%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU578^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n009.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:35:23 AM UTC 2026
% Result : Theorem 2.39s 1.59s
% Output : Refutation 2.39s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 6
% Syntax : Number of formulae : 37 ( 10 unt; 0 typ; 0 def)
% Number of atoms : 1687 ( 467 equ; 0 cnn)
% Maximal formula atoms : 95 ( 45 avg)
% Number of connectives : 905 ( 46 ~; 31 |; 298 &; 141 @)
% ( 4 <=>; 385 =>; 0 <=; 0 <~>)
% Maximal formula depth : 99 ( 22 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 209 ( 206 usr; 101 con; 0-4 aty)
% Number of variables : 59 ( 0 ^; 44 !; 15 ?; 59 :)
% 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_123,type,
sP1: $i > $i > $o ).
thf(func_def_124,type,
sP2: $i > $o ).
thf(func_def_125,type,
sP3: $i > $i > $o ).
thf(func_def_126,type,
sP4: $i > $i > $o ).
thf(func_def_133,type,
sK11: $i > $o ).
thf(func_def_135,type,
sK13: ( $i > $o ) > $i > $i ).
thf(func_def_136,type,
sK14: $i > $o ).
thf(func_def_139,type,
sK17: $i > ( $i > $o ) > $i ).
thf(func_def_142,type,
sK20: $i > $i > $i ).
thf(func_def_143,type,
sK21: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_144,type,
sK22: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_147,type,
sK25: $i > $o ).
thf(func_def_148,type,
sK26: $i > $o ).
thf(func_def_149,type,
sK27: $i > $o ).
thf(func_def_157,type,
sK35: $i > $i > $i ).
thf(func_def_158,type,
sK36: $i > $i > $i ).
thf(func_def_159,type,
sK37: $i > $o ).
thf(func_def_162,type,
sK40: $i > ( $i > $o ) > $i ).
thf(func_def_168,type,
sK46: $o > $i > $i > $i ).
thf(func_def_169,type,
sK47: $i > $i ).
thf(func_def_177,type,
sK55: $i > $o ).
thf(func_def_180,type,
sK58: $i > ( $i > $o ) > $i ).
thf(func_def_181,type,
sK59: $i > $o ).
thf(func_def_194,type,
sK72: $i > ( $i > $o ) > $i ).
thf(func_def_195,type,
sK73: $i > $o ).
thf(func_def_198,type,
sK76: $i > $i > $i ).
thf(func_def_210,type,
sK88: $i > $o ).
thf(func_def_220,type,
sK98: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_221,type,
sK99: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_222,type,
sK100: $i > $o ).
thf(func_def_223,type,
sK101: $i > $o ).
thf(func_def_224,type,
sK102: $i > $i ).
thf(func_def_227,type,
sK105: $i > $o ).
thf(func_def_236,type,
sK114: $i > $i ).
thf(func_def_238,type,
sK116: $i > $i > $o ).
thf(func_def_239,type,
sK117: $i > $i ).
thf(func_def_240,type,
sK118: $i > $i ).
thf(func_def_241,type,
sK119: ( $i > $i > $o ) > $i > $i ).
thf(func_def_242,type,
sK120: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_243,type,
sK121: ( $i > $i > $o ) > $i > $i ).
thf(func_def_244,type,
sK122: ( $i > $o ) > $i ).
thf(func_def_245,type,
sK123: $i > $o ).
thf(func_def_246,type,
sK124: $i > $i ).
thf(func_def_250,type,
sK128: $i > $i > $i ).
thf(func_def_261,type,
sK139: $i > $i > $i ).
thf(func_def_262,type,
sK140: $i > $i > $i ).
thf(func_def_263,type,
sK141: $i > $i > $i ).
thf(func_def_264,type,
sK142: $i > $i > $i ).
thf(func_def_265,type,
sK143: $i > $i > $i ).
thf(func_def_266,type,
sK144: $i > $i > $i > $i ).
thf(func_def_267,type,
sK145: $i > $i ).
thf(func_def_268,type,
sK146: $i > $i ).
thf(func_def_269,type,
sK147: $i > $i ).
thf(func_def_270,type,
sK148: $i > $i ).
thf(func_def_271,type,
sK149: $i > $i > $i > $i ).
thf(func_def_272,type,
sK150: $i > $i > $i > $i ).
thf(func_def_273,type,
sK151: $i > $i > $i ).
thf(func_def_274,type,
sK152: $i > $i > $i ).
thf(func_def_275,type,
sK153: $i > $i > $i ).
thf(func_def_276,type,
sK154: $i > $i ).
thf(func_def_278,type,
sK156: $i > $i ).
thf(func_def_279,type,
sK157: $i > $i ).
thf(func_def_283,type,
sK161: $i > $i > $i ).
thf(func_def_287,type,
sK165: $i > $o ).
thf(func_def_292,type,
sK170: $i > $i ).
thf(func_def_295,type,
sK173: $i > $i ).
thf(func_def_298,type,
sK176: $i > $i > $i ).
thf(func_def_299,type,
sK177: $i > $o ).
thf(func_def_300,type,
sK178: $i > $o ).
thf(func_def_303,type,
sK181: $i > ( $i > $o ) > $i ).
thf(func_def_308,type,
sK186: $i > $o ).
thf(func_def_310,type,
sK188: $i > $i ).
thf(func_def_329,type,
sK207: $i > $i ).
thf(func_def_343,type,
sK221: $i > $o ).
thf(func_def_344,type,
sK222: ( $i > $o ) > $i ).
thf(func_def_345,type,
sK223: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_346,type,
sK224: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_347,type,
sK225: $i > $o ).
thf(func_def_348,type,
sK226: $i > $o ).
thf(func_def_351,type,
sK229: ( $i > $o ) > $i > $i ).
thf(func_def_353,type,
sK231: $i > $o ).
thf(func_def_355,type,
sK233: $i > $o ).
thf(func_def_362,type,
sK240: $i > $o ).
thf(func_def_363,type,
sK241: $i > $i ).
thf(func_def_364,type,
sK242: ( $i > $o ) > $i ).
thf(func_def_369,type,
sK247: ( $i > $o ) > $i ).
thf(func_def_370,type,
sK248: ( $i > $o ) > $i ).
thf(func_def_371,type,
sK249: $i > $o ).
thf(func_def_373,type,
sK251: $i > $o ).
thf(func_def_375,type,
sK253: $i > ( $i > $o ) > $i ).
thf(func_def_376,type,
sK254: $i > $i > $i ).
thf(func_def_380,type,
sK258: $i > $o ).
thf(func_def_382,type,
sK260: $i > $o ).
thf(func_def_386,type,
ph264:
!>[X0: $tType] : X0 ).
thf(f2332,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f2328,f1892]) ).
thf(f1892,plain,
! [X3: $i] :
( ( subset @ X3 @ X3 )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f1706]) ).
thf(f1706,plain,
! [X3: $i] :
( ( ( subset @ X3 @ X3 )
= $true )
| ( $true != $true ) ),
inference(equality_resolution,[status(thm)],[f1417]) ).
thf(f1417,plain,
! [X2: $i,X3: $i] :
( ( $true
= ( subset @ X2 @ X3 ) )
| ( X2 != X3 )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f935,f1142]) ).
thf(f1142,plain,
eqimpsubset1 = $true,
inference(cnf_transformation,[status(thm)],[f649]) ).
thf(f649,plain,
( ( subsetTrans = $true )
& ( emptyinunitempty = $true )
& ( emptyset__Cong = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( exuEu = $true )
& ( notsubsetI = $true )
& ( noeltsimpempty = $true )
& ( exu__Cong = $true )
& ( foundationAx = $true )
& ( exuE2 = $true )
& ( omega__Cong = $true )
& ( eqimpsubset1 = $true )
& ( emptyI = $true )
& ( nonemptyImpWitness = $true )
& ( subsetRefl = $true )
& ( subsetE2 = $true )
& ( notdallE = $true )
& ( prop2setE = $true )
& ( omegaSAx = $true )
& ( powersetE = $true )
& ( subsetI1 = $true )
& ( exuE3e = $true )
& ( setadjoinOr = $true )
& ( exuE3u = $true )
& ( emptyinPowerset = $true )
& ( nonemptyI1 = $true )
& ( vacuousDall = $true )
& ( in__Cong = $true )
& ( nonemptyE1 = $true )
& ( emptysetsubset = $true )
& ( singletonsswitch = $true )
& ( powersetAx = $true )
& ( notequalI1 = $true )
& ( setunionI = $true )
& ( descr__Cong = $true )
& ( quantDeMorgan3 = $true )
& ( upairsetIR = $true )
& ( dsetconstrI = $true )
& ( dsetconstrEL = $true )
& ( nonemptyI = $true )
& ( inCongP = $true )
& ( dsetconstr__Cong = $true )
& ( emptyE1 = $true )
& ( quantDeMorgan1 = $true )
& ( setoftrueEq = $true )
& ( setadjoinE = $true )
& ( replAx = $true )
& ( quantDeMorgan4 = $true )
& ( subPowSU = $true )
& ( subset2powerset = $true )
& ( emptysetAx = $true )
& ( notinsingleton = $true )
& ( notinemptyset = $true )
& ( quantDeMorgan2 = $true )
& ( wellorderingAx = $true )
& ( setadjoin__Cong = $true )
& ( prop2set2propI = $true )
& ( setextsub = $true )
& ( omega0Ax = $true )
& ( powersetI1 = $true )
& ( descrp = $true )
& ( setunionE = $true )
& ( omegaIndAx = $true )
& ( eqimpsubset2 = $true )
& ( setadjoinSub2 = $true )
& ( prop2setI = $true )
& ( setadjoinIR = $true )
& ( setadjoinAx = $true )
& ( setadjoinIL = $true )
& ( setbeta = $true )
& ( emptysetimpfalse = $true )
& ( powersetE1 = $true )
& ( setextAx = $true )
& ( exuI1 = $true )
& ( uniqinunit = $true )
& ( setunionAx = $true )
& ( emptysetE = $true )
& ( upairsetE = $true )
& ( notequalI2 = $true )
& ( subsetemptysetimpeq = $true )
& ( emptyInPowerset = $true )
& ( powersetI = $true )
& ( setext = $true )
& ( subsetI2 = $true )
& ( upairsetIL = $true )
& ( setadjoinSub = $true )
& ( notdexE = $true )
& ( powerset__Cong = $true )
& ( setunion__Cong = $true )
& ( $true
!= ( in @ sK132 @ ( powerset @ sK132 ) ) )
& ( exuI2 = $true )
& ( subsetE = $true )
& ( eqinunit = $true )
& ( exuI3 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK132])],[f360,f648]) ).
thf(f648,plain,
( ? [X0: $i] :
( $true
!= ( in @ X0 @ ( powerset @ X0 ) ) )
=> ( $true
!= ( in @ sK132 @ ( powerset @ sK132 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f360,plain,
( ( subsetTrans = $true )
& ( emptyinunitempty = $true )
& ( emptyset__Cong = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( exuEu = $true )
& ( notsubsetI = $true )
& ( noeltsimpempty = $true )
& ( exu__Cong = $true )
& ( foundationAx = $true )
& ( exuE2 = $true )
& ( omega__Cong = $true )
& ( eqimpsubset1 = $true )
& ( emptyI = $true )
& ( nonemptyImpWitness = $true )
& ( subsetRefl = $true )
& ( subsetE2 = $true )
& ( notdallE = $true )
& ( prop2setE = $true )
& ( omegaSAx = $true )
& ( powersetE = $true )
& ( subsetI1 = $true )
& ( exuE3e = $true )
& ( setadjoinOr = $true )
& ( exuE3u = $true )
& ( emptyinPowerset = $true )
& ( nonemptyI1 = $true )
& ( vacuousDall = $true )
& ( in__Cong = $true )
& ( nonemptyE1 = $true )
& ( emptysetsubset = $true )
& ( singletonsswitch = $true )
& ( powersetAx = $true )
& ( notequalI1 = $true )
& ( setunionI = $true )
& ( descr__Cong = $true )
& ( quantDeMorgan3 = $true )
& ( upairsetIR = $true )
& ( dsetconstrI = $true )
& ( dsetconstrEL = $true )
& ( nonemptyI = $true )
& ( inCongP = $true )
& ( dsetconstr__Cong = $true )
& ( emptyE1 = $true )
& ( quantDeMorgan1 = $true )
& ( setoftrueEq = $true )
& ( setadjoinE = $true )
& ( replAx = $true )
& ( quantDeMorgan4 = $true )
& ( subPowSU = $true )
& ( subset2powerset = $true )
& ( emptysetAx = $true )
& ( notinsingleton = $true )
& ( notinemptyset = $true )
& ( quantDeMorgan2 = $true )
& ( wellorderingAx = $true )
& ( setadjoin__Cong = $true )
& ( prop2set2propI = $true )
& ( setextsub = $true )
& ( omega0Ax = $true )
& ( powersetI1 = $true )
& ( descrp = $true )
& ( setunionE = $true )
& ( omegaIndAx = $true )
& ( eqimpsubset2 = $true )
& ( setadjoinSub2 = $true )
& ( prop2setI = $true )
& ( setadjoinIR = $true )
& ( setadjoinAx = $true )
& ( setadjoinIL = $true )
& ( setbeta = $true )
& ( emptysetimpfalse = $true )
& ( powersetE1 = $true )
& ( setextAx = $true )
& ( exuI1 = $true )
& ( uniqinunit = $true )
& ( setunionAx = $true )
& ( emptysetE = $true )
& ( upairsetE = $true )
& ( notequalI2 = $true )
& ( subsetemptysetimpeq = $true )
& ( emptyInPowerset = $true )
& ( powersetI = $true )
& ( setext = $true )
& ( subsetI2 = $true )
& ( upairsetIL = $true )
& ( setadjoinSub = $true )
& ( notdexE = $true )
& ( powerset__Cong = $true )
& ( setunion__Cong = $true )
& ? [X0: $i] :
( $true
!= ( in @ X0 @ ( powerset @ X0 ) ) )
& ( exuI2 = $true )
& ( subsetE = $true )
& ( eqinunit = $true )
& ( exuI3 = $true ) ),
inference(flattening,[status(thm)],[f359]) ).
thf(f359,plain,
( ? [X0: $i] :
( $true
!= ( in @ X0 @ ( powerset @ X0 ) ) )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f193]) ).
thf(f193,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ! [X0: $i] :
( $true
= ( in @ X0 @ ( powerset @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f192]) ).
thf(f192,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ! [X0: $i] : ( in @ X0 @ ( powerset @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f98]) ).
thf(f98,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ! [X3: $i] : ( in @ X3 @ ( powerset @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f97]) ).
thf(f97,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ! [X3: $i] : ( in @ X3 @ ( powerset @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.EC5jtjE9C7/DTF2THF_10635.p',inPowerset) ).
thf(f935,plain,
! [X2: $i,X3: $i] :
( ( $true
= ( subset @ X2 @ X3 ) )
| ( X2 != X3 )
| ( eqimpsubset1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f495]) ).
thf(f495,plain,
( ( ( eqimpsubset1 = $true )
| ( ( $true
!= ( subset @ sK49 @ sK50 ) )
& ( sK49 = sK50 ) ) )
& ( ! [X2: $i,X3: $i] :
( ( $true
= ( subset @ X2 @ X3 ) )
| ( X2 != X3 ) )
| ( eqimpsubset1 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK49,sK50])],[f493,f494]) ).
thf(f494,plain,
( ? [X0: $i,X1: $i] :
( ( $true
!= ( subset @ X0 @ X1 ) )
& ( X0 = X1 ) )
=> ( ( $true
!= ( subset @ sK49 @ sK50 ) )
& ( sK49 = sK50 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f493,plain,
( ( ( eqimpsubset1 = $true )
| ? [X0: $i,X1: $i] :
( ( $true
!= ( subset @ X0 @ X1 ) )
& ( X0 = X1 ) ) )
& ( ! [X2: $i,X3: $i] :
( ( $true
= ( subset @ X2 @ X3 ) )
| ( X2 != X3 ) )
| ( eqimpsubset1 != $true ) ) ),
inference(rectify,[status(thm)],[f492]) ).
thf(f492,plain,
( ( ( eqimpsubset1 = $true )
| ? [X0: $i,X1: $i] :
( ( $true
!= ( subset @ X0 @ X1 ) )
& ( X0 = X1 ) ) )
& ( ! [X0: $i,X1: $i] :
( ( $true
= ( subset @ X0 @ X1 ) )
| ( X0 != X1 ) )
| ( eqimpsubset1 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f383]) ).
thf(f383,plain,
( ( eqimpsubset1 = $true )
<=> ! [X0: $i,X1: $i] :
( ( $true
= ( subset @ X0 @ X1 ) )
| ( X0 != X1 ) ) ),
inference(ennf_transformation,[status(thm)],[f283]) ).
thf(f283,plain,
( ! [X0: $i,X1: $i] :
( ( X0 = X1 )
=> ( $true
= ( subset @ X0 @ X1 ) ) )
<=> ( eqimpsubset1 = $true ) ),
inference(fool_elimination,[status(thm)],[f282]) ).
thf(f282,plain,
( eqimpsubset1
= ( ! [X0: $i,X1: $i] :
( ( X0 = X1 )
=> ( subset @ X0 @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f80]) ).
thf(f80,axiom,
( eqimpsubset1
= ( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( subset @ X3 @ X4 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.EC5jtjE9C7/DTF2THF_10635.p',eqimpsubset1) ).
