TSTP Solution File: SEU682^1 by E---3.1.00

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : E---3.1.00
% Problem  : SEU682^1 : TPTP v8.2.0. Released v3.7.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_E %s %d THM

% Computer : n025.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 : Tue May 21 03:29:06 EDT 2024

% Result   : Theorem 0.66s 0.59s
% Output   : CNFRefutation 0.95s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :   13
%            Number of leaves      :  248
% Syntax   : Number of formulae    :  291 (  27 unt; 232 typ;   0 def)
%            Number of atoms       :  986 (  66 equ;   0 cnn)
%            Maximal formula atoms :  305 (  16 avg)
%            Number of connectives : 2561 ( 104   ~; 128   |; 268   &;1529   @)
%                                         (  17 <=>; 515  =>;   0  <=;   0 <~>)
%            Maximal formula depth :  280 (  19 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :  147 ( 147   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  234 ( 232 usr; 204 con; 0-4 aty)
%            Number of variables   :  353 (  84   ^ 241   !;  28   ?; 353   :)

% Comments : 
%------------------------------------------------------------------------------
thf(decl_22,type,
    in: $i > $i > $o ).

thf(decl_24,type,
    setextAx: $o ).

thf(decl_25,type,
    emptyset: $i ).

thf(decl_26,type,
    emptysetAx: $o ).

thf(decl_27,type,
    setadjoin: $i > $i > $i ).

thf(decl_28,type,
    setadjoinAx: $o ).

thf(decl_30,type,
    powersetAx: $o ).

thf(decl_32,type,
    setunionAx: $o ).

thf(decl_34,type,
    omega0Ax: $o ).

thf(decl_35,type,
    omegaSAx: $o ).

thf(decl_36,type,
    omegaIndAx: $o ).

thf(decl_37,type,
    replAx: $o ).

thf(decl_38,type,
    foundationAx: $o ).

thf(decl_39,type,
    wellorderingAx: $o ).

thf(decl_41,type,
    descrp: $o ).

thf(decl_42,type,
    dsetconstr: $i > ( $i > $o ) > $i ).

thf(decl_43,type,
    dsetconstrI: $o ).

thf(decl_44,type,
    dsetconstrEL: $o ).

thf(decl_45,type,
    dsetconstrER: $o ).

thf(decl_46,type,
    exuE1: $o ).

thf(decl_48,type,
    prop2setE: $o ).

thf(decl_49,type,
    emptysetE: $o ).

thf(decl_50,type,
    emptysetimpfalse: $o ).

thf(decl_51,type,
    notinemptyset: $o ).

thf(decl_52,type,
    exuE3e: $o ).

thf(decl_53,type,
    setext: $o ).

thf(decl_54,type,
    emptyI: $o ).

thf(decl_55,type,
    noeltsimpempty: $o ).

thf(decl_56,type,
    setbeta: $o ).

thf(decl_58,type,
    nonemptyE1: $o ).

thf(decl_59,type,
    nonemptyI: $o ).

thf(decl_60,type,
    nonemptyI1: $o ).

thf(decl_61,type,
    setadjoinIL: $o ).

thf(decl_62,type,
    emptyinunitempty: $o ).

thf(decl_63,type,
    setadjoinIR: $o ).

thf(decl_64,type,
    setadjoinE: $o ).

thf(decl_65,type,
    setadjoinOr: $o ).

thf(decl_66,type,
    setoftrueEq: $o ).

thf(decl_67,type,
    powersetI: $o ).

thf(decl_68,type,
    emptyinPowerset: $o ).

thf(decl_69,type,
    emptyInPowerset: $o ).

thf(decl_70,type,
    powersetE: $o ).

thf(decl_71,type,
    setunionI: $o ).

thf(decl_72,type,
    setunionE: $o ).

thf(decl_73,type,
    subPowSU: $o ).

thf(decl_74,type,
    exuE2: $o ).

thf(decl_75,type,
    nonemptyImpWitness: $o ).

thf(decl_76,type,
    uniqinunit: $o ).

thf(decl_77,type,
    notinsingleton: $o ).

thf(decl_78,type,
    eqinunit: $o ).

thf(decl_79,type,
    singletonsswitch: $o ).

thf(decl_80,type,
    upairsetE: $o ).

thf(decl_81,type,
    upairsetIL: $o ).

thf(decl_82,type,
    upairsetIR: $o ).

thf(decl_83,type,
    emptyE1: $o ).

thf(decl_84,type,
    vacuousDall: $o ).

thf(decl_85,type,
    quantDeMorgan1: $o ).

thf(decl_86,type,
    quantDeMorgan2: $o ).

thf(decl_87,type,
    quantDeMorgan3: $o ).

thf(decl_88,type,
    quantDeMorgan4: $o ).

thf(decl_89,type,
    prop2setI: $o ).

thf(decl_91,type,
    prop2set2propI: $o ).

thf(decl_92,type,
    notdexE: $o ).

thf(decl_93,type,
    notdallE: $o ).

thf(decl_94,type,
    exuI1: $o ).

thf(decl_95,type,
    exuI3: $o ).

thf(decl_96,type,
    exuI2: $o ).

thf(decl_97,type,
    inCongP: $o ).

thf(decl_98,type,
    in__Cong: $o ).

thf(decl_99,type,
    exuE3u: $o ).

thf(decl_100,type,
    exu__Cong: $o ).

thf(decl_101,type,
    emptyset__Cong: $o ).

thf(decl_102,type,
    setadjoin__Cong: $o ).

thf(decl_103,type,
    powerset__Cong: $o ).

thf(decl_104,type,
    setunion__Cong: $o ).

thf(decl_105,type,
    omega__Cong: $o ).

thf(decl_106,type,
    exuEu: $o ).

thf(decl_107,type,
    descr__Cong: $o ).

thf(decl_108,type,
    dsetconstr__Cong: $o ).

thf(decl_109,type,
    subset: $i > $i > $o ).

thf(decl_112,type,
    subsetI1: $o ).

thf(decl_113,type,
    eqimpsubset2: $o ).

thf(decl_114,type,
    eqimpsubset1: $o ).

thf(decl_115,type,
    subsetI2: $o ).

thf(decl_116,type,
    emptysetsubset: $o ).

thf(decl_117,type,
    subsetE: $o ).

thf(decl_118,type,
    subsetE2: $o ).

thf(decl_119,type,
    notsubsetI: $o ).

thf(decl_120,type,
    notequalI1: $o ).

thf(decl_121,type,
    notequalI2: $o ).

thf(decl_122,type,
    subsetRefl: $o ).

thf(decl_123,type,
    subsetTrans: $o ).

thf(decl_124,type,
    setadjoinSub: $o ).

thf(decl_125,type,
    setadjoinSub2: $o ).

thf(decl_126,type,
    subset2powerset: $o ).

thf(decl_127,type,
    setextsub: $o ).

thf(decl_128,type,
    subsetemptysetimpeq: $o ).

thf(decl_129,type,
    powersetI1: $o ).

thf(decl_130,type,
    powersetE1: $o ).

thf(decl_131,type,
    inPowerset: $o ).

thf(decl_132,type,
    powersetsubset: $o ).

thf(decl_133,type,
    sepInPowerset: $o ).

thf(decl_134,type,
    sepSubset: $o ).

thf(decl_136,type,
    binunionIL: $o ).

thf(decl_137,type,
    upairset2IR: $o ).

thf(decl_138,type,
    binunionIR: $o ).

thf(decl_139,type,
    binunionEcases: $o ).

thf(decl_140,type,
    binunionE: $o ).

thf(decl_141,type,
    binunionLsub: $o ).

thf(decl_142,type,
    binunionRsub: $o ).

thf(decl_144,type,
    binintersectI: $o ).

thf(decl_145,type,
    binintersectSubset5: $o ).

thf(decl_146,type,
    binintersectEL: $o ).

thf(decl_147,type,
    binintersectLsub: $o ).

thf(decl_148,type,
    binintersectSubset2: $o ).

thf(decl_149,type,
    binintersectSubset3: $o ).

thf(decl_150,type,
    binintersectER: $o ).

thf(decl_151,type,
    disjointsetsI1: $o ).

thf(decl_152,type,
    binintersectRsub: $o ).

thf(decl_153,type,
    binintersectSubset4: $o ).

thf(decl_154,type,
    binintersectSubset1: $o ).

thf(decl_155,type,
    bs114d: $o ).

thf(decl_158,type,
    setminusI: $o ).

thf(decl_159,type,
    setminusEL: $o ).

thf(decl_160,type,
    setminusER: $o ).

thf(decl_161,type,
    setminusSubset2: $o ).

thf(decl_162,type,
    setminusERneg: $o ).

thf(decl_163,type,
    setminusELneg: $o ).

thf(decl_164,type,
    setminusILneg: $o ).

thf(decl_165,type,
    setminusIRneg: $o ).

thf(decl_166,type,
    setminusLsub: $o ).

thf(decl_167,type,
    setminusSubset1: $o ).

thf(decl_169,type,
    symdiffE: $o ).

thf(decl_170,type,
    symdiffI1: $o ).

thf(decl_171,type,
    symdiffI2: $o ).

thf(decl_172,type,
    symdiffIneg1: $o ).

thf(decl_173,type,
    symdiffIneg2: $o ).

thf(decl_175,type,
    secondinupair: $o ).

thf(decl_176,type,
    setukpairIL: $o ).

thf(decl_177,type,
    setukpairIR: $o ).

thf(decl_178,type,
    kpairiskpair: $o ).

thf(decl_179,type,
    kpair: $i > $i > $i ).

thf(decl_180,type,
    kpairp: $o ).

thf(decl_181,type,
    cartprod: $i > $i > $i ).

thf(decl_182,type,
    singletonsubset: $o ).

thf(decl_183,type,
    singletoninpowerset: $o ).

thf(decl_184,type,
    singletoninpowunion: $o ).

thf(decl_185,type,
    upairset2E: $o ).

thf(decl_186,type,
    upairsubunion: $o ).

thf(decl_187,type,
    upairinpowunion: $o ).

thf(decl_188,type,
    ubforcartprodlem1: $o ).

thf(decl_189,type,
    ubforcartprodlem2: $o ).

thf(decl_190,type,
    ubforcartprodlem3: $o ).

thf(decl_191,type,
    cartprodpairin: $o ).

thf(decl_192,type,
    cartprodmempair1: $o ).

thf(decl_193,type,
    cartprodmempair: $o ).

thf(decl_194,type,
    setunionE2: $o ).

thf(decl_195,type,
    setunionsingleton1: $o ).

thf(decl_196,type,
    setunionsingleton2: $o ).

thf(decl_197,type,
    setunionsingleton: $o ).

thf(decl_198,type,
    singleton: $i > $o ).

thf(decl_199,type,
    singletonprop: $o ).

thf(decl_200,type,
    ex1: $i > ( $i > $o ) > $o ).

thf(decl_201,type,
    ex1E1: $o ).

thf(decl_202,type,
    ex1I: $o ).

thf(decl_203,type,
    ex1I2: $o ).

thf(decl_204,type,
    singletonsuniq: $o ).

thf(decl_209,type,
    setukpairinjL1: $o ).

thf(decl_210,type,
    kfstsingleton: $o ).

thf(decl_211,type,
    theprop: $o ).

thf(decl_213,type,
    kfstpairEq: $o ).

thf(decl_214,type,
    cartprodfstin: $o ).

thf(decl_215,type,
    setukpairinjL2: $o ).

thf(decl_216,type,
    setukpairinjL: $o ).

thf(decl_217,type,
    setukpairinjR11: $o ).

thf(decl_218,type,
    setukpairinjR12: $o ).

thf(decl_219,type,
    setukpairinjR1: $o ).

thf(decl_220,type,
    upairequniteq: $o ).

thf(decl_221,type,
    setukpairinjR2: $o ).

thf(decl_222,type,
    setukpairinjR: $o ).

thf(decl_223,type,
    ksndsingleton: $o ).

thf(decl_225,type,
    ksndpairEq: $o ).

thf(decl_226,type,
    kpairsurjEq: $o ).

thf(decl_227,type,
    cartprodsndin: $o ).

thf(decl_228,type,
    cartprodpairmemEL: $o ).

thf(decl_229,type,
    cartprodpairmemER: $o ).

thf(decl_230,type,
    cartprodmempaircEq: $o ).

thf(decl_231,type,
    cartprodfstpairEq: $o ).

thf(decl_232,type,
    cartprodsndpairEq: $o ).

thf(decl_233,type,
    cartprodpairsurjEq: $o ).

thf(decl_234,type,
    breln: $i > $i > $i > $o ).

thf(decl_235,type,
    dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).

thf(decl_236,type,
    dpsetconstrI: $o ).

thf(decl_237,type,
    dpsetconstrSub: $o ).

thf(decl_238,type,
    setOfPairsIsBReln: $o ).

thf(decl_239,type,
    dpsetconstrERa: $o ).

thf(decl_240,type,
    dpsetconstrEL1: $o ).

thf(decl_241,type,
    dpsetconstrEL2: $o ).

thf(decl_242,type,
    dpsetconstrER: $o ).

thf(decl_243,type,
    func: $i > $i > $i > $o ).

thf(decl_244,type,
    funcSet: $i > $i > $i ).

thf(decl_245,type,
    funcImageSingleton: $o ).

thf(decl_246,type,
    apProp: $o ).

thf(decl_247,type,
    ap: $i > $i > $i > $i > $i ).

thf(decl_248,type,
    app: $o ).

thf(decl_249,type,
    infuncsetfunc: $o ).

