Xx @ A ) )
% 0.18/0.39 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(emptyinPowerset_type,type,
% 0.18/0.39 emptyinPowerset: $o ).
% 0.18/0.39
% 0.18/0.39 thf(emptyinPowerset,definition,
% 0.18/0.39 ( emptyinPowerset
% 0.18/0.39 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(emptyInPowerset_type,type,
% 0.18/0.39 emptyInPowerset: $o ).
% 0.18/0.39
% 0.18/0.39 thf(emptyInPowerset,definition,
% 0.18/0.39 ( emptyInPowerset
% 0.18/0.39 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(powersetE_type,type,
% 0.18/0.39 powersetE: $o ).
% 0.18/0.39
% 0.18/0.39 thf(powersetE,definition,
% 0.18/0.39 ( powersetE
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ B @ ( powerset @ A ) )
% 0.18/0.39 => ( ( in @ Xx @ B )
% 0.18/0.39 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setunionI_type,type,
% 0.18/0.39 setunionI: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setunionI,definition,
% 0.18/0.39 ( setunionI
% 0.18/0.39 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.18/0.39 ( ( in @ Xx @ B )
% 0.18/0.39 => ( ( in @ B @ A )
% 0.18/0.39 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setunionE_type,type,
% 0.18/0.39 setunionE: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setunionE,definition,
% 0.18/0.39 ( setunionE
% 0.18/0.39 = ( ! [A: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ ( setunion @ A ) )
% 0.18/0.39 => ! [Xphi: $o] :
% 0.18/0.39 ( ! [B: $i] :
% 0.18/0.39 ( ( in @ Xx @ B )
% 0.18/0.39 => ( ( in @ B @ A )
% 0.18/0.39 => Xphi ) )
% 0.18/0.39 => Xphi ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subPowSU_type,type,
% 0.18/0.39 subPowSU: $o ).
% 0.18/0.39
% 0.18/0.39 thf(subPowSU,definition,
% 0.18/0.39 ( subPowSU
% 0.18/0.39 = ( ! [A: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(exuE2_type,type,
% 0.18/0.39 exuE2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(exuE2,definition,
% 0.18/0.39 ( exuE2
% 0.18/0.39 = ( ! [Xphi: $i > $o] :
% 0.18/0.39 ( ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 => ? [Xx: $i] :
% 0.18/0.39 ! [Xy: $i] :
% 0.18/0.39 ( ( Xphi @ Xy )
% 0.18/0.39 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(nonemptyImpWitness_type,type,
% 0.18/0.39 nonemptyImpWitness: $o ).
% 0.18/0.39
% 0.18/0.39 thf(nonemptyImpWitness,definition,
% 0.18/0.39 ( nonemptyImpWitness
% 0.18/0.39 = ( ! [A: $i] :
% 0.18/0.39 ( ( nonempty @ A )
% 0.18/0.39 => ? [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 & $true ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(uniqinunit_type,type,
% 0.18/0.39 uniqinunit: $o ).
% 0.18/0.39
% 0.18/0.39 thf(uniqinunit,definition,
% 0.18/0.39 ( uniqinunit
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.18/0.39 => ( Xx = Xy ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(notinsingleton_type,type,
% 0.18/0.39 notinsingleton: $o ).
% 0.18/0.39
% 0.18/0.39 thf(notinsingleton,definition,
% 0.18/0.39 ( notinsingleton
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xx != Xy )
% 0.18/0.39 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(eqinunit_type,type,
% 0.18/0.39 eqinunit: $o ).
% 0.18/0.39
% 0.18/0.39 thf(eqinunit,definition,
% 0.18/0.39 ( eqinunit
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xx = Xy )
% 0.18/0.39 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(singletonsswitch_type,type,
% 0.18/0.39 singletonsswitch: $o ).
% 0.18/0.39
% 0.18/0.39 thf(singletonsswitch,definition,
% 0.18/0.39 ( singletonsswitch
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.18/0.39 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(upairsetE_type,type,
% 0.18/0.39 upairsetE: $o ).
% 0.18/0.39
% 0.18/0.39 thf(upairsetE,definition,
% 0.18/0.39 ( upairsetE
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.18/0.39 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.18/0.39 => ( ( Xz = Xx )
% 0.18/0.39 | ( Xz = Xy ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(upairsetIL_type,type,
% 0.18/0.39 upairsetIL: $o ).
% 0.18/0.39
% 0.18/0.39 thf(upairsetIL,definition,
% 0.18/0.39 ( upairsetIL
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(upairsetIR_type,type,
% 0.18/0.39 upairsetIR: $o ).
% 0.18/0.39
% 0.18/0.39 thf(upairsetIR,definition,
% 0.18/0.39 ( upairsetIR
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(emptyE1_type,type,
% 0.18/0.39 emptyE1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(emptyE1,definition,
% 0.18/0.39 ( emptyE1
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.39 ( ? [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 & ( Xphi @ Xx ) )
% 0.18/0.39 => ( ( ( dsetconstr @ A
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 = emptyset )
% 0.18/0.39 => $false ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(vacuousDall_type,type,
% 0.18/0.39 vacuousDall: $o ).
% 0.18/0.39
% 0.18/0.39 thf(vacuousDall,definition,
% 0.18/0.39 ( vacuousDall
% 0.18/0.39 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ emptyset )
% 0.18/0.39 => ( Xphi @ Xx ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(quantDeMorgan1_type,type,
% 0.18/0.39 quantDeMorgan1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(quantDeMorgan1,definition,
% 0.18/0.39 ( quantDeMorgan1
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.39 ( ~ ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( Xphi @ Xx ) )
% 0.18/0.39 => ? [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(quantDeMorgan2_type,type,
% 0.18/0.39 quantDeMorgan2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(quantDeMorgan2,definition,
% 0.18/0.39 ( quantDeMorgan2
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ~ ( Xphi @ Xx ) )
% 0.18/0.39 => ~ ? [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(quantDeMorgan3_type,type,
% 0.18/0.39 quantDeMorgan3: $o ).
% 0.18/0.39
% 0.18/0.39 thf(quantDeMorgan3,definition,
% 0.18/0.39 ( quantDeMorgan3
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.39 ( ~ ? [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 & ( Xphi @ Xx ) )
% 0.18/0.39 => ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(quantDeMorgan4_type,type,
% 0.18/0.39 quantDeMorgan4: $o ).
% 0.18/0.39
% 0.18/0.39 thf(quantDeMorgan4,definition,
% 0.18/0.39 ( quantDeMorgan4
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.39 ( ? [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 & ~ ( Xphi @ Xx ) )
% 0.18/0.39 => ~ ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(prop2setI_type,type,
% 0.18/0.39 prop2setI: $o ).
% 0.18/0.39
% 0.18/0.39 thf(prop2setI,definition,
% 0.18/0.39 ( prop2setI
% 0.18/0.39 = ( ! [Xphi: $o] :
% 0.18/0.39 ( Xphi
% 0.18/0.39 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(set2prop_type,type,
% 0.18/0.39 set2prop: $i > $o ).
% 0.18/0.39
% 0.18/0.39 thf(prop2set2propI_type,type,
% 0.18/0.39 prop2set2propI: $o ).
% 0.18/0.39
% 0.18/0.39 thf(prop2set2propI,definition,
% 0.18/0.39 ( prop2set2propI
% 0.18/0.39 = ( ! [Xphi: $o] :
% 0.18/0.39 ( Xphi
% 0.18/0.39 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(notdexE_type,type,
% 0.18/0.39 notdexE: $o ).
% 0.18/0.39
% 0.18/0.39 thf(notdexE,definition,
% 0.18/0.39 ( notdexE
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.39 ( ~ ? [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 & ( Xphi @ Xx ) )
% 0.18/0.39 => ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(notdallE_type,type,
% 0.18/0.39 notdallE: $o ).
% 0.18/0.39
% 0.18/0.39 thf(notdallE,definition,
% 0.18/0.39 ( notdallE
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.39 ( ~ ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( Xphi @ Xx ) )
% 0.18/0.39 => ? [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(exuI1_type,type,
% 0.18/0.39 exuI1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(exuI1,definition,
% 0.18/0.39 ( exuI1
% 0.18/0.39 = ( ! [Xphi: $i > $o] :
% 0.18/0.39 ( ? [Xx: $i] :
% 0.18/0.39 ( ( Xphi @ Xx )
% 0.18/0.39 & ! [Xy: $i] :
% 0.18/0.39 ( ( Xphi @ Xy )
% 0.18/0.39 => ( Xx = Xy ) ) )
% 0.18/0.39 => ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(exuI3_type,type,
% 0.18/0.39 exuI3: $o ).
% 0.18/0.39
% 0.18/0.39 thf(exuI3,definition,
% 0.18/0.39 ( exuI3
% 0.18/0.39 = ( ! [Xphi: $i > $o] :
% 0.18/0.39 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.18/0.39 => ( ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xphi @ Xx )
% 0.18/0.39 => ( ( Xphi @ Xy )
% 0.18/0.39 => ( Xx = Xy ) ) )
% 0.18/0.39 => ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(exuI2_type,type,
% 0.18/0.39 exuI2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(exuI2,definition,
% 0.18/0.39 ( exuI2
% 0.18/0.39 = ( ! [Xphi: $i > $o] :
% 0.18/0.39 ( ? [Xx: $i] :
% 0.18/0.39 ! [Xy: $i] :
% 0.18/0.39 ( ( Xphi @ Xy )
% 0.18/0.39 <=> ( Xy = Xx ) )
% 0.18/0.39 => ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(inCongP_type,type,
% 0.18/0.39 inCongP: $o ).
% 0.18/0.39
% 0.18/0.39 thf(inCongP,definition,
% 0.18/0.39 ( inCongP
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( A = B )
% 0.18/0.39 => ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xx = Xy )
% 0.18/0.39 => ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(in__Cong_type,type,
% 0.18/0.39 in__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(in__Cong,definition,
% 0.18/0.39 ( in__Cong
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( A = B )
% 0.18/0.39 => ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xx = Xy )
% 0.18/0.39 => ( ( in @ Xx @ A )
% 0.18/0.39 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(exuE3u_type,type,
% 0.18/0.39 exuE3u: $o ).
% 0.18/0.39
% 0.18/0.39 thf(exuE3u,definition,
% 0.18/0.39 ( exuE3u
% 0.18/0.39 = ( ! [Xphi: $i > $o] :
% 0.18/0.39 ( ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 => ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xphi @ Xx )
% 0.18/0.39 => ( ( Xphi @ Xy )
% 0.18/0.39 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(exu__Cong_type,type,
% 0.18/0.39 exu__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(exu__Cong,definition,
% 0.18/0.39 ( exu__Cong
% 0.18/0.39 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.18/0.39 ( ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xx = Xy )
% 0.18/0.39 => ( ( Xphi @ Xx )
% 0.18/0.39 <=> ( Xpsi @ Xy ) ) )
% 0.18/0.39 => ( ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 <=> ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(emptyset__Cong_type,type,
% 0.18/0.39 emptyset__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(emptyset__Cong,definition,
% 0.18/0.39 ( emptyset__Cong
% 0.18/0.39 = ( emptyset = emptyset ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoin__Cong_type,type,
% 0.18/0.39 setadjoin__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoin__Cong,definition,
% 0.18/0.39 ( setadjoin__Cong
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xx = Xy )
% 0.18/0.39 => ! [Xz: $i,Xu: $i] :
% 0.18/0.39 ( ( Xz = Xu )
% 0.18/0.39 => ( ( setadjoin @ Xx @ Xz )
% 0.18/0.39 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(powerset__Cong_type,type,
% 0.18/0.39 powerset__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(powerset__Cong,definition,
% 0.18/0.39 ( powerset__Cong
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( A = B )
% 0.18/0.39 => ( ( powerset @ A )
% 0.18/0.39 = ( powerset @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setunion__Cong_type,type,
% 0.18/0.39 setunion__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setunion__Cong,definition,
% 0.18/0.39 ( setunion__Cong
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( A = B )
% 0.18/0.39 => ( ( setunion @ A )
% 0.18/0.39 = ( setunion @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(omega__Cong_type,type,
% 0.18/0.39 omega__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(omega__Cong,definition,
% 0.18/0.39 ( omega__Cong
% 0.18/0.39 = ( omega = omega ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(exuEu_type,type,
% 0.18/0.39 exuEu: $o ).
% 0.18/0.39
% 0.18/0.39 thf(exuEu,definition,
% 0.18/0.39 ( exuEu
% 0.18/0.39 = ( ! [Xphi: $i > $o] :
% 0.18/0.39 ( ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 => ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xphi @ Xx )
% 0.18/0.39 => ( ( Xphi @ Xy )
% 0.18/0.39 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(descr__Cong_type,type,
% 0.18/0.39 descr__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(descr__Cong,definition,
% 0.18/0.39 ( descr__Cong
% 0.18/0.39 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.18/0.39 ( ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xx = Xy )
% 0.18/0.39 => ( ( Xphi @ Xx )
% 0.18/0.39 <=> ( Xpsi @ Xy ) ) )
% 0.18/0.39 => ( ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 => ( ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.18/0.39 => ( ( descr
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 = ( descr
% 0.18/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(dsetconstr__Cong_type,type,
% 0.18/0.39 dsetconstr__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(dsetconstr__Cong,definition,
% 0.18/0.39 ( dsetconstr__Cong
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( A = B )
% 0.18/0.39 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ B )
% 0.18/0.39 => ( ( Xx = Xy )
% 0.18/0.39 => ( ( Xphi @ Xx )
% 0.18/0.39 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.18/0.39 => ( ( dsetconstr @ A
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 = ( dsetconstr @ B
% 0.18/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subset_type,type,
% 0.18/0.39 subset: $i > $i > $o ).
% 0.18/0.39
% 0.18/0.39 thf(disjoint_type,type,
% 0.18/0.39 disjoint: $i > $i > $o ).
% 0.18/0.39
% 0.18/0.39 thf(setsmeet_type,type,
% 0.18/0.39 setsmeet: $i > $i > $o ).