thf(f2328,plain,
( $true
!= ( subset @ sK132 @ sK132 ) ),
inference(trivial_inequality_removal,[status(thm)],[f2321]) ).
thf(f2321,plain,
( ( $true
!= ( subset @ sK132 @ sK132 ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f1064,f1805]) ).
thf(f1805,plain,
! [X2: $i,X3: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X3 ) ) )
| ( $true
!= ( subset @ X2 @ X3 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f1360]) ).
thf(f1360,plain,
! [X2: $i,X3: $i] :
( ( $true != $true )
| ( $true
!= ( subset @ X2 @ X3 ) )
| ( $true
= ( in @ X2 @ ( powerset @ X3 ) ) ) ),
inference(definition_unfolding,[status(thm)],[f877,f1094]) ).
thf(f1094,plain,
powersetI1 = $true,
inference(cnf_transformation,[status(thm)],[f649]) ).
thf(f877,plain,
! [X2: $i,X3: $i] :
( ( $true
!= ( subset @ X2 @ X3 ) )
| ( $true
= ( in @ X2 @ ( powerset @ X3 ) ) )
| ( powersetI1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f428]) ).
thf(f428,plain,
( ( ( powersetI1 = $true )
| ( ( $true
= ( subset @ sK8 @ sK9 ) )
& ( $true
!= ( in @ sK8 @ ( powerset @ sK9 ) ) ) ) )
& ( ! [X2: $i,X3: $i] :
( ( $true
!= ( subset @ X2 @ X3 ) )
| ( $true
= ( in @ X2 @ ( powerset @ X3 ) ) ) )
| ( powersetI1 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK8,sK9])],[f426,f427]) ).
thf(f427,plain,
( ? [X0: $i,X1: $i] :
( ( $true
= ( subset @ X0 @ X1 ) )
& ( $true
!= ( in @ X0 @ ( powerset @ X1 ) ) ) )
=> ( ( $true
= ( subset @ sK8 @ sK9 ) )
& ( $true
!= ( in @ sK8 @ ( powerset @ sK9 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f426,plain,
( ( ( powersetI1 = $true )
| ? [X0: $i,X1: $i] :
( ( $true
= ( subset @ X0 @ X1 ) )
& ( $true
!= ( in @ X0 @ ( powerset @ X1 ) ) ) ) )
& ( ! [X2: $i,X3: $i] :
( ( $true
!= ( subset @ X2 @ X3 ) )
| ( $true
= ( in @ X2 @ ( powerset @ X3 ) ) ) )
| ( powersetI1 != $true ) ) ),
inference(rectify,[status(thm)],[f425]) ).
thf(f425,plain,
( ( ( powersetI1 = $true )
| ? [X0: $i,X1: $i] :
( ( $true
= ( subset @ X0 @ X1 ) )
& ( $true
!= ( in @ X0 @ ( powerset @ X1 ) ) ) ) )
& ( ! [X0: $i,X1: $i] :
( ( $true
!= ( subset @ X0 @ X1 ) )
| ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) ) )
| ( powersetI1 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f354]) ).
thf(f354,plain,
( ( powersetI1 = $true )
<=> ! [X0: $i,X1: $i] :
( ( $true
!= ( subset @ X0 @ X1 ) )
| ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) ) ) ),
inference(ennf_transformation,[status(thm)],[f145]) ).
thf(f145,plain,
( ( powersetI1 = $true )
<=> ! [X0: $i,X1: $i] :
( ( $true
= ( subset @ X0 @ X1 ) )
=> ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f144]) ).
thf(f144,plain,
( powersetI1
= ( ! [X0: $i,X1: $i] :
( ( subset @ X0 @ X1 )
=> ( in @ X0 @ ( powerset @ X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f95]) ).
thf(f95,axiom,
( powersetI1
= ( ! [X4: $i,X3: $i] :
( ( subset @ X4 @ X3 )
=> ( in @ X4 @ ( powerset @ X3 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.EC5jtjE9C7/DTF2THF_10635.p',powersetI1) ).
thf(f1064,plain,
( $true
!= ( in @ sK132 @ ( powerset @ sK132 ) ) ),
inference(cnf_transformation,[status(thm)],[f649]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU578^1 : TPTP v9.2.1. Released v3.7.0.
% 0.11/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n009.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 12:27:25 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.30/0.46 ---- Original DTF file ---
% 0.30/0.46 thf(spec,logic,$$dhol).
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % File : SEU578^1 : TPTP v9.2.1. Released v3.7.0.
% 0.30/0.46 % Domain : Set Theory
% 0.30/0.46 % Problem : Preliminary Notions - Relations on Sets - Subsets
% 0.30/0.46 % Version : Especial.
% 0.30/0.46 % English : (! A:i.in A (powerset A))
% 0.30/0.46
% 0.30/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.30/0.46 % Source : [Bro08]
% 0.30/0.46 % Names : ZFC080g [Bro08]
% 0.30/0.46
% 0.30/0.46 % Status : Theorem
% 0.30/0.46 % Rating : 0.11 v9.1.0, 0.25 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 v5.2.0, 0.60 v5.1.0, 0.80 v5.0.0, 0.40 v4.1.0, 0.67 v3.7.0
% 0.30/0.46 % Syntax : Number of formulae : 206 ( 96 unt; 109 typ; 96 def)
% 0.30/0.46 % Number of atoms : 522 ( 145 equ; 0 cnn)
% 0.30/0.46 % Maximal formula atoms : 95 ( 5 avg)
% 0.30/0.46 % Number of connectives : 816 ( 29 ~; 5 |; 27 &; 504 @)
% 0.30/0.46 % ( 14 <=>; 237 =>; 0 <=; 0 <~>)
% 0.30/0.46 % Maximal formula depth : 98 ( 2 avg)
% 0.30/0.46 % Number of types : 2 ( 0 usr)
% 0.30/0.46 % Number of type conns : 53 ( 53 >; 0 *; 0 +; 0 <<)
% 0.30/0.46 % Number of symbols : 112 ( 109 usr; 98 con; 0-2 aty)
% 0.30/0.46 % Number of variables : 286 ( 28 ^; 230 !; 28 ?; 286 :)
% 0.30/0.46 % SPC : TH0_THM_EQU_NAR
% 0.30/0.46
% 0.30/0.46 % Comments : http://mathgate.info/detsetitem.php?id=476
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 thf(in_type,type,
% 0.30/0.46 in: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(exu_type,type,
% 0.30/0.46 exu: ( $i > $o ) > $o ).
% 0.30/0.46
% 0.30/0.46 thf(exu,definition,
% 0.30/0.46 ( exu
% 0.30/0.46 = ( ^ [Xphi: $i > $o] :
% 0.30/0.46 ? [Xx: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 & ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setextAx_type,type,
% 0.30/0.46 setextAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setextAx,definition,
% 0.30/0.46 ( setextAx
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 <=> ( in @ Xx @ B ) )
% 0.30/0.46 => ( A = B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyset_type,type,
% 0.30/0.46 emptyset: $i ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetAx_type,type,
% 0.30/0.46 emptysetAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetAx,definition,
% 0.30/0.46 ( emptysetAx
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoin_type,type,
% 0.30/0.46 setadjoin: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinAx_type,type,
% 0.30/0.46 setadjoinAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinAx,definition,
% 0.30/0.46 ( setadjoinAx
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 <=> ( ( Xy = Xx )
% 0.30/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powerset_type,type,
% 0.30/0.46 powerset: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(powersetAx_type,type,
% 0.30/0.46 powersetAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetAx,definition,
% 0.30/0.46 ( powersetAx
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.46 <=> ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunion_type,type,
% 0.30/0.46 setunion: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(setunionAx_type,type,
% 0.30/0.46 setunionAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionAx,definition,
% 0.30/0.46 ( setunionAx
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.46 <=> ? [B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omega_type,type,
% 0.30/0.46 omega: $i ).
% 0.30/0.46
% 0.30/0.46 thf(omega0Ax_type,type,
% 0.30/0.46 omega0Ax: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omega0Ax,definition,
% 0.30/0.46 ( omega0Ax
% 0.30/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omegaSAx_type,type,
% 0.30/0.46 omegaSAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omegaSAx,definition,
% 0.30/0.46 ( omegaSAx
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ omega )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omegaIndAx_type,type,
% 0.30/0.46 omegaIndAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omegaIndAx,definition,
% 0.30/0.46 ( omegaIndAx
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( ( in @ emptyset @ A )
% 0.30/0.46 & ! [Xx: $i] :
% 0.30/0.46 ( ( ( in @ Xx @ omega )
% 0.30/0.46 & ( in @ Xx @ A ) )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ omega )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(replAx_type,type,
% 0.30/0.46 replAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(replAx,definition,
% 0.30/0.46 ( replAx
% 0.30/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.30/0.46 => ? [B: $i] :
% 0.30/0.46 ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 <=> ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(foundationAx_type,type,
% 0.30/0.46 foundationAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(foundationAx,definition,
% 0.30/0.46 ( foundationAx
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/0.46 => ? [B: $i] :
% 0.30/0.46 ( ( in @ B @ A )
% 0.30/0.46 & ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(wellorderingAx_type,type,
% 0.30/0.46 wellorderingAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(wellorderingAx,definition,
% 0.30/0.46 ( wellorderingAx
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ? [B: $i] :
% 0.30/0.46 ( ! [C: $i] :
% 0.30/0.46 ( ( in @ C @ B )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ C )
% 0.30/0.46 => ( in @ Xx @ A ) ) )
% 0.30/0.46 & ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( ( in @ Xx @ A )
% 0.30/0.46 & ( in @ Xy @ A ) )
% 0.30/0.46 => ( ! [C: $i] :
% 0.30/0.46 ( ( in @ C @ B )
% 0.30/0.46 => ( ( in @ Xx @ C )
% 0.30/0.46 <=> ( in @ Xy @ C ) ) )
% 0.30/0.46 => ( Xx = Xy ) ) )
% 0.30/0.46 & ! [C: $i,D: $i] :
% 0.30/0.46 ( ( ( in @ C @ B )
% 0.30/0.46 & ( in @ D @ B ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ C )
% 0.30/0.46 => ( in @ Xx @ D ) )
% 0.30/0.46 | ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ D )
% 0.30/0.46 => ( in @ Xx @ C ) ) ) )
% 0.30/0.46 & ! [C: $i] :
% 0.30/0.46 ( ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ C )
% 0.30/0.46 => ( in @ Xx @ A ) )
% 0.30/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.30/0.46 => ? [D: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ D @ B )
% 0.30/0.46 & ( in @ Xx @ C )
% 0.30/0.46 & ~ ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ D )
% 0.30/0.46 & ( in @ Xy @ C ) )
% 0.30/0.46 & ! [E: $i] :
% 0.30/0.46 ( ( in @ E @ B )
% 0.30/0.46 => ( ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ E )
% 0.30/0.46 => ( in @ Xy @ D ) )
% 0.30/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(descr_type,type,
% 0.30/0.46 descr: ( $i > $o ) > $i ).
% 0.30/0.46
% 0.30/0.46 thf(descrp_type,type,
% 0.30/0.46 descrp: $o ).
% 0.30/0.46
% 0.30/0.46 thf(descrp,definition,
% 0.30/0.46 ( descrp
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ( Xphi
% 0.30/0.46 @ ( descr
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstr_type,type,
% 0.30/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrI_type,type,
% 0.30/0.46 dsetconstrI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrI,definition,
% 0.30/0.46 ( dsetconstrI
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrEL_type,type,
% 0.30/0.46 dsetconstrEL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrEL,definition,
% 0.30/0.46 ( dsetconstrEL
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrER_type,type,
% 0.30/0.46 dsetconstrER: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrER,definition,
% 0.30/0.46 ( dsetconstrER
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE1_type,type,
% 0.30/0.46 exuE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE1,definition,
% 0.30/0.46 ( exuE1
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 & ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(prop2set_type,type,
% 0.30/0.46 prop2set: $o > $i ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setE_type,type,
% 0.30/0.46 prop2setE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setE,definition,
% 0.30/0.46 ( prop2setE
% 0.30/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.30/0.46 => Xphi ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetE_type,type,
% 0.30/0.46 emptysetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetE,definition,
% 0.30/0.46 ( emptysetE
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ emptyset )
% 0.30/0.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetimpfalse_type,type,
% 0.30/0.46 emptysetimpfalse: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetimpfalse,definition,
% 0.30/0.46 ( emptysetimpfalse
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ emptyset )
% 0.30/0.46 => $false ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notinemptyset_type,type,
% 0.30/0.46 notinemptyset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notinemptyset,definition,
% 0.30/0.46 ( notinemptyset
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3e_type,type,
% 0.30/0.46 exuE3e: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3e,definition,
% 0.30/0.46 ( exuE3e
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setext_type,type,
% 0.30/0.46 setext: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setext,definition,
% 0.30/0.46 ( setext
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) )
% 0.30/0.46 => ( A = B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyI_type,type,
% 0.30/0.46 emptyI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyI,definition,
% 0.30/0.46 ( emptyI
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ A )
% 0.30/0.46 => ( A = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(noeltsimpempty_type,type,
% 0.30/0.46 noeltsimpempty: $o ).
% 0.30/0.46
% 0.30/0.46 thf(noeltsimpempty,definition,
% 0.30/0.46 ( noeltsimpempty
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ A )
% 0.30/0.46 => ( A = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setbeta_type,type,
% 0.30/0.46 setbeta: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setbeta,definition,
% 0.30/0.46 ( setbeta
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonempty_type,type,
% 0.30/0.46 nonempty: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonempty,definition,
% 0.30/0.46 ( nonempty
% 0.30/0.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyE1_type,type,
% 0.30/0.46 nonemptyE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyE1,definition,
% 0.30/0.46 ( nonemptyE1
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( nonempty @ A )
% 0.30/0.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI_type,type,
% 0.30/0.46 nonemptyI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI,definition,
% 0.30/0.46 ( nonemptyI
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( nonempty
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI1_type,type,
% 0.30/0.46 nonemptyI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI1,definition,
% 0.30/0.46 ( nonemptyI1
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/0.46 => ( nonempty @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIL_type,type,
% 0.30/0.46 setadjoinIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIL,definition,
% 0.30/0.46 ( setadjoinIL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinunitempty_type,type,
% 0.30/0.46 emptyinunitempty: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinunitempty,definition,
% 0.30/0.46 ( emptyinunitempty
% 0.30/0.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIR_type,type,
% 0.30/0.46 setadjoinIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIR,definition,
% 0.30/0.46 ( setadjoinIR
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinE_type,type,
% 0.30/0.46 setadjoinE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinE,definition,
% 0.30/0.46 ( setadjoinE
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ( ( Xy = Xx )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => ( ( ( in @ Xy @ A )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinOr_type,type,
% 0.30/0.46 setadjoinOr: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinOr,definition,
% 0.30/0.46 ( setadjoinOr
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 => ( ( Xy = Xx )
% 0.30/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setoftrueEq_type,type,
% 0.30/0.46 setoftrueEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setoftrueEq,definition,
% 0.30/0.46 ( setoftrueEq
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : $true )
% 0.30/0.46 = A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI_type,type,
% 0.30/0.46 powersetI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI,definition,
% 0.30/0.46 ( powersetI
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) )
% 0.30/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinPowerset_type,type,
% 0.30/0.46 emptyinPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinPowerset,definition,
% 0.30/0.46 ( emptyinPowerset
% 0.30/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyInPowerset_type,type,
% 0.30/0.46 emptyInPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyInPowerset,definition,
% 0.30/0.46 ( emptyInPowerset
% 0.30/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE_type,type,
% 0.30/0.46 powersetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE,definition,
% 0.30/0.46 ( powersetE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionI_type,type,
% 0.30/0.46 setunionI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionI,definition,
% 0.30/0.46 ( setunionI
% 0.30/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( ( in @ B @ A )
% 0.30/0.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE_type,type,
% 0.30/0.46 setunionE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE,definition,
% 0.30/0.46 ( setunionE
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ! [B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( ( in @ B @ A )
% 0.30/0.46 => Xphi ) )
% 0.30/0.46 => Xphi ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subPowSU_type,type,
% 0.30/0.46 subPowSU: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subPowSU,definition,
% 0.30/0.46 ( subPowSU
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE2_type,type,
% 0.30/0.46 exuE2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE2,definition,
% 0.30/0.46 ( exuE2
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyImpWitness_type,type,
% 0.30/0.46 nonemptyImpWitness: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyImpWitness,definition,
% 0.30/0.46 ( nonemptyImpWitness
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( nonempty @ A )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & $true ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(uniqinunit_type,type,
% 0.30/0.46 uniqinunit: $o ).
% 0.30/0.46
% 0.30/0.46 thf(uniqinunit,definition,
% 0.30/0.46 ( uniqinunit
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notinsingleton_type,type,
% 0.30/0.46 notinsingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notinsingleton,definition,
% 0.30/0.46 ( notinsingleton
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx != Xy )
% 0.30/0.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqinunit_type,type,
% 0.30/0.46 eqinunit: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqinunit,definition,
% 0.30/0.46 ( eqinunit
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsswitch_type,type,
% 0.30/0.46 singletonsswitch: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsswitch,definition,
% 0.30/0.46 ( singletonsswitch
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetE_type,type,
% 0.30/0.46 upairsetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetE,definition,
% 0.30/0.46 ( upairsetE
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.30/0.46 => ( ( Xz = Xx )
% 0.30/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIL_type,type,
% 0.30/0.46 upairsetIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIL,definition,
% 0.30/0.46 ( upairsetIL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIR_type,type,
% 0.30/0.46 upairsetIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIR,definition,
% 0.30/0.46 ( upairsetIR
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyE1_type,type,
% 0.30/0.46 emptyE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyE1,definition,
% 0.30/0.46 ( emptyE1
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ( ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 = emptyset )
% 0.30/0.46 => $false ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(vacuousDall_type,type,
% 0.30/0.46 vacuousDall: $o ).
