0.11/0.34 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.11/0.37 % Command : run_portfolio.sh /export/starexec/sandbox/benchmark/theBenchmark.p /export/starexec/sandbox/tmp/tmp.Unx66yQy4z 0.20/0.58 % Computer : n026.cluster.edu 0.20/0.58 % Model : x86_64 x86_64 0.20/0.58 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.20/0.58 % Memory : 8042.1875MB 0.20/0.58 % OS : Linux 3.10.0-693.el7.x86_64 0.20/0.58 % CPULimit : 960 0.20/0.58 % DateTime : Thu Jul 2 09:09:10 EDT 2020 0.20/0.58 % CPUTime : 0.20/0.58 % Running portfolio for 120 s 0.20/0.58 % File : /export/starexec/sandbox/benchmark/theBenchmark.p 0.20/0.58 % Number of cores: 8 0.20/0.59 % Python version: Python 3.6.8 0.20/0.59 % Running in HO mode 0.23/0.87 % Estimated wc time : 1286 0.23/0.87 % Estimated cpu time (7 cpus) : 183.71428571428572 0.23/0.96 % /export/starexec/sandbox/solver/bin/lams/s6.sh running for 21s 0.23/0.96 % /export/starexec/sandbox/solver/bin/lams/new_b3.2.norw.sh running for 15s 0.23/1.00 % /export/starexec/sandbox/solver/bin/lams/b.rpo.sh running for 21s 0.23/1.00 % /export/starexec/sandbox/solver/bin/lams/s6.ds.sh running for 10s 0.23/1.00 % /export/starexec/sandbox/solver/bin/lams/c.s.2.sh running for 68s 0.23/1.00 % /export/starexec/sandbox/solver/bin/lams/old_b3.avatar.sh running for 15s 0.23/1.00 % /export/starexec/sandbox/solver/bin/lams/c.easy.sh running for 26s 71.27/11.07 % /export/starexec/sandbox/solver/bin/lams/old_sp2.sh running for 17s 105.54/16.08 % /export/starexec/sandbox/solver/bin/lams/b.sh running for 31s 105.54/16.11 % /export/starexec/sandbox/solver/bin/lams/sh1.sh running for 26s 146.76/22.08 % /export/starexec/sandbox/solver/bin/lams/b.lsimp.sh running for 17s 146.76/22.11 % /export/starexec/sandbox/solver/bin/lams/b.split.sh running for 31s 166.45/25.03 % Solved by lams/b.lsimp.sh. 166.45/25.03 % done 1163 iterations in 2.863s 166.45/25.03 % SZS status Theorem for '/export/starexec/sandbox/benchmark/theBenchmark.p' 166.45/25.03 % SZS output start Refutation 166.45/25.03 tff(dpsetconstrERa, conjecture, 166.45/25.03 (setextAx => 166.45/25.03 (emptysetAx => 166.45/25.03 (setadjoinAx => 166.45/25.03 (powersetAx => 166.45/25.03 (setunionAx => 166.45/25.03 (omega0Ax => 166.45/25.03 (omegaSAx => 166.45/25.03 (omegaIndAx => 166.45/25.03 (replAx => 166.45/25.03 (foundationAx => 166.45/25.03 (wellorderingAx => 166.45/25.03 (descrp => 166.45/25.03 (dsetconstrI => 166.45/25.03 (dsetconstrEL => 166.45/25.03 (dsetconstrER => 166.45/25.03 (exuE1 => 166.45/25.03 (prop2setE => 166.45/25.03 (emptysetE => 166.45/25.03 (emptysetimpfalse => 166.45/25.03 (notinemptyset => 166.45/25.03 (exuE3e => 166.45/25.03 (setext => 166.45/25.03 (emptyI => 166.45/25.03 (noeltsimpempty => 166.45/25.03 (setbeta => 166.45/25.03 (nonemptyE1 => 166.45/25.03 (nonemptyI => 166.45/25.03 (nonemptyI1 => 166.45/25.03 (setadjoinIL => 166.45/25.03 (emptyinunitempty => 166.45/25.03 (setadjoinIR => 166.45/25.03 (setadjoinE => 166.45/25.03 (setadjoinOr => 166.45/25.03 (setoftrueEq => 166.45/25.03 (powersetI => 166.45/25.03 (emptyinPowerset => 166.45/25.03 (emptyInPowerset => 166.45/25.03 (powersetE => 166.45/25.03 (setunionI => 166.45/25.03 (setunionE => 166.45/25.03 (subPowSU => 166.45/25.03 (exuE2 => 166.45/25.03 (nonemptyImpWitness => 166.45/25.03 (uniqinunit => 166.45/25.03 (notinsingleton => 166.45/25.03 (eqinunit => 166.45/25.03 (singletonsswitch => 166.45/25.03 (upairsetE => 166.45/25.03 (upairsetIL => 166.45/25.03 (upairsetIR => 166.45/25.03 (emptyE1 => 166.45/25.03 (vacuousDall => 166.45/25.03 (quantDeMorgan1 => 166.45/25.03 (quantDeMorgan2 => 166.45/25.03 (quantDeMorgan3 => 166.45/25.03 (quantDeMorgan4 => 166.45/25.03 (prop2setI => 166.45/25.03 (prop2set2propI => 166.45/25.03 (notdexE => 166.45/25.03 (notdallE => 166.45/25.03 (exuI1 => 166.45/25.03 (exuI3 => 166.45/25.03 (exuI2 => 166.45/25.03 (inCongP => 166.45/25.03 (in__Cong => 166.45/25.03 (exuE3u => 166.45/25.03 ( 166.45/25.03 exu__Cong => 166.45/25.03 ( 166.45/25.03 emptyset__Cong => 166.45/25.03 ( 166.45/25.03 setadjoin__Cong => 166.45/25.03 ( 166.45/25.03 powerset__Cong => 166.45/25.03 ( 166.45/25.03 setunion__Cong => 166.45/25.03 ( 166.45/25.03 omega__Cong => 166.45/25.03 ( 166.45/25.03 exuEu => 166.45/25.03 ( 166.45/25.03 descr__Cong => 166.45/25.03 ( 166.45/25.03 dsetconstr__Cong => 166.45/25.03 ( 166.45/25.03 subsetI1 => 166.45/25.03 ( 166.45/25.03 eqimpsubset2 => 166.45/25.03 ( 166.45/25.03 eqimpsubset1 => 166.45/25.03 ( 166.45/25.03 subsetI2 => 166.45/25.03 ( 166.45/25.03 emptysetsubset => 166.45/25.03 ( 166.45/25.03 subsetE => 166.45/25.03 ( 166.45/25.03 subsetE2 => 166.45/25.03 ( 166.45/25.03 notsubsetI => 166.45/25.03 ( 166.45/25.03 notequalI1 => 166.45/25.03 ( 166.45/25.03 notequalI2 => 166.45/25.03 ( 166.45/25.03 subsetRefl => 166.45/25.03 ( 166.45/25.03 subsetTrans => 166.45/25.03 ( 166.45/25.03 setadjoinSub => 166.45/25.03 ( 166.45/25.03 setadjoinSub2 => 166.45/25.03 ( 166.45/25.03 subset2powerset => 166.45/25.03 ( 166.45/25.03 setextsub => 166.45/25.03 ( 166.45/25.03 subsetemptysetimpeq => 166.45/25.03 ( 166.45/25.03 powersetI1 => 166.45/25.03 ( 166.45/25.03 powersetE1 => 166.45/25.03 ( 166.45/25.03 inPowerset => 166.45/25.03 ( 166.45/25.03 powersetsubset => 166.45/25.03 ( 166.45/25.03 sepInPowerset => 166.45/25.03 ( 166.45/25.03 sepSubset => 166.45/25.03 ( 166.45/25.03 binunionIL => 166.45/25.03 ( 166.45/25.03 upairset2IR => 166.45/25.03 ( 166.45/25.03 binunionIR => 166.45/25.03 ( 166.45/25.03 binunionEcases => 166.45/25.03 ( 166.45/25.03 binunionE => 166.45/25.03 ( 166.45/25.03 binunionLsub => 166.45/25.03 ( 166.45/25.03 binunionRsub => 166.45/25.03 ( 166.45/25.03 binintersectI => 166.45/25.03 ( 166.45/25.03 binintersectSubset5 => 166.45/25.03 ( 166.45/25.03 binintersectEL => 166.45/25.03 ( 166.45/25.03 binintersectLsub => 166.45/25.03 ( 166.45/25.03 binintersectSubset2 => 166.45/25.03 ( 166.45/25.03 binintersectSubset3 => 166.45/25.03 ( 166.45/25.03 binintersectER => 166.45/25.03 ( 166.45/25.03 disjointsetsI1 => 166.45/25.03 ( 166.45/25.03 binintersectRsub => 166.45/25.03 ( 166.45/25.03 binintersectSubset4 => 166.45/25.03 ( 166.45/25.03 binintersectSubset1 => 166.45/25.03 ( 166.45/25.03 bs114d => 166.45/25.03 ( 166.45/25.03 setminusI => 166.45/25.03 ( 166.45/25.03 setminusEL => 166.45/25.03 ( 166.45/25.03 setminusER => 166.45/25.03 ( 166.45/25.03 setminusSubset2 => 166.45/25.03 ( 166.45/25.03 setminusERneg => 166.45/25.03 ( 166.45/25.03 setminusELneg => 166.45/25.03 ( 166.45/25.03 setminusILneg => 166.45/25.03 ( 166.45/25.03 setminusIRneg => 166.45/25.03 ( 166.45/25.03 setminusLsub => 166.45/25.03 ( 166.45/25.03 setminusSubset1 => 166.45/25.03 ( 166.45/25.03 symdiffE => 166.45/25.03 ( 166.45/25.03 symdiffI1 => 166.45/25.03 ( 166.45/25.03 symdiffI2 => 166.45/25.03 ( 166.45/25.03 symdiffIneg1 => 166.45/25.03 ( 166.45/25.03 symdiffIneg2 => 166.45/25.03 ( 166.45/25.03 secondinupair => 166.45/25.03 ( 166.45/25.03 setukpairIL => 166.45/25.03 ( 166.45/25.03 setukpairIR => 166.45/25.03 ( 166.45/25.03 kpairiskpair => 166.45/25.03 ( 166.45/25.03 kpairp => 166.45/25.03 ( 166.45/25.03 singletonsubset => 166.45/25.03 ( 166.45/25.03 singletoninpowerset => 166.45/25.03 ( 166.45/25.03 singletoninpowunion => 166.45/25.03 ( 166.45/25.03 upairset2E => 166.45/25.03 ( 166.45/25.03 upairsubunion => 166.45/25.03 ( 166.45/25.03 upairinpowunion => 166.45/25.03 ( 166.45/25.03 ubforcartprodlem1 => 166.45/25.03 ( 166.45/25.03 ubforcartprodlem2 => 166.45/25.03 ( 166.45/25.03 ubforcartprodlem3 => 166.45/25.03 ( 166.45/25.03 cartprodpairin => 166.45/25.03 ( 166.45/25.03 cartprodmempair1 => 166.45/25.03 ( 166.45/25.03 cartprodmempair => 166.45/25.03 ( 166.45/25.03 setunionE2 => 166.45/25.03 ( 166.45/25.03 setunionsingleton1 => 166.45/25.03 ( 166.45/25.03 setunionsingleton2 => 166.45/25.03 ( 166.45/25.03 setunionsingleton => 166.45/25.03 ( 166.45/25.03 singletonprop => 166.45/25.03 ( 166.45/25.03 ex1E1 => 166.45/25.03 ( 166.45/25.03 ex1I => 166.45/25.03 ( 166.45/25.03 ex1I2 => 166.45/25.03 ( 166.45/25.03 singletonsuniq => 166.45/25.03 ( 166.45/25.03 setukpairinjL1 => 166.45/25.03 ( 166.45/25.03 kfstsingleton => 166.45/25.03 ( 166.45/25.03 theprop => 166.45/25.03 ( 166.45/25.03 kfstpairEq => 166.45/25.03 ( 166.45/25.03 cartprodfstin => 166.45/25.03 ( 166.45/25.03 setukpairinjL2 => 166.45/25.03 ( 166.45/25.03 setukpairinjL => 166.45/25.03 ( 166.45/25.03 setukpairinjR11 => 166.45/25.03 ( 166.45/25.03 setukpairinjR12 => 166.45/25.03 ( 166.45/25.03 setukpairinjR1 => 166.45/25.03 ( 166.45/25.03 upairequniteq => 166.45/25.03 ( 166.45/25.03 setukpairinjR2 => 166.45/25.03 ( 166.45/25.03 setukpairinjR => 166.45/25.03 ( 166.45/25.03 ksndsingleton => 166.45/25.03 ( 166.45/25.03 ksndpairEq => 166.45/25.03 ( 166.45/25.03 kpairsurjEq => 166.45/25.03 ( 166.45/25.03 cartprodsndin => 166.45/25.03 ( 166.45/25.03 cartprodpairmemEL => 166.45/25.03 ( 166.45/25.03 cartprodpairmemER => 166.45/25.03 ( 166.45/25.03 cartprodmempaircEq => 166.45/25.03 ( 166.45/25.03 cartprodfstpairEq => 166.45/25.03 ( 166.45/25.03 cartprodsndpairEq => 166.45/25.03 ( 166.45/25.03 cartprodpairsurjEq => 166.45/25.03 ( 166.45/25.03 dpsetconstrI => 166.45/25.03 ( 166.45/25.03 dpsetconstrSub => 166.45/25.03 ( 166.45/25.03 setOfPairsIsBReln => 166.45/25.03 ( 166.45/25.03 ![A,B, 166.45/25.03 Xphi:( 166.45/25.03 ($i * $i) > 166.45/25.03 $o),Xx]: 166.45/25.03 (in(Xx,A) => 166.45/25.03 (![Xy]: 166.45/25.03 (in(Xy,B) => 166.45/25.03 (in( 166.45/25.03 kpair( 166.45/25.03 Xx,Xy), 166.45/25.03 dpsetconstr( 166.45/25.03 A,B, 166.45/25.03 ^[Xz,Xu]: 166.45/25.03 Xphi( 166.45/25.03 Xz,Xu))) => 166.45/25.03 Xphi( 166.45/25.03 Xx,Xy))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))). 