% 0.18/0.39
% 0.18/0.39 thf(subsetI1_type,type,
% 0.18/0.39 subsetI1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(subsetI1,definition,
% 0.18/0.39 ( subsetI1
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ Xx @ B ) )
% 0.18/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(eqimpsubset2_type,type,
% 0.18/0.39 eqimpsubset2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(eqimpsubset2,definition,
% 0.18/0.39 ( eqimpsubset2
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( A = B )
% 0.18/0.39 => ( subset @ B @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(eqimpsubset1_type,type,
% 0.18/0.39 eqimpsubset1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(eqimpsubset1,definition,
% 0.18/0.39 ( eqimpsubset1
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( A = B )
% 0.18/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subsetI2_type,type,
% 0.18/0.39 subsetI2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(subsetI2,definition,
% 0.18/0.39 ( subsetI2
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ Xx @ B ) )
% 0.18/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(emptysetsubset_type,type,
% 0.18/0.39 emptysetsubset: $o ).
% 0.18/0.39
% 0.18/0.39 thf(emptysetsubset,definition,
% 0.18/0.39 ( emptysetsubset
% 0.18/0.39 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subsetE_type,type,
% 0.18/0.39 subsetE: $o ).
% 0.18/0.39
% 0.18/0.39 thf(subsetE,definition,
% 0.18/0.39 ( subsetE
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( subset @ A @ B )
% 0.18/0.39 => ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subsetE2_type,type,
% 0.18/0.39 subsetE2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(subsetE2,definition,
% 0.18/0.39 ( subsetE2
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( subset @ A @ B )
% 0.18/0.39 => ( ~ ( in @ Xx @ B )
% 0.18/0.39 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(notsubsetI_type,type,
% 0.18/0.39 notsubsetI: $o ).
% 0.18/0.39
% 0.18/0.39 thf(notsubsetI,definition,
% 0.18/0.39 ( notsubsetI
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( ~ ( in @ Xx @ B )
% 0.18/0.39 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(notequalI1_type,type,
% 0.18/0.39 notequalI1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(notequalI1,definition,
% 0.18/0.39 ( notequalI1
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ~ ( subset @ A @ B )
% 0.18/0.39 => ( A != B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(notequalI2_type,type,
% 0.18/0.39 notequalI2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(notequalI2,definition,
% 0.18/0.39 ( notequalI2
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( ~ ( in @ Xx @ B )
% 0.18/0.39 => ( A != B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subsetRefl_type,type,
% 0.18/0.39 subsetRefl: $o ).
% 0.18/0.39
% 0.18/0.39 thf(subsetRefl,definition,
% 0.18/0.39 ( subsetRefl
% 0.18/0.39 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subsetTrans_type,type,
% 0.18/0.39 subsetTrans: $o ).
% 0.18/0.39
% 0.18/0.39 thf(subsetTrans,definition,
% 0.18/0.39 ( subsetTrans
% 0.18/0.39 = ( ! [A: $i,B: $i,C: $i] :
% 0.18/0.39 ( ( subset @ A @ B )
% 0.18/0.39 => ( ( subset @ B @ C )
% 0.18/0.39 => ( subset @ A @ C ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinSub_type,type,
% 0.18/0.39 setadjoinSub: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinSub,definition,
% 0.18/0.39 ( setadjoinSub
% 0.18/0.39 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinSub2_type,type,
% 0.18/0.39 setadjoinSub2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinSub2,definition,
% 0.18/0.39 ( setadjoinSub2
% 0.18/0.39 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.18/0.39 ( ( subset @ A @ B )
% 0.18/0.39 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subset2powerset_type,type,
% 0.18/0.39 subset2powerset: $o ).
% 0.18/0.39
% 0.18/0.39 thf(subset2powerset,definition,
% 0.18/0.39 ( subset2powerset
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( subset @ A @ B )
% 0.18/0.39 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setextsub_type,type,
% 0.18/0.39 setextsub: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setextsub,definition,
% 0.18/0.39 ( setextsub
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( subset @ A @ B )
% 0.18/0.39 => ( ( subset @ B @ A )
% 0.18/0.39 => ( A = B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subsetemptysetimpeq,conjecture,
% 0.18/0.39 ( setextAx
% 0.18/0.39 => ( emptysetAx
% 0.18/0.39 => ( setadjoinAx
% 0.18/0.39 => ( powersetAx
% 0.18/0.39 => ( setunionAx
% 0.18/0.39 => ( omega0Ax
% 0.18/0.39 => ( omegaSAx
% 0.18/0.39 => ( omegaIndAx
% 0.18/0.39 => ( replAx
% 0.18/0.39 => ( foundationAx
% 0.18/0.39 => ( wellorderingAx
% 0.18/0.39 => ( descrp
% 0.18/0.39 => ( dsetconstrI
% 0.18/0.39 => ( dsetconstrEL
% 0.18/0.39 => ( dsetconstrER
% 0.18/0.39 => ( exuE1
% 0.18/0.39 => ( prop2setE
% 0.18/0.39 => ( emptysetE
% 0.18/0.39 => ( emptysetimpfalse
% 0.18/0.39 => ( notinemptyset
% 0.18/0.39 => ( exuE3e
% 0.18/0.39 => ( setext
% 0.18/0.39 => ( emptyI
% 0.18/0.39 => ( noeltsimpempty
% 0.18/0.39 => ( setbeta
% 0.18/0.39 => ( nonemptyE1
% 0.18/0.39 => ( nonemptyI
% 0.18/0.39 => ( nonemptyI1
% 0.18/0.39 => ( setadjoinIL
% 0.18/0.39 => ( emptyinunitempty
% 0.18/0.39 => ( setadjoinIR
% 0.18/0.39 => ( setadjoinE
% 0.18/0.39 => ( setadjoinOr
% 0.18/0.39 => ( setoftrueEq
% 0.18/0.39 => ( powersetI
% 0.18/0.39 => ( emptyinPowerset
% 0.18/0.39 => ( emptyInPowerset
% 0.18/0.39 => ( powersetE
% 0.18/0.39 => ( setunionI
% 0.18/0.39 => ( setunionE
% 0.18/0.39 => ( subPowSU
% 0.18/0.39 => ( exuE2
% 0.18/0.39 => ( nonemptyImpWitness
% 0.18/0.39 => ( uniqinunit
% 0.18/0.39 => ( notinsingleton
% 0.18/0.39 => ( eqinunit
% 0.18/0.39 => ( singletonsswitch
% 0.18/0.39 => ( upairsetE
% 0.18/0.39 => ( upairsetIL
% 0.18/0.39 => ( upairsetIR
% 0.18/0.39 => ( emptyE1
% 0.18/0.39 => ( vacuousDall
% 0.18/0.39 => ( quantDeMorgan1
% 0.18/0.39 => ( quantDeMorgan2
% 0.18/0.39 => ( quantDeMorgan3
% 0.18/0.39 => ( quantDeMorgan4
% 0.18/0.39 => ( prop2setI
% 0.18/0.39 => ( prop2set2propI
% 0.18/0.39 => ( notdexE
% 0.18/0.39 => ( notdallE
% 0.18/0.39 => ( exuI1
% 0.18/0.39 => ( exuI3
% 0.18/0.39 => ( exuI2
% 0.18/0.39 => ( inCongP
% 0.18/0.39 => ( in__Cong
% 0.18/0.39 => ( exuE3u
% 0.18/0.39 => ( exu__Cong
% 0.18/0.39 => ( emptyset__Cong
% 0.18/0.39 => ( setadjoin__Cong
% 0.18/0.39 => ( powerset__Cong
% 0.18/0.39 => ( setunion__Cong
% 0.18/0.39 => ( omega__Cong
% 0.18/0.39 => ( exuEu
% 0.18/0.39 => ( descr__Cong
% 0.18/0.39 => ( dsetconstr__Cong
% 0.18/0.39 => ( subsetI1
% 0.18/0.39 => ( eqimpsubset2
% 0.18/0.39 => ( eqimpsubset1
% 0.18/0.39 => ( subsetI2
% 0.18/0.39 => ( emptysetsubset
% 0.18/0.39 => ( subsetE
% 0.18/0.39 => ( subsetE2
% 0.18/0.39 => ( notsubsetI
% 0.18/0.39 => ( notequalI1
% 0.18/0.39 => ( notequalI2
% 0.18/0.39 => ( subsetRefl
% 0.18/0.39 => ( subsetTrans
% 0.18/0.39 => ( setadjoinSub
% 0.18/0.39 => ( setadjoinSub2
% 0.18/0.39 => ( subset2powerset
% 0.18/0.39 => ( setextsub
% 0.18/0.39 => ! [A: $i] :
% 0.18/0.39 ( ( subset @ A @ emptyset )
% 0.18/0.39 => ( A = emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 %------------------------------------------------------------------------------
% 0.18/0.39 ------------------------
% 0.22/1.50 ---- Embedded in THF ---
% 0.22/1.50 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.22/1.50 %%% Generated on Tue Feb 24 12:27:11 EST 2026
% 0.22/1.50 %%% using '$$dhol' embedding, version 1.3.0.
% 0.22/1.50 %%% Logic specification used:
% 0.22/1.50 %%% thf(spec, logic, $$dhol).
% 0.22/1.50
% 0.22/1.50 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.x7ZFQe8mUJ/DTF2DTF11764.p
% See solution above
% 0.22/1.50 ------------------------
% 0.22/1.52 ---- Cleaned THF ---
% 0.22/1.52 thf(in_type,type,
% 0.22/1.52 in: $i > $i > $o ).
% 0.22/1.52
% 0.22/1.52 thf(exu_type,type,
% 0.22/1.52 exu: ( $i > $o ) > $o ).
% 0.22/1.52
% 0.22/1.52 thf(exu,definition,
% 0.22/1.52 ( exu
% 0.22/1.52 = ( ^ [Xphi: $i > $o] :
% 0.22/1.52 ? [Xx: $i] :
% 0.22/1.52 ( ( Xphi @ Xx )
% 0.22/1.52 & ! [Xy: $i] :
% 0.22/1.52 ( ( Xphi @ Xy )
% 0.22/1.52 => ( Xx = Xy ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(setextAx_type,type,
% 0.22/1.52 setextAx: $o ).
% 0.22/1.52
% 0.22/1.52 thf(setextAx,definition,
% 0.22/1.52 ( setextAx
% 0.22/1.52 = ( ! [A: $i,B: $i] :
% 0.22/1.52 ( ! [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 <=> ( in @ Xx @ B ) )
% 0.22/1.52 => ( A = B ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(emptyset_type,type,
% 0.22/1.52 emptyset: $i ).
% 0.22/1.52
% 0.22/1.52 thf(emptysetAx_type,type,
% 0.22/1.52 emptysetAx: $o ).
% 0.22/1.52
% 0.22/1.52 thf(emptysetAx,definition,
% 0.22/1.52 ( emptysetAx
% 0.22/1.52 = ( ! [Xx: $i] :
% 0.22/1.52 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(setadjoin_type,type,
% 0.22/1.52 setadjoin: $i > $i > $i ).
% 0.22/1.52
% 0.22/1.52 thf(setadjoinAx_type,type,
% 0.22/1.52 setadjoinAx: $o ).
% 0.22/1.52
% 0.22/1.52 thf(setadjoinAx,definition,
% 0.22/1.52 ( setadjoinAx
% 0.22/1.52 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.22/1.52 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.22/1.52 <=> ( ( Xy = Xx )
% 0.22/1.52 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(powerset_type,type,
% 0.22/1.52 powerset: $i > $i ).
% 0.22/1.52
% 0.22/1.52 thf(powersetAx_type,type,
% 0.22/1.52 powersetAx: $o ).
% 0.22/1.52
% 0.22/1.52 thf(powersetAx,definition,
% 0.22/1.52 ( powersetAx
% 0.22/1.52 = ( ! [A: $i,B: $i] :
% 0.22/1.52 ( ( in @ B @ ( powerset @ A ) )
% 0.22/1.52 <=> ! [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ B )
% 0.22/1.52 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(setunion_type,type,
% 0.22/1.52 setunion: $i > $i ).
% 0.22/1.52
% 0.22/1.52 thf(setunionAx_type,type,
% 0.22/1.52 setunionAx: $o ).
% 0.22/1.52
% 0.22/1.52 thf(setunionAx,definition,
% 0.22/1.52 ( setunionAx
% 0.22/1.52 = ( ! [A: $i,Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ ( setunion @ A ) )
% 0.22/1.52 <=> ? [B: $i] :
% 0.22/1.52 ( ( in @ Xx @ B )
% 0.22/1.52 & ( in @ B @ A ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(omega_type,type,
% 0.22/1.52 omega: $i ).
% 0.22/1.52
% 0.22/1.52 thf(omega0Ax_type,type,
% 0.22/1.52 omega0Ax: $o ).
% 0.22/1.52
% 0.22/1.52 thf(omega0Ax,definition,
% 0.22/1.52 ( omega0Ax
% 0.22/1.52 = ( in @ emptyset @ omega ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(omegaSAx_type,type,
% 0.22/1.52 omegaSAx: $o ).
% 0.22/1.52
% 0.22/1.52 thf(omegaSAx,definition,
% 0.22/1.52 ( omegaSAx
% 0.22/1.52 = ( ! [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ omega )
% 0.22/1.52 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(omegaIndAx_type,type,
% 0.22/1.52 omegaIndAx: $o ).
% 0.22/1.52
% 0.22/1.52 thf(omegaIndAx,definition,
% 0.22/1.52 ( omegaIndAx
% 0.22/1.52 = ( ! [A: $i] :
% 0.22/1.52 ( ( ( in @ emptyset @ A )
% 0.22/1.52 & ! [Xx: $i] :
% 0.22/1.52 ( ( ( in @ Xx @ omega )
% 0.22/1.52 & ( in @ Xx @ A ) )
% 0.22/1.52 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.22/1.52 => ! [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ omega )
% 0.22/1.52 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(replAx_type,type,
% 0.22/1.52 replAx: $o ).
