%------------------------------------------------------------------------------
% File : Zipperpin---2.1.9999
% Problem : SEU669^1 : TPTP v9.2.0. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : python3 /export/starexec/sandbox/solver/bin/portfolio.lams.parallel.py %s %d /export/starexec/sandbox/tmp/tmp.QGsZ4qv9YQ true
% Computer : n031.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 : Thu Oct 2 04:55:39 PM UTC 2025
% Result : Theorem 132.32s 17.31s
% Output : Refutation 132.82s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SEU669^1 : TPTP v9.2.0. Released v3.7.0.
% 0.03/0.13 % Command : python3 /export/starexec/sandbox/solver/bin/portfolio.lams.parallel.py %s %d /export/starexec/sandbox/tmp/tmp.QGsZ4qv9YQ true
% 0.13/0.34 % Computer : n031.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 : Wed Oct 1 16:03:08 EDT 2025
% 0.13/0.34 % CPUTime :
% 0.13/0.34 % Running portfolio for 300 s
% 0.13/0.34 % File : /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.13/0.35 % Number of cores: 8
% 0.13/0.35 % Python version: Python 3.6.8
% 0.13/0.35 % Running in HO mode
% 0.56/0.67 % Total configuration time : 828
% 0.56/0.67 % Estimated wc time : 1656
% 0.56/0.67 % Estimated cpu time (8 cpus) : 207.0
% 0.57/0.73 % /export/starexec/sandbox/solver/bin/lams/40_c.s.sh running for 80s
% 0.57/0.74 % /export/starexec/sandbox/solver/bin/lams/35_full_unif4.sh running for 80s
% 0.57/0.74 % /export/starexec/sandbox/solver/bin/lams/15_e_short1.sh running for 30s
% 0.57/0.74 % /export/starexec/sandbox/solver/bin/lams/40_noforms.sh running for 90s
% 0.57/0.74 % /export/starexec/sandbox/solver/bin/lams/40_c_ic.sh running for 80s
% 0.57/0.76 % /export/starexec/sandbox/solver/bin/lams/20_acsne_simpl.sh running for 40s
% 0.57/0.77 % /export/starexec/sandbox/solver/bin/lams/40_b.comb.sh running for 70s
% 0.57/0.78 % /export/starexec/sandbox/solver/bin/lams/30_sp5.sh running for 60s
% 132.32/17.31 % Solved by lams/40_c.s.sh.
% 132.32/17.31 % done 678 iterations in 16.438s
% 132.32/17.31 % SZS status Theorem for '/export/starexec/sandbox/benchmark/theBenchmark.p'
% 132.32/17.31 % SZS output start Refutation
% 132.32/17.31 thf(setadjoinAx_type, type, setadjoinAx: $o).
% 132.32/17.31 thf(cartprodmempaircEq_type, type, cartprodmempaircEq: $o).
% 132.32/17.31 thf(notdexE_type, type, notdexE: $o).
% 132.32/17.31 thf(setunionAx_type, type, setunionAx: $o).
% 132.32/17.31 thf(setminusELneg_type, type, setminusELneg: $o).
% 132.32/17.31 thf(kpairsurjEq_type, type, kpairsurjEq: $o).
% 132.32/17.31 thf(powersetE_type, type, powersetE: $o).
% 132.32/17.31 thf(omegaSAx_type, type, omegaSAx: $o).
% 132.32/17.31 thf(exuI1_type, type, exuI1: $o).
% 132.32/17.31 thf(exuI2_type, type, exuI2: $o).
% 132.32/17.31 thf(notequalI2_type, type, notequalI2: $o).
% 132.32/17.31 thf(kpairiskpair_type, type, kpairiskpair: $o).
% 132.32/17.31 thf(upairsubunion_type, type, upairsubunion: $o).
% 132.32/17.31 thf(powerset__Cong_type, type, powerset__Cong: $o).
% 132.32/17.31 thf(zip_tseitin_0_type, type, zip_tseitin_0: $i > $i > $i > $o).
% 132.32/17.31 thf(setminusILneg_type, type, setminusILneg: $o).
% 132.32/17.31 thf(wellorderingAx_type, type, wellorderingAx: $o).
% 132.32/17.31 thf(setadjoinSub2_type, type, setadjoinSub2: $o).
% 132.32/17.31 thf(binintersectSubset5_type, type, binintersectSubset5: $o).
% 132.32/17.31 thf(upairinpowunion_type, type, upairinpowunion: $o).
% 132.32/17.31 thf(setoftrueEq_type, type, setoftrueEq: $o).
% 132.32/17.31 thf(zip_tseitin_2_type, type, zip_tseitin_2: $i > $i > $i > $o).
% 132.32/17.31 thf(emptyset__Cong_type, type, emptyset__Cong: $o).
% 132.32/17.31 thf(cartprodpairmemER_type, type, cartprodpairmemER: $o).
% 132.32/17.31 thf(breln_type, type, breln: $i > $i > $i > $o).
% 132.32/17.31 thf(sepInPowerset_type, type, sepInPowerset: $o).
% 132.32/17.31 thf(setukpairIR_type, type, setukpairIR: $o).
% 132.32/17.31 thf(kpair_type, type, kpair: $i > $i > $i).
% 132.32/17.31 thf(binunionE_type, type, binunionE: $o).
% 132.32/17.31 thf(setext_type, type, setext: $o).
% 132.32/17.31 thf(emptyinPowerset_type, type, emptyinPowerset: $o).
% 132.32/17.31 thf(ksnd_type, type, ksnd: $i > $i).
% 132.32/17.31 thf(ubforcartprodlem1_type, type, ubforcartprodlem1: $o).
% 132.32/17.31 thf(setukpairinjR12_type, type, setukpairinjR12: $o).
% 132.32/17.31 thf(descr__Cong_type, type, descr__Cong: $o).
% 132.32/17.31 thf(binintersectSubset1_type, type, binintersectSubset1: $o).
% 132.32/17.31 thf(in_type, type, in: $i > $i > $o).
% 132.32/17.31 thf(cartprodfstpairEq_type, type, cartprodfstpairEq: $o).
% 132.32/17.31 thf(quantDeMorgan1_type, type, quantDeMorgan1: $o).
% 132.32/17.31 thf(binunion_type, type, binunion: $i > $i > $i).
% 132.32/17.31 thf(setadjoinIR_type, type, setadjoinIR: $o).
% 132.32/17.31 thf(setadjoin__Cong_type, type, setadjoin__Cong: $o).
% 132.32/17.31 thf(symdiffI2_type, type, symdiffI2: $o).
% 132.32/17.31 thf(sk__560_type, type, sk__560: $i).
% 132.32/17.31 thf(upairset2IR_type, type, upairset2IR: $o).
% 132.32/17.31 thf(sk__559_type, type, sk__559: $i).
% 132.32/17.31 thf(subsetE_type, type, subsetE: $o).
% 132.32/17.31 thf(prop2set2propI_type, type, prop2set2propI: $o).
% 132.32/17.31 thf(dsetconstrEL_type, type, dsetconstrEL: $o).
% 132.32/17.31 thf(setbeta_type, type, setbeta: $o).
% 132.32/17.31 thf(eqinunit_type, type, eqinunit: $o).
% 132.32/17.31 thf(upairsetIL_type, type, upairsetIL: $o).
% 132.32/17.31 thf(powersetI1_type, type, powersetI1: $o).
% 132.32/17.31 thf(binintersectSubset3_type, type, binintersectSubset3: $o).
% 132.32/17.31 thf(zip_tseitin_4_type, type, zip_tseitin_4: $i > $i > ($i > $o) > $o).
% 132.32/17.31 thf(setukpairinjR_type, type, setukpairinjR: $o).
% 132.32/17.31 thf(cartprodfstin_type, type, cartprodfstin: $o).
% 132.32/17.31 thf(setminusIRneg_type, type, setminusIRneg: $o).
% 132.32/17.31 thf(kfst_type, type, kfst: $i > $i).
% 132.32/17.31 thf(powersetAx_type, type, powersetAx: $o).
% 132.32/17.31 thf(singletonprop_type, type, singletonprop: $o).
% 132.32/17.32 thf(binintersectSubset4_type, type, binintersectSubset4: $o).
% 132.32/17.32 thf(singletoninpowunion_type, type, singletoninpowunion: $o).
% 132.32/17.32 thf(powersetI_type, type, powersetI: $o).
% 132.32/17.32 thf(cartprodmempair_type, type, cartprodmempair: $o).
% 132.32/17.32 thf(upairsetIR_type, type, upairsetIR: $o).
% 132.32/17.32 thf(cartprodsndpairEq_type, type, cartprodsndpairEq: $o).
% 132.32/17.32 thf(dpsetconstrERa_type, type, dpsetconstrERa: $o).
% 132.32/17.32 thf(inCongP_type, type, inCongP: $o).
% 132.32/17.32 thf(setukpairinjL_type, type, setukpairinjL: $o).
% 132.32/17.32 thf(setadjoinSub_type, type, setadjoinSub: $o).
% 132.32/17.32 thf(exuE3e_type, type, exuE3e: $o).
% 132.32/17.32 thf(emptysetE_type, type, emptysetE: $o).
% 132.32/17.32 thf(upairequniteq_type, type, upairequniteq: $o).
% 132.32/17.32 thf(binintersectSubset2_type, type, binintersectSubset2: $o).
% 132.32/17.32 thf(setunionI_type, type, setunionI: $o).
% 132.32/17.32 thf(symdiff_type, type, symdiff: $i > $i > $i).
% 132.32/17.32 thf(setunion__Cong_type, type, setunion__Cong: $o).
% 132.32/17.32 thf(prop2setI_type, type, prop2setI: $o).
% 132.32/17.32 thf(nonemptyImpWitness_type, type, nonemptyImpWitness: $o).
% 132.32/17.32 thf(setminusER_type, type, setminusER: $o).
% 132.32/17.32 thf(exuE2_type, type, exuE2: $o).
% 132.32/17.32 thf(emptyset_type, type, emptyset: $i).
% 132.32/17.32 thf(setminus_type, type, setminus: $i > $i > $i).
% 132.32/17.32 thf(powersetsubset_type, type, powersetsubset: $o).
% 132.32/17.32 thf(setukpairinjR1_type, type, setukpairinjR1: $o).
% 132.32/17.32 thf(setOfPairsIsBReln_type, type, setOfPairsIsBReln: $o).
% 132.32/17.32 thf(subPowSU_type, type, subPowSU: $o).
% 132.32/17.32 thf(ex1_type, type, ex1: $i > ($i > $o) > $o).
% 132.32/17.32 thf(notequalI1_type, type, notequalI1: $o).
% 132.32/17.32 thf(notinemptyset_type, type, notinemptyset: $o).
% 132.32/17.32 thf(nonemptyI1_type, type, nonemptyI1: $o).
% 132.32/17.32 thf(subsetTrans_type, type, subsetTrans: $o).
% 132.32/17.32 thf(setukpairinjR2_type, type, setukpairinjR2: $o).
% 132.32/17.32 thf(kpairp_type, type, kpairp: $o).
% 132.32/17.32 thf(setminusI_type, type, setminusI: $o).
% 132.32/17.32 thf(binunionIL_type, type, binunionIL: $o).
% 132.32/17.32 thf(dpsetconstrI_type, type, dpsetconstrI: $o).
% 132.32/17.32 thf(setunionsingleton_type, type, setunionsingleton: $o).
% 132.32/17.32 thf(subsetE2_type, type, subsetE2: $o).
% 132.32/17.32 thf(singletonsswitch_type, type, singletonsswitch: $o).
% 132.32/17.32 thf(kfstpairEq_type, type, kfstpairEq: $o).
% 132.32/17.32 thf(ubforcartprodlem3_type, type, ubforcartprodlem3: $o).
% 132.32/17.32 thf(setextsub_type, type, setextsub: $o).
% 132.32/17.32 thf(binunionLsub_type, type, binunionLsub: $o).
% 132.32/17.32 thf(setunionsingleton1_type, type, setunionsingleton1: $o).
% 132.32/17.32 thf(binintersect_type, type, binintersect: $i > $i > $i).
% 132.32/17.32 thf(emptysetsubset_type, type, emptysetsubset: $o).
% 132.32/17.32 thf(quantDeMorgan3_type, type, quantDeMorgan3: $o).
% 132.32/17.32 thf(setunionE_type, type, setunionE: $o).
% 132.32/17.32 thf(setminusSubset1_type, type, setminusSubset1: $o).
% 132.32/17.32 thf(sk__556_type, type, sk__556: $i).
% 132.32/17.32 thf(emptyinunitempty_type, type, emptyinunitempty: $o).
% 132.32/17.32 thf(binunionIR_type, type, binunionIR: $o).
% 132.32/17.32 thf(sk__558_type, type, sk__558: $i > $i > $o).
% 132.32/17.32 thf(setadjoinE_type, type, setadjoinE: $o).
% 132.32/17.32 thf(descr_type, type, descr: ($i > $o) > $i).
% 132.32/17.32 thf(cartprodpairmemEL_type, type, cartprodpairmemEL: $o).
% 132.32/17.32 thf(singletonsuniq_type, type, singletonsuniq: $o).
% 132.32/17.32 thf(setadjoinIL_type, type, setadjoinIL: $o).
% 132.32/17.32 thf(exuE1_type, type, exuE1: $o).
% 132.32/17.32 thf(ubforcartprodlem2_type, type, ubforcartprodlem2: $o).
% 132.32/17.32 thf(binintersectRsub_type, type, binintersectRsub: $o).
% 132.32/17.32 thf(setminusLsub_type, type, setminusLsub: $o).
% 132.32/17.32 thf(prop2set_type, type, prop2set: $o > $i).
% 132.32/17.32 thf(sk__557_type, type, sk__557: $i).
% 132.32/17.32 thf(notinsingleton_type, type, notinsingleton: $o).
% 132.32/17.32 thf(eqimpsubset2_type, type, eqimpsubset2: $o).
% 132.32/17.32 thf(exuEu_type, type, exuEu: $o).
% 132.32/17.32 thf(emptysetimpfalse_type, type, emptysetimpfalse: $o).
% 132.32/17.32 thf(ex1I2_type, type, ex1I2: $o).
% 132.32/17.32 thf(upairsetE_type, type, upairsetE: $o).
% 132.32/17.32 thf(omega0Ax_type, type, omega0Ax: $o).
% 132.32/17.32 thf('#l_lift14081_type', type, '#l_lift14081': $i > $i > ($i > $i > $o) > $i > $o).
% 132.32/17.32 thf(notsubsetI_type, type, notsubsetI: $o).
% 132.32/17.32 thf(quantDeMorgan4_type, type, quantDeMorgan4: $o).
% 132.32/17.32 thf(setextAx_type, type, setextAx: $o).
% 132.32/17.32 thf(powersetE1_type, type, powersetE1: $o).
% 132.32/17.32 thf(bs114d_type, type, bs114d: $o).
% 132.32/17.32 thf(noeltsimpempty_type, type, noeltsimpempty: $o).
% 132.32/17.32 thf(setunion_type, type, setunion: $i > $i).
% 132.32/17.32 thf(iskpair_type, type, iskpair: $i > $o).
% 132.32/17.32 thf(binintersectLsub_type, type, binintersectLsub: $o).
% 132.32/17.32 thf(subsetI1_type, type, subsetI1: $o).
% 132.32/17.32 thf(symdiffE_type, type, symdiffE: $o).
% 132.32/17.32 thf(descrp_type, type, descrp: $o).
% 132.32/17.32 thf(dsetconstr__Cong_type, type, dsetconstr__Cong: $o).
% 132.32/17.32 thf(foundationAx_type, type, foundationAx: $o).
% 132.32/17.32 thf(emptysetAx_type, type, emptysetAx: $o).
% 132.32/17.32 thf(emptyI_type, type, emptyI: $o).
% 132.32/17.32 thf(setadjoinOr_type, type, setadjoinOr: $o).
% 132.32/17.32 thf(binintersectEL_type, type, binintersectEL: $o).
% 132.32/17.32 thf(emptyE1_type, type, emptyE1: $o).
% 132.32/17.32 thf(emptyInPowerset_type, type, emptyInPowerset: $o).
% 132.32/17.32 thf(vacuousDall_type, type, vacuousDall: $o).
% 132.32/17.32 thf(dsetconstr_type, type, dsetconstr: $i > ($i > $o) > $i).
% 132.32/17.32 thf(dsetconstrER_type, type, dsetconstrER: $o).
% 132.32/17.32 thf(exu_type, type, exu: ($i > $o) > $o).
% 132.32/17.32 thf(binintersectI_type, type, binintersectI: $o).
% 132.32/17.32 thf(cartprodpairin_type, type, cartprodpairin: $o).
% 132.32/17.32 thf(ex1I_type, type, ex1I: $o).
% 132.32/17.32 thf(cartprodmempair1_type, type, cartprodmempair1: $o).
% 132.32/17.32 thf(cartprodsndin_type, type, cartprodsndin: $o).
% 132.32/17.32 thf(setukpairinjR11_type, type, setukpairinjR11: $o).
% 132.32/17.32 thf(dpsetconstr_type, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 132.32/17.32 thf(singleton_type, type, singleton: $i > $o).
% 132.32/17.32 thf(symdiffI1_type, type, symdiffI1: $o).
% 132.32/17.32 thf(subsetI2_type, type, subsetI2: $o).
% 132.32/17.32 thf(dpsetconstrSub_type, type, dpsetconstrSub: $o).
% 132.32/17.32 thf(symdiffIneg1_type, type, symdiffIneg1: $o).
% 132.32/17.32 thf(theprop_type, type, theprop: $o).
% 132.32/17.32 thf(setminusERneg_type, type, setminusERneg: $o).
% 132.32/17.32 thf(symdiffIneg2_type, type, symdiffIneg2: $o).
% 132.32/17.32 thf(omega__Cong_type, type, omega__Cong: $o).
% 132.32/17.32 thf(subsetRefl_type, type, subsetRefl: $o).
% 132.32/17.32 thf(uniqinunit_type, type, uniqinunit: $o).
% 132.32/17.32 thf(zip_tseitin_3_type, type, zip_tseitin_3: $i > $i > $i > ($i > $i > $o) > $o).
% 132.32/17.32 thf(setminusSubset2_type, type, setminusSubset2: $o).
% 132.32/17.32 thf(prop2setE_type, type, prop2setE: $o).
% 132.32/17.32 thf(binunionEcases_type, type, binunionEcases: $o).
% 132.32/17.32 thf(zip_tseitin_5_type, type, zip_tseitin_5: $i > ($i > $o) > $o).
% 132.32/17.32 thf(omega_type, type, omega: $i).
% 132.32/17.32 thf(nonemptyI_type, type, nonemptyI: $o).
% 132.32/17.32 thf(disjointsetsI1_type, type, disjointsetsI1: $o).
% 132.32/17.32 thf(exuE3u_type, type, exuE3u: $o).
% 132.32/17.32 thf(setminusEL_type, type, setminusEL: $o).
% 132.32/17.32 thf(setunionsingleton2_type, type, setunionsingleton2: $o).
% 132.32/17.32 thf(sepSubset_type, type, sepSubset: $o).
% 132.32/17.32 thf(upairset2E_type, type, upairset2E: $o).
% 132.32/17.32 thf(eqimpsubset1_type, type, eqimpsubset1: $o).
% 132.32/17.32 thf(exuI3_type, type, exuI3: $o).
% 132.32/17.32 thf(notdallE_type, type, notdallE: $o).
% 132.32/17.32 thf(binintersectER_type, type, binintersectER: $o).
% 132.32/17.32 thf(ksndsingleton_type, type, ksndsingleton: $o).
% 132.32/17.32 thf(powerset_type, type, powerset: $i > $i).
% 132.32/17.32 thf(kfstsingleton_type, type, kfstsingleton: $o).
% 132.32/17.32 thf(dsetconstrI_type, type, dsetconstrI: $o).
% 132.32/17.32 thf(cartprodpairsurjEq_type, type, cartprodpairsurjEq: $o).
% 132.32/17.32 thf(nonempty_type, type, nonempty: $i > $o).
% 132.32/17.32 thf(set2prop_type, type, set2prop: $i > $o).
% 132.32/17.32 thf(zip_tseitin_1_type, type, zip_tseitin_1: $i > $i > $i > $o).
% 132.32/17.32 thf(setadjoin_type, type, setadjoin: $i > $i > $i).
% 132.32/17.32 thf(exu__Cong_type, type, exu__Cong: $o).
% 132.32/17.32 thf(ksndpairEq_type, type, ksndpairEq: $o).
% 132.32/17.32 thf(quantDeMorgan2_type, type, quantDeMorgan2: $o).
% 132.32/17.32 thf(ex1E1_type, type, ex1E1: $o).
% 132.32/17.32 thf(secondinupair_type, type, secondinupair: $o).
% 132.32/17.32 thf(singletonsubset_type, type, singletonsubset: $o).
% 132.32/17.32 thf(singletoninpowerset_type, type, singletoninpowerset: $o).
% 132.32/17.32 thf(subsetemptysetimpeq_type, type, subsetemptysetimpeq: $o).
% 132.32/17.32 thf(replAx_type, type, replAx: $o).
% 132.32/17.32 thf(nonemptyE1_type, type, nonemptyE1: $o).
% 132.32/17.32 thf(omegaIndAx_type, type, omegaIndAx: $o).
% 132.32/17.32 thf(setukpairIL_type, type, setukpairIL: $o).
% 132.32/17.32 thf(setunionE2_type, type, setunionE2: $o).
% 132.32/17.32 thf(inPowerset_type, type, inPowerset: $o).
% 132.32/17.32 thf(cartprod_type, type, cartprod: $i > $i > $i).
% 132.32/17.32 thf(subset_type, type, subset: $i > $i > $o).
% 132.32/17.32 thf(in__Cong_type, type, in__Cong: $o).
% 132.32/17.32 thf(setukpairinjL2_type, type, setukpairinjL2: $o).
% 132.32/17.32 thf(binunionRsub_type, type, binunionRsub: $o).
% 132.32/17.32 thf(subset2powerset_type, type, subset2powerset: $o).
% 132.32/17.32 thf(setukpairinjL1_type, type, setukpairinjL1: $o).
% 132.32/17.32 thf(dpsetconstrERa, axiom, dpsetconstrERa =
% 132.32/17.32 (![A:$i,B:$i,Xphi:( $i > $i > $o ),Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) =>
% 132.32/17.32 ( ![Xy:$i]:
% 132.32/17.32 ( ( in @ Xy @ B ) =>
% 132.32/17.32 ( ( in @
% 132.32/17.32 ( kpair @ Xx @ Xy ) @
% 132.32/17.32 ( dpsetconstr @ A @ B @ ( ^[Xz:$i,Xu:$i]: ( Xphi @ Xz @ Xu ) ) ) ) =>
% 132.32/17.32 ( Xphi @ Xx @ Xy ) ) ) ) ))).
% 132.32/17.32 thf('0', plain,
% 132.32/17.32 (( dpsetconstrERa ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:( $i > $i > $o ),X10:$i]:
% 132.32/17.32 ( ( in @ X10 @ X4 ) =>
% 132.32/17.32 ( ![X12:$i]:
% 132.32/17.32 ( ( in @ X12 @ X6 ) =>
% 132.32/17.32 ( ( in @
% 132.32/17.32 ( kpair @ X10 @ X12 ) @
% 132.32/17.32 ( dpsetconstr @
% 132.32/17.32 X4 @ X6 @ ( ^[V_1:$i,V_2:$i]: ( X8 @ V_1 @ V_2 ) ) ) ) =>
% 132.32/17.32 ( X8 @ X10 @ X12 ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setOfPairsIsBReln, axiom, setOfPairsIsBReln =
% 132.32/17.32 (![A:$i,B:$i,Xphi:( $i > $i > $o )]:
% 132.32/17.32 ( breln @
% 132.32/17.32 A @ B @ ( dpsetconstr @ A @ B @ ( ^[Xx:$i,Xy:$i]: ( Xphi @ Xx @ Xy ) ) ) ))).
% 132.32/17.32 thf('1', plain,
% 132.32/17.32 (( setOfPairsIsBReln ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:( $i > $i > $o )]:
% 132.32/17.32 ( breln @
% 132.32/17.32 X4 @ X6 @
% 132.32/17.32 ( dpsetconstr @ X4 @ X6 @ ( ^[V_1:$i,V_2:$i]: ( X8 @ V_1 @ V_2 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(dpsetconstrSub, axiom, dpsetconstrSub =
% 132.32/17.32 (![A:$i,B:$i,Xphi:( $i > $i > $o )]:
% 132.32/17.32 ( subset @
% 132.32/17.32 ( dpsetconstr @ A @ B @ ( ^[Xx:$i,Xy:$i]: ( Xphi @ Xx @ Xy ) ) ) @
% 132.32/17.32 ( cartprod @ A @ B ) ))).
% 132.32/17.32 thf('2', plain,
% 132.32/17.32 (( dpsetconstrSub ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:( $i > $i > $o )]:
% 132.32/17.32 ( subset @
% 132.32/17.32 ( dpsetconstr @ X4 @ X6 @ ( ^[V_1:$i,V_2:$i]: ( X8 @ V_1 @ V_2 ) ) ) @
% 132.32/17.32 ( cartprod @ X4 @ X6 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(dpsetconstrI, axiom, dpsetconstrI =
% 132.32/17.32 (![A:$i,B:$i,Xphi:( $i > $i > $o ),Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) =>
% 132.32/17.32 ( ![Xy:$i]:
% 132.32/17.32 ( ( in @ Xy @ B ) =>
% 132.32/17.32 ( ( Xphi @ Xx @ Xy ) =>
% 132.32/17.32 ( in @
% 132.32/17.32 ( kpair @ Xx @ Xy ) @
% 132.32/17.32 ( dpsetconstr @ A @ B @ ( ^[Xz:$i,Xu:$i]: ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ))).
% 132.32/17.32 thf('3', plain,
% 132.32/17.32 (( dpsetconstrI ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:( $i > $i > $o ),X10:$i]:
% 132.32/17.32 ( ( in @ X10 @ X4 ) =>
% 132.32/17.32 ( ![X12:$i]:
% 132.32/17.32 ( ( in @ X12 @ X6 ) =>
% 132.32/17.32 ( ( X8 @ X10 @ X12 ) =>
% 132.32/17.32 ( in @
% 132.32/17.32 ( kpair @ X10 @ X12 ) @
% 132.32/17.32 ( dpsetconstr @
% 132.32/17.32 X4 @ X6 @ ( ^[V_1:$i,V_2:$i]: ( X8 @ V_1 @ V_2 ) ) ) ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(dpsetconstr, axiom,(( dpsetconstr ) =
% 132.32/17.32 (^[A:$i,B:$i,Xphi:( $i > $i > $o )]:
% 132.32/17.32 ( dsetconstr @
% 132.32/17.32 ( cartprod @ A @ B ) @
% 132.32/17.32 ( ^[Xu:$i]:
% 132.32/17.32 ( ?[Xx:$i]:
% 132.32/17.32 ( ( ?[Xy:$i]:
% 132.32/17.32 ( ( ( Xu ) = ( kpair @ Xx @ Xy ) ) & ( Xphi @ Xx @ Xy ) &
% 132.32/17.32 ( in @ Xy @ B ) ) ) &
% 132.32/17.32 ( in @ Xx @ A ) ) ) ) )))).
% 132.32/17.32 thf('4', plain,
% 132.32/17.32 (( dpsetconstr ) =
% 132.32/17.32 ( ^[A:$i,B:$i,Xphi:( $i > $i > $o )]:
% 132.32/17.32 ( dsetconstr @
% 132.32/17.32 ( cartprod @ A @ B ) @
% 132.32/17.32 ( ^[Xu:$i]:
% 132.32/17.32 ( ?[Xx:$i]:
% 132.32/17.32 ( ( ?[Xy:$i]:
% 132.32/17.32 ( ( ( Xu ) = ( kpair @ Xx @ Xy ) ) & ( Xphi @ Xx @ Xy ) &
% 132.32/17.32 ( in @ Xy @ B ) ) ) &
% 132.32/17.32 ( in @ Xx @ A ) ) ) ) ) )),
% 132.32/17.32 inference('simplify_rw_rule', [status(thm)], [dpsetconstr])).
% 132.32/17.32 thf('5', plain,
% 132.32/17.32 (( dpsetconstr ) =
% 132.32/17.32 ( ^[V_1:$i,V_2:$i,V_3:( $i > $i > $o )]:
% 132.32/17.32 ( dsetconstr @
% 132.32/17.32 ( cartprod @ V_1 @ V_2 ) @
% 132.32/17.32 ( ^[V_4:$i]:
% 132.32/17.32 ( ?[X4:$i]:
% 132.32/17.32 ( ( ?[X6:$i]:
% 132.32/17.32 ( ( ( V_4 ) = ( kpair @ X4 @ X6 ) ) & ( V_3 @ X4 @ X6 ) &
% 132.32/17.32 ( in @ X6 @ V_2 ) ) ) &
% 132.32/17.32 ( in @ X4 @ V_1 ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(breln, axiom,(( breln ) =
% 132.32/17.32 (^[A:$i,B:$i,C:$i]: ( subset @ C @ ( cartprod @ A @ B ) )))).
% 132.32/17.32 thf('6', plain,
% 132.32/17.32 (( breln ) = ( ^[A:$i,B:$i,C:$i]: ( subset @ C @ ( cartprod @ A @ B ) ) )),
% 132.32/17.32 inference('simplify_rw_rule', [status(thm)], [breln])).