166.45/25.03 tff(zf_stmt_0, negated_conjecture, 166.45/25.03 (~ 166.45/25.03 (setextAx => 166.45/25.03 (emptysetAx => 166.45/25.03 (setadjoinAx => 166.45/25.03 (powersetAx => 166.45/25.03 (setunionAx => 166.45/25.03 (omega0Ax => 166.45/25.03 (omegaSAx => 166.45/25.03 (omegaIndAx => 166.45/25.03 (replAx => 166.45/25.03 (foundationAx => 166.45/25.03 (wellorderingAx => 166.45/25.03 (descrp => 166.45/25.03 (dsetconstrI => 166.45/25.03 (dsetconstrEL => 166.45/25.03 (dsetconstrER => 166.45/25.03 (exuE1 => 166.45/25.03 (prop2setE => 166.45/25.03 (emptysetE => 166.45/25.03 (emptysetimpfalse => 166.45/25.03 (notinemptyset => 166.45/25.03 (exuE3e => 166.45/25.03 (setext => 166.45/25.03 (emptyI => 166.45/25.03 (noeltsimpempty => 166.45/25.03 (setbeta => 166.45/25.03 (nonemptyE1 => 166.45/25.03 (nonemptyI => 166.45/25.03 (nonemptyI1 => 166.45/25.03 (setadjoinIL => 166.45/25.03 (emptyinunitempty => 166.45/25.03 (setadjoinIR => 166.45/25.03 (setadjoinE => 166.45/25.03 (setadjoinOr => 166.45/25.03 (setoftrueEq => 166.45/25.03 (powersetI => 166.45/25.03 (emptyinPowerset => 166.45/25.03 (emptyInPowerset => 166.45/25.03 (powersetE => 166.45/25.03 (setunionI => 166.45/25.03 (setunionE => 166.45/25.03 (subPowSU => 166.45/25.03 (exuE2 => 166.45/25.03 (nonemptyImpWitness => 166.45/25.03 (uniqinunit => 166.45/25.03 (notinsingleton => 166.45/25.03 (eqinunit => 166.45/25.03 (singletonsswitch => 166.45/25.03 (upairsetE => 166.45/25.03 (upairsetIL => 166.45/25.03 (upairsetIR => 166.45/25.03 (emptyE1 => 166.45/25.03 (vacuousDall => 166.45/25.03 (quantDeMorgan1 => 166.45/25.03 (quantDeMorgan2 => 166.45/25.03 (quantDeMorgan3 => 166.45/25.03 (quantDeMorgan4 => 166.45/25.03 (prop2setI => 166.45/25.03 (prop2set2propI => 166.45/25.03 (notdexE => 166.45/25.03 (notdallE => 166.45/25.03 (exuI1 => 166.45/25.03 (exuI3 => 166.45/25.03 (exuI2 => 166.45/25.03 (inCongP => 166.45/25.03 ( 166.45/25.03 in__Cong => 166.45/25.03 ( 166.45/25.03 exuE3u => 166.45/25.03 ( 166.45/25.03 exu__Cong => 166.45/25.03 ( 166.45/25.03 emptyset__Cong => 166.45/25.03 ( 166.45/25.03 setadjoin__Cong => 166.45/25.03 ( 166.45/25.03 powerset__Cong => 166.45/25.03 ( 166.45/25.03 setunion__Cong => 166.45/25.03 ( 166.45/25.03 omega__Cong => 166.45/25.03 ( 166.45/25.03 exuEu => 166.45/25.03 ( 166.45/25.03 descr__Cong => 166.45/25.03 ( 166.45/25.03 dsetconstr__Cong => 166.45/25.03 ( 166.45/25.03 subsetI1 => 166.45/25.03 ( 166.45/25.03 eqimpsubset2 => 166.45/25.03 ( 166.45/25.03 eqimpsubset1 => 166.45/25.03 ( 166.45/25.03 subsetI2 => 166.45/25.03 ( 166.45/25.03 emptysetsubset => 166.45/25.03 ( 166.45/25.03 subsetE => 166.45/25.03 ( 166.45/25.03 subsetE2 => 166.45/25.03 ( 166.45/25.03 notsubsetI => 166.45/25.03 ( 166.45/25.03 notequalI1 => 166.45/25.03 ( 166.45/25.03 notequalI2 => 166.45/25.03 ( 166.45/25.03 subsetRefl => 166.45/25.03 ( 166.45/25.03 subsetTrans => 166.45/25.03 ( 166.45/25.03 setadjoinSub => 166.45/25.03 ( 166.45/25.03 setadjoinSub2 => 166.45/25.03 ( 166.45/25.03 subset2powerset => 166.45/25.03 ( 166.45/25.03 setextsub => 166.45/25.03 ( 166.45/25.03 subsetemptysetimpeq => 166.45/25.03 ( 166.45/25.03 powersetI1 => 166.45/25.03 ( 166.45/25.03 powersetE1 => 166.45/25.03 ( 166.45/25.03 inPowerset => 166.45/25.03 ( 166.45/25.03 powersetsubset => 166.45/25.03 ( 166.45/25.03 sepInPowerset => 166.45/25.03 ( 166.45/25.03 sepSubset => 166.45/25.03 ( 166.45/25.03 binunionIL => 166.45/25.03 ( 166.45/25.03 upairset2IR => 166.45/25.03 ( 166.45/25.03 binunionIR => 166.45/25.03 ( 166.45/25.03 binunionEcases => 166.45/25.03 ( 166.45/25.03 binunionE => 166.45/25.03 ( 166.45/25.03 binunionLsub => 166.45/25.03 ( 166.45/25.03 binunionRsub => 166.45/25.03 ( 166.45/25.03 binintersectI => 166.45/25.03 ( 166.45/25.03 binintersectSubset5 => 166.45/25.03 ( 166.45/25.03 binintersectEL => 166.45/25.03 ( 166.45/25.03 binintersectLsub => 166.45/25.03 ( 166.45/25.03 binintersectSubset2 => 166.45/25.03 ( 166.45/25.03 binintersectSubset3 => 166.45/25.03 ( 166.45/25.03 binintersectER => 166.45/25.03 ( 166.45/25.03 disjointsetsI1 => 166.45/25.03 ( 166.45/25.03 binintersectRsub => 166.45/25.03 ( 166.45/25.03 binintersectSubset4 => 166.45/25.03 ( 166.45/25.03 binintersectSubset1 => 166.45/25.03 ( 166.45/25.03 bs114d => 166.45/25.03 ( 166.45/25.03 setminusI => 166.45/25.03 ( 166.45/25.03 setminusEL => 166.45/25.03 ( 166.45/25.03 setminusER => 166.45/25.03 ( 166.45/25.03 setminusSubset2 => 166.45/25.03 ( 166.45/25.03 setminusERneg => 166.45/25.03 ( 166.45/25.03 setminusELneg => 166.45/25.03 ( 166.45/25.03 setminusILneg => 166.45/25.03 ( 166.45/25.03 setminusIRneg => 166.45/25.03 ( 166.45/25.03 setminusLsub => 166.45/25.03 ( 166.45/25.03 setminusSubset1 => 166.45/25.03 ( 166.45/25.03 symdiffE => 166.45/25.03 ( 166.45/25.03 symdiffI1 => 166.45/25.03 ( 166.45/25.03 symdiffI2 => 166.45/25.03 ( 166.45/25.03 symdiffIneg1 => 166.45/25.03 ( 166.45/25.03 symdiffIneg2 => 166.45/25.03 ( 166.45/25.03 secondinupair => 166.45/25.03 ( 166.45/25.03 setukpairIL => 166.45/25.03 ( 166.45/25.03 setukpairIR => 166.45/25.03 ( 166.45/25.03 kpairiskpair => 166.45/25.03 ( 166.45/25.03 kpairp => 166.45/25.03 ( 166.45/25.03 singletonsubset => 166.45/25.03 ( 166.45/25.03 singletoninpowerset => 166.45/25.03 ( 166.45/25.03 singletoninpowunion => 166.45/25.03 ( 166.45/25.03 upairset2E => 166.45/25.03 ( 166.45/25.03 upairsubunion => 166.45/25.03 ( 166.45/25.03 upairinpowunion => 166.45/25.03 ( 166.45/25.03 ubforcartprodlem1 => 166.45/25.03 ( 166.45/25.03 ubforcartprodlem2 => 166.45/25.03 ( 166.45/25.03 ubforcartprodlem3 => 166.45/25.03 ( 166.45/25.03 cartprodpairin => 166.45/25.03 ( 166.45/25.03 cartprodmempair1 => 166.45/25.03 ( 166.45/25.03 cartprodmempair => 166.45/25.03 ( 166.45/25.03 setunionE2 => 166.45/25.03 ( 166.45/25.03 setunionsingleton1 => 166.45/25.03 ( 166.45/25.03 setunionsingleton2 => 166.45/25.03 ( 166.45/25.03 setunionsingleton => 166.45/25.03 ( 166.45/25.03 singletonprop => 166.45/25.03 ( 166.45/25.03 ex1E1 => 166.45/25.03 ( 166.45/25.03 ex1I => 166.45/25.03 ( 166.45/25.03 ex1I2 => 166.45/25.03 ( 166.45/25.03 singletonsuniq => 166.45/25.03 ( 166.45/25.03 setukpairinjL1 => 166.45/25.03 ( 166.45/25.03 kfstsingleton => 166.45/25.03 ( 166.45/25.03 theprop => 166.45/25.03 ( 166.45/25.03 kfstpairEq => 166.45/25.03 ( 166.45/25.03 cartprodfstin => 166.45/25.03 ( 166.45/25.03 setukpairinjL2 => 166.45/25.03 ( 166.45/25.03 setukpairinjL => 166.45/25.03 ( 166.45/25.03 setukpairinjR11 => 166.45/25.03 ( 166.45/25.03 setukpairinjR12 => 166.45/25.03 ( 166.45/25.03 setukpairinjR1 => 166.45/25.03 ( 166.45/25.03 upairequniteq => 166.45/25.03 ( 166.45/25.03 setukpairinjR2 => 166.45/25.03 ( 166.45/25.03 setukpairinjR => 166.45/25.03 ( 166.45/25.03 ksndsingleton => 166.45/25.03 ( 166.45/25.03 ksndpairEq => 166.45/25.03 ( 166.45/25.03 kpairsurjEq => 166.45/25.03 ( 166.45/25.03 cartprodsndin => 166.45/25.03 ( 166.45/25.03 cartprodpairmemEL => 166.45/25.03 ( 166.45/25.03 cartprodpairmemER => 166.45/25.03 ( 166.45/25.03 cartprodmempaircEq => 166.45/25.03 ( 166.45/25.03 cartprodfstpairEq => 166.45/25.03 ( 166.45/25.03 cartprodsndpairEq => 166.45/25.03 ( 166.45/25.03 cartprodpairsurjEq => 166.45/25.03 ( 166.45/25.03 dpsetconstrI => 166.45/25.03 ( 166.45/25.03 dpsetconstrSub => 166.45/25.03 ( 166.45/25.03 setOfPairsIsBReln => 166.45/25.03 ( 166.45/25.03 ![A,B, 166.45/25.03 Xphi:( 166.45/25.03 ($i * $i) > 166.45/25.03 $o),Xx]: 166.45/25.03 (in(Xx,A) => 166.45/25.03 (![Xy]: 166.45/25.03 (in(Xy,B) => 166.45/25.03 (in( 166.45/25.03 kpair( 166.45/25.03 Xx,Xy), 166.45/25.03 dpsetconstr( 166.45/25.03 A,B, 166.45/25.03 ^[Xz,Xu]: 166.45/25.03 Xphi( 166.45/25.03 Xz,Xu))) => 166.45/25.03 Xphi( 166.45/25.03 Xx,Xy)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))). 