% 0.30/0.46
% 0.30/0.46 thf(vacuousDall,definition,
% 0.30/0.46 ( vacuousDall
% 0.30/0.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ emptyset )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan1_type,type,
% 0.30/0.46 quantDeMorgan1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan1,definition,
% 0.30/0.46 ( quantDeMorgan1
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan2_type,type,
% 0.30/0.46 quantDeMorgan2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan2,definition,
% 0.30/0.46 ( quantDeMorgan2
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ~ ( Xphi @ Xx ) )
% 0.30/0.46 => ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan3_type,type,
% 0.30/0.46 quantDeMorgan3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan3,definition,
% 0.30/0.46 ( quantDeMorgan3
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan4_type,type,
% 0.30/0.46 quantDeMorgan4: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan4,definition,
% 0.30/0.46 ( quantDeMorgan4
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ~ ( Xphi @ Xx ) )
% 0.30/0.46 => ~ ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setI_type,type,
% 0.30/0.46 prop2setI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setI,definition,
% 0.30/0.46 ( prop2setI
% 0.30/0.46 = ( ! [Xphi: $o] :
% 0.30/0.46 ( Xphi
% 0.30/0.46 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(set2prop_type,type,
% 0.30/0.46 set2prop: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2set2propI_type,type,
% 0.30/0.46 prop2set2propI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2set2propI,definition,
% 0.30/0.46 ( prop2set2propI
% 0.30/0.46 = ( ! [Xphi: $o] :
% 0.30/0.46 ( Xphi
% 0.30/0.46 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notdexE_type,type,
% 0.30/0.46 notdexE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notdexE,definition,
% 0.30/0.46 ( notdexE
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notdallE_type,type,
% 0.30/0.46 notdallE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notdallE,definition,
% 0.30/0.46 ( notdallE
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuI1_type,type,
% 0.30/0.46 exuI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuI1,definition,
% 0.30/0.46 ( exuI1
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 & ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuI3_type,type,
% 0.30/0.46 exuI3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuI3,definition,
% 0.30/0.46 ( exuI3
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.30/0.46 => ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuI2_type,type,
% 0.30/0.46 exuI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuI2,definition,
% 0.30/0.46 ( exuI2
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 <=> ( Xy = Xx ) )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inCongP_type,type,
% 0.30/0.46 inCongP: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inCongP,definition,
% 0.30/0.46 ( inCongP
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(in__Cong_type,type,
% 0.30/0.46 in__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(in__Cong,definition,
% 0.30/0.46 ( in__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3u_type,type,
% 0.30/0.46 exuE3u: $o ).
% 0.30/0.47
% 0.30/0.47 thf(exuE3u,definition,
% 0.30/0.47 ( exuE3u
% 0.30/0.47 = ( ! [Xphi: $i > $o] :
% 0.30/0.47 ( ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 => ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xphi @ Xx )
% 0.30/0.47 => ( ( Xphi @ Xy )
% 0.30/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(exu__Cong_type,type,
% 0.30/0.47 exu__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(exu__Cong,definition,
% 0.30/0.47 ( exu__Cong
% 0.30/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xx = Xy )
% 0.30/0.47 => ( ( Xphi @ Xx )
% 0.30/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.30/0.47 => ( ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 <=> ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(emptyset__Cong_type,type,
% 0.30/0.47 emptyset__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(emptyset__Cong,definition,
% 0.30/0.47 ( emptyset__Cong
% 0.30/0.47 = ( emptyset = emptyset ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoin__Cong_type,type,
% 0.30/0.47 setadjoin__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoin__Cong,definition,
% 0.30/0.47 ( setadjoin__Cong
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xx = Xy )
% 0.30/0.47 => ! [Xz: $i,Xu: $i] :
% 0.30/0.47 ( ( Xz = Xu )
% 0.30/0.47 => ( ( setadjoin @ Xx @ Xz )
% 0.30/0.47 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(powerset__Cong_type,type,
% 0.30/0.47 powerset__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(powerset__Cong,definition,
% 0.30/0.47 ( powerset__Cong
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 => ( ( powerset @ A )
% 0.30/0.47 = ( powerset @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setunion__Cong_type,type,
% 0.30/0.47 setunion__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setunion__Cong,definition,
% 0.30/0.47 ( setunion__Cong
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 => ( ( setunion @ A )
% 0.30/0.47 = ( setunion @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(omega__Cong_type,type,
% 0.30/0.47 omega__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(omega__Cong,definition,
% 0.30/0.47 ( omega__Cong
% 0.30/0.47 = ( omega = omega ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(exuEu_type,type,
% 0.30/0.47 exuEu: $o ).
% 0.30/0.47
% 0.30/0.47 thf(exuEu,definition,
% 0.30/0.47 ( exuEu
% 0.30/0.47 = ( ! [Xphi: $i > $o] :
% 0.30/0.47 ( ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 => ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xphi @ Xx )
% 0.30/0.47 => ( ( Xphi @ Xy )
% 0.30/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(descr__Cong_type,type,
% 0.30/0.47 descr__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(descr__Cong,definition,
% 0.30/0.47 ( descr__Cong
% 0.30/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xx = Xy )
% 0.30/0.47 => ( ( Xphi @ Xx )
% 0.30/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.30/0.47 => ( ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 => ( ( exu
% 0.30/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.30/0.47 => ( ( descr
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 = ( descr
% 0.30/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(dsetconstr__Cong_type,type,
% 0.30/0.47 dsetconstr__Cong: $o ).
% 0.30/0.47
% 0.30/0.47 thf(dsetconstr__Cong,definition,
% 0.30/0.47 ( dsetconstr__Cong
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( Xx = Xy )
% 0.30/0.47 => ( ( Xphi @ Xx )
% 0.30/0.47 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.30/0.47 => ( ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 = ( dsetconstr @ B
% 0.30/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subset_type,type,
% 0.30/0.47 subset: $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(disjoint_type,type,
% 0.30/0.47 disjoint: $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(setsmeet_type,type,
% 0.30/0.47 setsmeet: $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetI1_type,type,
% 0.30/0.47 subsetI1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetI1,definition,
% 0.30/0.47 ( subsetI1
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ Xx @ B ) )
% 0.30/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(eqimpsubset2_type,type,
% 0.30/0.47 eqimpsubset2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(eqimpsubset2,definition,
% 0.30/0.47 ( eqimpsubset2
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(eqimpsubset1_type,type,
% 0.30/0.47 eqimpsubset1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(eqimpsubset1,definition,
% 0.30/0.47 ( eqimpsubset1
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( A = B )
% 0.30/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subsetI2_type,type,
% 0.30/0.47 subsetI2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetI2,definition,
% 0.30/0.47 ( subsetI2
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ Xx @ B ) )
% 0.30/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(emptysetsubset_type,type,
% 0.30/0.47 emptysetsubset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(emptysetsubset,definition,
% 0.30/0.47 ( emptysetsubset
% 0.30/0.47 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subsetE_type,type,
% 0.30/0.47 subsetE: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetE,definition,
% 0.30/0.47 ( subsetE
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( subset @ A @ B )
% 0.30/0.47 => ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subsetE2_type,type,
% 0.30/0.47 subsetE2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetE2,definition,
% 0.30/0.47 ( subsetE2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( subset @ A @ B )
% 0.30/0.47 => ( ~ ( in @ Xx @ B )
% 0.30/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(notsubsetI_type,type,
% 0.30/0.47 notsubsetI: $o ).
% 0.30/0.47
% 0.30/0.47 thf(notsubsetI,definition,
% 0.30/0.47 ( notsubsetI
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ B )
% 0.30/0.47 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(notequalI1_type,type,
% 0.30/0.47 notequalI1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(notequalI1,definition,
% 0.30/0.47 ( notequalI1
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ~ ( subset @ A @ B )
% 0.30/0.47 => ( A != B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(notequalI2_type,type,
% 0.30/0.47 notequalI2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(notequalI2,definition,
% 0.30/0.47 ( notequalI2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ B )
% 0.30/0.47 => ( A != B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subsetRefl_type,type,
% 0.30/0.47 subsetRefl: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetRefl,definition,
% 0.30/0.47 ( subsetRefl
% 0.30/0.47 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subsetTrans_type,type,
% 0.30/0.47 subsetTrans: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetTrans,definition,
% 0.30/0.47 ( subsetTrans
% 0.30/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/0.47 ( ( subset @ A @ B )
% 0.30/0.47 => ( ( subset @ B @ C )
% 0.30/0.47 => ( subset @ A @ C ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoinSub_type,type,
% 0.30/0.47 setadjoinSub: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoinSub,definition,
% 0.30/0.47 ( setadjoinSub
% 0.30/0.47 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoinSub2_type,type,
% 0.30/0.47 setadjoinSub2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setadjoinSub2,definition,
% 0.30/0.47 ( setadjoinSub2
% 0.30/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/0.47 ( ( subset @ A @ B )
% 0.30/0.47 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subset2powerset_type,type,
% 0.30/0.47 subset2powerset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subset2powerset,definition,
% 0.30/0.47 ( subset2powerset
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( subset @ A @ B )
% 0.30/0.47 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setextsub_type,type,
% 0.30/0.47 setextsub: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setextsub,definition,
% 0.30/0.47 ( setextsub
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( subset @ A @ B )
% 0.30/0.47 => ( ( subset @ B @ A )
% 0.30/0.47 => ( A = B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subsetemptysetimpeq_type,type,
% 0.30/0.47 subsetemptysetimpeq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetemptysetimpeq,definition,
% 0.30/0.47 ( subsetemptysetimpeq
% 0.30/0.47 = ( ! [A: $i] :
% 0.30/0.47 ( ( subset @ A @ emptyset )
% 0.30/0.47 => ( A = emptyset ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(powersetI1_type,type,
% 0.30/0.47 powersetI1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(powersetI1,definition,
% 0.30/0.47 ( powersetI1
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( subset @ B @ A )
% 0.30/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(powersetE1_type,type,
% 0.30/0.47 powersetE1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(powersetE1,definition,
% 0.30/0.47 ( powersetE1
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(inPowerset,conjecture,
% 0.30/0.47 ( setextAx
% 0.30/0.47 => ( emptysetAx
% 0.30/0.47 => ( setadjoinAx
% 0.30/0.47 => ( powersetAx
% 0.30/0.47 => ( setunionAx
% 0.30/0.47 => ( omega0Ax
% 0.30/0.47 => ( omegaSAx
% 0.30/0.47 => ( omegaIndAx
% 0.30/0.47 => ( replAx
% 0.30/0.47 => ( foundationAx
% 0.30/0.47 => ( wellorderingAx
% 0.30/0.47 => ( descrp
% 0.30/0.47 => ( dsetconstrI
% 0.30/0.47 => ( dsetconstrEL
% 0.30/0.47 => ( dsetconstrER
% 0.30/0.47 => ( exuE1
% 0.30/0.47 => ( prop2setE
% 0.30/0.47 => ( emptysetE
% 0.30/0.47 => ( emptysetimpfalse
% 0.30/0.47 => ( notinemptyset
% 0.30/0.47 => ( exuE3e
% 0.30/0.47 => ( setext
% 0.30/0.47 => ( emptyI
% 0.30/0.47 => ( noeltsimpempty
% 0.30/0.47 => ( setbeta
% 0.30/0.47 => ( nonemptyE1
% 0.30/0.47 => ( nonemptyI
% 0.30/0.47 => ( nonemptyI1
% 0.30/0.47 => ( setadjoinIL
% 0.30/0.47 => ( emptyinunitempty
% 0.30/0.47 => ( setadjoinIR
% 0.30/0.47 => ( setadjoinE
% 0.30/0.47 => ( setadjoinOr
% 0.30/0.47 => ( setoftrueEq
% 0.30/0.47 => ( powersetI
% 0.30/0.47 => ( emptyinPowerset
% 0.30/0.47 => ( emptyInPowerset
% 0.30/0.47 => ( powersetE
% 0.30/0.47 => ( setunionI
% 0.30/0.47 => ( setunionE
% 0.30/0.47 => ( subPowSU
% 0.30/0.47 => ( exuE2
% 0.30/0.47 => ( nonemptyImpWitness
% 0.30/0.47 => ( uniqinunit
% 0.30/0.47 => ( notinsingleton
% 0.30/0.47 => ( eqinunit
% 0.30/0.47 => ( singletonsswitch
% 0.30/0.47 => ( upairsetE
% 0.30/0.47 => ( upairsetIL
% 0.30/0.47 => ( upairsetIR
% 0.30/0.47 => ( emptyE1
% 0.30/0.47 => ( vacuousDall
% 0.30/0.47 => ( quantDeMorgan1
% 0.30/0.47 => ( quantDeMorgan2
% 0.30/0.47 => ( quantDeMorgan3
% 0.30/0.47 => ( quantDeMorgan4
% 0.30/0.47 => ( prop2setI
% 0.30/0.47 => ( prop2set2propI
% 0.30/0.47 => ( notdexE
% 0.30/0.47 => ( notdallE
% 0.30/0.47 => ( exuI1
% 0.30/0.47 => ( exuI3
% 0.30/0.47 => ( exuI2
% 0.30/0.47 => ( inCongP
% 0.30/0.47 => ( in__Cong
% 0.30/0.47 => ( exuE3u
% 0.30/0.47 => ( exu__Cong
% 0.30/0.47 => ( emptyset__Cong
% 0.30/0.47 => ( setadjoin__Cong
% 0.30/0.47 => ( powerset__Cong
% 0.30/0.47 => ( setunion__Cong
% 0.30/0.47 => ( omega__Cong
% 0.30/0.47 => ( exuEu
% 0.30/0.47 => ( descr__Cong
% 0.30/0.47 => ( dsetconstr__Cong
% 0.30/0.47 => ( subsetI1
% 0.30/0.47 => ( eqimpsubset2
% 0.30/0.47 => ( eqimpsubset1
% 0.30/0.47 => ( subsetI2
% 0.30/0.47 => ( emptysetsubset
% 0.30/0.47 => ( subsetE
% 0.30/0.47 => ( subsetE2
% 0.30/0.47 => ( notsubsetI
% 0.30/0.47 => ( notequalI1
% 0.30/0.47 => ( notequalI2
% 0.30/0.47 => ( subsetRefl
% 0.30/0.47 => ( subsetTrans
% 0.30/0.47 => ( setadjoinSub
% 0.30/0.47 => ( setadjoinSub2
% 0.30/0.47 => ( subset2powerset
% 0.30/0.47 => ( setextsub
% 0.30/0.47 => ( subsetemptysetimpeq
% 0.30/0.47 => ( powersetI1
% 0.30/0.47 => ( powersetE1
% 0.30/0.47 => ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 ------------------------
% 2.18/1.46 ---- Embedded in THF ---
% 2.18/1.46 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.18/1.46 %%% Generated on Tue Feb 24 12:27:26 EST 2026
% 2.18/1.46 %%% using '$$dhol' embedding, version 1.3.0.
% 2.18/1.46 %%% Logic specification used:
% 2.18/1.46 %%% thf(spec, logic, $$dhol).
% 2.18/1.46
% 2.18/1.46 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.EC5jtjE9C7/DTF2DTF10635.p
% See solution above
% 2.18/1.46 ------------------------
% 2.18/1.49 ---- Cleaned THF ---
% 2.18/1.49 thf(in_type,type,
% 2.18/1.49 in: $i > $i > $o ).
% 2.18/1.49
% 2.18/1.49 thf(exu_type,type,
% 2.18/1.49 exu: ( $i > $o ) > $o ).
% 2.18/1.49
% 2.18/1.49 thf(exu,definition,
% 2.18/1.49 ( exu
% 2.18/1.49 = ( ^ [Xphi: $i > $o] :
% 2.18/1.49 ? [Xx: $i] :
% 2.18/1.49 ( ( Xphi @ Xx )
% 2.18/1.49 & ! [Xy: $i] :
% 2.18/1.49 ( ( Xphi @ Xy )
% 2.18/1.49 => ( Xx = Xy ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(setextAx_type,type,
% 2.18/1.49 setextAx: $o ).
% 2.18/1.49
% 2.18/1.49 thf(setextAx,definition,
% 2.18/1.49 ( setextAx
% 2.18/1.49 = ( ! [A: $i,B: $i] :
% 2.18/1.49 ( ! [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 <=> ( in @ Xx @ B ) )
% 2.18/1.49 => ( A = B ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(emptyset_type,type,
% 2.18/1.49 emptyset: $i ).
% 2.18/1.49
% 2.18/1.49 thf(emptysetAx_type,type,
% 2.18/1.49 emptysetAx: $o ).
% 2.18/1.49
% 2.18/1.49 thf(emptysetAx,definition,
% 2.18/1.49 ( emptysetAx
% 2.18/1.49 = ( ! [Xx: $i] :
% 2.18/1.49 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(setadjoin_type,type,
% 2.18/1.49 setadjoin: $i > $i > $i ).