% 132.32/17.32 thf('7', plain,
% 132.32/17.32 (( breln ) =
% 132.32/17.32 ( ^[V_1:$i,V_2:$i,V_3:$i]: ( subset @ V_3 @ ( cartprod @ V_1 @ V_2 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(cartprodpairsurjEq, axiom, cartprodpairsurjEq =
% 132.32/17.32 (![A:$i,B:$i,Xu:$i]:
% 132.32/17.32 ( ( in @ Xu @ ( cartprod @ A @ B ) ) =>
% 132.32/17.32 ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) ) = ( Xu ) ) ))).
% 132.32/17.32 thf('8', plain,
% 132.32/17.32 (( cartprodpairsurjEq ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ ( cartprod @ X4 @ X6 ) ) =>
% 132.32/17.32 ( ( kpair @ ( kfst @ X8 ) @ ( ksnd @ X8 ) ) = ( X8 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(cartprodsndpairEq, axiom, cartprodsndpairEq =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) =>
% 132.32/17.32 ( ![Xy:$i]:
% 132.32/17.32 ( ( in @ Xy @ B ) => ( ( ksnd @ ( kpair @ Xx @ Xy ) ) = ( Xy ) ) ) ) ))).
% 132.32/17.32 thf('9', plain,
% 132.32/17.32 (( cartprodsndpairEq ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X4 ) =>
% 132.32/17.32 ( ![X10:$i]:
% 132.32/17.32 ( ( in @ X10 @ X6 ) =>
% 132.32/17.32 ( ( ksnd @ ( kpair @ X8 @ X10 ) ) = ( X10 ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(cartprodfstpairEq, axiom, cartprodfstpairEq =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) =>
% 132.32/17.32 ( ![Xy:$i]:
% 132.32/17.32 ( ( in @ Xy @ B ) => ( ( kfst @ ( kpair @ Xx @ Xy ) ) = ( Xx ) ) ) ) ))).
% 132.32/17.32 thf('10', plain,
% 132.32/17.32 (( cartprodfstpairEq ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X4 ) =>
% 132.32/17.32 ( ![X10:$i]:
% 132.32/17.32 ( ( in @ X10 @ X6 ) =>
% 132.32/17.32 ( ( kfst @ ( kpair @ X8 @ X10 ) ) = ( X8 ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(cartprodmempaircEq, axiom, cartprodmempaircEq =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) =>
% 132.32/17.32 ( ![Xy:$i]:
% 132.32/17.32 ( ( in @ Xy @ B ) => ( ( kpair @ Xx @ Xy ) = ( kpair @ Xx @ Xy ) ) ) ) ))).
% 132.32/17.32 thf('11', plain,
% 132.32/17.32 (( cartprodmempaircEq ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X4 ) =>
% 132.32/17.32 ( ![X10:$i]:
% 132.32/17.32 ( ( in @ X10 @ X6 ) =>
% 132.32/17.32 ( ( kpair @ X8 @ X10 ) = ( kpair @ X8 @ X10 ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(cartprodpairmemER, axiom, cartprodpairmemER =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i,Xy:$i]:
% 132.32/17.32 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) => ( in @ Xy @ B ) ))).
% 132.32/17.32 thf('12', plain,
% 132.32/17.32 (( cartprodpairmemER ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i,X10:$i]:
% 132.32/17.32 ( ( in @ ( kpair @ X8 @ X10 ) @ ( cartprod @ X4 @ X6 ) ) =>
% 132.32/17.32 ( in @ X10 @ X6 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(cartprodpairmemEL, axiom, cartprodpairmemEL =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i,Xy:$i]:
% 132.32/17.32 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) => ( in @ Xx @ A ) ))).
% 132.32/17.32 thf('13', plain,
% 132.32/17.32 (( cartprodpairmemEL ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i,X10:$i]:
% 132.32/17.32 ( ( in @ ( kpair @ X8 @ X10 ) @ ( cartprod @ X4 @ X6 ) ) =>
% 132.32/17.32 ( in @ X8 @ X4 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(cartprodsndin, axiom, cartprodsndin =
% 132.32/17.32 (![A:$i,B:$i,Xu:$i]:
% 132.32/17.32 ( ( in @ Xu @ ( cartprod @ A @ B ) ) => ( in @ ( ksnd @ Xu ) @ B ) ))).
% 132.32/17.32 thf('14', plain,
% 132.32/17.32 (( cartprodsndin ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ ( cartprod @ X4 @ X6 ) ) => ( in @ ( ksnd @ X8 ) @ X6 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(kpairsurjEq, axiom, kpairsurjEq =
% 132.32/17.32 (![Xu:$i]:
% 132.32/17.32 ( ( iskpair @ Xu ) =>
% 132.32/17.32 ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) ) = ( Xu ) ) ))).
% 132.32/17.32 thf('15', plain,
% 132.32/17.32 (( kpairsurjEq ) =
% 132.32/17.32 ( ![X4:$i]:
% 132.32/17.32 ( ( iskpair @ X4 ) =>
% 132.32/17.32 ( ( kpair @ ( kfst @ X4 ) @ ( ksnd @ X4 ) ) = ( X4 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(ksndpairEq, axiom, ksndpairEq =
% 132.32/17.32 (![Xx:$i,Xy:$i]: ( ( ksnd @ ( kpair @ Xx @ Xy ) ) = ( Xy ) ))).
% 132.32/17.32 thf('16', plain,
% 132.32/17.32 (( ksndpairEq ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]: ( ( ksnd @ ( kpair @ X4 @ X6 ) ) = ( X6 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(ksndsingleton, axiom, ksndsingleton =
% 132.32/17.32 (![Xu:$i]:
% 132.32/17.32 ( ( iskpair @ Xu ) =>
% 132.32/17.32 ( singleton @
% 132.32/17.32 ( dsetconstr @
% 132.32/17.32 ( setunion @ Xu ) @
% 132.32/17.32 ( ^[Xx:$i]: ( ( Xu ) = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ))).
% 132.32/17.32 thf('17', plain,
% 132.32/17.32 (( ksndsingleton ) =
% 132.32/17.32 ( ![X4:$i]:
% 132.32/17.32 ( ( iskpair @ X4 ) =>
% 132.32/17.32 ( singleton @
% 132.32/17.32 ( dsetconstr @
% 132.32/17.32 ( setunion @ X4 ) @
% 132.32/17.32 ( ^[V_1:$i]: ( ( X4 ) = ( kpair @ ( kfst @ X4 ) @ V_1 ) ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setukpairinjR, axiom, setukpairinjR =
% 132.32/17.32 (![Xx:$i,Xy:$i,Xz:$i,Xu:$i]:
% 132.32/17.32 ( ( ( kpair @ Xx @ Xy ) = ( kpair @ Xz @ Xu ) ) => ( ( Xy ) = ( Xu ) ) ))).
% 132.32/17.32 thf('18', plain,
% 132.32/17.32 (( setukpairinjR ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i,X10:$i]:
% 132.32/17.32 ( ( ( kpair @ X4 @ X6 ) = ( kpair @ X8 @ X10 ) ) =>
% 132.32/17.32 ( ( X6 ) = ( X10 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setukpairinjR2, axiom, setukpairinjR2 =
% 132.32/17.32 (![Xx:$i,Xy:$i,Xz:$i,Xu:$i]:
% 132.32/17.32 ( ( ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xx @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) =
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xz @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) ) =>
% 132.32/17.32 ( ( Xy ) = ( Xu ) ) ))).
% 132.32/17.32 thf('19', plain,
% 132.32/17.32 (( setukpairinjR2 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i,X10:$i]:
% 132.32/17.32 ( ( ( setadjoin @
% 132.32/17.32 ( setadjoin @ X4 @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) @ emptyset ) ) =
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ X8 @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ X8 @ ( setadjoin @ X10 @ emptyset ) ) @ emptyset ) ) ) =>
% 132.32/17.32 ( ( X6 ) = ( X10 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(upairequniteq, axiom, upairequniteq =
% 132.32/17.32 (![Xx:$i,Xy:$i,Xz:$i]:
% 132.32/17.32 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) =
% 132.32/17.32 ( setadjoin @ Xz @ emptyset ) ) =>
% 132.32/17.32 ( ( Xx ) = ( Xy ) ) ))).
% 132.32/17.32 thf('20', plain,
% 132.32/17.32 (( upairequniteq ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) =
% 132.32/17.32 ( setadjoin @ X8 @ emptyset ) ) =>
% 132.32/17.32 ( ( X4 ) = ( X6 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setukpairinjR1, axiom, setukpairinjR1 =
% 132.32/17.32 (![Xx:$i,Xy:$i,Xz:$i,Xu:$i]:
% 132.32/17.32 ( ( ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xx @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) =
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xz @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) ) =>
% 132.32/17.32 ( ( ( Xz ) = ( Xu ) ) => ( ( Xy ) = ( Xu ) ) ) ))).
% 132.32/17.32 thf('21', plain,
% 132.32/17.32 (( setukpairinjR1 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i,X10:$i]:
% 132.32/17.32 ( ( ( setadjoin @
% 132.32/17.32 ( setadjoin @ X4 @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) @ emptyset ) ) =
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ X8 @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ X8 @ ( setadjoin @ X10 @ emptyset ) ) @ emptyset ) ) ) =>
% 132.32/17.32 ( ( ( X8 ) = ( X10 ) ) => ( ( X6 ) = ( X10 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setukpairinjR12, axiom, setukpairinjR12 =
% 132.32/17.32 (![Xx:$i,Xy:$i]:
% 132.32/17.32 ( ( ( Xx ) = ( Xy ) ) =>
% 132.32/17.32 ( ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xx @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) =
% 132.32/17.32 ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ))).
% 132.32/17.32 thf('22', plain,
% 132.32/17.32 (( setukpairinjR12 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( ( X4 ) = ( X6 ) ) =>
% 132.32/17.32 ( ( setadjoin @
% 132.32/17.32 ( setadjoin @ X4 @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) @ emptyset ) ) =
% 132.32/17.32 ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ emptyset ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setukpairinjR11, axiom, setukpairinjR11 =
% 132.32/17.32 (![Xx:$i,Xy:$i]:
% 132.32/17.32 ( ( ( Xx ) = ( Xy ) ) =>
% 132.32/17.32 ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) =
% 132.32/17.32 ( setadjoin @ Xx @ emptyset ) ) ))).
% 132.32/17.32 thf('23', plain,
% 132.32/17.32 (( setukpairinjR11 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( ( X4 ) = ( X6 ) ) =>
% 132.32/17.32 ( ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) =
% 132.32/17.32 ( setadjoin @ X4 @ emptyset ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setukpairinjL, axiom, setukpairinjL =
% 132.32/17.32 (![Xx:$i,Xy:$i,Xz:$i,Xu:$i]:
% 132.32/17.32 ( ( ( kpair @ Xx @ Xy ) = ( kpair @ Xz @ Xu ) ) => ( ( Xx ) = ( Xz ) ) ))).
% 132.32/17.32 thf('24', plain,
% 132.32/17.32 (( setukpairinjL ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i,X10:$i]:
% 132.32/17.32 ( ( ( kpair @ X4 @ X6 ) = ( kpair @ X8 @ X10 ) ) =>
% 132.32/17.32 ( ( X4 ) = ( X8 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setukpairinjL2, axiom, setukpairinjL2 =
% 132.32/17.32 (![Xx:$i,Xy:$i,Xz:$i,Xu:$i]:
% 132.32/17.32 ( ( ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xx @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) =
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xz @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) ) =>
% 132.32/17.32 ( ( Xx ) = ( Xz ) ) ))).
% 132.32/17.32 thf('25', plain,
% 132.32/17.32 (( setukpairinjL2 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i,X10:$i]:
% 132.32/17.32 ( ( ( setadjoin @
% 132.32/17.32 ( setadjoin @ X4 @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) @ emptyset ) ) =
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ X8 @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ X8 @ ( setadjoin @ X10 @ emptyset ) ) @ emptyset ) ) ) =>
% 132.32/17.32 ( ( X4 ) = ( X8 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(cartprodfstin, axiom, cartprodfstin =
% 132.32/17.32 (![A:$i,B:$i,Xu:$i]:
% 132.32/17.32 ( ( in @ Xu @ ( cartprod @ A @ B ) ) => ( in @ ( kfst @ Xu ) @ A ) ))).
% 132.32/17.32 thf('26', plain,
% 132.32/17.32 (( cartprodfstin ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ ( cartprod @ X4 @ X6 ) ) => ( in @ ( kfst @ X8 ) @ X4 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(kfstpairEq, axiom, kfstpairEq =
% 132.32/17.32 (![Xx:$i,Xy:$i]: ( ( kfst @ ( kpair @ Xx @ Xy ) ) = ( Xx ) ))).
% 132.32/17.32 thf('27', plain,
% 132.32/17.32 (( kfstpairEq ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]: ( ( kfst @ ( kpair @ X4 @ X6 ) ) = ( X4 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(theprop, axiom, theprop =
% 132.32/17.32 (![X:$i]: ( ( singleton @ X ) => ( in @ ( setunion @ X ) @ X ) ))).
% 132.32/17.32 thf('28', plain,
% 132.32/17.32 (( theprop ) =
% 132.32/17.32 ( ![X4:$i]: ( ( singleton @ X4 ) => ( in @ ( setunion @ X4 ) @ X4 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(kfstsingleton, axiom, kfstsingleton =
% 132.32/17.32 (![Xu:$i]:
% 132.32/17.32 ( ( iskpair @ Xu ) =>
% 132.32/17.32 ( singleton @
% 132.32/17.32 ( dsetconstr @
% 132.32/17.32 ( setunion @ Xu ) @
% 132.32/17.32 ( ^[Xx:$i]: ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ))).
% 132.32/17.32 thf('29', plain,
% 132.32/17.32 (( kfstsingleton ) =
% 132.32/17.32 ( ![X4:$i]:
% 132.32/17.32 ( ( iskpair @ X4 ) =>
% 132.32/17.32 ( singleton @
% 132.32/17.32 ( dsetconstr @
% 132.32/17.32 ( setunion @ X4 ) @
% 132.32/17.32 ( ^[V_1:$i]: ( in @ ( setadjoin @ V_1 @ emptyset ) @ X4 ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setukpairinjL1, axiom, setukpairinjL1 =
% 132.32/17.32 (![Xx:$i,Xy:$i,Xz:$i]:
% 132.32/17.32 ( ( in @
% 132.32/17.32 ( setadjoin @ Xz @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xx @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) =>
% 132.32/17.32 ( ( Xx ) = ( Xz ) ) ))).
% 132.32/17.32 thf('30', plain,
% 132.32/17.32 (( setukpairinjL1 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @
% 132.32/17.32 ( setadjoin @ X8 @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ X4 @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) @ emptyset ) ) ) =>
% 132.32/17.32 ( ( X4 ) = ( X8 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(singletonsuniq, axiom, singletonsuniq =
% 132.32/17.32 (![Xx:$i,Xy:$i]:
% 132.32/17.32 ( ( ( setadjoin @ Xx @ emptyset ) = ( setadjoin @ Xy @ emptyset ) ) =>
% 132.32/17.32 ( ( Xx ) = ( Xy ) ) ))).
% 132.32/17.32 thf('31', plain,
% 132.32/17.32 (( singletonsuniq ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( ( setadjoin @ X4 @ emptyset ) = ( setadjoin @ X6 @ emptyset ) ) =>
% 132.32/17.32 ( ( X4 ) = ( X6 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(ex1I2, axiom, ex1I2 =
% 132.32/17.32 (![A:$i,Xphi:( $i > $o )]:
% 132.32/17.32 ( ( ![Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) =>
% 132.32/17.32 ( ![Xy:$i]:
% 132.32/17.32 ( ( in @ Xy @ A ) =>
% 132.32/17.32 ( ( Xphi @ Xx ) => ( ( Xphi @ Xy ) => ( ( Xx ) = ( Xy ) ) ) ) ) ) ) ) =>
% 132.32/17.32 ( ( ?[Xx:$i]: ( ( Xphi @ Xx ) & ( in @ Xx @ A ) ) ) =>
% 132.32/17.32 ( ex1 @ A @ ( ^[Xx:$i]: ( Xphi @ Xx ) ) ) ) ))).
% 132.32/17.32 thf('32', plain,
% 132.32/17.32 (( ex1I2 ) =
% 132.32/17.32 ( ![X4:$i,X6:( $i > $o )]:
% 132.32/17.32 ( ( ![X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X4 ) =>
% 132.32/17.32 ( ![X10:$i]:
% 132.32/17.32 ( ( in @ X10 @ X4 ) =>
% 132.32/17.32 ( ( X6 @ X8 ) => ( ( X6 @ X10 ) => ( ( X8 ) = ( X10 ) ) ) ) ) ) ) ) =>
% 132.32/17.32 ( ( ?[X12:$i]: ( ( X6 @ X12 ) & ( in @ X12 @ X4 ) ) ) =>
% 132.32/17.32 ( ex1 @ X4 @ ( ^[V_1:$i]: ( X6 @ V_1 ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(ex1I, axiom, ex1I =
% 132.32/17.32 (![A:$i,Xphi:( $i > $o ),Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) =>
% 132.32/17.32 ( ( Xphi @ Xx ) =>
% 132.32/17.32 ( ( ![Xy:$i]:
% 132.32/17.32 ( ( in @ Xy @ A ) => ( ( Xphi @ Xy ) => ( ( Xy ) = ( Xx ) ) ) ) ) =>
% 132.32/17.32 ( ex1 @ A @ ( ^[Xy:$i]: ( Xphi @ Xy ) ) ) ) ) ))).
% 132.32/17.32 thf('33', plain,
% 132.32/17.32 (( ex1I ) =
% 132.32/17.32 ( ![X4:$i,X6:( $i > $o ),X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X4 ) =>
% 132.32/17.32 ( ( X6 @ X8 ) =>
% 132.32/17.32 ( ( ![X10:$i]:
% 132.32/17.32 ( ( in @ X10 @ X4 ) =>
% 132.32/17.32 ( ( X6 @ X10 ) => ( ( X10 ) = ( X8 ) ) ) ) ) =>
% 132.32/17.32 ( ex1 @ X4 @ ( ^[V_1:$i]: ( X6 @ V_1 ) ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(ex1E1, axiom, ex1E1 =
% 132.32/17.32 (![A:$i,Xphi:( $i > $o )]:
% 132.32/17.32 ( ( ex1 @ A @ ( ^[Xx:$i]: ( Xphi @ Xx ) ) ) =>
% 132.32/17.32 ( ?[Xx:$i]: ( ( Xphi @ Xx ) & ( in @ Xx @ A ) ) ) ))).
% 132.32/17.32 thf('34', plain,
% 132.32/17.32 (( ex1E1 ) =
% 132.32/17.32 ( ![X4:$i,X6:( $i > $o )]:
% 132.32/17.32 ( ( ex1 @ X4 @ ( ^[V_1:$i]: ( X6 @ V_1 ) ) ) =>
% 132.32/17.32 ( ?[X8:$i]: ( ( X6 @ X8 ) & ( in @ X8 @ X4 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(ex1, axiom,(( ex1 ) =
% 132.32/17.32 (^[A:$i,Xphi:( $i > $o )]:
% 132.32/17.32 ( singleton @ ( dsetconstr @ A @ ( ^[Xx:$i]: ( Xphi @ Xx ) ) ) )))).
% 132.32/17.32 thf(singleton, axiom,(( singleton ) =
% 132.32/17.32 (^[A:$i]:
% 132.32/17.32 ( ?[Xx:$i]:
% 132.32/17.32 ( ( ( A ) = ( setadjoin @ Xx @ emptyset ) ) & ( in @ Xx @ A ) ) )))).
% 132.32/17.32 thf('35', plain,
% 132.32/17.32 (( singleton ) =
% 132.32/17.32 ( ^[A:$i]:
% 132.32/17.32 ( ?[Xx:$i]:
% 132.32/17.32 ( ( ( A ) = ( setadjoin @ Xx @ emptyset ) ) & ( in @ Xx @ A ) ) ) )),
% 132.32/17.32 inference('simplify_rw_rule', [status(thm)], [singleton])).
% 132.32/17.32 thf('36', plain,
% 132.32/17.32 (( singleton ) =
% 132.32/17.32 ( ^[V_1:$i]:
% 132.32/17.32 ( ?[X4:$i]:
% 132.32/17.32 ( ( ( V_1 ) = ( setadjoin @ X4 @ emptyset ) ) & ( in @ X4 @ V_1 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf('37', plain,
% 132.32/17.32 (( ex1 ) =
% 132.32/17.32 ( ^[A:$i,Xphi:( $i > $o )]:
% 132.32/17.32 ( singleton @ ( dsetconstr @ A @ ( ^[Xx:$i]: ( Xphi @ Xx ) ) ) ) )),
% 132.32/17.32 inference('simplify_rw_rule', [status(thm)], [ex1, '36'])).
% 132.32/17.32 thf('38', plain,
% 132.32/17.32 (( ex1 ) =
% 132.32/17.32 ( ^[V_1:$i,V_2:( $i > $o )]:
% 132.32/17.32 ( singleton @ ( dsetconstr @ V_1 @ ( ^[V_3:$i]: ( V_2 @ V_3 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(singletonprop, axiom, singletonprop =
% 132.32/17.32 (![A:$i,Xphi:( $i > $o )]:
% 132.32/17.32 ( ( ![Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) =>
% 132.32/17.32 ( ![Xy:$i]:
% 132.32/17.32 ( ( in @ Xy @ A ) =>
% 132.32/17.32 ( ( Xphi @ Xx ) => ( ( Xphi @ Xy ) => ( ( Xx ) = ( Xy ) ) ) ) ) ) ) ) =>
% 132.32/17.32 ( ( ?[Xx:$i]: ( ( Xphi @ Xx ) & ( in @ Xx @ A ) ) ) =>
% 132.32/17.32 ( singleton @ ( dsetconstr @ A @ ( ^[Xx:$i]: ( Xphi @ Xx ) ) ) ) ) ))).
% 132.32/17.32 thf('39', plain,
% 132.32/17.32 (( singletonprop ) =
% 132.32/17.32 ( ![X4:$i,X6:( $i > $o )]:
% 132.32/17.32 ( ( ![X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X4 ) =>
% 132.32/17.32 ( ![X10:$i]:
% 132.32/17.32 ( ( in @ X10 @ X4 ) =>
% 132.32/17.32 ( ( X6 @ X8 ) => ( ( X6 @ X10 ) => ( ( X8 ) = ( X10 ) ) ) ) ) ) ) ) =>
% 132.32/17.32 ( ( ?[X12:$i]: ( ( X6 @ X12 ) & ( in @ X12 @ X4 ) ) ) =>
% 132.32/17.32 ( singleton @ ( dsetconstr @ X4 @ ( ^[V_1:$i]: ( X6 @ V_1 ) ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setunionsingleton, axiom, setunionsingleton =
% 132.32/17.32 (![Xx:$i]: ( ( setunion @ ( setadjoin @ Xx @ emptyset ) ) = ( Xx ) ))).
% 132.32/17.32 thf('40', plain,
% 132.32/17.32 (( setunionsingleton ) =
% 132.32/17.32 ( ![X4:$i]: ( ( setunion @ ( setadjoin @ X4 @ emptyset ) ) = ( X4 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setunionsingleton2, axiom, setunionsingleton2 =
% 132.32/17.32 (![A:$i]: ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ))).
% 132.32/17.32 thf('41', plain,
% 132.32/17.32 (( setunionsingleton2 ) =
% 132.32/17.32 ( ![X4:$i]:
% 132.32/17.32 ( subset @ X4 @ ( setunion @ ( setadjoin @ X4 @ emptyset ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setunionsingleton1, axiom, setunionsingleton1 =
% 132.32/17.32 (![A:$i]: ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ))).
% 132.32/17.32 thf('42', plain,
% 132.32/17.32 (( setunionsingleton1 ) =
% 132.32/17.32 ( ![X4:$i]:
% 132.32/17.32 ( subset @ ( setunion @ ( setadjoin @ X4 @ emptyset ) ) @ X4 ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setunionE2, axiom, setunionE2 =
% 132.32/17.32 (![A:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ ( setunion @ A ) ) =>
% 132.32/17.32 ( ?[X:$i]: ( ( in @ Xx @ X ) & ( in @ X @ A ) ) ) ))).
% 132.32/17.32 thf('43', plain,
% 132.32/17.32 (( setunionE2 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( in @ X6 @ ( setunion @ X4 ) ) =>
% 132.32/17.32 ( ?[X8:$i]: ( ( in @ X6 @ X8 ) & ( in @ X8 @ X4 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(cartprodmempair, axiom, cartprodmempair =
% 132.32/17.32 (![A:$i,B:$i,Xu:$i]:
% 132.32/17.32 ( ( in @ Xu @ ( cartprod @ A @ B ) ) => ( iskpair @ Xu ) ))).
% 132.32/17.32 thf('44', plain,
% 132.32/17.32 (( cartprodmempair ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ ( cartprod @ X4 @ X6 ) ) => ( iskpair @ X8 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(cartprodmempair1, axiom, cartprodmempair1 =
% 132.32/17.32 (![A:$i,B:$i,Xu:$i]:
% 132.32/17.32 ( ( in @ Xu @ ( cartprod @ A @ B ) ) =>
% 132.32/17.32 ( ?[Xx:$i]:
% 132.32/17.32 ( ( ?[Xy:$i]: ( ( ( Xu ) = ( kpair @ Xx @ Xy ) ) & ( in @ Xy @ B ) ) ) &
% 132.32/17.32 ( in @ Xx @ A ) ) ) ))).
% 132.32/17.32 thf('45', plain,
% 132.32/17.32 (( cartprodmempair1 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ ( cartprod @ X4 @ X6 ) ) =>
% 132.32/17.32 ( ?[X10:$i]:
% 132.32/17.32 ( ( ?[X12:$i]:
% 132.32/17.32 ( ( ( X8 ) = ( kpair @ X10 @ X12 ) ) & ( in @ X12 @ X6 ) ) ) &
% 132.32/17.32 ( in @ X10 @ X4 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(cartprodpairin, axiom, cartprodpairin =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) =>
% 132.32/17.32 ( ![Xy:$i]:
% 132.32/17.32 ( ( in @ Xy @ B ) =>
% 132.32/17.32 ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ))).
% 132.32/17.32 thf('46', plain,
% 132.32/17.32 (( cartprodpairin ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X4 ) =>
% 132.32/17.32 ( ![X10:$i]:
% 132.32/17.32 ( ( in @ X10 @ X6 ) =>
% 132.32/17.32 ( in @ ( kpair @ X8 @ X10 ) @ ( cartprod @ X4 @ X6 ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(ubforcartprodlem3, axiom, ubforcartprodlem3 =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) =>
% 132.32/17.32 ( ![Xy:$i]:
% 132.32/17.32 ( ( in @ Xy @ B ) =>
% 132.32/17.32 ( in @
% 132.32/17.32 ( kpair @ Xx @ Xy ) @
% 132.32/17.32 ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ))).
% 132.32/17.32 thf('47', plain,
% 132.32/17.32 (( ubforcartprodlem3 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X4 ) =>
% 132.32/17.32 ( ![X10:$i]:
% 132.32/17.32 ( ( in @ X10 @ X6 ) =>
% 132.32/17.32 ( in @
% 132.32/17.32 ( kpair @ X8 @ X10 ) @
% 132.32/17.32 ( powerset @ ( powerset @ ( binunion @ X4 @ X6 ) ) ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(ubforcartprodlem2, axiom, ubforcartprodlem2 =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) =>
% 132.32/17.32 ( ![Xy:$i]:
% 132.32/17.32 ( ( in @ Xy @ B ) =>
% 132.32/17.32 ( in @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xx @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @
% 132.32/17.32 ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ))).
% 132.32/17.32 thf('48', plain,
% 132.32/17.32 (( ubforcartprodlem2 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X4 ) =>
% 132.32/17.32 ( ![X10:$i]:
% 132.32/17.32 ( ( in @ X10 @ X6 ) =>
% 132.32/17.32 ( in @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ X8 @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ X8 @ ( setadjoin @ X10 @ emptyset ) ) @
% 132.32/17.32 emptyset ) ) @
% 132.32/17.32 ( powerset @ ( powerset @ ( binunion @ X4 @ X6 ) ) ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(ubforcartprodlem1, axiom, ubforcartprodlem1 =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) =>
% 132.32/17.32 ( ![Xy:$i]:
% 132.32/17.32 ( ( in @ Xy @ B ) =>
% 132.32/17.32 ( subset @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xx @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @
% 132.32/17.32 ( powerset @ ( binunion @ A @ B ) ) ) ) ) ))).
% 132.32/17.32 thf('49', plain,
% 132.32/17.32 (( ubforcartprodlem1 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X4 ) =>
% 132.32/17.32 ( ![X10:$i]:
% 132.32/17.32 ( ( in @ X10 @ X6 ) =>
% 132.32/17.32 ( subset @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ X8 @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ X8 @ ( setadjoin @ X10 @ emptyset ) ) @
% 132.32/17.32 emptyset ) ) @
% 132.32/17.32 ( powerset @ ( binunion @ X4 @ X6 ) ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(upairinpowunion, axiom, upairinpowunion =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) =>
% 132.32/17.32 ( ![Xy:$i]:
% 132.32/17.32 ( ( in @ Xy @ B ) =>
% 132.32/17.32 ( in @
% 132.32/17.32 ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @
% 132.32/17.32 ( powerset @ ( binunion @ A @ B ) ) ) ) ) ))).