% 0.22/1.52
% 0.22/1.52 thf(replAx,definition,
% 0.22/1.52 ( replAx
% 0.22/1.52 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.22/1.52 ( ! [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 => ( exu
% 0.22/1.52 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.22/1.52 => ? [B: $i] :
% 0.22/1.52 ! [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ B )
% 0.22/1.52 <=> ? [Xy: $i] :
% 0.22/1.52 ( ( in @ Xy @ A )
% 0.22/1.52 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(foundationAx_type,type,
% 0.22/1.52 foundationAx: $o ).
% 0.22/1.52
% 0.22/1.52 thf(foundationAx,definition,
% 0.22/1.52 ( foundationAx
% 0.22/1.52 = ( ! [A: $i] :
% 0.22/1.52 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.22/1.52 => ? [B: $i] :
% 0.22/1.52 ( ( in @ B @ A )
% 0.22/1.52 & ~ ? [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ B )
% 0.22/1.52 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(wellorderingAx_type,type,
% 0.22/1.52 wellorderingAx: $o ).
% 0.22/1.52
% 0.22/1.52 thf(wellorderingAx,definition,
% 0.22/1.52 ( wellorderingAx
% 0.22/1.52 = ( ! [A: $i] :
% 0.22/1.52 ? [B: $i] :
% 0.22/1.52 ( ! [C: $i] :
% 0.22/1.52 ( ( in @ C @ B )
% 0.22/1.52 => ! [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ C )
% 0.22/1.52 => ( in @ Xx @ A ) ) )
% 0.22/1.52 & ! [Xx: $i,Xy: $i] :
% 0.22/1.52 ( ( ( in @ Xx @ A )
% 0.22/1.52 & ( in @ Xy @ A ) )
% 0.22/1.52 => ( ! [C: $i] :
% 0.22/1.52 ( ( in @ C @ B )
% 0.22/1.52 => ( ( in @ Xx @ C )
% 0.22/1.52 <=> ( in @ Xy @ C ) ) )
% 0.22/1.52 => ( Xx = Xy ) ) )
% 0.22/1.52 & ! [C: $i,D: $i] :
% 0.22/1.52 ( ( ( in @ C @ B )
% 0.22/1.52 & ( in @ D @ B ) )
% 0.22/1.52 => ( ! [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ C )
% 0.22/1.52 => ( in @ Xx @ D ) )
% 0.22/1.52 | ! [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ D )
% 0.22/1.52 => ( in @ Xx @ C ) ) ) )
% 0.22/1.52 & ! [C: $i] :
% 0.22/1.52 ( ( ! [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ C )
% 0.22/1.52 => ( in @ Xx @ A ) )
% 0.22/1.52 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.22/1.52 => ? [D: $i,Xx: $i] :
% 0.22/1.52 ( ( in @ D @ B )
% 0.22/1.52 & ( in @ Xx @ C )
% 0.22/1.52 & ~ ? [Xy: $i] :
% 0.22/1.52 ( ( in @ Xy @ D )
% 0.22/1.52 & ( in @ Xy @ C ) )
% 0.22/1.52 & ! [E: $i] :
% 0.22/1.52 ( ( in @ E @ B )
% 0.22/1.52 => ( ! [Xy: $i] :
% 0.22/1.52 ( ( in @ Xy @ E )
% 0.22/1.52 => ( in @ Xy @ D ) )
% 0.22/1.52 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(descr_type,type,
% 0.22/1.52 descr: ( $i > $o ) > $i ).
% 0.22/1.52
% 0.22/1.52 thf(descrp_type,type,
% 0.22/1.52 descrp: $o ).
% 0.22/1.52
% 0.22/1.52 thf(descrp,definition,
% 0.22/1.52 ( descrp
% 0.22/1.52 = ( ! [Xphi: $i > $o] :
% 0.22/1.52 ( ( exu
% 0.22/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.22/1.52 => ( Xphi
% 0.22/1.52 @ ( descr
% 0.22/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(dsetconstr_type,type,
% 0.22/1.52 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.22/1.52
% 0.22/1.52 thf(dsetconstrI_type,type,
% 0.22/1.52 dsetconstrI: $o ).
% 0.22/1.52
% 0.22/1.52 thf(dsetconstrI,definition,
% 0.22/1.52 ( dsetconstrI
% 0.22/1.52 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 => ( ( Xphi @ Xx )
% 0.22/1.52 => ( in @ Xx
% 0.22/1.52 @ ( dsetconstr @ A
% 0.22/1.52 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(dsetconstrEL_type,type,
% 0.22/1.52 dsetconstrEL: $o ).
% 0.22/1.52
% 0.22/1.52 thf(dsetconstrEL,definition,
% 0.22/1.52 ( dsetconstrEL
% 0.22/1.52 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.22/1.52 ( ( in @ Xx
% 0.22/1.52 @ ( dsetconstr @ A
% 0.22/1.52 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.22/1.52 => ( in @ Xx @ A ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(dsetconstrER_type,type,
% 0.22/1.52 dsetconstrER: $o ).
% 0.22/1.52
% 0.22/1.52 thf(dsetconstrER,definition,
% 0.22/1.52 ( dsetconstrER
% 0.22/1.52 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.22/1.52 ( ( in @ Xx
% 0.22/1.52 @ ( dsetconstr @ A
% 0.22/1.52 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.22/1.52 => ( Xphi @ Xx ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(exuE1_type,type,
% 0.22/1.52 exuE1: $o ).
% 0.22/1.52
% 0.22/1.52 thf(exuE1,definition,
% 0.22/1.52 ( exuE1
% 0.22/1.52 = ( ! [Xphi: $i > $o] :
% 0.22/1.52 ( ( exu
% 0.22/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.22/1.52 => ? [Xx: $i] :
% 0.22/1.52 ( ( Xphi @ Xx )
% 0.22/1.52 & ! [Xy: $i] :
% 0.22/1.52 ( ( Xphi @ Xy )
% 0.22/1.52 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(prop2set_type,type,
% 0.22/1.52 prop2set: $o > $i ).
% 0.22/1.52
% 0.22/1.52 thf(prop2setE_type,type,
% 0.22/1.52 prop2setE: $o ).
% 0.22/1.52
% 0.22/1.52 thf(prop2setE,definition,
% 0.22/1.52 ( prop2setE
% 0.22/1.52 = ( ! [Xphi: $o,Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.22/1.52 => Xphi ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(emptysetE_type,type,
% 0.22/1.52 emptysetE: $o ).
% 0.22/1.52
% 0.22/1.52 thf(emptysetE,definition,
% 0.22/1.52 ( emptysetE
% 0.22/1.52 = ( ! [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ emptyset )
% 0.22/1.52 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(emptysetimpfalse_type,type,
% 0.22/1.52 emptysetimpfalse: $o ).
% 0.22/1.52
% 0.22/1.52 thf(emptysetimpfalse,definition,
% 0.22/1.52 ( emptysetimpfalse
% 0.22/1.52 = ( ! [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ emptyset )
% 0.22/1.52 => $false ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(notinemptyset_type,type,
% 0.22/1.52 notinemptyset: $o ).
% 0.22/1.52
% 0.22/1.52 thf(notinemptyset,definition,
% 0.22/1.52 ( notinemptyset
% 0.22/1.52 = ( ! [Xx: $i] :
% 0.22/1.52 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(exuE3e_type,type,
% 0.22/1.52 exuE3e: $o ).
% 0.22/1.52
% 0.22/1.52 thf(exuE3e,definition,
% 0.22/1.52 ( exuE3e
% 0.22/1.52 = ( ! [Xphi: $i > $o] :
% 0.22/1.52 ( ( exu
% 0.22/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.22/1.52 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(setext_type,type,
% 0.22/1.52 setext: $o ).
% 0.22/1.52
% 0.22/1.52 thf(setext,definition,
% 0.22/1.52 ( setext
% 0.22/1.52 = ( ! [A: $i,B: $i] :
% 0.22/1.52 ( ! [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 => ( in @ Xx @ B ) )
% 0.22/1.52 => ( ! [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ B )
% 0.22/1.52 => ( in @ Xx @ A ) )
% 0.22/1.52 => ( A = B ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(emptyI_type,type,
% 0.22/1.52 emptyI: $o ).
% 0.22/1.52
% 0.22/1.52 thf(emptyI,definition,
% 0.22/1.52 ( emptyI
% 0.22/1.52 = ( ! [A: $i] :
% 0.22/1.52 ( ! [Xx: $i] :
% 0.22/1.52 ~ ( in @ Xx @ A )
% 0.22/1.52 => ( A = emptyset ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(noeltsimpempty_type,type,
% 0.22/1.52 noeltsimpempty: $o ).
% 0.22/1.52
% 0.22/1.52 thf(noeltsimpempty,definition,
% 0.22/1.52 ( noeltsimpempty
% 0.22/1.52 = ( ! [A: $i] :
% 0.22/1.52 ( ! [Xx: $i] :
% 0.22/1.52 ~ ( in @ Xx @ A )
% 0.22/1.52 => ( A = emptyset ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(setbeta_type,type,
% 0.22/1.52 setbeta: $o ).
% 0.22/1.52
% 0.22/1.52 thf(setbeta,definition,
% 0.22/1.52 ( setbeta
% 0.22/1.52 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 => ( ( in @ Xx
% 0.22/1.52 @ ( dsetconstr @ A
% 0.22/1.52 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.22/1.52 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(nonempty_type,type,
% 0.22/1.52 nonempty: $i > $o ).
% 0.22/1.52
% 0.22/1.52 thf(nonempty,definition,
% 0.22/1.52 ( nonempty
% 0.22/1.52 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(nonemptyE1_type,type,
% 0.22/1.52 nonemptyE1: $o ).
% 0.22/1.52
% 0.22/1.52 thf(nonemptyE1,definition,
% 0.22/1.52 ( nonemptyE1
% 0.22/1.52 = ( ! [A: $i] :
% 0.22/1.52 ( ( nonempty @ A )
% 0.22/1.52 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(nonemptyI_type,type,
% 0.22/1.52 nonemptyI: $o ).
% 0.22/1.52
% 0.22/1.52 thf(nonemptyI,definition,
% 0.22/1.52 ( nonemptyI
% 0.22/1.52 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 => ( ( Xphi @ Xx )
% 0.22/1.52 => ( nonempty
% 0.22/1.52 @ ( dsetconstr @ A
% 0.22/1.52 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(nonemptyI1_type,type,
% 0.22/1.52 nonemptyI1: $o ).
% 0.22/1.52
% 0.22/1.52 thf(nonemptyI1,definition,
% 0.22/1.52 ( nonemptyI1
% 0.22/1.52 = ( ! [A: $i] :
% 0.22/1.52 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.22/1.52 => ( nonempty @ A ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(setadjoinIL_type,type,
% 0.22/1.52 setadjoinIL: $o ).
% 0.22/1.52
% 0.22/1.52 thf(setadjoinIL,definition,
% 0.22/1.52 ( setadjoinIL
% 0.22/1.52 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(emptyinunitempty_type,type,
% 0.22/1.52 emptyinunitempty: $o ).
% 0.22/1.52
% 0.22/1.52 thf(emptyinunitempty,definition,
% 0.22/1.52 ( emptyinunitempty
% 0.22/1.52 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(setadjoinIR_type,type,
% 0.22/1.52 setadjoinIR: $o ).
% 0.22/1.52
% 0.22/1.52 thf(setadjoinIR,definition,
% 0.22/1.52 ( setadjoinIR
% 0.22/1.52 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.22/1.52 ( ( in @ Xy @ A )
% 0.22/1.52 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(setadjoinE_type,type,
% 0.22/1.52 setadjoinE: $o ).
% 0.22/1.52
% 0.22/1.52 thf(setadjoinE,definition,
% 0.22/1.52 ( setadjoinE
% 0.22/1.52 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.22/1.52 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.22/1.52 => ! [Xphi: $o] :
% 0.22/1.52 ( ( ( Xy = Xx )
% 0.22/1.52 => Xphi )
% 0.22/1.52 => ( ( ( in @ Xy @ A )
% 0.22/1.52 => Xphi )
% 0.22/1.52 => Xphi ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(setadjoinOr_type,type,
% 0.22/1.52 setadjoinOr: $o ).
% 0.22/1.52
% 0.22/1.52 thf(setadjoinOr,definition,
% 0.22/1.52 ( setadjoinOr
% 0.22/1.52 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.22/1.52 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.22/1.52 => ( ( Xy = Xx )
% 0.22/1.52 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(setoftrueEq_type,type,
% 0.22/1.52 setoftrueEq: $o ).
% 0.22/1.52
% 0.22/1.52 thf(setoftrueEq,definition,
% 0.22/1.52 ( setoftrueEq
% 0.22/1.52 = ( ! [A: $i] :
% 0.22/1.52 ( ( dsetconstr @ A
% 0.22/1.52 @ ^ [Xx: $i] : $true )
% 0.22/1.52 = A ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(powersetI_type,type,
% 0.22/1.52 powersetI: $o ).
% 0.22/1.52
% 0.22/1.52 thf(powersetI,definition,
% 0.22/1.52 ( powersetI
% 0.22/1.52 = ( ! [A: $i,B: $i] :
% 0.22/1.52 ( ! [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ B )
% 0.22/1.52 => ( in @ Xx @ A ) )
% 0.22/1.52 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(emptyinPowerset_type,type,
% 0.22/1.52 emptyinPowerset: $o ).
% 0.22/1.52
% 0.22/1.52 thf(emptyinPowerset,definition,
% 0.22/1.52 ( emptyinPowerset
% 0.22/1.52 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(emptyInPowerset_type,type,
% 0.22/1.52 emptyInPowerset: $o ).
% 0.22/1.52
% 0.22/1.52 thf(emptyInPowerset,definition,
% 0.22/1.52 ( emptyInPowerset
% 0.22/1.52 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(powersetE_type,type,
% 0.22/1.52 powersetE: $o ).
% 0.22/1.52
% 0.22/1.52 thf(powersetE,definition,
% 0.22/1.52 ( powersetE
% 0.22/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/1.52 ( ( in @ B @ ( powerset @ A ) )
% 0.22/1.52 => ( ( in @ Xx @ B )
% 0.22/1.52 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(setunionI_type,type,
% 0.22/1.52 setunionI: $o ).