thf(decl_250,type,
    ap2p: $o ).

thf(decl_251,type,
    funcinfuncset: $o ).

thf(decl_252,type,
    lamProp: $o ).

thf(decl_253,type,
    lam: $i > $i > ( $i > $i ) > $i ).

thf(decl_254,type,
    lamp: $o ).

thf(decl_255,type,
    lam2p: $o ).

thf(decl_256,type,
    brelnall1: $o ).

thf(decl_257,type,
    esk1_2: $i > ( $i > $o ) > $i ).

thf(decl_258,type,
    esk2_3: $i > ( $i > $o ) > $i > $i ).

thf(decl_259,type,
    esk3_3: $i > ( $i > $o ) > $i > $i ).

thf(decl_260,type,
    esk4_2: $i > ( $i > $o ) > $i ).

thf(decl_261,type,
    esk5_2: $i > ( $i > $o ) > $i ).

thf(decl_262,type,
    esk6_2: $i > ( $i > $o ) > $i ).

thf(decl_263,type,
    esk7_3: $i > $i > $i > $i ).

thf(decl_264,type,
    esk8_4: $i > $i > $i > $i > $i ).

thf(decl_265,type,
    esk9_3: $i > $i > $i > $i ).

thf(decl_266,type,
    esk10_3: $i > $i > ( $i > $i ) > $i ).

thf(decl_267,type,
    esk11_4: $i > $i > ( $i > $i ) > $i > $i ).

thf(decl_268,type,
    esk12_3: $i > $i > ( $i > $i ) > $i ).

thf(decl_269,type,
    esk13_4: $i > $i > $i > ( $i > $o ) > $i ).

thf(decl_270,type,
    esk14_4: $i > $i > $i > ( $i > $o ) > $i ).

thf(decl_271,type,
    esk15_0: $i ).

thf(decl_272,type,
    esk16_0: $i ).

thf(decl_273,type,
    esk17_0: $i ).

thf(decl_274,type,
    epred1_0: $i > $o ).

thf(decl_275,type,
    esk18_0: $i ).

thf(ex1,axiom,
    ( ex1
    = ( ^ [X4: $i,X1: $i > $o] :
          ( singleton
          @ ( dsetconstr @ X4
            @ ^ [X2: $i] : ( X1 @ X2 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ex1) ).

thf(singleton,axiom,
    ( singleton
    = ( ^ [X4: $i] :
        ? [X2: $i] :
          ( ( in @ X2 @ X4 )
          & ( X4
            = ( setadjoin @ X2 @ emptyset ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',singleton) ).

thf(func,axiom,
    ( func
    = ( ^ [X4: $i,X5: $i,X29: $i] :
          ( ( breln @ X4 @ X5 @ X29 )
          & ! [X2: $i] :
              ( ( in @ X2 @ X4 )
             => ( ex1 @ X5
                @ ^ [X3: $i] : ( in @ ( kpair @ X2 @ X3 ) @ X29 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',func) ).

thf(breln,axiom,
    ( breln
    = ( ^ [X4: $i,X5: $i,X7: $i] : ( subset @ X7 @ ( cartprod @ X4 @ X5 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',breln) ).

thf(lamp,axiom,
    ( lamp
  <=> ! [X4: $i,X5: $i,X33: $i > $i] :
        ( ! [X2: $i] :
            ( ( in @ X2 @ X4 )
           => ( in @ ( X33 @ X2 ) @ X5 ) )
       => ( func @ X4 @ X5
          @ ( lam @ X4 @ X5
            @ ^ [X2: $i] : ( X33 @ X2 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',lamp) ).

thf(ex1I2,axiom,
    ( ex1I2
  <=> ! [X4: $i,X1: $i > $o] :
        ( ! [X2: $i] :
            ( ( in @ X2 @ X4 )
           => ! [X3: $i] :
                ( ( in @ X3 @ X4 )
               => ( ( X1 @ X2 )
                 => ( ( X1 @ X3 )
                   => ( X2 = X3 ) ) ) ) )
       => ( ? [X2: $i] :
              ( ( in @ X2 @ X4 )
              & ( X1 @ X2 ) )
         => ( ex1 @ X4
            @ ^ [X2: $i] : ( X1 @ X2 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ex1I2) ).

thf(ex1I,axiom,
    ( ex1I
  <=> ! [X4: $i,X1: $i > $o,X2: $i] :
        ( ( in @ X2 @ X4 )
       => ( ( X1 @ X2 )
         => ( ! [X3: $i] :
                ( ( in @ X3 @ X4 )
               => ( ( X1 @ X3 )
                 => ( X3 = X2 ) ) )
           => ( ex1 @ X4
              @ ^ [X3: $i] : ( X1 @ X3 ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ex1I) ).

thf(ex1E1,axiom,
    ( ex1E1
  <=> ! [X4: $i,X1: $i > $o] :
        ( ( ex1 @ X4
          @ ^ [X2: $i] : ( X1 @ X2 ) )
       => ? [X2: $i] :
            ( ( in @ X2 @ X4 )
            & ( X1 @ X2 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ex1E1) ).

thf(setOfPairsIsBReln,axiom,
    ( setOfPairsIsBReln
  <=> ! [X4: $i,X5: $i,X24: $i > $i > $o] :
        ( breln @ X4 @ X5
        @ ( dpsetconstr @ X4 @ X5
          @ ^ [X2: $i,X3: $i] : ( X24 @ X2 @ X3 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',setOfPairsIsBReln) ).

thf(app,axiom,
    ( app
  <=> ! [X4: $i,X5: $i,X30: $i] :
        ( ( func @ X4 @ X5 @ X30 )
       => ! [X2: $i] :
            ( ( in @ X2 @ X4 )
           => ( in @ ( ap @ X4 @ X5 @ X30 @ X2 ) @ X5 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',app) ).

thf(funcinfuncset,axiom,
    ( funcinfuncset
  <=> ! [X4: $i,X5: $i,X30: $i] :
        ( ( func @ X4 @ X5 @ X30 )
       => ( in @ X30 @ ( funcSet @ X4 @ X5 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',funcinfuncset) ).

thf(infuncsetfunc,axiom,
    ( infuncsetfunc
  <=> ! [X4: $i,X5: $i,X30: $i] :
        ( ( in @ X30 @ ( funcSet @ X4 @ X5 ) )
       => ( func @ X4 @ X5 @ X30 ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',infuncsetfunc) ).

thf(lam2p,axiom,
    ( lam2p
  <=> ! [X4: $i,X5: $i,X34: $i > $i] :
        ( ! [X2: $i] :
            ( ( in @ X2 @ X4 )
           => ( in @ ( X34 @ X2 ) @ X5 ) )
       => ( in
          @ ( lam @ X4 @ X5
            @ ^ [X2: $i] : ( X34 @ X2 ) )
          @ ( funcSet @ X4 @ X5 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',lam2p) ).

thf(brelnall1,axiom,
    ( brelnall1
  <=> ! [X4: $i,X5: $i,X29: $i] :
        ( ( breln @ X4 @ X5 @ X29 )
       => ! [X1: $i > $o] :
            ( ! [X2: $i] :
                ( ( in @ X2 @ X4 )
               => ! [X3: $i] :
                    ( ( in @ X3 @ X5 )
                   => ( ( in @ ( kpair @ X2 @ X3 ) @ X29 )
                     => ( X1 @ ( kpair @ X2 @ X3 ) ) ) ) )
           => ! [X2: $i] :
                ( ( in @ X2 @ X29 )
               => ( X1 @ X2 ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',brelnall1) ).