% 132.32/17.32 thf('50', plain,
% 132.32/17.32 (( upairinpowunion ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X4 ) =>
% 132.32/17.32 ( ![X10:$i]:
% 132.32/17.32 ( ( in @ X10 @ X6 ) =>
% 132.32/17.32 ( in @
% 132.32/17.32 ( setadjoin @ X8 @ ( setadjoin @ X10 @ emptyset ) ) @
% 132.32/17.32 ( powerset @ ( binunion @ X4 @ X6 ) ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(upairsubunion, axiom, upairsubunion =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) =>
% 132.32/17.32 ( ![Xy:$i]:
% 132.32/17.32 ( ( in @ Xy @ B ) =>
% 132.32/17.32 ( subset @
% 132.32/17.32 ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @
% 132.32/17.32 ( binunion @ A @ B ) ) ) ) ))).
% 132.32/17.32 thf('51', plain,
% 132.32/17.32 (( upairsubunion ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X4 ) =>
% 132.32/17.32 ( ![X10:$i]:
% 132.32/17.32 ( ( in @ X10 @ X6 ) =>
% 132.32/17.32 ( subset @
% 132.32/17.32 ( setadjoin @ X8 @ ( setadjoin @ X10 @ emptyset ) ) @
% 132.32/17.32 ( binunion @ X4 @ X6 ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(upairset2E, axiom, upairset2E =
% 132.32/17.32 (![Xx:$i,Xy:$i,Xz:$i]:
% 132.32/17.32 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) =>
% 132.32/17.32 ( ( ( Xz ) = ( Xx ) ) | ( ( Xz ) = ( Xy ) ) ) ))).
% 132.32/17.32 thf('52', plain,
% 132.32/17.32 (( upairset2E ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) ) =>
% 132.32/17.32 ( ( ( X8 ) = ( X4 ) ) | ( ( X8 ) = ( X6 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(singletoninpowunion, axiom, singletoninpowunion =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) =>
% 132.32/17.32 ( in @
% 132.32/17.32 ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ))).
% 132.32/17.32 thf('53', plain,
% 132.32/17.32 (( singletoninpowunion ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X4 ) =>
% 132.32/17.32 ( in @
% 132.32/17.32 ( setadjoin @ X8 @ emptyset ) @
% 132.32/17.32 ( powerset @ ( binunion @ X4 @ X6 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(singletoninpowerset, axiom, singletoninpowerset =
% 132.32/17.32 (![A:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) =>
% 132.32/17.32 ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ))).
% 132.32/17.32 thf('54', plain,
% 132.32/17.32 (( singletoninpowerset ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( in @ X6 @ X4 ) =>
% 132.32/17.32 ( in @ ( setadjoin @ X6 @ emptyset ) @ ( powerset @ X4 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(singletonsubset, axiom, singletonsubset =
% 132.32/17.32 (![A:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ))).
% 132.32/17.32 thf('55', plain,
% 132.32/17.32 (( singletonsubset ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( in @ X6 @ X4 ) => ( subset @ ( setadjoin @ X6 @ emptyset ) @ X4 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(kpairp, axiom, kpairp =
% 132.32/17.32 (![Xx:$i,Xy:$i]: ( iskpair @ ( kpair @ Xx @ Xy ) ))).
% 132.32/17.32 thf('56', plain,
% 132.32/17.32 (( kpairp ) = ( ![X4:$i,X6:$i]: ( iskpair @ ( kpair @ X4 @ X6 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(kpairiskpair, axiom, kpairiskpair =
% 132.32/17.32 (![Xx:$i,Xy:$i]:
% 132.32/17.32 ( iskpair @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xx @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ))).
% 132.32/17.32 thf('57', plain,
% 132.32/17.32 (( kpairiskpair ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( iskpair @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ X4 @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) @ emptyset ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setukpairIR, axiom, setukpairIR =
% 132.32/17.32 (![Xx:$i,Xy:$i]:
% 132.32/17.32 ( in @
% 132.32/17.32 Xy @
% 132.32/17.32 ( setunion @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xx @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ))).
% 132.32/17.32 thf('58', plain,
% 132.32/17.32 (( setukpairIR ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( in @
% 132.32/17.32 X6 @
% 132.32/17.32 ( setunion @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ X4 @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) @ emptyset ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setukpairIL, axiom, setukpairIL =
% 132.32/17.32 (![Xx:$i,Xy:$i]:
% 132.32/17.32 ( in @
% 132.32/17.32 Xx @
% 132.32/17.32 ( setunion @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xx @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ))).
% 132.32/17.32 thf('59', plain,
% 132.32/17.32 (( setukpairIL ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( in @
% 132.32/17.32 X4 @
% 132.32/17.32 ( setunion @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ X4 @ emptyset ) @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) @ emptyset ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(secondinupair, axiom, secondinupair =
% 132.32/17.32 (![Xx:$i,Xy:$i]:
% 132.32/17.32 ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ))).
% 132.32/17.32 thf('60', plain,
% 132.32/17.32 (( secondinupair ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( in @ X6 @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(symdiffIneg2, axiom, symdiffIneg2 =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( ~( in @ Xx @ A ) ) =>
% 132.32/17.32 ( ( ~( in @ Xx @ B ) ) => ( ~( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ))).
% 132.32/17.32 thf('61', plain,
% 132.32/17.32 (( symdiffIneg2 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( ~( in @ X8 @ X4 ) ) =>
% 132.32/17.32 ( ( ~( in @ X8 @ X6 ) ) => ( ~( in @ X8 @ ( symdiff @ X4 @ X6 ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(symdiffIneg1, axiom, symdiffIneg1 =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) =>
% 132.32/17.32 ( ( in @ Xx @ B ) => ( ~( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ))).
% 132.32/17.32 thf('62', plain,
% 132.32/17.32 (( symdiffIneg1 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X4 ) =>
% 132.32/17.32 ( ( in @ X8 @ X6 ) => ( ~( in @ X8 @ ( symdiff @ X4 @ X6 ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(symdiffI2, axiom, symdiffI2 =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( ~( in @ Xx @ A ) ) =>
% 132.32/17.32 ( ( in @ Xx @ B ) => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ))).
% 132.32/17.32 thf('63', plain,
% 132.32/17.32 (( symdiffI2 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( ~( in @ X8 @ X4 ) ) =>
% 132.32/17.32 ( ( in @ X8 @ X6 ) => ( in @ X8 @ ( symdiff @ X4 @ X6 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(symdiffI1, axiom, symdiffI1 =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) =>
% 132.32/17.32 ( ( ~( in @ Xx @ B ) ) => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ))).
% 132.32/17.32 thf('64', plain,
% 132.32/17.32 (( symdiffI1 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X4 ) =>
% 132.32/17.32 ( ( ~( in @ X8 @ X6 ) ) => ( in @ X8 @ ( symdiff @ X4 @ X6 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(symdiffE, axiom, symdiffE =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ ( symdiff @ A @ B ) ) =>
% 132.32/17.32 ( ![Xphi:$o]:
% 132.32/17.32 ( ( ( in @ Xx @ A ) => ( ( ~( in @ Xx @ B ) ) => ( Xphi ) ) ) =>
% 132.32/17.32 ( ( ( ~( in @ Xx @ A ) ) => ( ( in @ Xx @ B ) => ( Xphi ) ) ) =>
% 132.32/17.32 ( Xphi ) ) ) ) ))).
% 132.32/17.32 thf('65', plain,
% 132.32/17.32 (( symdiffE ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ ( symdiff @ X4 @ X6 ) ) =>
% 132.32/17.32 ( ![X10:$o]:
% 132.32/17.32 ( ( ( in @ X8 @ X4 ) => ( ( ~( in @ X8 @ X6 ) ) => ( X10 ) ) ) =>
% 132.32/17.32 ( ( ( ~( in @ X8 @ X4 ) ) => ( ( in @ X8 @ X6 ) => ( X10 ) ) ) =>
% 132.32/17.32 ( X10 ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setminusSubset1, axiom, setminusSubset1 =
% 132.32/17.32 (![A:$i,B:$i]:
% 132.32/17.32 ( ( ( setminus @ A @ B ) = ( emptyset ) ) => ( subset @ A @ B ) ))).
% 132.32/17.32 thf('66', plain,
% 132.32/17.32 (( setminusSubset1 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( ( setminus @ X4 @ X6 ) = ( emptyset ) ) => ( subset @ X4 @ X6 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setminusLsub, axiom, setminusLsub =
% 132.32/17.32 (![A:$i,B:$i]: ( subset @ ( setminus @ A @ B ) @ A ))).
% 132.32/17.32 thf('67', plain,
% 132.32/17.32 (( setminusLsub ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]: ( subset @ ( setminus @ X4 @ X6 ) @ X4 ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setminusIRneg, axiom, setminusIRneg =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ B ) => ( ~( in @ Xx @ ( setminus @ A @ B ) ) ) ))).
% 132.32/17.32 thf('68', plain,
% 132.32/17.32 (( setminusIRneg ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X6 ) => ( ~( in @ X8 @ ( setminus @ X4 @ X6 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setminusILneg, axiom, setminusILneg =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( ~( in @ Xx @ A ) ) => ( ~( in @ Xx @ ( setminus @ A @ B ) ) ) ))).
% 132.32/17.32 thf('69', plain,
% 132.32/17.32 (( setminusILneg ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( ~( in @ X8 @ X4 ) ) => ( ~( in @ X8 @ ( setminus @ X4 @ X6 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setminusELneg, axiom, setminusELneg =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( ~( in @ Xx @ ( setminus @ A @ B ) ) ) =>
% 132.32/17.32 ( ( ~( in @ Xx @ B ) ) => ( ~( in @ Xx @ A ) ) ) ))).
% 132.32/17.32 thf('70', plain,
% 132.32/17.32 (( setminusELneg ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( ~( in @ X8 @ ( setminus @ X4 @ X6 ) ) ) =>
% 132.32/17.32 ( ( ~( in @ X8 @ X6 ) ) => ( ~( in @ X8 @ X4 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setminusERneg, axiom, setminusERneg =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( ~( in @ Xx @ ( setminus @ A @ B ) ) ) =>
% 132.32/17.32 ( ( in @ Xx @ A ) => ( in @ Xx @ B ) ) ))).
% 132.32/17.32 thf('71', plain,
% 132.32/17.32 (( setminusERneg ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( ~( in @ X8 @ ( setminus @ X4 @ X6 ) ) ) =>
% 132.32/17.32 ( ( in @ X8 @ X4 ) => ( in @ X8 @ X6 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setminusSubset2, axiom, setminusSubset2 =
% 132.32/17.32 (![A:$i,B:$i]:
% 132.32/17.32 ( ( subset @ A @ B ) => ( ( setminus @ A @ B ) = ( emptyset ) ) ))).
% 132.32/17.32 thf('72', plain,
% 132.32/17.32 (( setminusSubset2 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( subset @ X4 @ X6 ) => ( ( setminus @ X4 @ X6 ) = ( emptyset ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setminusER, axiom, setminusER =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ ( setminus @ A @ B ) ) => ( ~( in @ Xx @ B ) ) ))).
% 132.32/17.32 thf('73', plain,
% 132.32/17.32 (( setminusER ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ ( setminus @ X4 @ X6 ) ) => ( ~( in @ X8 @ X6 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setminusEL, axiom, setminusEL =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ ( setminus @ A @ B ) ) => ( in @ Xx @ A ) ))).
% 132.32/17.32 thf('74', plain,
% 132.32/17.32 (( setminusEL ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ ( setminus @ X4 @ X6 ) ) => ( in @ X8 @ X4 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setminusI, axiom, setminusI =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) =>
% 132.32/17.32 ( ( ~( in @ Xx @ B ) ) => ( in @ Xx @ ( setminus @ A @ B ) ) ) ))).
% 132.32/17.32 thf('75', plain,
% 132.32/17.32 (( setminusI ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X4 ) =>
% 132.32/17.32 ( ( ~( in @ X8 @ X6 ) ) => ( in @ X8 @ ( setminus @ X4 @ X6 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(bs114d, axiom, bs114d =
% 132.32/17.32 (![A:$i,B:$i,C:$i]:
% 132.32/17.32 ( ( binintersect @ A @ ( binunion @ B @ C ) ) =
% 132.32/17.32 ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ))).
% 132.32/17.32 thf('76', plain,
% 132.32/17.32 (( bs114d ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( binintersect @ X4 @ ( binunion @ X6 @ X8 ) ) =
% 132.32/17.32 ( binunion @ ( binintersect @ X4 @ X6 ) @ ( binintersect @ X4 @ X8 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(binintersectSubset1, axiom, binintersectSubset1 =
% 132.32/17.32 (![A:$i,B:$i]:
% 132.32/17.32 ( ( ( binintersect @ A @ B ) = ( A ) ) => ( subset @ A @ B ) ))).
% 132.32/17.32 thf('77', plain,
% 132.32/17.32 (( binintersectSubset1 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( ( binintersect @ X4 @ X6 ) = ( X4 ) ) => ( subset @ X4 @ X6 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(binintersectSubset4, axiom, binintersectSubset4 =
% 132.32/17.32 (![A:$i,B:$i]:
% 132.32/17.32 ( ( subset @ B @ A ) => ( ( binintersect @ A @ B ) = ( B ) ) ))).
% 132.32/17.32 thf('78', plain,
% 132.32/17.32 (( binintersectSubset4 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( subset @ X6 @ X4 ) => ( ( binintersect @ X4 @ X6 ) = ( X6 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(binintersectRsub, axiom, binintersectRsub =
% 132.32/17.32 (![A:$i,B:$i]: ( subset @ ( binintersect @ A @ B ) @ B ))).
% 132.32/17.32 thf('79', plain,
% 132.32/17.32 (( binintersectRsub ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]: ( subset @ ( binintersect @ X4 @ X6 ) @ X6 ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(disjointsetsI1, axiom, disjointsetsI1 =
% 132.32/17.32 (![A:$i,B:$i]:
% 132.32/17.32 ( ( ~( ?[Xx:$i]: ( ( in @ Xx @ B ) & ( in @ Xx @ A ) ) ) ) =>
% 132.32/17.32 ( ( binintersect @ A @ B ) = ( emptyset ) ) ))).
% 132.32/17.32 thf('80', plain,
% 132.32/17.32 (( disjointsetsI1 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( ~( ?[X8:$i]: ( ( in @ X8 @ X6 ) & ( in @ X8 @ X4 ) ) ) ) =>
% 132.32/17.32 ( ( binintersect @ X4 @ X6 ) = ( emptyset ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(binintersectER, axiom, binintersectER =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ ( binintersect @ A @ B ) ) => ( in @ Xx @ B ) ))).
% 132.32/17.32 thf('81', plain,
% 132.32/17.32 (( binintersectER ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ ( binintersect @ X4 @ X6 ) ) => ( in @ X8 @ X6 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(binintersectSubset3, axiom, binintersectSubset3 =
% 132.32/17.32 (![A:$i,B:$i]:
% 132.32/17.32 ( ( ( binintersect @ A @ B ) = ( B ) ) => ( subset @ B @ A ) ))).
% 132.32/17.32 thf('82', plain,
% 132.32/17.32 (( binintersectSubset3 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( ( binintersect @ X4 @ X6 ) = ( X6 ) ) => ( subset @ X6 @ X4 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(binintersectSubset2, axiom, binintersectSubset2 =
% 132.32/17.32 (![A:$i,B:$i]:
% 132.32/17.32 ( ( subset @ A @ B ) => ( ( binintersect @ A @ B ) = ( A ) ) ))).
% 132.32/17.32 thf('83', plain,
% 132.32/17.32 (( binintersectSubset2 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( subset @ X4 @ X6 ) => ( ( binintersect @ X4 @ X6 ) = ( X4 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(binintersectLsub, axiom, binintersectLsub =
% 132.32/17.32 (![A:$i,B:$i]: ( subset @ ( binintersect @ A @ B ) @ A ))).
% 132.32/17.32 thf('84', plain,
% 132.32/17.32 (( binintersectLsub ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]: ( subset @ ( binintersect @ X4 @ X6 ) @ X4 ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(binintersectEL, axiom, binintersectEL =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ ( binintersect @ A @ B ) ) => ( in @ Xx @ A ) ))).
% 132.32/17.32 thf('85', plain,
% 132.32/17.32 (( binintersectEL ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ ( binintersect @ X4 @ X6 ) ) => ( in @ X8 @ X4 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(binintersectSubset5, axiom, binintersectSubset5 =
% 132.32/17.32 (![A:$i,B:$i,C:$i]:
% 132.32/17.32 ( ( subset @ C @ A ) =>
% 132.32/17.32 ( ( subset @ C @ B ) => ( subset @ C @ ( binintersect @ A @ B ) ) ) ))).
% 132.32/17.32 thf('86', plain,
% 132.32/17.32 (( binintersectSubset5 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( subset @ X8 @ X4 ) =>
% 132.32/17.32 ( ( subset @ X8 @ X6 ) =>
% 132.32/17.32 ( subset @ X8 @ ( binintersect @ X4 @ X6 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(binintersectI, axiom, binintersectI =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) =>
% 132.32/17.32 ( ( in @ Xx @ B ) => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ))).
% 132.32/17.32 thf('87', plain,
% 132.32/17.32 (( binintersectI ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X4 ) =>
% 132.32/17.32 ( ( in @ X8 @ X6 ) => ( in @ X8 @ ( binintersect @ X4 @ X6 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(binunionRsub, axiom, binunionRsub =
% 132.32/17.32 (![A:$i,B:$i]: ( subset @ B @ ( binunion @ A @ B ) ))).
% 132.32/17.32 thf('88', plain,
% 132.32/17.32 (( binunionRsub ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]: ( subset @ X6 @ ( binunion @ X4 @ X6 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(binunionLsub, axiom, binunionLsub =
% 132.32/17.32 (![A:$i,B:$i]: ( subset @ A @ ( binunion @ A @ B ) ))).
% 132.32/17.32 thf('89', plain,
% 132.32/17.32 (( binunionLsub ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]: ( subset @ X4 @ ( binunion @ X4 @ X6 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(binunionE, axiom, binunionE =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ ( binunion @ A @ B ) ) =>
% 132.32/17.32 ( ( in @ Xx @ A ) | ( in @ Xx @ B ) ) ))).
% 132.32/17.32 thf('90', plain,
% 132.32/17.32 (( binunionE ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ ( binunion @ X4 @ X6 ) ) =>
% 132.32/17.32 ( ( in @ X8 @ X4 ) | ( in @ X8 @ X6 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(binunionEcases, axiom, binunionEcases =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i,Xphi:$o]:
% 132.32/17.32 ( ( in @ Xx @ ( binunion @ A @ B ) ) =>
% 132.32/17.32 ( ( ( in @ Xx @ A ) => ( Xphi ) ) =>
% 132.32/17.32 ( ( ( in @ Xx @ B ) => ( Xphi ) ) => ( Xphi ) ) ) ))).
% 132.32/17.32 thf('91', plain,
% 132.32/17.32 (( binunionEcases ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i,X10:$o]:
% 132.32/17.32 ( ( in @ X8 @ ( binunion @ X4 @ X6 ) ) =>
% 132.32/17.32 ( ( ( in @ X8 @ X4 ) => ( X10 ) ) =>
% 132.32/17.32 ( ( ( in @ X8 @ X6 ) => ( X10 ) ) => ( X10 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(binunionIR, axiom, binunionIR =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ B ) => ( in @ Xx @ ( binunion @ A @ B ) ) ))).
% 132.32/17.32 thf('92', plain,
% 132.32/17.32 (( binunionIR ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X6 ) => ( in @ X8 @ ( binunion @ X4 @ X6 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(upairset2IR, axiom, upairset2IR =
% 132.32/17.32 (![Xx:$i,Xy:$i]:
% 132.32/17.32 ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ))).
% 132.32/17.32 thf('93', plain,
% 132.32/17.32 (( upairset2IR ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( in @ X6 @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(binunionIL, axiom, binunionIL =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) => ( in @ Xx @ ( binunion @ A @ B ) ) ))).
% 132.32/17.32 thf('94', plain,
% 132.32/17.32 (( binunionIL ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X4 ) => ( in @ X8 @ ( binunion @ X4 @ X6 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(sepSubset, axiom, sepSubset =
% 132.32/17.32 (![A:$i,Xphi:( $i > $o )]:
% 132.32/17.32 ( subset @ ( dsetconstr @ A @ ( ^[Xx:$i]: ( Xphi @ Xx ) ) ) @ A ))).
% 132.32/17.32 thf('95', plain,
% 132.32/17.32 (( sepSubset ) =
% 132.32/17.32 ( ![X4:$i,X6:( $i > $o )]:
% 132.32/17.32 ( subset @ ( dsetconstr @ X4 @ ( ^[V_1:$i]: ( X6 @ V_1 ) ) ) @ X4 ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(sepInPowerset, axiom, sepInPowerset =
% 132.32/17.32 (![A:$i,Xphi:( $i > $o )]:
% 132.32/17.32 ( in @ ( dsetconstr @ A @ ( ^[Xx:$i]: ( Xphi @ Xx ) ) ) @ ( powerset @ A ) ))).
% 132.32/17.32 thf('96', plain,
% 132.32/17.32 (( sepInPowerset ) =
% 132.32/17.32 ( ![X4:$i,X6:( $i > $o )]:
% 132.32/17.32 ( in @
% 132.32/17.32 ( dsetconstr @ X4 @ ( ^[V_1:$i]: ( X6 @ V_1 ) ) ) @
% 132.32/17.32 ( powerset @ X4 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(powersetsubset, axiom, powersetsubset =
% 132.32/17.32 (![A:$i,B:$i]:
% 132.32/17.32 ( ( subset @ A @ B ) => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ))).
% 132.32/17.32 thf('97', plain,
% 132.32/17.32 (( powersetsubset ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( subset @ X4 @ X6 ) =>
% 132.32/17.32 ( subset @ ( powerset @ X4 ) @ ( powerset @ X6 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(inPowerset, axiom, inPowerset = (![A:$i]: ( in @ A @ ( powerset @ A ) ))).
% 132.32/17.32 thf('98', plain,
% 132.32/17.32 (( inPowerset ) = ( ![X4:$i]: ( in @ X4 @ ( powerset @ X4 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(powersetE1, axiom, powersetE1 =
% 132.32/17.32 (![A:$i,B:$i]: ( ( in @ B @ ( powerset @ A ) ) => ( subset @ B @ A ) ))).
% 132.32/17.32 thf('99', plain,
% 132.32/17.32 (( powersetE1 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( in @ X6 @ ( powerset @ X4 ) ) => ( subset @ X6 @ X4 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(powersetI1, axiom, powersetI1 =
% 132.32/17.32 (![A:$i,B:$i]: ( ( subset @ B @ A ) => ( in @ B @ ( powerset @ A ) ) ))).
% 132.32/17.32 thf('100', plain,
% 132.32/17.32 (( powersetI1 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( subset @ X6 @ X4 ) => ( in @ X6 @ ( powerset @ X4 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(subsetemptysetimpeq, axiom, subsetemptysetimpeq =
% 132.32/17.32 (![A:$i]: ( ( subset @ A @ emptyset ) => ( ( A ) = ( emptyset ) ) ))).
% 132.32/17.32 thf('101', plain,
% 132.32/17.32 (( subsetemptysetimpeq ) =
% 132.32/17.32 ( ![X4:$i]: ( ( subset @ X4 @ emptyset ) => ( ( X4 ) = ( emptyset ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setextsub, axiom, setextsub =
% 132.32/17.32 (![A:$i,B:$i]:
% 132.32/17.32 ( ( subset @ A @ B ) => ( ( subset @ B @ A ) => ( ( A ) = ( B ) ) ) ))).
% 132.32/17.32 thf('102', plain,
% 132.32/17.32 (( setextsub ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( subset @ X4 @ X6 ) =>
% 132.32/17.32 ( ( subset @ X6 @ X4 ) => ( ( X4 ) = ( X6 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(subset2powerset, axiom, subset2powerset =
% 132.32/17.32 (![A:$i,B:$i]: ( ( subset @ A @ B ) => ( in @ A @ ( powerset @ B ) ) ))).
% 132.32/17.32 thf('103', plain,
% 132.32/17.32 (( subset2powerset ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( subset @ X4 @ X6 ) => ( in @ X4 @ ( powerset @ X6 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setadjoinSub2, axiom, setadjoinSub2 =
% 132.32/17.32 (![A:$i,Xx:$i,B:$i]:
% 132.32/17.32 ( ( subset @ A @ B ) => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ))).
% 132.32/17.32 thf('104', plain,
% 132.32/17.32 (( setadjoinSub2 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( subset @ X4 @ X8 ) => ( subset @ X4 @ ( setadjoin @ X6 @ X8 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setadjoinSub, axiom, setadjoinSub =
% 132.32/17.32 (![Xx:$i,A:$i]: ( subset @ A @ ( setadjoin @ Xx @ A ) ))).
% 132.32/17.32 thf('105', plain,
% 132.32/17.32 (( setadjoinSub ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]: ( subset @ X6 @ ( setadjoin @ X4 @ X6 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(subsetTrans, axiom, subsetTrans =
% 132.32/17.32 (![A:$i,B:$i,C:$i]:
% 132.32/17.32 ( ( subset @ A @ B ) => ( ( subset @ B @ C ) => ( subset @ A @ C ) ) ))).
% 132.32/17.32 thf('106', plain,
% 132.32/17.32 (( subsetTrans ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( subset @ X4 @ X6 ) =>
% 132.32/17.32 ( ( subset @ X6 @ X8 ) => ( subset @ X4 @ X8 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(subsetRefl, axiom, subsetRefl = (![A:$i]: ( subset @ A @ A ))).
% 132.32/17.32 thf('107', plain, (( subsetRefl ) = ( ![X4:$i]: ( subset @ X4 @ X4 ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(notequalI2, axiom, notequalI2 =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) => ( ( ~( in @ Xx @ B ) ) => ( ( A ) != ( B ) ) ) ))).
% 132.32/17.32 thf('108', plain,
% 132.32/17.32 (( notequalI2 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X4 ) =>
% 132.32/17.32 ( ( ~( in @ X8 @ X6 ) ) => ( ( X4 ) != ( X6 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(notequalI1, axiom, notequalI1 =
% 132.32/17.32 (![A:$i,B:$i]: ( ( ~( subset @ A @ B ) ) => ( ( A ) != ( B ) ) ))).
% 132.32/17.32 thf('109', plain,
% 132.32/17.32 (( notequalI1 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]: ( ( ~( subset @ X4 @ X6 ) ) => ( ( X4 ) != ( X6 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(notsubsetI, axiom, notsubsetI =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) => ( ( ~( in @ Xx @ B ) ) => ( ~( subset @ A @ B ) ) ) ))).
% 132.32/17.32 thf('110', plain,
% 132.32/17.32 (( notsubsetI ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X4 ) =>
% 132.32/17.32 ( ( ~( in @ X8 @ X6 ) ) => ( ~( subset @ X4 @ X6 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(subsetE2, axiom, subsetE2 =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( subset @ A @ B ) => ( ( ~( in @ Xx @ B ) ) => ( ~( in @ Xx @ A ) ) ) ))).
% 132.32/17.32 thf('111', plain,
% 132.32/17.32 (( subsetE2 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( subset @ X4 @ X6 ) =>
% 132.32/17.32 ( ( ~( in @ X8 @ X6 ) ) => ( ~( in @ X8 @ X4 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(subsetE, axiom, subsetE =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( subset @ A @ B ) => ( ( in @ Xx @ A ) => ( in @ Xx @ B ) ) ))).
% 132.32/17.32 thf('112', plain,
% 132.32/17.32 (( subsetE ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( subset @ X4 @ X6 ) => ( ( in @ X8 @ X4 ) => ( in @ X8 @ X6 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(emptysetsubset, axiom, emptysetsubset =
% 132.32/17.32 (![A:$i]: ( subset @ emptyset @ A ))).
% 132.32/17.32 thf('113', plain,
% 132.32/17.32 (( emptysetsubset ) = ( ![X4:$i]: ( subset @ emptyset @ X4 ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(subsetI2, axiom, subsetI2 =
% 132.32/17.32 (![A:$i,B:$i]:
% 132.32/17.32 ( ( ![Xx:$i]: ( ( in @ Xx @ A ) => ( in @ Xx @ B ) ) ) =>
% 132.32/17.32 ( subset @ A @ B ) ))).
% 132.32/17.32 thf('114', plain,
% 132.32/17.32 (( subsetI2 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( ![X8:$i]: ( ( in @ X8 @ X4 ) => ( in @ X8 @ X6 ) ) ) =>
% 132.32/17.32 ( subset @ X4 @ X6 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(eqimpsubset1, axiom, eqimpsubset1 =
% 132.32/17.32 (![A:$i,B:$i]: ( ( ( A ) = ( B ) ) => ( subset @ A @ B ) ))).
% 132.32/17.32 thf('115', plain,
% 132.32/17.32 (( eqimpsubset1 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]: ( ( ( X4 ) = ( X6 ) ) => ( subset @ X4 @ X6 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(eqimpsubset2, axiom, eqimpsubset2 =
% 132.32/17.32 (![A:$i,B:$i]: ( ( ( A ) = ( B ) ) => ( subset @ B @ A ) ))).
% 132.32/17.32 thf('116', plain,
% 132.32/17.32 (( eqimpsubset2 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]: ( ( ( X4 ) = ( X6 ) ) => ( subset @ X6 @ X4 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(subsetI1, axiom, subsetI1 =
% 132.32/17.32 (![A:$i,B:$i]:
% 132.32/17.32 ( ( ![Xx:$i]: ( ( in @ Xx @ A ) => ( in @ Xx @ B ) ) ) =>
% 132.32/17.32 ( subset @ A @ B ) ))).