% 0.22/1.52
% 0.22/1.52 thf(setunionI,definition,
% 0.22/1.52 ( setunionI
% 0.22/1.52 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.22/1.52 ( ( in @ Xx @ B )
% 0.22/1.52 => ( ( in @ B @ A )
% 0.22/1.52 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(setunionE_type,type,
% 0.22/1.52 setunionE: $o ).
% 0.22/1.52
% 0.22/1.52 thf(setunionE,definition,
% 0.22/1.52 ( setunionE
% 0.22/1.52 = ( ! [A: $i,Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ ( setunion @ A ) )
% 0.22/1.52 => ! [Xphi: $o] :
% 0.22/1.52 ( ! [B: $i] :
% 0.22/1.52 ( ( in @ Xx @ B )
% 0.22/1.52 => ( ( in @ B @ A )
% 0.22/1.52 => Xphi ) )
% 0.22/1.52 => Xphi ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(subPowSU_type,type,
% 0.22/1.52 subPowSU: $o ).
% 0.22/1.52
% 0.22/1.52 thf(subPowSU,definition,
% 0.22/1.52 ( subPowSU
% 0.22/1.52 = ( ! [A: $i,Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(exuE2_type,type,
% 0.22/1.52 exuE2: $o ).
% 0.22/1.52
% 0.22/1.52 thf(exuE2,definition,
% 0.22/1.52 ( exuE2
% 0.22/1.52 = ( ! [Xphi: $i > $o] :
% 0.22/1.52 ( ( exu
% 0.22/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.22/1.52 => ? [Xx: $i] :
% 0.22/1.52 ! [Xy: $i] :
% 0.22/1.52 ( ( Xphi @ Xy )
% 0.22/1.52 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(nonemptyImpWitness_type,type,
% 0.22/1.52 nonemptyImpWitness: $o ).
% 0.22/1.52
% 0.22/1.52 thf(nonemptyImpWitness,definition,
% 0.22/1.52 ( nonemptyImpWitness
% 0.22/1.52 = ( ! [A: $i] :
% 0.22/1.52 ( ( nonempty @ A )
% 0.22/1.52 => ? [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 & $true ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(uniqinunit_type,type,
% 0.22/1.52 uniqinunit: $o ).
% 0.22/1.52
% 0.22/1.52 thf(uniqinunit,definition,
% 0.22/1.52 ( uniqinunit
% 0.22/1.52 = ( ! [Xx: $i,Xy: $i] :
% 0.22/1.52 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.22/1.52 => ( Xx = Xy ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(notinsingleton_type,type,
% 0.22/1.52 notinsingleton: $o ).
% 0.22/1.52
% 0.22/1.52 thf(notinsingleton,definition,
% 0.22/1.52 ( notinsingleton
% 0.22/1.52 = ( ! [Xx: $i,Xy: $i] :
% 0.22/1.52 ( ( Xx != Xy )
% 0.22/1.52 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(eqinunit_type,type,
% 0.22/1.52 eqinunit: $o ).
% 0.22/1.52
% 0.22/1.52 thf(eqinunit,definition,
% 0.22/1.52 ( eqinunit
% 0.22/1.52 = ( ! [Xx: $i,Xy: $i] :
% 0.22/1.52 ( ( Xx = Xy )
% 0.22/1.52 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(singletonsswitch_type,type,
% 0.22/1.52 singletonsswitch: $o ).
% 0.22/1.52
% 0.22/1.52 thf(singletonsswitch,definition,
% 0.22/1.52 ( singletonsswitch
% 0.22/1.52 = ( ! [Xx: $i,Xy: $i] :
% 0.22/1.52 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.22/1.52 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(upairsetE_type,type,
% 0.22/1.52 upairsetE: $o ).
% 0.22/1.52
% 0.22/1.52 thf(upairsetE,definition,
% 0.22/1.52 ( upairsetE
% 0.22/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.22/1.52 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.22/1.52 => ( ( Xz = Xx )
% 0.22/1.52 | ( Xz = Xy ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(upairsetIL_type,type,
% 0.22/1.52 upairsetIL: $o ).
% 0.22/1.52
% 0.22/1.52 thf(upairsetIL,definition,
% 0.22/1.52 ( upairsetIL
% 0.22/1.52 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(upairsetIR_type,type,
% 0.22/1.52 upairsetIR: $o ).
% 0.22/1.52
% 0.22/1.52 thf(upairsetIR,definition,
% 0.22/1.52 ( upairsetIR
% 0.22/1.52 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(emptyE1_type,type,
% 0.22/1.52 emptyE1: $o ).
% 0.22/1.52
% 0.22/1.52 thf(emptyE1,definition,
% 0.22/1.52 ( emptyE1
% 0.22/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 0.22/1.52 ( ? [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 & ( Xphi @ Xx ) )
% 0.22/1.52 => ( ( ( dsetconstr @ A
% 0.22/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.22/1.52 = emptyset )
% 0.22/1.52 => $false ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(vacuousDall_type,type,
% 0.22/1.52 vacuousDall: $o ).
% 0.22/1.52
% 0.22/1.52 thf(vacuousDall,definition,
% 0.22/1.52 ( vacuousDall
% 0.22/1.52 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ emptyset )
% 0.22/1.52 => ( Xphi @ Xx ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(quantDeMorgan1_type,type,
% 0.22/1.52 quantDeMorgan1: $o ).
% 0.22/1.52
% 0.22/1.52 thf(quantDeMorgan1,definition,
% 0.22/1.52 ( quantDeMorgan1
% 0.22/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 0.22/1.52 ( ~ ! [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 => ( Xphi @ Xx ) )
% 0.22/1.52 => ? [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(quantDeMorgan2_type,type,
% 0.22/1.52 quantDeMorgan2: $o ).
% 0.22/1.52
% 0.22/1.52 thf(quantDeMorgan2,definition,
% 0.22/1.52 ( quantDeMorgan2
% 0.22/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 0.22/1.52 ( ! [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 => ~ ( Xphi @ Xx ) )
% 0.22/1.52 => ~ ? [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(quantDeMorgan3_type,type,
% 0.22/1.52 quantDeMorgan3: $o ).
% 0.22/1.52
% 0.22/1.52 thf(quantDeMorgan3,definition,
% 0.22/1.52 ( quantDeMorgan3
% 0.22/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 0.22/1.52 ( ~ ? [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 & ( Xphi @ Xx ) )
% 0.22/1.52 => ! [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(quantDeMorgan4_type,type,
% 0.22/1.52 quantDeMorgan4: $o ).
% 0.22/1.52
% 0.22/1.52 thf(quantDeMorgan4,definition,
% 0.22/1.52 ( quantDeMorgan4
% 0.22/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 0.22/1.52 ( ? [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 & ~ ( Xphi @ Xx ) )
% 0.22/1.52 => ~ ! [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(prop2setI_type,type,
% 0.22/1.52 prop2setI: $o ).
% 0.22/1.52
% 0.22/1.52 thf(prop2setI,definition,
% 0.22/1.52 ( prop2setI
% 0.22/1.52 = ( ! [Xphi: $o] :
% 0.22/1.52 ( Xphi
% 0.22/1.52 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(set2prop_type,type,
% 0.22/1.52 set2prop: $i > $o ).
% 0.22/1.52
% 0.22/1.52 thf(prop2set2propI_type,type,
% 0.22/1.52 prop2set2propI: $o ).
% 0.22/1.52
% 0.22/1.52 thf(prop2set2propI,definition,
% 0.22/1.52 ( prop2set2propI
% 0.22/1.52 = ( ! [Xphi: $o] :
% 0.22/1.52 ( Xphi
% 0.22/1.52 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(notdexE_type,type,
% 0.22/1.52 notdexE: $o ).
% 0.22/1.52
% 0.22/1.52 thf(notdexE,definition,
% 0.22/1.52 ( notdexE
% 0.22/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 0.22/1.52 ( ~ ? [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 & ( Xphi @ Xx ) )
% 0.22/1.52 => ! [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(notdallE_type,type,
% 0.22/1.52 notdallE: $o ).
% 0.22/1.52
% 0.22/1.52 thf(notdallE,definition,
% 0.22/1.52 ( notdallE
% 0.22/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 0.22/1.52 ( ~ ! [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 => ( Xphi @ Xx ) )
% 0.22/1.52 => ? [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(exuI1_type,type,
% 0.22/1.52 exuI1: $o ).
% 0.22/1.52
% 0.22/1.52 thf(exuI1,definition,
% 0.22/1.52 ( exuI1
% 0.22/1.52 = ( ! [Xphi: $i > $o] :
% 0.22/1.52 ( ? [Xx: $i] :
% 0.22/1.52 ( ( Xphi @ Xx )
% 0.22/1.52 & ! [Xy: $i] :
% 0.22/1.52 ( ( Xphi @ Xy )
% 0.22/1.52 => ( Xx = Xy ) ) )
% 0.22/1.52 => ( exu
% 0.22/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(exuI3_type,type,
% 0.22/1.52 exuI3: $o ).
% 0.22/1.52
% 0.22/1.52 thf(exuI3,definition,
% 0.22/1.52 ( exuI3
% 0.22/1.52 = ( ! [Xphi: $i > $o] :
% 0.22/1.52 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.22/1.52 => ( ! [Xx: $i,Xy: $i] :
% 0.22/1.52 ( ( Xphi @ Xx )
% 0.22/1.52 => ( ( Xphi @ Xy )
% 0.22/1.52 => ( Xx = Xy ) ) )
% 0.22/1.52 => ( exu
% 0.22/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(exuI2_type,type,
% 0.22/1.52 exuI2: $o ).
% 0.22/1.52
% 0.22/1.52 thf(exuI2,definition,
% 0.22/1.52 ( exuI2
% 0.22/1.52 = ( ! [Xphi: $i > $o] :
% 0.22/1.52 ( ? [Xx: $i] :
% 0.22/1.52 ! [Xy: $i] :
% 0.22/1.52 ( ( Xphi @ Xy )
% 0.22/1.52 <=> ( Xy = Xx ) )
% 0.22/1.52 => ( exu
% 0.22/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(inCongP_type,type,
% 0.22/1.52 inCongP: $o ).
% 0.22/1.52
% 0.22/1.52 thf(inCongP,definition,
% 0.22/1.52 ( inCongP
% 0.22/1.52 = ( ! [A: $i,B: $i] :
% 0.22/1.52 ( ( A = B )
% 0.22/1.52 => ! [Xx: $i,Xy: $i] :
% 0.22/1.52 ( ( Xx = Xy )
% 0.22/1.52 => ( ( in @ Xx @ A )
% 0.22/1.52 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(in__Cong_type,type,
% 0.22/1.52 in__Cong: $o ).
% 0.22/1.52
% 0.22/1.52 thf(in__Cong,definition,
% 0.22/1.52 ( in__Cong
% 0.22/1.52 = ( ! [A: $i,B: $i] :
% 0.22/1.52 ( ( A = B )
% 0.22/1.52 => ! [Xx: $i,Xy: $i] :
% 0.22/1.52 ( ( Xx = Xy )
% 0.22/1.52 => ( ( in @ Xx @ A )
% 0.22/1.52 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(exuE3u_type,type,
% 0.22/1.52 exuE3u: $o ).
% 0.22/1.52
% 0.22/1.52 thf(exuE3u,definition,
% 0.22/1.52 ( exuE3u
% 0.22/1.52 = ( ! [Xphi: $i > $o] :
% 0.22/1.52 ( ( exu
% 0.22/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.22/1.52 => ! [Xx: $i,Xy: $i] :
% 0.22/1.52 ( ( Xphi @ Xx )
% 0.22/1.52 => ( ( Xphi @ Xy )
% 0.22/1.52 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(exu__Cong_type,type,
% 0.22/1.52 exu__Cong: $o ).
% 0.22/1.52
% 0.22/1.52 thf(exu__Cong,definition,
% 0.22/1.52 ( exu__Cong
% 0.22/1.52 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.22/1.52 ( ! [Xx: $i,Xy: $i] :
% 0.22/1.52 ( ( Xx = Xy )
% 0.22/1.52 => ( ( Xphi @ Xx )
% 0.22/1.52 <=> ( Xpsi @ Xy ) ) )
% 0.22/1.52 => ( ( exu
% 0.22/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.22/1.52 <=> ( exu
% 0.22/1.52 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(emptyset__Cong_type,type,
% 0.22/1.52 emptyset__Cong: $o ).
% 0.22/1.52
% 0.22/1.52 thf(emptyset__Cong,definition,
% 0.22/1.52 ( emptyset__Cong
% 0.22/1.52 = ( emptyset = emptyset ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(setadjoin__Cong_type,type,
% 0.22/1.52 setadjoin__Cong: $o ).
% 0.22/1.52
% 0.22/1.52 thf(setadjoin__Cong,definition,
% 0.22/1.52 ( setadjoin__Cong
% 0.22/1.52 = ( ! [Xx: $i,Xy: $i] :
% 0.22/1.52 ( ( Xx = Xy )
% 0.22/1.52 => ! [Xz: $i,Xu: $i] :
% 0.22/1.52 ( ( Xz = Xu )
% 0.22/1.52 => ( ( setadjoin @ Xx @ Xz )
% 0.22/1.52 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(powerset__Cong_type,type,
% 0.22/1.52 powerset__Cong: $o ).
% 0.22/1.52
% 0.22/1.52 thf(powerset__Cong,definition,
% 0.22/1.52 ( powerset__Cong
% 0.22/1.52 = ( ! [A: $i,B: $i] :
% 0.22/1.52 ( ( A = B )
% 0.22/1.52 => ( ( powerset @ A )
% 0.22/1.52 = ( powerset @ B ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(setunion__Cong_type,type,
% 0.22/1.52 setunion__Cong: $o ).
% 0.22/1.52
% 0.22/1.52 thf(setunion__Cong,definition,
% 0.22/1.52 ( setunion__Cong
% 0.22/1.52 = ( ! [A: $i,B: $i] :
% 0.22/1.52 ( ( A = B )
% 0.22/1.52 => ( ( setunion @ A )
% 0.22/1.52 = ( setunion @ B ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(omega__Cong_type,type,
% 0.22/1.52 omega__Cong: $o ).