thf(brelnall2,conjecture,
    ( setextAx
   => ( emptysetAx
     => ( setadjoinAx
       => ( powersetAx
         => ( setunionAx
           => ( omega0Ax
             => ( omegaSAx
               => ( omegaIndAx
                 => ( replAx
                   => ( foundationAx
                     => ( wellorderingAx
                       => ( descrp
                         => ( dsetconstrI
                           => ( dsetconstrEL
                             => ( dsetconstrER
                               => ( exuE1
                                 => ( prop2setE
                                   => ( emptysetE
                                     => ( emptysetimpfalse
                                       => ( notinemptyset
                                         => ( exuE3e
                                           => ( setext
                                             => ( emptyI
                                               => ( noeltsimpempty
                                                 => ( setbeta
                                                   => ( nonemptyE1
                                                     => ( nonemptyI
                                                       => ( nonemptyI1
                                                         => ( setadjoinIL
                                                           => ( emptyinunitempty
                                                             => ( setadjoinIR
                                                               => ( setadjoinE
                                                                 => ( setadjoinOr
                                                                   => ( setoftrueEq
                                                                     => ( powersetI
                                                                       => ( emptyinPowerset
                                                                         => ( emptyInPowerset
                                                                           => ( powersetE
                                                                             => ( setunionI
                                                                               => ( setunionE
                                                                                 => ( subPowSU
                                                                                   => ( exuE2
                                                                                     => ( nonemptyImpWitness
                                                                                       => ( uniqinunit
                                                                                         => ( notinsingleton
                                                                                           => ( eqinunit
                                                                                             => ( singletonsswitch
                                                                                               => ( upairsetE
                                                                                                 => ( upairsetIL
                                                                                                   => ( upairsetIR
                                                                                                     => ( emptyE1
                                                                                                       => ( vacuousDall
                                                                                                         => ( quantDeMorgan1
                                                                                                           => ( quantDeMorgan2
                                                                                                             => ( quantDeMorgan3
                                                                                                               => ( quantDeMorgan4
                                                                                                                 => ( prop2setI
                                                                                                                   => ( prop2set2propI
                                                                                                                     => ( notdexE
                                                                                                                       => ( notdallE
                                                                                                                         => ( exuI1
                                                                                                                           => ( exuI3
                                                                                                                             => ( exuI2
                                                                                                                               => ( inCongP
                                                                                                                                 => ( in__Cong
                                                                                                                                   => ( exuE3u
                                                                                                                                     => ( exu__Cong
                                                                                                                                       => ( emptyset__Cong
                                                                                                                                         => ( setadjoin__Cong
                                                                                                                                           => ( powerset__Cong
                                                                                                                                             => ( setunion__Cong
                                                                                                                                               => ( omega__Cong
                                                                                                                                                 => ( exuEu
                                                                                                                                                   => ( descr__Cong
                                                                                                                                                     => ( dsetconstr__Cong
                                                                                                                                                       => ( subsetI1
                                                                                                                                                         => ( eqimpsubset2
                                                                                                                                                           => ( eqimpsubset1
                                                                                                                                                             => ( subsetI2
                                                                                                                                                               => ( emptysetsubset
                                                                                                                                                                 => ( subsetE
                                                                                                                                                                   => ( subsetE2
                                                                                                                                                                     => ( notsubsetI
                                                                                                                                                                       => ( notequalI1
                                                                                                                                                                         => ( notequalI2
                                                                                                                                                                           => ( subsetRefl
                                                                                                                                                                             => ( subsetTrans
                                                                                                                                                                               => ( setadjoinSub
                                                                                                                                                                                 => ( setadjoinSub2
                                                                                                                                                                                   => ( subset2powerset
                                                                                                                                                                                     => ( setextsub
                                                                                                                                                                                       => ( subsetemptysetimpeq
                                                                                                                                                                                         => ( powersetI1
                                                                                                                                                                                           => ( powersetE1
                                                                                                                                                                                             => ( inPowerset
                                                                                                                                                                                               => ( powersetsubset
                                                                                                                                                                                                 => ( sepInPowerset
                                                                                                                                                                                                   => ( sepSubset
                                                                                                                                                                                                     => ( binunionIL
                                                                                                                                                                                                       => ( upairset2IR
                                                                                                                                                                                                         => ( binunionIR
                                                                                                                                                                                                           => ( binunionEcases
                                                                                                                                                                                                             => ( binunionE
                                                                                                                                                                                                               => ( binunionLsub
                                                                                                                                                                                                                 => ( binunionRsub
                                                                                                                                                                                                                   => ( binintersectI
                                                                                                                                                                                                                     => ( binintersectSubset5
                                                                                                                                                                                                                       => ( binintersectEL
                                                                                                                                                                                                                         => ( binintersectLsub
                                                                                                                                                                                                                           => ( binintersectSubset2
                                                                                                                                                                                                                             => ( binintersectSubset3
                                                                                                                                                                                                                               => ( binintersectER
                                                                                                                                                                                                                                 => ( disjointsetsI1
                                                                                                                                                                                                                                   => ( binintersectRsub
                                                                                                                                                                                                                                     => ( binintersectSubset4
                                                                                                                                                                                                                                       => ( binintersectSubset1
                                                                                                                                                                                                                                         => ( bs114d
                                                                                                                                                                                                                                           => ( setminusI
                                                                                                                                                                                                                                             => ( setminusEL
                                                                                                                                                                                                                                               => ( setminusER
                                                                                                                                                                                                                                                 => ( setminusSubset2
                                                                                                                                                                                                                                                   => ( setminusERneg
                                                                                                                                                                                                                                                     => ( setminusELneg
                                                                                                                                                                                                                                                       => ( setminusILneg
                                                                                                                                                                                                                                                         => ( setminusIRneg
                                                                                                                                                                                                                                                           => ( setminusLsub
                                                                                                                                                                                                                                                             => ( setminusSubset1
                                                                                                                                                                                                                                                               => ( symdiffE
                                                                                                                                                                                                                                                                 => ( symdiffI1
                                                                                                                                                                                                                                                                   => ( symdiffI2
                                                                                                                                                                                                                                                                     => ( symdiffIneg1
                                                                                                                                                                                                                                                                       => ( symdiffIneg2
                                                                                                                                                                                                                                                                         => ( secondinupair
                                                                                                                                                                                                                                                                           => ( setukpairIL
                                                                                                                                                                                                                                                                             => ( setukpairIR
                                                                                                                                                                                                                                                                               => ( kpairiskpair
                                                                                                                                                                                                                                                                                 => ( kpairp
                                                                                                                                                                                                                                                                                   => ( singletonsubset
                                                                                                                                                                                                                                                                                     => ( singletoninpowerset
                                                                                                                                                                                                                                                                                       => ( singletoninpowunion
                                                                                                                                                                                                                                                                                         => ( upairset2E
                                                                                                                                                                                                                                                                                           => ( upairsubunion
                                                                                                                                                                                                                                                                                             => ( upairinpowunion
                                                                                                                                                                                                                                                                                               => ( ubforcartprodlem1
                                                                                                                                                                                                                                                                                                 => ( ubforcartprodlem2
                                                                                                                                                                                                                                                                                                   => ( ubforcartprodlem3
                                                                                                                                                                                                                                                                                                     => ( cartprodpairin
                                                                                                                                                                                                                                                                                                       => ( cartprodmempair1
                                                                                                                                                                                                                                                                                                         => ( cartprodmempair
                                                                                                                                                                                                                                                                                                           => ( setunionE2
                                                                                                                                                                                                                                                                                                             => ( setunionsingleton1
                                                                                                                                                                                                                                                                                                               => ( setunionsingleton2
                                                                                                                                                                                                                                                                                                                 => ( setunionsingleton
                                                                                                                                                                                                                                                                                                                   => ( singletonprop
                                                                                                                                                                                                                                                                                                                     => ( ex1E1
                                                                                                                                                                                                                                                                                                                       => ( ex1I
                                                                                                                                                                                                                                                                                                                         => ( ex1I2
                                                                                                                                                                                                                                                                                                                           => ( singletonsuniq
                                                                                                                                                                                                                                                                                                                             => ( setukpairinjL1
                                                                                                                                                                                                                                                                                                                               => ( kfstsingleton
                                                                                                                                                                                                                                                                                                                                 => ( theprop
                                                                                                                                                                                                                                                                                                                                   => ( kfstpairEq
                                                                                                                                                                                                                                                                                                                                     => ( cartprodfstin
                                                                                                                                                                                                                                                                                                                                       => ( setukpairinjL2
                                                                                                                                                                                                                                                                                                                                         => ( setukpairinjL
                                                                                                                                                                                                                                                                                                                                           => ( setukpairinjR11
                                                                                                                                                                                                                                                                                                                                             => ( setukpairinjR12
                                                                                                                                                                                                                                                                                                                                               => ( setukpairinjR1
                                                                                                                                                                                                                                                                                                                                                 => ( upairequniteq
                                                                                                                                                                                                                                                                                                                                                   => ( setukpairinjR2
                                                                                                                                                                                                                                                                                                                                                     => ( setukpairinjR
                                                                                                                                                                                                                                                                                                                                                       => ( ksndsingleton
                                                                                                                                                                                                                                                                                                                                                         => ( ksndpairEq
                                                                                                                                                                                                                                                                                                                                                           => ( kpairsurjEq
                                                                                                                                                                                                                                                                                                                                                             => ( cartprodsndin
                                                                                                                                                                                                                                                                                                                                                               => ( cartprodpairmemEL
                                                                                                                                                                                                                                                                                                                                                                 => ( cartprodpairmemER
                                                                                                                                                                                                                                                                                                                                                                   => ( cartprodmempaircEq
                                                                                                                                                                                                                                                                                                                                                                     => ( cartprodfstpairEq
                                                                                                                                                                                                                                                                                                                                                                       => ( cartprodsndpairEq
                                                                                                                                                                                                                                                                                                                                                                         => ( cartprodpairsurjEq
                                                                                                                                                                                                                                                                                                                                                                           => ( dpsetconstrI
                                                                                                                                                                                                                                                                                                                                                                             => ( dpsetconstrSub
                                                                                                                                                                                                                                                                                                                                                                               => ( setOfPairsIsBReln
                                                                                                                                                                                                                                                                                                                                                                                 => ( dpsetconstrERa
                                                                                                                                                                                                                                                                                                                                                                                   => ( dpsetconstrEL1
                                                                                                                                                                                                                                                                                                                                                                                     => ( dpsetconstrEL2
                                                                                                                                                                                                                                                                                                                                                                                       => ( dpsetconstrER
                                                                                                                                                                                                                                                                                                                                                                                         => ( funcImageSingleton
                                                                                                                                                                                                                                                                                                                                                                                           => ( apProp
                                                                                                                                                                                                                                                                                                                                                                                             => ( app
                                                                                                                                                                                                                                                                                                                                                                                               => ( infuncsetfunc
                                                                                                                                                                                                                                                                                                                                                                                                 => ( ap2p
                                                                                                                                                                                                                                                                                                                                                                                                   => ( funcinfuncset
                                                                                                                                                                                                                                                                                                                                                                                                     => ( lamProp
                                                                                                                                                                                                                                                                                                                                                                                                       => ( lamp
                                                                                                                                                                                                                                                                                                                                                                                                         => ( lam2p
                                                                                                                                                                                                                                                                                                                                                                                                           => ( brelnall1
                                                                                                                                                                                                                                                                                                                                                                                                             => ! [X4: $i,X5: $i,X29: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                                  ( ( breln @ X4 @ X5 @ X29 )
                                                                                                                                                                                                                                                                                                                                                                                                                 => ! [X1: $i > $o] :
                                                                                                                                                                                                                                                                                                                                                                                                                      ( ! [X2: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                                          ( ( in @ X2 @ X4 )
                                                                                                                                                                                                                                                                                                                                                                                                                         => ! [X3: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                                              ( ( in @ X3 @ X5 )
                                                                                                                                                                                                                                                                                                                                                                                                                             => ( ( in @ ( kpair @ X2 @ X3 ) @ X29 )
                                                                                                                                                                                                                                                                                                                                                                                                                               => ( X1 @ ( kpair @ X2 @ X3 ) ) ) ) )
                                                                                                                                                                                                                                                                                                                                                                                                                     => ! [X2: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                                          ( ( in @ X2 @ X29 )
                                                                                                                                                                                                                                                                                                                                                                                                                         => ( X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',brelnall2) ).

thf(ap2p,axiom,
    ( ap2p
  <=> ! [X4: $i,X5: $i,X30: $i] :
        ( ( in @ X30 @ ( funcSet @ X4 @ X5 ) )
       => ! [X2: $i] :
            ( ( in @ X2 @ X4 )
           => ( in @ ( ap @ X4 @ X5 @ X30 @ X2 ) @ X5 ) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',ap2p) ).

thf(c_0_16,plain,
    ( ex1
    = ( ^ [Z0: $i,Z1: $i > $o] :
        ? [X86: $i] :
          ( ( in @ X86
            @ ( dsetconstr @ Z0
              @ ^ [Z2: $i] : ( Z1 @ Z2 ) ) )
          & ( ( dsetconstr @ Z0
              @ ^ [Z2: $i] : ( Z1 @ Z2 ) )
            = ( setadjoin @ X86 @ emptyset ) ) ) ) ),
    inference(fof_simplification,[status(thm)],[ex1]) ).

thf(c_0_17,plain,
    ( singleton
    = ( ^ [Z0: $i] :
        ? [X2: $i] :
          ( ( in @ X2 @ Z0 )
          & ( Z0
            = ( setadjoin @ X2 @ emptyset ) ) ) ) ),
    inference(fof_simplification,[status(thm)],[singleton]) ).

thf(c_0_18,plain,
    ( func
    = ( ^ [Z0: $i,Z1: $i,Z2: $i] :
          ( ( subset @ Z2 @ ( cartprod @ Z0 @ Z1 ) )
          & ! [X2: $i] :
              ( ( in @ X2 @ Z0 )
             => ? [X90: $i] :
                  ( ( in @ X90
                    @ ( dsetconstr @ Z1
                      @ ^ [Z3: $i] : ( in @ ( kpair @ X2 @ Z3 ) @ Z2 ) ) )
                  & ( ( dsetconstr @ Z1
                      @ ^ [Z3: $i] : ( in @ ( kpair @ X2 @ Z3 ) @ Z2 ) )
                    = ( setadjoin @ X90 @ emptyset ) ) ) ) ) ) ),
    inference(fof_simplification,[status(thm)],[func]) ).

thf(c_0_19,plain,
    ( breln
    = ( ^ [Z0: $i,Z1: $i,Z2: $i] : ( subset @ Z2 @ ( cartprod @ Z0 @ Z1 ) ) ) ),
    inference(fof_simplification,[status(thm)],[breln]) ).

thf(c_0_20,plain,
    ( ex1
    = ( ^ [Z0: $i,Z1: $i > $o] :
        ? [X86: $i] :
          ( ( in @ X86
            @ ( dsetconstr @ Z0
              @ ^ [Z2: $i] : ( Z1 @ Z2 ) ) )
          & ( ( dsetconstr @ Z0
              @ ^ [Z2: $i] : ( Z1 @ Z2 ) )
            = ( setadjoin @ X86 @ emptyset ) ) ) ) ),
    inference(apply_def,[status(thm)],[c_0_16,c_0_17]) ).

thf(c_0_21,plain,
    ( func
    = ( ^ [Z0: $i,Z1: $i,Z2: $i] :
          ( ( subset @ Z2 @ ( cartprod @ Z0 @ Z1 ) )
          & ! [X2: $i] :
              ( ( in @ X2 @ Z0 )
             => ? [X90: $i] :
                  ( ( in @ X90
                    @ ( dsetconstr @ Z1
                      @ ^ [Z3: $i] : ( in @ ( kpair @ X2 @ Z3 ) @ Z2 ) ) )
                  & ( ( dsetconstr @ Z1
                      @ ^ [Z3: $i] : ( in @ ( kpair @ X2 @ Z3 ) @ Z2 ) )
                    = ( setadjoin @ X90 @ emptyset ) ) ) ) ) ) ),
    inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[c_0_18,c_0_19]),c_0_20]) ).

thf(c_0_22,plain,
    ( lamp
  <=> ! [X4: $i,X5: $i,X33: $i > $i] :
        ( ! [X2: $i] :
            ( ( in @ X2 @ X4 )
           => ( in @ ( X33 @ X2 ) @ X5 ) )
       => ( func @ X4 @ X5
          @ ( lam @ X4 @ X5
            @ ^ [Z0: $i] : ( X33 @ Z0 ) ) ) ) ),
    inference(fof_simplification,[status(thm)],[lamp]) ).

thf(c_0_23,plain,
    ( ex1I2
  <=> ! [X4: $i,X1: $i > $o] :
        ( ! [X2: $i] :
            ( ( in @ X2 @ X4 )
           => ! [X3: $i] :
                ( ( in @ X3 @ X4 )
               => ( ( X1 @ X2 )
                 => ( ( X1 @ X3 )
                   => ( X2 = X3 ) ) ) ) )
       => ( ? [X2: $i] :
              ( ( in @ X2 @ X4 )
              & ( X1 @ X2 ) )
         => ( ex1 @ X4
            @ ^ [Z0: $i] : ( X1 @ Z0 ) ) ) ) ),
    inference(fof_simplification,[status(thm)],[ex1I2]) ).

thf(c_0_24,plain,
    ( ex1I
  <=> ! [X4: $i,X1: $i > $o,X2: $i] :
        ( ( in @ X2 @ X4 )
       => ( ( X1 @ X2 )
         => ( ! [X3: $i] :
                ( ( in @ X3 @ X4 )
               => ( ( X1 @ X3 )
                 => ( X3 = X2 ) ) )
           => ( ex1 @ X4
              @ ^ [Z0: $i] : ( X1 @ Z0 ) ) ) ) ) ),
    inference(fof_simplification,[status(thm)],[ex1I]) ).

thf(c_0_25,plain,
    ( ex1E1
  <=> ! [X4: $i,X1: $i > $o] :
        ( ( ex1 @ X4
          @ ^ [Z0: $i] : ( X1 @ Z0 ) )
       => ? [X2: $i] :
            ( ( in @ X2 @ X4 )
            & ( X1 @ X2 ) ) ) ),
    inference(fof_simplification,[status(thm)],[ex1E1]) ).

thf(c_0_26,plain,
    ( setOfPairsIsBReln
  <=> ! [X4: $i,X5: $i,X24: $i > $i > $o] :
        ( breln @ X4 @ X5
        @ ( dpsetconstr @ X4 @ X5
          @ ^ [Z0: $i,Z1: $i] : ( X24 @ Z0 @ Z1 ) ) ) ),
    inference(fof_simplification,[status(thm)],[setOfPairsIsBReln]) ).

thf(c_0_27,axiom,
    ( app
    = ( ! [X4: $i,X5: $i,X30: $i] :
          ( ( ( subset @ X30 @ ( cartprod @ X4 @ X5 ) )
            & ! [X91: $i] :
                ( ( in @ X91 @ X4 )
               => ? [X92: $i] :
                    ( ( in @ X92
                      @ ( dsetconstr @ X5
                        @ ^ [Z0: $i] : ( in @ ( kpair @ X91 @ Z0 ) @ X30 ) ) )
                    & ( ( dsetconstr @ X5
                        @ ^ [Z0: $i] : ( in @ ( kpair @ X91 @ Z0 ) @ X30 ) )
                      = ( setadjoin @ X92 @ emptyset ) ) ) ) )
         => ! [X2: $i] :
              ( ( in @ X2 @ X4 )
             => ( in @ ( ap @ X4 @ X5 @ X30 @ X2 ) @ X5 ) ) ) ) ),
    inference(apply_def,[status(thm)],[app,c_0_21]) ).