% 132.32/17.32 thf('117', plain,
% 132.32/17.32 (( subsetI1 ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( ![X8:$i]: ( ( in @ X8 @ X4 ) => ( in @ X8 @ X6 ) ) ) =>
% 132.32/17.32 ( subset @ X4 @ X6 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(dsetconstr__Cong, axiom, dsetconstr__Cong =
% 132.32/17.32 (![A:$i,B:$i]:
% 132.32/17.32 ( ( ( A ) = ( B ) ) =>
% 132.32/17.32 ( ![Xphi:( $i > $o ),Xpsi:( $i > $o )]:
% 132.32/17.32 ( ( ![Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) =>
% 132.32/17.32 ( ![Xy:$i]:
% 132.32/17.32 ( ( in @ Xy @ B ) =>
% 132.32/17.32 ( ( ( Xx ) = ( Xy ) ) => ( ( Xphi @ Xx ) <=> ( Xpsi @ Xy ) ) ) ) ) ) ) =>
% 132.32/17.32 ( ( dsetconstr @ A @ ( ^[Xx:$i]: ( Xphi @ Xx ) ) ) =
% 132.32/17.32 ( dsetconstr @ B @ ( ^[Xx:$i]: ( Xpsi @ Xx ) ) ) ) ) ) ))).
% 132.32/17.32 thf('118', plain,
% 132.32/17.32 (( dsetconstr__Cong ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( ( X4 ) = ( X6 ) ) =>
% 132.32/17.32 ( ![X8:( $i > $o ),X10:( $i > $o )]:
% 132.32/17.32 ( ( ![X12:$i]:
% 132.32/17.32 ( ( in @ X12 @ X4 ) =>
% 132.32/17.32 ( ![X14:$i]:
% 132.32/17.32 ( ( in @ X14 @ X6 ) =>
% 132.32/17.32 ( ( ( X12 ) = ( X14 ) ) =>
% 132.32/17.32 ( ( X8 @ X12 ) <=> ( X10 @ X14 ) ) ) ) ) ) ) =>
% 132.32/17.32 ( ( dsetconstr @ X4 @ ( ^[V_1:$i]: ( X8 @ V_1 ) ) ) =
% 132.32/17.32 ( dsetconstr @ X6 @ ( ^[V_2:$i]: ( X10 @ V_2 ) ) ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(descr__Cong, axiom, descr__Cong =
% 132.32/17.32 (![Xphi:( $i > $o ),Xpsi:( $i > $o )]:
% 132.32/17.32 ( ( ![Xx:$i,Xy:$i]:
% 132.32/17.32 ( ( ( Xx ) = ( Xy ) ) => ( ( Xphi @ Xx ) <=> ( Xpsi @ Xy ) ) ) ) =>
% 132.32/17.32 ( ( exu @ ( ^[Xx:$i]: ( Xphi @ Xx ) ) ) =>
% 132.32/17.32 ( ( exu @ ( ^[Xx:$i]: ( Xpsi @ Xx ) ) ) =>
% 132.32/17.32 ( ( descr @ ( ^[Xx:$i]: ( Xphi @ Xx ) ) ) =
% 132.32/17.32 ( descr @ ( ^[Xx:$i]: ( Xpsi @ Xx ) ) ) ) ) ) ))).
% 132.32/17.32 thf('119', plain,
% 132.32/17.32 (( descr__Cong ) =
% 132.32/17.32 ( ![X4:( $i > $o ),X6:( $i > $o )]:
% 132.32/17.32 ( ( ![X8:$i,X10:$i]:
% 132.32/17.32 ( ( ( X8 ) = ( X10 ) ) => ( ( X4 @ X8 ) <=> ( X6 @ X10 ) ) ) ) =>
% 132.32/17.32 ( ( exu @ ( ^[V_1:$i]: ( X4 @ V_1 ) ) ) =>
% 132.32/17.32 ( ( exu @ ( ^[V_2:$i]: ( X6 @ V_2 ) ) ) =>
% 132.32/17.32 ( ( descr @ ( ^[V_3:$i]: ( X4 @ V_3 ) ) ) =
% 132.32/17.32 ( descr @ ( ^[V_4:$i]: ( X6 @ V_4 ) ) ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(exuEu, axiom, exuEu =
% 132.32/17.32 (![Xphi:( $i > $o )]:
% 132.32/17.32 ( ( exu @ ( ^[Xx:$i]: ( Xphi @ Xx ) ) ) =>
% 132.32/17.32 ( ![Xx:$i,Xy:$i]:
% 132.32/17.32 ( ( Xphi @ Xx ) => ( ( Xphi @ Xy ) => ( ( Xx ) = ( Xy ) ) ) ) ) ))).
% 132.32/17.32 thf('120', plain,
% 132.32/17.32 (( exuEu ) =
% 132.32/17.32 ( ![X4:( $i > $o )]:
% 132.32/17.32 ( ( exu @ ( ^[V_1:$i]: ( X4 @ V_1 ) ) ) =>
% 132.32/17.32 ( ![X6:$i,X8:$i]:
% 132.32/17.32 ( ( X4 @ X6 ) => ( ( X4 @ X8 ) => ( ( X6 ) = ( X8 ) ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(omega__Cong, axiom, omega__Cong = (( omega ) = ( omega ))).
% 132.32/17.32 thf('121', plain, (( omega__Cong ) = ( ( omega ) = ( omega ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setunion__Cong, axiom, setunion__Cong =
% 132.32/17.32 (![A:$i,B:$i]:
% 132.32/17.32 ( ( ( A ) = ( B ) ) => ( ( setunion @ A ) = ( setunion @ B ) ) ))).
% 132.32/17.32 thf('122', plain,
% 132.32/17.32 (( setunion__Cong ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( ( X4 ) = ( X6 ) ) => ( ( setunion @ X4 ) = ( setunion @ X6 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(powerset__Cong, axiom, powerset__Cong =
% 132.32/17.32 (![A:$i,B:$i]:
% 132.32/17.32 ( ( ( A ) = ( B ) ) => ( ( powerset @ A ) = ( powerset @ B ) ) ))).
% 132.32/17.32 thf('123', plain,
% 132.32/17.32 (( powerset__Cong ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( ( X4 ) = ( X6 ) ) => ( ( powerset @ X4 ) = ( powerset @ X6 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setadjoin__Cong, axiom, setadjoin__Cong =
% 132.32/17.32 (![Xx:$i,Xy:$i]:
% 132.32/17.32 ( ( ( Xx ) = ( Xy ) ) =>
% 132.32/17.32 ( ![Xz:$i,Xu:$i]:
% 132.32/17.32 ( ( ( Xz ) = ( Xu ) ) =>
% 132.32/17.32 ( ( setadjoin @ Xx @ Xz ) = ( setadjoin @ Xy @ Xu ) ) ) ) ))).
% 132.32/17.32 thf('124', plain,
% 132.32/17.32 (( setadjoin__Cong ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( ( X4 ) = ( X6 ) ) =>
% 132.32/17.32 ( ![X8:$i,X10:$i]:
% 132.32/17.32 ( ( ( X8 ) = ( X10 ) ) =>
% 132.32/17.32 ( ( setadjoin @ X4 @ X8 ) = ( setadjoin @ X6 @ X10 ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(emptyset__Cong, axiom, emptyset__Cong = (( emptyset ) = ( emptyset ))).
% 132.32/17.32 thf('125', plain, (( emptyset__Cong ) = ( ( emptyset ) = ( emptyset ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(exu__Cong, axiom, exu__Cong =
% 132.32/17.32 (![Xphi:( $i > $o ),Xpsi:( $i > $o )]:
% 132.32/17.32 ( ( ![Xx:$i,Xy:$i]:
% 132.32/17.32 ( ( ( Xx ) = ( Xy ) ) => ( ( Xphi @ Xx ) <=> ( Xpsi @ Xy ) ) ) ) =>
% 132.32/17.32 ( ( exu @ ( ^[Xx:$i]: ( Xphi @ Xx ) ) ) <=>
% 132.32/17.32 ( exu @ ( ^[Xx:$i]: ( Xpsi @ Xx ) ) ) ) ))).
% 132.32/17.32 thf('126', plain,
% 132.32/17.32 (( exu__Cong ) =
% 132.32/17.32 ( ![X4:( $i > $o ),X6:( $i > $o )]:
% 132.32/17.32 ( ( ![X8:$i,X10:$i]:
% 132.32/17.32 ( ( ( X8 ) = ( X10 ) ) => ( ( X4 @ X8 ) <=> ( X6 @ X10 ) ) ) ) =>
% 132.32/17.32 ( ( exu @ ( ^[V_1:$i]: ( X4 @ V_1 ) ) ) <=>
% 132.32/17.32 ( exu @ ( ^[V_2:$i]: ( X6 @ V_2 ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(exuE3u, axiom, exuE3u =
% 132.32/17.32 (![Xphi:( $i > $o )]:
% 132.32/17.32 ( ( exu @ ( ^[Xx:$i]: ( Xphi @ Xx ) ) ) =>
% 132.32/17.32 ( ![Xx:$i,Xy:$i]:
% 132.32/17.32 ( ( Xphi @ Xx ) => ( ( Xphi @ Xy ) => ( ( Xx ) = ( Xy ) ) ) ) ) ))).
% 132.32/17.32 thf('127', plain,
% 132.32/17.32 (( exuE3u ) =
% 132.32/17.32 ( ![X4:( $i > $o )]:
% 132.32/17.32 ( ( exu @ ( ^[V_1:$i]: ( X4 @ V_1 ) ) ) =>
% 132.32/17.32 ( ![X6:$i,X8:$i]:
% 132.32/17.32 ( ( X4 @ X6 ) => ( ( X4 @ X8 ) => ( ( X6 ) = ( X8 ) ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(in__Cong, axiom, in__Cong =
% 132.32/17.32 (![A:$i,B:$i]:
% 132.32/17.32 ( ( ( A ) = ( B ) ) =>
% 132.32/17.32 ( ![Xx:$i,Xy:$i]:
% 132.32/17.32 ( ( ( Xx ) = ( Xy ) ) => ( ( in @ Xx @ A ) <=> ( in @ Xy @ B ) ) ) ) ))).
% 132.32/17.32 thf('128', plain,
% 132.32/17.32 (( in__Cong ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( ( X4 ) = ( X6 ) ) =>
% 132.32/17.32 ( ![X8:$i,X10:$i]:
% 132.32/17.32 ( ( ( X8 ) = ( X10 ) ) =>
% 132.32/17.32 ( ( in @ X8 @ X4 ) <=> ( in @ X10 @ X6 ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(inCongP, axiom, inCongP =
% 132.32/17.32 (![A:$i,B:$i]:
% 132.32/17.32 ( ( ( A ) = ( B ) ) =>
% 132.32/17.32 ( ![Xx:$i,Xy:$i]:
% 132.32/17.32 ( ( ( Xx ) = ( Xy ) ) => ( ( in @ Xx @ A ) => ( in @ Xy @ B ) ) ) ) ))).
% 132.32/17.32 thf('129', plain,
% 132.32/17.32 (( inCongP ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( ( X4 ) = ( X6 ) ) =>
% 132.32/17.32 ( ![X8:$i,X10:$i]:
% 132.32/17.32 ( ( ( X8 ) = ( X10 ) ) =>
% 132.32/17.32 ( ( in @ X8 @ X4 ) => ( in @ X10 @ X6 ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(exuI2, axiom, exuI2 =
% 132.32/17.32 (![Xphi:( $i > $o )]:
% 132.32/17.32 ( ( ?[Xx:$i]: ( ![Xy:$i]: ( ( Xphi @ Xy ) <=> ( ( Xy ) = ( Xx ) ) ) ) ) =>
% 132.32/17.32 ( exu @ ( ^[Xx:$i]: ( Xphi @ Xx ) ) ) ))).
% 132.32/17.32 thf('130', plain,
% 132.32/17.32 (( exuI2 ) =
% 132.32/17.32 ( ![X4:( $i > $o )]:
% 132.32/17.32 ( ( ?[X6:$i]: ( ![X8:$i]: ( ( X4 @ X8 ) <=> ( ( X8 ) = ( X6 ) ) ) ) ) =>
% 132.32/17.32 ( exu @ ( ^[V_1:$i]: ( X4 @ V_1 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(exuI3, axiom, exuI3 =
% 132.32/17.32 (![Xphi:( $i > $o )]:
% 132.32/17.32 ( ( ?[Xx:$i]: ( Xphi @ Xx ) ) =>
% 132.32/17.32 ( ( ![Xx:$i,Xy:$i]:
% 132.32/17.32 ( ( Xphi @ Xx ) => ( ( Xphi @ Xy ) => ( ( Xx ) = ( Xy ) ) ) ) ) =>
% 132.32/17.32 ( exu @ ( ^[Xx:$i]: ( Xphi @ Xx ) ) ) ) ))).
% 132.32/17.32 thf('131', plain,
% 132.32/17.32 (( exuI3 ) =
% 132.32/17.32 ( ![X4:( $i > $o )]:
% 132.32/17.32 ( ( ?[X6:$i]: ( X4 @ X6 ) ) =>
% 132.32/17.32 ( ( ![X8:$i,X10:$i]:
% 132.32/17.32 ( ( X4 @ X8 ) => ( ( X4 @ X10 ) => ( ( X8 ) = ( X10 ) ) ) ) ) =>
% 132.32/17.32 ( exu @ ( ^[V_1:$i]: ( X4 @ V_1 ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(exuI1, axiom, exuI1 =
% 132.32/17.32 (![Xphi:( $i > $o )]:
% 132.32/17.32 ( ( ?[Xx:$i]:
% 132.32/17.32 ( ( ![Xy:$i]: ( ( Xphi @ Xy ) => ( ( Xx ) = ( Xy ) ) ) ) &
% 132.32/17.32 ( Xphi @ Xx ) ) ) =>
% 132.32/17.32 ( exu @ ( ^[Xx:$i]: ( Xphi @ Xx ) ) ) ))).
% 132.32/17.32 thf('132', plain,
% 132.32/17.32 (( exuI1 ) =
% 132.32/17.32 ( ![X4:( $i > $o )]:
% 132.32/17.32 ( ( ?[X6:$i]:
% 132.32/17.32 ( ( ![X8:$i]: ( ( X4 @ X8 ) => ( ( X6 ) = ( X8 ) ) ) ) &
% 132.32/17.32 ( X4 @ X6 ) ) ) =>
% 132.32/17.32 ( exu @ ( ^[V_1:$i]: ( X4 @ V_1 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(notdallE, axiom, notdallE =
% 132.32/17.32 (![A:$i,Xphi:( $i > $o )]:
% 132.32/17.32 ( ( ~( ![Xx:$i]: ( ( in @ Xx @ A ) => ( Xphi @ Xx ) ) ) ) =>
% 132.32/17.32 ( ?[Xx:$i]: ( ( ~( Xphi @ Xx ) ) & ( in @ Xx @ A ) ) ) ))).
% 132.32/17.32 thf('133', plain,
% 132.32/17.32 (( notdallE ) =
% 132.32/17.32 ( ![X4:$i,X6:( $i > $o )]:
% 132.32/17.32 ( ( ~( ![X8:$i]: ( ( in @ X8 @ X4 ) => ( X6 @ X8 ) ) ) ) =>
% 132.32/17.32 ( ?[X10:$i]: ( ( ~( X6 @ X10 ) ) & ( in @ X10 @ X4 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(notdexE, axiom, notdexE =
% 132.32/17.32 (![A:$i,Xphi:( $i > $o )]:
% 132.32/17.32 ( ( ~( ?[Xx:$i]: ( ( Xphi @ Xx ) & ( in @ Xx @ A ) ) ) ) =>
% 132.32/17.32 ( ![Xx:$i]: ( ( in @ Xx @ A ) => ( ~( Xphi @ Xx ) ) ) ) ))).
% 132.32/17.32 thf('134', plain,
% 132.32/17.32 (( notdexE ) =
% 132.32/17.32 ( ![X4:$i,X6:( $i > $o )]:
% 132.32/17.32 ( ( ~( ?[X8:$i]: ( ( X6 @ X8 ) & ( in @ X8 @ X4 ) ) ) ) =>
% 132.32/17.32 ( ![X10:$i]: ( ( in @ X10 @ X4 ) => ( ~( X6 @ X10 ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(prop2set2propI, axiom, prop2set2propI =
% 132.32/17.32 (![Xphi:$o]: ( ( Xphi ) => ( set2prop @ ( prop2set @ Xphi ) ) ))).
% 132.32/17.32 thf('135', plain,
% 132.32/17.32 (( prop2set2propI ) =
% 132.32/17.32 ( ![X4:$o]: ( ( X4 ) => ( set2prop @ ( prop2set @ X4 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(prop2setI, axiom, prop2setI =
% 132.32/17.32 (![Xphi:$o]: ( ( Xphi ) => ( in @ emptyset @ ( prop2set @ Xphi ) ) ))).
% 132.32/17.32 thf('136', plain,
% 132.32/17.32 (( prop2setI ) =
% 132.32/17.32 ( ![X4:$o]: ( ( X4 ) => ( in @ emptyset @ ( prop2set @ X4 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(quantDeMorgan4, axiom, quantDeMorgan4 =
% 132.32/17.32 (![A:$i,Xphi:( $i > $o )]:
% 132.32/17.32 ( ( ?[Xx:$i]: ( ( ~( Xphi @ Xx ) ) & ( in @ Xx @ A ) ) ) =>
% 132.32/17.32 ( ~( ![Xx:$i]: ( ( in @ Xx @ A ) => ( Xphi @ Xx ) ) ) ) ))).
% 132.32/17.32 thf('137', plain,
% 132.32/17.32 (( quantDeMorgan4 ) =
% 132.32/17.32 ( ![X4:$i,X6:( $i > $o )]:
% 132.32/17.32 ( ( ?[X8:$i]: ( ( ~( X6 @ X8 ) ) & ( in @ X8 @ X4 ) ) ) =>
% 132.32/17.32 ( ~( ![X10:$i]: ( ( in @ X10 @ X4 ) => ( X6 @ X10 ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(quantDeMorgan3, axiom, quantDeMorgan3 =
% 132.32/17.32 (![A:$i,Xphi:( $i > $o )]:
% 132.32/17.32 ( ( ~( ?[Xx:$i]: ( ( Xphi @ Xx ) & ( in @ Xx @ A ) ) ) ) =>
% 132.32/17.32 ( ![Xx:$i]: ( ( in @ Xx @ A ) => ( ~( Xphi @ Xx ) ) ) ) ))).
% 132.32/17.32 thf('138', plain,
% 132.32/17.32 (( quantDeMorgan3 ) =
% 132.32/17.32 ( ![X4:$i,X6:( $i > $o )]:
% 132.32/17.32 ( ( ~( ?[X8:$i]: ( ( X6 @ X8 ) & ( in @ X8 @ X4 ) ) ) ) =>
% 132.32/17.32 ( ![X10:$i]: ( ( in @ X10 @ X4 ) => ( ~( X6 @ X10 ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(quantDeMorgan2, axiom, quantDeMorgan2 =
% 132.32/17.32 (![A:$i,Xphi:( $i > $o )]:
% 132.32/17.32 ( ( ![Xx:$i]: ( ( in @ Xx @ A ) => ( ~( Xphi @ Xx ) ) ) ) =>
% 132.32/17.32 ( ~( ?[Xx:$i]: ( ( Xphi @ Xx ) & ( in @ Xx @ A ) ) ) ) ))).
% 132.32/17.32 thf('139', plain,
% 132.32/17.32 (( quantDeMorgan2 ) =
% 132.32/17.32 ( ![X4:$i,X6:( $i > $o )]:
% 132.32/17.32 ( ( ![X8:$i]: ( ( in @ X8 @ X4 ) => ( ~( X6 @ X8 ) ) ) ) =>
% 132.32/17.32 ( ~( ?[X10:$i]: ( ( X6 @ X10 ) & ( in @ X10 @ X4 ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(quantDeMorgan1, axiom, quantDeMorgan1 =
% 132.32/17.32 (![A:$i,Xphi:( $i > $o )]:
% 132.32/17.32 ( ( ~( ![Xx:$i]: ( ( in @ Xx @ A ) => ( Xphi @ Xx ) ) ) ) =>
% 132.32/17.32 ( ?[Xx:$i]: ( ( ~( Xphi @ Xx ) ) & ( in @ Xx @ A ) ) ) ))).
% 132.32/17.32 thf('140', plain,
% 132.32/17.32 (( quantDeMorgan1 ) =
% 132.32/17.32 ( ![X4:$i,X6:( $i > $o )]:
% 132.32/17.32 ( ( ~( ![X8:$i]: ( ( in @ X8 @ X4 ) => ( X6 @ X8 ) ) ) ) =>
% 132.32/17.32 ( ?[X10:$i]: ( ( ~( X6 @ X10 ) ) & ( in @ X10 @ X4 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(vacuousDall, axiom, vacuousDall =
% 132.32/17.32 (![Xphi:( $i > $o ),Xx:$i]: ( ( in @ Xx @ emptyset ) => ( Xphi @ Xx ) ))).
% 132.32/17.32 thf('141', plain,
% 132.32/17.32 (( vacuousDall ) =
% 132.32/17.32 ( ![X4:( $i > $o ),X6:$i]: ( ( in @ X6 @ emptyset ) => ( X4 @ X6 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(emptyE1, axiom, emptyE1 =
% 132.32/17.32 (![A:$i,Xphi:( $i > $o )]:
% 132.32/17.32 ( ( ?[Xx:$i]: ( ( Xphi @ Xx ) & ( in @ Xx @ A ) ) ) =>
% 132.32/17.32 ( ( ( dsetconstr @ A @ ( ^[Xx:$i]: ( Xphi @ Xx ) ) ) = ( emptyset ) ) =>
% 132.32/17.32 ( $false ) ) ))).
% 132.32/17.32 thf('142', plain,
% 132.32/17.32 (( emptyE1 ) =
% 132.32/17.32 ( ![X4:$i,X6:( $i > $o )]:
% 132.32/17.32 ( ( ?[X8:$i]: ( ( X6 @ X8 ) & ( in @ X8 @ X4 ) ) ) =>
% 132.32/17.32 ( ( ( dsetconstr @ X4 @ ( ^[V_1:$i]: ( X6 @ V_1 ) ) ) = ( emptyset ) ) =>
% 132.32/17.32 ( $false ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(upairsetIR, axiom, upairsetIR =
% 132.32/17.32 (![Xx:$i,Xy:$i]:
% 132.32/17.32 ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ))).
% 132.32/17.32 thf('143', plain,
% 132.32/17.32 (( upairsetIR ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( in @ X6 @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(upairsetIL, axiom, upairsetIL =
% 132.32/17.32 (![Xx:$i,Xy:$i]:
% 132.32/17.32 ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ))).
% 132.32/17.32 thf('144', plain,
% 132.32/17.32 (( upairsetIL ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( in @ X4 @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(upairsetE, axiom, upairsetE =
% 132.32/17.32 (![Xx:$i,Xy:$i,Xz:$i]:
% 132.32/17.32 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) =>
% 132.32/17.32 ( ( ( Xz ) = ( Xx ) ) | ( ( Xz ) = ( Xy ) ) ) ))).
% 132.32/17.32 thf('145', plain,
% 132.32/17.32 (( upairsetE ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) ) =>
% 132.32/17.32 ( ( ( X8 ) = ( X4 ) ) | ( ( X8 ) = ( X6 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(singletonsswitch, axiom, singletonsswitch =
% 132.32/17.32 (![Xx:$i,Xy:$i]:
% 132.32/17.32 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) =>
% 132.32/17.32 ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ))).
% 132.32/17.32 thf('146', plain,
% 132.32/17.32 (( singletonsswitch ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( in @ X4 @ ( setadjoin @ X6 @ emptyset ) ) =>
% 132.32/17.32 ( in @ X6 @ ( setadjoin @ X4 @ emptyset ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(eqinunit, axiom, eqinunit =
% 132.32/17.32 (![Xx:$i,Xy:$i]:
% 132.32/17.32 ( ( ( Xx ) = ( Xy ) ) => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ))).
% 132.32/17.32 thf('147', plain,
% 132.32/17.32 (( eqinunit ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( ( X4 ) = ( X6 ) ) => ( in @ X4 @ ( setadjoin @ X6 @ emptyset ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(notinsingleton, axiom, notinsingleton =
% 132.32/17.32 (![Xx:$i,Xy:$i]:
% 132.32/17.32 ( ( ( Xx ) != ( Xy ) ) => ( ~( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ))).
% 132.32/17.32 thf('148', plain,
% 132.32/17.32 (( notinsingleton ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( ( X4 ) != ( X6 ) ) =>
% 132.32/17.32 ( ~( in @ X6 @ ( setadjoin @ X4 @ emptyset ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(uniqinunit, axiom, uniqinunit =
% 132.32/17.32 (![Xx:$i,Xy:$i]:
% 132.32/17.32 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) => ( ( Xx ) = ( Xy ) ) ))).
% 132.32/17.32 thf('149', plain,
% 132.32/17.32 (( uniqinunit ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( in @ X4 @ ( setadjoin @ X6 @ emptyset ) ) => ( ( X4 ) = ( X6 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(nonemptyImpWitness, axiom, nonemptyImpWitness =
% 132.32/17.32 (![A:$i]: ( ( nonempty @ A ) => ( ?[Xx:$i]: ( in @ Xx @ A ) ) ))).
% 132.32/17.32 thf('150', plain,
% 132.32/17.32 (( nonemptyImpWitness ) =
% 132.32/17.32 ( ![X4:$i]: ( ( nonempty @ X4 ) => ( ?[X6:$i]: ( in @ X6 @ X4 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(exuE2, axiom, exuE2 =
% 132.32/17.32 (![Xphi:( $i > $o )]:
% 132.32/17.32 ( ( exu @ ( ^[Xx:$i]: ( Xphi @ Xx ) ) ) =>
% 132.32/17.32 ( ?[Xx:$i]: ( ![Xy:$i]: ( ( Xphi @ Xy ) <=> ( ( Xy ) = ( Xx ) ) ) ) ) ))).
% 132.32/17.32 thf('151', plain,
% 132.32/17.32 (( exuE2 ) =
% 132.32/17.32 ( ![X4:( $i > $o )]:
% 132.32/17.32 ( ( exu @ ( ^[V_1:$i]: ( X4 @ V_1 ) ) ) =>
% 132.32/17.32 ( ?[X6:$i]: ( ![X8:$i]: ( ( X4 @ X8 ) <=> ( ( X8 ) = ( X6 ) ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(subPowSU, axiom, subPowSU =
% 132.32/17.32 (![A:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ))).
% 132.32/17.32 thf('152', plain,
% 132.32/17.32 (( subPowSU ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( in @ X6 @ X4 ) => ( in @ X6 @ ( powerset @ ( setunion @ X4 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setunionE, axiom, setunionE =
% 132.32/17.32 (![A:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ ( setunion @ A ) ) =>
% 132.32/17.32 ( ![Xphi:$o]:
% 132.32/17.32 ( ( ![B:$i]: ( ( in @ Xx @ B ) => ( ( in @ B @ A ) => ( Xphi ) ) ) ) =>
% 132.32/17.32 ( Xphi ) ) ) ))).
% 132.32/17.32 thf('153', plain,
% 132.32/17.32 (( setunionE ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( in @ X6 @ ( setunion @ X4 ) ) =>
% 132.32/17.32 ( ![X8:$o]:
% 132.32/17.32 ( ( ![X10:$i]:
% 132.32/17.32 ( ( in @ X6 @ X10 ) => ( ( in @ X10 @ X4 ) => ( X8 ) ) ) ) =>
% 132.32/17.32 ( X8 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setunionI, axiom, setunionI =
% 132.32/17.32 (![A:$i,Xx:$i,B:$i]:
% 132.32/17.32 ( ( in @ Xx @ B ) => ( ( in @ B @ A ) => ( in @ Xx @ ( setunion @ A ) ) ) ))).
% 132.32/17.32 thf('154', plain,
% 132.32/17.32 (( setunionI ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X6 @ X8 ) =>
% 132.32/17.32 ( ( in @ X8 @ X4 ) => ( in @ X6 @ ( setunion @ X4 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(powersetE, axiom, powersetE =
% 132.32/17.32 (![A:$i,B:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ B @ ( powerset @ A ) ) => ( ( in @ Xx @ B ) => ( in @ Xx @ A ) ) ))).
% 132.32/17.32 thf('155', plain,
% 132.32/17.32 (( powersetE ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X6 @ ( powerset @ X4 ) ) =>
% 132.32/17.32 ( ( in @ X8 @ X6 ) => ( in @ X8 @ X4 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(emptyInPowerset, axiom, emptyInPowerset =
% 132.32/17.32 (![A:$i]: ( in @ emptyset @ ( powerset @ A ) ))).
% 132.32/17.32 thf('156', plain,
% 132.32/17.32 (( emptyInPowerset ) =
% 132.32/17.32 ( ![X4:$i]: ( in @ emptyset @ ( powerset @ X4 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(emptyinPowerset, axiom, emptyinPowerset =
% 132.32/17.32 (![A:$i]: ( in @ emptyset @ ( powerset @ A ) ))).