% 2.18/1.49
% 2.18/1.49 thf(setadjoinAx_type,type,
% 2.18/1.49 setadjoinAx: $o ).
% 2.18/1.49
% 2.18/1.49 thf(setadjoinAx,definition,
% 2.18/1.49 ( setadjoinAx
% 2.18/1.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.18/1.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.18/1.49 <=> ( ( Xy = Xx )
% 2.18/1.49 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(powerset_type,type,
% 2.18/1.49 powerset: $i > $i ).
% 2.18/1.49
% 2.18/1.49 thf(powersetAx_type,type,
% 2.18/1.49 powersetAx: $o ).
% 2.18/1.49
% 2.18/1.49 thf(powersetAx,definition,
% 2.18/1.49 ( powersetAx
% 2.18/1.49 = ( ! [A: $i,B: $i] :
% 2.18/1.49 ( ( in @ B @ ( powerset @ A ) )
% 2.18/1.49 <=> ! [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ B )
% 2.18/1.49 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(setunion_type,type,
% 2.18/1.49 setunion: $i > $i ).
% 2.18/1.49
% 2.18/1.49 thf(setunionAx_type,type,
% 2.18/1.49 setunionAx: $o ).
% 2.18/1.49
% 2.18/1.49 thf(setunionAx,definition,
% 2.18/1.49 ( setunionAx
% 2.18/1.49 = ( ! [A: $i,Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ ( setunion @ A ) )
% 2.18/1.49 <=> ? [B: $i] :
% 2.18/1.49 ( ( in @ Xx @ B )
% 2.18/1.49 & ( in @ B @ A ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(omega_type,type,
% 2.18/1.49 omega: $i ).
% 2.18/1.49
% 2.18/1.49 thf(omega0Ax_type,type,
% 2.18/1.49 omega0Ax: $o ).
% 2.18/1.49
% 2.18/1.49 thf(omega0Ax,definition,
% 2.18/1.49 ( omega0Ax
% 2.18/1.49 = ( in @ emptyset @ omega ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(omegaSAx_type,type,
% 2.18/1.49 omegaSAx: $o ).
% 2.18/1.49
% 2.18/1.49 thf(omegaSAx,definition,
% 2.18/1.49 ( omegaSAx
% 2.18/1.49 = ( ! [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ omega )
% 2.18/1.49 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(omegaIndAx_type,type,
% 2.18/1.49 omegaIndAx: $o ).
% 2.18/1.49
% 2.18/1.49 thf(omegaIndAx,definition,
% 2.18/1.49 ( omegaIndAx
% 2.18/1.49 = ( ! [A: $i] :
% 2.18/1.49 ( ( ( in @ emptyset @ A )
% 2.18/1.49 & ! [Xx: $i] :
% 2.18/1.49 ( ( ( in @ Xx @ omega )
% 2.18/1.49 & ( in @ Xx @ A ) )
% 2.18/1.49 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.18/1.49 => ! [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ omega )
% 2.18/1.49 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(replAx_type,type,
% 2.18/1.49 replAx: $o ).
% 2.18/1.49
% 2.18/1.49 thf(replAx,definition,
% 2.18/1.49 ( replAx
% 2.18/1.49 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.18/1.49 ( ! [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 => ( exu
% 2.18/1.49 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.18/1.49 => ? [B: $i] :
% 2.18/1.49 ! [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ B )
% 2.18/1.49 <=> ? [Xy: $i] :
% 2.18/1.49 ( ( in @ Xy @ A )
% 2.18/1.49 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(foundationAx_type,type,
% 2.18/1.49 foundationAx: $o ).
% 2.18/1.49
% 2.18/1.49 thf(foundationAx,definition,
% 2.18/1.49 ( foundationAx
% 2.18/1.49 = ( ! [A: $i] :
% 2.18/1.49 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.18/1.49 => ? [B: $i] :
% 2.18/1.49 ( ( in @ B @ A )
% 2.18/1.49 & ~ ? [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ B )
% 2.18/1.49 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(wellorderingAx_type,type,
% 2.18/1.49 wellorderingAx: $o ).
% 2.18/1.49
% 2.18/1.49 thf(wellorderingAx,definition,
% 2.18/1.49 ( wellorderingAx
% 2.18/1.49 = ( ! [A: $i] :
% 2.18/1.49 ? [B: $i] :
% 2.18/1.49 ( ! [C: $i] :
% 2.18/1.49 ( ( in @ C @ B )
% 2.18/1.49 => ! [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ C )
% 2.18/1.49 => ( in @ Xx @ A ) ) )
% 2.18/1.49 & ! [Xx: $i,Xy: $i] :
% 2.18/1.49 ( ( ( in @ Xx @ A )
% 2.18/1.49 & ( in @ Xy @ A ) )
% 2.18/1.49 => ( ! [C: $i] :
% 2.18/1.49 ( ( in @ C @ B )
% 2.18/1.49 => ( ( in @ Xx @ C )
% 2.18/1.49 <=> ( in @ Xy @ C ) ) )
% 2.18/1.49 => ( Xx = Xy ) ) )
% 2.18/1.49 & ! [C: $i,D: $i] :
% 2.18/1.49 ( ( ( in @ C @ B )
% 2.18/1.49 & ( in @ D @ B ) )
% 2.18/1.49 => ( ! [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ C )
% 2.18/1.49 => ( in @ Xx @ D ) )
% 2.18/1.49 | ! [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ D )
% 2.18/1.49 => ( in @ Xx @ C ) ) ) )
% 2.18/1.49 & ! [C: $i] :
% 2.18/1.49 ( ( ! [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ C )
% 2.18/1.49 => ( in @ Xx @ A ) )
% 2.18/1.49 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.18/1.49 => ? [D: $i,Xx: $i] :
% 2.18/1.49 ( ( in @ D @ B )
% 2.18/1.49 & ( in @ Xx @ C )
% 2.18/1.49 & ~ ? [Xy: $i] :
% 2.18/1.49 ( ( in @ Xy @ D )
% 2.18/1.49 & ( in @ Xy @ C ) )
% 2.18/1.49 & ! [E: $i] :
% 2.18/1.49 ( ( in @ E @ B )
% 2.18/1.49 => ( ! [Xy: $i] :
% 2.18/1.49 ( ( in @ Xy @ E )
% 2.18/1.49 => ( in @ Xy @ D ) )
% 2.18/1.49 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(descr_type,type,
% 2.18/1.49 descr: ( $i > $o ) > $i ).
% 2.18/1.49
% 2.18/1.49 thf(descrp_type,type,
% 2.18/1.49 descrp: $o ).
% 2.18/1.49
% 2.18/1.49 thf(descrp,definition,
% 2.18/1.49 ( descrp
% 2.18/1.49 = ( ! [Xphi: $i > $o] :
% 2.18/1.49 ( ( exu
% 2.18/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.49 => ( Xphi
% 2.18/1.49 @ ( descr
% 2.18/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(dsetconstr_type,type,
% 2.18/1.49 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.18/1.49
% 2.18/1.49 thf(dsetconstrI_type,type,
% 2.18/1.49 dsetconstrI: $o ).
% 2.18/1.49
% 2.18/1.49 thf(dsetconstrI,definition,
% 2.18/1.49 ( dsetconstrI
% 2.18/1.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 => ( ( Xphi @ Xx )
% 2.18/1.49 => ( in @ Xx
% 2.18/1.49 @ ( dsetconstr @ A
% 2.18/1.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(dsetconstrEL_type,type,
% 2.18/1.49 dsetconstrEL: $o ).
% 2.18/1.49
% 2.18/1.49 thf(dsetconstrEL,definition,
% 2.18/1.49 ( dsetconstrEL
% 2.18/1.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.18/1.49 ( ( in @ Xx
% 2.18/1.49 @ ( dsetconstr @ A
% 2.18/1.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.18/1.49 => ( in @ Xx @ A ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(dsetconstrER_type,type,
% 2.18/1.49 dsetconstrER: $o ).
% 2.18/1.49
% 2.18/1.49 thf(dsetconstrER,definition,
% 2.18/1.49 ( dsetconstrER
% 2.18/1.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.18/1.49 ( ( in @ Xx
% 2.18/1.49 @ ( dsetconstr @ A
% 2.18/1.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.18/1.49 => ( Xphi @ Xx ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(exuE1_type,type,
% 2.18/1.49 exuE1: $o ).
% 2.18/1.49
% 2.18/1.49 thf(exuE1,definition,
% 2.18/1.49 ( exuE1
% 2.18/1.49 = ( ! [Xphi: $i > $o] :
% 2.18/1.49 ( ( exu
% 2.18/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.49 => ? [Xx: $i] :
% 2.18/1.49 ( ( Xphi @ Xx )
% 2.18/1.49 & ! [Xy: $i] :
% 2.18/1.49 ( ( Xphi @ Xy )
% 2.18/1.49 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(prop2set_type,type,
% 2.18/1.49 prop2set: $o > $i ).
% 2.18/1.49
% 2.18/1.49 thf(prop2setE_type,type,
% 2.18/1.49 prop2setE: $o ).
% 2.18/1.49
% 2.18/1.49 thf(prop2setE,definition,
% 2.18/1.49 ( prop2setE
% 2.18/1.49 = ( ! [Xphi: $o,Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.18/1.49 => Xphi ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(emptysetE_type,type,
% 2.18/1.49 emptysetE: $o ).
% 2.18/1.49
% 2.18/1.49 thf(emptysetE,definition,
% 2.18/1.49 ( emptysetE
% 2.18/1.49 = ( ! [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ emptyset )
% 2.18/1.49 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(emptysetimpfalse_type,type,
% 2.18/1.49 emptysetimpfalse: $o ).
% 2.18/1.49
% 2.18/1.49 thf(emptysetimpfalse,definition,
% 2.18/1.49 ( emptysetimpfalse
% 2.18/1.49 = ( ! [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ emptyset )
% 2.18/1.49 => $false ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(notinemptyset_type,type,
% 2.18/1.49 notinemptyset: $o ).
% 2.18/1.49
% 2.18/1.49 thf(notinemptyset,definition,
% 2.18/1.49 ( notinemptyset
% 2.18/1.49 = ( ! [Xx: $i] :
% 2.18/1.49 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(exuE3e_type,type,
% 2.18/1.49 exuE3e: $o ).
% 2.18/1.49
% 2.18/1.49 thf(exuE3e,definition,
% 2.18/1.49 ( exuE3e
% 2.18/1.49 = ( ! [Xphi: $i > $o] :
% 2.18/1.49 ( ( exu
% 2.18/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.49 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(setext_type,type,
% 2.18/1.49 setext: $o ).
% 2.18/1.49
% 2.18/1.49 thf(setext,definition,
% 2.18/1.49 ( setext
% 2.18/1.49 = ( ! [A: $i,B: $i] :
% 2.18/1.49 ( ! [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 => ( in @ Xx @ B ) )
% 2.18/1.49 => ( ! [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ B )
% 2.18/1.49 => ( in @ Xx @ A ) )
% 2.18/1.49 => ( A = B ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(emptyI_type,type,
% 2.18/1.49 emptyI: $o ).
% 2.18/1.49
% 2.18/1.49 thf(emptyI,definition,
% 2.18/1.49 ( emptyI
% 2.18/1.49 = ( ! [A: $i] :
% 2.18/1.49 ( ! [Xx: $i] :
% 2.18/1.49 ~ ( in @ Xx @ A )
% 2.18/1.49 => ( A = emptyset ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(noeltsimpempty_type,type,
% 2.18/1.49 noeltsimpempty: $o ).
% 2.18/1.49
% 2.18/1.49 thf(noeltsimpempty,definition,
% 2.18/1.49 ( noeltsimpempty
% 2.18/1.49 = ( ! [A: $i] :
% 2.18/1.49 ( ! [Xx: $i] :
% 2.18/1.49 ~ ( in @ Xx @ A )
% 2.18/1.49 => ( A = emptyset ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(setbeta_type,type,
% 2.18/1.49 setbeta: $o ).
% 2.18/1.49
% 2.18/1.49 thf(setbeta,definition,
% 2.18/1.49 ( setbeta
% 2.18/1.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 => ( ( in @ Xx
% 2.18/1.49 @ ( dsetconstr @ A
% 2.18/1.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.18/1.49 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(nonempty_type,type,
% 2.18/1.49 nonempty: $i > $o ).
% 2.18/1.49
% 2.18/1.49 thf(nonempty,definition,
% 2.18/1.49 ( nonempty
% 2.18/1.49 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(nonemptyE1_type,type,
% 2.18/1.49 nonemptyE1: $o ).
% 2.18/1.49
% 2.18/1.49 thf(nonemptyE1,definition,
% 2.18/1.49 ( nonemptyE1
% 2.18/1.49 = ( ! [A: $i] :
% 2.18/1.49 ( ( nonempty @ A )
% 2.18/1.49 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(nonemptyI_type,type,
% 2.18/1.49 nonemptyI: $o ).
% 2.18/1.49
% 2.18/1.49 thf(nonemptyI,definition,
% 2.18/1.49 ( nonemptyI
% 2.18/1.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 => ( ( Xphi @ Xx )
% 2.18/1.49 => ( nonempty
% 2.18/1.49 @ ( dsetconstr @ A
% 2.18/1.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(nonemptyI1_type,type,
% 2.18/1.49 nonemptyI1: $o ).
% 2.18/1.49
% 2.18/1.49 thf(nonemptyI1,definition,
% 2.18/1.49 ( nonemptyI1
% 2.18/1.49 = ( ! [A: $i] :
% 2.18/1.49 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.18/1.49 => ( nonempty @ A ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(setadjoinIL_type,type,
% 2.18/1.49 setadjoinIL: $o ).
% 2.18/1.49
% 2.18/1.49 thf(setadjoinIL,definition,
% 2.18/1.49 ( setadjoinIL
% 2.18/1.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(emptyinunitempty_type,type,
% 2.18/1.49 emptyinunitempty: $o ).
% 2.18/1.49
% 2.18/1.49 thf(emptyinunitempty,definition,
% 2.18/1.49 ( emptyinunitempty
% 2.18/1.49 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(setadjoinIR_type,type,
% 2.18/1.49 setadjoinIR: $o ).
% 2.18/1.49
% 2.18/1.49 thf(setadjoinIR,definition,
% 2.18/1.49 ( setadjoinIR
% 2.18/1.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.18/1.49 ( ( in @ Xy @ A )
% 2.18/1.49 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(setadjoinE_type,type,
% 2.18/1.49 setadjoinE: $o ).
% 2.18/1.49
% 2.18/1.49 thf(setadjoinE,definition,
% 2.18/1.49 ( setadjoinE
% 2.18/1.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.18/1.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.18/1.49 => ! [Xphi: $o] :
% 2.18/1.49 ( ( ( Xy = Xx )
% 2.18/1.49 => Xphi )
% 2.18/1.49 => ( ( ( in @ Xy @ A )
% 2.18/1.49 => Xphi )
% 2.18/1.49 => Xphi ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(setadjoinOr_type,type,
% 2.18/1.49 setadjoinOr: $o ).
% 2.18/1.49
% 2.18/1.49 thf(setadjoinOr,definition,
% 2.18/1.49 ( setadjoinOr
% 2.18/1.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.18/1.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.18/1.49 => ( ( Xy = Xx )
% 2.18/1.49 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(setoftrueEq_type,type,
% 2.18/1.49 setoftrueEq: $o ).
% 2.18/1.49
% 2.18/1.49 thf(setoftrueEq,definition,
% 2.18/1.49 ( setoftrueEq
% 2.18/1.49 = ( ! [A: $i] :
% 2.18/1.49 ( ( dsetconstr @ A
% 2.18/1.49 @ ^ [Xx: $i] : $true )
% 2.18/1.49 = A ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(powersetI_type,type,
% 2.18/1.49 powersetI: $o ).
% 2.18/1.49
% 2.18/1.49 thf(powersetI,definition,
% 2.18/1.49 ( powersetI
% 2.18/1.49 = ( ! [A: $i,B: $i] :
% 2.18/1.49 ( ! [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ B )
% 2.18/1.49 => ( in @ Xx @ A ) )
% 2.18/1.49 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(emptyinPowerset_type,type,
% 2.18/1.49 emptyinPowerset: $o ).
% 2.18/1.49
% 2.18/1.49 thf(emptyinPowerset,definition,
% 2.18/1.49 ( emptyinPowerset
% 2.18/1.49 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(emptyInPowerset_type,type,
% 2.18/1.49 emptyInPowerset: $o ).
% 2.18/1.49
% 2.18/1.49 thf(emptyInPowerset,definition,
% 2.18/1.49 ( emptyInPowerset
% 2.18/1.49 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(powersetE_type,type,
% 2.18/1.49 powersetE: $o ).
% 2.18/1.49
% 2.18/1.49 thf(powersetE,definition,
% 2.18/1.49 ( powersetE
% 2.18/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.49 ( ( in @ B @ ( powerset @ A ) )
% 2.18/1.49 => ( ( in @ Xx @ B )
% 2.18/1.49 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(setunionI_type,type,
% 2.18/1.49 setunionI: $o ).