thf(c_0_28,axiom,
    ( funcinfuncset
    = ( ! [X4: $i,X5: $i,X30: $i] :
          ( ( ( subset @ X30 @ ( cartprod @ X4 @ X5 ) )
            & ! [X95: $i] :
                ( ( in @ X95 @ X4 )
               => ? [X96: $i] :
                    ( ( in @ X96
                      @ ( dsetconstr @ X5
                        @ ^ [Z0: $i] : ( in @ ( kpair @ X95 @ Z0 ) @ X30 ) ) )
                    & ( ( dsetconstr @ X5
                        @ ^ [Z0: $i] : ( in @ ( kpair @ X95 @ Z0 ) @ X30 ) )
                      = ( setadjoin @ X96 @ emptyset ) ) ) ) )
         => ( in @ X30 @ ( funcSet @ X4 @ X5 ) ) ) ) ),
    inference(apply_def,[status(thm)],[funcinfuncset,c_0_21]) ).

thf(c_0_29,axiom,
    ( infuncsetfunc
    = ( ! [X4: $i,X5: $i,X30: $i] :
          ( ( in @ X30 @ ( funcSet @ X4 @ X5 ) )
         => ( ( subset @ X30 @ ( cartprod @ X4 @ X5 ) )
            & ! [X93: $i] :
                ( ( in @ X93 @ X4 )
               => ? [X94: $i] :
                    ( ( in @ X94
                      @ ( dsetconstr @ X5
                        @ ^ [Z0: $i] : ( in @ ( kpair @ X93 @ Z0 ) @ X30 ) ) )
                    & ( ( dsetconstr @ X5
                        @ ^ [Z0: $i] : ( in @ ( kpair @ X93 @ Z0 ) @ X30 ) )
                      = ( setadjoin @ X94 @ emptyset ) ) ) ) ) ) ) ),
    inference(apply_def,[status(thm)],[infuncsetfunc,c_0_21]) ).

thf(c_0_30,plain,
    ( lam2p
  <=> ! [X4: $i,X5: $i,X34: $i > $i] :
        ( ! [X2: $i] :
            ( ( in @ X2 @ X4 )
           => ( in @ ( X34 @ X2 ) @ X5 ) )
       => ( in
          @ ( lam @ X4 @ X5
            @ ^ [Z0: $i] : ( X34 @ Z0 ) )
          @ ( funcSet @ X4 @ X5 ) ) ) ),
    inference(fof_simplification,[status(thm)],[lam2p]) ).

thf(c_0_31,plain,
    ( lamp
    = ( ! [X4: $i,X5: $i,X33: $i > $i] :
          ( ! [X2: $i] :
              ( ( in @ X2 @ X4 )
             => ( in @ ( X33 @ X2 ) @ X5 ) )
         => ( ( subset
              @ ( lam @ X4 @ X5
                @ ^ [Z0: $i] : ( X33 @ Z0 ) )
              @ ( cartprod @ X4 @ X5 ) )
            & ! [X97: $i] :
                ( ( in @ X97 @ X4 )
               => ? [X98: $i] :
                    ( ( in @ X98
                      @ ( dsetconstr @ X5
                        @ ^ [Z0: $i] :
                            ( in @ ( kpair @ X97 @ Z0 )
                            @ ( lam @ X4 @ X5
                              @ ^ [Z1: $i] : ( X33 @ Z1 ) ) ) ) )
                    & ( ( dsetconstr @ X5
                        @ ^ [Z0: $i] :
                            ( in @ ( kpair @ X97 @ Z0 )
                            @ ( lam @ X4 @ X5
                              @ ^ [Z1: $i] : ( X33 @ Z1 ) ) ) )
                      = ( setadjoin @ X98 @ emptyset ) ) ) ) ) ) ) ),
    inference(apply_def,[status(thm)],[c_0_22,c_0_21]) ).

thf(c_0_32,plain,
    ( ex1I2
    = ( ! [X4: $i,X1: $i > $o] :
          ( ! [X2: $i] :
              ( ( in @ X2 @ X4 )
             => ! [X3: $i] :
                  ( ( in @ X3 @ X4 )
                 => ( ( X1 @ X2 )
                   => ( ( X1 @ X3 )
                     => ( X2 = X3 ) ) ) ) )
         => ( ? [X2: $i] :
                ( ( in @ X2 @ X4 )
                & ( X1 @ X2 ) )
           => ? [X89: $i] :
                ( ( in @ X89
                  @ ( dsetconstr @ X4
                    @ ^ [Z0: $i] : ( X1 @ Z0 ) ) )
                & ( ( dsetconstr @ X4
                    @ ^ [Z0: $i] : ( X1 @ Z0 ) )
                  = ( setadjoin @ X89 @ emptyset ) ) ) ) ) ) ),
    inference(apply_def,[status(thm)],[c_0_23,c_0_20]) ).

thf(c_0_33,plain,
    ( ex1I
    = ( ! [X4: $i,X1: $i > $o,X2: $i] :
          ( ( in @ X2 @ X4 )
         => ( ( X1 @ X2 )
           => ( ! [X3: $i] :
                  ( ( in @ X3 @ X4 )
                 => ( ( X1 @ X3 )
                   => ( X3 = X2 ) ) )
             => ? [X88: $i] :
                  ( ( in @ X88
                    @ ( dsetconstr @ X4
                      @ ^ [Z0: $i] : ( X1 @ Z0 ) ) )
                  & ( ( dsetconstr @ X4
                      @ ^ [Z0: $i] : ( X1 @ Z0 ) )
                    = ( setadjoin @ X88 @ emptyset ) ) ) ) ) ) ) ),
    inference(apply_def,[status(thm)],[c_0_24,c_0_20]) ).

thf(c_0_34,plain,
    ( ex1E1
    = ( ! [X4: $i,X1: $i > $o] :
          ( ? [X87: $i] :
              ( ( in @ X87
                @ ( dsetconstr @ X4
                  @ ^ [Z0: $i] : ( X1 @ Z0 ) ) )
              & ( ( dsetconstr @ X4
                  @ ^ [Z0: $i] : ( X1 @ Z0 ) )
                = ( setadjoin @ X87 @ emptyset ) ) )
         => ? [X2: $i] :
              ( ( in @ X2 @ X4 )
              & ( X1 @ X2 ) ) ) ) ),
    inference(apply_def,[status(thm)],[c_0_25,c_0_20]) ).

thf(c_0_35,axiom,
    ( brelnall1
    = ( ! [X4: $i,X5: $i,X29: $i] :
          ( ( subset @ X29 @ ( cartprod @ X4 @ X5 ) )
         => ! [X1: $i > $o] :
              ( ! [X2: $i] :
                  ( ( in @ X2 @ X4 )
                 => ! [X3: $i] :
                      ( ( in @ X3 @ X5 )
                     => ( ( in @ ( kpair @ X2 @ X3 ) @ X29 )
                       => ( X1 @ ( kpair @ X2 @ X3 ) ) ) ) )
             => ! [X2: $i] :
                  ( ( in @ X2 @ X29 )
                 => ( X1 @ X2 ) ) ) ) ) ),
    inference(apply_def,[status(thm)],[brelnall1,c_0_19]) ).

thf(c_0_36,plain,
    ( setOfPairsIsBReln
    = ( ! [X4: $i,X5: $i,X24: $i > $i > $o] :
          ( subset
          @ ( dpsetconstr @ X4 @ X5
            @ ^ [Z0: $i,Z1: $i] : ( X24 @ Z0 @ Z1 ) )
          @ ( cartprod @ X4 @ X5 ) ) ) ),
    inference(apply_def,[status(thm)],[c_0_26,c_0_19]) ).