% 132.32/17.32 thf('157', plain,
% 132.32/17.32 (( emptyinPowerset ) =
% 132.32/17.32 ( ![X4:$i]: ( in @ emptyset @ ( powerset @ X4 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(powersetI, axiom, powersetI =
% 132.32/17.32 (![A:$i,B:$i]:
% 132.32/17.32 ( ( ![Xx:$i]: ( ( in @ Xx @ B ) => ( in @ Xx @ A ) ) ) =>
% 132.32/17.32 ( in @ B @ ( powerset @ A ) ) ))).
% 132.32/17.32 thf('158', plain,
% 132.32/17.32 (( powersetI ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( ![X8:$i]: ( ( in @ X8 @ X6 ) => ( in @ X8 @ X4 ) ) ) =>
% 132.32/17.32 ( in @ X6 @ ( powerset @ X4 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setoftrueEq, axiom, setoftrueEq =
% 132.32/17.32 (![A:$i]: ( ( dsetconstr @ A @ ( ^[Xx:$i]: ( $true ) ) ) = ( A ) ))).
% 132.32/17.32 thf('159', plain,
% 132.32/17.32 (( setoftrueEq ) =
% 132.32/17.32 ( ![X4:$i]: ( ( dsetconstr @ X4 @ ( ^[V_1:$i]: ( $true ) ) ) = ( X4 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setadjoinOr, axiom, setadjoinOr =
% 132.32/17.32 (![Xx:$i,A:$i,Xy:$i]:
% 132.32/17.32 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) ) =>
% 132.32/17.32 ( ( ( Xy ) = ( Xx ) ) | ( in @ Xy @ A ) ) ))).
% 132.32/17.32 thf('160', plain,
% 132.32/17.32 (( setadjoinOr ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ ( setadjoin @ X4 @ X6 ) ) =>
% 132.32/17.32 ( ( ( X8 ) = ( X4 ) ) | ( in @ X8 @ X6 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setadjoinE, axiom, setadjoinE =
% 132.32/17.32 (![Xx:$i,A:$i,Xy:$i]:
% 132.32/17.32 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) ) =>
% 132.32/17.32 ( ![Xphi:$o]:
% 132.32/17.32 ( ( ( ( Xy ) = ( Xx ) ) => ( Xphi ) ) =>
% 132.32/17.32 ( ( ( in @ Xy @ A ) => ( Xphi ) ) => ( Xphi ) ) ) ) ))).
% 132.32/17.32 thf('161', plain,
% 132.32/17.32 (( setadjoinE ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ ( setadjoin @ X4 @ X6 ) ) =>
% 132.32/17.32 ( ![X10:$o]:
% 132.32/17.32 ( ( ( ( X8 ) = ( X4 ) ) => ( X10 ) ) =>
% 132.32/17.32 ( ( ( in @ X8 @ X6 ) => ( X10 ) ) => ( X10 ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setadjoinIR, axiom, setadjoinIR =
% 132.32/17.32 (![Xx:$i,A:$i,Xy:$i]:
% 132.32/17.32 ( ( in @ Xy @ A ) => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ))).
% 132.32/17.32 thf('162', plain,
% 132.32/17.32 (( setadjoinIR ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X6 ) => ( in @ X8 @ ( setadjoin @ X4 @ X6 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(emptyinunitempty, axiom,(( emptyinunitempty ) =
% 132.32/17.32 (in @ emptyset @ ( setadjoin @ emptyset @ emptyset )))).
% 132.32/17.32 thf('163', plain,
% 132.32/17.32 (( emptyinunitempty ) =
% 132.32/17.32 ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )),
% 132.32/17.32 inference('simplify_rw_rule', [status(thm)], [emptyinunitempty])).
% 132.32/17.32 thf('164', plain,
% 132.32/17.32 (( emptyinunitempty ) =
% 132.32/17.32 ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setadjoinIL, axiom, setadjoinIL =
% 132.32/17.32 (![Xx:$i,Xy:$i]: ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ))).
% 132.32/17.32 thf('165', plain,
% 132.32/17.32 (( setadjoinIL ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]: ( in @ X4 @ ( setadjoin @ X4 @ X6 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(nonemptyI1, axiom, nonemptyI1 =
% 132.32/17.32 (![A:$i]: ( ( ?[Xx:$i]: ( in @ Xx @ A ) ) => ( nonempty @ A ) ))).
% 132.32/17.32 thf('166', plain,
% 132.32/17.32 (( nonemptyI1 ) =
% 132.32/17.32 ( ![X4:$i]: ( ( ?[X6:$i]: ( in @ X6 @ X4 ) ) => ( nonempty @ X4 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(nonemptyI, axiom, nonemptyI =
% 132.32/17.32 (![A:$i,Xphi:( $i > $o ),Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) =>
% 132.32/17.32 ( ( Xphi @ Xx ) =>
% 132.32/17.32 ( nonempty @ ( dsetconstr @ A @ ( ^[Xy:$i]: ( Xphi @ Xy ) ) ) ) ) ))).
% 132.32/17.32 thf('167', plain,
% 132.32/17.32 (( nonemptyI ) =
% 132.32/17.32 ( ![X4:$i,X6:( $i > $o ),X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X4 ) =>
% 132.32/17.32 ( ( X6 @ X8 ) =>
% 132.32/17.32 ( nonempty @ ( dsetconstr @ X4 @ ( ^[V_1:$i]: ( X6 @ V_1 ) ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(nonemptyE1, axiom, nonemptyE1 =
% 132.32/17.32 (![A:$i]: ( ( nonempty @ A ) => ( ?[Xx:$i]: ( in @ Xx @ A ) ) ))).
% 132.32/17.32 thf('168', plain,
% 132.32/17.32 (( nonemptyE1 ) =
% 132.32/17.32 ( ![X4:$i]: ( ( nonempty @ X4 ) => ( ?[X6:$i]: ( in @ X6 @ X4 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(nonempty, axiom,(( nonempty ) = (^[Xx:$i]: ( ( Xx ) != ( emptyset ) )))).
% 132.32/17.32 thf('169', plain, (( nonempty ) = ( ^[Xx:$i]: ( ( Xx ) != ( emptyset ) ) )),
% 132.32/17.32 inference('simplify_rw_rule', [status(thm)], [nonempty])).
% 132.32/17.32 thf('170', plain,
% 132.32/17.32 (( nonempty ) = ( ^[V_1:$i]: ( ( V_1 ) != ( emptyset ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setbeta, axiom, setbeta =
% 132.32/17.32 (![A:$i,Xphi:( $i > $o ),Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) =>
% 132.32/17.32 ( ( in @ Xx @ ( dsetconstr @ A @ ( ^[Xy:$i]: ( Xphi @ Xy ) ) ) ) <=>
% 132.32/17.32 ( Xphi @ Xx ) ) ))).
% 132.32/17.32 thf('171', plain,
% 132.32/17.32 (( setbeta ) =
% 132.32/17.32 ( ![X4:$i,X6:( $i > $o ),X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X4 ) =>
% 132.32/17.32 ( ( in @ X8 @ ( dsetconstr @ X4 @ ( ^[V_1:$i]: ( X6 @ V_1 ) ) ) ) <=>
% 132.32/17.32 ( X6 @ X8 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(noeltsimpempty, axiom, noeltsimpempty =
% 132.32/17.32 (![A:$i]:
% 132.32/17.32 ( ( ![Xx:$i]: ( ~( in @ Xx @ A ) ) ) => ( ( A ) = ( emptyset ) ) ))).
% 132.32/17.32 thf('172', plain,
% 132.32/17.32 (( noeltsimpempty ) =
% 132.32/17.32 ( ![X4:$i]:
% 132.32/17.32 ( ( ![X6:$i]: ( ~( in @ X6 @ X4 ) ) ) => ( ( X4 ) = ( emptyset ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(emptyI, axiom, emptyI =
% 132.32/17.32 (![A:$i]:
% 132.32/17.32 ( ( ![Xx:$i]: ( ~( in @ Xx @ A ) ) ) => ( ( A ) = ( emptyset ) ) ))).
% 132.32/17.32 thf('173', plain,
% 132.32/17.32 (( emptyI ) =
% 132.32/17.32 ( ![X4:$i]:
% 132.32/17.32 ( ( ![X6:$i]: ( ~( in @ X6 @ X4 ) ) ) => ( ( X4 ) = ( emptyset ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setext, axiom, setext =
% 132.32/17.32 (![A:$i,B:$i]:
% 132.32/17.32 ( ( ![Xx:$i]: ( ( in @ Xx @ A ) => ( in @ Xx @ B ) ) ) =>
% 132.32/17.32 ( ( ![Xx:$i]: ( ( in @ Xx @ B ) => ( in @ Xx @ A ) ) ) =>
% 132.32/17.32 ( ( A ) = ( B ) ) ) ))).
% 132.32/17.32 thf('174', plain,
% 132.32/17.32 (( setext ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( ![X8:$i]: ( ( in @ X8 @ X4 ) => ( in @ X8 @ X6 ) ) ) =>
% 132.32/17.32 ( ( ![X10:$i]: ( ( in @ X10 @ X6 ) => ( in @ X10 @ X4 ) ) ) =>
% 132.32/17.32 ( ( X4 ) = ( X6 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(exuE3e, axiom, exuE3e =
% 132.32/17.32 (![Xphi:( $i > $o )]:
% 132.32/17.32 ( ( exu @ ( ^[Xx:$i]: ( Xphi @ Xx ) ) ) => ( ?[Xx:$i]: ( Xphi @ Xx ) ) ))).
% 132.32/17.32 thf('175', plain,
% 132.32/17.32 (( exuE3e ) =
% 132.32/17.32 ( ![X4:( $i > $o )]:
% 132.32/17.32 ( ( exu @ ( ^[V_1:$i]: ( X4 @ V_1 ) ) ) => ( ?[X6:$i]: ( X4 @ X6 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(notinemptyset, axiom, notinemptyset =
% 132.32/17.32 (![Xx:$i]: ( ~( in @ Xx @ emptyset ) ))).
% 132.32/17.32 thf('176', plain,
% 132.32/17.32 (( notinemptyset ) = ( ![X4:$i]: ( ~( in @ X4 @ emptyset ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(emptysetimpfalse, axiom, emptysetimpfalse =
% 132.32/17.32 (![Xx:$i]: ( ( in @ Xx @ emptyset ) => ( $false ) ))).
% 132.32/17.32 thf('177', plain,
% 132.32/17.32 (( emptysetimpfalse ) =
% 132.32/17.32 ( ![X4:$i]: ( ( in @ X4 @ emptyset ) => ( $false ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(emptysetE, axiom, emptysetE =
% 132.32/17.32 (![Xx:$i]: ( ( in @ Xx @ emptyset ) => ( ![Xphi:$o]: ( Xphi ) ) ))).
% 132.32/17.32 thf('178', plain,
% 132.32/17.32 (( emptysetE ) =
% 132.32/17.32 ( ![X4:$i]: ( ( in @ X4 @ emptyset ) => ( ![X6:$o]: ( X6 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(prop2setE, axiom, prop2setE =
% 132.32/17.32 (![Xphi:$o,Xx:$i]: ( ( in @ Xx @ ( prop2set @ Xphi ) ) => ( Xphi ) ))).
% 132.32/17.32 thf('179', plain,
% 132.32/17.32 (( prop2setE ) =
% 132.32/17.32 ( ![X4:$o,X6:$i]: ( ( in @ X6 @ ( prop2set @ X4 ) ) => ( X4 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(exuE1, axiom, exuE1 =
% 132.32/17.32 (![Xphi:( $i > $o )]:
% 132.32/17.32 ( ( exu @ ( ^[Xx:$i]: ( Xphi @ Xx ) ) ) =>
% 132.32/17.32 ( ?[Xx:$i]:
% 132.32/17.32 ( ( ![Xy:$i]: ( ( Xphi @ Xy ) => ( ( Xx ) = ( Xy ) ) ) ) &
% 132.32/17.32 ( Xphi @ Xx ) ) ) ))).
% 132.32/17.32 thf('180', plain,
% 132.32/17.32 (( exuE1 ) =
% 132.32/17.32 ( ![X4:( $i > $o )]:
% 132.32/17.32 ( ( exu @ ( ^[V_1:$i]: ( X4 @ V_1 ) ) ) =>
% 132.32/17.32 ( ?[X6:$i]:
% 132.32/17.32 ( ( ![X8:$i]: ( ( X4 @ X8 ) => ( ( X6 ) = ( X8 ) ) ) ) &
% 132.32/17.32 ( X4 @ X6 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(dsetconstrER, axiom, dsetconstrER =
% 132.32/17.32 (![A:$i,Xphi:( $i > $o ),Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ ( dsetconstr @ A @ ( ^[Xy:$i]: ( Xphi @ Xy ) ) ) ) =>
% 132.32/17.32 ( Xphi @ Xx ) ))).
% 132.32/17.32 thf('181', plain,
% 132.32/17.32 (( dsetconstrER ) =
% 132.32/17.32 ( ![X4:$i,X6:( $i > $o ),X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ ( dsetconstr @ X4 @ ( ^[V_1:$i]: ( X6 @ V_1 ) ) ) ) =>
% 132.32/17.32 ( X6 @ X8 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(dsetconstrEL, axiom, dsetconstrEL =
% 132.32/17.32 (![A:$i,Xphi:( $i > $o ),Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ ( dsetconstr @ A @ ( ^[Xy:$i]: ( Xphi @ Xy ) ) ) ) =>
% 132.32/17.32 ( in @ Xx @ A ) ))).
% 132.32/17.32 thf('182', plain,
% 132.32/17.32 (( dsetconstrEL ) =
% 132.32/17.32 ( ![X4:$i,X6:( $i > $o ),X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ ( dsetconstr @ X4 @ ( ^[V_1:$i]: ( X6 @ V_1 ) ) ) ) =>
% 132.32/17.32 ( in @ X8 @ X4 ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(dsetconstrI, axiom, dsetconstrI =
% 132.32/17.32 (![A:$i,Xphi:( $i > $o ),Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) =>
% 132.32/17.32 ( ( Xphi @ Xx ) =>
% 132.32/17.32 ( in @ Xx @ ( dsetconstr @ A @ ( ^[Xy:$i]: ( Xphi @ Xy ) ) ) ) ) ))).
% 132.32/17.32 thf('183', plain,
% 132.32/17.32 (( dsetconstrI ) =
% 132.32/17.32 ( ![X4:$i,X6:( $i > $o ),X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X4 ) =>
% 132.32/17.32 ( ( X6 @ X8 ) =>
% 132.32/17.32 ( in @ X8 @ ( dsetconstr @ X4 @ ( ^[V_1:$i]: ( X6 @ V_1 ) ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(descrp, axiom, descrp =
% 132.32/17.32 (![Xphi:( $i > $o )]:
% 132.32/17.32 ( ( exu @ ( ^[Xx:$i]: ( Xphi @ Xx ) ) ) =>
% 132.32/17.32 ( Xphi @ ( descr @ ( ^[Xx:$i]: ( Xphi @ Xx ) ) ) ) ))).
% 132.32/17.32 thf('184', plain,
% 132.32/17.32 (( descrp ) =
% 132.32/17.32 ( ![X4:( $i > $o )]:
% 132.32/17.32 ( ( exu @ ( ^[V_1:$i]: ( X4 @ V_1 ) ) ) =>
% 132.32/17.32 ( X4 @ ( descr @ ( ^[V_2:$i]: ( X4 @ V_2 ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(wellorderingAx, axiom, wellorderingAx =
% 132.32/17.32 (![A:$i]:
% 132.32/17.32 ( ?[B:$i]:
% 132.32/17.32 ( ( ![C:$i]:
% 132.32/17.32 ( ( ( ![Xx:$i]: ( ( in @ Xx @ C ) => ( in @ Xx @ A ) ) ) &
% 132.32/17.32 ( ?[Xx:$i]: ( in @ Xx @ C ) ) ) =>
% 132.32/17.32 ( ?[D:$i,Xx:$i]:
% 132.32/17.32 ( ( ![E:$i]:
% 132.32/17.32 ( ( in @ E @ B ) =>
% 132.32/17.32 ( ( ![Xy:$i]: ( ( in @ Xy @ E ) => ( in @ Xy @ D ) ) ) |
% 132.32/17.32 ( in @ Xx @ E ) ) ) ) &
% 132.32/17.32 ( ~( ?[Xy:$i]: ( ( in @ Xy @ C ) & ( in @ Xy @ D ) ) ) ) &
% 132.32/17.32 ( in @ Xx @ C ) & ( in @ D @ B ) ) ) ) ) &
% 132.32/17.32 ( ![C:$i,D:$i]:
% 132.32/17.32 ( ( ( in @ C @ B ) & ( in @ D @ B ) ) =>
% 132.32/17.32 ( ( ![Xx:$i]: ( ( in @ Xx @ C ) => ( in @ Xx @ D ) ) ) |
% 132.32/17.32 ( ![Xx:$i]: ( ( in @ Xx @ D ) => ( in @ Xx @ C ) ) ) ) ) ) &
% 132.32/17.32 ( ![Xx:$i,Xy:$i]:
% 132.32/17.32 ( ( ( in @ Xx @ A ) & ( in @ Xy @ A ) ) =>
% 132.32/17.32 ( ( ![C:$i]:
% 132.32/17.32 ( ( in @ C @ B ) => ( ( in @ Xx @ C ) <=> ( in @ Xy @ C ) ) ) ) =>
% 132.32/17.32 ( ( Xx ) = ( Xy ) ) ) ) ) &
% 132.32/17.32 ( ![C:$i]:
% 132.32/17.32 ( ( in @ C @ B ) =>
% 132.32/17.32 ( ![Xx:$i]: ( ( in @ Xx @ C ) => ( in @ Xx @ A ) ) ) ) ) ) ))).
% 132.32/17.32 thf('185', plain,
% 132.32/17.32 (( wellorderingAx ) =
% 132.32/17.32 ( ![X4:$i]:
% 132.32/17.32 ( ?[X6:$i]:
% 132.32/17.32 ( ( ![X8:$i]:
% 132.32/17.32 ( ( ( ![X10:$i]: ( ( in @ X10 @ X8 ) => ( in @ X10 @ X4 ) ) ) &
% 132.32/17.32 ( ?[X12:$i]: ( in @ X12 @ X8 ) ) ) =>
% 132.32/17.32 ( ?[X14:$i,X16:$i]:
% 132.32/17.32 ( ( ![X18:$i]:
% 132.32/17.32 ( ( in @ X18 @ X6 ) =>
% 132.32/17.32 ( ( ![X20:$i]:
% 132.32/17.32 ( ( in @ X20 @ X18 ) => ( in @ X20 @ X14 ) ) ) |
% 132.32/17.32 ( in @ X16 @ X18 ) ) ) ) &
% 132.32/17.32 ( ~( ?[X22:$i]: ( ( in @ X22 @ X8 ) & ( in @ X22 @ X14 ) ) ) ) &
% 132.32/17.32 ( in @ X16 @ X8 ) & ( in @ X14 @ X6 ) ) ) ) ) &
% 132.32/17.32 ( ![X24:$i,X26:$i]:
% 132.32/17.32 ( ( ( in @ X24 @ X6 ) & ( in @ X26 @ X6 ) ) =>
% 132.32/17.32 ( ( ![X28:$i]: ( ( in @ X28 @ X24 ) => ( in @ X28 @ X26 ) ) ) |
% 132.32/17.32 ( ![X30:$i]: ( ( in @ X30 @ X26 ) => ( in @ X30 @ X24 ) ) ) ) ) ) &
% 132.32/17.32 ( ![X32:$i,X34:$i]:
% 132.32/17.32 ( ( ( in @ X32 @ X4 ) & ( in @ X34 @ X4 ) ) =>
% 132.32/17.32 ( ( ![X36:$i]:
% 132.32/17.32 ( ( in @ X36 @ X6 ) =>
% 132.32/17.32 ( ( in @ X32 @ X36 ) <=> ( in @ X34 @ X36 ) ) ) ) =>
% 132.32/17.32 ( ( X32 ) = ( X34 ) ) ) ) ) &
% 132.32/17.32 ( ![X38:$i]:
% 132.32/17.32 ( ( in @ X38 @ X6 ) =>
% 132.32/17.32 ( ![X40:$i]: ( ( in @ X40 @ X38 ) => ( in @ X40 @ X4 ) ) ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(foundationAx, axiom, foundationAx =
% 132.32/17.32 (![A:$i]:
% 132.32/17.32 ( ( ?[Xx:$i]: ( in @ Xx @ A ) ) =>
% 132.32/17.32 ( ?[B:$i]:
% 132.32/17.32 ( ( ~( ?[Xx:$i]: ( ( in @ Xx @ A ) & ( in @ Xx @ B ) ) ) ) &
% 132.32/17.32 ( in @ B @ A ) ) ) ))).
% 132.32/17.32 thf('186', plain,
% 132.32/17.32 (( foundationAx ) =
% 132.32/17.32 ( ![X4:$i]:
% 132.32/17.32 ( ( ?[X6:$i]: ( in @ X6 @ X4 ) ) =>
% 132.32/17.32 ( ?[X8:$i]:
% 132.32/17.32 ( ( ~( ?[X10:$i]: ( ( in @ X10 @ X4 ) & ( in @ X10 @ X8 ) ) ) ) &
% 132.32/17.32 ( in @ X8 @ X4 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(replAx, axiom, replAx =
% 132.32/17.32 (![Xphi:( $i > $i > $o ),A:$i]:
% 132.32/17.32 ( ( ![Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ A ) => ( exu @ ( ^[Xy:$i]: ( Xphi @ Xx @ Xy ) ) ) ) ) =>
% 132.32/17.32 ( ?[B:$i]:
% 132.32/17.32 ( ![Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ B ) <=>
% 132.32/17.32 ( ?[Xy:$i]: ( ( Xphi @ Xy @ Xx ) & ( in @ Xy @ A ) ) ) ) ) ) ))).
% 132.32/17.32 thf('187', plain,
% 132.32/17.32 (( replAx ) =
% 132.32/17.32 ( ![X4:( $i > $i > $o ),X6:$i]:
% 132.32/17.32 ( ( ![X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ X6 ) => ( exu @ ( ^[V_1:$i]: ( X4 @ X8 @ V_1 ) ) ) ) ) =>
% 132.32/17.32 ( ?[X10:$i]:
% 132.32/17.32 ( ![X12:$i]:
% 132.32/17.32 ( ( in @ X12 @ X10 ) <=>
% 132.32/17.32 ( ?[X14:$i]: ( ( X4 @ X14 @ X12 ) & ( in @ X14 @ X6 ) ) ) ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(omegaIndAx, axiom, omegaIndAx =
% 132.32/17.32 (![A:$i]:
% 132.32/17.32 ( ( ( in @ emptyset @ A ) &
% 132.32/17.32 ( ![Xx:$i]:
% 132.32/17.32 ( ( ( in @ Xx @ omega ) & ( in @ Xx @ A ) ) =>
% 132.32/17.32 ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) ) ) =>
% 132.32/17.32 ( ![Xx:$i]: ( ( in @ Xx @ omega ) => ( in @ Xx @ A ) ) ) ))).
% 132.32/17.32 thf('188', plain,
% 132.32/17.32 (( omegaIndAx ) =
% 132.32/17.32 ( ![X4:$i]:
% 132.32/17.32 ( ( ( in @ emptyset @ X4 ) &
% 132.32/17.32 ( ![X6:$i]:
% 132.32/17.32 ( ( ( in @ X6 @ omega ) & ( in @ X6 @ X4 ) ) =>
% 132.32/17.32 ( in @ ( setadjoin @ X6 @ X6 ) @ X4 ) ) ) ) =>
% 132.32/17.32 ( ![X8:$i]: ( ( in @ X8 @ omega ) => ( in @ X8 @ X4 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(omegaSAx, axiom, omegaSAx =
% 132.32/17.32 (![Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ omega ) => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ))).
% 132.32/17.32 thf('189', plain,
% 132.32/17.32 (( omegaSAx ) =
% 132.32/17.32 ( ![X4:$i]:
% 132.32/17.32 ( ( in @ X4 @ omega ) => ( in @ ( setadjoin @ X4 @ X4 ) @ omega ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(omega0Ax, axiom,(( omega0Ax ) = (in @ emptyset @ omega))).
% 132.32/17.32 thf('190', plain, (( omega0Ax ) = ( in @ emptyset @ omega )),
% 132.32/17.32 inference('simplify_rw_rule', [status(thm)], [omega0Ax])).
% 132.32/17.32 thf('191', plain, (( omega0Ax ) = ( in @ emptyset @ omega )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setunionAx, axiom, setunionAx =
% 132.32/17.32 (![A:$i,Xx:$i]:
% 132.32/17.32 ( ( in @ Xx @ ( setunion @ A ) ) <=>
% 132.32/17.32 ( ?[B:$i]: ( ( in @ B @ A ) & ( in @ Xx @ B ) ) ) ))).
% 132.32/17.32 thf('192', plain,
% 132.32/17.32 (( setunionAx ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( in @ X6 @ ( setunion @ X4 ) ) <=>
% 132.32/17.32 ( ?[X8:$i]: ( ( in @ X8 @ X4 ) & ( in @ X6 @ X8 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(powersetAx, axiom, powersetAx =
% 132.32/17.32 (![A:$i,B:$i]:
% 132.32/17.32 ( ( in @ B @ ( powerset @ A ) ) <=>
% 132.32/17.32 ( ![Xx:$i]: ( ( in @ Xx @ B ) => ( in @ Xx @ A ) ) ) ))).
% 132.32/17.32 thf('193', plain,
% 132.32/17.32 (( powersetAx ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( in @ X6 @ ( powerset @ X4 ) ) <=>
% 132.32/17.32 ( ![X8:$i]: ( ( in @ X8 @ X6 ) => ( in @ X8 @ X4 ) ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setadjoinAx, axiom, setadjoinAx =
% 132.32/17.32 (![Xx:$i,A:$i,Xy:$i]:
% 132.32/17.32 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) ) <=>
% 132.32/17.32 ( ( ( Xy ) = ( Xx ) ) | ( in @ Xy @ A ) ) ))).
% 132.32/17.32 thf('194', plain,
% 132.32/17.32 (( setadjoinAx ) =
% 132.32/17.32 ( ![X4:$i,X6:$i,X8:$i]:
% 132.32/17.32 ( ( in @ X8 @ ( setadjoin @ X4 @ X6 ) ) <=>
% 132.32/17.32 ( ( ( X8 ) = ( X4 ) ) | ( in @ X8 @ X6 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(emptysetAx, axiom, emptysetAx = (![Xx:$i]: ( ~( in @ Xx @ emptyset ) ))).
% 132.32/17.32 thf('195', plain,
% 132.32/17.32 (( emptysetAx ) = ( ![X4:$i]: ( ~( in @ X4 @ emptyset ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(setextAx, axiom, setextAx =
% 132.32/17.32 (![A:$i,B:$i]:
% 132.32/17.32 ( ( ![Xx:$i]: ( ( in @ Xx @ A ) <=> ( in @ Xx @ B ) ) ) =>
% 132.32/17.32 ( ( A ) = ( B ) ) ))).
% 132.32/17.32 thf('196', plain,
% 132.32/17.32 (( setextAx ) =
% 132.32/17.32 ( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( ![X8:$i]: ( ( in @ X8 @ X4 ) <=> ( in @ X8 @ X6 ) ) ) =>
% 132.32/17.32 ( ( X4 ) = ( X6 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(exu, axiom,(( exu ) =
% 132.32/17.32 (^[Xphi:( $i > $o )]:
% 132.32/17.32 ( ?[Xx:$i]:
% 132.32/17.32 ( ( ![Xy:$i]: ( ( Xphi @ Xy ) => ( ( Xx ) = ( Xy ) ) ) ) & ( Xphi @ Xx ) ) )))).
% 132.32/17.32 thf('197', plain,
% 132.32/17.32 (( exu ) =
% 132.32/17.32 ( ^[Xphi:( $i > $o )]:
% 132.32/17.32 ( ?[Xx:$i]:
% 132.32/17.32 ( ( ![Xy:$i]: ( ( Xphi @ Xy ) => ( ( Xx ) = ( Xy ) ) ) ) &
% 132.32/17.32 ( Xphi @ Xx ) ) ) )),
% 132.32/17.32 inference('simplify_rw_rule', [status(thm)], [exu])).
% 132.32/17.32 thf('198', plain,
% 132.32/17.32 (( exu ) =
% 132.32/17.32 ( ^[V_1:( $i > $o )]:
% 132.32/17.32 ( ?[X4:$i]:
% 132.32/17.32 ( ( ![X6:$i]: ( ( V_1 @ X6 ) => ( ( X4 ) = ( X6 ) ) ) ) &
% 132.32/17.32 ( V_1 @ X4 ) ) ) )),
% 132.32/17.32 define([status(thm)])).