166.45/25.03 tff('0', plain, 166.45/25.03 ~ (setextAx => 166.45/25.03 (emptysetAx => 166.45/25.03 (setadjoinAx => 166.45/25.03 (powersetAx => 166.45/25.03 (setunionAx => 166.45/25.03 (omega0Ax => 166.45/25.03 (omegaSAx => 166.45/25.03 (omegaIndAx => 166.45/25.03 (replAx => 166.45/25.03 (foundationAx => 166.45/25.03 (wellorderingAx => 166.45/25.03 (descrp => 166.45/25.03 (dsetconstrI => 166.45/25.03 (dsetconstrEL => 166.45/25.03 (dsetconstrER => 166.45/25.03 (exuE1 => 166.45/25.03 (prop2setE => 166.45/25.03 (emptysetE => 166.45/25.03 (emptysetimpfalse => 166.45/25.03 (notinemptyset => 166.45/25.03 (exuE3e => 166.45/25.03 (setext => 166.45/25.03 (emptyI => 166.45/25.03 (noeltsimpempty => 166.45/25.03 (setbeta => 166.45/25.03 (nonemptyE1 => 166.45/25.03 (nonemptyI => 166.45/25.03 (nonemptyI1 => 166.45/25.03 (setadjoinIL => 166.45/25.03 (emptyinunitempty => 166.45/25.03 (setadjoinIR => 166.45/25.03 (setadjoinE => 166.45/25.03 (setadjoinOr => 166.45/25.03 (setoftrueEq => 166.45/25.03 (powersetI => 166.45/25.03 (emptyinPowerset => 166.45/25.03 (emptyInPowerset => 166.45/25.03 (powersetE => 166.45/25.03 (setunionI => 166.45/25.03 (setunionE => 166.45/25.03 (subPowSU => 166.45/25.03 (exuE2 => 166.45/25.03 (nonemptyImpWitness => 166.45/25.03 (uniqinunit => 166.45/25.03 (notinsingleton => 166.45/25.03 (eqinunit => 166.45/25.03 (singletonsswitch => 166.45/25.03 (upairsetE => 166.45/25.03 (upairsetIL => 166.45/25.03 (upairsetIR => 166.45/25.03 (emptyE1 => 166.45/25.03 (vacuousDall => 166.45/25.03 (quantDeMorgan1 => 166.45/25.03 (quantDeMorgan2 => 166.45/25.03 (quantDeMorgan3 => 166.45/25.03 (quantDeMorgan4 => 166.45/25.03 (prop2setI => 166.45/25.03 (prop2set2propI => 166.45/25.03 (notdexE => 166.45/25.03 (notdallE => 166.45/25.03 (exuI1 => 166.45/25.03 (exuI3 => 166.45/25.03 ( 166.45/25.03 exuI2 => 166.45/25.03 ( 166.45/25.03 inCongP => 166.45/25.03 ( 166.45/25.03 in__Cong => 166.45/25.03 ( 166.45/25.03 exuE3u => 166.45/25.03 ( 166.45/25.03 exu__Cong => 166.45/25.03 ( 166.45/25.03 emptyset__Cong => 166.45/25.03 ( 166.45/25.03 setadjoin__Cong => 166.45/25.03 ( 166.45/25.03 powerset__Cong => 166.45/25.03 ( 166.45/25.03 setunion__Cong => 166.45/25.03 ( 166.45/25.03 omega__Cong => 166.45/25.03 ( 166.45/25.03 exuEu => 166.45/25.03 ( 166.45/25.03 descr__Cong => 166.45/25.03 ( 166.45/25.03 dsetconstr__Cong => 166.45/25.03 ( 166.45/25.03 subsetI1 => 166.45/25.03 ( 166.45/25.03 eqimpsubset2 => 166.45/25.03 ( 166.45/25.03 eqimpsubset1 => 166.45/25.03 ( 166.45/25.03 subsetI2 => 166.45/25.03 ( 166.45/25.03 emptysetsubset => 166.45/25.03 ( 166.45/25.03 subsetE => 166.45/25.03 ( 166.45/25.03 subsetE2 => 166.45/25.03 ( 166.45/25.03 notsubsetI => 166.45/25.03 ( 166.45/25.03 notequalI1 => 166.45/25.03 ( 166.45/25.03 notequalI2 => 166.45/25.03 ( 166.45/25.03 subsetRefl => 166.45/25.03 ( 166.45/25.03 subsetTrans => 166.45/25.03 ( 166.45/25.03 setadjoinSub => 166.45/25.03 ( 166.45/25.03 setadjoinSub2 => 166.45/25.03 ( 166.45/25.03 subset2powerset => 166.45/25.03 ( 166.45/25.03 setextsub => 166.45/25.03 ( 166.45/25.03 subsetemptysetimpeq => 166.45/25.03 ( 166.45/25.03 powersetI1 => 166.45/25.03 ( 166.45/25.03 powersetE1 => 166.45/25.03 ( 166.45/25.03 inPowerset => 166.45/25.03 ( 166.45/25.03 powersetsubset => 166.45/25.03 ( 166.45/25.03 sepInPowerset => 166.45/25.03 ( 166.45/25.03 sepSubset => 166.45/25.03 ( 166.45/25.03 binunionIL => 166.45/25.03 ( 166.45/25.03 upairset2IR => 166.45/25.03 ( 166.45/25.03 binunionIR => 166.45/25.03 ( 166.45/25.03 binunionEcases => 166.45/25.03 ( 166.45/25.03 binunionE => 166.45/25.03 ( 166.45/25.03 binunionLsub => 166.45/25.03 ( 166.45/25.03 binunionRsub => 166.45/25.03 ( 166.45/25.03 binintersectI => 166.45/25.03 ( 166.45/25.03 binintersectSubset5 => 166.45/25.03 ( 166.45/25.03 binintersectEL => 166.45/25.03 ( 166.45/25.03 binintersectLsub => 166.45/25.03 ( 166.45/25.03 binintersectSubset2 => 166.45/25.03 ( 166.45/25.03 binintersectSubset3 => 166.45/25.03 ( 166.45/25.03 binintersectER => 166.45/25.03 ( 166.45/25.03 disjointsetsI1 => 166.45/25.03 ( 166.45/25.03 binintersectRsub => 166.45/25.03 ( 166.45/25.03 binintersectSubset4 => 166.45/25.03 ( 166.45/25.03 binintersectSubset1 => 166.45/25.03 ( 166.45/25.03 bs114d => 166.45/25.03 ( 166.45/25.03 setminusI => 166.45/25.03 ( 166.45/25.03 setminusEL => 166.45/25.03 ( 166.45/25.03 setminusER => 166.45/25.03 ( 166.45/25.03 setminusSubset2 => 166.45/25.03 ( 166.45/25.03 setminusERneg => 166.45/25.03 ( 166.45/25.03 setminusELneg => 166.45/25.03 ( 166.45/25.03 setminusILneg => 166.45/25.03 ( 166.45/25.03 setminusIRneg => 166.45/25.03 ( 166.45/25.03 setminusLsub => 166.45/25.03 ( 166.45/25.03 setminusSubset1 => 166.45/25.03 ( 166.45/25.03 symdiffE => 166.45/25.03 ( 166.45/25.03 symdiffI1 => 166.45/25.03 ( 166.45/25.03 symdiffI2 => 166.45/25.03 ( 166.45/25.03 symdiffIneg1 => 166.45/25.03 ( 166.45/25.03 symdiffIneg2 => 166.45/25.03 ( 166.45/25.03 secondinupair => 166.45/25.03 ( 166.45/25.03 setukpairIL => 166.45/25.03 ( 166.45/25.03 setukpairIR => 166.45/25.03 ( 166.45/25.03 kpairiskpair => 166.45/25.03 ( 166.45/25.03 kpairp => 166.45/25.03 ( 166.45/25.03 singletonsubset => 166.45/25.03 ( 166.45/25.03 singletoninpowerset => 166.45/25.03 ( 166.45/25.03 singletoninpowunion => 166.45/25.03 ( 166.45/25.03 upairset2E => 166.45/25.03 ( 166.45/25.03 upairsubunion => 166.45/25.03 ( 166.45/25.03 upairinpowunion => 166.45/25.03 ( 166.45/25.03 ubforcartprodlem1 => 166.45/25.03 ( 166.45/25.03 ubforcartprodlem2 => 166.45/25.03 ( 166.45/25.03 ubforcartprodlem3 => 166.45/25.03 ( 166.45/25.03 cartprodpairin => 166.45/25.03 ( 166.45/25.03 cartprodmempair1 => 166.45/25.03 ( 166.45/25.03 cartprodmempair => 166.45/25.03 ( 166.45/25.03 setunionE2 => 166.45/25.03 ( 166.45/25.03 setunionsingleton1 => 166.45/25.03 ( 166.45/25.03 setunionsingleton2 => 166.45/25.03 ( 166.45/25.03 setunionsingleton => 166.45/25.03 ( 166.45/25.03 singletonprop => 166.45/25.03 ( 166.45/25.03 ex1E1 => 166.45/25.03 ( 166.45/25.03 ex1I => 166.45/25.03 ( 166.45/25.03 ex1I2 => 166.45/25.03 ( 166.45/25.03 singletonsuniq => 166.45/25.03 ( 166.45/25.03 setukpairinjL1 => 166.45/25.03 ( 166.45/25.03 kfstsingleton => 166.45/25.03 ( 166.45/25.03 theprop => 166.45/25.03 ( 166.45/25.03 kfstpairEq => 166.45/25.03 ( 166.45/25.03 cartprodfstin => 166.45/25.03 ( 166.45/25.03 setukpairinjL2 => 166.45/25.03 ( 166.45/25.03 setukpairinjL => 166.45/25.03 ( 166.45/25.03 setukpairinjR11 => 166.45/25.03 ( 166.45/25.03 setukpairinjR12 => 166.45/25.03 ( 166.45/25.03 setukpairinjR1 => 166.45/25.03 ( 166.45/25.03 upairequniteq => 166.45/25.03 ( 166.45/25.03 setukpairinjR2 => 166.45/25.03 ( 166.45/25.03 setukpairinjR => 166.45/25.03 ( 166.45/25.03 ksndsingleton => 166.45/25.03 ( 166.45/25.03 ksndpairEq => 166.45/25.03 ( 166.45/25.03 kpairsurjEq => 166.45/25.03 ( 166.45/25.03 cartprodsndin => 166.45/25.03 ( 166.45/25.03 cartprodpairmemEL => 166.45/25.03 ( 166.45/25.03 cartprodpairmemER => 166.45/25.03 ( 166.45/25.03 cartprodmempaircEq => 166.45/25.03 ( 166.45/25.03 cartprodfstpairEq => 166.45/25.03 ( 166.45/25.03 cartprodsndpairEq => 166.45/25.03 ( 166.45/25.03 cartprodpairsurjEq => 166.45/25.03 ( 166.45/25.03 dpsetconstrI => 166.45/25.03 ( 166.45/25.03 dpsetconstrSub => 166.45/25.03 ( 166.45/25.03 setOfPairsIsBReln => 166.45/25.03 ( 166.45/25.03 !!(( 166.45/25.03 ^[Y0 : $i]: 166.45/25.03 (!!(( 166.45/25.03 ^[Y1 : $i]: 166.45/25.03 (!!(( 166.45/25.03 ^[Y2 : ( 166.45/25.03 $i * $i) > $o]: 166.45/25.03 (!!(( 166.45/25.03 ^[Y3 : $i]: 166.45/25.03 (in(Y3, 166.45/25.03 Y0) => 166.45/25.03 (!!(( 166.45/25.03 ^[Y4 : $i]: 166.45/25.03 (in(Y4, 166.45/25.03 Y1) => 166.45/25.03 (in( 166.45/25.03 kpair( 166.45/25.03 Y3, Y4), 166.45/25.03 dpsetconstr( 166.45/25.03 Y0, Y1, 166.45/25.03 (^[Y5 : $i, 166.45/25.03 Y6 : $i]: 166.45/25.03 Y2(Y5, 166.45/25.03 Y6)))) => 166.45/25.03 Y2(Y3, 166.45/25.03 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.45/25.03 inference('cnf', [status(esa)], [zf_stmt_0])). 166.45/25.03 tff('1', plain, 166.45/25.03 ~ (setextAx => 166.45/25.03 (emptysetAx => 166.45/25.03 (setadjoinAx => 166.45/25.03 (powersetAx => 166.45/25.03 (setunionAx => 166.45/25.03 (omega0Ax => 166.45/25.03 (omegaSAx => 166.45/25.03 (omegaIndAx => 166.45/25.03 (replAx => 166.45/25.03 (foundationAx => 166.45/25.03 (wellorderingAx => 166.45/25.03 (descrp => 166.45/25.03 (dsetconstrI => 166.45/25.03 (dsetconstrEL => 166.45/25.03 (dsetconstrER => 166.45/25.03 (exuE1 => 166.45/25.03 (prop2setE => 166.45/25.03 (emptysetE => 166.45/25.03 (emptysetimpfalse => 166.45/25.03 (notinemptyset => 166.45/25.03 (exuE3e => 166.45/25.03 (setext => 166.45/25.03 (emptyI => 166.45/25.03 (noeltsimpempty => 166.45/25.03 (setbeta => 166.45/25.03 (nonemptyE1 => 166.45/25.03 (nonemptyI => 166.45/25.03 (nonemptyI1 => 166.45/25.03 (setadjoinIL => 166.45/25.03 (emptyinunitempty => 166.45/25.03 (setadjoinIR => 166.45/25.03 (setadjoinE => 166.45/25.03 (setadjoinOr => 166.45/25.03 (setoftrueEq => 166.45/25.03 (powersetI => 166.45/25.03 (emptyinPowerset => 166.45/25.03 (emptyInPowerset => 166.45/25.03 (powersetE => 166.45/25.03 (setunionI => 166.45/25.03 (setunionE => 166.45/25.03 (subPowSU => 166.45/25.03 (exuE2 => 166.45/25.03 (nonemptyImpWitness => 166.45/25.03 (uniqinunit => 166.45/25.03 (notinsingleton => 166.45/25.03 (eqinunit => 166.45/25.03 (singletonsswitch => 166.45/25.03 (upairsetE => 166.45/25.03 (upairsetIL => 166.45/25.03 (upairsetIR => 166.45/25.03 (emptyE1 => 166.45/25.03 (vacuousDall => 166.45/25.03 (quantDeMorgan1 => 166.45/25.03 (quantDeMorgan2 => 166.45/25.03 (quantDeMorgan3 => 166.45/25.03 (quantDeMorgan4 => 166.45/25.03 (prop2setI => 166.45/25.03 (prop2set2propI => 166.45/25.03 (notdexE => 166.45/25.03 (notdallE => 166.45/25.03 (exuI1 => 166.45/25.03 (exuI3 => 166.45/25.03 ( 166.45/25.03 exuI2 => 166.45/25.03 ( 166.45/25.03 inCongP => 166.45/25.03 ( 166.45/25.03 in__Cong => 166.45/25.03 ( 166.45/25.03 exuE3u => 166.45/25.03 ( 166.45/25.03 exu__Cong => 166.45/25.03 ( 166.45/25.03 emptyset__Cong => 166.45/25.03 ( 166.45/25.03 setadjoin__Cong => 166.45/25.03 ( 166.45/25.03 powerset__Cong => 166.45/25.03 ( 166.45/25.03 setunion__Cong => 166.45/25.03 ( 166.45/25.03 omega__Cong => 166.45/25.03 ( 166.45/25.03 exuEu => 166.45/25.03 ( 166.45/25.03 descr__Cong => 166.45/25.03 ( 166.45/25.03 dsetconstr__Cong => 166.45/25.03 ( 166.45/25.03 subsetI1 => 166.45/25.03 ( 166.45/25.03 