thf(c_0_37,negated_conjecture,
    ~ ( setextAx
     => ( emptysetAx
       => ( setadjoinAx
         => ( powersetAx
           => ( setunionAx
             => ( omega0Ax
               => ( omegaSAx
                 => ( omegaIndAx
                   => ( replAx
                     => ( foundationAx
                       => ( wellorderingAx
                         => ( descrp
                           => ( dsetconstrI
                             => ( dsetconstrEL
                               => ( dsetconstrER
                                 => ( exuE1
                                   => ( prop2setE
                                     => ( emptysetE
                                       => ( emptysetimpfalse
                                         => ( notinemptyset
                                           => ( exuE3e
                                             => ( setext
                                               => ( emptyI
                                                 => ( noeltsimpempty
                                                   => ( setbeta
                                                     => ( nonemptyE1
                                                       => ( nonemptyI
                                                         => ( nonemptyI1
                                                           => ( setadjoinIL
                                                             => ( emptyinunitempty
                                                               => ( setadjoinIR
                                                                 => ( setadjoinE
                                                                   => ( setadjoinOr
                                                                     => ( setoftrueEq
                                                                       => ( powersetI
                                                                         => ( emptyinPowerset
                                                                           => ( emptyInPowerset
                                                                             => ( powersetE
                                                                               => ( setunionI
                                                                                 => ( setunionE
                                                                                   => ( subPowSU
                                                                                     => ( exuE2
                                                                                       => ( nonemptyImpWitness
                                                                                         => ( uniqinunit
                                                                                           => ( notinsingleton
                                                                                             => ( eqinunit
                                                                                               => ( singletonsswitch
                                                                                                 => ( upairsetE
                                                                                                   => ( upairsetIL
                                                                                                     => ( upairsetIR
                                                                                                       => ( emptyE1
                                                                                                         => ( vacuousDall
                                                                                                           => ( quantDeMorgan1
                                                                                                             => ( quantDeMorgan2
                                                                                                               => ( quantDeMorgan3
                                                                                                                 => ( quantDeMorgan4
                                                                                                                   => ( prop2setI
                                                                                                                     => ( prop2set2propI
                                                                                                                       => ( notdexE
                                                                                                                         => ( notdallE
                                                                                                                           => ( exuI1
                                                                                                                             => ( exuI3
                                                                                                                               => ( exuI2
                                                                                                                                 => ( inCongP
                                                                                                                                   => ( in__Cong
                                                                                                                                     => ( exuE3u
                                                                                                                                       => ( exu__Cong
                                                                                                                                         => ( emptyset__Cong
                                                                                                                                           => ( setadjoin__Cong
                                                                                                                                             => ( powerset__Cong
                                                                                                                                               => ( setunion__Cong
                                                                                                                                                 => ( omega__Cong
                                                                                                                                                   => ( exuEu
                                                                                                                                                     => ( descr__Cong
                                                                                                                                                       => ( dsetconstr__Cong
                                                                                                                                                         => ( subsetI1
                                                                                                                                                           => ( eqimpsubset2
                                                                                                                                                             => ( eqimpsubset1
                                                                                                                                                               => ( subsetI2
                                                                                                                                                                 => ( emptysetsubset
                                                                                                                                                                   => ( subsetE
                                                                                                                                                                     => ( subsetE2
                                                                                                                                                                       => ( notsubsetI
                                                                                                                                                                         => ( notequalI1
                                                                                                                                                                           => ( notequalI2
                                                                                                                                                                             => ( subsetRefl
                                                                                                                                                                               => ( subsetTrans
                                                                                                                                                                                 => ( setadjoinSub
                                                                                                                                                                                   => ( setadjoinSub2
                                                                                                                                                                                     => ( subset2powerset
                                                                                                                                                                                       => ( setextsub
                                                                                                                                                                                         => ( subsetemptysetimpeq
                                                                                                                                                                                           => ( powersetI1
                                                                                                                                                                                             => ( powersetE1
                                                                                                                                                                                               => ( inPowerset
                                                                                                                                                                                                 => ( powersetsubset
                                                                                                                                                                                                   => ( sepInPowerset
                                                                                                                                                                                                     => ( sepSubset
                                                                                                                                                                                                       => ( binunionIL
                                                                                                                                                                                                         => ( upairset2IR
                                                                                                                                                                                                           => ( binunionIR
                                                                                                                                                                                                             => ( binunionEcases
                                                                                                                                                                                                               => ( binunionE
                                                                                                                                                                                                                 => ( binunionLsub
                                                                                                                                                                                                                   => ( binunionRsub
                                                                                                                                                                                                                     => ( binintersectI
                                                                                                                                                                                                                       => ( binintersectSubset5
                                                                                                                                                                                                                         => ( binintersectEL
                                                                                                                                                                                                                           => ( binintersectLsub
                                                                                                                                                                                                                             => ( binintersectSubset2
                                                                                                                                                                                                                               => ( binintersectSubset3
                                                                                                                                                                                                                                 => ( binintersectER
                                                                                                                                                                                                                                   => ( disjointsetsI1
                                                                                                                                                                                                                                     => ( binintersectRsub
                                                                                                                                                                                                                                       => ( binintersectSubset4
                                                                                                                                                                                                                                         => ( binintersectSubset1
                                                                                                                                                                                                                                           => ( bs114d
                                                                                                                                                                                                                                             => ( setminusI
                                                                                                                                                                                                                                               => ( setminusEL
                                                                                                                                                                                                                                                 => ( setminusER
                                                                                                                                                                                                                                                   => ( setminusSubset2
                                                                                                                                                                                                                                                     => ( setminusERneg
                                                                                                                                                                                                                                                       => ( setminusELneg
                                                                                                                                                                                                                                                         => ( setminusILneg
                                                                                                                                                                                                                                                           => ( setminusIRneg
                                                                                                                                                                                                                                                             => ( setminusLsub
                                                                                                                                                                                                                                                               => ( setminusSubset1
                                                                                                                                                                                                                                                                 => ( symdiffE
                                                                                                                                                                                                                                                                   => ( symdiffI1
                                                                                                                                                                                                                                                                     => ( symdiffI2
                                                                                                                                                                                                                                                                       => ( symdiffIneg1
                                                                                                                                                                                                                                                                         => ( symdiffIneg2
                                                                                                                                                                                                                                                                           => ( secondinupair
                                                                                                                                                                                                                                                                             => ( setukpairIL
                                                                                                                                                                                                                                                                               => ( setukpairIR
                                                                                                                                                                                                                                                                                 => ( kpairiskpair
                                                                                                                                                                                                                                                                                   => ( kpairp
                                                                                                                                                                                                                                                                                     => ( singletonsubset
                                                                                                                                                                                                                                                                                       => ( singletoninpowerset
                                                                                                                                                                                                                                                                                         => ( singletoninpowunion
                                                                                                                                                                                                                                                                                           => ( upairset2E
                                                                                                                                                                                                                                                                                             => ( upairsubunion
                                                                                                                                                                                                                                                                                               => ( upairinpowunion
                                                                                                                                                                                                                                                                                                 => ( ubforcartprodlem1
                                                                                                                                                                                                                                                                                                   => ( ubforcartprodlem2
                                                                                                                                                                                                                                                                                                     => ( ubforcartprodlem3
                                                                                                                                                                                                                                                                                                       => ( cartprodpairin
                                                                                                                                                                                                                                                                                                         => ( cartprodmempair1
                                                                                                                                                                                                                                                                                                           => ( cartprodmempair
                                                                                                                                                                                                                                                                                                             => ( setunionE2
                                                                                                                                                                                                                                                                                                               => ( setunionsingleton1
                                                                                                                                                                                                                                                                                                                 => ( setunionsingleton2
                                                                                                                                                                                                                                                                                                                   => ( setunionsingleton
                                                                                                                                                                                                                                                                                                                     => ( singletonprop
                                                                                                                                                                                                                                                                                                                       => ( ! [X99: $i,X100: $i > $o] :
                                                                                                                                                                                                                                                                                                                              ( ? [X101: $i] :
                                                                                                                                                                                                                                                                                                                                  ( ( in @ X101 @ ( dsetconstr @ X99 @ X100 ) )
                                                                                                                                                                                                                                                                                                                                  & ( ( dsetconstr @ X99 @ X100 )
                                                                                                                                                                                                                                                                                                                                    = ( setadjoin @ X101 @ emptyset ) ) )
                                                                                                                                                                                                                                                                                                                             => ? [X102: $i] :
                                                                                                                                                                                                                                                                                                                                  ( ( in @ X102 @ X99 )
                                                                                                                                                                                                                                                                                                                                  & ( X100 @ X102 ) ) )
                                                                                                                                                                                                                                                                                                                         => ( ! [X103: $i,X104: $i > $o,X105: $i] :
                                                                                                                                                                                                                                                                                                                                ( ( in @ X105 @ X103 )
                                                                                                                                                                                                                                                                                                                               => ( ( X104 @ X105 )
                                                                                                                                                                                                                                                                                                                                 => ( ! [X106: $i] :
                                                                                                                                                                                                                                                                                                                                        ( ( in @ X106 @ X103 )
                                                                                                                                                                                                                                                                                                                                       => ( ( X104 @ X106 )
                                                                                                                                                                                                                                                                                                                                         => ( X106 = X105 ) ) )
                                                                                                                                                                                                                                                                                                                                   => ? [X107: $i] :
                                                                                                                                                                                                                                                                                                                                        ( ( in @ X107 @ ( dsetconstr @ X103 @ X104 ) )
                                                                                                                                                                                                                                                                                                                                        & ( ( dsetconstr @ X103 @ X104 )
                                                                                                                                                                                                                                                                                                                                          = ( setadjoin @ X107 @ emptyset ) ) ) ) ) )
                                                                                                                                                                                                                                                                                                                           => ( ! [X108: $i,X109: $i > $o] :
                                                                                                                                                                                                                                                                                                                                  ( ! [X110: $i] :
                                                                                                                                                                                                                                                                                                                                      ( ( in @ X110 @ X108 )
                                                                                                                                                                                                                                                                                                                                     => ! [X111: $i] :
                                                                                                                                                                                                                                                                                                                                          ( ( in @ X111 @ X108 )
                                                                                                                                                                                                                                                                                                                                         => ( ( X109 @ X110 )
                                                                                                                                                                                                                                                                                                                                           => ( ( X109 @ X111 )
                                                                                                                                                                                                                                                                                                                                             => ( X110 = X111 ) ) ) ) )
                                                                                                                                                                                                                                                                                                                                 => ( ? [X112: $i] :
                                                                                                                                                                                                                                                                                                                                        ( ( in @ X112 @ X108 )
                                                                                                                                                                                                                                                                                                                                        & ( X109 @ X112 ) )
                                                                                                                                                                                                                                                                                                                                   => ? [X113: $i] :
                                                                                                                                                                                                                                                                                                                                        ( ( in @ X113 @ ( dsetconstr @ X108 @ X109 ) )
                                                                                                                                                                                                                                                                                                                                        & ( ( dsetconstr @ X108 @ X109 )
                                                                                                                                                                                                                                                                                                                                          = ( setadjoin @ X113 @ emptyset ) ) ) ) )
                                                                                                                                                                                                                                                                                                                             => ( singletonsuniq
                                                                                                                                                                                                                                                                                                                               => ( setukpairinjL1
                                                                                                                                                                                                                                                                                                                                 => ( kfstsingleton
                                                                                                                                                                                                                                                                                                                                   => ( theprop
                                                                                                                                                                                                                                                                                                                                     => ( kfstpairEq
                                                                                                                                                                                                                                                                                                                                       => ( cartprodfstin
                                                                                                                                                                                                                                                                                                                                         => ( setukpairinjL2
                                                                                                                                                                                                                                                                                                                                           => ( setukpairinjL
                                                                                                                                                                                                                                                                                                                                             => ( setukpairinjR11
                                                                                                                                                                                                                                                                                                                                               => ( setukpairinjR12
                                                                                                                                                                                                                                                                                                                                                 => ( setukpairinjR1
                                                                                                                                                                                                                                                                                                                                                   => ( upairequniteq
                                                                                                                                                                                                                                                                                                                                                     => ( setukpairinjR2
                                                                                                                                                                                                                                                                                                                                                       => ( setukpairinjR
                                                                                                                                                                                                                                                                                                                                                         => ( ksndsingleton
                                                                                                                                                                                                                                                                                                                                                           => ( ksndpairEq
                                                                                                                                                                                                                                                                                                                                                             => ( kpairsurjEq
                                                                                                                                                                                                                                                                                                                                                               => ( cartprodsndin
                                                                                                                                                                                                                                                                                                                                                                 => ( cartprodpairmemEL
                                                                                                                                                                                                                                                                                                                                                                   => ( cartprodpairmemER
                                                                                                                                                                                                                                                                                                                                                                     => ( cartprodmempaircEq
                                                                                                                                                                                                                                                                                                                                                                       => ( cartprodfstpairEq
                                                                                                                                                                                                                                                                                                                                                                         => ( cartprodsndpairEq
                                                                                                                                                                                                                                                                                                                                                                           => ( cartprodpairsurjEq
                                                                                                                                                                                                                                                                                                                                                                             => ( dpsetconstrI
                                                                                                                                                                                                                                                                                                                                                                               => ( dpsetconstrSub
                                                                                                                                                                                                                                                                                                                                                                                 => ( ! [X114: $i,X115: $i,X116: $i > $i > $o] : ( subset @ ( dpsetconstr @ X114 @ X115 @ X116 ) @ ( cartprod @ X114 @ X115 ) )
                                                                                                                                                                                                                                                                                                                                                                                   => ( dpsetconstrERa
                                                                                                                                                                                                                                                                                                                                                                                     => ( dpsetconstrEL1
                                                                                                                                                                                                                                                                                                                                                                                       => ( dpsetconstrEL2
                                                                                                                                                                                                                                                                                                                                                                                         => ( dpsetconstrER
                                                                                                                                                                                                                                                                                                                                                                                           => ( funcImageSingleton
                                                                                                                                                                                                                                                                                                                                                                                             => ( apProp
                                                                                                                                                                                                                                                                                                                                                                                               => ( ! [X117: $i,X118: $i,X119: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                      ( ( ( subset @ X119 @ ( cartprod @ X117 @ X118 ) )
                                                                                                                                                                                                                                                                                                                                                                                                        & ! [X120: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                            ( ( in @ X120 @ X117 )
                                                                                                                                                                                                                                                                                                                                                                                                           => ? [X121: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                                ( ( in @ X121
                                                                                                                                                                                                                                                                                                                                                                                                                  @ ( dsetconstr @ X118
                                                                                                                                                                                                                                                                                                                                                                                                                    @ ^ [Z0: $i] : ( in @ ( kpair @ X120 @ Z0 ) @ X119 ) ) )
                                                                                                                                                                                                                                                                                                                                                                                                                & ( ( dsetconstr @ X118
                                                                                                                                                                                                                                                                                                                                                                                                                    @ ^ [Z0: $i] : ( in @ ( kpair @ X120 @ Z0 ) @ X119 ) )
                                                                                                                                                                                                                                                                                                                                                                                                                  = ( setadjoin @ X121 @ emptyset ) ) ) ) )
                                                                                                                                                                                                                                                                                                                                                                                                     => ! [X122: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                          ( ( in @ X122 @ X117 )
                                                                                                                                                                                                                                                                                                                                                                                                         => ( in @ ( ap @ X117 @ X118 @ X119 @ X122 ) @ X118 ) ) )
                                                                                                                                                                                                                                                                                                                                                                                                 => ( ! [X123: $i,X124: $i,X125: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                        ( ( in @ X125 @ ( funcSet @ X123 @ X124 ) )
                                                                                                                                                                                                                                                                                                                                                                                                       => ( ( subset @ X125 @ ( cartprod @ X123 @ X124 ) )
                                                                                                                                                                                                                                                                                                                                                                                                          & ! [X126: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                              ( ( in @ X126 @ X123 )
                                                                                                                                                                                                                                                                                                                                                                                                             => ? [X127: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                                  ( ( in @ X127
                                                                                                                                                                                                                                                                                                                                                                                                                    @ ( dsetconstr @ X124
                                                                                                                                                                                                                                                                                                                                                                                                                      @ ^ [Z0: $i] : ( in @ ( kpair @ X126 @ Z0 ) @ X125 ) ) )
                                                                                                                                                                                                                                                                                                                                                                                                                  & ( ( dsetconstr @ X124
                                                                                                                                                                                                                                                                                                                                                                                                                      @ ^ [Z0: $i] : ( in @ ( kpair @ X126 @ Z0 ) @ X125 ) )
                                                                                                                                                                                                                                                                                                                                                                                                                    = ( setadjoin @ X127 @ emptyset ) ) ) ) ) )
                                                                                                                                                                                                                                                                                                                                                                                                   => ( ! [X128: $i,X129: $i,X130: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                          ( ( in @ X130 @ ( funcSet @ X128 @ X129 ) )
                                                                                                                                                                                                                                                                                                                                                                                                         => ! [X131: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                              ( ( in @ X131 @ X128 )
                                                                                                                                                                                                                                                                                                                                                                                                             => ( in @ ( ap @ X128 @ X129 @ X130 @ X131 ) @ X129 ) ) )
                                                                                                                                                                                                                                                                                                                                                                                                     => ( ! [X132: $i,X133: $i,X134: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                            ( ( ( subset @ X134 @ ( cartprod @ X132 @ X133 ) )
                                                                                                                                                                                                                                                                                                                                                                                                              & ! [X135: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                                  ( ( in @ X135 @ X132 )
                                                                                                                                                                                                                                                                                                                                                                                                                 => ? [X136: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                                      ( ( in @ X136
                                                                                                                                                                                                                                                                                                                                                                                                                        @ ( dsetconstr @ X133
                                                                                                                                                                                                                                                                                                                                                                                                                          @ ^ [Z0: $i] : ( in @ ( kpair @ X135 @ Z0 ) @ X134 ) ) )
                                                                                                                                                                                                                                                                                                                                                                                                                      & ( ( dsetconstr @ X133
                                                                                                                                                                                                                                                                                                                                                                                                                          @ ^ [Z0: $i] : ( in @ ( kpair @ X135 @ Z0 ) @ X134 ) )
                                                                                                                                                                                                                                                                                                                                                                                                                        = ( setadjoin @ X136 @ emptyset ) ) ) ) )
                                                                                                                                                                                                                                                                                                                                                                                                           => ( in @ X134 @ ( funcSet @ X132 @ X133 ) ) )
                                                                                                                                                                                                                                                                                                                                                                                                       => ( lamProp
                                                                                                                                                                                                                                                                                                                                                                                                         => ( ! [X137: $i,X138: $i,X139: $i > $i] :
                                                                                                                                                                                                                                                                                                                                                                                                                ( ! [X140: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                                    ( ( in @ X140 @ X137 )
                                                                                                                                                                                                                                                                                                                                                                                                                   => ( in @ ( X139 @ X140 ) @ X138 ) )
                                                                                                                                                                                                                                                                                                                                                                                                               => ( ( subset @ ( lam @ X137 @ X138 @ X139 ) @ ( cartprod @ X137 @ X138 ) )
                                                                                                                                                                                                                                                                                                                                                                                                                  & ! [X141: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                                      ( ( in @ X141 @ X137 )
                                                                                                                                                                                                                                                                                                                                                                                                                     => ? [X142: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                                          ( ( in @ X142
                                                                                                                                                                                                                                                                                                                                                                                                                            @ ( dsetconstr @ X138
                                                                                                                                                                                                                                                                                                                                                                                                                              @ ^ [Z0: $i] : ( in @ ( kpair @ X141 @ Z0 ) @ ( lam @ X137 @ X138 @ X139 ) ) ) )
                                                                                                                                                                                                                                                                                                                                                                                                                          & ( ( dsetconstr @ X138
                                                                                                                                                                                                                                                                                                                                                                                                                              @ ^ [Z0: $i] : ( in @ ( kpair @ X141 @ Z0 ) @ ( lam @ X137 @ X138 @ X139 ) ) )
                                                                                                                                                                                                                                                                                                                                                                                                                            = ( setadjoin @ X142 @ emptyset ) ) ) ) ) )
                                                                                                                                                                                                                                                                                                                                                                                                           => ( ! [X143: $i,X144: $i,X145: $i > $i] :
                                                                                                                                                                                                                                                                                                                                                                                                                  ( ! [X146: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                                      ( ( in @ X146 @ X143 )
                                                                                                                                                                                                                                                                                                                                                                                                                     => ( in @ ( X145 @ X146 ) @ X144 ) )
                                                                                                                                                                                                                                                                                                                                                                                                                 => ( in @ ( lam @ X143 @ X144 @ X145 ) @ ( funcSet @ X143 @ X144 ) ) )
                                                                                                                                                                                                                                                                                                                                                                                                             => ( ! [X147: $i,X148: $i,X149: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                                    ( ( subset @ X149 @ ( cartprod @ X147 @ X148 ) )
                                                                                                                                                                                                                                                                                                                                                                                                                   => ! [X150: $i > $o] :
                                                                                                                                                                                                                                                                                                                                                                                                                        ( ! [X151: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                                            ( ( in @ X151 @ X147 )
                                                                                                                                                                                                                                                                                                                                                                                                                           => ! [X152: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                                                ( ( in @ X152 @ X148 )
                                                                                                                                                                                                                                                                                                                                                                                                                               => ( ( in @ ( kpair @ X151 @ X152 ) @ X149 )
                                                                                                                                                                                                                                                                                                                                                                                                                                 => ( X150 @ ( kpair @ X151 @ X152 ) ) ) ) )
                                                                                                                                                                                                                                                                                                                                                                                                                       => ! [X153: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                                            ( ( in @ X153 @ X149 )
                                                                                                                                                                                                                                                                                                                                                                                                                           => ( X150 @ X153 ) ) ) )
                                                                                                                                                                                                                                                                                                                                                                                                               => ! [X4: $i,X5: $i,X29: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                                    ( ( subset @ X29 @ ( cartprod @ X4 @ X5 ) )
                                                                                                                                                                                                                                                                                                                                                                                                                   => ! [X1: $i > $o] :
                                                                                                                                                                                                                                                                                                                                                                                                                        ( ! [X2: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                                            ( ( in @ X2 @ X4 )
                                                                                                                                                                                                                                                                                                                                                                                                                           => ! [X3: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                                                ( ( in @ X3 @ X5 )
                                                                                                                                                                                                                                                                                                                                                                                                                               => ( ( in @ ( kpair @ X2 @ X3 ) @ X29 )
                                                                                                                                                                                                                                                                                                                                                                                                                                 => ( X1 @ ( kpair @ X2 @ X3 ) ) ) ) )
                                                                                                                                                                                                                                                                                                                                                                                                                       => ! [X2: $i] :
                                                                                                                                                                                                                                                                                                                                                                                                                            ( ( in @ X2 @ X29 )
                                                                                                                                                                                                                                                                                                                                                                                                                           => ( X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    inference(fool_unroll,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(assume_negation,[status(cth)],[brelnall2]),c_0_19]),c_0_27]),c_0_28]),ap2p]),c_0_29]),c_0_30]),c_0_31]),c_0_32]),c_0_33]),c_0_34]),c_0_35]),c_0_36])]) ).