% 132.32/17.32 thf(dpsetconstrEL1, conjecture,
% 132.32/17.32 (( setextAx ) =>
% 132.32/17.32 ( ( emptysetAx ) =>
% 132.32/17.32 ( ( setadjoinAx ) =>
% 132.32/17.32 ( ( powersetAx ) =>
% 132.32/17.32 ( ( setunionAx ) =>
% 132.32/17.32 ( ( omega0Ax ) =>
% 132.32/17.32 ( ( omegaSAx ) =>
% 132.32/17.32 ( ( omegaIndAx ) =>
% 132.32/17.32 ( ( replAx ) =>
% 132.32/17.32 ( ( foundationAx ) =>
% 132.32/17.32 ( ( wellorderingAx ) =>
% 132.32/17.32 ( ( descrp ) =>
% 132.32/17.32 ( ( dsetconstrI ) =>
% 132.32/17.32 ( ( dsetconstrEL ) =>
% 132.32/17.32 ( ( dsetconstrER ) =>
% 132.32/17.32 ( ( exuE1 ) =>
% 132.32/17.32 ( ( prop2setE ) =>
% 132.32/17.32 ( ( emptysetE ) =>
% 132.32/17.32 ( ( emptysetimpfalse ) =>
% 132.32/17.32 ( ( notinemptyset ) =>
% 132.32/17.32 ( ( exuE3e ) =>
% 132.32/17.32 ( ( setext ) =>
% 132.32/17.32 ( ( emptyI ) =>
% 132.32/17.32 ( ( noeltsimpempty ) =>
% 132.32/17.32 ( ( setbeta ) =>
% 132.32/17.32 ( ( nonemptyE1 ) =>
% 132.32/17.32 ( ( nonemptyI ) =>
% 132.32/17.32 ( ( nonemptyI1 ) =>
% 132.32/17.32 ( ( setadjoinIL ) =>
% 132.32/17.32 ( ( emptyinunitempty ) =>
% 132.32/17.32 ( ( setadjoinIR ) =>
% 132.32/17.32 ( ( setadjoinE ) =>
% 132.32/17.32 ( (
% 132.32/17.32 setadjoinOr ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setoftrueEq ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 powersetI ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 emptyinPowerset ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 emptyInPowerset ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 powersetE ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setunionI ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setunionE ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 subPowSU ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 exuE2 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 nonemptyImpWitness ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 uniqinunit ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 notinsingleton ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 eqinunit ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 singletonsswitch ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 upairsetE ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 upairsetIL ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 upairsetIR ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 emptyE1 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 vacuousDall ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 quantDeMorgan1 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 quantDeMorgan2 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 quantDeMorgan3 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 quantDeMorgan4 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 prop2setI ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 prop2set2propI ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 notdexE ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 notdallE ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 exuI1 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 exuI3 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 exuI2 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 inCongP ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 in__Cong ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 exuE3u ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 exu__Cong ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 emptyset__Cong ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setadjoin__Cong ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 powerset__Cong ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setunion__Cong ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 omega__Cong ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 exuEu ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 descr__Cong ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 dsetconstr__Cong ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 subsetI1 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 eqimpsubset2 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 eqimpsubset1 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 subsetI2 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 emptysetsubset ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 subsetE ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 subsetE2 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 notsubsetI ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 notequalI1 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 notequalI2 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 subsetRefl ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 subsetTrans ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setadjoinSub ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setadjoinSub2 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 subset2powerset ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setextsub ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 subsetemptysetimpeq ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 powersetI1 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 powersetE1 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 inPowerset ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 powersetsubset ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 sepInPowerset ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 sepSubset ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 binunionIL ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 upairset2IR ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 binunionIR ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 binunionEcases ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 binunionE ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 binunionLsub ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 binunionRsub ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 binintersectI ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 binintersectSubset5 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 binintersectEL ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 binintersectLsub ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 binintersectSubset2 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 binintersectSubset3 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 binintersectER ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 disjointsetsI1 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 binintersectRsub ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 binintersectSubset4 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 binintersectSubset1 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 bs114d ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setminusI ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setminusEL ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setminusER ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setminusSubset2 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setminusERneg ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setminusELneg ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setminusILneg ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setminusIRneg ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setminusLsub ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setminusSubset1 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 symdiffE ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 symdiffI1 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 symdiffI2 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 symdiffIneg1 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 symdiffIneg2 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 secondinupair ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setukpairIL ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setukpairIR ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 kpairiskpair ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 kpairp ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 singletonsubset ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 singletoninpowerset ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 singletoninpowunion ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 upairset2E ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 upairsubunion ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 upairinpowunion ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 ubforcartprodlem1 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 ubforcartprodlem2 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 ubforcartprodlem3 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 cartprodpairin ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 cartprodmempair1 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 cartprodmempair ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setunionE2 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setunionsingleton1 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setunionsingleton2 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setunionsingleton ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 singletonprop ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 ex1E1 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 ex1I ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 ex1I2 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 singletonsuniq ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setukpairinjL1 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 kfstsingleton ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 theprop ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 kfstpairEq ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 cartprodfstin ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setukpairinjL2 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setukpairinjL ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setukpairinjR11 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setukpairinjR12 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setukpairinjR1 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 upairequniteq ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setukpairinjR2 ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setukpairinjR ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 ksndsingleton ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 ksndpairEq ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 kpairsurjEq ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 cartprodsndin ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 cartprodpairmemEL ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 cartprodpairmemER ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 cartprodmempaircEq ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 cartprodfstpairEq ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 cartprodsndpairEq ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 cartprodpairsurjEq ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 dpsetconstrI ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 dpsetconstrSub ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 setOfPairsIsBReln ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 dpsetconstrERa ) =>
% 132.32/17.32 (
% 132.32/17.32 ![
% 132.32/17.32 A:$i,B:$i,
% 132.32/17.32 Xphi:(
% 132.32/17.32 $i > $i >
% 132.32/17.32 $o ),
% 132.32/17.32 Xx:$i,
% 132.32/17.32 Xy:$i]:
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 in @
% 132.32/17.32 (
% 132.32/17.32 kpair @
% 132.32/17.32 Xx @ Xy ) @
% 132.32/17.32 (
% 132.32/17.32 dpsetconstr
% 132.32/17.32 @
% 132.32/17.32 A @ B @
% 132.32/17.32 (
% 132.32/17.32 ^[
% 132.32/17.32 Xz:$i,
% 132.32/17.32 Xu:$i]:
% 132.32/17.32 (
% 132.32/17.32 Xphi @
% 132.32/17.32 Xz @ Xu ) ) ) ) =>
% 132.32/17.32 (
% 132.32/17.32 in @
% 132.32/17.32 Xx @ A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ))).
% 132.32/17.32 thf(zf_stmt_0, conjecture,
% 132.32/17.32 (( ![X4:$i,X6:$i]:
% 132.32/17.32 ( ( ![X8:$i]: ( ( in @ X8 @ X4 ) <=> ( in @ X8 @ X6 ) ) ) =>
% 132.32/17.32 ( ( X4 ) = ( X6 ) ) ) ) =>
% 132.32/17.32 ( ( ![X10:$i]: ( ~( in @ X10 @ emptyset ) ) ) =>
% 132.32/17.32 ( ( ![X12:$i,X14:$i,X16:$i]:
% 132.32/17.32 ( ( in @ X16 @ ( setadjoin @ X12 @ X14 ) ) <=>
% 132.32/17.32 ( ( in @ X16 @ X14 ) | ( ( X16 ) = ( X12 ) ) ) ) ) =>
% 132.32/17.32 ( ( ![X18:$i,X20:$i]:
% 132.32/17.32 ( ( in @ X20 @ ( powerset @ X18 ) ) <=>
% 132.32/17.32 ( ![X22:$i]: ( ( in @ X22 @ X20 ) => ( in @ X22 @ X18 ) ) ) ) ) =>
% 132.32/17.32 ( ( ![X24:$i,X26:$i]:
% 132.32/17.32 ( ( in @ X26 @ ( setunion @ X24 ) ) <=>
% 132.32/17.32 ( ?[X28:$i]: ( ( in @ X26 @ X28 ) & ( in @ X28 @ X24 ) ) ) ) ) =>
% 132.32/17.32 ( ( in @ emptyset @ omega ) =>
% 132.32/17.32 ( ( ![X30:$i]:
% 132.32/17.32 ( ( in @ X30 @ omega ) =>
% 132.32/17.32 ( in @ ( setadjoin @ X30 @ X30 ) @ omega ) ) ) =>
% 132.32/17.32 ( ( ![X32:$i]:
% 132.32/17.32 ( ( ( ![X34:$i]:
% 132.32/17.32 ( ( ( in @ X34 @ X32 ) & ( in @ X34 @ omega ) ) =>
% 132.32/17.32 ( in @ ( setadjoin @ X34 @ X34 ) @ X32 ) ) ) &
% 132.32/17.32 ( in @ emptyset @ X32 ) ) =>
% 132.32/17.32 ( ![X36:$i]:
% 132.32/17.32 ( ( in @ X36 @ omega ) => ( in @ X36 @ X32 ) ) ) ) ) =>
% 132.32/17.32 ( ( ![X38:( $i > $i > $o ),X40:$i]:
% 132.32/17.32 ( ( ![X42:$i]:
% 132.32/17.32 ( ( in @ X42 @ X40 ) =>
% 132.32/17.32 ( ?[X44:$i]:
% 132.32/17.32 ( ( X38 @ X42 @ X44 ) &
% 132.32/17.32 ( ![X46:$i]:
% 132.32/17.32 ( ( X38 @ X42 @ X46 ) =>
% 132.32/17.32 ( ( X44 ) = ( X46 ) ) ) ) ) ) ) ) =>
% 132.32/17.32 ( ?[X48:$i]:
% 132.32/17.32 ( ![X50:$i]:
% 132.32/17.32 ( ( in @ X50 @ X48 ) <=>
% 132.32/17.32 ( ?[X52:$i]:
% 132.32/17.32 ( ( in @ X52 @ X40 ) & ( X38 @ X52 @ X50 ) ) ) ) ) ) ) ) =>
% 132.32/17.32 ( ( ![X54:$i]:
% 132.32/17.32 ( ( ?[X56:$i]: ( in @ X56 @ X54 ) ) =>
% 132.32/17.32 ( ?[X58:$i]:
% 132.32/17.32 ( ( in @ X58 @ X54 ) &
% 132.32/17.32 ( ~( ?[X60:$i]:
% 132.32/17.32 ( ( in @ X60 @ X58 ) & ( in @ X60 @ X54 ) ) ) ) ) ) ) ) =>
% 132.32/17.32 ( ( ![X62:$i]:
% 132.32/17.32 ( ?[X64:$i]:
% 132.32/17.32 ( ( ![X96:$i]:
% 132.32/17.32 ( ( in @ X96 @ X64 ) =>
% 132.32/17.32 ( ![X98:$i]:
% 132.32/17.32 ( ( in @ X98 @ X96 ) => ( in @ X98 @ X62 ) ) ) ) ) &
% 132.32/17.32 ( ![X90:$i,X92:$i]:
% 132.32/17.32 ( ( ( in @ X92 @ X62 ) & ( in @ X90 @ X62 ) ) =>
% 132.32/17.32 ( ( ![X94:$i]:
% 132.32/17.32 ( ( in @ X94 @ X64 ) =>
% 132.32/17.32 ( ( in @ X90 @ X94 ) <=>
% 132.32/17.32 ( in @ X92 @ X94 ) ) ) ) =>
% 132.32/17.32 ( ( X90 ) = ( X92 ) ) ) ) ) &
% 132.32/17.32 ( ![X82:$i,X84:$i]:
% 132.32/17.32 ( ( ( in @ X84 @ X64 ) & ( in @ X82 @ X64 ) ) =>
% 132.32/17.32 ( ( ![X88:$i]:
% 132.32/17.32 ( ( in @ X88 @ X84 ) =>
% 132.32/17.32 ( in @ X88 @ X82 ) ) ) |
% 132.32/17.32 ( ![X86:$i]:
% 132.32/17.32 ( ( in @ X86 @ X82 ) =>
% 132.32/17.32 ( in @ X86 @ X84 ) ) ) ) ) ) &
% 132.32/17.32 ( ![X66:$i]:
% 132.32/17.32 ( ( ( ?[X70:$i]: ( in @ X70 @ X66 ) ) &
% 132.32/17.32 ( ![X68:$i]:
% 132.32/17.32 ( ( in @ X68 @ X66 ) =>
% 132.32/17.32 ( in @ X68 @ X62 ) ) ) ) =>
% 132.32/17.32 ( ?[X72:$i,X74:$i]:
% 132.32/17.32 ( ( in @ X72 @ X64 ) &
% 132.32/17.32 ( in @ X74 @ X66 ) &
% 132.32/17.32 ( ~( ?[X80:$i]:
% 132.32/17.32 ( ( in @ X80 @ X72 ) &
% 132.32/17.32 ( in @ X80 @ X66 ) ) ) ) &
% 132.32/17.32 ( ![X76:$i]:
% 132.32/17.32 ( ( in @ X76 @ X64 ) =>
% 132.32/17.32 ( ( in @ X74 @ X76 ) |
% 132.32/17.32 ( ![X78:$i]:
% 132.32/17.32 ( ( in @ X78 @ X76 ) =>
% 132.32/17.32 ( in @ X78 @ X72 ) ) ) ) ) ) ) ) ) ) ) ) ) =>
% 132.32/17.32 ( ( ![X100:( $i > $o )]:
% 132.32/17.32 ( ( ?[X102:$i]:
% 132.32/17.32 ( ( X100 @ X102 ) &
% 132.32/17.32 ( ![X104:$i]:
% 132.32/17.32 ( ( X100 @ X104 ) =>
% 132.32/17.32 ( ( X102 ) = ( X104 ) ) ) ) ) ) =>
% 132.32/17.32 ( X100 @
% 132.32/17.32 ( descr @ ( ^[V_1:$i]: ( X100 @ V_1 ) ) ) ) ) ) =>
% 132.32/17.32 ( ( ![X106:$i,X108:( $i > $o ),X110:$i]:
% 132.32/17.32 ( ( in @ X110 @ X106 ) =>
% 132.32/17.32 ( ( X108 @ X110 ) =>
% 132.32/17.32 ( in @
% 132.32/17.32 X110 @
% 132.32/17.32 ( dsetconstr @
% 132.32/17.32 X106 @ ( ^[V_2:$i]: ( X108 @ V_2 ) ) ) ) ) ) ) =>
% 132.32/17.32 ( ( ![X112:$i,X114:( $i > $o ),X116:$i]:
% 132.32/17.32 ( ( in @
% 132.32/17.32 X116 @
% 132.32/17.32 ( dsetconstr @
% 132.32/17.32 X112 @ ( ^[V_3:$i]: ( X114 @ V_3 ) ) ) ) =>
% 132.32/17.32 ( in @ X116 @ X112 ) ) ) =>
% 132.32/17.32 ( ( ![X118:$i,X120:( $i > $o ),X122:$i]:
% 132.32/17.32 ( ( in @
% 132.32/17.32 X122 @
% 132.32/17.32 ( dsetconstr @
% 132.32/17.32 X118 @ ( ^[V_4:$i]: ( X120 @ V_4 ) ) ) ) =>
% 132.32/17.32 ( X120 @ X122 ) ) ) =>
% 132.32/17.32 ( ( ![X124:( $i > $o )]:
% 132.32/17.32 ( ( ?[X126:$i]:
% 132.32/17.32 ( ( X124 @ X126 ) &
% 132.32/17.32 ( ![X128:$i]:
% 132.32/17.32 ( ( X124 @ X128 ) =>
% 132.32/17.32 ( ( X126 ) = ( X128 ) ) ) ) ) ) =>
% 132.32/17.32 ( ?[X130:$i]:
% 132.32/17.32 ( ( X124 @ X130 ) &
% 132.32/17.32 ( ![X132:$i]:
% 132.32/17.32 ( ( X124 @ X132 ) =>
% 132.32/17.32 ( ( X130 ) = ( X132 ) ) ) ) ) ) ) ) =>
% 132.32/17.32 ( ( ![X134:$o,X136:$i]:
% 132.32/17.32 ( ( in @ X136 @ ( prop2set @ X134 ) ) =>
% 132.32/17.32 ( X134 ) ) ) =>
% 132.32/17.32 ( ( ![X138:$i]:
% 132.32/17.32 ( ( in @ X138 @ emptyset ) =>
% 132.32/17.32 ( ![X140:$o]: ( X140 ) ) ) ) =>
% 132.32/17.32 ( ( ![X142:$i]:
% 132.32/17.32 ( ( in @ X142 @ emptyset ) =>
% 132.32/17.32 ( $false ) ) ) =>
% 132.32/17.32 ( ( ![X144:$i]:
% 132.32/17.32 ( ~( in @ X144 @ emptyset ) ) ) =>
% 132.32/17.32 ( ( ![X146:( $i > $o )]:
% 132.32/17.32 ( ( ?[X148:$i]:
% 132.32/17.32 ( ( X146 @ X148 ) &
% 132.32/17.32 ( ![X150:$i]:
% 132.32/17.32 ( ( X146 @ X150 ) =>
% 132.32/17.32 ( ( X148 ) = ( X150 ) ) ) ) ) ) =>
% 132.32/17.32 ( ?[X152:$i]: ( X146 @ X152 ) ) ) ) =>
% 132.32/17.32 ( ( ![X154:$i,X156:$i]:
% 132.32/17.32 ( ( ![X158:$i]:
% 132.32/17.32 ( ( in @ X158 @ X154 ) =>
% 132.32/17.32 ( in @ X158 @ X156 ) ) ) =>
% 132.32/17.32 ( ( ![X160:$i]:
% 132.32/17.32 ( ( in @ X160 @ X156 ) =>
% 132.32/17.32 ( in @ X160 @ X154 ) ) ) =>
% 132.32/17.32 ( ( X154 ) = ( X156 ) ) ) ) ) =>
% 132.32/17.32 ( ( ![X162:$i]:
% 132.32/17.32 ( ( ![X164:$i]:
% 132.32/17.32 ( ~( in @ X164 @ X162 ) ) ) =>
% 132.32/17.32 ( ( X162 ) = ( emptyset ) ) ) ) =>
% 132.32/17.32 ( ( ![X166:$i]:
% 132.32/17.32 ( ( ![X168:$i]:
% 132.32/17.32 ( ~( in @ X168 @ X166 ) ) ) =>
% 132.32/17.32 ( ( X166 ) =
% 132.32/17.32 ( emptyset ) ) ) ) =>
% 132.32/17.32 ( ( ![X170:$i,
% 132.32/17.32 X172:( $i > $o ),
% 132.32/17.32 X174:$i]:
% 132.32/17.32 ( ( in @ X174 @ X170 ) =>
% 132.32/17.32 ( ( in @
% 132.32/17.32 X174 @
% 132.32/17.32 ( dsetconstr @
% 132.32/17.32 X170 @
% 132.32/17.32 ( ^[V_5:$i]:
% 132.32/17.32 ( X172 @ V_5 ) ) ) ) <=>
% 132.32/17.32 ( X172 @ X174 ) ) ) ) =>
% 132.32/17.32 ( ( ![X176:$i]:
% 132.32/17.32 ( ( ( X176 ) !=
% 132.32/17.32 ( emptyset ) ) =>
% 132.32/17.32 ( ?[X178:$i]:
% 132.32/17.32 ( in @ X178 @ X176 ) ) ) ) =>
% 132.32/17.32 ( ( ![X180:$i,
% 132.32/17.32 X182:( $i > $o ),
% 132.32/17.32 X184:$i]:
% 132.32/17.32 ( ( in @ X184 @ X180 ) =>
% 132.32/17.32 ( ( X182 @ X184 ) =>
% 132.32/17.32 ( ( dsetconstr @
% 132.32/17.32 X180 @
% 132.32/17.32 ( ^[
% 132.32/17.32 V_6:$i]:
% 132.32/17.32 (
% 132.32/17.32 X182 @ V_6 ) ) ) !=
% 132.32/17.32 ( emptyset ) ) ) ) ) =>
% 132.32/17.32 ( ( ![X186:$i]:
% 132.32/17.32 ( ( ?[X188:$i]:
% 132.32/17.32 ( in @
% 132.32/17.32 X188 @ X186 ) ) =>
% 132.32/17.32 ( ( X186 ) !=
% 132.32/17.32 ( emptyset ) ) ) ) =>
% 132.32/17.32 ( ( ![X190:$i,
% 132.32/17.32 X192:$i]:
% 132.32/17.32 ( in @
% 132.32/17.32 X190 @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 X190 @ X192 ) ) ) =>
% 132.32/17.32 ( ( in @
% 132.32/17.32 emptyset @
% 132.32/17.32 ( setadjoin @
% 132.32/17.32 emptyset @
% 132.32/17.32 emptyset ) ) =>
% 132.32/17.32 ( ( ![X194:$i,
% 132.32/17.32 X196:$i,
% 132.32/17.32 X198:$i]:
% 132.32/17.32 ( (
% 132.32/17.32 in @
% 132.32/17.32 X198 @
% 132.32/17.32 X196 ) =>
% 132.32/17.32 (
% 132.32/17.32 in @
% 132.32/17.32 X198 @
% 132.32/17.32 (
% 132.32/17.32 setadjoin
% 132.32/17.32 @
% 132.32/17.32 X194 @
% 132.32/17.32 X196 ) ) ) ) =>
% 132.32/17.32 ( ( ![
% 132.32/17.32 X200:$i,
% 132.32/17.32 X202:$i,
% 132.32/17.32 X204:$i]:
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 in @
% 132.32/17.32 X204 @
% 132.32/17.32 (
% 132.32/17.32 setadjoin
% 132.32/17.32 @
% 132.32/17.32 X200 @
% 132.32/17.32 X202 ) ) =>
% 132.32/17.32 (
% 132.32/17.32 ![
% 132.32/17.32 X206:$o]:
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 X204 ) =
% 132.32/17.32 (
% 132.32/17.32 X200 ) ) =>
% 132.32/17.32 (
% 132.32/17.32 X206 ) ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 in @
% 132.32/17.32 X204 @
% 132.32/17.32 X202 ) =>
% 132.32/17.32 (
% 132.32/17.32 X206 ) ) =>
% 132.32/17.32 (
% 132.32/17.32 X206 ) ) ) ) ) ) =>
% 132.32/17.32 ( (
% 132.32/17.32 ![
% 132.32/17.32 X208:$i,
% 132.32/17.32 X210:$i,
% 132.32/17.32 X212:$i]:
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 in @
% 132.32/17.32 X212 @
% 132.32/17.32 (
% 132.32/17.32 setadjoin
% 132.32/17.32 @
% 132.32/17.32 X208 @
% 132.32/17.32 X210 ) ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 in @
% 132.32/17.32 X212 @
% 132.32/17.32 X210 ) |
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 X212 ) =
% 132.32/17.32 (
% 132.32/17.32 X208 ) ) ) ) ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 ![
% 132.32/17.32 X214:$i]:
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 dsetconstr
% 132.32/17.32 @
% 132.32/17.32 X214 @
% 132.32/17.32 (
% 132.32/17.32 ^[
% 132.32/17.32 V_7:$i]:
% 132.32/17.32 (
% 132.32/17.32 $true ) ) ) =
% 132.32/17.32 (
% 132.32/17.32 X214 ) ) ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 ![
% 132.32/17.32 X216:$i,
% 132.32/17.32 X218:$i]:
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 ![
% 132.32/17.32 X220:$i]:
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 in @
% 132.32/17.32 X220 @
% 132.32/17.32 X218 ) =>
% 132.32/17.32 (
% 132.32/17.32 in @
% 132.32/17.32 X220 @
% 132.32/17.32 X216 ) ) ) =>
% 132.32/17.32 (
% 132.32/17.32 in @
% 132.32/17.32 X218 @
% 132.32/17.32 (
% 132.32/17.32 powerset @
% 132.32/17.32 X216 ) ) ) ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 ![
% 132.32/17.32 X222:$i]:
% 132.32/17.32 (
% 132.32/17.32 in @
% 132.32/17.32 emptyset @
% 132.32/17.32 (
% 132.32/17.32 powerset @
% 132.32/17.32 X222 ) ) ) =>
% 132.32/17.32 (
% 132.32/17.32 (
% 132.32/17.32 ![
% 132.32/17.32 X224:$i]:
% 132.32/17.32 (
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% 132.32/17.33 X628:$i,
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% 132.32/17.33 X634:$i,
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% 132.32/17.33 X1012:$i,
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% 132.32/17.33 X1016:$i]:
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% 132.32/17.33 (
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% 132.32/17.33 X1018:$i,
% 132.32/17.33 X1020:$i,
% 132.32/17.33 X1022:$i]:
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% 132.32/17.33 X1022 @
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% 132.32/17.33 (
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% 132.32/17.33 (
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% 132.32/17.33 X1026:$i,
% 132.32/17.33 X1028:$i,
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% 132.32/17.33 X1032:$i,
% 132.32/17.33 X1034:$i,
% 132.32/17.33 X1036:$i,
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% 132.32/17.33 (
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% 132.32/17.33 (
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% 132.32/17.33 X1056:$i,
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% 132.32/17.33 X1058 ) =>
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% 132.32/17.33 (
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% 132.32/17.33 $i > $i >
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% 132.32/17.33 X1076:$i]:
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% 132.32/17.33 kpair @
% 132.32/17.33 X1076 @
% 132.32/17.33 X1078 ) @
% 132.32/17.33 (
% 132.32/17.33 dsetconstr
% 132.32/17.33 @
% 132.32/17.33 (
% 132.32/17.33 cartprod @
% 132.32/17.33 X1070 @
% 132.32/17.33 X1072 ) @
% 132.32/17.33 (
% 132.32/17.33 ^[
% 132.32/17.33 V_27:$i]:
% 132.32/17.33 (
% 132.32/17.33 ?[
% 132.32/17.33 X1080:$i]:
% 132.32/17.33 (
% 132.32/17.33 (
% 132.32/17.33 in @
% 132.32/17.33 X1080 @
% 132.32/17.33 X1070 ) &
% 132.32/17.33 (
% 132.32/17.33 ?[
% 132.32/17.33 X1082:$i]:
% 132.32/17.33 (
% 132.32/17.33 (
% 132.32/17.33 in @
% 132.32/17.33 X1082 @
% 132.32/17.33 X1072 ) &
% 132.32/17.33 (
% 132.32/17.33 X1074 @
% 132.32/17.33 X1080 @
% 132.32/17.33 X1082 ) &
% 132.32/17.33 (
% 132.32/17.33 (
% 132.32/17.33 V_27 ) =
% 132.32/17.33 (
% 132.32/17.33 kpair @
% 132.32/17.33 X1080 @
% 132.32/17.33 X1082 ) ) ) ) ) ) ) ) ) ) ) ) ) ) =>
% 132.32/17.33 (
% 132.32/17.33 (
% 132.32/17.33 ![
% 132.32/17.33 X1084:$i,
% 132.32/17.33 X1086:$i,
% 132.32/17.33 X1088:(
% 132.32/17.33 $i > $i >
% 132.32/17.33 $o )]:
% 132.32/17.33 (
% 132.32/17.33 subset @
% 132.32/17.33 (
% 132.32/17.33 dsetconstr
% 132.32/17.33 @
% 132.32/17.33 (
% 132.32/17.33 cartprod @
% 132.32/17.33 X1084 @
% 132.32/17.33 X1086 ) @
% 132.32/17.33 (
% 132.32/17.33 ^[
% 132.32/17.33 V_28:$i]:
% 132.32/17.33 (
% 132.32/17.33 ?[
% 132.32/17.33 X1090:$i]:
% 132.32/17.33 (
% 132.32/17.33 (
% 132.32/17.33 in @
% 132.32/17.33 X1090 @
% 132.32/17.33 X1084 ) &
% 132.32/17.33 (
% 132.32/17.33 ?[
% 132.32/17.33 X1092:$i]:
% 132.32/17.33 (
% 132.32/17.33 (
% 132.32/17.33 in @
% 132.32/17.33 X1092 @
% 132.32/17.33 X1086 ) &
% 132.32/17.33 (
% 132.32/17.33 X1088 @
% 132.32/17.33 X1090 @
% 132.32/17.33 X1092 ) &
% 132.32/17.33 (
% 132.32/17.33 (
% 132.32/17.33 V_28 ) =
% 132.32/17.33 (
% 132.32/17.33 kpair @
% 132.32/17.33 X1090 @
% 132.32/17.33 X1092 ) ) ) ) ) ) ) ) @
% 132.32/17.33 (
% 132.32/17.33 cartprod @
% 132.32/17.33 X1084 @
% 132.32/17.33 X1086 ) ) ) =>
% 132.32/17.33 (
% 132.32/17.33 (
% 132.32/17.33 ![
% 132.32/17.33 X1094:$i,
% 132.32/17.33 X1096:$i,
% 132.32/17.33 X1098:(
% 132.32/17.33 $i > $i >
% 132.32/17.33 $o )]:
% 132.32/17.33 (
% 132.32/17.33 subset @
% 132.32/17.33 (
% 132.32/17.33 dsetconstr
% 132.32/17.33 @
% 132.32/17.33 (
% 132.32/17.33 cartprod @
% 132.32/17.33 X1094 @
% 132.32/17.33 X1096 ) @
% 132.32/17.33 (
% 132.32/17.33 ^[
% 132.32/17.33 V_29:$i]:
% 132.32/17.33 (
% 132.32/17.33 ?[
% 132.32/17.33 X1100:$i]:
% 132.32/17.33 (
% 132.32/17.33 (
% 132.32/17.33 in @
% 132.32/17.33 X1100 @
% 132.32/17.33 X1094 ) &
% 132.32/17.33 (
% 132.32/17.33 ?[
% 132.32/17.33 X1102:$i]:
% 132.32/17.33 (
% 132.32/17.33 (
% 132.32/17.33 in @
% 132.32/17.33 X1102 @
% 132.32/17.33 X1096 ) &
% 132.32/17.33 (
% 132.32/17.33 X1098 @
% 132.32/17.33 X1100 @
% 132.32/17.33 X1102 ) &
% 132.32/17.33 (
% 132.32/17.33 (
% 132.32/17.33 V_29 ) =
% 132.32/17.33 (
% 132.32/17.33 kpair @
% 132.32/17.33 X1100 @
% 132.32/17.33 X1102 ) ) ) ) ) ) ) ) @
% 132.32/17.33 (
% 132.32/17.33 cartprod @
% 132.32/17.33 X1094 @
% 132.32/17.33 X1096 ) ) ) =>
% 132.32/17.33 (
% 132.32/17.33 (
% 132.32/17.33 ![
% 132.32/17.33 X1104:$i,
% 132.32/17.33 X1106:$i,
% 132.32/17.33 X1108:(
% 132.32/17.33 $i > $i >
% 132.32/17.33 $o ),
% 132.32/17.33 X1110:$i]:
% 132.32/17.33 (
% 132.32/17.33 (
% 132.82/17.33 in @
% 132.82/17.33 X1110 @
% 132.82/17.33 X1104 ) =>
% 132.82/17.33 (
% 132.82/17.33 ![
% 132.82/17.33 X1112:$i]:
% 132.82/17.33 (
% 132.82/17.33 (
% 132.82/17.33 in @
% 132.82/17.33 X1112 @
% 132.82/17.33 X1106 ) =>
% 132.82/17.33 (
% 132.82/17.33 (
% 132.82/17.33 in @
% 132.82/17.33 (
% 132.82/17.33 kpair @
% 132.82/17.33 X1110 @
% 132.82/17.33 X1112 ) @
% 132.82/17.33 (
% 132.82/17.33 dsetconstr
% 132.82/17.33 @
% 132.82/17.33 (
% 132.82/17.33 cartprod @
% 132.82/17.33 X1104 @
% 132.82/17.33 X1106 ) @
% 132.82/17.33 (
% 132.82/17.33 ^[
% 132.82/17.33 V_30:$i]:
% 132.82/17.33 (
% 132.82/17.33 ?[
% 132.82/17.33 X1114:$i]:
% 132.82/17.33 (
% 132.82/17.33 (
% 132.82/17.33 in @
% 132.82/17.33 X1114 @
% 132.82/17.33 X1104 ) &
% 132.82/17.33 (
% 132.82/17.33 ?[
% 132.82/17.33 X1116:$i]:
% 132.82/17.33 (
% 132.82/17.33 (
% 132.82/17.33 in @
% 132.82/17.33 X1116 @
% 132.82/17.33 X1106 ) &
% 132.82/17.33 (
% 132.82/17.33 X1108 @
% 132.82/17.33 X1114 @
% 132.82/17.33 X1116 ) &
% 132.82/17.33 (
% 132.82/17.33 (
% 132.82/17.33 V_30 ) =
% 132.82/17.33 (
% 132.82/17.33 kpair @
% 132.82/17.33 X1114 @
% 132.82/17.33 X1116 ) ) ) ) ) ) ) ) ) =>
% 132.82/17.33 (
% 132.82/17.33 X1108 @
% 132.82/17.33 X1110 @
% 132.82/17.33 X1112 ) ) ) ) ) ) =>
% 132.82/17.33 (
% 132.82/17.33 ![
% 132.82/17.33 X1118:$i,
% 132.82/17.33 X1120:$i,
% 132.82/17.33 X1122:(
% 132.82/17.33 $i > $i >
% 132.82/17.33 $o ),
% 132.82/17.33 X1124:$i,
% 132.82/17.33 X1126:$i]:
% 132.82/17.33 (
% 132.82/17.33 (
% 132.82/17.33 in @
% 132.82/17.33 (
% 132.82/17.33 kpair @
% 132.82/17.33 X1124 @
% 132.82/17.33 X1126 ) @
% 132.82/17.33 (
% 132.82/17.33 dsetconstr
% 132.82/17.33 @
% 132.82/17.33 (
% 132.82/17.33 cartprod @
% 132.82/17.33 X1118 @
% 132.82/17.33 X1120 ) @
% 132.82/17.33 (
% 132.82/17.33 ^[
% 132.82/17.33 V_31:$i]:
% 132.82/17.33 (
% 132.82/17.33 ?[
% 132.82/17.33 X1128:$i]:
% 132.82/17.33 (
% 132.82/17.33 (
% 132.82/17.33 in @
% 132.82/17.33 X1128 @
% 132.82/17.33 X1118 ) &
% 132.82/17.33 (
% 132.82/17.33 ?[
% 132.82/17.33 X1130:$i]:
% 132.82/17.33 (
% 132.82/17.33 (
% 132.82/17.33 in @
% 132.82/17.33 X1130 @
% 132.82/17.33 X1120 ) &
% 132.82/17.33 (
% 132.82/17.33 X1122 @
% 132.82/17.33 X1128 @
% 132.82/17.33 X1130 ) &
% 132.82/17.33 (
% 132.82/17.33 (
% 132.82/17.33 V_31 ) =
% 132.82/17.33 (
% 132.82/17.33 kpair @
% 132.82/17.33 X1128 @
% 132.82/17.33 X1130 ) ) ) ) ) ) ) ) ) =>
% 132.82/17.33 (
% 132.82/17.33 in @
% 132.82/17.33 X1124 @
% 132.82/17.33 X1118 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ))).