% 2.18/1.49
% 2.18/1.49 thf(setunionI,definition,
% 2.18/1.49 ( setunionI
% 2.18/1.49 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.18/1.49 ( ( in @ Xx @ B )
% 2.18/1.49 => ( ( in @ B @ A )
% 2.18/1.49 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(setunionE_type,type,
% 2.18/1.49 setunionE: $o ).
% 2.18/1.49
% 2.18/1.49 thf(setunionE,definition,
% 2.18/1.49 ( setunionE
% 2.18/1.49 = ( ! [A: $i,Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ ( setunion @ A ) )
% 2.18/1.49 => ! [Xphi: $o] :
% 2.18/1.49 ( ! [B: $i] :
% 2.18/1.49 ( ( in @ Xx @ B )
% 2.18/1.49 => ( ( in @ B @ A )
% 2.18/1.49 => Xphi ) )
% 2.18/1.49 => Xphi ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(subPowSU_type,type,
% 2.18/1.49 subPowSU: $o ).
% 2.18/1.49
% 2.18/1.49 thf(subPowSU,definition,
% 2.18/1.49 ( subPowSU
% 2.18/1.49 = ( ! [A: $i,Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(exuE2_type,type,
% 2.18/1.49 exuE2: $o ).
% 2.18/1.49
% 2.18/1.49 thf(exuE2,definition,
% 2.18/1.49 ( exuE2
% 2.18/1.49 = ( ! [Xphi: $i > $o] :
% 2.18/1.49 ( ( exu
% 2.18/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.49 => ? [Xx: $i] :
% 2.18/1.49 ! [Xy: $i] :
% 2.18/1.49 ( ( Xphi @ Xy )
% 2.18/1.49 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(nonemptyImpWitness_type,type,
% 2.18/1.49 nonemptyImpWitness: $o ).
% 2.18/1.49
% 2.18/1.49 thf(nonemptyImpWitness,definition,
% 2.18/1.49 ( nonemptyImpWitness
% 2.18/1.49 = ( ! [A: $i] :
% 2.18/1.49 ( ( nonempty @ A )
% 2.18/1.49 => ? [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 & $true ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(uniqinunit_type,type,
% 2.18/1.49 uniqinunit: $o ).
% 2.18/1.49
% 2.18/1.49 thf(uniqinunit,definition,
% 2.18/1.49 ( uniqinunit
% 2.18/1.49 = ( ! [Xx: $i,Xy: $i] :
% 2.18/1.49 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.18/1.49 => ( Xx = Xy ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(notinsingleton_type,type,
% 2.18/1.49 notinsingleton: $o ).
% 2.18/1.49
% 2.18/1.49 thf(notinsingleton,definition,
% 2.18/1.49 ( notinsingleton
% 2.18/1.49 = ( ! [Xx: $i,Xy: $i] :
% 2.18/1.49 ( ( Xx != Xy )
% 2.18/1.49 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(eqinunit_type,type,
% 2.18/1.49 eqinunit: $o ).
% 2.18/1.49
% 2.18/1.49 thf(eqinunit,definition,
% 2.18/1.49 ( eqinunit
% 2.18/1.49 = ( ! [Xx: $i,Xy: $i] :
% 2.18/1.49 ( ( Xx = Xy )
% 2.18/1.49 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(singletonsswitch_type,type,
% 2.18/1.49 singletonsswitch: $o ).
% 2.18/1.49
% 2.18/1.49 thf(singletonsswitch,definition,
% 2.18/1.49 ( singletonsswitch
% 2.18/1.49 = ( ! [Xx: $i,Xy: $i] :
% 2.18/1.49 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.18/1.49 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(upairsetE_type,type,
% 2.18/1.49 upairsetE: $o ).
% 2.18/1.49
% 2.18/1.49 thf(upairsetE,definition,
% 2.18/1.49 ( upairsetE
% 2.18/1.49 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.18/1.49 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.18/1.49 => ( ( Xz = Xx )
% 2.18/1.49 | ( Xz = Xy ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(upairsetIL_type,type,
% 2.18/1.49 upairsetIL: $o ).
% 2.18/1.49
% 2.18/1.49 thf(upairsetIL,definition,
% 2.18/1.49 ( upairsetIL
% 2.18/1.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(upairsetIR_type,type,
% 2.18/1.49 upairsetIR: $o ).
% 2.18/1.49
% 2.18/1.49 thf(upairsetIR,definition,
% 2.18/1.49 ( upairsetIR
% 2.18/1.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(emptyE1_type,type,
% 2.18/1.49 emptyE1: $o ).
% 2.18/1.49
% 2.18/1.49 thf(emptyE1,definition,
% 2.18/1.49 ( emptyE1
% 2.18/1.49 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.49 ( ? [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 & ( Xphi @ Xx ) )
% 2.18/1.49 => ( ( ( dsetconstr @ A
% 2.18/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.49 = emptyset )
% 2.18/1.49 => $false ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(vacuousDall_type,type,
% 2.18/1.49 vacuousDall: $o ).
% 2.18/1.49
% 2.18/1.49 thf(vacuousDall,definition,
% 2.18/1.49 ( vacuousDall
% 2.18/1.49 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ emptyset )
% 2.18/1.49 => ( Xphi @ Xx ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(quantDeMorgan1_type,type,
% 2.18/1.49 quantDeMorgan1: $o ).
% 2.18/1.49
% 2.18/1.49 thf(quantDeMorgan1,definition,
% 2.18/1.49 ( quantDeMorgan1
% 2.18/1.49 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.49 ( ~ ! [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 => ( Xphi @ Xx ) )
% 2.18/1.49 => ? [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(quantDeMorgan2_type,type,
% 2.18/1.49 quantDeMorgan2: $o ).
% 2.18/1.49
% 2.18/1.49 thf(quantDeMorgan2,definition,
% 2.18/1.49 ( quantDeMorgan2
% 2.18/1.49 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.49 ( ! [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 => ~ ( Xphi @ Xx ) )
% 2.18/1.49 => ~ ? [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(quantDeMorgan3_type,type,
% 2.18/1.49 quantDeMorgan3: $o ).
% 2.18/1.49
% 2.18/1.49 thf(quantDeMorgan3,definition,
% 2.18/1.49 ( quantDeMorgan3
% 2.18/1.49 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.49 ( ~ ? [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 & ( Xphi @ Xx ) )
% 2.18/1.49 => ! [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(quantDeMorgan4_type,type,
% 2.18/1.49 quantDeMorgan4: $o ).
% 2.18/1.49
% 2.18/1.49 thf(quantDeMorgan4,definition,
% 2.18/1.49 ( quantDeMorgan4
% 2.18/1.49 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.49 ( ? [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 & ~ ( Xphi @ Xx ) )
% 2.18/1.49 => ~ ! [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(prop2setI_type,type,
% 2.18/1.49 prop2setI: $o ).
% 2.18/1.49
% 2.18/1.49 thf(prop2setI,definition,
% 2.18/1.49 ( prop2setI
% 2.18/1.49 = ( ! [Xphi: $o] :
% 2.18/1.49 ( Xphi
% 2.18/1.49 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(set2prop_type,type,
% 2.18/1.49 set2prop: $i > $o ).
% 2.18/1.49
% 2.18/1.49 thf(prop2set2propI_type,type,
% 2.18/1.49 prop2set2propI: $o ).
% 2.18/1.49
% 2.18/1.49 thf(prop2set2propI,definition,
% 2.18/1.49 ( prop2set2propI
% 2.18/1.49 = ( ! [Xphi: $o] :
% 2.18/1.49 ( Xphi
% 2.18/1.49 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(notdexE_type,type,
% 2.18/1.49 notdexE: $o ).
% 2.18/1.49
% 2.18/1.49 thf(notdexE,definition,
% 2.18/1.49 ( notdexE
% 2.18/1.49 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.49 ( ~ ? [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 & ( Xphi @ Xx ) )
% 2.18/1.49 => ! [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(notdallE_type,type,
% 2.18/1.49 notdallE: $o ).
% 2.18/1.49
% 2.18/1.49 thf(notdallE,definition,
% 2.18/1.49 ( notdallE
% 2.18/1.49 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.49 ( ~ ! [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 => ( Xphi @ Xx ) )
% 2.18/1.49 => ? [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(exuI1_type,type,
% 2.18/1.49 exuI1: $o ).
% 2.18/1.49
% 2.18/1.49 thf(exuI1,definition,
% 2.18/1.49 ( exuI1
% 2.18/1.49 = ( ! [Xphi: $i > $o] :
% 2.18/1.49 ( ? [Xx: $i] :
% 2.18/1.49 ( ( Xphi @ Xx )
% 2.18/1.49 & ! [Xy: $i] :
% 2.18/1.49 ( ( Xphi @ Xy )
% 2.18/1.49 => ( Xx = Xy ) ) )
% 2.18/1.49 => ( exu
% 2.18/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(exuI3_type,type,
% 2.18/1.49 exuI3: $o ).
% 2.18/1.49
% 2.18/1.49 thf(exuI3,definition,
% 2.18/1.49 ( exuI3
% 2.18/1.49 = ( ! [Xphi: $i > $o] :
% 2.18/1.49 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.18/1.49 => ( ! [Xx: $i,Xy: $i] :
% 2.18/1.49 ( ( Xphi @ Xx )
% 2.18/1.49 => ( ( Xphi @ Xy )
% 2.18/1.49 => ( Xx = Xy ) ) )
% 2.18/1.49 => ( exu
% 2.18/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(exuI2_type,type,
% 2.18/1.49 exuI2: $o ).
% 2.18/1.49
% 2.18/1.49 thf(exuI2,definition,
% 2.18/1.49 ( exuI2
% 2.18/1.49 = ( ! [Xphi: $i > $o] :
% 2.18/1.49 ( ? [Xx: $i] :
% 2.18/1.49 ! [Xy: $i] :
% 2.18/1.49 ( ( Xphi @ Xy )
% 2.18/1.49 <=> ( Xy = Xx ) )
% 2.18/1.49 => ( exu
% 2.18/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(inCongP_type,type,
% 2.18/1.49 inCongP: $o ).
% 2.18/1.49
% 2.18/1.49 thf(inCongP,definition,
% 2.18/1.49 ( inCongP
% 2.18/1.49 = ( ! [A: $i,B: $i] :
% 2.18/1.49 ( ( A = B )
% 2.18/1.49 => ! [Xx: $i,Xy: $i] :
% 2.18/1.49 ( ( Xx = Xy )
% 2.18/1.49 => ( ( in @ Xx @ A )
% 2.18/1.49 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(in__Cong_type,type,
% 2.18/1.49 in__Cong: $o ).
% 2.18/1.49
% 2.18/1.49 thf(in__Cong,definition,
% 2.18/1.49 ( in__Cong
% 2.18/1.49 = ( ! [A: $i,B: $i] :
% 2.18/1.49 ( ( A = B )
% 2.18/1.49 => ! [Xx: $i,Xy: $i] :
% 2.18/1.49 ( ( Xx = Xy )
% 2.18/1.49 => ( ( in @ Xx @ A )
% 2.18/1.49 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(exuE3u_type,type,
% 2.18/1.49 exuE3u: $o ).
% 2.18/1.49
% 2.18/1.49 thf(exuE3u,definition,
% 2.18/1.49 ( exuE3u
% 2.18/1.49 = ( ! [Xphi: $i > $o] :
% 2.18/1.49 ( ( exu
% 2.18/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.49 => ! [Xx: $i,Xy: $i] :
% 2.18/1.49 ( ( Xphi @ Xx )
% 2.18/1.49 => ( ( Xphi @ Xy )
% 2.18/1.49 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(exu__Cong_type,type,
% 2.18/1.49 exu__Cong: $o ).
% 2.18/1.49
% 2.18/1.49 thf(exu__Cong,definition,
% 2.18/1.49 ( exu__Cong
% 2.18/1.49 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.18/1.49 ( ! [Xx: $i,Xy: $i] :
% 2.18/1.49 ( ( Xx = Xy )
% 2.18/1.49 => ( ( Xphi @ Xx )
% 2.18/1.49 <=> ( Xpsi @ Xy ) ) )
% 2.18/1.49 => ( ( exu
% 2.18/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.49 <=> ( exu
% 2.18/1.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(emptyset__Cong_type,type,
% 2.18/1.49 emptyset__Cong: $o ).
% 2.18/1.49
% 2.18/1.49 thf(emptyset__Cong,definition,
% 2.18/1.49 ( emptyset__Cong
% 2.18/1.49 = ( emptyset = emptyset ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(setadjoin__Cong_type,type,
% 2.18/1.49 setadjoin__Cong: $o ).
% 2.18/1.49
% 2.18/1.49 thf(setadjoin__Cong,definition,
% 2.18/1.49 ( setadjoin__Cong
% 2.18/1.49 = ( ! [Xx: $i,Xy: $i] :
% 2.18/1.49 ( ( Xx = Xy )
% 2.18/1.49 => ! [Xz: $i,Xu: $i] :
% 2.18/1.49 ( ( Xz = Xu )
% 2.18/1.49 => ( ( setadjoin @ Xx @ Xz )
% 2.18/1.49 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(powerset__Cong_type,type,
% 2.18/1.49 powerset__Cong: $o ).
% 2.18/1.49
% 2.18/1.49 thf(powerset__Cong,definition,
% 2.18/1.49 ( powerset__Cong
% 2.18/1.49 = ( ! [A: $i,B: $i] :
% 2.18/1.49 ( ( A = B )
% 2.18/1.49 => ( ( powerset @ A )
% 2.18/1.49 = ( powerset @ B ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(setunion__Cong_type,type,
% 2.18/1.49 setunion__Cong: $o ).
% 2.18/1.49
% 2.18/1.49 thf(setunion__Cong,definition,
% 2.18/1.49 ( setunion__Cong
% 2.18/1.49 = ( ! [A: $i,B: $i] :
% 2.18/1.49 ( ( A = B )
% 2.18/1.49 => ( ( setunion @ A )
% 2.18/1.49 = ( setunion @ B ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(omega__Cong_type,type,
% 2.18/1.49 omega__Cong: $o ).
% 2.18/1.49
% 2.18/1.49 thf(omega__Cong,definition,
% 2.18/1.49 ( omega__Cong
% 2.18/1.49 = ( omega = omega ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(exuEu_type,type,
% 2.18/1.49 exuEu: $o ).
% 2.18/1.49
% 2.18/1.49 thf(exuEu,definition,
% 2.18/1.49 ( exuEu
% 2.18/1.49 = ( ! [Xphi: $i > $o] :
% 2.18/1.49 ( ( exu
% 2.18/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.49 => ! [Xx: $i,Xy: $i] :
% 2.18/1.49 ( ( Xphi @ Xx )
% 2.18/1.49 => ( ( Xphi @ Xy )
% 2.18/1.49 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(descr__Cong_type,type,
% 2.18/1.49 descr__Cong: $o ).
% 2.18/1.49
% 2.18/1.49 thf(descr__Cong,definition,
% 2.18/1.49 ( descr__Cong
% 2.18/1.49 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.18/1.49 ( ! [Xx: $i,Xy: $i] :
% 2.18/1.49 ( ( Xx = Xy )
% 2.18/1.49 => ( ( Xphi @ Xx )
% 2.18/1.49 <=> ( Xpsi @ Xy ) ) )
% 2.18/1.49 => ( ( exu
% 2.18/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.49 => ( ( exu
% 2.18/1.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.18/1.49 => ( ( descr
% 2.18/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.49 = ( descr
% 2.18/1.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(dsetconstr__Cong_type,type,
% 2.18/1.49 dsetconstr__Cong: $o ).
% 2.18/1.49
% 2.18/1.49 thf(dsetconstr__Cong,definition,
% 2.18/1.49 ( dsetconstr__Cong
% 2.18/1.49 = ( ! [A: $i,B: $i] :
% 2.18/1.49 ( ( A = B )
% 2.18/1.49 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.18/1.49 ( ! [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 => ! [Xy: $i] :
% 2.18/1.49 ( ( in @ Xy @ B )
% 2.18/1.49 => ( ( Xx = Xy )
% 2.18/1.49 => ( ( Xphi @ Xx )
% 2.18/1.49 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.18/1.49 => ( ( dsetconstr @ A
% 2.18/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.49 = ( dsetconstr @ B
% 2.18/1.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(subset_type,type,
% 2.18/1.49 subset: $i > $i > $o ).
% 2.18/1.49
% 2.18/1.49 thf(disjoint_type,type,
% 2.18/1.49 disjoint: $i > $i > $o ).
% 2.18/1.49
% 2.18/1.49 thf(setsmeet_type,type,
% 2.18/1.49 setsmeet: $i > $i > $o ).
% 2.18/1.49
% 2.18/1.49 thf(subsetI1_type,type,
% 2.18/1.49 subsetI1: $o ).
% 2.18/1.49
% 2.18/1.49 thf(subsetI1,definition,
% 2.18/1.49 ( subsetI1
% 2.18/1.49 = ( ! [A: $i,B: $i] :
% 2.18/1.49 ( ! [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 => ( in @ Xx @ B ) )
% 2.18/1.49 => ( subset @ A @ B ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(eqimpsubset2_type,type,
% 2.18/1.49 eqimpsubset2: $o ).
% 2.18/1.49
% 2.18/1.49 thf(eqimpsubset2,definition,
% 2.18/1.49 ( eqimpsubset2
% 2.18/1.49 = ( ! [A: $i,B: $i] :
% 2.18/1.49 ( ( A = B )
% 2.18/1.49 => ( subset @ B @ A ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(eqimpsubset1_type,type,
% 2.18/1.49 eqimpsubset1: $o ).