thf(c_0_38,negated_conjecture,
    ! [X154: $i,X155: $i > $o,X156: $i,X158: $i,X159: $i > $o,X160: $i,X163: $i,X164: $i > $o,X167: $i,X169: $i,X170: $i,X171: $i > $i > $o,X172: $i,X173: $i,X174: $i,X176: $i,X177: $i,X178: $i,X179: $i,X180: $i,X181: $i,X183: $i,X184: $i,X185: $i,X186: $i,X187: $i,X188: $i,X189: $i,X191: $i,X192: $i,X193: $i,X194: $i > $i,X196: $i,X198: $i,X199: $i,X200: $i > $i,X202: $i,X203: $i,X204: $i,X205: $i > $o,X208: $i,X213: $i,X214: $i] :
      ( setextAx
      & emptysetAx
      & setadjoinAx
      & powersetAx
      & setunionAx
      & omega0Ax
      & omegaSAx
      & omegaIndAx
      & replAx
      & foundationAx
      & wellorderingAx
      & descrp
      & dsetconstrI
      & dsetconstrEL
      & dsetconstrER
      & exuE1
      & prop2setE
      & emptysetE
      & emptysetimpfalse
      & notinemptyset
      & exuE3e
      & setext
      & emptyI
      & noeltsimpempty
      & setbeta
      & nonemptyE1
      & nonemptyI
      & nonemptyI1
      & setadjoinIL
      & emptyinunitempty
      & setadjoinIR
      & setadjoinE
      & setadjoinOr
      & setoftrueEq
      & powersetI
      & emptyinPowerset
      & emptyInPowerset
      & powersetE
      & setunionI
      & setunionE
      & subPowSU
      & exuE2
      & nonemptyImpWitness
      & uniqinunit
      & notinsingleton
      & eqinunit
      & singletonsswitch
      & upairsetE
      & upairsetIL
      & upairsetIR
      & emptyE1
      & vacuousDall
      & quantDeMorgan1
      & quantDeMorgan2
      & quantDeMorgan3
      & quantDeMorgan4
      & prop2setI
      & prop2set2propI
      & notdexE
      & notdallE
      & exuI1
      & exuI3
      & exuI2
      & inCongP
      & in__Cong
      & exuE3u
      & exu__Cong
      & emptyset__Cong
      & setadjoin__Cong
      & powerset__Cong
      & setunion__Cong
      & omega__Cong
      & exuEu
      & descr__Cong
      & dsetconstr__Cong
      & subsetI1
      & eqimpsubset2
      & eqimpsubset1
      & subsetI2
      & emptysetsubset
      & subsetE
      & subsetE2
      & notsubsetI
      & notequalI1
      & notequalI2
      & subsetRefl
      & subsetTrans
      & setadjoinSub
      & setadjoinSub2
      & subset2powerset
      & setextsub
      & subsetemptysetimpeq
      & powersetI1
      & powersetE1
      & inPowerset
      & powersetsubset
      & sepInPowerset
      & sepSubset
      & binunionIL
      & upairset2IR
      & binunionIR
      & binunionEcases
      & binunionE
      & binunionLsub
      & binunionRsub
      & binintersectI
      & binintersectSubset5
      & binintersectEL
      & binintersectLsub
      & binintersectSubset2
      & binintersectSubset3
      & binintersectER
      & disjointsetsI1
      & binintersectRsub
      & binintersectSubset4
      & binintersectSubset1
      & bs114d
      & setminusI
      & setminusEL
      & setminusER
      & setminusSubset2
      & setminusERneg
      & setminusELneg
      & setminusILneg
      & setminusIRneg
      & setminusLsub
      & setminusSubset1
      & symdiffE
      & symdiffI1
      & symdiffI2
      & symdiffIneg1
      & symdiffIneg2
      & secondinupair
      & setukpairIL
      & setukpairIR
      & kpairiskpair
      & kpairp
      & singletonsubset
      & singletoninpowerset
      & singletoninpowunion
      & upairset2E
      & upairsubunion
      & upairinpowunion
      & ubforcartprodlem1
      & ubforcartprodlem2
      & ubforcartprodlem3
      & cartprodpairin
      & cartprodmempair1
      & cartprodmempair
      & setunionE2
      & setunionsingleton1
      & setunionsingleton2
      & setunionsingleton
      & singletonprop
      & ( ( in @ ( esk1_2 @ X154 @ X155 ) @ X154 )
        | ~ ( in @ X156 @ ( dsetconstr @ X154 @ X155 ) )
        | ( ( dsetconstr @ X154 @ X155 )
         != ( setadjoin @ X156 @ emptyset ) ) )
      & ( ( X155 @ ( esk1_2 @ X154 @ X155 ) )
        | ~ ( in @ X156 @ ( dsetconstr @ X154 @ X155 ) )
        | ( ( dsetconstr @ X154 @ X155 )
         != ( setadjoin @ X156 @ emptyset ) ) )
      & ( ( in @ ( esk3_3 @ X158 @ X159 @ X160 ) @ ( dsetconstr @ X158 @ X159 ) )
        | ( in @ ( esk2_3 @ X158 @ X159 @ X160 ) @ X158 )
        | ~ ( X159 @ X160 )
        | ~ ( in @ X160 @ X158 ) )
      & ( ( ( dsetconstr @ X158 @ X159 )
          = ( setadjoin @ ( esk3_3 @ X158 @ X159 @ X160 ) @ emptyset ) )
        | ( in @ ( esk2_3 @ X158 @ X159 @ X160 ) @ X158 )
        | ~ ( X159 @ X160 )
        | ~ ( in @ X160 @ X158 ) )
      & ( ( in @ ( esk3_3 @ X158 @ X159 @ X160 ) @ ( dsetconstr @ X158 @ X159 ) )
        | ( X159 @ ( esk2_3 @ X158 @ X159 @ X160 ) )
        | ~ ( X159 @ X160 )
        | ~ ( in @ X160 @ X158 ) )
      & ( ( ( dsetconstr @ X158 @ X159 )
          = ( setadjoin @ ( esk3_3 @ X158 @ X159 @ X160 ) @ emptyset ) )
        | ( X159 @ ( esk2_3 @ X158 @ X159 @ X160 ) )
        | ~ ( X159 @ X160 )
        | ~ ( in @ X160 @ X158 ) )
      & ( ( in @ ( esk3_3 @ X158 @ X159 @ X160 ) @ ( dsetconstr @ X158 @ X159 ) )
        | ( ( esk2_3 @ X158 @ X159 @ X160 )
         != X160 )
        | ~ ( X159 @ X160 )
        | ~ ( in @ X160 @ X158 ) )
      & ( ( ( dsetconstr @ X158 @ X159 )
          = ( setadjoin @ ( esk3_3 @ X158 @ X159 @ X160 ) @ emptyset ) )
        | ( ( esk2_3 @ X158 @ X159 @ X160 )
         != X160 )
        | ~ ( X159 @ X160 )
        | ~ ( in @ X160 @ X158 ) )
      & ( ( in @ ( esk6_2 @ X163 @ X164 ) @ ( dsetconstr @ X163 @ X164 ) )
        | ~ ( in @ X167 @ X163 )
        | ~ ( X164 @ X167 )
        | ( in @ ( esk4_2 @ X163 @ X164 ) @ X163 ) )
      & ( ( ( dsetconstr @ X163 @ X164 )
          = ( setadjoin @ ( esk6_2 @ X163 @ X164 ) @ emptyset ) )
        | ~ ( in @ X167 @ X163 )
        | ~ ( X164 @ X167 )
        | ( in @ ( esk4_2 @ X163 @ X164 ) @ X163 ) )
      & ( ( in @ ( esk6_2 @ X163 @ X164 ) @ ( dsetconstr @ X163 @ X164 ) )
        | ~ ( in @ X167 @ X163 )
        | ~ ( X164 @ X167 )
        | ( in @ ( esk5_2 @ X163 @ X164 ) @ X163 ) )
      & ( ( ( dsetconstr @ X163 @ X164 )
          = ( setadjoin @ ( esk6_2 @ X163 @ X164 ) @ emptyset ) )
        | ~ ( in @ X167 @ X163 )
        | ~ ( X164 @ X167 )
        | ( in @ ( esk5_2 @ X163 @ X164 ) @ X163 ) )
      & ( ( in @ ( esk6_2 @ X163 @ X164 ) @ ( dsetconstr @ X163 @ X164 ) )
        | ~ ( in @ X167 @ X163 )
        | ~ ( X164 @ X167 )
        | ( X164 @ ( esk4_2 @ X163 @ X164 ) ) )
      & ( ( ( dsetconstr @ X163 @ X164 )
          = ( setadjoin @ ( esk6_2 @ X163 @ X164 ) @ emptyset ) )
        | ~ ( in @ X167 @ X163 )
        | ~ ( X164 @ X167 )
        | ( X164 @ ( esk4_2 @ X163 @ X164 ) ) )
      & ( ( in @ ( esk6_2 @ X163 @ X164 ) @ ( dsetconstr @ X163 @ X164 ) )
        | ~ ( in @ X167 @ X163 )
        | ~ ( X164 @ X167 )
        | ( X164 @ ( esk5_2 @ X163 @ X164 ) ) )
      & ( ( ( dsetconstr @ X163 @ X164 )
          = ( setadjoin @ ( esk6_2 @ X163 @ X164 ) @ emptyset ) )
        | ~ ( in @ X167 @ X163 )
        | ~ ( X164 @ X167 )
        | ( X164 @ ( esk5_2 @ X163 @ X164 ) ) )
      & ( ( in @ ( esk6_2 @ X163 @ X164 ) @ ( dsetconstr @ X163 @ X164 ) )
        | ~ ( in @ X167 @ X163 )
        | ~ ( X164 @ X167 )
        | ( ( esk4_2 @ X163 @ X164 )
         != ( esk5_2 @ X163 @ X164 ) ) )
      & ( ( ( dsetconstr @ X163 @ X164 )
          = ( setadjoin @ ( esk6_2 @ X163 @ X164 ) @ emptyset ) )
        | ~ ( in @ X167 @ X163 )
        | ~ ( X164 @ X167 )
        | ( ( esk4_2 @ X163 @ X164 )
         != ( esk5_2 @ X163 @ X164 ) ) )
      & singletonsuniq
      & setukpairinjL1
      & kfstsingleton
      & theprop
      & kfstpairEq
      & cartprodfstin
      & setukpairinjL2
      & setukpairinjL
      & setukpairinjR11
      & setukpairinjR12
      & setukpairinjR1
      & upairequniteq
      & setukpairinjR2
      & setukpairinjR
      & ksndsingleton
      & ksndpairEq
      & kpairsurjEq
      & cartprodsndin
      & cartprodpairmemEL
      & cartprodpairmemER
      & cartprodmempaircEq
      & cartprodfstpairEq
      & cartprodsndpairEq
      & cartprodpairsurjEq
      & dpsetconstrI
      & dpsetconstrSub
      & ( subset @ ( dpsetconstr @ X169 @ X170 @ X171 ) @ ( cartprod @ X169 @ X170 ) )
      & dpsetconstrERa
      & dpsetconstrEL1
      & dpsetconstrEL2
      & dpsetconstrER
      & funcImageSingleton
      & apProp
      & ( ( in @ ( esk7_3 @ X172 @ X173 @ X174 ) @ X172 )
        | ~ ( subset @ X174 @ ( cartprod @ X172 @ X173 ) )
        | ~ ( in @ X177 @ X172 )
        | ( in @ ( ap @ X172 @ X173 @ X174 @ X177 ) @ X173 ) )
      & ( ~ ( in @ X176
            @ ( dsetconstr @ X173
              @ ^ [Z0: $i] : ( in @ ( kpair @ ( esk7_3 @ X172 @ X173 @ X174 ) @ Z0 ) @ X174 ) ) )
        | ( ( dsetconstr @ X173
            @ ^ [Z0: $i] : ( in @ ( kpair @ ( esk7_3 @ X172 @ X173 @ X174 ) @ Z0 ) @ X174 ) )
         != ( setadjoin @ X176 @ emptyset ) )
        | ~ ( subset @ X174 @ ( cartprod @ X172 @ X173 ) )
        | ~ ( in @ X177 @ X172 )
        | ( in @ ( ap @ X172 @ X173 @ X174 @ X177 ) @ X173 ) )
      & ( ( subset @ X180 @ ( cartprod @ X178 @ X179 ) )
        | ~ ( in @ X180 @ ( funcSet @ X178 @ X179 ) ) )
      & ( ( in @ ( esk8_4 @ X178 @ X179 @ X180 @ X181 )
          @ ( dsetconstr @ X179
            @ ^ [Z0: $i] : ( in @ ( kpair @ X181 @ Z0 ) @ X180 ) ) )
        | ~ ( in @ X181 @ X178 )
        | ~ ( in @ X180 @ ( funcSet @ X178 @ X179 ) ) )
      & ( ( ( dsetconstr @ X179
            @ ^ [Z0: $i] : ( in @ ( kpair @ X181 @ Z0 ) @ X180 ) )
          = ( setadjoin @ ( esk8_4 @ X178 @ X179 @ X180 @ X181 ) @ emptyset ) )
        | ~ ( in @ X181 @ X178 )
        | ~ ( in @ X180 @ ( funcSet @ X178 @ X179 ) ) )
      & ( ~ ( in @ X185 @ ( funcSet @ X183 @ X184 ) )
        | ~ ( in @ X186 @ X183 )
        | ( in @ ( ap @ X183 @ X184 @ X185 @ X186 ) @ X184 ) )
      & ( ( in @ ( esk9_3 @ X187 @ X188 @ X189 ) @ X187 )
        | ~ ( subset @ X189 @ ( cartprod @ X187 @ X188 ) )
        | ( in @ X189 @ ( funcSet @ X187 @ X188 ) ) )
      & ( ~ ( in @ X191
            @ ( dsetconstr @ X188
              @ ^ [Z0: $i] : ( in @ ( kpair @ ( esk9_3 @ X187 @ X188 @ X189 ) @ Z0 ) @ X189 ) ) )
        | ( ( dsetconstr @ X188
            @ ^ [Z0: $i] : ( in @ ( kpair @ ( esk9_3 @ X187 @ X188 @ X189 ) @ Z0 ) @ X189 ) )
         != ( setadjoin @ X191 @ emptyset ) )
        | ~ ( subset @ X189 @ ( cartprod @ X187 @ X188 ) )
        | ( in @ X189 @ ( funcSet @ X187 @ X188 ) ) )
      & lamProp
      & ( ( subset @ ( lam @ X192 @ X193 @ X194 ) @ ( cartprod @ X192 @ X193 ) )
        | ( in @ ( esk10_3 @ X192 @ X193 @ X194 ) @ X192 ) )
      & ( ( in @ ( esk11_4 @ X192 @ X193 @ X194 @ X196 )
          @ ( dsetconstr @ X193
            @ ^ [Z0: $i] : ( in @ ( kpair @ X196 @ Z0 ) @ ( lam @ X192 @ X193 @ X194 ) ) ) )
        | ~ ( in @ X196 @ X192 )
        | ( in @ ( esk10_3 @ X192 @ X193 @ X194 ) @ X192 ) )
      & ( ( ( dsetconstr @ X193
            @ ^ [Z0: $i] : ( in @ ( kpair @ X196 @ Z0 ) @ ( lam @ X192 @ X193 @ X194 ) ) )
          = ( setadjoin @ ( esk11_4 @ X192 @ X193 @ X194 @ X196 ) @ emptyset ) )
        | ~ ( in @ X196 @ X192 )
        | ( in @ ( esk10_3 @ X192 @ X193 @ X194 ) @ X192 ) )
      & ( ( subset @ ( lam @ X192 @ X193 @ X194 ) @ ( cartprod @ X192 @ X193 ) )
        | ~ ( in @ ( X194 @ ( esk10_3 @ X192 @ X193 @ X194 ) ) @ X193 ) )
      & ( ( in @ ( esk11_4 @ X192 @ X193 @ X194 @ X196 )
          @ ( dsetconstr @ X193
            @ ^ [Z0: $i] : ( in @ ( kpair @ X196 @ Z0 ) @ ( lam @ X192 @ X193 @ X194 ) ) ) )
        | ~ ( in @ X196 @ X192 )
        | ~ ( in @ ( X194 @ ( esk10_3 @ X192 @ X193 @ X194 ) ) @ X193 ) )
      & ( ( ( dsetconstr @ X193
            @ ^ [Z0: $i] : ( in @ ( kpair @ X196 @ Z0 ) @ ( lam @ X192 @ X193 @ X194 ) ) )
          = ( setadjoin @ ( esk11_4 @ X192 @ X193 @ X194 @ X196 ) @ emptyset ) )
        | ~ ( in @ X196 @ X192 )
        | ~ ( in @ ( X194 @ ( esk10_3 @ X192 @ X193 @ X194 ) ) @ X193 ) )
      & ( ( in @ ( esk12_3 @ X198 @ X199 @ X200 ) @ X198 )
        | ( in @ ( lam @ X198 @ X199 @ X200 ) @ ( funcSet @ X198 @ X199 ) ) )
      & ( ~ ( in @ ( X200 @ ( esk12_3 @ X198 @ X199 @ X200 ) ) @ X199 )
        | ( in @ ( lam @ X198 @ X199 @ X200 ) @ ( funcSet @ X198 @ X199 ) ) )
      & ( ( in @ ( esk13_4 @ X202 @ X203 @ X204 @ X205 ) @ X202 )
        | ~ ( in @ X208 @ X204 )
        | ( X205 @ X208 )
        | ~ ( subset @ X204 @ ( cartprod @ X202 @ X203 ) ) )
      & ( ( in @ ( esk14_4 @ X202 @ X203 @ X204 @ X205 ) @ X203 )
        | ~ ( in @ X208 @ X204 )
        | ( X205 @ X208 )
        | ~ ( subset @ X204 @ ( cartprod @ X202 @ X203 ) ) )
      & ( ( in @ ( kpair @ ( esk13_4 @ X202 @ X203 @ X204 @ X205 ) @ ( esk14_4 @ X202 @ X203 @ X204 @ X205 ) ) @ X204 )
        | ~ ( in @ X208 @ X204 )
        | ( X205 @ X208 )
        | ~ ( subset @ X204 @ ( cartprod @ X202 @ X203 ) ) )
      & ( ~ ( X205 @ ( kpair @ ( esk13_4 @ X202 @ X203 @ X204 @ X205 ) @ ( esk14_4 @ X202 @ X203 @ X204 @ X205 ) ) )
        | ~ ( in @ X208 @ X204 )
        | ( X205 @ X208 )
        | ~ ( subset @ X204 @ ( cartprod @ X202 @ X203 ) ) )
      & ( subset @ esk17_0 @ ( cartprod @ esk15_0 @ esk16_0 ) )
      & ( ~ ( in @ X213 @ esk15_0 )
        | ~ ( in @ X214 @ esk16_0 )
        | ~ ( in @ ( kpair @ X213 @ X214 ) @ esk17_0 )
        | ( epred1_0 @ ( kpair @ X213 @ X214 ) ) )
      & ( in @ esk18_0 @ esk17_0 )
      & ~ ( epred1_0 @ esk18_0 ) ),
    inference(distribute,[status(thm)],[inference(fof_nnf,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(skolemize,[status(esa)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_37])])])])])]) ).