% 0.22/1.52
% 0.22/1.52 thf(omega__Cong,definition,
% 0.22/1.52 ( omega__Cong
% 0.22/1.52 = ( omega = omega ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(exuEu_type,type,
% 0.22/1.52 exuEu: $o ).
% 0.22/1.52
% 0.22/1.52 thf(exuEu,definition,
% 0.22/1.52 ( exuEu
% 0.22/1.52 = ( ! [Xphi: $i > $o] :
% 0.22/1.52 ( ( exu
% 0.22/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.22/1.52 => ! [Xx: $i,Xy: $i] :
% 0.22/1.52 ( ( Xphi @ Xx )
% 0.22/1.52 => ( ( Xphi @ Xy )
% 0.22/1.52 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(descr__Cong_type,type,
% 0.22/1.52 descr__Cong: $o ).
% 0.22/1.52
% 0.22/1.52 thf(descr__Cong,definition,
% 0.22/1.52 ( descr__Cong
% 0.22/1.52 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.22/1.52 ( ! [Xx: $i,Xy: $i] :
% 0.22/1.52 ( ( Xx = Xy )
% 0.22/1.52 => ( ( Xphi @ Xx )
% 0.22/1.52 <=> ( Xpsi @ Xy ) ) )
% 0.22/1.52 => ( ( exu
% 0.22/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.22/1.52 => ( ( exu
% 0.22/1.52 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.22/1.52 => ( ( descr
% 0.22/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.22/1.52 = ( descr
% 0.22/1.52 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(dsetconstr__Cong_type,type,
% 0.22/1.52 dsetconstr__Cong: $o ).
% 0.22/1.52
% 0.22/1.52 thf(dsetconstr__Cong,definition,
% 0.22/1.52 ( dsetconstr__Cong
% 0.22/1.52 = ( ! [A: $i,B: $i] :
% 0.22/1.52 ( ( A = B )
% 0.22/1.52 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.22/1.52 ( ! [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 => ! [Xy: $i] :
% 0.22/1.52 ( ( in @ Xy @ B )
% 0.22/1.52 => ( ( Xx = Xy )
% 0.22/1.52 => ( ( Xphi @ Xx )
% 0.22/1.52 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.22/1.52 => ( ( dsetconstr @ A
% 0.22/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.22/1.52 = ( dsetconstr @ B
% 0.22/1.52 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(subset_type,type,
% 0.22/1.52 subset: $i > $i > $o ).
% 0.22/1.52
% 0.22/1.52 thf(disjoint_type,type,
% 0.22/1.52 disjoint: $i > $i > $o ).
% 0.22/1.52
% 0.22/1.52 thf(setsmeet_type,type,
% 0.22/1.52 setsmeet: $i > $i > $o ).
% 0.22/1.52
% 0.22/1.52 thf(subsetI1_type,type,
% 0.22/1.52 subsetI1: $o ).
% 0.22/1.52
% 0.22/1.52 thf(subsetI1,definition,
% 0.22/1.52 ( subsetI1
% 0.22/1.52 = ( ! [A: $i,B: $i] :
% 0.22/1.52 ( ! [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 => ( in @ Xx @ B ) )
% 0.22/1.52 => ( subset @ A @ B ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(eqimpsubset2_type,type,
% 0.22/1.52 eqimpsubset2: $o ).
% 0.22/1.52
% 0.22/1.52 thf(eqimpsubset2,definition,
% 0.22/1.52 ( eqimpsubset2
% 0.22/1.52 = ( ! [A: $i,B: $i] :
% 0.22/1.52 ( ( A = B )
% 0.22/1.52 => ( subset @ B @ A ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(eqimpsubset1_type,type,
% 0.22/1.52 eqimpsubset1: $o ).
% 0.22/1.52
% 0.22/1.52 thf(eqimpsubset1,definition,
% 0.22/1.52 ( eqimpsubset1
% 0.22/1.52 = ( ! [A: $i,B: $i] :
% 0.22/1.52 ( ( A = B )
% 0.22/1.52 => ( subset @ A @ B ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(subsetI2_type,type,
% 0.22/1.52 subsetI2: $o ).
% 0.22/1.52
% 0.22/1.52 thf(subsetI2,definition,
% 0.22/1.52 ( subsetI2
% 0.22/1.52 = ( ! [A: $i,B: $i] :
% 0.22/1.52 ( ! [Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 => ( in @ Xx @ B ) )
% 0.22/1.52 => ( subset @ A @ B ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(emptysetsubset_type,type,
% 0.22/1.52 emptysetsubset: $o ).
% 0.22/1.52
% 0.22/1.52 thf(emptysetsubset,definition,
% 0.22/1.52 ( emptysetsubset
% 0.22/1.52 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(subsetE_type,type,
% 0.22/1.52 subsetE: $o ).
% 0.22/1.52
% 0.22/1.52 thf(subsetE,definition,
% 0.22/1.52 ( subsetE
% 0.22/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/1.52 ( ( subset @ A @ B )
% 0.22/1.52 => ( ( in @ Xx @ A )
% 0.22/1.52 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(subsetE2_type,type,
% 0.22/1.52 subsetE2: $o ).
% 0.22/1.52
% 0.22/1.52 thf(subsetE2,definition,
% 0.22/1.52 ( subsetE2
% 0.22/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/1.52 ( ( subset @ A @ B )
% 0.22/1.52 => ( ~ ( in @ Xx @ B )
% 0.22/1.52 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(notsubsetI_type,type,
% 0.22/1.52 notsubsetI: $o ).
% 0.22/1.52
% 0.22/1.52 thf(notsubsetI,definition,
% 0.22/1.52 ( notsubsetI
% 0.22/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 => ( ~ ( in @ Xx @ B )
% 0.22/1.52 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(notequalI1_type,type,
% 0.22/1.52 notequalI1: $o ).
% 0.22/1.52
% 0.22/1.52 thf(notequalI1,definition,
% 0.22/1.52 ( notequalI1
% 0.22/1.52 = ( ! [A: $i,B: $i] :
% 0.22/1.52 ( ~ ( subset @ A @ B )
% 0.22/1.52 => ( A != B ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(notequalI2_type,type,
% 0.22/1.52 notequalI2: $o ).
% 0.22/1.52
% 0.22/1.52 thf(notequalI2,definition,
% 0.22/1.52 ( notequalI2
% 0.22/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/1.52 ( ( in @ Xx @ A )
% 0.22/1.52 => ( ~ ( in @ Xx @ B )
% 0.22/1.52 => ( A != B ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(subsetRefl_type,type,
% 0.22/1.52 subsetRefl: $o ).
% 0.22/1.52
% 0.22/1.52 thf(subsetRefl,definition,
% 0.22/1.52 ( subsetRefl
% 0.22/1.52 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(subsetTrans_type,type,
% 0.22/1.52 subsetTrans: $o ).
% 0.22/1.52
% 0.22/1.52 thf(subsetTrans,definition,
% 0.22/1.52 ( subsetTrans
% 0.22/1.52 = ( ! [A: $i,B: $i,C: $i] :
% 0.22/1.52 ( ( subset @ A @ B )
% 0.22/1.52 => ( ( subset @ B @ C )
% 0.22/1.52 => ( subset @ A @ C ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(setadjoinSub_type,type,
% 0.22/1.52 setadjoinSub: $o ).
% 0.22/1.52
% 0.22/1.52 thf(setadjoinSub,definition,
% 0.22/1.52 ( setadjoinSub
% 0.22/1.52 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(setadjoinSub2_type,type,
% 0.22/1.52 setadjoinSub2: $o ).
% 0.22/1.52
% 0.22/1.52 thf(setadjoinSub2,definition,
% 0.22/1.52 ( setadjoinSub2
% 0.22/1.52 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.22/1.52 ( ( subset @ A @ B )
% 0.22/1.52 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(subset2powerset_type,type,
% 0.22/1.52 subset2powerset: $o ).
% 0.22/1.52
% 0.22/1.52 thf(subset2powerset,definition,
% 0.22/1.52 ( subset2powerset
% 0.22/1.52 = ( ! [A: $i,B: $i] :
% 0.22/1.52 ( ( subset @ A @ B )
% 0.22/1.52 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(setextsub_type,type,
% 0.22/1.52 setextsub: $o ).
% 0.22/1.52
% 0.22/1.52 thf(setextsub,definition,
% 0.22/1.52 ( setextsub
% 0.22/1.52 = ( ! [A: $i,B: $i] :
% 0.22/1.52 ( ( subset @ A @ B )
% 0.22/1.52 => ( ( subset @ B @ A )
% 0.22/1.52 => ( A = B ) ) ) ) ) ).
% 0.22/1.52
% 0.22/1.52 thf(subsetemptysetimpeq,conjecture,
% 0.22/1.52 ( setextAx
% 0.22/1.52 => ( emptysetAx
% 0.22/1.52 => ( setadjoinAx
% 0.22/1.52 => ( powersetAx
% 0.22/1.52 => ( setunionAx
% 0.22/1.52 => ( omega0Ax
% 0.22/1.52 => ( omegaSAx
% 0.22/1.52 => ( omegaIndAx
% 0.22/1.52 => ( replAx
% 0.22/1.52 => ( foundationAx
% 0.22/1.52 => ( wellorderingAx
% 0.22/1.52 => ( descrp
% 0.22/1.52 => ( dsetconstrI
% 0.22/1.52 => ( dsetconstrEL
% 0.22/1.52 => ( dsetconstrER
% 0.22/1.52 => ( exuE1
% 0.22/1.52 => ( prop2setE
% 0.22/1.52 => ( emptysetE
% 0.22/1.52 => ( emptysetimpfalse
% 0.22/1.52 => ( notinemptyset
% 0.22/1.52 => ( exuE3e
% 0.22/1.52 => ( setext
% 0.22/1.52 => ( emptyI
% 0.22/1.52 => ( noeltsimpempty
% 0.22/1.52 => ( setbeta
% 0.22/1.52 => ( nonemptyE1
% 0.22/1.52 => ( nonemptyI
% 0.22/1.52 => ( nonemptyI1
% 0.22/1.52 => ( setadjoinIL
% 0.22/1.52 => ( emptyinunitempty
% 0.22/1.52 => ( setadjoinIR
% 0.22/1.52 => ( setadjoinE
% 0.22/1.52 => ( setadjoinOr
% 0.22/1.52 => ( setoftrueEq
% 0.22/1.52 => ( powersetI
% 0.22/1.52 => ( emptyinPowerset
% 0.22/1.52 => ( emptyInPowerset
% 0.22/1.52 => ( powersetE
% 0.22/1.52 => ( setunionI
% 0.22/1.52 => ( setunionE
% 0.22/1.52 => ( subPowSU
% 0.22/1.52 => ( exuE2
% 0.22/1.52 => ( nonemptyImpWitness
% 0.22/1.52 => ( uniqinunit
% 0.22/1.52 => ( notinsingleton
% 0.22/1.52 => ( eqinunit
% 0.22/1.52 => ( singletonsswitch
% 0.22/1.52 => ( upairsetE
% 0.22/1.52 => ( upairsetIL
% 0.22/1.52 => ( upairsetIR
% 0.22/1.52 => ( emptyE1
% 0.22/1.52 => ( vacuousDall
% 0.22/1.52 => ( quantDeMorgan1
% 0.22/1.52 => ( quantDeMorgan2
% 0.22/1.52 => ( quantDeMorgan3
% 0.22/1.52 => ( quantDeMorgan4
% 0.22/1.52 => ( prop2setI
% 0.22/1.52 => ( prop2set2propI
% 0.22/1.52 => ( notdexE
% 0.22/1.52 => ( notdallE
% 0.22/1.52 => ( exuI1
% 0.22/1.52 => ( exuI3
% 0.22/1.52 => ( exuI2
% 0.22/1.52 => ( inCongP
% 0.22/1.52 => ( in__Cong
% 0.22/1.52 => ( exuE3u
% 0.22/1.52 => ( exu__Cong
% 0.22/1.52 => ( emptyset__Cong
% 0.22/1.52 => ( setadjoin__Cong
% 0.22/1.52 => ( powerset__Cong
% 0.22/1.52 => ( setunion__Cong
% 0.22/1.52 => ( omega__Cong
% 0.22/1.52 => ( exuEu
% 0.22/1.52 => ( descr__Cong
% 0.22/1.52 => ( dsetconstr__Cong
% 0.22/1.52 => ( subsetI1
% 0.22/1.52 => ( eqimpsubset2
% 0.22/1.52 => ( eqimpsubset1
% 0.22/1.52 => ( subsetI2
% 0.22/1.52 => ( emptysetsubset
% 0.22/1.52 => ( subsetE
% 0.22/1.52 => ( subsetE2
% 0.22/1.52 => ( notsubsetI
% 0.22/1.52 => ( notequalI1
% 0.22/1.52 => ( notequalI2
% 0.22/1.52 => ( subsetRefl
% 0.22/1.52 => ( subsetTrans
% 0.22/1.52 => ( setadjoinSub
% 0.22/1.52 => ( setadjoinSub2
% 0.22/1.52 => ( subset2powerset
% 0.22/1.52 => ( setextsub
% 0.22/1.52 => ! [A: $i] :
% 0.22/1.52 ( ( subset @ A @ emptyset )
% 0.22/1.52 => ( A = emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/1.52 ------------------------
% 0.22/1.55 % (11863)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_11764 for (2999ds/4Mi)
% 0.22/1.55 % (11862)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_11764 for (2999ds/183Mi)
% 0.22/1.55 % (11865)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_11764 for (2999ds/2Mi)
% 0.22/1.55 % (11866)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_11764 for (2999ds/2Mi)
% 0.22/1.55 % (11867)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_11764 for (2999ds/275Mi)
% 0.22/1.55 % (11868)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_11764 for (2999ds/18Mi)
% 0.22/1.55 % (11864)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_11764 for (2999ds/27Mi)
% 0.22/1.55 % (11865)Instruction limit reached!