eqimpsubset2 => 166.45/25.03 ( 166.45/25.03 eqimpsubset1 => 166.45/25.03 ( 166.45/25.03 subsetI2 => 166.45/25.03 ( 166.45/25.03 emptysetsubset => 166.45/25.03 ( 166.45/25.03 subsetE => 166.45/25.03 ( 166.45/25.03 subsetE2 => 166.45/25.03 ( 166.45/25.03 notsubsetI => 166.45/25.03 ( 166.45/25.03 notequalI1 => 166.45/25.03 ( 166.45/25.03 notequalI2 => 166.45/25.03 ( 166.45/25.03 subsetRefl => 166.45/25.03 ( 166.45/25.03 subsetTrans => 166.45/25.03 ( 166.45/25.03 setadjoinSub => 166.45/25.03 ( 166.45/25.03 setadjoinSub2 => 166.45/25.03 ( 166.45/25.03 subset2powerset => 166.45/25.03 ( 166.45/25.03 setextsub => 166.45/25.03 ( 166.45/25.03 subsetemptysetimpeq => 166.45/25.03 ( 166.45/25.03 powersetI1 => 166.45/25.03 ( 166.45/25.03 powersetE1 => 166.45/25.03 ( 166.45/25.03 inPowerset => 166.45/25.03 ( 166.45/25.03 powersetsubset => 166.45/25.03 ( 166.45/25.03 sepInPowerset => 166.45/25.03 ( 166.45/25.03 sepSubset => 166.45/25.03 ( 166.45/25.03 binunionIL => 166.45/25.03 ( 166.45/25.03 upairset2IR => 166.45/25.03 ( 166.45/25.03 binunionIR => 166.45/25.03 ( 166.45/25.03 binunionEcases => 166.45/25.03 ( 166.45/25.03 binunionE => 166.45/25.03 ( 166.45/25.03 binunionLsub => 166.45/25.03 ( 166.45/25.03 binunionRsub => 166.45/25.03 ( 166.45/25.03 binintersectI => 166.45/25.03 ( 166.45/25.03 binintersectSubset5 => 166.45/25.03 ( 166.45/25.03 binintersectEL => 166.45/25.03 ( 166.45/25.03 binintersectLsub => 166.45/25.03 ( 166.45/25.03 binintersectSubset2 => 166.45/25.03 ( 166.45/25.03 binintersectSubset3 => 166.45/25.03 ( 166.45/25.03 binintersectER => 166.45/25.03 ( 166.45/25.03 disjointsetsI1 => 166.45/25.03 ( 166.45/25.03 binintersectRsub => 166.45/25.03 ( 166.45/25.03 binintersectSubset4 => 166.45/25.03 ( 166.45/25.03 binintersectSubset1 => 166.45/25.03 ( 166.45/25.03 bs114d => 166.45/25.03 ( 166.45/25.03 setminusI => 166.97/25.03 ( 166.97/25.03 setminusEL => 166.97/25.03 ( 166.97/25.03 setminusER => 166.97/25.03 ( 166.97/25.03 setminusSubset2 => 166.97/25.03 ( 166.97/25.03 setminusERneg => 166.97/25.03 ( 166.97/25.03 setminusELneg => 166.97/25.03 ( 166.97/25.03 setminusILneg => 166.97/25.03 ( 166.97/25.03 setminusIRneg => 166.97/25.03 ( 166.97/25.03 setminusLsub => 166.97/25.03 ( 166.97/25.03 setminusSubset1 => 166.97/25.03 ( 166.97/25.03 symdiffE => 166.97/25.03 ( 166.97/25.03 symdiffI1 => 166.97/25.03 ( 166.97/25.03 symdiffI2 => 166.97/25.03 ( 166.97/25.03 symdiffIneg1 => 166.97/25.03 ( 166.97/25.03 symdiffIneg2 => 166.97/25.03 ( 166.97/25.03 secondinupair => 166.97/25.03 ( 166.97/25.03 setukpairIL => 166.97/25.03 ( 166.97/25.03 setukpairIR => 166.97/25.03 ( 166.97/25.03 kpairiskpair => 166.97/25.03 ( 166.97/25.03 kpairp => 166.97/25.03 ( 166.97/25.03 singletonsubset => 166.97/25.03 ( 166.97/25.03 singletoninpowerset => 166.97/25.03 ( 166.97/25.03 singletoninpowunion => 166.97/25.03 ( 166.97/25.03 upairset2E => 166.97/25.03 ( 166.97/25.03 upairsubunion => 166.97/25.03 ( 166.97/25.03 upairinpowunion => 166.97/25.03 ( 166.97/25.03 ubforcartprodlem1 => 166.97/25.03 ( 166.97/25.03 ubforcartprodlem2 => 166.97/25.03 ( 166.97/25.03 ubforcartprodlem3 => 166.97/25.03 ( 166.97/25.03 cartprodpairin => 166.97/25.03 ( 166.97/25.03 cartprodmempair1 => 166.97/25.03 ( 166.97/25.03 cartprodmempair => 166.97/25.03 ( 166.97/25.03 setunionE2 => 166.97/25.03 ( 166.97/25.03 setunionsingleton1 => 166.97/25.03 ( 166.97/25.03 setunionsingleton2 => 166.97/25.03 ( 166.97/25.03 setunionsingleton => 166.97/25.03 ( 166.97/25.03 singletonprop => 166.97/25.03 ( 166.97/25.03 ex1E1 => 166.97/25.03 ( 166.97/25.03 ex1I => 166.97/25.03 ( 166.97/25.03 ex1I2 => 166.97/25.03 ( 166.97/25.03 singletonsuniq => 166.97/25.03 ( 166.97/25.03 setukpairinjL1 => 166.97/25.03 ( 166.97/25.03 kfstsingleton => 166.97/25.03 ( 166.97/25.03 theprop => 166.97/25.03 ( 166.97/25.03 kfstpairEq => 166.97/25.03 ( 166.97/25.03 cartprodfstin => 166.97/25.03 ( 166.97/25.03 setukpairinjL2 => 166.97/25.03 ( 166.97/25.03 setukpairinjL => 166.97/25.03 ( 166.97/25.03 setukpairinjR11 => 166.97/25.03 ( 166.97/25.03 setukpairinjR12 => 166.97/25.03 ( 166.97/25.03 setukpairinjR1 => 166.97/25.03 ( 166.97/25.03 upairequniteq => 166.97/25.03 ( 166.97/25.03 setukpairinjR2 => 166.97/25.03 ( 166.97/25.03 setukpairinjR => 166.97/25.03 ( 166.97/25.03 ksndsingleton => 166.97/25.03 ( 166.97/25.03 ksndpairEq => 166.97/25.03 ( 166.97/25.03 kpairsurjEq => 166.97/25.03 ( 166.97/25.03 cartprodsndin => 166.97/25.03 ( 166.97/25.03 cartprodpairmemEL => 166.97/25.03 ( 166.97/25.03 cartprodpairmemER => 166.97/25.03 ( 166.97/25.03 cartprodmempaircEq => 166.97/25.03 ( 166.97/25.03 cartprodfstpairEq => 166.97/25.03 ( 166.97/25.03 cartprodsndpairEq => 166.97/25.03 ( 166.97/25.03 cartprodpairsurjEq => 166.97/25.03 ( 166.97/25.03 dpsetconstrI => 166.97/25.03 ( 166.97/25.03 dpsetconstrSub => 166.97/25.03 ( 166.97/25.03 setOfPairsIsBReln => 166.97/25.03 ( 166.97/25.03 !!(( 166.97/25.03 ^[Y0 : $i]: 166.97/25.03 (!!(( 166.97/25.03 ^[Y1 : $i]: 166.97/25.03 (!!(( 166.97/25.03 ^[Y2 : ( 166.97/25.03 $i * $i) > $o]: 166.97/25.03 (!!(( 166.97/25.03 ^[Y3 : $i]: 166.97/25.03 (in(Y3, 166.97/25.03 Y0) => 166.97/25.03 (!!(( 166.97/25.03 ^[Y4 : $i]: 166.97/25.03 (in(Y4, 166.97/25.03 Y1) => 166.97/25.03 (in( 166.97/25.03 kpair( 166.97/25.03 Y3, Y4), 166.97/25.03 dpsetconstr( 166.97/25.03 Y0, Y1, 166.97/25.03 Y2)) => 166.97/25.03 Y2(Y3, 166.97/25.03 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.03 inference('ho_norm', [status(thm)], ['0'])). 166.97/25.03 tff('2', plain, 166.97/25.03 ~ (emptysetAx => 166.97/25.03 (setadjoinAx => 166.97/25.03 (powersetAx => 166.97/25.03 (setunionAx => 166.97/25.03 (omega0Ax => 166.97/25.03 (omegaSAx => 166.97/25.03 (omegaIndAx => 166.97/25.03 (replAx => 166.97/25.03 (foundationAx => 166.97/25.03 (wellorderingAx => 166.97/25.03 (descrp => 166.97/25.03 (dsetconstrI => 166.97/25.03 (dsetconstrEL => 166.97/25.03 (dsetconstrER => 166.97/25.03 (exuE1 => 166.97/25.03 (prop2setE => 166.97/25.03 (emptysetE => 166.97/25.03 (emptysetimpfalse => 166.97/25.03 (notinemptyset => 166.97/25.03 (exuE3e => 166.97/25.03 (setext => 166.97/25.03 (emptyI => 166.97/25.03 (noeltsimpempty => 166.97/25.03 (setbeta => 166.97/25.03 (nonemptyE1 => 166.97/25.03 (nonemptyI => 166.97/25.03 (nonemptyI1 => 166.97/25.03 (setadjoinIL => 166.97/25.03 (emptyinunitempty => 166.97/25.03 (setadjoinIR => 166.97/25.03 (setadjoinE => 166.97/25.03 (setadjoinOr => 166.97/25.03 (setoftrueEq => 166.97/25.03 (powersetI => 166.97/25.03 (emptyinPowerset => 166.97/25.03 (emptyInPowerset => 166.97/25.03 (powersetE => 166.97/25.03 (setunionI => 166.97/25.03 (setunionE => 166.97/25.03 (subPowSU => 166.97/25.03 (exuE2 => 166.97/25.03 (nonemptyImpWitness => 166.97/25.03 (uniqinunit => 166.97/25.03 (notinsingleton => 166.97/25.03 (eqinunit => 166.97/25.03 (singletonsswitch => 166.97/25.03 (upairsetE => 166.97/25.03 (upairsetIL => 166.97/25.03 (upairsetIR => 166.97/25.03 (emptyE1 => 166.97/25.03 (vacuousDall => 166.97/25.03 (quantDeMorgan1 => 166.97/25.03 (quantDeMorgan2 => 166.97/25.03 (quantDeMorgan3 => 166.97/25.03 (quantDeMorgan4 => 166.97/25.03 (prop2setI => 166.97/25.03 (prop2set2propI => 166.97/25.03 (notdexE => 166.97/25.03 (notdallE => 166.97/25.03 (exuI1 => 166.97/25.03 (exuI3 => 166.97/25.03 (exuI2 => 166.97/25.03 ( 166.97/25.03 inCongP => 166.97/25.03 ( 166.97/25.03 in__Cong => 166.97/25.03 ( 166.97/25.03 exuE3u => 166.97/25.03 ( 166.97/25.03 exu__Cong => 166.97/25.03 ( 166.97/25.03 emptyset__Cong => 166.97/25.03 ( 166.97/25.03 setadjoin__Cong => 166.97/25.03 ( 166.97/25.03 powerset__Cong => 166.97/25.03 ( 166.97/25.03 setunion__Cong => 166.97/25.03 ( 166.97/25.03 omega__Cong => 166.97/25.03 ( 166.97/25.03 exuEu => 166.97/25.03 ( 166.97/25.03 descr__Cong => 166.97/25.03 ( 166.97/25.03 dsetconstr__Cong => 166.97/25.03 ( 166.97/25.03 subsetI1 => 166.97/25.03 ( 166.97/25.03 eqimpsubset2 => 166.97/25.03 ( 166.97/25.03 eqimpsubset1 => 166.97/25.03 ( 166.97/25.03 subsetI2 => 166.97/25.03 ( 166.97/25.03 emptysetsubset => 166.97/25.03 ( 166.97/25.03 subsetE => 166.97/25.03 ( 166.97/25.03 subsetE2 => 166.97/25.03 ( 166.97/25.03 notsubsetI => 166.97/25.03 ( 166.97/25.03 notequalI1 => 166.97/25.03 ( 166.97/25.03 notequalI2 => 166.97/25.03 ( 166.97/25.03 subsetRefl => 166.97/25.03 ( 166.97/25.03 subsetTrans => 166.97/25.03 ( 166.97/25.03 setadjoinSub => 166.97/25.03 ( 166.97/25.03 setadjoinSub2 => 166.97/25.03 ( 166.97/25.03 subset2powerset => 166.97/25.03 ( 166.97/25.03 setextsub => 166.97/25.03 ( 166.97/25.03 subsetemptysetimpeq => 166.97/25.03 ( 166.97/25.03 powersetI1 => 166.97/25.03 ( 166.97/25.03 powersetE1 => 166.97/25.03 ( 166.97/25.03 inPowerset => 166.97/25.03 ( 166.97/25.03 powersetsubset => 166.97/25.03 ( 166.97/25.03 sepInPowerset => 166.97/25.03 ( 166.97/25.03 sepSubset => 166.97/25.03 ( 166.97/25.03 binunionIL => 166.97/25.03 ( 166.97/25.03 upairset2IR => 166.97/25.03 ( 166.97/25.03 binunionIR => 166.97/25.03 ( 166.97/25.03 binunionEcases => 166.97/25.03 ( 166.97/25.03 binunionE => 166.97/25.03 ( 166.97/25.03 binunionLsub => 166.97/25.03 ( 166.97/25.03 binunionRsub => 166.97/25.03 ( 166.97/25.03 binintersectI => 166.97/25.03 ( 166.97/25.03 binintersectSubset5 => 166.97/25.03 ( 166.97/25.03 binintersectEL => 166.97/25.03 ( 166.97/25.03 binintersectLsub => 166.97/25.03 ( 166.97/25.03 binintersectSubset2 => 166.97/25.03 ( 166.97/25.03 binintersectSubset3 => 166.97/25.03 ( 166.97/25.03 binintersectER => 166.97/25.03 ( 166.97/25.03 disjointsetsI1 => 166.97/25.03 ( 166.97/25.03 binintersectRsub => 166.97/25.03 ( 166.97/25.03 binintersectSubset4 => 166.97/25.03 ( 166.97/25.03 binintersectSubset1 => 166.97/25.03 ( 166.97/25.03 bs114d => 166.97/25.03 ( 166.97/25.03 setminusI => 166.97/25.03 ( 166.97/25.03 setminusEL => 166.97/25.03 ( 166.97/25.03 setminusER => 166.97/25.03 ( 166.97/25.03 setminusSubset2 => 166.97/25.03 ( 166.97/25.03 setminusERneg => 166.97/25.03 ( 166.97/25.03 setminusELneg => 166.97/25.03 ( 166.97/25.03 setminusILneg => 166.97/25.03 ( 166.97/25.03 setminusIRneg => 166.97/25.03 ( 166.97/25.03 setminusLsub => 166.97/25.03 ( 166.97/25.03 setminusSubset1 => 166.97/25.03 ( 166.97/25.03 symdiffE => 166.97/25.03 ( 166.97/25.03 symdiffI1 => 166.97/25.03 ( 166.97/25.03 symdiffI2 => 166.97/25.03 ( 166.97/25.03 symdiffIneg1 => 166.97/25.03 ( 166.97/25.03 symdiffIneg2 => 166.97/25.03 ( 166.97/25.03 secondinupair => 166.97/25.03 ( 166.97/25.03 setukpairIL => 166.97/25.03 ( 166.97/25.03 setukpairIR => 166.97/25.03 ( 166.97/25.03 kpairiskpair => 166.97/25.03 ( 166.97/25.03 kpairp => 166.97/25.03 ( 166.97/25.03 singletonsubset => 166.97/25.03 ( 166.97/25.03 singletoninpowerset => 166.97/25.03 ( 166.97/25.03 singletoninpowunion => 166.97/25.03 ( 166.97/25.03 upairset2E => 166.97/25.03 ( 166.97/25.03 upairsubunion => 166.97/25.03 ( 166.97/25.03 upairinpowunion => 166.97/25.03 ( 166.97/25.03 ubforcartprodlem1 => 166.97/25.03 ( 166.97/25.03 ubforcartprodlem2 => 166.97/25.03 ( 166.97/25.03 ubforcartprodlem3 => 166.97/25.03 ( 166.97/25.03 cartprodpairin => 166.97/25.03 ( 166.97/25.03 cartprodmempair1 => 166.97/25.03 ( 166.97/25.03 cartprodmempair => 166.97/25.03 ( 166.97/25.03 setunionE2 => 166.97/25.03 ( 166.97/25.03 setunionsingleton1 => 166.97/25.03 ( 166.97/25.03 setunionsingleton2 => 166.97/25.03 ( 166.97/25.03 setunionsingleton => 166.97/25.03 ( 166.97/25.03 singletonprop => 166.97/25.03 ( 166.97/25.03 ex1E1 => 166.97/25.03 ( 166.97/25.03 ex1I => 166.97/25.03 ( 166.97/25.03 ex1I2 => 166.97/25.03 ( 166.97/25.03 singletonsuniq => 166.97/25.03 ( 166.97/25.03 setukpairinjL1 => 166.97/25.03 ( 166.97/25.03 kfstsingleton => 166.97/25.03 ( 166.97/25.03 theprop => 166.97/25.03 ( 166.97/25.03 kfstpairEq => 166.97/25.03 ( 166.97/25.03 cartprodfstin => 166.97/25.03 ( 166.97/25.03 setukpairinjL2 => 166.97/25.03 ( 166.97/25.03 setukpairinjL => 166.97/25.03 ( 166.97/25.03 setukpairinjR11 => 166.97/25.03 ( 166.97/25.03 setukpairinjR12 => 166.97/25.03 ( 166.97/25.03 setukpairinjR1 => 166.97/25.03 ( 166.97/25.03 upairequniteq => 166.97/25.03 ( 166.97/25.03 setukpairinjR2 => 166.97/25.03 ( 166.97/25.03 setukpairinjR => 166.97/25.03 ( 166.97/25.03 ksndsingleton => 166.97/25.03 ( 166.97/25.03 ksndpairEq => 166.97/25.03 ( 166.97/25.03 kpairsurjEq => 166.97/25.03 ( 166.97/25.03 cartprodsndin => 166.97/25.03 ( 166.97/25.03 cartprodpairmemEL => 166.97/25.03 ( 166.97/25.03 cartprodpairmemER => 166.97/25.03 ( 166.97/25.03 cartprodmempaircEq => 166.97/25.03 ( 166.97/25.03 cartprodfstpairEq => 166.97/25.03 ( 166.97/25.03 cartprodsndpairEq => 166.97/25.03 ( 166.97/25.03 cartprodpairsurjEq => 166.97/25.03 ( 166.97/25.03 dpsetconstrI => 166.97/25.03 ( 166.97/25.03 dpsetconstrSub => 166.97/25.03 ( 166.97/25.03 setOfPairsIsBReln => 166.97/25.03 ( 166.97/25.03 !!(( 166.97/25.03 ^[Y0 : $i]: 166.97/25.03 (!!(( 166.97/25.03 ^[Y1 : $i]: 166.97/25.03 (!!(( 166.97/25.03 ^[Y2 : ( 166.97/25.03 $i * $i) > $o]: 166.97/25.03 (!!(( 166.97/25.03 ^[Y3 : $i]: 166.97/25.03 (in(Y3, 166.97/25.03 Y0) => 166.97/25.03 (!!(( 166.97/25.03 ^[Y4 : $i]: 166.97/25.03 (in(Y4, 166.97/25.03 Y1) => 166.97/25.03 (in( 166.97/25.03 kpair( 166.97/25.03 Y3, Y4), 166.97/25.03 dpsetconstr( 166.97/25.03 Y0, Y1, 166.97/25.03 Y2)) => 166.97/25.03 Y2(Y3, 166.97/25.03 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.03 inference('lazy_cnf_imply', [status(thm)], ['1'])). 166.97/25.03 tff('3', plain, 166.97/25.03 ~ (setadjoinAx => 166.97/25.03 (powersetAx => 166.97/25.03 (setunionAx => 166.97/25.03 (omega0Ax => 166.97/25.03 (omegaSAx => 166.97/25.03 (omegaIndAx => 166.97/25.03 (replAx => 166.97/25.03 (foundationAx => 166.97/25.03 (wellorderingAx => 166.97/25.03 (descrp => 166.97/25.03 (dsetconstrI => 166.97/25.03 (dsetconstrEL => 166.97/25.03 (dsetconstrER => 166.97/25.03 (exuE1 => 166.97/25.03 (prop2setE => 166.97/25.03 (emptysetE => 166.97/25.03 (emptysetimpfalse => 166.97/25.03 (notinemptyset => 166.97/25.03 (exuE3e => 166.97/25.03 (setext => 166.97/25.03 (emptyI => 166.97/25.03 (noeltsimpempty => 166.97/25.03 (setbeta => 166.97/25.03 (nonemptyE1 => 166.97/25.03 (nonemptyI => 166.97/25.03 (nonemptyI1 => 166.97/25.03 (setadjoinIL => 166.97/25.03 (emptyinunitempty => 166.97/25.03 (setadjoinIR => 166.97/25.03 (setadjoinE => 166.97/25.03 (setadjoinOr => 166.97/25.03 (setoftrueEq => 166.97/25.03 (powersetI => 166.97/25.03 (emptyinPowerset => 166.97/25.03 (emptyInPowerset => 166.97/25.03 (powersetE => 166.97/25.03 (setunionI => 166.97/25.03 (setunionE => 166.97/25.03 (subPowSU => 166.97/25.03 (exuE2 => 166.97/25.03 (nonemptyImpWitness => 166.97/25.03 (uniqinunit => 166.97/25.03 (notinsingleton => 166.97/25.03 (eqinunit => 166.97/25.03 (singletonsswitch => 166.97/25.03 (upairsetE => 166.97/25.03 (upairsetIL => 166.97/25.03 (upairsetIR => 166.97/25.03 (emptyE1 => 166.97/25.03 (vacuousDall => 166.97/25.03 (quantDeMorgan1 => 166.97/25.03 (quantDeMorgan2 => 166.97/25.03 (quantDeMorgan3 => 166.97/25.03 (quantDeMorgan4 => 166.97/25.03 (prop2setI => 166.97/25.03 (prop2set2propI => 166.97/25.03 (notdexE => 166.97/25.03 (notdallE => 166.97/25.03 (exuI1 => 166.97/25.03 (exuI3 => 166.97/25.03 (exuI2 => 166.97/25.03 (inCongP => 166.97/25.03 ( 166.97/25.03 in__Cong => 166.97/25.03 ( 166.97/25.03 exuE3u => 166.97/25.03 ( 166.97/25.03 exu__Cong => 166.97/25.03 ( 166.97/25.03 emptyset__Cong => 166.97/25.03 ( 166.97/25.03 setadjoin__Cong => 166.97/25.03 ( 166.97/25.03 powerset__Cong => 166.97/25.03 ( 166.97/25.03 setunion__Cong => 166.97/25.03 ( 166.97/25.03 omega__Cong => 166.97/25.03 ( 166.97/25.03 exuEu => 166.97/25.03 ( 166.97/25.03 descr__Cong => 166.97/25.03 ( 166.97/25.03 dsetconstr__Cong => 166.97/25.03 ( 166.97/25.03 subsetI1 => 166.97/25.03 ( 166.97/25.03 eqimpsubset2 => 166.97/25.03 ( 166.97/25.03 eqimpsubset1 => 166.97/25.03 ( 166.97/25.03 subsetI2 => 166.97/25.03 ( 166.97/25.03 emptysetsubset => 166.97/25.03 ( 166.97/25.03 subsetE => 166.97/25.03 ( 166.97/25.03 subsetE2 => 166.97/25.03 ( 166.97/25.03 notsubsetI => 166.97/25.03 ( 166.97/25.03 notequalI1 => 166.97/25.03 ( 166.97/25.03 notequalI2 => 166.97/25.03 ( 166.97/25.03 subsetRefl => 166.97/25.03 ( 166.97/25.03 subsetTrans => 166.97/25.03 ( 166.97/25.03 setadjoinSub => 166.97/25.03 ( 166.97/25.03 setadjoinSub2 => 166.97/25.03 ( 166.97/25.03 subset2powerset => 166.97/25.03 ( 166.97/25.03 setextsub => 166.97/25.03 ( 166.97/25.03 subsetemptysetimpeq => 166.97/25.03 ( 166.97/25.03 powersetI1 => 166.97/25.03 ( 166.97/25.03 powersetE1 => 166.97/25.03 ( 166.97/25.03 inPowerset => 166.97/25.03 ( 166.97/25.03 powersetsubset => 166.97/25.03 ( 166.97/25.03 sepInPowerset => 166.97/25.03 ( 166.97/25.03 sepSubset => 166.97/25.03 ( 166.97/25.03 binunionIL => 166.97/25.03 ( 166.97/25.03 upairset2IR => 166.97/25.03 ( 166.97/25.03 binunionIR => 166.97/25.03 ( 166.97/25.03 binunionEcases => 166.97/25.03 ( 166.97/25.03 binunionE => 166.97/25.03 ( 166.97/25.03 binunionLsub => 166.97/25.03 ( 166.97/25.03 binunionRsub => 166.97/25.03 ( 166.97/25.03 binintersectI => 166.97/25.03 ( 166.97/25.03 binintersectSubset5 => 166.97/25.03 ( 166.97/25.03 binintersectEL => 166.97/25.03 ( 166.97/25.03 binintersectLsub => 166.97/25.03 ( 166.97/25.03 binintersectSubset2 => 166.97/25.03 ( 166.97/25.03 binintersectSubset3 => 166.97/25.03 ( 166.97/25.03 binintersectER => 166.97/25.03 ( 166.97/25.03 disjointsetsI1 => 166.97/25.03 ( 166.97/25.03 binintersectRsub => 166.97/25.03 ( 166.97/25.03 binintersectSubset4 => 166.97/25.03 ( 166.97/25.03 binintersectSubset1 => 166.97/25.03 ( 166.97/25.03 bs114d => 166.97/25.03 ( 166.97/25.03 setminusI => 166.97/25.03 ( 166.97/25.03 setminusEL => 166.97/25.03 ( 166.97/25.03 setminusER => 166.97/25.03 ( 166.97/25.03 setminusSubset2 => 166.97/25.03 ( 166.97/25.03 setminusERneg => 166.97/25.03 ( 166.97/25.03 setminusELneg => 166.97/25.03 ( 166.97/25.03 setminusILneg => 166.97/25.03 ( 166.97/25.03 setminusIRneg => 166.97/25.03 ( 166.97/25.03 setminusLsub => 166.97/25.03 ( 166.97/25.03 setminusSubset1 => 166.97/25.03 ( 166.97/25.03 symdiffE => 166.97/25.03 ( 166.97/25.03 symdiffI1 => 166.97/25.03 ( 166.97/25.03 symdiffI2 => 166.97/25.03 ( 166.97/25.03 symdiffIneg1 => 166.97/25.03 ( 166.97/25.03 symdiffIneg2 => 166.97/25.03 ( 166.97/25.03 secondinupair => 166.97/25.03 ( 166.97/25.03 setukpairIL => 166.97/25.03 ( 166.97/25.03 setukpairIR => 166.97/25.03 ( 166.97/25.03 kpairiskpair => 166.97/25.03 ( 166.97/25.03 kpairp => 166.97/25.03 ( 166.97/25.03 singletonsubset => 166.97/25.03 ( 166.97/25.03 singletoninpowerset => 166.97/25.03 ( 166.97/25.03 singletoninpowunion => 166.97/25.03 ( 166.97/25.03 upairset2E => 