% 2.18/1.49
% 2.18/1.49 thf(eqimpsubset1,definition,
% 2.18/1.49 ( eqimpsubset1
% 2.18/1.49 = ( ! [A: $i,B: $i] :
% 2.18/1.49 ( ( A = B )
% 2.18/1.49 => ( subset @ A @ B ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(subsetI2_type,type,
% 2.18/1.49 subsetI2: $o ).
% 2.18/1.49
% 2.18/1.49 thf(subsetI2,definition,
% 2.18/1.49 ( subsetI2
% 2.18/1.49 = ( ! [A: $i,B: $i] :
% 2.18/1.49 ( ! [Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 => ( in @ Xx @ B ) )
% 2.18/1.49 => ( subset @ A @ B ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(emptysetsubset_type,type,
% 2.18/1.49 emptysetsubset: $o ).
% 2.18/1.49
% 2.18/1.49 thf(emptysetsubset,definition,
% 2.18/1.49 ( emptysetsubset
% 2.18/1.49 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(subsetE_type,type,
% 2.18/1.49 subsetE: $o ).
% 2.18/1.49
% 2.18/1.49 thf(subsetE,definition,
% 2.18/1.49 ( subsetE
% 2.18/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.49 ( ( subset @ A @ B )
% 2.18/1.49 => ( ( in @ Xx @ A )
% 2.18/1.49 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(subsetE2_type,type,
% 2.18/1.49 subsetE2: $o ).
% 2.18/1.49
% 2.18/1.49 thf(subsetE2,definition,
% 2.18/1.49 ( subsetE2
% 2.18/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.49 ( ( subset @ A @ B )
% 2.18/1.49 => ( ~ ( in @ Xx @ B )
% 2.18/1.49 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(notsubsetI_type,type,
% 2.18/1.49 notsubsetI: $o ).
% 2.18/1.49
% 2.18/1.49 thf(notsubsetI,definition,
% 2.18/1.49 ( notsubsetI
% 2.18/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 => ( ~ ( in @ Xx @ B )
% 2.18/1.49 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(notequalI1_type,type,
% 2.18/1.49 notequalI1: $o ).
% 2.18/1.49
% 2.18/1.49 thf(notequalI1,definition,
% 2.18/1.49 ( notequalI1
% 2.18/1.49 = ( ! [A: $i,B: $i] :
% 2.18/1.49 ( ~ ( subset @ A @ B )
% 2.18/1.49 => ( A != B ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(notequalI2_type,type,
% 2.18/1.49 notequalI2: $o ).
% 2.18/1.49
% 2.18/1.49 thf(notequalI2,definition,
% 2.18/1.49 ( notequalI2
% 2.18/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.49 ( ( in @ Xx @ A )
% 2.18/1.49 => ( ~ ( in @ Xx @ B )
% 2.18/1.49 => ( A != B ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(subsetRefl_type,type,
% 2.18/1.49 subsetRefl: $o ).
% 2.18/1.49
% 2.18/1.49 thf(subsetRefl,definition,
% 2.18/1.49 ( subsetRefl
% 2.18/1.49 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(subsetTrans_type,type,
% 2.18/1.49 subsetTrans: $o ).
% 2.18/1.49
% 2.18/1.49 thf(subsetTrans,definition,
% 2.18/1.49 ( subsetTrans
% 2.18/1.49 = ( ! [A: $i,B: $i,C: $i] :
% 2.18/1.49 ( ( subset @ A @ B )
% 2.18/1.49 => ( ( subset @ B @ C )
% 2.18/1.49 => ( subset @ A @ C ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(setadjoinSub_type,type,
% 2.18/1.49 setadjoinSub: $o ).
% 2.18/1.49
% 2.18/1.49 thf(setadjoinSub,definition,
% 2.18/1.49 ( setadjoinSub
% 2.18/1.49 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(setadjoinSub2_type,type,
% 2.18/1.49 setadjoinSub2: $o ).
% 2.18/1.49
% 2.18/1.49 thf(setadjoinSub2,definition,
% 2.18/1.49 ( setadjoinSub2
% 2.18/1.49 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.18/1.49 ( ( subset @ A @ B )
% 2.18/1.49 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(subset2powerset_type,type,
% 2.18/1.49 subset2powerset: $o ).
% 2.18/1.49
% 2.18/1.49 thf(subset2powerset,definition,
% 2.18/1.49 ( subset2powerset
% 2.18/1.49 = ( ! [A: $i,B: $i] :
% 2.18/1.49 ( ( subset @ A @ B )
% 2.18/1.49 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(setextsub_type,type,
% 2.18/1.49 setextsub: $o ).
% 2.18/1.49
% 2.18/1.49 thf(setextsub,definition,
% 2.18/1.49 ( setextsub
% 2.18/1.49 = ( ! [A: $i,B: $i] :
% 2.18/1.49 ( ( subset @ A @ B )
% 2.18/1.49 => ( ( subset @ B @ A )
% 2.18/1.49 => ( A = B ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(subsetemptysetimpeq_type,type,
% 2.18/1.49 subsetemptysetimpeq: $o ).
% 2.18/1.49
% 2.18/1.49 thf(subsetemptysetimpeq,definition,
% 2.18/1.49 ( subsetemptysetimpeq
% 2.18/1.49 = ( ! [A: $i] :
% 2.18/1.49 ( ( subset @ A @ emptyset )
% 2.18/1.49 => ( A = emptyset ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(powersetI1_type,type,
% 2.18/1.49 powersetI1: $o ).
% 2.18/1.49
% 2.18/1.49 thf(powersetI1,definition,
% 2.18/1.49 ( powersetI1
% 2.18/1.49 = ( ! [A: $i,B: $i] :
% 2.18/1.49 ( ( subset @ B @ A )
% 2.18/1.49 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(powersetE1_type,type,
% 2.18/1.49 powersetE1: $o ).
% 2.18/1.49
% 2.18/1.49 thf(powersetE1,definition,
% 2.18/1.49 ( powersetE1
% 2.18/1.49 = ( ! [A: $i,B: $i] :
% 2.18/1.49 ( ( in @ B @ ( powerset @ A ) )
% 2.18/1.49 => ( subset @ B @ A ) ) ) ) ).
% 2.18/1.49
% 2.18/1.49 thf(inPowerset,conjecture,
% 2.18/1.49 ( setextAx
% 2.18/1.49 => ( emptysetAx
% 2.18/1.49 => ( setadjoinAx
% 2.18/1.49 => ( powersetAx
% 2.18/1.49 => ( setunionAx
% 2.18/1.49 => ( omega0Ax
% 2.18/1.49 => ( omegaSAx
% 2.18/1.49 => ( omegaIndAx
% 2.18/1.49 => ( replAx
% 2.18/1.49 => ( foundationAx
% 2.18/1.49 => ( wellorderingAx
% 2.18/1.49 => ( descrp
% 2.18/1.49 => ( dsetconstrI
% 2.18/1.49 => ( dsetconstrEL
% 2.18/1.49 => ( dsetconstrER
% 2.18/1.49 => ( exuE1
% 2.18/1.49 => ( prop2setE
% 2.18/1.49 => ( emptysetE
% 2.18/1.49 => ( emptysetimpfalse
% 2.18/1.49 => ( notinemptyset
% 2.18/1.49 => ( exuE3e
% 2.18/1.49 => ( setext
% 2.18/1.49 => ( emptyI
% 2.18/1.49 => ( noeltsimpempty
% 2.18/1.49 => ( setbeta
% 2.18/1.49 => ( nonemptyE1
% 2.18/1.49 => ( nonemptyI
% 2.18/1.49 => ( nonemptyI1
% 2.18/1.49 => ( setadjoinIL
% 2.18/1.49 => ( emptyinunitempty
% 2.18/1.49 => ( setadjoinIR
% 2.18/1.49 => ( setadjoinE
% 2.18/1.49 => ( setadjoinOr
% 2.18/1.49 => ( setoftrueEq
% 2.18/1.49 => ( powersetI
% 2.18/1.49 => ( emptyinPowerset
% 2.18/1.49 => ( emptyInPowerset
% 2.18/1.49 => ( powersetE
% 2.18/1.49 => ( setunionI
% 2.18/1.49 => ( setunionE
% 2.18/1.49 => ( subPowSU
% 2.18/1.49 => ( exuE2
% 2.18/1.49 => ( nonemptyImpWitness
% 2.18/1.49 => ( uniqinunit
% 2.18/1.49 => ( notinsingleton
% 2.18/1.49 => ( eqinunit
% 2.18/1.49 => ( singletonsswitch
% 2.18/1.49 => ( upairsetE
% 2.18/1.49 => ( upairsetIL
% 2.18/1.49 => ( upairsetIR
% 2.18/1.49 => ( emptyE1
% 2.18/1.49 => ( vacuousDall
% 2.18/1.49 => ( quantDeMorgan1
% 2.18/1.49 => ( quantDeMorgan2
% 2.18/1.49 => ( quantDeMorgan3
% 2.18/1.49 => ( quantDeMorgan4
% 2.18/1.49 => ( prop2setI
% 2.18/1.49 => ( prop2set2propI
% 2.18/1.49 => ( notdexE
% 2.18/1.49 => ( notdallE
% 2.18/1.49 => ( exuI1
% 2.18/1.49 => ( exuI3
% 2.18/1.49 => ( exuI2
% 2.18/1.49 => ( inCongP
% 2.18/1.49 => ( in__Cong
% 2.18/1.49 => ( exuE3u
% 2.18/1.49 => ( exu__Cong
% 2.18/1.49 => ( emptyset__Cong
% 2.18/1.49 => ( setadjoin__Cong
% 2.18/1.49 => ( powerset__Cong
% 2.18/1.49 => ( setunion__Cong
% 2.18/1.49 => ( omega__Cong
% 2.18/1.49 => ( exuEu
% 2.18/1.49 => ( descr__Cong
% 2.18/1.49 => ( dsetconstr__Cong
% 2.18/1.49 => ( subsetI1
% 2.18/1.49 => ( eqimpsubset2
% 2.18/1.49 => ( eqimpsubset1
% 2.18/1.49 => ( subsetI2
% 2.18/1.49 => ( emptysetsubset
% 2.18/1.49 => ( subsetE
% 2.18/1.49 => ( subsetE2
% 2.18/1.49 => ( notsubsetI
% 2.18/1.49 => ( notequalI1
% 2.18/1.49 => ( notequalI2
% 2.18/1.49 => ( subsetRefl
% 2.18/1.49 => ( subsetTrans
% 2.18/1.49 => ( setadjoinSub
% 2.18/1.49 => ( setadjoinSub2
% 2.18/1.49 => ( subset2powerset
% 2.18/1.49 => ( setextsub
% 2.18/1.49 => ( subsetemptysetimpeq
% 2.18/1.49 => ( powersetI1
% 2.18/1.49 => ( powersetE1
% 2.18/1.49 => ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.18/1.49 ------------------------
% 2.39/1.51 % (10755)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_10635 for (2999ds/2Mi)
% 2.39/1.51 % (10755)Instruction limit reached!
% 2.39/1.51 % (10755)------------------------------
% 2.39/1.51 % (10755)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.51 % (10755)Termination reason: Unknown
% 2.39/1.51 % (10755)Termination phase: shuffling
% 2.39/1.51
% 2.39/1.51 % (10755)Memory used [KB]: 1151
% 2.39/1.51 % (10755)Time elapsed: 0.002 s
% 2.39/1.51 % (10755)Instructions burned: 2 (million)
% 2.39/1.51 % (10755)------------------------------
% 2.39/1.51 % (10755)------------------------------
% 2.39/1.51 % (10753)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_10635 for (2999ds/27Mi)
% 2.39/1.51 % (10754)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_10635 for (2999ds/2Mi)
% 2.39/1.51 % (10754)Instruction limit reached!
% 2.39/1.51 % (10754)------------------------------
% 2.39/1.51 % (10754)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.51 % (10754)Termination reason: Unknown
% 2.39/1.51 % (10754)Termination phase: shuffling
% 2.39/1.51
% 2.39/1.51 % (10754)Memory used [KB]: 1151
% 2.39/1.51 % (10754)Time elapsed: 0.002 s
% 2.39/1.51 % (10754)Instructions burned: 2 (million)
% 2.39/1.51 % (10754)------------------------------
% 2.39/1.51 % (10754)------------------------------
% 2.39/1.51 % (10756)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_10635 for (2999ds/275Mi)
% 2.39/1.51 % (10751)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_10635 for (2999ds/183Mi)
% 2.39/1.51 % (10752)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_10635 for (2999ds/4Mi)
% 2.39/1.51 % (10757)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_10635 for (2999ds/18Mi)
% 2.39/1.52 % (10753)Instruction limit reached!
% 2.39/1.52 % (10753)------------------------------
% 2.39/1.52 % (10753)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.52 % (10753)Termination reason: Unknown
% 2.39/1.52 % (10753)Termination phase: Property scanning
% 2.39/1.52
% 2.39/1.52 % (10753)Memory used [KB]: 1535
% 2.39/1.52 % (10753)Time elapsed: 0.009 s
% 2.39/1.52 % (10753)Instructions burned: 27 (million)
% 2.39/1.52 % (10753)------------------------------
% 2.39/1.52 % (10753)------------------------------
% 2.39/1.52 % (10752)Instruction limit reached!
% 2.39/1.52 % (10752)------------------------------
% 2.39/1.52 % (10752)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.52 % (10752)Termination reason: Unknown
% 2.39/1.52 % (10752)Termination phase: shuffling
% 2.39/1.52
% 2.39/1.52 % (10752)Memory used [KB]: 1151
% 2.39/1.52 % (10752)Time elapsed: 0.004 s
% 2.39/1.52 % (10752)Instructions burned: 4 (million)
% 2.39/1.52 % (10752)------------------------------
% 2.39/1.52 % (10752)------------------------------
% 2.39/1.52 % (10758)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_10635 for (2999ds/3Mi)
% 2.39/1.52 % (10758)Instruction limit reached!
% 2.39/1.52 % (10758)------------------------------
% 2.39/1.52 % (10758)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.52 % (10758)Termination reason: Unknown
% 2.39/1.52 % (10758)Termination phase: shuffling
% 2.39/1.52
% 2.39/1.52 % (10758)Memory used [KB]: 1151
% 2.39/1.52 % (10758)Time elapsed: 0.002 s
% 2.39/1.52 % (10758)Instructions burned: 4 (million)
% 2.39/1.52 % (10758)------------------------------
% 2.39/1.52 % (10758)------------------------------
% 2.39/1.52 % (10759)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_10635 for (2999ds/37Mi)
% 2.39/1.53 % (10760)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_10635 for (2999ds/15Mi)
% 2.39/1.53 % (10757)Instruction limit reached!
% 2.39/1.53 % (10757)------------------------------
% 2.39/1.53 % (10757)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.53 % (10757)Termination reason: Unknown
% 2.39/1.53 % (10757)Termination phase: Preprocessing 3
% 2.39/1.53
% 2.39/1.53 % (10757)Memory used [KB]: 1535
% 2.39/1.53 % (10757)Time elapsed: 0.013 s
% 2.39/1.53 % (10757)Instructions burned: 19 (million)
% 2.39/1.53 % (10757)------------------------------
% 2.39/1.53 % (10757)------------------------------
% 2.39/1.53 % (10762)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_10635 for (2999ds/1041Mi)
% 2.39/1.53 % (10760)Instruction limit reached!
% 2.39/1.53 % (10760)------------------------------
% 2.39/1.53 % (10760)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.53 % (10760)Termination reason: Unknown
% 2.39/1.53 % (10760)Termination phase: Preprocessing 3
% 2.39/1.53
% 2.39/1.53 % (10760)Memory used [KB]: 1535
% 2.39/1.53 % (10760)Time elapsed: 0.007 s
% 2.39/1.53 % (10760)Instructions burned: 16 (million)
% 2.39/1.53 % (10760)------------------------------
% 2.39/1.53 % (10760)------------------------------
% 2.39/1.53 % (10759)Instruction limit reached!
% 2.39/1.53 % (10759)------------------------------
% 2.39/1.53 % (10759)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.53 % (10759)Termination reason: Unknown
% 2.39/1.53 % (10759)Termination phase: Property scanning
% 2.39/1.53
% 2.39/1.53 % (10759)Memory used [KB]: 1535
% 2.39/1.53 % (10759)Time elapsed: 0.012 s
% 2.39/1.53 % (10759)Instructions burned: 39 (million)
% 2.39/1.53 % (10759)------------------------------
% 2.39/1.53 % (10759)------------------------------
% 2.39/1.53 % (10761)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_10635 for (2999ds/3Mi)
% 2.39/1.54 % (10761)Instruction limit reached!
% 2.39/1.54 % (10761)------------------------------
% 2.39/1.54 % (10761)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.54 % (10761)Termination reason: Unknown
% 2.39/1.54 % (10761)Termination phase: shuffling
% 2.39/1.54
% 2.39/1.54 % (10761)Memory used [KB]: 1151
% 2.39/1.54 % (10761)Time elapsed: 0.004 s
% 2.39/1.54 % (10761)Instructions burned: 4 (million)
% 2.39/1.54 % (10761)------------------------------
% 2.39/1.54 % (10761)------------------------------
% 2.39/1.54 % (10764)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_10635 for (2999ds/16Mi)
% 2.39/1.54 % (10765)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_10635 for (2999ds/3Mi)
% 2.39/1.54 % (10763)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_10635 for (2999ds/7Mi)
% 2.39/1.54 % (10765)Instruction limit reached!