thf(c_0_39,negated_conjecture,
    ! [X1: $i > $o,X2: $i,X5: $i,X4: $i,X3: $i] :
      ( ( X1 @ X5 )
      | ~ ( X1 @ ( kpair @ ( esk13_4 @ X2 @ X3 @ X4 @ X1 ) @ ( esk14_4 @ X2 @ X3 @ X4 @ X1 ) ) )
      | ~ ( in @ X5 @ X4 )
      | ~ ( subset @ X4 @ ( cartprod @ X2 @ X3 ) ) ),
    inference(split_conjunct,[status(thm)],[c_0_38]) ).

thf(c_0_40,negated_conjecture,
    subset @ esk17_0 @ ( cartprod @ esk15_0 @ esk16_0 ),
    inference(split_conjunct,[status(thm)],[c_0_38]) ).

thf(c_0_41,negated_conjecture,
    ! [X1: $i > $o,X2: $i] :
      ( ( X1 @ X2 )
      | ~ ( X1 @ ( kpair @ ( esk13_4 @ esk15_0 @ esk16_0 @ esk17_0 @ X1 ) @ ( esk14_4 @ esk15_0 @ esk16_0 @ esk17_0 @ X1 ) ) )
      | ~ ( in @ X2 @ esk17_0 ) ),
    inference(spm,[status(thm)],[c_0_39,c_0_40]) ).

thf(c_0_42,negated_conjecture,
    in @ esk18_0 @ esk17_0,
    inference(split_conjunct,[status(thm)],[c_0_38]) ).

thf(c_0_43,negated_conjecture,
    ! [X1: $i > $o,X2: $i,X5: $i,X4: $i,X3: $i] :
      ( ( in @ ( kpair @ ( esk13_4 @ X2 @ X3 @ X4 @ X1 ) @ ( esk14_4 @ X2 @ X3 @ X4 @ X1 ) ) @ X4 )
      | ( X1 @ X5 )
      | ~ ( in @ X5 @ X4 )
      | ~ ( subset @ X4 @ ( cartprod @ X2 @ X3 ) ) ),
    inference(split_conjunct,[status(thm)],[c_0_38]) ).

thf(c_0_44,negated_conjecture,
    ! [X1: $i > $o] :
      ( ( X1 @ esk18_0 )
      | ~ ( X1 @ ( kpair @ ( esk13_4 @ esk15_0 @ esk16_0 @ esk17_0 @ X1 ) @ ( esk14_4 @ esk15_0 @ esk16_0 @ esk17_0 @ X1 ) ) ) ),
    inference(spm,[status(thm)],[c_0_41,c_0_42]) ).

thf(c_0_45,negated_conjecture,
    ! [X2: $i,X3: $i] :
      ( ( epred1_0 @ ( kpair @ X2 @ X3 ) )
      | ~ ( in @ X2 @ esk15_0 )
      | ~ ( in @ X3 @ esk16_0 )
      | ~ ( in @ ( kpair @ X2 @ X3 ) @ esk17_0 ) ),
    inference(split_conjunct,[status(thm)],[c_0_38]) ).

thf(c_0_46,negated_conjecture,
    ~ ( epred1_0 @ esk18_0 ),
    inference(split_conjunct,[status(thm)],[c_0_38]) ).