% 132.82/17.33 thf(zf_stmt_1, type, zip_tseitin_5 : $i > ( $i > $o ) > $o).
% 132.82/17.33 thf(zf_stmt_2, axiom,
% 132.82/17.33 (![X422:$i,X414:( $i > $o )]:
% 132.82/17.33 ( ( zip_tseitin_5 @ X422 @ X414 ) <=>
% 132.82/17.33 ( ( ![X424:$i]: ( zip_tseitin_4 @ X424 @ X422 @ X414 ) ) &
% 132.82/17.33 ( X414 @ X422 ) ) ))).
% 132.82/17.33 thf(zf_stmt_3, type, zip_tseitin_4 : $i > $i > ( $i > $o ) > $o).
% 132.82/17.33 thf(zf_stmt_4, axiom,
% 132.82/17.33 (![X424:$i,X422:$i,X414:( $i > $o )]:
% 132.82/17.33 ( ( zip_tseitin_4 @ X424 @ X422 @ X414 ) <=>
% 132.82/17.33 ( ( X414 @ X424 ) => ( ( X422 ) = ( X424 ) ) ) ))).
% 132.82/17.33 thf(zf_stmt_5, type, zip_tseitin_3 : $i > $i > $i > ( $i > $i > $o ) > $o).
% 132.82/17.33 thf(zf_stmt_6, axiom,
% 132.82/17.33 (![X52:$i,X50:$i,X40:$i,X38:( $i > $i > $o )]:
% 132.82/17.33 ( ( zip_tseitin_3 @ X52 @ X50 @ X40 @ X38 ) <=>
% 132.82/17.33 ( ( X38 @ X52 @ X50 ) & ( in @ X52 @ X40 ) ) ))).
% 132.82/17.33 thf(zf_stmt_7, type, zip_tseitin_2 : $i > $i > $i > $o).
% 132.82/17.33 thf(zf_stmt_8, axiom,
% 132.82/17.33 (![X28:$i,X26:$i,X24:$i]:
% 132.82/17.33 ( ( zip_tseitin_2 @ X28 @ X26 @ X24 ) <=>
% 132.82/17.33 ( ( in @ X28 @ X24 ) & ( in @ X26 @ X28 ) ) ))).
% 132.82/17.33 thf(zf_stmt_9, type, zip_tseitin_1 : $i > $i > $i > $o).
% 132.82/17.33 thf(zf_stmt_10, axiom,
% 132.82/17.33 (![X22:$i,X20:$i,X18:$i]:
% 132.82/17.33 ( ( zip_tseitin_1 @ X22 @ X20 @ X18 ) <=>
% 132.82/17.33 ( ( in @ X22 @ X20 ) => ( in @ X22 @ X18 ) ) ))).
% 132.82/17.33 thf(zf_stmt_11, type, zip_tseitin_0 : $i > $i > $i > $o).
% 132.82/17.33 thf(zf_stmt_12, axiom,
% 132.82/17.33 (![X16:$i,X14:$i,X12:$i]:
% 132.82/17.33 ( ( zip_tseitin_0 @ X16 @ X14 @ X12 ) <=>
% 132.82/17.33 ( ( ( X16 ) = ( X12 ) ) | ( in @ X16 @ X14 ) ) ))).
% 132.82/17.33 thf(zf_stmt_13, conjecture,
% 132.82/17.33 (( ![X4:$i,X6:$i]:
% 132.82/17.33 ( ( ![X8:$i]: ( ( in @ X8 @ X4 ) <=> ( in @ X8 @ X6 ) ) ) =>
% 132.82/17.33 ( ( X4 ) = ( X6 ) ) ) ) =>
% 132.82/17.33 ( ( ![X10:$i]: ( ~( in @ X10 @ emptyset ) ) ) =>
% 132.82/17.33 ( ( ![X12:$i,X14:$i,X16:$i]:
% 132.82/17.33 ( ( in @ X16 @ ( setadjoin @ X12 @ X14 ) ) <=>
% 132.82/17.33 ( zip_tseitin_0 @ X16 @ X14 @ X12 ) ) ) =>
% 132.82/17.33 ( ( ![X18:$i,X20:$i]:
% 132.82/17.33 ( ( in @ X20 @ ( powerset @ X18 ) ) <=>
% 132.82/17.33 ( ![X22:$i]: ( zip_tseitin_1 @ X22 @ X20 @ X18 ) ) ) ) =>
% 132.82/17.33 ( ( ![X24:$i,X26:$i]:
% 132.82/17.33 ( ( in @ X26 @ ( setunion @ X24 ) ) <=>
% 132.82/17.33 ( ?[X28:$i]: ( zip_tseitin_2 @ X28 @ X26 @ X24 ) ) ) ) =>
% 132.82/17.33 ( ( in @ emptyset @ omega ) =>
% 132.82/17.33 ( ( ![X30:$i]:
% 132.82/17.33 ( ( in @ X30 @ omega ) =>
% 132.82/17.33 ( in @ ( setadjoin @ X30 @ X30 ) @ omega ) ) ) =>
% 132.82/17.33 ( ( ![X32:$i]:
% 132.82/17.33 ( ( ( in @ emptyset @ X32 ) &
% 132.82/17.33 ( ![X34:$i]:
% 132.82/17.33 ( ( ( in @ X34 @ omega ) & ( in @ X34 @ X32 ) ) =>
% 132.82/17.33 ( in @ ( setadjoin @ X34 @ X34 ) @ X32 ) ) ) ) =>
% 132.82/17.33 ( ![X36:$i]:
% 132.82/17.33 ( ( in @ X36 @ omega ) => ( in @ X36 @ X32 ) ) ) ) ) =>
% 132.82/17.33 ( ( ![X38:( $i > $i > $o ),X40:$i]:
% 132.82/17.33 ( ( ![X42:$i]:
% 132.82/17.33 ( ( in @ X42 @ X40 ) =>
% 132.82/17.33 ( ?[X44:$i]:
% 132.82/17.33 ( ( ![X46:$i]:
% 132.82/17.33 ( ( X38 @ X42 @ X46 ) =>
% 132.82/17.33 ( ( X44 ) = ( X46 ) ) ) ) &
% 132.82/17.33 ( X38 @ X42 @ X44 ) ) ) ) ) =>
% 132.82/17.33 ( ?[X48:$i]:
% 132.82/17.33 ( ![X50:$i]:
% 132.82/17.33 ( ( in @ X50 @ X48 ) <=>
% 132.82/17.33 ( ?[X52:$i]:
% 132.82/17.33 ( zip_tseitin_3 @ X52 @ X50 @ X40 @ X38 ) ) ) ) ) ) ) =>
% 132.82/17.33 ( ( ![X54:$i]:
% 132.82/17.33 ( ( ?[X56:$i]: ( in @ X56 @ X54 ) ) =>
% 132.82/17.33 ( ?[X58:$i]:
% 132.82/17.33 ( ( ~( ?[X60:$i]:
% 132.82/17.33 ( ( in @ X60 @ X54 ) & ( in @ X60 @ X58 ) ) ) ) &
% 132.82/17.33 ( in @ X58 @ X54 ) ) ) ) ) =>
% 132.82/17.33 ( ( ![X62:$i]:
% 132.82/17.33 ( ?[X64:$i]:
% 132.82/17.33 ( ( ![X66:$i]:
% 132.82/17.33 ( ( ( ![X68:$i]:
% 132.82/17.33 ( ( in @ X68 @ X66 ) =>
% 132.82/17.33 ( in @ X68 @ X62 ) ) ) &
% 132.82/17.33 ( ?[X70:$i]: ( in @ X70 @ X66 ) ) ) =>
% 132.82/17.33 ( ?[X72:$i,X74:$i]:
% 132.82/17.33 ( ( ![X76:$i]:
% 132.82/17.33 ( ( in @ X76 @ X64 ) =>
% 132.82/17.33 ( ( ![X78:$i]:
% 132.82/17.33 ( ( in @ X78 @ X76 ) =>
% 132.82/17.33 ( in @ X78 @ X72 ) ) ) |
% 132.82/17.33 ( in @ X74 @ X76 ) ) ) ) &
% 132.82/17.33 ( ~( ?[X80:$i]:
% 132.82/17.33 ( ( in @ X80 @ X66 ) &
% 132.82/17.33 ( in @ X80 @ X72 ) ) ) ) &
% 132.82/17.33 ( in @ X74 @ X66 ) & ( in @ X72 @ X64 ) ) ) ) ) &
% 132.82/17.33 ( ![X82:$i,X84:$i]:
% 132.82/17.33 ( ( ( in @ X82 @ X64 ) & ( in @ X84 @ X64 ) ) =>
% 132.82/17.33 ( ( ![X86:$i]:
% 132.82/17.33 ( ( in @ X86 @ X82 ) =>
% 132.82/17.33 ( in @ X86 @ X84 ) ) ) |
% 132.82/17.33 ( ![X88:$i]:
% 132.82/17.33 ( ( in @ X88 @ X84 ) =>
% 132.82/17.33 ( in @ X88 @ X82 ) ) ) ) ) ) &
% 132.82/17.33 ( ![X90:$i,X92:$i]:
% 132.82/17.33 ( ( ( in @ X90 @ X62 ) & ( in @ X92 @ X62 ) ) =>
% 132.82/17.33 ( ( ![X94:$i]:
% 132.82/17.33 ( ( in @ X94 @ X64 ) =>
% 132.82/17.33 ( ( in @ X90 @ X94 ) <=>
% 132.82/17.33 ( in @ X92 @ X94 ) ) ) ) =>
% 132.82/17.33 ( ( X90 ) = ( X92 ) ) ) ) ) &
% 132.82/17.33 ( ![X96:$i]:
% 132.82/17.33 ( ( in @ X96 @ X64 ) =>
% 132.82/17.33 ( ![X98:$i]:
% 132.82/17.33 ( ( in @ X98 @ X96 ) => ( in @ X98 @ X62 ) ) ) ) ) ) ) ) =>
% 132.82/17.33 ( ( ![X100:( $i > $o )]:
% 132.82/17.33 ( ( ?[X102:$i]:
% 132.82/17.33 ( ( ![X104:$i]:
% 132.82/17.33 ( ( X100 @ X104 ) =>
% 132.82/17.33 ( ( X102 ) = ( X104 ) ) ) ) &
% 132.82/17.33 ( X100 @ X102 ) ) ) =>
% 132.82/17.33 ( X100 @
% 132.82/17.33 ( descr @ ( ^[V_1:$i]: ( X100 @ V_1 ) ) ) ) ) ) =>
% 132.82/17.33 ( ( ![X106:$i,X108:( $i > $o ),X110:$i]:
% 132.82/17.33 ( ( in @ X110 @ X106 ) =>
% 132.82/17.33 ( ( X108 @ X110 ) =>
% 132.82/17.33 ( in @
% 132.82/17.33 X110 @
% 132.82/17.33 ( dsetconstr @
% 132.82/17.33 X106 @ ( ^[V_2:$i]: ( X108 @ V_2 ) ) ) ) ) ) ) =>
% 132.82/17.33 ( ( ![X112:$i,X114:( $i > $o ),X116:$i]:
% 132.82/17.33 ( ( in @
% 132.82/17.33 X116 @
% 132.82/17.33 ( dsetconstr @
% 132.82/17.33 X112 @ ( ^[V_3:$i]: ( X114 @ V_3 ) ) ) ) =>
% 132.82/17.33 ( in @ X116 @ X112 ) ) ) =>
% 132.82/17.33 ( ( ![X118:$i,X120:( $i > $o ),X122:$i]:
% 132.82/17.33 ( ( in @
% 132.82/17.33 X122 @
% 132.82/17.33 ( dsetconstr @
% 132.82/17.33 X118 @ ( ^[V_4:$i]: ( X120 @ V_4 ) ) ) ) =>
% 132.82/17.33 ( X120 @ X122 ) ) ) =>
% 132.82/17.33 ( ( ![X124:( $i > $o )]:
% 132.82/17.33 ( ( ?[X126:$i]:
% 132.82/17.33 ( ( ![X128:$i]:
% 132.82/17.33 ( ( X124 @ X128 ) =>
% 132.82/17.33 ( ( X126 ) = ( X128 ) ) ) ) &
% 132.82/17.33 ( X124 @ X126 ) ) ) =>
% 132.82/17.33 ( ?[X130:$i]:
% 132.82/17.33 ( ( ![X132:$i]:
% 132.82/17.33 ( ( X124 @ X132 ) =>
% 132.82/17.33 ( ( X130 ) = ( X132 ) ) ) ) &
% 132.82/17.33 ( X124 @ X130 ) ) ) ) ) =>
% 132.82/17.33 ( ( ![X134:$o,X136:$i]:
% 132.82/17.33 ( ( in @ X136 @ ( prop2set @ X134 ) ) =>
% 132.82/17.33 ( X134 ) ) ) =>
% 132.82/17.33 ( ( ![X138:$i]:
% 132.82/17.33 ( ( in @ X138 @ emptyset ) =>
% 132.82/17.33 ( ![X140:$o]: ( X140 ) ) ) ) =>
% 132.82/17.33 ( ( ![X142:$i]:
% 132.82/17.33 ( ~( in @ X142 @ emptyset ) ) ) =>
% 132.82/17.33 ( ( ![X144:$i]:
% 132.82/17.33 ( ~( in @ X144 @ emptyset ) ) ) =>
% 132.82/17.33 ( ( ![X146:( $i > $o )]:
% 132.82/17.33 ( ( ?[X148:$i]:
% 132.82/17.33 ( ( ![X150:$i]:
% 132.82/17.33 ( ( X146 @ X150 ) =>
% 132.82/17.33 ( ( X148 ) = ( X150 ) ) ) ) &
% 132.82/17.33 ( X146 @ X148 ) ) ) =>
% 132.82/17.33 ( ?[X152:$i]: ( X146 @ X152 ) ) ) ) =>
% 132.82/17.33 ( ( ![X154:$i,X156:$i]:
% 132.82/17.33 ( ( ![X158:$i]:
% 132.82/17.33 ( ( in @ X158 @ X154 ) =>
% 132.82/17.33 ( in @ X158 @ X156 ) ) ) =>
% 132.82/17.33 ( ( ![X160:$i]:
% 132.82/17.33 ( ( in @ X160 @ X156 ) =>
% 132.82/17.33 ( in @ X160 @ X154 ) ) ) =>
% 132.82/17.33 ( ( X154 ) = ( X156 ) ) ) ) ) =>
% 132.82/17.33 ( ( ![X162:$i]:
% 132.82/17.33 ( ( ![X164:$i]:
% 132.82/17.33 ( ~( in @ X164 @ X162 ) ) ) =>
% 132.82/17.33 ( ( X162 ) = ( emptyset ) ) ) ) =>
% 132.82/17.33 ( ( ![X166:$i]:
% 132.82/17.33 ( ( ![X168:$i]:
% 132.82/17.33 ( ~( in @ X168 @ X166 ) ) ) =>
% 132.82/17.33 ( ( X166 ) =
% 132.82/17.33 ( emptyset ) ) ) ) =>
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% 132.82/17.34 X1032:$i,
% 132.82/17.34 X1034:$i,
% 132.82/17.34 X1036:$i,
% 132.82/17.34 X1038:$i]:
% 132.82/17.34 (
% 132.82/17.34 (
% 132.82/17.34 in @
% 132.82/17.34 (
% 132.82/17.34 kpair @
% 132.82/17.34 X1036 @
% 132.82/17.34 X1038 ) @
% 132.82/17.34 (
% 132.82/17.34 cartprod @
% 132.82/17.34 X1032 @
% 132.82/17.34 X1034 ) ) =>
% 132.82/17.34 (
% 132.82/17.34 in @
% 132.82/17.34 X1038 @
% 132.82/17.34 X1034 ) ) ) =>
% 132.82/17.34 (
% 132.82/17.34 (
% 132.82/17.34 ![
% 132.82/17.34 X1040:$i,
% 132.82/17.34 X1042:$i,
% 132.82/17.34 X1044:$i]:
% 132.82/17.34 (
% 132.82/17.34 (
% 132.82/17.34 in @
% 132.82/17.34 X1044 @
% 132.82/17.34 X1040 ) =>
% 132.82/17.34 (
% 132.82/17.34 ![
% 132.82/17.34 X1046:$i]:
% 132.82/17.34 (
% 132.82/17.34 (
% 132.82/17.34 in @
% 132.82/17.34 X1046 @
% 132.82/17.34 X1042 ) =>
% 132.82/17.34 (
% 132.82/17.34 (
% 132.82/17.34 kpair @
% 132.82/17.34 X1044 @
% 132.82/17.34 X1046 ) =
% 132.82/17.34 (
% 132.82/17.34 kpair @
% 132.82/17.34 X1044 @
% 132.82/17.34 X1046 ) ) ) ) ) ) =>
% 132.82/17.34 (
% 132.82/17.34 (
% 132.82/17.34 ![
% 132.82/17.34 X1048:$i,
% 132.82/17.34 X1050:$i,
% 132.82/17.34 X1052:$i]:
% 132.82/17.34 (
% 132.82/17.34 (
% 132.82/17.34 in @
% 132.82/17.34 X1052 @
% 132.82/17.34 X1048 ) =>
% 132.82/17.34 (
% 132.82/17.34 ![
% 132.82/17.34 X1054:$i]:
% 132.82/17.34 (
% 132.82/17.34 (
% 132.82/17.34 in @
% 132.82/17.34 X1054 @
% 132.82/17.34 X1050 ) =>
% 132.82/17.34 (
% 132.82/17.34 (
% 132.82/17.34 kfst @
% 132.82/17.34 (
% 132.82/17.34 kpair @
% 132.82/17.34 X1052 @
% 132.82/17.34 X1054 ) ) =
% 132.82/17.34 (
% 132.82/17.34 X1052 ) ) ) ) ) ) =>
% 132.82/17.34 (
% 132.82/17.34 (
% 132.82/17.34 ![
% 132.82/17.34 X1056:$i,
% 132.82/17.34 X1058:$i,
% 132.82/17.34 X1060:$i]:
% 132.82/17.34 (
% 132.82/17.34 (
% 132.82/17.34 in @
% 132.82/17.34 X1060 @
% 132.82/17.34 X1056 ) =>
% 132.82/17.34 (
% 132.82/17.34 ![
% 132.82/17.34 X1062:$i]:
% 132.82/17.34 (
% 132.82/17.34 (
% 132.82/17.34 in @
% 132.82/17.34 X1062 @
% 132.82/17.34 X1058 ) =>
% 132.82/17.34 (
% 132.82/17.34 (
% 132.82/17.34 ksnd @
% 132.82/17.34 (
% 132.82/17.34 kpair @
% 132.82/17.34 X1060 @
% 132.82/17.34 X1062 ) ) =
% 132.82/17.34 (
% 132.82/17.34 X1062 ) ) ) ) ) ) =>
% 132.82/17.34 (
% 132.82/17.34 (
% 132.82/17.34 ![
% 132.82/17.34 X1064:$i,
% 132.82/17.34 X1066:$i,
% 132.82/17.34 X1068:$i]:
% 132.82/17.34 (
% 132.82/17.34 (
% 132.82/17.34 in @
% 132.82/17.34 X1068 @
% 132.82/17.34 (
% 132.82/17.34 cartprod @
% 132.82/17.34 X1064 @
% 132.82/17.34 X1066 ) ) =>
% 132.82/17.34 (
% 132.82/17.34 (
% 132.82/17.34 kpair @
% 132.82/17.34 (
% 132.82/17.34 kfst @
% 132.82/17.34 X1068 ) @
% 132.82/17.34 (
% 132.82/17.34 ksnd @
% 132.82/17.34 X1068 ) ) =
% 132.82/17.34 (
% 132.82/17.34 X1068 ) ) ) ) =>
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 ![
% 132.82/17.35 X1070:$i,
% 132.82/17.35 X1072:$i,
% 132.82/17.35 X1074:(
% 132.82/17.35 $i > $i >
% 132.82/17.35 $o ),
% 132.82/17.35 X1076:$i]:
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 in @
% 132.82/17.35 X1076 @
% 132.82/17.35 X1070 ) =>
% 132.82/17.35 (
% 132.82/17.35 ![
% 132.82/17.35 X1078:$i]:
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 in @
% 132.82/17.35 X1078 @
% 132.82/17.35 X1072 ) =>
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 X1074 @
% 132.82/17.35 X1076 @
% 132.82/17.35 X1078 ) =>
% 132.82/17.35 (
% 132.82/17.35 in @
% 132.82/17.35 (
% 132.82/17.35 kpair @
% 132.82/17.35 X1076 @
% 132.82/17.35 X1078 ) @
% 132.82/17.35 (
% 132.82/17.35 dsetconstr
% 132.82/17.35 @
% 132.82/17.35 (
% 132.82/17.35 cartprod @
% 132.82/17.35 X1070 @
% 132.82/17.35 X1072 ) @
% 132.82/17.35 (
% 132.82/17.35 ^[
% 132.82/17.35 V_27:$i]:
% 132.82/17.35 (
% 132.82/17.35 ?[
% 132.82/17.35 X1080:$i]:
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 in @
% 132.82/17.35 X1080 @
% 132.82/17.35 X1070 ) &
% 132.82/17.35 (
% 132.82/17.35 ?[
% 132.82/17.35 X1082:$i]:
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 in @
% 132.82/17.35 X1082 @
% 132.82/17.35 X1072 ) &
% 132.82/17.35 (
% 132.82/17.35 X1074 @
% 132.82/17.35 X1080 @
% 132.82/17.35 X1082 ) &
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 V_27 ) =
% 132.82/17.35 (
% 132.82/17.35 kpair @
% 132.82/17.35 X1080 @
% 132.82/17.35 X1082 ) ) ) ) ) ) ) ) ) ) ) ) ) ) =>
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 ![
% 132.82/17.35 X1084:$i,
% 132.82/17.35 X1086:$i,
% 132.82/17.35 X1088:(
% 132.82/17.35 $i > $i >
% 132.82/17.35 $o )]:
% 132.82/17.35 (
% 132.82/17.35 subset @
% 132.82/17.35 (
% 132.82/17.35 dsetconstr
% 132.82/17.35 @
% 132.82/17.35 (
% 132.82/17.35 cartprod @
% 132.82/17.35 X1084 @
% 132.82/17.35 X1086 ) @
% 132.82/17.35 (
% 132.82/17.35 ^[
% 132.82/17.35 V_28:$i]:
% 132.82/17.35 (
% 132.82/17.35 ?[
% 132.82/17.35 X1090:$i]:
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 in @
% 132.82/17.35 X1090 @
% 132.82/17.35 X1084 ) &
% 132.82/17.35 (
% 132.82/17.35 ?[
% 132.82/17.35 X1092:$i]:
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 in @
% 132.82/17.35 X1092 @
% 132.82/17.35 X1086 ) &
% 132.82/17.35 (
% 132.82/17.35 X1088 @
% 132.82/17.35 X1090 @
% 132.82/17.35 X1092 ) &
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 V_28 ) =
% 132.82/17.35 (
% 132.82/17.35 kpair @
% 132.82/17.35 X1090 @
% 132.82/17.35 X1092 ) ) ) ) ) ) ) ) @
% 132.82/17.35 (
% 132.82/17.35 cartprod @
% 132.82/17.35 X1084 @
% 132.82/17.35 X1086 ) ) ) =>
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 ![
% 132.82/17.35 X1094:$i,
% 132.82/17.35 X1096:$i,
% 132.82/17.35 X1098:(
% 132.82/17.35 $i > $i >
% 132.82/17.35 $o )]:
% 132.82/17.35 (
% 132.82/17.35 subset @
% 132.82/17.35 (
% 132.82/17.35 dsetconstr
% 132.82/17.35 @
% 132.82/17.35 (
% 132.82/17.35 cartprod @
% 132.82/17.35 X1094 @
% 132.82/17.35 X1096 ) @
% 132.82/17.35 (
% 132.82/17.35 ^[
% 132.82/17.35 V_29:$i]:
% 132.82/17.35 (
% 132.82/17.35 ?[
% 132.82/17.35 X1100:$i]:
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 in @
% 132.82/17.35 X1100 @
% 132.82/17.35 X1094 ) &
% 132.82/17.35 (
% 132.82/17.35 ?[
% 132.82/17.35 X1102:$i]:
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 in @
% 132.82/17.35 X1102 @
% 132.82/17.35 X1096 ) &
% 132.82/17.35 (
% 132.82/17.35 X1098 @
% 132.82/17.35 X1100 @
% 132.82/17.35 X1102 ) &
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 V_29 ) =
% 132.82/17.35 (
% 132.82/17.35 kpair @
% 132.82/17.35 X1100 @
% 132.82/17.35 X1102 ) ) ) ) ) ) ) ) @
% 132.82/17.35 (
% 132.82/17.35 cartprod @
% 132.82/17.35 X1094 @
% 132.82/17.35 X1096 ) ) ) =>
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 ![
% 132.82/17.35 X1104:$i,
% 132.82/17.35 X1106:$i,
% 132.82/17.35 X1108:(
% 132.82/17.35 $i > $i >
% 132.82/17.35 $o ),
% 132.82/17.35 X1110:$i]:
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 in @
% 132.82/17.35 X1110 @
% 132.82/17.35 X1104 ) =>
% 132.82/17.35 (
% 132.82/17.35 ![
% 132.82/17.35 X1112:$i]:
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 in @
% 132.82/17.35 X1112 @
% 132.82/17.35 X1106 ) =>
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 in @
% 132.82/17.35 (
% 132.82/17.35 kpair @
% 132.82/17.35 X1110 @
% 132.82/17.35 X1112 ) @
% 132.82/17.35 (
% 132.82/17.35 dsetconstr
% 132.82/17.35 @
% 132.82/17.35 (
% 132.82/17.35 cartprod @
% 132.82/17.35 X1104 @
% 132.82/17.35 X1106 ) @
% 132.82/17.35 (
% 132.82/17.35 ^[
% 132.82/17.35 V_30:$i]:
% 132.82/17.35 (
% 132.82/17.35 ?[
% 132.82/17.35 X1114:$i]:
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 in @
% 132.82/17.35 X1114 @
% 132.82/17.35 X1104 ) &
% 132.82/17.35 (
% 132.82/17.35 ?[
% 132.82/17.35 X1116:$i]:
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 in @
% 132.82/17.35 X1116 @
% 132.82/17.35 X1106 ) &
% 132.82/17.35 (
% 132.82/17.35 X1108 @
% 132.82/17.35 X1114 @
% 132.82/17.35 X1116 ) &
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 V_30 ) =
% 132.82/17.35 (
% 132.82/17.35 kpair @
% 132.82/17.35 X1114 @
% 132.82/17.35 X1116 ) ) ) ) ) ) ) ) ) =>
% 132.82/17.35 (
% 132.82/17.35 X1108 @
% 132.82/17.35 X1110 @
% 132.82/17.35 X1112 ) ) ) ) ) ) =>
% 132.82/17.35 (
% 132.82/17.35 ![
% 132.82/17.35 X1118:$i,
% 132.82/17.35 X1120:$i,
% 132.82/17.35 X1122:(
% 132.82/17.35 $i > $i >
% 132.82/17.35 $o ),
% 132.82/17.35 X1124:$i,
% 132.82/17.35 X1126:$i]:
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 in @
% 132.82/17.35 (
% 132.82/17.35 kpair @
% 132.82/17.35 X1124 @
% 132.82/17.35 X1126 ) @
% 132.82/17.35 (
% 132.82/17.35 dsetconstr
% 132.82/17.35 @
% 132.82/17.35 (
% 132.82/17.35 cartprod @
% 132.82/17.35 X1118 @
% 132.82/17.35 X1120 ) @
% 132.82/17.35 (
% 132.82/17.35 ^[
% 132.82/17.35 V_31:$i]:
% 132.82/17.35 (
% 132.82/17.35 ?[
% 132.82/17.35 X1128:$i]:
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 in @
% 132.82/17.35 X1128 @
% 132.82/17.35 X1118 ) &
% 132.82/17.35 (
% 132.82/17.35 ?[
% 132.82/17.35 X1130:$i]:
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 in @
% 132.82/17.35 X1130 @
% 132.82/17.35 X1120 ) &
% 132.82/17.35 (
% 132.82/17.35 X1122 @
% 132.82/17.35 X1128 @
% 132.82/17.35 X1130 ) &
% 132.82/17.35 (
% 132.82/17.35 (
% 132.82/17.35 V_31 ) =
% 132.82/17.35 (
% 132.82/17.35 kpair @
% 132.82/17.35 X1128 @
% 132.82/17.35 X1130 ) ) ) ) ) ) ) ) ) =>
% 132.82/17.35 (
% 132.82/17.35 in @
% 132.82/17.35 X1124 @
% 132.82/17.35 X1118 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ))).