% 2.39/1.54 % (10765)------------------------------
% 2.39/1.54 % (10765)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.54 % (10765)Termination reason: Unknown
% 2.39/1.54 % (10765)Termination phase: shuffling
% 2.39/1.54
% 2.39/1.54 % (10765)Memory used [KB]: 1151
% 2.39/1.54 % (10765)Time elapsed: 0.003 s
% 2.39/1.54 % (10765)Instructions burned: 4 (million)
% 2.39/1.54 % (10765)------------------------------
% 2.39/1.54 % (10765)------------------------------
% 2.39/1.54 % (10763)Instruction limit reached!
% 2.39/1.54 % (10763)------------------------------
% 2.39/1.54 % (10763)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.54 % (10763)Termination reason: Unknown
% 2.39/1.54 % (10763)Termination phase: shuffling
% 2.39/1.54
% 2.39/1.54 % (10763)Memory used [KB]: 1279
% 2.39/1.54 % (10763)Time elapsed: 0.005 s
% 2.39/1.54 % (10763)Instructions burned: 8 (million)
% 2.39/1.54 % (10763)------------------------------
% 2.39/1.54 % (10763)------------------------------
% 2.39/1.55 % (10764)Instruction limit reached!
% 2.39/1.55 % (10764)------------------------------
% 2.39/1.55 % (10764)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.55 % (10764)Termination reason: Unknown
% 2.39/1.55 % (10764)Termination phase: Property scanning
% 2.39/1.55
% 2.39/1.55 % (10764)Memory used [KB]: 1407
% 2.39/1.55 % (10764)Time elapsed: 0.007 s
% 2.39/1.55 % (10764)Instructions burned: 19 (million)
% 2.39/1.55 % (10764)------------------------------
% 2.39/1.55 % (10764)------------------------------
% 2.39/1.55 % (10767)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_10635 for (2999ds/7Mi)
% 2.39/1.55 % (10766)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_10635 for (2999ds/3Mi)
% 2.39/1.55 % (10767)Instruction limit reached!
% 2.39/1.55 % (10767)------------------------------
% 2.39/1.55 % (10767)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.55 % (10767)Termination reason: Unknown
% 2.39/1.55 % (10767)Termination phase: shuffling
% 2.39/1.55
% 2.39/1.55 % (10767)Memory used [KB]: 1279
% 2.39/1.55 % (10767)Time elapsed: 0.004 s
% 2.39/1.55 % (10767)Instructions burned: 8 (million)
% 2.39/1.55 % (10767)------------------------------
% 2.39/1.55 % (10767)------------------------------
% 2.39/1.55 % (10766)Instruction limit reached!
% 2.39/1.55 % (10766)------------------------------
% 2.39/1.55 % (10766)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.55 % (10766)Termination reason: Unknown
% 2.39/1.55 % (10766)Termination phase: shuffling
% 2.39/1.55
% 2.39/1.55 % (10766)Memory used [KB]: 1151
% 2.39/1.55 % (10766)Time elapsed: 0.004 s
% 2.39/1.55 % (10766)Instructions burned: 4 (million)
% 2.39/1.55 % (10766)------------------------------
% 2.39/1.55 % (10766)------------------------------
% 2.39/1.56 % (10769)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_10635 for (2999ds/4Mi)
% 2.39/1.56 % (10769)Instruction limit reached!
% 2.39/1.56 % (10769)------------------------------
% 2.39/1.56 % (10769)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.56 % (10769)Termination reason: Unknown
% 2.39/1.56 % (10769)Termination phase: shuffling
% 2.39/1.56 % (10768)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_10635 for (2999ds/3Mi)
% 2.39/1.56
% 2.39/1.56 % (10769)Memory used [KB]: 1151
% 2.39/1.56 % (10769)Time elapsed: 0.003 s
% 2.39/1.56 % (10769)Instructions burned: 4 (million)
% 2.39/1.56 % (10769)------------------------------
% 2.39/1.56 % (10769)------------------------------
% 2.39/1.56 % (10768)Instruction limit reached!
% 2.39/1.56 % (10768)------------------------------
% 2.39/1.56 % (10768)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.56 % (10768)Termination reason: Unknown
% 2.39/1.56 % (10768)Termination phase: shuffling
% 2.39/1.56
% 2.39/1.56 % (10768)Memory used [KB]: 1151
% 2.39/1.56 % (10768)Time elapsed: 0.003 s
% 2.39/1.56 % (10768)Instructions burned: 5 (million)
% 2.39/1.56 % (10768)------------------------------
% 2.39/1.56 % (10768)------------------------------
% 2.39/1.56 % (10770)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_10635 for (2999ds/18Mi)
% 2.39/1.57 % (10772)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_10635 for (2999ds/6Mi)
% 2.39/1.57 % (10773)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_10635 for (2999ds/902Mi)
% 2.39/1.57 % (10772)Instruction limit reached!
% 2.39/1.57 % (10772)------------------------------
% 2.39/1.57 % (10772)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.57 % (10772)Termination reason: Unknown
% 2.39/1.57 % (10772)Termination phase: shuffling
% 2.39/1.57
% 2.39/1.57 % (10772)Memory used [KB]: 1279
% 2.39/1.57 % (10772)Time elapsed: 0.004 s
% 2.39/1.57 % (10772)Instructions burned: 8 (million)
% 2.39/1.57 % (10772)------------------------------
% 2.39/1.57 % (10772)------------------------------
% 2.39/1.57 % (10770)Instruction limit reached!
% 2.39/1.57 % (10770)------------------------------
% 2.39/1.57 % (10770)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.57 % (10770)Termination reason: Unknown
% 2.39/1.57 % (10770)Termination phase: Property scanning
% 2.39/1.57
% 2.39/1.57 % (10770)Memory used [KB]: 1407
% 2.39/1.57 % (10770)Time elapsed: 0.007 s
% 2.39/1.57 % (10770)Instructions burned: 18 (million)
% 2.39/1.57 % (10770)------------------------------
% 2.39/1.57 % (10770)------------------------------
% 2.39/1.57 % (10771)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_10635 for (2999ds/710Mi)
% 2.39/1.58 % (10774)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_10635 for (2999ds/21Mi)
% 2.39/1.58 % (10775)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_10635 for (2999ds/5Mi)
% 2.39/1.58 % (10775)Instruction limit reached!
% 2.39/1.58 % (10775)------------------------------
% 2.39/1.58 % (10775)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.58 % (10775)Termination reason: Unknown
% 2.39/1.58 % (10775)Termination phase: shuffling
% 2.39/1.58
% 2.39/1.58 % (10775)Memory used [KB]: 1279
% 2.39/1.58 % (10775)Time elapsed: 0.004 s
% 2.39/1.58 % (10775)Instructions burned: 8 (million)
% 2.39/1.58 % (10775)------------------------------
% 2.39/1.58 % (10775)------------------------------
% 2.39/1.59 % (10774)Instruction limit reached!
% 2.39/1.59 % (10774)------------------------------
% 2.39/1.59 % (10774)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.59 % (10774)Termination reason: Unknown
% 2.39/1.59 % (10774)Termination phase: Property scanning
% 2.39/1.59
% 2.39/1.59 % (10774)Memory used [KB]: 1407
% 2.39/1.59 % (10774)Time elapsed: 0.008 s
% 2.39/1.59 % (10774)Instructions burned: 22 (million)
% 2.39/1.59 % (10774)------------------------------
% 2.39/1.59 % (10774)------------------------------
% 2.39/1.59 % (10756)First to succeed.
% 2.39/1.59 % (10776)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_10635 for (2999ds/6Mi)
% 2.39/1.59 % (10756)Refutation found. Thanks to Tanya!
% 2.39/1.59 % SZS status Theorem for DTF2THF_10635
% 2.39/1.59 % SZS output start Proof for DTF2THF_10635
% 2.39/1.59 thf(func_def_0, type, in: $i > $i > $o).
% 2.39/1.59 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.39/1.59 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.39/1.59 thf(func_def_8, type, powerset: $i > $i).
% 2.39/1.59 thf(func_def_10, type, setunion: $i > $i).
% 2.39/1.59 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.39/1.59 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.39/1.59 thf(func_def_26, type, prop2set: $o > $i).
% 2.39/1.59 thf(func_def_36, type, nonempty: $i > $o).
% 2.39/1.59 thf(func_def_69, type, set2prop: $i > $o).
% 2.39/1.59 thf(func_def_88, type, subset: $i > $i > $o).
% 2.39/1.59 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.39/1.59 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.39/1.59 thf(func_def_123, type, sP1: $i > $i > $o).
% 2.39/1.59 thf(func_def_124, type, sP2: $i > $o).
% 2.39/1.59 thf(func_def_125, type, sP3: $i > $i > $o).
% 2.39/1.59 thf(func_def_126, type, sP4: $i > $i > $o).
% 2.39/1.59 thf(func_def_133, type, sK11: $i > $o).
% 2.39/1.59 thf(func_def_135, type, sK13: ($i > $o) > $i > $i).
% 2.39/1.59 thf(func_def_136, type, sK14: $i > $o).
% 2.39/1.59 thf(func_def_139, type, sK17: $i > ($i > $o) > $i).
% 2.39/1.59 thf(func_def_142, type, sK20: $i > $i > $i).
% 2.39/1.59 thf(func_def_143, type, sK21: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.39/1.59 thf(func_def_144, type, sK22: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.39/1.59 thf(func_def_147, type, sK25: $i > $o).
% 2.39/1.59 thf(func_def_148, type, sK26: $i > $o).
% 2.39/1.59 thf(func_def_149, type, sK27: $i > $o).
% 2.39/1.59 thf(func_def_157, type, sK35: $i > $i > $i).
% 2.39/1.59 thf(func_def_158, type, sK36: $i > $i > $i).
% 2.39/1.59 thf(func_def_159, type, sK37: $i > $o).
% 2.39/1.59 thf(func_def_162, type, sK40: $i > ($i > $o) > $i).
% 2.39/1.59 thf(func_def_168, type, sK46: $o > $i > $i > $i).
% 2.39/1.59 thf(func_def_169, type, sK47: $i > $i).
% 2.39/1.59 thf(func_def_177, type, sK55: $i > $o).
% 2.39/1.59 thf(func_def_180, type, sK58: $i > ($i > $o) > $i).
% 2.39/1.59 thf(func_def_181, type, sK59: $i > $o).
% 2.39/1.59 thf(func_def_194, type, sK72: $i > ($i > $o) > $i).
% 2.39/1.59 thf(func_def_195, type, sK73: $i > $o).
% 2.39/1.59 thf(func_def_198, type, sK76: $i > $i > $i).
% 2.39/1.59 thf(func_def_210, type, sK88: $i > $o).
% 2.39/1.59 thf(func_def_220, type, sK98: ($i > $o) > ($i > $o) > $i).
% 2.39/1.59 thf(func_def_221, type, sK99: ($i > $o) > ($i > $o) > $i).
% 2.39/1.59 thf(func_def_222, type, sK100: $i > $o).
% 2.39/1.59 thf(func_def_223, type, sK101: $i > $o).
% 2.39/1.59 thf(func_def_224, type, sK102: $i > $i).
% 2.39/1.59 thf(func_def_227, type, sK105: $i > $o).
% 2.39/1.59 thf(func_def_236, type, sK114: $i > $i).
% 2.39/1.59 thf(func_def_238, type, sK116: $i > $i > $o).
% 2.39/1.59 thf(func_def_239, type, sK117: $i > $i).
% 2.39/1.59 thf(func_def_240, type, sK118: $i > $i).
% 2.39/1.59 thf(func_def_241, type, sK119: ($i > $i > $o) > $i > $i).
% 2.39/1.59 thf(func_def_242, type, sK120: $i > ($i > $i > $o) > $i > $i).
% 2.39/1.59 thf(func_def_243, type, sK121: ($i > $i > $o) > $i > $i).
% 2.39/1.59 thf(func_def_244, type, sK122: ($i > $o) > $i).
% 2.39/1.59 thf(func_def_245, type, sK123: $i > $o).
% 2.39/1.59 thf(func_def_246, type, sK124: $i > $i).
% 2.39/1.59 thf(func_def_250, type, sK128: $i > $i > $i).
% 2.39/1.59 thf(func_def_261, type, sK139: $i > $i > $i).
% 2.39/1.59 thf(func_def_262, type, sK140: $i > $i > $i).
% 2.39/1.59 thf(func_def_263, type, sK141: $i > $i > $i).
% 2.39/1.59 thf(func_def_264, type, sK142: $i > $i > $i).
% 2.39/1.59 thf(func_def_265, type, sK143: $i > $i > $i).
% 2.39/1.59 thf(func_def_266, type, sK144: $i > $i > $i > $i).
% 2.39/1.59 thf(func_def_267, type, sK145: $i > $i).
% 2.39/1.59 thf(func_def_268, type, sK146: $i > $i).
% 2.39/1.59 thf(func_def_269, type, sK147: $i > $i).
% 2.39/1.59 thf(func_def_270, type, sK148: $i > $i).
% 2.39/1.59 thf(func_def_271, type, sK149: $i > $i > $i > $i).
% 2.39/1.59 thf(func_def_272, type, sK150: $i > $i > $i > $i).
% 2.39/1.59 thf(func_def_273, type, sK151: $i > $i > $i).
% 2.39/1.59 thf(func_def_274, type, sK152: $i > $i > $i).
% 2.39/1.59 thf(func_def_275, type, sK153: $i > $i > $i).
% 2.39/1.59 thf(func_def_276, type, sK154: $i > $i).
% 2.39/1.59 thf(func_def_278, type, sK156: $i > $i).
% 2.39/1.59 thf(func_def_279, type, sK157: $i > $i).
% 2.39/1.59 thf(func_def_283, type, sK161: $i > $i > $i).
% 2.39/1.59 thf(func_def_287, type, sK165: $i > $o).
% 2.39/1.59 thf(func_def_292, type, sK170: $i > $i).
% 2.39/1.59 thf(func_def_295, type, sK173: $i > $i).
% 2.39/1.59 thf(func_def_298, type, sK176: $i > $i > $i).
% 2.39/1.59 thf(func_def_299, type, sK177: $i > $o).
% 2.39/1.59 thf(func_def_300, type, sK178: $i > $o).
% 2.39/1.59 thf(func_def_303, type, sK181: $i > ($i > $o) > $i).
% 2.39/1.59 thf(func_def_308, type, sK186: $i > $o).
% 2.39/1.59 thf(func_def_310, type, sK188: $i > $i).
% 2.39/1.59 thf(func_def_329, type, sK207: $i > $i).
% 2.39/1.59 thf(func_def_343, type, sK221: $i > $o).
% 2.39/1.59 thf(func_def_344, type, sK222: ($i > $o) > $i).
% 2.39/1.59 thf(func_def_345, type, sK223: ($i > $o) > ($i > $o) > $i).
% 2.39/1.59 thf(func_def_346, type, sK224: ($i > $o) > ($i > $o) > $i).
% 2.39/1.59 thf(func_def_347, type, sK225: $i > $o).
% 2.39/1.59 thf(func_def_348, type, sK226: $i > $o).
% 2.39/1.59 thf(func_def_351, type, sK229: ($i > $o) > $i > $i).
% 2.39/1.59 thf(func_def_353, type, sK231: $i > $o).
% 2.39/1.59 thf(func_def_355, type, sK233: $i > $o).
% 2.39/1.59 thf(func_def_362, type, sK240: $i > $o).
% 2.39/1.59 thf(func_def_363, type, sK241: $i > $i).
% 2.39/1.59 thf(func_def_364, type, sK242: ($i > $o) > $i).
% 2.39/1.59 thf(func_def_369, type, sK247: ($i > $o) > $i).
% 2.39/1.59 thf(func_def_370, type, sK248: ($i > $o) > $i).
% 2.39/1.59 thf(func_def_371, type, sK249: $i > $o).
% 2.39/1.59 thf(func_def_373, type, sK251: $i > $o).
% 2.39/1.59 thf(func_def_375, type, sK253: $i > ($i > $o) > $i).
% 2.39/1.59 thf(func_def_376, type, sK254: $i > $i > $i).
% 2.39/1.59 thf(func_def_380, type, sK258: $i > $o).
% 2.39/1.59 thf(func_def_382, type, sK260: $i > $o).
% 2.39/1.59 thf(func_def_386, type, ph264: !>[X0: $tType]:(X0)).
% 2.39/1.59 thf(f2332,plain,(
% 2.39/1.59 $false),
% 2.39/1.59 inference(subsumption_resolution,[status(thm)],[f2328,f1892])).
% 2.39/1.59 thf(f1892,plain,(
% 2.39/1.59 ( ! [X3 : $i] : (((subset @ X3 @ X3) = $true)) )),
% 2.39/1.59 inference(trivial_inequality_removal,[status(thm)],[f1706])).
% 2.39/1.59 thf(f1706,plain,(
% 2.39/1.59 ( ! [X3 : $i] : (((subset @ X3 @ X3) = $true) | ($true != $true)) )),
% 2.39/1.59 inference(equality_resolution,[status(thm)],[f1417])).