thf(c_0_47,negated_conjecture,
    ! [X1: $i > $o,X2: $i] :
      ( ( in @ ( kpair @ ( esk13_4 @ esk15_0 @ esk16_0 @ esk17_0 @ X1 ) @ ( esk14_4 @ esk15_0 @ esk16_0 @ esk17_0 @ X1 ) ) @ esk17_0 )
      | ( X1 @ X2 )
      | ~ ( in @ X2 @ esk17_0 ) ),
    inference(spm,[status(thm)],[c_0_43,c_0_40]) ).

thf(c_0_48,negated_conjecture,
    ! [X1: $i > $o,X2: $i,X5: $i,X4: $i,X3: $i] :
      ( ( in @ ( esk14_4 @ X2 @ X3 @ X4 @ X1 ) @ X3 )
      | ( X1 @ X5 )
      | ~ ( in @ X5 @ X4 )
      | ~ ( subset @ X4 @ ( cartprod @ X2 @ X3 ) ) ),
    inference(split_conjunct,[status(thm)],[c_0_38]) ).

thf(c_0_49,negated_conjecture,
    ( ~ ( in @ ( kpair @ ( esk13_4 @ esk15_0 @ esk16_0 @ esk17_0 @ epred1_0 ) @ ( esk14_4 @ esk15_0 @ esk16_0 @ esk17_0 @ epred1_0 ) ) @ esk17_0 )
    | ~ ( in @ ( esk14_4 @ esk15_0 @ esk16_0 @ esk17_0 @ epred1_0 ) @ esk16_0 )
    | ~ ( in @ ( esk13_4 @ esk15_0 @ esk16_0 @ esk17_0 @ epred1_0 ) @ esk15_0 ) ),
    inference(sr,[status(thm)],[inference(spm,[status(thm)],[c_0_44,c_0_45]),c_0_46]) ).

thf(c_0_50,negated_conjecture,
    ! [X1: $i > $o] :
      ( ( in @ ( kpair @ ( esk13_4 @ esk15_0 @ esk16_0 @ esk17_0 @ X1 ) @ ( esk14_4 @ esk15_0 @ esk16_0 @ esk17_0 @ X1 ) ) @ esk17_0 )
      | ( X1 @ esk18_0 ) ),
    inference(spm,[status(thm)],[c_0_47,c_0_42]) ).

thf(c_0_51,negated_conjecture,
    ! [X1: $i > $o,X2: $i] :
      ( ( in @ ( esk14_4 @ esk15_0 @ esk16_0 @ esk17_0 @ X1 ) @ esk16_0 )
      | ( X1 @ X2 )
      | ~ ( in @ X2 @ esk17_0 ) ),
    inference(spm,[status(thm)],[c_0_48,c_0_40]) ).

thf(c_0_52,negated_conjecture,
    ! [X1: $i > $o,X2: $i,X5: $i,X4: $i,X3: $i] :
      ( ( in @ ( esk13_4 @ X2 @ X3 @ X4 @ X1 ) @ X2 )
      | ( X1 @ X5 )
      | ~ ( in @ X5 @ X4 )
      | ~ ( subset @ X4 @ ( cartprod @ X2 @ X3 ) ) ),
    inference(split_conjunct,[status(thm)],[c_0_38]) ).

thf(c_0_53,negated_conjecture,
    ( ~ ( in @ ( esk14_4 @ esk15_0 @ esk16_0 @ esk17_0 @ epred1_0 ) @ esk16_0 )
    | ~ ( in @ ( esk13_4 @ esk15_0 @ esk16_0 @ esk17_0 @ epred1_0 ) @ esk15_0 ) ),
    inference(sr,[status(thm)],[inference(spm,[status(thm)],[c_0_49,c_0_50]),c_0_46]) ).

thf(c_0_54,negated_conjecture,
    ! [X1: $i > $o] :
      ( ( in @ ( esk14_4 @ esk15_0 @ esk16_0 @ esk17_0 @ X1 ) @ esk16_0 )
      | ( X1 @ esk18_0 ) ),
    inference(spm,[status(thm)],[c_0_51,c_0_42]) ).

thf(c_0_55,negated_conjecture,
    ! [X1: $i > $o,X2: $i] :
      ( ( in @ ( esk13_4 @ esk15_0 @ esk16_0 @ esk17_0 @ X1 ) @ esk15_0 )
      | ( X1 @ X2 )
      | ~ ( in @ X2 @ esk17_0 ) ),
    inference(spm,[status(thm)],[c_0_52,c_0_40]) ).

thf(c_0_56,negated_conjecture,
    ~ ( in @ ( esk13_4 @ esk15_0 @ esk16_0 @ esk17_0 @ epred1_0 ) @ esk15_0 ),
    inference(sr,[status(thm)],[inference(spm,[status(thm)],[c_0_53,c_0_54]),c_0_46]) ).

thf(c_0_57,negated_conjecture,
    ! [X1: $i > $o] :
      ( ( in @ ( esk13_4 @ esk15_0 @ esk16_0 @ esk17_0 @ X1 ) @ esk15_0 )
      | ( X1 @ esk18_0 ) ),
    inference(spm,[status(thm)],[c_0_55,c_0_42]) ).

thf(c_0_58,negated_conjecture,
    $false,
    inference(sr,[status(thm)],[inference(spm,[status(thm)],[c_0_56,c_0_57]),c_0_46]),
    [proof] ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12  % Problem    : SEU682^1 : TPTP v8.2.0. Released v3.7.0.
% 0.12/0.13  % Command    : run_E %s %d THM
% 0.13/0.34  % Computer : n025.cluster.edu
% 0.13/0.34  % Model    : x86_64 x86_64
% 0.13/0.34  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34  % Memory   : 8042.1875MB
% 0.13/0.34  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34  % CPULimit   : 300
% 0.13/0.34  % WCLimit    : 300
% 0.13/0.34  % DateTime   : Sun May 19 17:28:08 EDT 2024
% 0.20/0.34  % CPUTime    : 
% 0.20/0.47  Running higher-order theorem proving
% 0.20/0.47  Running: /export/starexec/sandbox2/solver/bin/eprover-ho --delete-bad-limit=2000000000 --definitional-cnf=24 -s --print-statistics -R --print-version --proof-object --auto-schedule=8 --cpu-limit=300 /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.66/0.59  # Version: 3.1.0-ho
% 0.66/0.59  # Preprocessing class: HSLMSLSSLLLCHSA.
% 0.66/0.59  # Scheduled 5 strats onto 8 cores with 300 seconds (2400 total)
% 0.66/0.59  # Starting ho_unfolding_5 with 1200s (4) cores
% 0.66/0.59  # Starting pre_casc_3 with 300s (1) cores
% 0.66/0.59  # Starting new_ho_10_cnf2 with 300s (1) cores
% 0.66/0.59  # Starting full_lambda_10 with 300s (1) cores
% 0.66/0.59  # Starting ehoh_best_nonlift_rwall with 300s (1) cores
% 0.66/0.59  # full_lambda_10 with pid 7834 completed with status 0
% 0.66/0.59  # Result found by full_lambda_10
% 0.66/0.59  # Preprocessing class: HSLMSLSSLLLCHSA.
% 0.66/0.59  # Scheduled 5 strats onto 8 cores with 300 seconds (2400 total)
% 0.66/0.59  # Starting ho_unfolding_5 with 1200s (4) cores
% 0.66/0.59  # Starting pre_casc_3 with 300s (1) cores
% 0.66/0.59  # Starting new_ho_10_cnf2 with 300s (1) cores
% 0.66/0.59  # Starting full_lambda_10 with 300s (1) cores
% 0.66/0.59  # SinE strategy is GSinE(CountFormulas,hypos,5,,5,20000,3.0,true)
% 0.66/0.59  # Search class: HGUSF-FFMM32-DHSMMFBN
% 0.66/0.59  # partial match(5): HGHSM-FMLM32-MHSMMFBN
% 0.66/0.59  # Scheduled 6 strats onto 1 cores with 300 seconds (300 total)
% 0.66/0.59  # Starting sh10 with 160s (1) cores
% 0.66/0.59  # sh10 with pid 7836 completed with status 0
% 0.66/0.59  # Result found by sh10
% 0.66/0.59  # Preprocessing class: HSLMSLSSLLLCHSA.
% 0.66/0.59  # Scheduled 5 strats onto 8 cores with 300 seconds (2400 total)
% 0.66/0.59  # Starting ho_unfolding_5 with 1200s (4) cores
% 0.66/0.59  # Starting pre_casc_3 with 300s (1) cores
% 0.66/0.59  # Starting new_ho_10_cnf2 with 300s (1) cores
% 0.66/0.59  # Starting full_lambda_10 with 300s (1) cores
% 0.66/0.59  # SinE strategy is GSinE(CountFormulas,hypos,5,,5,20000,3.0,true)
% 0.66/0.59  # Search class: HGUSF-FFMM32-DHSMMFBN
% 0.66/0.59  # partial match(5): HGHSM-FMLM32-MHSMMFBN
% 0.66/0.59  # Scheduled 6 strats onto 1 cores with 300 seconds (300 total)
% 0.66/0.59  # Starting sh10 with 160s (1) cores
% 0.66/0.59  # Preprocessing time       : 0.019 s
% 0.66/0.59  
% 0.66/0.59  # Proof found!
% 0.66/0.59  # SZS status Theorem
% 0.66/0.59  # SZS output start CNFRefutation
% See solution above
% 0.95/0.60  # Parsed axioms                        : 442
% 0.95/0.60  # Removed by relevancy pruning/SinE    : 424
% 0.95/0.60  # Initial clauses                      : 232
% 0.95/0.60  # Removed in clause preprocessing      : 0
% 0.95/0.60  # Initial clauses in saturation        : 232
% 0.95/0.60  # Processed clauses                    : 305
% 0.95/0.60  # ...of these trivial                  : 0
% 0.95/0.60  # ...subsumed                          : 16
% 0.95/0.60  # ...remaining for further processing  : 289
% 0.95/0.60  # Other redundant clauses eliminated   : 39
% 0.95/0.60  # Clauses deleted for lack of memory   : 0
% 0.95/0.60  # Backward-subsumed                    : 2
% 0.95/0.60  # Backward-rewritten                   : 0
% 0.95/0.60  # Generated clauses                    : 729
% 0.95/0.60  # ...of the previous two non-redundant : 685
% 0.95/0.60  # ...aggressively subsumed             : 0
% 0.95/0.60  # Contextual simplify-reflections      : 3
% 0.95/0.60  # Paramodulations                      : 620
% 0.95/0.60  # Factorizations                       : 0
% 0.95/0.60  # NegExts                              : 7
% 0.95/0.60  # Equation resolutions                 : 39
% 0.95/0.60  # Disequality decompositions           : 0
% 0.95/0.60  # Total rewrite steps                  : 2
% 0.95/0.60  # ...of those cached                   : 1
% 0.95/0.60  # Propositional unsat checks           : 0
% 0.95/0.60  #    Propositional check models        : 0
% 0.95/0.60  #    Propositional check unsatisfiable : 0
% 0.95/0.60  #    Propositional clauses             : 0
% 0.95/0.60  #    Propositional clauses after purity: 0
% 0.95/0.60  #    Propositional unsat core size     : 0
% 0.95/0.60  #    Propositional preprocessing time  : 0.000
% 0.95/0.60  #    Propositional encoding time       : 0.000
% 0.95/0.60  #    Propositional solver time         : 0.000
% 0.95/0.60  #    Success case prop preproc time    : 0.000
% 0.95/0.60  #    Success case prop encoding time   : 0.000
% 0.95/0.60  #    Success case prop solver time     : 0.000
% 0.95/0.60  # Current number of processed clauses  : 287
% 0.95/0.60  #    Positive orientable unit clauses  : 195
% 0.95/0.60  #    Positive unorientable unit clauses: 7
% 0.95/0.60  #    Negative unit clauses             : 2
% 0.95/0.60  #    Non-unit-clauses                  : 83
% 0.95/0.60  # Current number of unprocessed clauses: 612
% 0.95/0.60  # ...number of literals in the above   : 3027
% 0.95/0.60  # Current number of archived formulas  : 0
% 0.95/0.60  # Current number of archived clauses   : 2
% 0.95/0.60  # Clause-clause subsumption calls (NU) : 1901
% 0.95/0.60  # Rec. Clause-clause subsumption calls : 261
% 0.95/0.60  # Non-unit clause-clause subsumptions  : 5
% 0.95/0.60  # Unit Clause-clause subsumption calls : 666
% 0.95/0.60  # Rewrite failures with RHS unbound    : 0
% 0.95/0.60  # BW rewrite match attempts            : 93
% 0.95/0.60  # BW rewrite match successes           : 11
% 0.95/0.60  # Condensation attempts                : 0
% 0.95/0.60  # Condensation successes               : 0
% 0.95/0.60  # Termbank termtop insertions          : 44832
% 0.95/0.60  # Search garbage collected termcells   : 6652
% 0.95/0.60  
% 0.95/0.60  # -------------------------------------------------
% 0.95/0.60  # User time                : 0.077 s
% 0.95/0.60  # System time              : 0.010 s
% 0.95/0.60  # Total time               : 0.087 s
% 0.95/0.60  # Maximum resident set size: 3836 pages
% 0.95/0.60  
% 0.95/0.60  # -------------------------------------------------
% 0.95/0.60  # User time                : 0.087 s
% 0.95/0.60  # System time              : 0.014 s
% 0.95/0.60  # Total time               : 0.101 s
% 0.95/0.60  # Maximum resident set size: 2196 pages
% 0.95/0.60  % E---3.1 exiting
% 0.95/0.60  % E exiting
%------------------------------------------------------------------------------