166.97/25.03 ( 166.97/25.03 upairsubunion => 166.97/25.03 ( 166.97/25.03 upairinpowunion => 166.97/25.03 ( 166.97/25.03 ubforcartprodlem1 => 166.97/25.03 ( 166.97/25.03 ubforcartprodlem2 => 166.97/25.03 ( 166.97/25.03 ubforcartprodlem3 => 166.97/25.03 ( 166.97/25.03 cartprodpairin => 166.97/25.03 ( 166.97/25.03 cartprodmempair1 => 166.97/25.03 ( 166.97/25.03 cartprodmempair => 166.97/25.03 ( 166.97/25.03 setunionE2 => 166.97/25.03 ( 166.97/25.03 setunionsingleton1 => 166.97/25.03 ( 166.97/25.03 setunionsingleton2 => 166.97/25.03 ( 166.97/25.03 setunionsingleton => 166.97/25.03 ( 166.97/25.03 singletonprop => 166.97/25.03 ( 166.97/25.03 ex1E1 => 166.97/25.03 ( 166.97/25.03 ex1I => 166.97/25.03 ( 166.97/25.03 ex1I2 => 166.97/25.03 ( 166.97/25.03 singletonsuniq => 166.97/25.03 ( 166.97/25.03 setukpairinjL1 => 166.97/25.03 ( 166.97/25.03 kfstsingleton => 166.97/25.03 ( 166.97/25.03 theprop => 166.97/25.03 ( 166.97/25.03 kfstpairEq => 166.97/25.03 ( 166.97/25.03 cartprodfstin => 166.97/25.03 ( 166.97/25.03 setukpairinjL2 => 166.97/25.03 ( 166.97/25.03 setukpairinjL => 166.97/25.03 ( 166.97/25.03 setukpairinjR11 => 166.97/25.03 ( 166.97/25.03 setukpairinjR12 => 166.97/25.03 ( 166.97/25.03 setukpairinjR1 => 166.97/25.03 ( 166.97/25.03 upairequniteq => 166.97/25.03 ( 166.97/25.03 setukpairinjR2 => 166.97/25.03 ( 166.97/25.03 setukpairinjR => 166.97/25.03 ( 166.97/25.03 ksndsingleton => 166.97/25.03 ( 166.97/25.03 ksndpairEq => 166.97/25.03 ( 166.97/25.03 kpairsurjEq => 166.97/25.03 ( 166.97/25.03 cartprodsndin => 166.97/25.03 ( 166.97/25.03 cartprodpairmemEL => 166.97/25.03 ( 166.97/25.03 cartprodpairmemER => 166.97/25.03 ( 166.97/25.03 cartprodmempaircEq => 166.97/25.03 ( 166.97/25.03 cartprodfstpairEq => 166.97/25.03 ( 166.97/25.03 cartprodsndpairEq => 166.97/25.03 ( 166.97/25.03 cartprodpairsurjEq => 166.97/25.03 ( 166.97/25.03 dpsetconstrI => 166.97/25.03 ( 166.97/25.03 dpsetconstrSub => 166.97/25.03 ( 166.97/25.03 setOfPairsIsBReln => 166.97/25.03 ( 166.97/25.03 !!(( 166.97/25.03 ^[Y0 : $i]: 166.97/25.03 (!!(( 166.97/25.03 ^[Y1 : $i]: 166.97/25.03 (!!(( 166.97/25.03 ^[Y2 : ( 166.97/25.03 $i * $i) > $o]: 166.97/25.03 (!!(( 166.97/25.03 ^[Y3 : $i]: 166.97/25.03 (in(Y3, 166.97/25.03 Y0) => 166.97/25.03 (!!(( 166.97/25.03 ^[Y4 : $i]: 166.97/25.03 (in(Y4, 166.97/25.03 Y1) => 166.97/25.03 (in( 166.97/25.03 kpair( 166.97/25.03 Y3, Y4), 166.97/25.03 dpsetconstr( 166.97/25.03 Y0, Y1, 166.97/25.03 Y2)) => 166.97/25.03 Y2(Y3, 166.97/25.03 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.03 inference('lazy_cnf_imply', [status(thm)], ['2'])). 166.97/25.03 tff('4', plain, 166.97/25.03 ~ (powersetAx => 166.97/25.03 (setunionAx => 166.97/25.03 (omega0Ax => 166.97/25.03 (omegaSAx => 166.97/25.03 (omegaIndAx => 166.97/25.03 (replAx => 166.97/25.03 (foundationAx => 166.97/25.03 (wellorderingAx => 166.97/25.03 (descrp => 166.97/25.03 (dsetconstrI => 166.97/25.03 (dsetconstrEL => 166.97/25.03 (dsetconstrER => 166.97/25.03 (exuE1 => 166.97/25.03 (prop2setE => 166.97/25.03 (emptysetE => 166.97/25.03 (emptysetimpfalse => 166.97/25.03 (notinemptyset => 166.97/25.03 (exuE3e => 166.97/25.03 (setext => 166.97/25.03 (emptyI => 166.97/25.03 (noeltsimpempty => 166.97/25.03 (setbeta => 166.97/25.03 (nonemptyE1 => 166.97/25.03 (nonemptyI => 166.97/25.03 (nonemptyI1 => 166.97/25.03 (setadjoinIL => 166.97/25.03 (emptyinunitempty => 166.97/25.03 (setadjoinIR => 166.97/25.03 (setadjoinE => 166.97/25.03 (setadjoinOr => 166.97/25.03 (setoftrueEq => 166.97/25.03 (powersetI => 166.97/25.03 (emptyinPowerset => 166.97/25.03 (emptyInPowerset => 166.97/25.03 (powersetE => 166.97/25.03 (setunionI => 166.97/25.03 (setunionE => 166.97/25.03 (subPowSU => 166.97/25.03 (exuE2 => 166.97/25.03 (nonemptyImpWitness => 166.97/25.03 (uniqinunit => 166.97/25.03 (notinsingleton => 166.97/25.03 (eqinunit => 166.97/25.03 (singletonsswitch => 166.97/25.03 (upairsetE => 166.97/25.03 (upairsetIL => 166.97/25.03 (upairsetIR => 166.97/25.03 (emptyE1 => 166.97/25.03 (vacuousDall => 166.97/25.03 (quantDeMorgan1 => 166.97/25.03 (quantDeMorgan2 => 166.97/25.03 (quantDeMorgan3 => 166.97/25.03 (quantDeMorgan4 => 166.97/25.03 (prop2setI => 166.97/25.03 (prop2set2propI => 166.97/25.03 (notdexE => 166.97/25.03 (notdallE => 166.97/25.03 (exuI1 => 166.97/25.03 (exuI3 => 166.97/25.03 (exuI2 => 166.97/25.03 (inCongP => 166.97/25.03 (in__Cong => 166.97/25.03 ( 166.97/25.03 exuE3u => 166.97/25.03 ( 166.97/25.03 exu__Cong => 166.97/25.03 ( 166.97/25.03 emptyset__Cong => 166.97/25.03 ( 166.97/25.03 setadjoin__Cong => 166.97/25.03 ( 166.97/25.03 powerset__Cong => 166.97/25.03 ( 166.97/25.03 setunion__Cong => 166.97/25.03 ( 166.97/25.03 omega__Cong => 166.97/25.03 ( 166.97/25.03 exuEu => 166.97/25.03 ( 166.97/25.03 descr__Cong => 166.97/25.03 ( 166.97/25.03 dsetconstr__Cong => 166.97/25.03 ( 166.97/25.03 subsetI1 => 166.97/25.03 ( 166.97/25.03 eqimpsubset2 => 166.97/25.03 ( 166.97/25.03 eqimpsubset1 => 166.97/25.03 ( 166.97/25.03 subsetI2 => 166.97/25.03 ( 166.97/25.03 emptysetsubset => 166.97/25.03 ( 166.97/25.03 subsetE => 166.97/25.03 ( 166.97/25.03 subsetE2 => 166.97/25.03 ( 166.97/25.03 notsubsetI => 166.97/25.03 ( 166.97/25.03 notequalI1 => 166.97/25.03 ( 166.97/25.03 notequalI2 => 166.97/25.03 ( 166.97/25.03 subsetRefl => 166.97/25.03 ( 166.97/25.03 subsetTrans => 166.97/25.03 ( 166.97/25.03 setadjoinSub => 166.97/25.03 ( 166.97/25.03 setadjoinSub2 => 166.97/25.03 ( 166.97/25.03 subset2powerset => 166.97/25.03 ( 166.97/25.03 setextsub => 166.97/25.03 ( 166.97/25.03 subsetemptysetimpeq => 166.97/25.03 ( 166.97/25.04 powersetI1 => 166.97/25.04 ( 166.97/25.04 powersetE1 => 166.97/25.04 ( 166.97/25.04 inPowerset => 166.97/25.04 ( 166.97/25.04 powersetsubset => 166.97/25.04 ( 166.97/25.04 sepInPowerset => 166.97/25.04 ( 166.97/25.04 sepSubset => 166.97/25.04 ( 166.97/25.04 binunionIL => 166.97/25.04 ( 166.97/25.04 upairset2IR => 166.97/25.04 ( 166.97/25.04 binunionIR => 166.97/25.04 ( 166.97/25.04 binunionEcases => 166.97/25.04 ( 166.97/25.04 binunionE => 166.97/25.04 ( 166.97/25.04 binunionLsub => 166.97/25.04 ( 166.97/25.04 binunionRsub => 166.97/25.04 ( 166.97/25.04 binintersectI => 166.97/25.04 ( 166.97/25.04 binintersectSubset5 => 166.97/25.04 ( 166.97/25.04 binintersectEL => 166.97/25.04 ( 166.97/25.04 binintersectLsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset2 => 166.97/25.04 ( 166.97/25.04 binintersectSubset3 => 166.97/25.04 ( 166.97/25.04 binintersectER => 166.97/25.04 ( 166.97/25.04 disjointsetsI1 => 166.97/25.04 ( 166.97/25.04 binintersectRsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset4 => 166.97/25.04 ( 166.97/25.04 binintersectSubset1 => 166.97/25.04 ( 166.97/25.04 bs114d => 166.97/25.04 ( 166.97/25.04 setminusI => 166.97/25.04 ( 166.97/25.04 setminusEL => 166.97/25.04 ( 166.97/25.04 setminusER => 166.97/25.04 ( 166.97/25.04 setminusSubset2 => 166.97/25.04 ( 166.97/25.04 setminusERneg => 166.97/25.04 ( 166.97/25.04 setminusELneg => 166.97/25.04 ( 166.97/25.04 setminusILneg => 166.97/25.04 ( 166.97/25.04 setminusIRneg => 166.97/25.04 ( 166.97/25.04 setminusLsub => 166.97/25.04 ( 166.97/25.04 setminusSubset1 => 166.97/25.04 ( 166.97/25.04 symdiffE => 166.97/25.04 ( 166.97/25.04 symdiffI1 => 166.97/25.04 ( 166.97/25.04 symdiffI2 => 166.97/25.04 ( 166.97/25.04 symdiffIneg1 => 166.97/25.04 ( 166.97/25.04 symdiffIneg2 => 166.97/25.04 ( 166.97/25.04 secondinupair => 166.97/25.04 ( 166.97/25.04 setukpairIL => 166.97/25.04 ( 166.97/25.04 setukpairIR => 166.97/25.04 ( 166.97/25.04 kpairiskpair => 166.97/25.04 ( 166.97/25.04 kpairp => 166.97/25.04 ( 166.97/25.04 singletonsubset => 166.97/25.04 ( 166.97/25.04 singletoninpowerset => 166.97/25.04 ( 166.97/25.04 singletoninpowunion => 166.97/25.04 ( 166.97/25.04 upairset2E => 166.97/25.04 ( 166.97/25.04 upairsubunion => 166.97/25.04 ( 166.97/25.04 upairinpowunion => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem1 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem2 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem3 => 166.97/25.04 ( 166.97/25.04 cartprodpairin => 166.97/25.04 ( 166.97/25.04 cartprodmempair1 => 166.97/25.04 ( 166.97/25.04 cartprodmempair => 166.97/25.04 ( 166.97/25.04 setunionE2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton1 => 166.97/25.04 ( 166.97/25.04 setunionsingleton2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton => 166.97/25.04 ( 166.97/25.04 singletonprop => 166.97/25.04 ( 166.97/25.04 ex1E1 => 166.97/25.04 ( 166.97/25.04 ex1I => 166.97/25.04 ( 166.97/25.04 ex1I2 => 166.97/25.04 ( 166.97/25.04 singletonsuniq => 166.97/25.04 ( 166.97/25.04 setukpairinjL1 => 166.97/25.04 ( 166.97/25.04 kfstsingleton => 166.97/25.04 ( 166.97/25.04 theprop => 166.97/25.04 ( 166.97/25.04 kfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodfstin => 166.97/25.04 ( 166.97/25.04 setukpairinjL2 => 166.97/25.04 ( 166.97/25.04 setukpairinjL => 166.97/25.04 ( 166.97/25.04 setukpairinjR11 => 166.97/25.04 ( 166.97/25.04 setukpairinjR12 => 166.97/25.04 ( 166.97/25.04 setukpairinjR1 => 166.97/25.04 ( 166.97/25.04 upairequniteq => 166.97/25.04 ( 166.97/25.04 setukpairinjR2 => 166.97/25.04 ( 166.97/25.04 setukpairinjR => 166.97/25.04 ( 166.97/25.04 ksndsingleton => 166.97/25.04 ( 166.97/25.04 ksndpairEq => 166.97/25.04 ( 166.97/25.04 kpairsurjEq => 166.97/25.04 ( 166.97/25.04 cartprodsndin => 166.97/25.04 ( 166.97/25.04 cartprodpairmemEL => 166.97/25.04 ( 166.97/25.04 cartprodpairmemER => 166.97/25.04 ( 166.97/25.04 cartprodmempaircEq => 166.97/25.04 ( 166.97/25.04 cartprodfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodsndpairEq => 166.97/25.04 ( 166.97/25.04 cartprodpairsurjEq => 166.97/25.04 ( 166.97/25.04 dpsetconstrI => 166.97/25.04 ( 166.97/25.04 dpsetconstrSub => 166.97/25.04 ( 166.97/25.04 setOfPairsIsBReln => 166.97/25.04 ( 166.97/25.04 !!