% 2.39/1.59 thf(f1417,plain,(
% 2.39/1.59 ( ! [X2 : $i,X3 : $i] : (($true = (subset @ X2 @ X3)) | (X2 != X3) | ($true != $true)) )),
% 2.39/1.59 inference(definition_unfolding,[status(thm)],[f935,f1142])).
% 2.39/1.59 thf(f1142,plain,(
% 2.39/1.59 (eqimpsubset1 = $true)),
% 2.39/1.59 inference(cnf_transformation,[status(thm)],[f649])).
% 2.39/1.59 thf(f649,plain,(
% 2.39/1.59 (subsetTrans = $true) & (emptyinunitempty = $true) & (emptyset__Cong = $true) & (exuE1 = $true) & (dsetconstrER = $true) & (exuEu = $true) & (notsubsetI = $true) & (noeltsimpempty = $true) & (exu__Cong = $true) & (foundationAx = $true) & (exuE2 = $true) & (omega__Cong = $true) & (eqimpsubset1 = $true) & (emptyI = $true) & (nonemptyImpWitness = $true) & (subsetRefl = $true) & (subsetE2 = $true) & (notdallE = $true) & (prop2setE = $true) & (omegaSAx = $true) & (powersetE = $true) & (subsetI1 = $true) & (exuE3e = $true) & (setadjoinOr = $true) & (exuE3u = $true) & (emptyinPowerset = $true) & (nonemptyI1 = $true) & (vacuousDall = $true) & (in__Cong = $true) & (nonemptyE1 = $true) & (emptysetsubset = $true) & (singletonsswitch = $true) & (powersetAx = $true) & (notequalI1 = $true) & (setunionI = $true) & (descr__Cong = $true) & (quantDeMorgan3 = $true) & (upairsetIR = $true) & (dsetconstrI = $true) & (dsetconstrEL = $true) & (nonemptyI = $true) & (inCongP = $true) & (dsetconstr__Cong = $true) & (emptyE1 = $true) & (quantDeMorgan1 = $true) & (setoftrueEq = $true) & (setadjoinE = $true) & (replAx = $true) & (quantDeMorgan4 = $true) & (subPowSU = $true) & (subset2powerset = $true) & (emptysetAx = $true) & (notinsingleton = $true) & (notinemptyset = $true) & (quantDeMorgan2 = $true) & (wellorderingAx = $true) & (setadjoin__Cong = $true) & (prop2set2propI = $true) & (setextsub = $true) & (omega0Ax = $true) & (powersetI1 = $true) & (descrp = $true) & (setunionE = $true) & (omegaIndAx = $true) & (eqimpsubset2 = $true) & (setadjoinSub2 = $true) & (prop2setI = $true) & (setadjoinIR = $true) & (setadjoinAx = $true) & (setadjoinIL = $true) & (setbeta = $true) & (emptysetimpfalse = $true) & (powersetE1 = $true) & (setextAx = $true) & (exuI1 = $true) & (uniqinunit = $true) & (setunionAx = $true) & (emptysetE = $true) & (upairsetE = $true) & (notequalI2 = $true) & (subsetemptysetimpeq = $true) & (emptyInPowerset = $true) & (powersetI = $true) & (setext = $true) & (subsetI2 = $true) & (upairsetIL = $true) & (setadjoinSub = $true) & (notdexE = $true) & (powerset__Cong = $true) & (setunion__Cong = $true) & ($true != (in @ sK132 @ (powerset @ sK132))) & (exuI2 = $true) & (subsetE = $true) & (eqinunit = $true) & (exuI3 = $true)),
% 2.39/1.59 inference(skolemisation,[status(esa),new_symbols(skolem,[sK132])],[f360,f648])).
% 2.39/1.59 thf(f648,plain,(
% 2.39/1.59 ? [X0 : $i] : ($true != (in @ X0 @ (powerset @ X0))) => ($true != (in @ sK132 @ (powerset @ sK132)))),
% 2.39/1.59 introduced(definition,[],[choice_axiom])).
% 2.39/1.59 thf(f360,plain,(
% 2.39/1.59 (subsetTrans = $true) & (emptyinunitempty = $true) & (emptyset__Cong = $true) & (exuE1 = $true) & (dsetconstrER = $true) & (exuEu = $true) & (notsubsetI = $true) & (noeltsimpempty = $true) & (exu__Cong = $true) & (foundationAx = $true) & (exuE2 = $true) & (omega__Cong = $true) & (eqimpsubset1 = $true) & (emptyI = $true) & (nonemptyImpWitness = $true) & (subsetRefl = $true) & (subsetE2 = $true) & (notdallE = $true) & (prop2setE = $true) & (omegaSAx = $true) & (powersetE = $true) & (subsetI1 = $true) & (exuE3e = $true) & (setadjoinOr = $true) & (exuE3u = $true) & (emptyinPowerset = $true) & (nonemptyI1 = $true) & (vacuousDall = $true) & (in__Cong = $true) & (nonemptyE1 = $true) & (emptysetsubset = $true) & (singletonsswitch = $true) & (powersetAx = $true) & (notequalI1 = $true) & (setunionI = $true) & (descr__Cong = $true) & (quantDeMorgan3 = $true) & (upairsetIR = $true) & (dsetconstrI = $true) & (dsetconstrEL = $true) & (nonemptyI = $true) & (inCongP = $true) & (dsetconstr__Cong = $true) & (emptyE1 = $true) & (quantDeMorgan1 = $true) & (setoftrueEq = $true) & (setadjoinE = $true) & (replAx = $true) & (quantDeMorgan4 = $true) & (subPowSU = $true) & (subset2powerset = $true) & (emptysetAx = $true) & (notinsingleton = $true) & (notinemptyset = $true) & (quantDeMorgan2 = $true) & (wellorderingAx = $true) & (setadjoin__Cong = $true) & (prop2set2propI = $true) & (setextsub = $true) & (omega0Ax = $true) & (powersetI1 = $true) & (descrp = $true) & (setunionE = $true) & (omegaIndAx = $true) & (eqimpsubset2 = $true) & (setadjoinSub2 = $true) & (prop2setI = $true) & (setadjoinIR = $true) & (setadjoinAx = $true) & (setadjoinIL = $true) & (setbeta = $true) & (emptysetimpfalse = $true) & (powersetE1 = $true) & (setextAx = $true) & (exuI1 = $true) & (uniqinunit = $true) & (setunionAx = $true) & (emptysetE = $true) & (upairsetE = $true) & (notequalI2 = $true) & (subsetemptysetimpeq = $true) & (emptyInPowerset = $true) & (powersetI = $true) & (setext = $true) & (subsetI2 = $true) & (upairsetIL = $true) & (setadjoinSub = $true) & (notdexE = $true) & (powerset__Cong = $true) & (setunion__Cong = $true) & ? [X0 : $i] : ($true != (in @ X0 @ (powerset @ X0))) & (exuI2 = $true) & (subsetE = $true) & (eqinunit = $true) & (exuI3 = $true)),
% 2.39/1.59 inference(flattening,[status(thm)],[f359])).
% 2.39/1.59 thf(f359,plain,(
% 2.39/1.59 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i] : ($true != (in @ X0 @ (powerset @ X0))) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.39/1.59 inference(ennf_transformation,[status(thm)],[f193])).
% 2.39/1.59 thf(f193,plain,(
% 2.39/1.59 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ! [X0 : $i] : ($true = (in @ X0 @ (powerset @ X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.39/1.59 inference(fool_elimination,[status(thm)],[f192])).
% 2.39/1.59 thf(f192,plain,(
% 2.39/1.59 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => ! [X0 : $i] : (in @ X0 @ (powerset @ X0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.39/1.59 inference(rectify,[status(thm)],[f98])).
% 2.39/1.59 thf(f98,negated_conjecture,(
% 2.39/1.59 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => ! [X3 : $i] : (in @ X3 @ (powerset @ X3))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.39/1.59 inference(negated_conjecture,[status(cth)],[f97])).
% 2.39/1.59 thf(f97,conjecture,(
% 2.39/1.59 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => ! [X3 : $i] : (in @ X3 @ (powerset @ X3)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.39/1.59 file('/export/starexec/sandbox/tmp/tmp.EC5jtjE9C7/DTF2THF_10635.p',inPowerset)).
% 2.39/1.59 thf(f935,plain,(
% 2.39/1.59 ( ! [X2 : $i,X3 : $i] : (($true = (subset @ X2 @ X3)) | (X2 != X3) | (eqimpsubset1 != $true)) )),
% 2.39/1.59 inference(cnf_transformation,[status(thm)],[f495])).
% 2.39/1.59 thf(f495,plain,(
% 2.39/1.59 ((eqimpsubset1 = $true) | (($true != (subset @ sK49 @ sK50)) & (sK49 = sK50))) & (! [X2 : $i,X3 : $i] : (($true = (subset @ X2 @ X3)) | (X2 != X3)) | (eqimpsubset1 != $true))),
% 2.39/1.59 inference(skolemisation,[status(esa),new_symbols(skolem,[sK49,sK50])],[f493,f494])).
% 2.39/1.59 thf(f494,plain,(
% 2.39/1.59 ? [X0 : $i,X1 : $i] : (($true != (subset @ X0 @ X1)) & (X0 = X1)) => (($true != (subset @ sK49 @ sK50)) & (sK49 = sK50))),
% 2.39/1.59 introduced(definition,[],[choice_axiom])).
% 2.39/1.59 thf(f493,plain,(
% 2.39/1.59 ((eqimpsubset1 = $true) | ? [X0 : $i,X1 : $i] : (($true != (subset @ X0 @ X1)) & (X0 = X1))) & (! [X2 : $i,X3 : $i] : (($true = (subset @ X2 @ X3)) | (X2 != X3)) | (eqimpsubset1 != $true))),
% 2.39/1.59 inference(rectify,[status(thm)],[f492])).
% 2.39/1.59 thf(f492,plain,(
% 2.39/1.59 ((eqimpsubset1 = $true) | ? [X0 : $i,X1 : $i] : (($true != (subset @ X0 @ X1)) & (X0 = X1))) & (! [X0 : $i,X1 : $i] : (($true = (subset @ X0 @ X1)) | (X0 != X1)) | (eqimpsubset1 != $true))),
% 2.39/1.59 inference(nnf_transformation,[status(thm)],[f383])).
% 2.39/1.59 thf(f383,plain,(
% 2.39/1.59 (eqimpsubset1 = $true) <=> ! [X0 : $i,X1 : $i] : (($true = (subset @ X0 @ X1)) | (X0 != X1))),
% 2.39/1.59 inference(ennf_transformation,[status(thm)],[f283])).
% 2.39/1.59 thf(f283,plain,(
% 2.39/1.59 ! [X0 : $i,X1 : $i] : ((X0 = X1) => ($true = (subset @ X0 @ X1))) <=> (eqimpsubset1 = $true)),
% 2.39/1.59 inference(fool_elimination,[status(thm)],[f282])).
% 2.39/1.59 thf(f282,plain,(
% 2.39/1.59 (eqimpsubset1 = ! [X0 : $i,X1 : $i] : ((X0 = X1) => (subset @ X0 @ X1)))),
% 2.39/1.59 inference(rectify,[status(thm)],[f80])).
% 2.39/1.59 thf(f80,axiom,(
% 2.39/1.59 (eqimpsubset1 = ! [X3 : $i,X4 : $i] : ((X3 = X4) => (subset @ X3 @ X4)))),
% 2.39/1.59 file('/export/starexec/sandbox/tmp/tmp.EC5jtjE9C7/DTF2THF_10635.p',eqimpsubset1)).
% 2.39/1.59 thf(f2328,plain,(
% 2.39/1.59 ($true != (subset @ sK132 @ sK132))),
% 2.39/1.59 inference(trivial_inequality_removal,[status(thm)],[f2321])).
% 2.39/1.59 thf(f2321,plain,(
% 2.39/1.59 ($true != (subset @ sK132 @ sK132)) | ($true != $true)),
% 2.39/1.59 inference(superposition,[status(thm)],[f1064,f1805])).
% 2.39/1.59 thf(f1805,plain,(
% 2.39/1.59 ( ! [X2 : $i,X3 : $i] : (($true = (in @ X2 @ (powerset @ X3))) | ($true != (subset @ X2 @ X3))) )),
% 2.39/1.59 inference(trivial_inequality_removal,[status(thm)],[f1360])).
% 2.39/1.59 thf(f1360,plain,(
% 2.39/1.59 ( ! [X2 : $i,X3 : $i] : (($true != $true) | ($true != (subset @ X2 @ X3)) | ($true = (in @ X2 @ (powerset @ X3)))) )),
% 2.39/1.59 inference(definition_unfolding,[status(thm)],[f877,f1094])).
% 2.39/1.59 thf(f1094,plain,(
% 2.39/1.59 (powersetI1 = $true)),
% 2.39/1.59 inference(cnf_transformation,[status(thm)],[f649])).
% 2.39/1.59 thf(f877,plain,(
% 2.39/1.59 ( ! [X2 : $i,X3 : $i] : (($true != (subset @ X2 @ X3)) | ($true = (in @ X2 @ (powerset @ X3))) | (powersetI1 != $true)) )),
% 2.39/1.59 inference(cnf_transformation,[status(thm)],[f428])).
% 2.39/1.59 thf(f428,plain,(
% 2.39/1.59 ((powersetI1 = $true) | (($true = (subset @ sK8 @ sK9)) & ($true != (in @ sK8 @ (powerset @ sK9))))) & (! [X2 : $i,X3 : $i] : (($true != (subset @ X2 @ X3)) | ($true = (in @ X2 @ (powerset @ X3)))) | (powersetI1 != $true))),
% 2.39/1.59 inference(skolemisation,[status(esa),new_symbols(skolem,[sK8,sK9])],[f426,f427])).
% 2.39/1.59 thf(f427,plain,(
% 2.39/1.59 ? [X0 : $i,X1 : $i] : (($true = (subset @ X0 @ X1)) & ($true != (in @ X0 @ (powerset @ X1)))) => (($true = (subset @ sK8 @ sK9)) & ($true != (in @ sK8 @ (powerset @ sK9))))),
% 2.39/1.59 introduced(definition,[],[choice_axiom])).
% 2.39/1.59 thf(f426,plain,(
% 2.39/1.59 ((powersetI1 = $true) | ? [X0 : $i,X1 : $i] : (($true = (subset @ X0 @ X1)) & ($true != (in @ X0 @ (powerset @ X1))))) & (! [X2 : $i,X3 : $i] : (($true != (subset @ X2 @ X3)) | ($true = (in @ X2 @ (powerset @ X3)))) | (powersetI1 != $true))),
% 2.39/1.59 inference(rectify,[status(thm)],[f425])).
% 2.39/1.59 thf(f425,plain,(
% 2.39/1.59 ((powersetI1 = $true) | ? [X0 : $i,X1 : $i] : (($true = (subset @ X0 @ X1)) & ($true != (in @ X0 @ (powerset @ X1))))) & (! [X0 : $i,X1 : $i] : (($true != (subset @ X0 @ X1)) | ($true = (in @ X0 @ (powerset @ X1)))) | (powersetI1 != $true))),
% 2.39/1.59 inference(nnf_transformation,[status(thm)],[f354])).
% 2.39/1.59 thf(f354,plain,(
% 2.39/1.59 (powersetI1 = $true) <=> ! [X0 : $i,X1 : $i] : (($true != (subset @ X0 @ X1)) | ($true = (in @ X0 @ (powerset @ X1))))),
% 2.39/1.59 inference(ennf_transformation,[status(thm)],[f145])).
% 2.39/1.59 thf(f145,plain,(
% 2.39/1.59 (powersetI1 = $true) <=> ! [X0 : $i,X1 : $i] : (($true = (subset @ X0 @ X1)) => ($true = (in @ X0 @ (powerset @ X1))))),
% 2.39/1.59 inference(fool_elimination,[status(thm)],[f144])).
% 2.39/1.59 thf(f144,plain,(
% 2.39/1.59 (powersetI1 = ! [X0 : $i,X1 : $i] : ((subset @ X0 @ X1) => (in @ X0 @ (powerset @ X1))))),
% 2.39/1.59 inference(rectify,[status(thm)],[f95])).
% 2.39/1.59 thf(f95,axiom,(
% 2.39/1.59 (powersetI1 = ! [X4 : $i,X3 : $i] : ((subset @ X4 @ X3) => (in @ X4 @ (powerset @ X3))))),
% 2.39/1.59 file('/export/starexec/sandbox/tmp/tmp.EC5jtjE9C7/DTF2THF_10635.p',powersetI1)).
% 2.39/1.59 thf(f1064,plain,(
% 2.39/1.59 ($true != (in @ sK132 @ (powerset @ sK132)))),
% 2.39/1.59 inference(cnf_transformation,[status(thm)],[f649])).
% 2.39/1.59 % SZS output end Proof for DTF2THF_10635
% 2.39/1.59 % (10756)------------------------------
% 2.39/1.59 % (10756)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.39/1.59 % (10756)Termination reason: Refutation
% 2.39/1.59
% 2.39/1.59 % (10756)Memory used [KB]: 7419
% 2.39/1.59 % (10756)Time elapsed: 0.081 s
% 2.39/1.59 % (10756)Instructions burned: 225 (million)
% 2.39/1.59 % (10756)------------------------------
% 2.39/1.59 % (10756)------------------------------
% 2.39/1.59 % (10750)Success in time 0.091 s
%------------------------------------------------------------------------------