% 0.22/1.55 % (11865)------------------------------
% 0.22/1.55 % (11865)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.22/1.55 % (11865)Termination reason: Unknown
% 0.22/1.55 % (11865)Termination phase: shuffling
% 0.22/1.55 % (11866)Instruction limit reached!
% 0.22/1.55 % (11866)------------------------------
% 0.22/1.55 % (11866)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.22/1.55
% 0.22/1.55 % (11865)Memory used [KB]: 1151
% 0.22/1.55 % (11865)Time elapsed: 0.003 s
% 0.22/1.55 % (11865)Instructions burned: 3 (million)
% 0.22/1.55 % (11865)------------------------------
% 0.22/1.55 % (11865)------------------------------
% 0.22/1.55 % (11866)Termination reason: Unknown
% 0.22/1.55 % (11866)Termination phase: shuffling
% 0.22/1.55
% 0.22/1.55 % (11866)Memory used [KB]: 1151
% 0.22/1.55 % (11866)Time elapsed: 0.003 s
% 0.22/1.55 % (11866)Instructions burned: 3 (million)
% 0.22/1.55 % (11866)------------------------------
% 0.22/1.55 % (11866)------------------------------
% 0.22/1.55 % (11863)Instruction limit reached!
% 0.22/1.55 % (11863)------------------------------
% 0.22/1.55 % (11863)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.22/1.55 % (11863)Termination reason: Unknown
% 0.22/1.55 % (11863)Termination phase: shuffling
% 0.22/1.55
% 0.22/1.55 % (11863)Memory used [KB]: 1151
% 0.22/1.55 % (11863)Time elapsed: 0.004 s
% 0.22/1.55 % (11863)Instructions burned: 4 (million)
% 0.22/1.55 % (11863)------------------------------
% 0.22/1.55 % (11863)------------------------------
% 0.22/1.56 % (11868)Instruction limit reached!
% 0.22/1.56 % (11868)------------------------------
% 0.22/1.56 % (11868)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.22/1.56 % (11868)Termination reason: Unknown
% 0.22/1.56 % (11868)Termination phase: Preprocessing 3
% 0.22/1.56
% 0.22/1.56 % (11868)Memory used [KB]: 1663
% 0.22/1.56 % (11868)Time elapsed: 0.011 s
% 0.22/1.56 % (11868)Instructions burned: 19 (million)
% 0.22/1.56 % (11868)------------------------------
% 0.22/1.56 % (11868)------------------------------
% 0.22/1.56 % (11864)Instruction limit reached!
% 0.22/1.56 % (11864)------------------------------
% 0.22/1.56 % (11864)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.22/1.56 % (11864)Termination reason: Unknown
% 0.22/1.56 % (11864)Termination phase: Property scanning
% 0.22/1.56
% 0.22/1.56 % (11864)Memory used [KB]: 1407
% 0.22/1.56 % (11864)Time elapsed: 0.014 s
% 0.22/1.56 % (11864)Instructions burned: 27 (million)
% 0.22/1.56 % (11864)------------------------------
% 0.22/1.56 % (11864)------------------------------
% 0.22/1.56 % (11869)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_11764 for (2999ds/3Mi)
% 0.22/1.56 % (11870)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_11764 for (2999ds/37Mi)
% 0.22/1.56 % (11871)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_11764 for (2999ds/15Mi)
% 0.22/1.56 % (11869)Instruction limit reached!
% 0.22/1.56 % (11869)------------------------------
% 0.22/1.56 % (11869)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.22/1.56 % (11869)Termination reason: Unknown
% 0.22/1.56 % (11869)Termination phase: shuffling
% 0.22/1.56
% 0.22/1.56 % (11869)Memory used [KB]: 1151
% 0.22/1.57 % (11869)Time elapsed: 0.003 s
% 0.22/1.57 % (11869)Instructions burned: 3 (million)
% 0.22/1.57 % (11869)------------------------------
% 0.22/1.57 % (11869)------------------------------
% 0.22/1.57 % (11871)Instruction limit reached!
% 0.22/1.57 % (11871)------------------------------
% 0.22/1.57 % (11871)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.22/1.57 % (11871)Termination reason: Unknown
% 0.22/1.57 % (11871)Termination phase: Preprocessing 3
% 0.22/1.57
% 0.22/1.57 % (11871)Memory used [KB]: 1407
% 0.22/1.57 % (11871)Time elapsed: 0.009 s
% 0.22/1.57 % (11872)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_11764 for (2999ds/3Mi)
% 0.22/1.57 % (11871)Instructions burned: 15 (million)
% 0.22/1.57 % (11871)------------------------------
% 0.22/1.57 % (11871)------------------------------
% 0.22/1.57 % (11872)Instruction limit reached!
% 0.22/1.57 % (11872)------------------------------
% 0.22/1.57 % (11872)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.22/1.57 % (11872)Termination reason: Unknown
% 0.22/1.57 % (11872)Termination phase: shuffling
% 0.22/1.57
% 0.22/1.57 % (11872)Memory used [KB]: 1151
% 0.22/1.57 % (11872)Time elapsed: 0.003 s
% 0.22/1.57 % (11872)Instructions burned: 3 (million)
% 0.22/1.57 % (11872)------------------------------
% 0.22/1.57 % (11872)------------------------------
% 0.22/1.57 % (11873)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_11764 for (2999ds/1041Mi)
% 0.22/1.58 % (11870)Instruction limit reached!
% 0.22/1.58 % (11870)------------------------------
% 0.22/1.58 % (11870)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.22/1.58 % (11870)Termination reason: Unknown
% 0.22/1.58 % (11870)Termination phase: Property scanning
% 0.22/1.58
% 0.22/1.58 % (11870)Memory used [KB]: 1535
% 0.22/1.58 % (11870)Time elapsed: 0.018 s
% 0.22/1.58 % (11870)Instructions burned: 38 (million)
% 0.22/1.58 % (11870)------------------------------
% 0.22/1.58 % (11870)------------------------------
% 0.22/1.59 % (11874)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_11764 for (2999ds/7Mi)
% 0.22/1.59 % (11875)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_11764 for (2999ds/16Mi)
% 0.22/1.59 % (11876)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_11764 for (2999ds/3Mi)
% 0.22/1.59 % (11874)Instruction limit reached!
% 0.22/1.59 % (11874)------------------------------
% 0.22/1.59 % (11874)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.22/1.59 % (11874)Termination reason: Unknown
% 0.22/1.59 % (11874)Termination phase: shuffling
% 0.22/1.59
% 0.22/1.59 % (11874)Memory used [KB]: 1279
% 0.22/1.59 % (11874)Time elapsed: 0.005 s
% 0.22/1.59 % (11874)Instructions burned: 8 (million)
% 0.22/1.59 % (11874)------------------------------
% 0.22/1.59 % (11874)------------------------------
% 0.22/1.59 % (11876)Instruction limit reached!
% 0.22/1.59 % (11876)------------------------------
% 0.22/1.59 % (11876)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.22/1.59 % (11876)Termination reason: Unknown
% 0.22/1.59 % (11876)Termination phase: shuffling
% 0.22/1.59
% 0.22/1.59 % (11876)Memory used [KB]: 1151
% 0.22/1.59 % (11876)Time elapsed: 0.003 s
% 0.22/1.59 % (11876)Instructions burned: 3 (million)
% 0.22/1.59 % (11876)------------------------------
% 0.22/1.59 % (11876)------------------------------
% 0.22/1.60 % (11875)Instruction limit reached!
% 0.22/1.60 % (11875)------------------------------
% 0.22/1.60 % (11875)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.22/1.60 % (11875)Termination reason: Unknown
% 0.22/1.60 % (11875)Termination phase: Property scanning
% 0.22/1.60
% 0.22/1.60 % (11875)Memory used [KB]: 1407
% 0.22/1.60 % (11875)Time elapsed: 0.009 s
% 0.22/1.60 % (11875)Instructions burned: 17 (million)
% 0.22/1.60 % (11875)------------------------------
% 0.22/1.60 % (11875)------------------------------
% 2.43/1.60 % (11877)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_11764 for (2999ds/3Mi)
% 2.43/1.60 % (11877)Instruction limit reached!
% 2.43/1.60 % (11877)------------------------------
% 2.43/1.60 % (11877)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.60 % (11877)Termination reason: Unknown
% 2.43/1.60 % (11877)Termination phase: shuffling
% 2.43/1.60
% 2.43/1.60 % (11877)Memory used [KB]: 1151
% 2.43/1.60 % (11877)Time elapsed: 0.002 s
% 2.43/1.60 % (11877)Instructions burned: 3 (million)
% 2.43/1.60 % (11877)------------------------------
% 2.43/1.60 % (11877)------------------------------
% 2.43/1.61 % (11878)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_11764 for (2999ds/7Mi)
% 2.43/1.61 % (11879)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_11764 for (2999ds/3Mi)
% 2.43/1.61 % (11879)Instruction limit reached!
% 2.43/1.61 % (11879)------------------------------
% 2.43/1.61 % (11879)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.61 % (11879)Termination reason: Unknown
% 2.43/1.61 % (11879)Termination phase: shuffling
% 2.43/1.61
% 2.43/1.61 % (11879)Memory used [KB]: 1151
% 2.43/1.61 % (11879)Time elapsed: 0.003 s
% 2.43/1.61 % (11879)Instructions burned: 4 (million)
% 2.43/1.61 % (11879)------------------------------
% 2.43/1.61 % (11879)------------------------------
% 2.43/1.61 % (11878)Instruction limit reached!
% 2.43/1.61 % (11878)------------------------------
% 2.43/1.61 % (11878)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.61 % (11878)Termination reason: Unknown
% 2.43/1.61 % (11878)Termination phase: shuffling
% 2.43/1.61
% 2.43/1.61 % (11878)Memory used [KB]: 1279
% 2.43/1.61 % (11878)Time elapsed: 0.005 s
% 2.43/1.61 % (11878)Instructions burned: 8 (million)
% 2.43/1.61 % (11878)------------------------------
% 2.43/1.61 % (11878)------------------------------
% 2.43/1.61 % (11880)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_11764 for (2999ds/4Mi)
% 2.43/1.61 % (11880)Instruction limit reached!
% 2.43/1.61 % (11880)------------------------------
% 2.43/1.61 % (11880)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.61 % (11880)Termination reason: Unknown
% 2.43/1.61 % (11880)Termination phase: shuffling
% 2.43/1.61
% 2.43/1.61 % (11880)Memory used [KB]: 1151
% 2.43/1.61 % (11880)Time elapsed: 0.004 s
% 2.43/1.61 % (11880)Instructions burned: 5 (million)
% 2.43/1.61 % (11880)------------------------------
% 2.43/1.61 % (11880)------------------------------
% 2.43/1.62 % (11862)Instruction limit reached!
% 2.43/1.62 % (11862)------------------------------
% 2.43/1.62 % (11862)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.62 % (11862)Termination reason: Unknown
% 2.43/1.62 % (11862)Termination phase: Saturation
% 2.43/1.62
% 2.43/1.62 % (11862)Memory used [KB]: 6908
% 2.43/1.62 % (11862)Time elapsed: 0.066 s
% 2.43/1.62 % (11862)Instructions burned: 183 (million)
% 2.43/1.62 % (11862)------------------------------
% 2.43/1.62 % (11862)------------------------------
% 2.43/1.62 % (11881)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_11764 for (2999ds/18Mi)
% 2.43/1.63 % (11882)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_11764 for (2999ds/710Mi)
% 2.43/1.63 % (11883)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_11764 for (2999ds/6Mi)
% 2.43/1.63 % (11881)Instruction limit reached!
% 2.43/1.63 % (11881)------------------------------
% 2.43/1.63 % (11881)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.63 % (11881)Termination reason: Unknown
% 2.43/1.63 % (11881)Termination phase: Property scanning
% 2.43/1.63
% 2.43/1.63 % (11881)Memory used [KB]: 1407
% 2.43/1.63 % (11881)Time elapsed: 0.010 s
% 2.43/1.63 % (11881)Instructions burned: 19 (million)
% 2.43/1.63 % (11881)------------------------------
% 2.43/1.63 % (11881)------------------------------
% 2.43/1.63 % (11883)Instruction limit reached!
% 2.43/1.63 % (11883)------------------------------
% 2.43/1.63 % (11883)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.63 % (11883)Termination reason: Unknown
% 2.43/1.63 % (11883)Termination phase: shuffling
% 2.43/1.63
% 2.43/1.63 % (11883)Memory used [KB]: 1279
% 2.43/1.63 % (11883)Time elapsed: 0.004 s
% 2.43/1.63 % (11883)Instructions burned: 6 (million)
% 2.43/1.63 % (11883)------------------------------
% 2.43/1.63 % (11883)------------------------------
% 2.43/1.63 % (11884)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_11764 for (2999ds/902Mi)
% 2.43/1.64 % (11885)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_11764 for (2999ds/21Mi)
% 2.43/1.64 % (11886)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_11764 for (2999ds/5Mi)
% 2.43/1.64 % (11885)Instruction limit reached!
% 2.43/1.64 % (11885)------------------------------
% 2.43/1.64 % (11885)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.64 % (11885)Termination reason: Unknown
% 2.43/1.64 % (11885)Termination phase: Property scanning
% 2.43/1.64
% 2.43/1.64 % (11885)Memory used [KB]: 1407
% 2.43/1.64 % (11885)Time elapsed: 0.011 s
% 2.43/1.64 % (11885)Instructions burned: 21 (million)
% 2.43/1.64 % (11885)------------------------------
% 2.43/1.64 % (11885)------------------------------
% 2.43/1.65 % (11887)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_11764 for (2999ds/6Mi)
% 2.43/1.65 % (11886)Instruction limit reached!