% 132.82/17.35 thf(zf_stmt_14, negated_conjecture,
% 132.82/17.35 (~( ( ![X4:$i,X6:$i]:
% 132.82/17.35 ( ( ![X8:$i]: ( ( in @ X8 @ X4 ) <=> ( in @ X8 @ X6 ) ) ) =>
% 132.82/17.35 ( ( X4 ) = ( X6 ) ) ) ) =>
% 132.82/17.35 ( ( ![X10:$i]: ( ~( in @ X10 @ emptyset ) ) ) =>
% 132.82/17.35 ( ( ![X12:$i,X14:$i,X16:$i]:
% 132.82/17.35 ( ( in @ X16 @ ( setadjoin @ X12 @ X14 ) ) <=>
% 132.82/17.35 ( zip_tseitin_0 @ X16 @ X14 @ X12 ) ) ) =>
% 132.82/17.35 ( ( ![X18:$i,X20:$i]:
% 132.82/17.35 ( ( in @ X20 @ ( powerset @ X18 ) ) <=>
% 132.82/17.35 ( ![X22:$i]: ( zip_tseitin_1 @ X22 @ X20 @ X18 ) ) ) ) =>
% 132.82/17.35 ( ( ![X24:$i,X26:$i]:
% 132.82/17.35 ( ( in @ X26 @ ( setunion @ X24 ) ) <=>
% 132.82/17.35 ( ?[X28:$i]: ( zip_tseitin_2 @ X28 @ X26 @ X24 ) ) ) ) =>
% 132.82/17.35 ( ( in @ emptyset @ omega ) =>
% 132.82/17.35 ( ( ![X30:$i]:
% 132.82/17.35 ( ( in @ X30 @ omega ) =>
% 132.82/17.35 ( in @ ( setadjoin @ X30 @ X30 ) @ omega ) ) ) =>
% 132.82/17.35 ( ( ![X32:$i]:
% 132.82/17.35 ( ( ( in @ emptyset @ X32 ) &
% 132.82/17.35 ( ![X34:$i]:
% 132.82/17.35 ( ( ( in @ X34 @ omega ) & ( in @ X34 @ X32 ) ) =>
% 132.82/17.35 ( in @ ( setadjoin @ X34 @ X34 ) @ X32 ) ) ) ) =>
% 132.82/17.35 ( ![X36:$i]:
% 132.82/17.35 ( ( in @ X36 @ omega ) => ( in @ X36 @ X32 ) ) ) ) ) =>
% 132.82/17.35 ( ( ![X38:( $i > $i > $o ),X40:$i]:
% 132.82/17.35 ( ( ![X42:$i]:
% 132.82/17.35 ( ( in @ X42 @ X40 ) =>
% 132.82/17.35 ( ?[X44:$i]:
% 132.82/17.35 ( ( ![X46:$i]:
% 132.82/17.35 ( ( X38 @ X42 @ X46 ) =>
% 132.82/17.35 ( ( X44 ) = ( X46 ) ) ) ) &
% 132.82/17.35 ( X38 @ X42 @ X44 ) ) ) ) ) =>
% 132.82/17.35 ( ?[X48:$i]:
% 132.82/17.35 ( ![X50:$i]:
% 132.82/17.35 ( ( in @ X50 @ X48 ) <=>
% 132.82/17.35 ( ?[X52:$i]:
% 132.82/17.35 ( zip_tseitin_3 @ X52 @ X50 @ X40 @ X38 ) ) ) ) ) ) ) =>
% 132.82/17.35 ( ( ![X54:$i]:
% 132.82/17.35 ( ( ?[X56:$i]: ( in @ X56 @ X54 ) ) =>
% 132.82/17.35 ( ?[X58:$i]:
% 132.82/17.35 ( ( ~( ?[X60:$i]:
% 132.82/17.35 ( ( in @ X60 @ X54 ) &
% 132.82/17.35 ( in @ X60 @ X58 ) ) ) ) &
% 132.82/17.35 ( in @ X58 @ X54 ) ) ) ) ) =>
% 132.82/17.35 ( ( ![X62:$i]:
% 132.82/17.35 ( ?[X64:$i]:
% 132.82/17.35 ( ( ![X66:$i]:
% 132.82/17.35 ( ( ( ![X68:$i]:
% 132.82/17.35 ( ( in @ X68 @ X66 ) =>
% 132.82/17.35 ( in @ X68 @ X62 ) ) ) &
% 132.82/17.35 ( ?[X70:$i]: ( in @ X70 @ X66 ) ) ) =>
% 132.82/17.35 ( ?[X72:$i,X74:$i]:
% 132.82/17.35 ( ( ![X76:$i]:
% 132.82/17.35 ( ( in @ X76 @ X64 ) =>
% 132.82/17.35 ( ( ![X78:$i]:
% 132.82/17.35 ( ( in @ X78 @ X76 ) =>
% 132.82/17.35 ( in @ X78 @ X72 ) ) ) |
% 132.82/17.35 ( in @ X74 @ X76 ) ) ) ) &
% 132.82/17.35 ( ~( ?[X80:$i]:
% 132.82/17.35 ( ( in @ X80 @ X66 ) &
% 132.82/17.35 ( in @ X80 @ X72 ) ) ) ) &
% 132.82/17.35 ( in @ X74 @ X66 ) &
% 132.82/17.35 ( in @ X72 @ X64 ) ) ) ) ) &
% 132.82/17.35 ( ![X82:$i,X84:$i]:
% 132.82/17.35 ( ( ( in @ X82 @ X64 ) & ( in @ X84 @ X64 ) ) =>
% 132.82/17.35 ( ( ![X86:$i]:
% 132.82/17.35 ( ( in @ X86 @ X82 ) =>
% 132.82/17.35 ( in @ X86 @ X84 ) ) ) |
% 132.82/17.35 ( ![X88:$i]:
% 132.82/17.35 ( ( in @ X88 @ X84 ) =>
% 132.82/17.35 ( in @ X88 @ X82 ) ) ) ) ) ) &
% 132.82/17.35 ( ![X90:$i,X92:$i]:
% 132.82/17.35 ( ( ( in @ X90 @ X62 ) & ( in @ X92 @ X62 ) ) =>
% 132.82/17.35 ( ( ![X94:$i]:
% 132.82/17.35 ( ( in @ X94 @ X64 ) =>
% 132.82/17.35 ( ( in @ X90 @ X94 ) <=>
% 132.82/17.35 ( in @ X92 @ X94 ) ) ) ) =>
% 132.82/17.35 ( ( X90 ) = ( X92 ) ) ) ) ) &
% 132.82/17.35 ( ![X96:$i]:
% 132.82/17.35 ( ( in @ X96 @ X64 ) =>
% 132.82/17.35 ( ![X98:$i]:
% 132.82/17.35 ( ( in @ X98 @ X96 ) =>
% 132.82/17.35 ( in @ X98 @ X62 ) ) ) ) ) ) ) ) =>
% 132.82/17.35 ( ( ![X100:( $i > $o )]:
% 132.82/17.35 ( ( ?[X102:$i]:
% 132.82/17.35 ( ( ![X104:$i]:
% 132.82/17.35 ( ( X100 @ X104 ) =>
% 132.82/17.35 ( ( X102 ) = ( X104 ) ) ) ) &
% 132.82/17.35 ( X100 @ X102 ) ) ) =>
% 132.82/17.35 ( X100 @
% 132.82/17.35 ( descr @ ( ^[V_1:$i]: ( X100 @ V_1 ) ) ) ) ) ) =>
% 132.82/17.35 ( ( ![X106:$i,X108:( $i > $o ),X110:$i]:
% 132.82/17.35 ( ( in @ X110 @ X106 ) =>
% 132.82/17.35 ( ( X108 @ X110 ) =>
% 132.82/17.35 ( in @
% 132.82/17.35 X110 @
% 132.82/17.35 ( dsetconstr @
% 132.82/17.35 X106 @ ( ^[V_2:$i]: ( X108 @ V_2 ) ) ) ) ) ) ) =>
% 132.82/17.35 ( ( ![X112:$i,X114:( $i > $o ),X116:$i]:
% 132.82/17.35 ( ( in @
% 132.82/17.35 X116 @
% 132.82/17.35 ( dsetconstr @
% 132.82/17.35 X112 @ ( ^[V_3:$i]: ( X114 @ V_3 ) ) ) ) =>
% 132.82/17.35 ( in @ X116 @ X112 ) ) ) =>
% 132.82/17.35 ( ( ![X118:$i,X120:( $i > $o ),X122:$i]:
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% 132.82/17.36 (
% 132.82/17.36 setadjoin
% 132.82/17.36 @
% 132.82/17.36 X996 @
% 132.82/17.36 emptyset ) @
% 132.82/17.36 (
% 132.82/17.36 setadjoin
% 132.82/17.36 @
% 132.82/17.36 (
% 132.82/17.36 setadjoin
% 132.82/17.36 @
% 132.82/17.36 X996 @
% 132.82/17.36 (
% 132.82/17.36 setadjoin
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% 132.82/17.36 X998 @
% 132.82/17.36 emptyset ) ) @
% 132.82/17.36 emptyset ) ) ) =>
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 X994 ) =
% 132.82/17.36 (
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% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 ![
% 132.82/17.36 X1000:$i,
% 132.82/17.36 X1002:$i,
% 132.82/17.36 X1004:$i,
% 132.82/17.36 X1006:$i]:
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 kpair @
% 132.82/17.36 X1000 @
% 132.82/17.36 X1002 ) =
% 132.82/17.36 (
% 132.82/17.36 kpair @
% 132.82/17.36 X1004 @
% 132.82/17.36 X1006 ) ) =>
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 X1002 ) =
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% 132.82/17.36 X1006 ) ) ) ) =>
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 ![
% 132.82/17.36 X1008:$i]:
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 iskpair @
% 132.82/17.36 X1008 ) =>
% 132.82/17.36 (
% 132.82/17.36 ?[
% 132.82/17.36 X1010:$i]:
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 dsetconstr
% 132.82/17.36 @
% 132.82/17.36 (
% 132.82/17.36 setunion
% 132.82/17.36 @
% 132.82/17.36 X1008 ) @
% 132.82/17.36 (
% 132.82/17.36 ^[
% 132.82/17.36 V_25:$i]:
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 X1008 ) =
% 132.82/17.36 (
% 132.82/17.36 kpair @
% 132.82/17.36 (
% 132.82/17.36 kfst @
% 132.82/17.36 X1008 ) @
% 132.82/17.36 V_25 ) ) ) ) =
% 132.82/17.36 (
% 132.82/17.36 setadjoin
% 132.82/17.36 @
% 132.82/17.36 X1010 @
% 132.82/17.36 emptyset ) ) &
% 132.82/17.36 (
% 132.82/17.36 in @
% 132.82/17.36 X1010 @
% 132.82/17.36 (
% 132.82/17.36 dsetconstr
% 132.82/17.36 @
% 132.82/17.36 (
% 132.82/17.36 setunion
% 132.82/17.36 @
% 132.82/17.36 X1008 ) @
% 132.82/17.36 (
% 132.82/17.36 ^[
% 132.82/17.36 V_26:$i]:
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 X1008 ) =
% 132.82/17.36 (
% 132.82/17.36 kpair @
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% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 ![
% 132.82/17.36 X1012:$i,
% 132.82/17.36 X1014:$i]:
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 ksnd @
% 132.82/17.36 (
% 132.82/17.36 kpair @
% 132.82/17.36 X1012 @
% 132.82/17.36 X1014 ) ) =
% 132.82/17.36 (
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% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 ![
% 132.82/17.36 X1016:$i]:
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 iskpair @
% 132.82/17.36 X1016 ) =>
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 kpair @
% 132.82/17.36 (
% 132.82/17.36 kfst @
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% 132.82/17.36 (
% 132.82/17.36 ksnd @
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% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 ![
% 132.82/17.36 X1018:$i,
% 132.82/17.36 X1020:$i,
% 132.82/17.36 X1022:$i]:
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 in @
% 132.82/17.36 X1022 @
% 132.82/17.36 (
% 132.82/17.36 cartprod
% 132.82/17.36 @
% 132.82/17.36 X1018 @
% 132.82/17.36 X1020 ) ) =>
% 132.82/17.36 (
% 132.82/17.36 in @
% 132.82/17.36 (
% 132.82/17.36 ksnd @
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% 132.82/17.36 X1020 ) ) ) =>
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 ![
% 132.82/17.36 X1024:$i,
% 132.82/17.36 X1026:$i,
% 132.82/17.36 X1028:$i,
% 132.82/17.36 X1030:$i]:
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 in @
% 132.82/17.36 (
% 132.82/17.36 kpair @
% 132.82/17.36 X1028 @
% 132.82/17.36 X1030 ) @
% 132.82/17.36 (
% 132.82/17.36 cartprod
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% 132.82/17.36 X1024 @
% 132.82/17.36 X1026 ) ) =>
% 132.82/17.36 (
% 132.82/17.36 in @
% 132.82/17.36 X1028 @
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% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 ![
% 132.82/17.36 X1032:$i,
% 132.82/17.36 X1034:$i,
% 132.82/17.36 X1036:$i,
% 132.82/17.36 X1038:$i]:
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 in @
% 132.82/17.36 (
% 132.82/17.36 kpair @
% 132.82/17.36 X1036 @
% 132.82/17.36 X1038 ) @
% 132.82/17.36 (
% 132.82/17.36 cartprod
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% 132.82/17.36 X1032 @
% 132.82/17.36 X1034 ) ) =>
% 132.82/17.36 (
% 132.82/17.36 in @
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% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 ![
% 132.82/17.36 X1040:$i,
% 132.82/17.36 X1042:$i,
% 132.82/17.36 X1044:$i]:
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 in @
% 132.82/17.36 X1044 @
% 132.82/17.36 X1040 ) =>
% 132.82/17.36 (
% 132.82/17.36 ![
% 132.82/17.36 X1046:$i]:
% 132.82/17.36 (
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% 132.82/17.36 X1042 ) =>
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 kpair @
% 132.82/17.36 X1044 @
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% 132.82/17.36 (
% 132.82/17.36 kpair @
% 132.82/17.36 X1044 @
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% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 ![
% 132.82/17.36 X1048:$i,
% 132.82/17.36 X1050:$i,
% 132.82/17.36 X1052:$i]:
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 in @
% 132.82/17.36 X1052 @
% 132.82/17.36 X1048 ) =>
% 132.82/17.36 (
% 132.82/17.36 ![
% 132.82/17.36 X1054:$i]:
% 132.82/17.36 (
% 132.82/17.36 (
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% 132.82/17.36 X1054 @
% 132.82/17.36 X1050 ) =>
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 kfst @
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% 132.82/17.36 X1052 @
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% 132.82/17.36 (
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% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 ![
% 132.82/17.36 X1056:$i,
% 132.82/17.36 X1058:$i,
% 132.82/17.36 X1060:$i]:
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 in @
% 132.82/17.36 X1060 @
% 132.82/17.36 X1056 ) =>
% 132.82/17.36 (
% 132.82/17.36 ![
% 132.82/17.36 X1062:$i]:
% 132.82/17.36 (
% 132.82/17.36 (
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% 132.82/17.36 X1062 @
% 132.82/17.36 X1058 ) =>
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 ksnd @
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% 132.82/17.36 (
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% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 ![
% 132.82/17.36 X1064:$i,
% 132.82/17.36 X1066:$i,
% 132.82/17.36 X1068:$i]:
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 in @
% 132.82/17.36 X1068 @
% 132.82/17.36 (
% 132.82/17.36 cartprod
% 132.82/17.36 @
% 132.82/17.36 X1064 @
% 132.82/17.36 X1066 ) ) =>
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 kpair @
% 132.82/17.36 (
% 132.82/17.36 kfst @
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% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 ![
% 132.82/17.36 X1070:$i,
% 132.82/17.36 X1072:$i,
% 132.82/17.36 X1074:(
% 132.82/17.36 $i > $i >
% 132.82/17.36 $o ),
% 132.82/17.36 X1076:$i]:
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 in @
% 132.82/17.36 X1076 @
% 132.82/17.36 X1070 ) =>
% 132.82/17.36 (
% 132.82/17.36 ![
% 132.82/17.36 X1078:$i]:
% 132.82/17.36 (
% 132.82/17.36 (
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% 132.82/17.36 X1078 @
% 132.82/17.36 X1072 ) =>
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 X1074 @
% 132.82/17.36 X1076 @
% 132.82/17.36 X1078 ) =>
% 132.82/17.36 (
% 132.82/17.36 in @
% 132.82/17.36 (
% 132.82/17.36 kpair @
% 132.82/17.36 X1076 @
% 132.82/17.36 X1078 ) @
% 132.82/17.36 (
% 132.82/17.36 dsetconstr
% 132.82/17.36 @
% 132.82/17.36 (
% 132.82/17.36 cartprod
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% 132.82/17.36 X1070 @
% 132.82/17.36 X1072 ) @
% 132.82/17.36 (
% 132.82/17.36 ^[
% 132.82/17.36 V_27:$i]:
% 132.82/17.36 (
% 132.82/17.36 ?[
% 132.82/17.36 X1080:$i]:
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 in @
% 132.82/17.36 X1080 @
% 132.82/17.36 X1070 ) &
% 132.82/17.36 (
% 132.82/17.36 ?[
% 132.82/17.36 X1082:$i]:
% 132.82/17.36 (
% 132.82/17.36 (
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% 132.82/17.36 X1082 @
% 132.82/17.36 X1072 ) &
% 132.82/17.36 (
% 132.82/17.36 X1074 @
% 132.82/17.36 X1080 @
% 132.82/17.36 X1082 ) &
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 V_27 ) =
% 132.82/17.36 (
% 132.82/17.36 kpair @
% 132.82/17.36 X1080 @
% 132.82/17.36 X1082 ) ) ) ) ) ) ) ) ) ) ) ) ) ) =>
% 132.82/17.36 (
% 132.82/17.36 (
% 132.82/17.36 ![
% 132.82/17.36 X1084:$i,
% 132.82/17.36 X1086:$i,
% 132.82/17.36 X1088:(
% 132.82/17.36 $i > $i >
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% 132.82/17.36 (
% 132.82/17.36 subset @
% 132.82/17.36 (
% 132.82/17.36 dsetconstr
% 132.82/17.36 @
% 132.82/17.36 (
% 132.82/17.36 cartprod
% 132.82/17.36 @
% 132.82/17.36 X1084 @
% 132.82/17.36 X1086 ) @
% 132.82/17.36 (
% 132.82/17.36 ^[
% 132.82/17.36 V_28:$i]:
% 132.82/17.36 (
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% 132.82/17.36 X1090:$i]:
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% 132.82/17.36 X1090 @
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% 132.82/17.36 (
% 132.82/17.36 ?[
% 132.82/17.36 X1092:$i]:
% 132.82/17.36 (
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% 132.82/17.36 X1092 @
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% 132.82/17.36 X1090 @
% 132.82/17.36 X1092 ) &
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% 132.82/17.36 (
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% 132.82/17.36 (
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% 132.82/17.36 X1094:$i,
% 132.82/17.36 X1096:$i,
% 132.82/17.36 X1098:(
% 132.82/17.36 $i > $i >
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% 132.82/17.36 (
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% 132.82/17.36 (
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% 132.82/17.36 (
% 132.82/17.36 ^[
% 132.82/17.36 V_29:$i]:
% 132.82/17.36 (
% 132.82/17.36 ?[
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% 132.82/17.36 X1100 @
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% 132.82/17.36 (
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% 132.82/17.36 X1104:$i,
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% 132.82/17.36 $i > $i >
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% 132.82/17.36 X1110:$i]:
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% 132.82/17.36 X1112:$i]:
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% 132.82/17.36 (
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% 132.82/17.36 ?[
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% 132.82/17.36 X1114 @
% 132.82/17.36 X1116 ) &
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% 132.82/17.36 (
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% 132.82/17.36 (
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% 132.82/17.36 X1118:$i,
% 132.82/17.36 X1120:$i,
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% 132.82/17.36 $i > $i >
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% 132.82/17.36 X1124:$i,
% 132.82/17.36 X1126:$i]:
% 132.82/17.36 (
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% 132.82/17.36 kpair @
% 132.82/17.36 X1124 @
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% 132.82/17.36 (
% 132.82/17.36 ^[
% 132.82/17.36 V_31:$i]:
% 132.82/17.36 (
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% 132.82/17.36 X1128:$i]:
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% 132.82/17.36 ?[
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% 132.82/17.36 (
% 132.82/17.36 in @
% 132.82/17.36 X1124 @
% 132.82/17.36 X1118 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )),
% 132.82/17.36 inference('cnf.neg', [status(esa)], [zf_stmt_13])).
% 132.82/17.36 thf(zip_derived_cl166, plain, (~ (in @ sk__559 @ sk__556)),
% 132.82/17.36 inference('cnf', [status(esa)], [zf_stmt_14])).
% 132.82/17.36 thf(zip_derived_cl167, plain,
% 132.82/17.36 ( (in @ (kpair @ sk__559 @ sk__560) @
% 132.82/17.36 (dsetconstr @ (cartprod @ sk__556 @ sk__557) @
% 132.82/17.36 (^[Y0 : $i]:
% 132.82/17.36 (((??) @ (^[Y1 : $i]:
% 132.82/17.36 (((in @ Y1 @ sk__556) &
% 132.82/17.36 (((??) @ (^[Y2 : $i]:
% 132.82/17.36 (((in @ Y2 @ sk__557) &
% 132.82/17.36 (sk__558 @ Y1 @ Y2) &
% 132.82/17.36 (((Y0) = (kpair @ Y1 @ Y2))))))))))))))))),
% 132.82/17.36 inference('cnf', [status(esa)], [zf_stmt_14])).
% 132.82/17.36 thf(zip_derived_cl164, plain,
% 132.82/17.36 (![X254 : $i, X255 : $i, X256 : $i > $i > $o]:
% 132.82/17.36 (subset @
% 132.82/17.36 (dsetconstr @ (cartprod @ X254 @ X255) @
% 132.82/17.36 (^[Y0 : $i]:
% 132.82/17.36 (((??) @ (^[Y1 : $i]:
% 132.82/17.36 (((in @ Y1 @ X254) &
% 132.82/17.36 (((??) @ (^[Y2 : $i]:
% 132.82/17.36 (((in @ Y2 @ X255) &
% 132.82/17.36 (X256 @ Y1 @ Y2) &
% 132.82/17.36 (((Y0) = (kpair @ Y1 @ Y2))))))))))))))) @
% 132.82/17.36 (cartprod @ X254 @ X255))),
% 132.82/17.36 inference('cnf', [status(esa)], [zf_stmt_14])).
% 132.82/17.36 thf(zip_derived_cl10935, plain,
% 132.82/17.36 (![X254 : $i, X255 : $i, X256 : $i > $i > $o, X257 : $i]:
% 132.82/17.36 (('#l_lift14081' @ X254 @ X255 @ X256 @ X257)
% 132.82/17.36 = (((??) @ (^[Y0 : $i]:
% 132.82/17.36 (((in @ Y0 @ X254) &
% 132.82/17.36 (((??) @ (^[Y1 : $i]:
% 132.82/17.36 (((in @ Y1 @ X255) & (X256 @ Y0 @ Y1) &
% 132.82/17.36 (((X257) = (kpair @ Y0 @ Y1))))))))))))))),
% 132.82/17.36 define([status(thm)])).
% 132.82/17.36 thf(zip_derived_cl10938, plain,
% 132.82/17.36 ( (in @ (kpair @ sk__559 @ sk__560) @
% 132.82/17.36 (dsetconstr @ (cartprod @ sk__556 @ sk__557) @
% 132.82/17.36 ('#l_lift14081' @ sk__556 @ sk__557 @ sk__558)))),
% 132.82/17.36 inference('lambda_lifting', [status(thm)],
% 132.82/17.36 [zip_derived_cl167, zip_derived_cl10935])).
% 132.82/17.36 thf(zip_derived_cl172, plain,
% 132.82/17.36 (![X278 : $i, X279 : $i, X280 : $i, X281 : $i]:
% 132.82/17.36 ( (in @ X278 @ X279)
% 132.82/17.36 | ~ (in @ (kpair @ X278 @ X280) @ (cartprod @ X279 @ X281)))),
% 132.82/17.36 inference('cnf', [status(esa)], [zf_stmt_14])).
% 132.82/17.36 thf(zip_derived_cl10978, plain,
% 132.82/17.36 (![X471 : $i, X472 : $i, X473 : $i > $o]:
% 132.82/17.36 ( (in @ X471 @ X472) | ~ (in @ X471 @ (dsetconstr @ X472 @ X473)))),
% 132.82/17.36 inference('cnf', [status(esa)], [zf_stmt_14])).
% 132.82/17.36 thf(zip_derived_cl10981, plain, ($false),
% 132.82/17.36 inference('eprover', [status(thm)],
% 132.82/17.36 [zip_derived_cl166, zip_derived_cl10938, zip_derived_cl172,
% 132.82/17.36 zip_derived_cl10978])).
% 132.82/17.36
% 132.82/17.36 % SZS output end Refutation
% 132.82/17.36
% 132.82/17.36
% 132.82/17.36 % Terminating...
% 133.08/17.45 % Runner terminated.
% 133.12/17.47 % Zipperpin 1.5 exiting
%------------------------------------------------------------------------------