(( 166.97/25.04 ^[Y0 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y1 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y2 : ( 166.97/25.04 $i * $i) > $o]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y3 : $i]: 166.97/25.04 (in(Y3, 166.97/25.04 Y0) => 166.97/25.04 (!!(( 166.97/25.04 ^[Y4 : $i]: 166.97/25.04 (in(Y4, 166.97/25.04 Y1) => 166.97/25.04 (in( 166.97/25.04 kpair( 166.97/25.04 Y3, Y4), 166.97/25.04 dpsetconstr( 166.97/25.04 Y0, Y1, 166.97/25.04 Y2)) => 166.97/25.04 Y2(Y3, 166.97/25.04 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.04 inference('lazy_cnf_imply', [status(thm)], ['3'])). 166.97/25.04 tff('5', plain, 166.97/25.04 ~ (setunionAx => 166.97/25.04 (omega0Ax => 166.97/25.04 (omegaSAx => 166.97/25.04 (omegaIndAx => 166.97/25.04 (replAx => 166.97/25.04 (foundationAx => 166.97/25.04 (wellorderingAx => 166.97/25.04 (descrp => 166.97/25.04 (dsetconstrI => 166.97/25.04 (dsetconstrEL => 166.97/25.04 (dsetconstrER => 166.97/25.04 (exuE1 => 166.97/25.04 (prop2setE => 166.97/25.04 (emptysetE => 166.97/25.04 (emptysetimpfalse => 166.97/25.04 (notinemptyset => 166.97/25.04 (exuE3e => 166.97/25.04 (setext => 166.97/25.04 (emptyI => 166.97/25.04 (noeltsimpempty => 166.97/25.04 (setbeta => 166.97/25.04 (nonemptyE1 => 166.97/25.04 (nonemptyI => 166.97/25.04 (nonemptyI1 => 166.97/25.04 (setadjoinIL => 166.97/25.04 (emptyinunitempty => 166.97/25.04 (setadjoinIR => 166.97/25.04 (setadjoinE => 166.97/25.04 (setadjoinOr => 166.97/25.04 (setoftrueEq => 166.97/25.04 (powersetI => 166.97/25.04 (emptyinPowerset => 166.97/25.04 (emptyInPowerset => 166.97/25.04 (powersetE => 166.97/25.04 (setunionI => 166.97/25.04 (setunionE => 166.97/25.04 (subPowSU => 166.97/25.04 (exuE2 => 166.97/25.04 (nonemptyImpWitness => 166.97/25.04 (uniqinunit => 166.97/25.04 (notinsingleton => 166.97/25.04 (eqinunit => 166.97/25.04 (singletonsswitch => 166.97/25.04 (upairsetE => 166.97/25.04 (upairsetIL => 166.97/25.04 (upairsetIR => 166.97/25.04 (emptyE1 => 166.97/25.04 (vacuousDall => 166.97/25.04 (quantDeMorgan1 => 166.97/25.04 (quantDeMorgan2 => 166.97/25.04 (quantDeMorgan3 => 166.97/25.04 (quantDeMorgan4 => 166.97/25.04 (prop2setI => 166.97/25.04 (prop2set2propI => 166.97/25.04 (notdexE => 166.97/25.04 (notdallE => 166.97/25.04 (exuI1 => 166.97/25.04 (exuI3 => 166.97/25.04 (exuI2 => 166.97/25.04 (inCongP => 166.97/25.04 (in__Cong => 166.97/25.04 (exuE3u => 166.97/25.04 ( 166.97/25.04 exu__Cong => 166.97/25.04 ( 166.97/25.04 emptyset__Cong => 166.97/25.04 ( 166.97/25.04 setadjoin__Cong => 166.97/25.04 ( 166.97/25.04 powerset__Cong => 166.97/25.04 ( 166.97/25.04 setunion__Cong => 166.97/25.04 ( 166.97/25.04 omega__Cong => 166.97/25.04 ( 166.97/25.04 exuEu => 166.97/25.04 ( 166.97/25.04 descr__Cong => 166.97/25.04 ( 166.97/25.04 dsetconstr__Cong => 166.97/25.04 ( 166.97/25.04 subsetI1 => 166.97/25.04 ( 166.97/25.04 eqimpsubset2 => 166.97/25.04 ( 166.97/25.04 eqimpsubset1 => 166.97/25.04 ( 166.97/25.04 subsetI2 => 166.97/25.04 ( 166.97/25.04 emptysetsubset => 166.97/25.04 ( 166.97/25.04 subsetE => 166.97/25.04 ( 166.97/25.04 subsetE2 => 166.97/25.04 ( 166.97/25.04 notsubsetI => 166.97/25.04 ( 166.97/25.04 notequalI1 => 166.97/25.04 ( 166.97/25.04 notequalI2 => 166.97/25.04 ( 166.97/25.04 subsetRefl => 166.97/25.04 ( 166.97/25.04 subsetTrans => 166.97/25.04 ( 166.97/25.04 setadjoinSub => 166.97/25.04 ( 166.97/25.04 setadjoinSub2 => 166.97/25.04 ( 166.97/25.04 subset2powerset => 166.97/25.04 ( 166.97/25.04 setextsub => 166.97/25.04 ( 166.97/25.04 subsetemptysetimpeq => 166.97/25.04 ( 166.97/25.04 powersetI1 => 166.97/25.04 ( 166.97/25.04 powersetE1 => 166.97/25.04 ( 166.97/25.04 inPowerset => 166.97/25.04 ( 166.97/25.04 powersetsubset => 166.97/25.04 ( 166.97/25.04 sepInPowerset => 166.97/25.04 ( 166.97/25.04 sepSubset => 166.97/25.04 ( 166.97/25.04 binunionIL => 166.97/25.04 ( 166.97/25.04 upairset2IR => 166.97/25.04 ( 166.97/25.04 binunionIR => 166.97/25.04 ( 166.97/25.04 binunionEcases => 166.97/25.04 ( 166.97/25.04 binunionE => 166.97/25.04 ( 166.97/25.04 binunionLsub => 166.97/25.04 ( 166.97/25.04 binunionRsub => 166.97/25.04 ( 166.97/25.04 binintersectI => 166.97/25.04 ( 166.97/25.04 binintersectSubset5 => 166.97/25.04 ( 166.97/25.04 binintersectEL => 166.97/25.04 ( 166.97/25.04 binintersectLsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset2 => 166.97/25.04 ( 166.97/25.04 binintersectSubset3 => 166.97/25.04 ( 166.97/25.04 binintersectER => 166.97/25.04 ( 166.97/25.04 disjointsetsI1 => 166.97/25.04 ( 166.97/25.04 binintersectRsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset4 => 166.97/25.04 ( 166.97/25.04 binintersectSubset1 => 166.97/25.04 ( 166.97/25.04 bs114d => 166.97/25.04 ( 166.97/25.04 setminusI => 166.97/25.04 ( 166.97/25.04 setminusEL => 166.97/25.04 ( 166.97/25.04 setminusER => 166.97/25.04 ( 166.97/25.04 setminusSubset2 => 166.97/25.04 ( 166.97/25.04 setminusERneg => 166.97/25.04 ( 166.97/25.04 setminusELneg => 166.97/25.04 ( 166.97/25.04 setminusILneg => 166.97/25.04 ( 166.97/25.04 setminusIRneg => 166.97/25.04 ( 166.97/25.04 setminusLsub => 166.97/25.04 ( 166.97/25.04 setminusSubset1 => 166.97/25.04 ( 166.97/25.04 symdiffE => 166.97/25.04 ( 166.97/25.04 symdiffI1 => 166.97/25.04 ( 166.97/25.04 symdiffI2 => 166.97/25.04 ( 166.97/25.04 symdiffIneg1 => 166.97/25.04 ( 166.97/25.04 symdiffIneg2 => 166.97/25.04 ( 166.97/25.04 secondinupair => 166.97/25.04 ( 166.97/25.04 setukpairIL => 166.97/25.04 ( 166.97/25.04 setukpairIR => 166.97/25.04 ( 166.97/25.04 kpairiskpair => 166.97/25.04 ( 166.97/25.04 kpairp => 166.97/25.04 ( 166.97/25.04 singletonsubset => 166.97/25.04 ( 166.97/25.04 singletoninpowerset => 166.97/25.04 ( 166.97/25.04 singletoninpowunion => 166.97/25.04 ( 166.97/25.04 upairset2E => 166.97/25.04 ( 166.97/25.04 upairsubunion => 166.97/25.04 ( 166.97/25.04 upairinpowunion => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem1 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem2 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem3 => 166.97/25.04 ( 166.97/25.04 cartprodpairin => 166.97/25.04 ( 166.97/25.04 cartprodmempair1 => 166.97/25.04 ( 166.97/25.04 cartprodmempair => 166.97/25.04 ( 166.97/25.04 setunionE2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton1 => 166.97/25.04 ( 166.97/25.04 setunionsingleton2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton => 166.97/25.04 ( 166.97/25.04 singletonprop => 166.97/25.04 ( 166.97/25.04 ex1E1 => 166.97/25.04 ( 166.97/25.04 ex1I => 166.97/25.04 ( 166.97/25.04 ex1I2 => 166.97/25.04 ( 166.97/25.04 singletonsuniq => 166.97/25.04 ( 166.97/25.04 setukpairinjL1 => 166.97/25.04 ( 166.97/25.04 kfstsingleton => 166.97/25.04 ( 166.97/25.04 theprop => 166.97/25.04 ( 166.97/25.04 kfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodfstin => 166.97/25.04 ( 166.97/25.04 setukpairinjL2 => 166.97/25.04 ( 166.97/25.04 setukpairinjL => 166.97/25.04 ( 166.97/25.04 setukpairinjR11 => 166.97/25.04 ( 166.97/25.04 setukpairinjR12 => 166.97/25.04 ( 166.97/25.04 setukpairinjR1 => 166.97/25.04 ( 166.97/25.04 upairequniteq => 166.97/25.04 ( 166.97/25.04 setukpairinjR2 => 166.97/25.04 ( 166.97/25.04 setukpairinjR => 166.97/25.04 ( 166.97/25.04 ksndsingleton => 166.97/25.04 ( 166.97/25.04 ksndpairEq => 166.97/25.04 ( 166.97/25.04 kpairsurjEq => 166.97/25.04 ( 166.97/25.04 cartprodsndin => 166.97/25.04 ( 166.97/25.04 cartprodpairmemEL => 166.97/25.04 ( 166.97/25.04 cartprodpairmemER => 166.97/25.04 ( 166.97/25.04 cartprodmempaircEq => 166.97/25.04 ( 166.97/25.04 cartprodfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodsndpairEq => 166.97/25.04 ( 166.97/25.04 cartprodpairsurjEq => 166.97/25.04 ( 166.97/25.04 dpsetconstrI => 166.97/25.04 ( 166.97/25.04 dpsetconstrSub => 166.97/25.04 ( 166.97/25.04 setOfPairsIsBReln => 166.97/25.04 ( 166.97/25.04 !!(( 166.97/25.04 ^[Y0 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y1 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y2 : ( 166.97/25.04 $i * $i) > $o]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y3 : $i]: 166.97/25.04 (in(Y3, 166.97/25.04 Y0) => 166.97/25.04 (!!(( 166.97/25.04 ^[Y4 : $i]: 166.97/25.04 (in(Y4, 166.97/25.04 Y1) => 166.97/25.04 (in( 166.97/25.04 kpair( 166.97/25.04 Y3, Y4), 166.97/25.04 dpsetconstr( 166.97/25.04 Y0, Y1, 166.97/25.04 Y2)) => 166.97/25.04 Y2(Y3, 166.97/25.04 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.04 inference('lazy_cnf_imply', [status(thm)], ['4'])). 