% 2.43/1.65 % (11886)------------------------------
% 2.43/1.65 % (11886)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.65 % (11886)Termination reason: Unknown
% 2.43/1.65 % (11886)Termination phase: shuffling
% 2.43/1.65
% 2.43/1.65 % (11886)Memory used [KB]: 1279
% 2.43/1.65 % (11886)Time elapsed: 0.004 s
% 2.43/1.65 % (11886)Instructions burned: 6 (million)
% 2.43/1.65 % (11886)------------------------------
% 2.43/1.65 % (11886)------------------------------
% 2.43/1.65 % (11887)Instruction limit reached!
% 2.43/1.65 % (11887)------------------------------
% 2.43/1.65 % (11887)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.65 % (11887)Termination reason: Unknown
% 2.43/1.65 % (11887)Termination phase: shuffling
% 2.43/1.65
% 2.43/1.65 % (11887)Memory used [KB]: 1279
% 2.43/1.65 % (11887)Time elapsed: 0.004 s
% 2.43/1.65 % (11887)Instructions burned: 6 (million)
% 2.43/1.65 % (11887)------------------------------
% 2.43/1.65 % (11887)------------------------------
% 2.43/1.66 % (11888)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_11764 for (2998ds/377Mi)
% 2.43/1.66 % (11889)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_11764 for (2998ds/779Mi)
% 2.43/1.66 % (11867)First to succeed.
% 2.43/1.66 % (11890)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_11764 for (2998ds/19Mi)
% 2.43/1.67 % (11867)Refutation found. Thanks to Tanya!
% 2.43/1.67 % SZS status Theorem for DTF2THF_11764
% 2.43/1.67 % SZS output start Proof for DTF2THF_11764
% 2.43/1.67 thf(func_def_0, type, in: $i > $i > $o).
% 2.43/1.67 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.43/1.67 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.43/1.67 thf(func_def_8, type, powerset: $i > $i).
% 2.43/1.67 thf(func_def_10, type, setunion: $i > $i).
% 2.43/1.67 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.43/1.67 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.43/1.67 thf(func_def_26, type, prop2set: $o > $i).
% 2.43/1.67 thf(func_def_36, type, nonempty: $i > $o).
% 2.43/1.67 thf(func_def_69, type, set2prop: $i > $o).
% 2.43/1.67 thf(func_def_88, type, subset: $i > $i > $o).
% 2.43/1.67 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.43/1.67 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.43/1.67 thf(func_def_120, type, sP1: $i > $i > $o).
% 2.43/1.67 thf(func_def_121, type, sP2: $i > $o).
% 2.43/1.67 thf(func_def_122, type, sP3: $i > $i > $o).
% 2.43/1.67 thf(func_def_123, type, sP4: $i > $i > $o).
% 2.43/1.67 thf(func_def_124, type, sK5: $i > $o).
% 2.43/1.67 thf(func_def_125, type, sK6: ($i > $o) > $i).
% 2.43/1.67 thf(func_def_135, type, sK16: $i > $i > $i).
% 2.43/1.67 thf(func_def_138, type, sK19: $i > $i > $i).
% 2.43/1.67 thf(func_def_143, type, sK24: $i > $o).
% 2.43/1.67 thf(func_def_145, type, sK26: ($i > $o) > $i > $i).
% 2.43/1.67 thf(func_def_147, type, sK28: $i > $o).
% 2.43/1.67 thf(func_def_149, type, sK30: ($i > $o) > $i > $i).
% 2.43/1.67 thf(func_def_155, type, sK36: $i > $o).
% 2.43/1.67 thf(func_def_157, type, sK38: ($i > $o) > $i > $i).
% 2.43/1.67 thf(func_def_159, type, sK40: $i > $i > $o).
% 2.43/1.67 thf(func_def_160, type, sK41: $i > $i).
% 2.43/1.67 thf(func_def_161, type, sK42: $i > $i).
% 2.43/1.67 thf(func_def_162, type, sK43: ($i > $i > $o) > $i > $i).
% 2.43/1.67 thf(func_def_163, type, sK44: $i > ($i > $i > $o) > $i > $i).
% 2.43/1.67 thf(func_def_164, type, sK45: ($i > $i > $o) > $i > $i).
% 2.43/1.67 thf(func_def_166, type, sK47: $i > $o).
% 2.43/1.67 thf(func_def_169, type, sK50: $i > $i).
% 2.43/1.67 thf(func_def_180, type, sK61: $i > $i > $i).
% 2.43/1.67 thf(func_def_188, type, sK69: $i > $o).
% 2.43/1.67 thf(func_def_190, type, sK71: $i > ($i > $o) > $i).
% 2.43/1.67 thf(func_def_193, type, sK74: $i > $i > $i).
% 2.43/1.67 thf(func_def_198, type, sK79: $i > $o).
% 2.43/1.67 thf(func_def_199, type, sK80: $i > $o).
% 2.43/1.67 thf(func_def_200, type, sK81: ($i > $o) > ($i > $o) > $i).
% 2.43/1.67 thf(func_def_201, type, sK82: ($i > $o) > ($i > $o) > $i).
% 2.43/1.67 thf(func_def_205, type, sK86: $i > $i).
% 2.43/1.67 thf(func_def_208, type, sK89: $i > $o).
% 2.43/1.67 thf(func_def_219, type, sK100: $i > $o).
% 2.43/1.67 thf(func_def_223, type, sK104: $i > $i).
% 2.43/1.67 thf(func_def_225, type, sK106: $i > $o).
% 2.43/1.67 thf(func_def_230, type, sK111: $i > ($i > $o) > $i).
% 2.43/1.67 thf(func_def_231, type, sK112: $i > $o).
% 2.43/1.67 thf(func_def_242, type, sK123: $i > $i).
% 2.43/1.67 thf(func_def_243, type, sK124: $i > $i).
% 2.43/1.67 thf(func_def_244, type, sK125: ($i > $o) > $i).
% 2.43/1.67 thf(func_def_245, type, sK126: ($i > $o) > $i).
% 2.43/1.67 thf(func_def_246, type, sK127: $i > $o).
% 2.43/1.67 thf(func_def_252, type, sK133: $i > $o).
% 2.43/1.67 thf(func_def_256, type, sK137: $i > $o).
% 2.43/1.67 thf(func_def_260, type, sK141: ($i > $o) > $i).
% 2.43/1.67 thf(func_def_261, type, sK142: $i > $o).
% 2.43/1.67 thf(func_def_262, type, sK143: $i > $i).
% 2.43/1.67 thf(func_def_264, type, sK145: $i > $i).
% 2.43/1.67 thf(func_def_269, type, sK150: $i > $o).
% 2.43/1.67 thf(func_def_270, type, sK151: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.43/1.67 thf(func_def_271, type, sK152: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.43/1.67 thf(func_def_274, type, sK155: $i > $o).
% 2.43/1.67 thf(func_def_275, type, sK156: $i > $o).
% 2.43/1.67 thf(func_def_280, type, sK161: $i > $i > $i).
% 2.43/1.67 thf(func_def_281, type, sK162: $i > $i > $i).
% 2.43/1.67 thf(func_def_284, type, sK165: ($i > $o) > $i).
% 2.43/1.67 thf(func_def_285, type, sK166: $i > $o).
% 2.43/1.67 thf(func_def_286, type, sK167: $i > $i).
% 2.43/1.67 thf(func_def_287, type, sK168: $i > $o).
% 2.43/1.67 thf(func_def_292, type, sK173: ($i > $o) > ($i > $o) > $i).
% 2.43/1.67 thf(func_def_293, type, sK174: ($i > $o) > ($i > $o) > $i).
% 2.43/1.67 thf(func_def_294, type, sK175: $i > $o).
% 2.43/1.67 thf(func_def_295, type, sK176: $i > $o).
% 2.43/1.67 thf(func_def_297, type, sK178: $i > $i).
% 2.43/1.67 thf(func_def_301, type, sK182: $i > $o).
% 2.43/1.67 thf(func_def_304, type, sK185: $i > ($i > $o) > $i).
% 2.43/1.67 thf(func_def_309, type, sK190: ($i > $o) > $i > $i).
% 2.43/1.67 thf(func_def_311, type, sK192: $i > $o).
% 2.43/1.67 thf(func_def_325, type, sK206: $i > $i > $i).
% 2.43/1.67 thf(func_def_329, type, sK210: $i > $i > $i).
% 2.43/1.67 thf(func_def_330, type, sK211: $i > $i > $i).
% 2.43/1.67 thf(func_def_331, type, sK212: $i > $i > $i).
% 2.43/1.67 thf(func_def_332, type, sK213: $i > $i > $i).
% 2.43/1.67 thf(func_def_333, type, sK214: $i > $i > $i > $i).
% 2.43/1.67 thf(func_def_334, type, sK215: $i > $i).
% 2.43/1.67 thf(func_def_335, type, sK216: $i > $i).
% 2.43/1.67 thf(func_def_336, type, sK217: $i > $i).
% 2.43/1.67 thf(func_def_337, type, sK218: $i > $i).
% 2.43/1.67 thf(func_def_338, type, sK219: $i > $i > $i).
% 2.43/1.67 thf(func_def_339, type, sK220: $i > $i > $i > $i).
% 2.43/1.67 thf(func_def_340, type, sK221: $i > $i > $i > $i).
% 2.43/1.67 thf(func_def_341, type, sK222: $i > $i > $i).
% 2.43/1.67 thf(func_def_342, type, sK223: $i > $i > $i).
% 2.43/1.67 thf(func_def_343, type, sK224: $i > $i).
% 2.43/1.67 thf(func_def_344, type, sK225: $i > $i > $i).
% 2.43/1.67 thf(func_def_346, type, sK227: $i > $i).
% 2.43/1.67 thf(func_def_347, type, sK228: $i > $i).
% 2.43/1.67 thf(func_def_353, type, sK234: $i > $i > $i).
% 2.43/1.67 thf(func_def_354, type, sK235: $i > ($i > $o) > $i).
% 2.43/1.67 thf(func_def_355, type, sK236: $i > $o).
% 2.43/1.67 thf(func_def_361, type, sK242: $i > $o).
% 2.43/1.67 thf(func_def_365, type, sK246: $o > $i > $i > $i).
% 2.43/1.67 thf(func_def_369, type, sK250: $i > $o).
% 2.43/1.67 thf(func_def_378, type, ph259: !>[X0: $tType]:(X0)).
% 2.43/1.67 thf(f2333,plain,(
% 2.43/1.67 $false),
% 2.43/1.67 inference(subsumption_resolution,[status(thm)],[f2332,f1723])).
% 2.43/1.67 thf(f1723,plain,(
% 2.43/1.67 ( ! [X0 : $i] : (((subset @ emptyset @ X0) = $true)) )),
% 2.43/1.67 inference(trivial_inequality_removal,[status(thm)],[f1423])).
% 2.43/1.67 thf(f1423,plain,(
% 2.43/1.67 ( ! [X0 : $i] : (($true != $true) | ((subset @ emptyset @ X0) = $true)) )),
% 2.43/1.67 inference(definition_unfolding,[status(thm)],[f1057,f909])).
% 2.43/1.67 thf(f909,plain,(
% 2.43/1.67 (emptysetsubset = $true)),
% 2.43/1.67 inference(cnf_transformation,[status(thm)],[f415])).
% 2.43/1.67 thf(f415,plain,(
% 2.43/1.67 (subsetI1 = $true) & (emptyI = $true) & (descr__Cong = $true) & (notdallE = $true) & (emptyinPowerset = $true) & (nonemptyI = $true) & (upairsetE = $true) & (exuE1 = $true) & (powersetI = $true) & (exuI3 = $true) & (descrp = $true) & (notsubsetI = $true) & (vacuousDall = $true) & (subPowSU = $true) & (nonemptyE1 = $true) & (powersetAx = $true) & (eqimpsubset1 = $true) & (subset2powerset = $true) & (dsetconstrEL = $true) & (omegaSAx = $true) & (notinsingleton = $true) & (eqinunit = $true) & (uniqinunit = $true) & (exuE3e = $true) & (emptysetAx = $true) & (dsetconstrI = $true) & (quantDeMorgan3 = $true) & (notdexE = $true) & (powersetE = $true) & (emptyInPowerset = $true) & (in__Cong = $true) & (setadjoinAx = $true) & (prop2setE = $true) & (setadjoinOr = $true) & (quantDeMorgan4 = $true) & (emptysetsubset = $true) & (setoftrueEq = $true) & (prop2set2propI = $true) & (subsetE = $true) & (nonemptyImpWitness = $true) & (setadjoin__Cong = $true) & (setadjoinIR = $true) & (setextsub = $true) & (emptysetimpfalse = $true) & (emptyE1 = $true) & (exuI2 = $true) & (subsetTrans = $true) & (singletonsswitch = $true) & (upairsetIR = $true) & (exuE2 = $true) & (powerset__Cong = $true) & (emptysetE = $true) & (setunionI = $true) & (setadjoinE = $true) & (eqimpsubset2 = $true) & (subsetRefl = $true) & (notinemptyset = $true) & (dsetconstrER = $true) & (wellorderingAx = $true) & (setunionAx = $true) & (notequalI1 = $true) & (exuI1 = $true) & (omega__Cong = $true) & (setadjoinSub = $true) & (quantDeMorgan1 = $true) & (setunion__Cong = $true) & (foundationAx = $true) & (inCongP = $true) & (setunionE = $true) & (quantDeMorgan2 = $true) & (exu__Cong = $true) & (setadjoinSub2 = $true) & (notequalI2 = $true) & (emptyinunitempty = $true) & (upairsetIL = $true) & (setbeta = $true) & (omega0Ax = $true) & (emptyset__Cong = $true) & (setadjoinIL = $true) & (noeltsimpempty = $true) & (setextAx = $true) & (exuE3u = $true) & (subsetI2 = $true) & (dsetconstr__Cong = $true) & (subsetE2 = $true) & (exuEu = $true) & (omegaIndAx = $true) & (replAx = $true) & (nonemptyI1 = $true) & (setext = $true) & (((subset @ sK7 @ emptyset) = $true) & (emptyset != sK7)) & (prop2setI = $true)),
% 2.43/1.67 inference(skolemisation,[status(esa),new_symbols(skolem,[sK7])],[f359,f414])).
% 2.43/1.67 thf(f414,plain,(
% 2.43/1.67 ? [X0 : $i] : (((subset @ X0 @ emptyset) = $true) & (emptyset != X0)) => (((subset @ sK7 @ emptyset) = $true) & (emptyset != sK7))),
% 2.43/1.67 introduced(definition,[],[choice_axiom])).
% 2.43/1.67 thf(f359,plain,(
% 2.43/1.67 (subsetI1 = $true) & (emptyI = $true) & (descr__Cong = $true) & (notdallE = $true) & (emptyinPowerset = $true) & (nonemptyI = $true) & (upairsetE = $true) & (exuE1 = $true) & (powersetI = $true) & (exuI3 = $true) & (descrp = $true) & (notsubsetI = $true) & (vacuousDall = $true) & (subPowSU = $true) & (nonemptyE1 = $true) & (powersetAx = $true) & (eqimpsubset1 = $true) & (subset2powerset = $true) & (dsetconstrEL = $true) & (omegaSAx = $true) & (notinsingleton = $true) & (eqinunit = $true) & (uniqinunit = $true) & (exuE3e = $true) & (emptysetAx = $true) & (dsetconstrI = $true) & (quantDeMorgan3 = $true) & (notdexE = $true) & (powersetE = $true) & (emptyInPowerset = $true) & (in__Cong = $true) & (setadjoinAx = $true) & (prop2setE = $true) & (setadjoinOr = $true) & (quantDeMorgan4 = $true) & (emptysetsubset = $true) & (setoftrueEq = $true) & (prop2set2propI = $true) & (subsetE = $true) & (nonemptyImpWitness = $true) & (setadjoin__Cong = $true) & (setadjoinIR = $true) & (setextsub = $true) & (emptysetimpfalse = $true) & (emptyE1 = $true) & (exuI2 = $true) & (subsetTrans = $true) & (singletonsswitch = $true) & (upairsetIR = $true) & (exuE2 = $true) & (powerset__Cong = $true) & (emptysetE = $true) & (setunionI = $true) & (setadjoinE = $true) & (eqimpsubset2 = $true) & (subsetRefl = $true) & (notinemptyset = $true) & (dsetconstrER = $true) & (wellorderingAx = $true) & (setunionAx = $true) & (notequalI1 = $true) & (exuI1 = $true) & (omega__Cong = $true) & (setadjoinSub = $true) & (quantDeMorgan1 = $true) & (setunion__Cong = $true) & (foundationAx = $true) & (inCongP = $true) & (setunionE = $true) & (quantDeMorgan2 = $true) & (exu__Cong = $true) & (setadjoinSub2 = $true) & (notequalI2 = $true) & (emptyinunitempty = $true) & (upairsetIL = $true) & (setbeta = $true) & (omega0Ax = $true) & (emptyset__Cong = $true) & (setadjoinIL = $true) & (noeltsimpempty = $true) & (setextAx = $true) & (exuE3u = $true) & (subsetI2 = $true) & (dsetconstr__Cong = $true) & (subsetE2 = $true) & (exuEu = $true) & (omegaIndAx = $true) & (replAx = $true) & (nonemptyI1 = $true) & (setext = $true) & ? [X0 : $i] : (((subset @ X0 @ emptyset) = $true) & (emptyset != X0)) & (prop2setI = $true)),
% 2.43/1.67 inference(flattening,[status(thm)],[f358])).
% 2.43/1.67 thf(f358,plain,(
% 2.43/1.67 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i] : (((subset @ X0 @ emptyset) = $true) & (emptyset != X0)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.43/1.67 inference(ennf_transformation,[status(thm)],[f134])).
% 2.43/1.67 thf(f134,plain,(
% 2.43/1.67 ~((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) => ! [X0 : $i] : (((subset @ X0 @ emptyset) = $true) => (emptyset = X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.43/1.67 inference(fool_elimination,[status(thm)],[f133])).
% 2.43/1.67 thf(f133,plain,(
% 2.43/1.67 ~(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 => ! [X0 : $i] : ((subset @ X0 @ emptyset) => (emptyset = X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.43/1.67 inference(rectify,[status(thm)],[f95])).
% 2.43/1.67 thf(f95,negated_conjecture,(
% 2.43/1.67 ~(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 => ! [X3 : $i] : ((subset @ X3 @ emptyset) => (emptyset = X3)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.43/1.67 inference(negated_conjecture,[status(cth)],[f94])).
% 2.43/1.67 thf(f94,conjecture,(
% 2.43/1.67 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 => ! [X3 : $i] : ((subset @ X3 @ emptyset) => (emptyset = X3))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.43/1.67 file('/export/starexec/sandbox/tmp/tmp.x7ZFQe8mUJ/DTF2THF_11764.p',subsetemptysetimpeq)).
% 2.43/1.67 thf(f1057,plain,(
% 2.43/1.67 ( ! [X0 : $i] : (((subset @ emptyset @ X0) = $true) | (emptysetsubset != $true)) )),
% 2.43/1.67 inference(cnf_transformation,[status(thm)],[f542])).
% 2.43/1.67 thf(f542,plain,(
% 2.43/1.67 (! [X0 : $i] : ((subset @ emptyset @ X0) = $true) | (emptysetsubset != $true)) & ((emptysetsubset = $true) | ((subset @ emptyset @ sK85) != $true))),
% 2.43/1.67 inference(skolemisation,[status(esa),new_symbols(skolem,[sK85])],[f540,f541])).
% 2.43/1.67 thf(f541,plain,(
% 2.43/1.67 ? [X1 : $i] : ((subset @ emptyset @ X1) != $true) => ((subset @ emptyset @ sK85) != $true)),
% 2.43/1.67 introduced(definition,[],[choice_axiom])).
% 2.43/1.67 thf(f540,plain,(
% 2.43/1.67 (! [X0 : $i] : ((subset @ emptyset @ X0) = $true) | (emptysetsubset != $true)) & ((emptysetsubset = $true) | ? [X1 : $i] : ((subset @ emptyset @ X1) != $true))),
% 2.43/1.67 inference(rectify,[status(thm)],[f539])).
% 2.43/1.67 thf(f539,plain,(
% 2.43/1.67 (! [X0 : $i] : ((subset @ emptyset @ X0) = $true) | (emptysetsubset != $true)) & ((emptysetsubset = $true) | ? [X0 : $i] : ((subset @ emptyset @ X0) != $true))),
% 2.43/1.67 inference(nnf_transformation,[status(thm)],[f268])).
% 2.43/1.67 thf(f268,plain,(
% 2.43/1.67 ! [X0 : $i] : ((subset @ emptyset @ X0) = $true) <=> (emptysetsubset = $true)),
% 2.43/1.67 inference(fool_elimination,[status(thm)],[f267])).
% 2.43/1.67 thf(f267,plain,(
% 2.43/1.67 (! [X0 : $i] : (subset @ emptyset @ X0) = emptysetsubset)),
% 2.43/1.67 inference(rectify,[status(thm)],[f82])).
% 2.43/1.67 thf(f82,axiom,(
% 2.43/1.67 (! [X3 : $i] : (subset @ emptyset @ X3) = emptysetsubset)),
% 2.43/1.67 file('/export/starexec/sandbox/tmp/tmp.x7ZFQe8mUJ/DTF2THF_11764.p',emptysetsubset)).
% 2.43/1.67 thf(f2332,plain,(
% 2.43/1.67 ((subset @ emptyset @ sK7) != $true)),
% 2.43/1.67 inference(subsumption_resolution,[status(thm)],[f2329,f853])).
% 2.43/1.67 thf(f853,plain,(
% 2.43/1.67 (emptyset != sK7)),
% 2.43/1.67 inference(cnf_transformation,[status(thm)],[f415])).
% 2.43/1.67 thf(f2329,plain,(
% 2.43/1.67 (emptyset = sK7) | ((subset @ emptyset @ sK7) != $true)),
% 2.43/1.67 inference(trivial_inequality_removal,[status(thm)],[f2309])).
% 2.43/1.67 thf(f2309,plain,(
% 2.43/1.67 (emptyset = sK7) | ($true != $true) | ((subset @ emptyset @ sK7) != $true)),
% 2.43/1.67 inference(superposition,[status(thm)],[f1840,f854])).
% 2.43/1.67 thf(f854,plain,(
% 2.43/1.67 ((subset @ sK7 @ emptyset) = $true)),
% 2.43/1.67 inference(cnf_transformation,[status(thm)],[f415])).
% 2.43/1.67 thf(f1840,plain,(
% 2.43/1.67 ( ! [X2 : $i,X3 : $i] : (((subset @ X3 @ X2) != $true) | ((subset @ X2 @ X3) != $true) | (X2 = X3)) )),
% 2.43/1.67 inference(trivial_inequality_removal,[status(thm)],[f1396])).
% 2.43/1.67 thf(f1396,plain,(
% 2.43/1.67 ( ! [X2 : $i,X3 : $i] : (((subset @ X3 @ X2) != $true) | ($true != $true) | ((subset @ X2 @ X3) != $true) | (X2 = X3)) )),
% 2.43/1.67 inference(definition_unfolding,[status(thm)],[f1026,f902])).
% 2.43/1.67 thf(f902,plain,(
% 2.43/1.67 (setextsub = $true)),
% 2.43/1.67 inference(cnf_transformation,[status(thm)],[f415])).
% 2.43/1.67 thf(f1026,plain,(
% 2.43/1.67 ( ! [X2 : $i,X3 : $i] : ((X2 = X3) | ((subset @ X3 @ X2) != $true) | ((subset @ X2 @ X3) != $true) | (setextsub != $true)) )),
% 2.43/1.67 inference(cnf_transformation,[status(thm)],[f508])).
% 2.43/1.67 thf(f508,plain,(
% 2.43/1.67 ((setextsub = $true) | ((sK67 != sK68) & ((subset @ sK68 @ sK67) = $true) & ((subset @ sK67 @ sK68) = $true))) & (! [X2 : $i,X3 : $i] : ((X2 = X3) | ((subset @ X3 @ X2) != $true) | ((subset @ X2 @ X3) != $true)) | (setextsub != $true))),
% 2.43/1.67 inference(skolemisation,[status(esa),new_symbols(skolem,[sK67,sK68])],[f506,f507])).
% 2.43/1.67 thf(f507,plain,(
% 2.43/1.67 ? [X0 : $i,X1 : $i] : ((X0 != X1) & ($true = (subset @ X1 @ X0)) & ((subset @ X0 @ X1) = $true)) => ((sK67 != sK68) & ((subset @ sK68 @ sK67) = $true) & ((subset @ sK67 @ sK68) = $true))),
% 2.43/1.67 introduced(definition,[],[choice_axiom])).
% 2.43/1.67 thf(f506,plain,(
% 2.43/1.67 ((setextsub = $true) | ? [X0 : $i,X1 : $i] : ((X0 != X1) & ($true = (subset @ X1 @ X0)) & ((subset @ X0 @ X1) = $true))) & (! [X2 : $i,X3 : $i] : ((X2 = X3) | ((subset @ X3 @ X2) != $true) | ((subset @ X2 @ X3) != $true)) | (setextsub != $true))),
% 2.43/1.67 inference(rectify,[status(thm)],[f505])).
% 2.43/1.67 thf(f505,plain,(
% 2.43/1.67 ((setextsub = $true) | ? [X0 : $i,X1 : $i] : ((X0 != X1) & ($true = (subset @ X1 @ X0)) & ((subset @ X0 @ X1) = $true))) & (! [X0 : $i,X1 : $i] : ((X0 = X1) | ($true != (subset @ X1 @ X0)) | ((subset @ X0 @ X1) != $true)) | (setextsub != $true))),
% 2.43/1.67 inference(nnf_transformation,[status(thm)],[f373])).
% 2.43/1.67 thf(f373,plain,(
% 2.43/1.67 (setextsub = $true) <=> ! [X0 : $i,X1 : $i] : ((X0 = X1) | ($true != (subset @ X1 @ X0)) | ((subset @ X0 @ X1) != $true))),
% 2.43/1.67 inference(flattening,[status(thm)],[f372])).
% 2.43/1.67 thf(f372,plain,(
% 2.43/1.67 (setextsub = $true) <=> ! [X0 : $i,X1 : $i] : (((X0 = X1) | ($true != (subset @ X1 @ X0))) | ((subset @ X0 @ X1) != $true))),
% 2.43/1.67 inference(ennf_transformation,[status(thm)],[f192])).
% 2.43/1.67 thf(f192,plain,(
% 2.43/1.67 (setextsub = $true) <=> ! [X0 : $i,X1 : $i] : (((subset @ X0 @ X1) = $true) => (($true = (subset @ X1 @ X0)) => (X0 = X1)))),
% 2.43/1.67 inference(fool_elimination,[status(thm)],[f191])).
% 2.43/1.67 thf(f191,plain,(
% 2.43/1.67 (setextsub = ! [X0 : $i,X1 : $i] : ((subset @ X0 @ X1) => ((subset @ X1 @ X0) => (X0 = X1))))),
% 2.43/1.67 inference(rectify,[status(thm)],[f93])).
% 2.43/1.67 thf(f93,axiom,(
% 2.43/1.67 (setextsub = ! [X3 : $i,X4 : $i] : ((subset @ X3 @ X4) => ((subset @ X4 @ X3) => (X3 = X4))))),
% 2.43/1.67 file('/export/starexec/sandbox/tmp/tmp.x7ZFQe8mUJ/DTF2THF_11764.p',setextsub)).
% 2.43/1.67 % SZS output end Proof for DTF2THF_11764
% 2.43/1.67 % (11867)------------------------------
% 2.43/1.67 % (11867)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.43/1.67 % (11867)Termination reason: Refutation
% 2.43/1.67
% 2.43/1.67 % (11867)Memory used [KB]: 7547
% 2.43/1.67 % (11867)Time elapsed: 0.121 s
% 2.43/1.67 % (11867)Instructions burned: 232 (million)
% 2.43/1.67 % (11867)------------------------------
% 2.43/1.67 % (11867)------------------------------
% 2.43/1.67 % (11861)Success in time 0.13 s
%------------------------------------------------------------------------------