166.97/25.04 tff('6', plain, 166.97/25.04 ~ (omega0Ax => 166.97/25.04 (omegaSAx => 166.97/25.04 (omegaIndAx => 166.97/25.04 (replAx => 166.97/25.04 (foundationAx => 166.97/25.04 (wellorderingAx => 166.97/25.04 (descrp => 166.97/25.04 (dsetconstrI => 166.97/25.04 (dsetconstrEL => 166.97/25.04 (dsetconstrER => 166.97/25.04 (exuE1 => 166.97/25.04 (prop2setE => 166.97/25.04 (emptysetE => 166.97/25.04 (emptysetimpfalse => 166.97/25.04 (notinemptyset => 166.97/25.04 (exuE3e => 166.97/25.04 (setext => 166.97/25.04 (emptyI => 166.97/25.04 (noeltsimpempty => 166.97/25.04 (setbeta => 166.97/25.04 (nonemptyE1 => 166.97/25.04 (nonemptyI => 166.97/25.04 (nonemptyI1 => 166.97/25.04 (setadjoinIL => 166.97/25.04 (emptyinunitempty => 166.97/25.04 (setadjoinIR => 166.97/25.04 (setadjoinE => 166.97/25.04 (setadjoinOr => 166.97/25.04 (setoftrueEq => 166.97/25.04 (powersetI => 166.97/25.04 (emptyinPowerset => 166.97/25.04 (emptyInPowerset => 166.97/25.04 (powersetE => 166.97/25.04 (setunionI => 166.97/25.04 (setunionE => 166.97/25.04 (subPowSU => 166.97/25.04 (exuE2 => 166.97/25.04 (nonemptyImpWitness => 166.97/25.04 (uniqinunit => 166.97/25.04 (notinsingleton => 166.97/25.04 (eqinunit => 166.97/25.04 (singletonsswitch => 166.97/25.04 (upairsetE => 166.97/25.04 (upairsetIL => 166.97/25.04 (upairsetIR => 166.97/25.04 (emptyE1 => 166.97/25.04 (vacuousDall => 166.97/25.04 (quantDeMorgan1 => 166.97/25.04 (quantDeMorgan2 => 166.97/25.04 (quantDeMorgan3 => 166.97/25.04 (quantDeMorgan4 => 166.97/25.04 (prop2setI => 166.97/25.04 (prop2set2propI => 166.97/25.04 (notdexE => 166.97/25.04 (notdallE => 166.97/25.04 (exuI1 => 166.97/25.04 (exuI3 => 166.97/25.04 (exuI2 => 166.97/25.04 (inCongP => 166.97/25.04 (in__Cong => 166.97/25.04 (exuE3u => 166.97/25.04 (exu__Cong => 166.97/25.04 ( 166.97/25.04 emptyset__Cong => 166.97/25.04 ( 166.97/25.04 setadjoin__Cong => 166.97/25.04 ( 166.97/25.04 powerset__Cong => 166.97/25.04 ( 166.97/25.04 setunion__Cong => 166.97/25.04 ( 166.97/25.04 omega__Cong => 166.97/25.04 ( 166.97/25.04 exuEu => 166.97/25.04 ( 166.97/25.04 descr__Cong => 166.97/25.04 ( 166.97/25.04 dsetconstr__Cong => 166.97/25.04 ( 166.97/25.04 subsetI1 => 166.97/25.04 ( 166.97/25.04 eqimpsubset2 => 166.97/25.04 ( 166.97/25.04 eqimpsubset1 => 166.97/25.04 ( 166.97/25.04 subsetI2 => 166.97/25.04 ( 166.97/25.04 emptysetsubset => 166.97/25.04 ( 166.97/25.04 subsetE => 166.97/25.04 ( 166.97/25.04 subsetE2 => 166.97/25.04 ( 166.97/25.04 notsubsetI => 166.97/25.04 ( 166.97/25.04 notequalI1 => 166.97/25.04 ( 166.97/25.04 notequalI2 => 166.97/25.04 ( 166.97/25.04 subsetRefl => 166.97/25.04 ( 166.97/25.04 subsetTrans => 166.97/25.04 ( 166.97/25.04 setadjoinSub => 166.97/25.04 ( 166.97/25.04 setadjoinSub2 => 166.97/25.04 ( 166.97/25.04 subset2powerset => 166.97/25.04 ( 166.97/25.04 setextsub => 166.97/25.04 ( 166.97/25.04 subsetemptysetimpeq => 166.97/25.04 ( 166.97/25.04 powersetI1 => 166.97/25.04 ( 166.97/25.04 powersetE1 => 166.97/25.04 ( 166.97/25.04 inPowerset => 166.97/25.04 ( 166.97/25.04 powersetsubset => 166.97/25.04 ( 166.97/25.04 sepInPowerset => 166.97/25.04 ( 166.97/25.04 sepSubset => 166.97/25.04 ( 166.97/25.04 binunionIL => 166.97/25.04 ( 166.97/25.04 upairset2IR => 166.97/25.04 ( 166.97/25.04 binunionIR => 166.97/25.04 ( 166.97/25.04 binunionEcases => 166.97/25.04 ( 166.97/25.04 binunionE => 166.97/25.04 ( 166.97/25.04 binunionLsub => 166.97/25.04 ( 166.97/25.04 binunionRsub => 166.97/25.04 ( 166.97/25.04 binintersectI => 166.97/25.04 ( 166.97/25.04 binintersectSubset5 => 166.97/25.04 ( 166.97/25.04 binintersectEL => 166.97/25.04 ( 166.97/25.04 binintersectLsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset2 => 166.97/25.04 ( 166.97/25.04 binintersectSubset3 => 166.97/25.04 ( 166.97/25.04 binintersectER => 166.97/25.04 ( 166.97/25.04 disjointsetsI1 => 166.97/25.04 ( 166.97/25.04 binintersectRsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset4 => 166.97/25.04 ( 166.97/25.04 binintersectSubset1 => 166.97/25.04 ( 166.97/25.04 bs114d => 166.97/25.04 ( 166.97/25.04 setminusI => 166.97/25.04 ( 166.97/25.04 setminusEL => 166.97/25.04 ( 166.97/25.04 setminusER => 166.97/25.04 ( 166.97/25.04 setminusSubset2 => 166.97/25.04 ( 166.97/25.04 setminusERneg => 166.97/25.04 ( 166.97/25.04 setminusELneg => 166.97/25.04 ( 166.97/25.04 setminusILneg => 166.97/25.04 ( 166.97/25.04 setminusIRneg => 166.97/25.04 ( 166.97/25.04 setminusLsub => 166.97/25.04 ( 166.97/25.04 setminusSubset1 => 166.97/25.04 ( 166.97/25.04 symdiffE => 166.97/25.04 ( 166.97/25.04 symdiffI1 => 166.97/25.04 ( 166.97/25.04 symdiffI2 => 166.97/25.04 ( 166.97/25.04 symdiffIneg1 => 166.97/25.04 ( 166.97/25.04 symdiffIneg2 => 166.97/25.04 ( 166.97/25.04 secondinupair => 166.97/25.04 ( 166.97/25.04 setukpairIL => 166.97/25.04 ( 166.97/25.04 setukpairIR => 166.97/25.04 ( 166.97/25.04 kpairiskpair => 166.97/25.04 ( 166.97/25.04 kpairp => 166.97/25.04 ( 166.97/25.04 singletonsubset => 166.97/25.04 ( 166.97/25.04 singletoninpowerset => 166.97/25.04 ( 166.97/25.04 singletoninpowunion => 166.97/25.04 ( 166.97/25.04 upairset2E => 166.97/25.04 ( 166.97/25.04 upairsubunion => 166.97/25.04 ( 166.97/25.04 upairinpowunion => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem1 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem2 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem3 => 166.97/25.04 ( 166.97/25.04 cartprodpairin => 166.97/25.04 ( 166.97/25.04 cartprodmempair1 => 166.97/25.04 ( 166.97/25.04 cartprodmempair => 166.97/25.04 ( 166.97/25.04 setunionE2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton1 => 166.97/25.04 ( 166.97/25.04 setunionsingleton2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton => 166.97/25.04 ( 166.97/25.04 singletonprop => 166.97/25.04 ( 166.97/25.04 ex1E1 => 166.97/25.04 ( 166.97/25.04 ex1I => 166.97/25.04 ( 166.97/25.04 ex1I2 => 166.97/25.04 ( 166.97/25.04 singletonsuniq => 166.97/25.04 ( 166.97/25.04 setukpairinjL1 => 166.97/25.04 ( 166.97/25.04 kfstsingleton => 166.97/25.04 ( 166.97/25.04 theprop => 166.97/25.04 ( 166.97/25.04 kfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodfstin => 166.97/25.04 ( 166.97/25.04 setukpairinjL2 => 166.97/25.04 ( 166.97/25.04 setukpairinjL => 166.97/25.04 ( 166.97/25.04 setukpairinjR11 => 166.97/25.04 ( 166.97/25.04 setukpairinjR12 => 166.97/25.04 ( 166.97/25.04 setukpairinjR1 => 166.97/25.04 ( 166.97/25.04 upairequniteq => 166.97/25.04 ( 166.97/25.04 setukpairinjR2 => 166.97/25.04 ( 166.97/25.04 setukpairinjR => 166.97/25.04 ( 166.97/25.04 ksndsingleton => 166.97/25.04 ( 166.97/25.04 ksndpairEq => 166.97/25.04 ( 166.97/25.04 kpairsurjEq => 166.97/25.04 ( 166.97/25.04 cartprodsndin => 166.97/25.04 ( 166.97/25.04 cartprodpairmemEL => 166.97/25.04 ( 166.97/25.04 cartprodpairmemER => 166.97/25.04 ( 166.97/25.04 cartprodmempaircEq => 166.97/25.04 ( 166.97/25.04 cartprodfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodsndpairEq => 166.97/25.04 ( 166.97/25.04 cartprodpairsurjEq => 166.97/25.04 ( 166.97/25.04 dpsetconstrI => 166.97/25.04 ( 166.97/25.04 dpsetconstrSub => 166.97/25.04 ( 166.97/25.04 setOfPairsIsBReln => 166.97/25.04 ( 166.97/25.04 !!(( 166.97/25.04 ^[Y0 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y1 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y2 : ( 166.97/25.04 $i * $i) > $o]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y3 : $i]: 166.97/25.04 (in(Y3, 166.97/25.04 Y0) => 166.97/25.04 (!!(( 166.97/25.04 ^[Y4 : $i]: 166.97/25.04 (in(Y4, 166.97/25.04 Y1) => 166.97/25.04 (in( 166.97/25.04 kpair( 166.97/25.04 Y3, Y4), 166.97/25.04 dpsetconstr( 166.97/25.04 Y0, Y1, 166.97/25.04 Y2)) => 166.97/25.04 Y2(Y3, 166.97/25.04 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.04 inference('lazy_cnf_imply', [status(thm)], ['5'])). 166.97/25.04 tff('7', plain, 166.97/25.04 ~ (omegaSAx => 166.97/25.04 (omegaIndAx => 166.97/25.04 (replAx => 166.97/25.04 (foundationAx => 166.97/25.04 (wellorderingAx => 166.97/25.04 (descrp => 166.97/25.04 (dsetconstrI => 166.97/25.04 (dsetconstrEL => 166.97/25.04 (dsetconstrER => 166.97/25.04 (exuE1 => 166.97/25.04 (prop2setE => 166.97/25.04 (emptysetE => 166.97/25.04 (emptysetimpfalse => 166.97/25.04 (notinemptyset => 166.97/25.04 (exuE3e => 166.97/25.04 (setext => 166.97/25.04 (emptyI => 166.97/25.04 (noeltsimpempty => 166.97/25.04 (setbeta => 166.97/25.04 (nonemptyE1 => 166.97/25.04 (nonemptyI => 166.97/25.04 (nonemptyI1 => 166.97/25.04 (setadjoinIL => 166.97/25.04 (emptyinunitempty => 166.97/25.04 (setadjoinIR => 166.97/25.04 (setadjoinE => 166.97/25.04 (setadjoinOr => 166.97/25.04 (setoftrueEq => 166.97/25.04 (powersetI => 166.97/25.04 (emptyinPowerset => 166.97/25.04 (emptyInPowerset => 166.97/25.04 (powersetE => 166.97/25.04 (setunionI => 166.97/25.04 (setunionE => 166.97/25.04 (subPowSU => 166.97/25.04 (exuE2 => 166.97/25.04 (nonemptyImpWitness => 166.97/25.04 (uniqinunit => 166.97/25.04 (notinsingleton => 166.97/25.04 (eqinunit => 166.97/25.04 (singletonsswitch => 166.97/25.04 (upairsetE => 166.97/25.04 (upairsetIL => 166.97/25.04 (upairsetIR => 166.97/25.04 (emptyE1 => 166.97/25.04 (vacuousDall => 166.97/25.04 (quantDeMorgan1 => 166.97/25.04 (quantDeMorgan2 => 166.97/25.04 (quantDeMorgan3 => 166.97/25.04 (quantDeMorgan4 => 166.97/25.04 (prop2setI => 166.97/25.04 (prop2set2propI => 166.97/25.04 (notdexE => 166.97/25.04 (notdallE => 166.97/25.04 (exuI1 => 166.97/25.04 (exuI3 => 166.97/25.04 (exuI2 => 166.97/25.04 (inCongP => 166.97/25.04 (in__Cong => 166.97/25.04 (exuE3u => 166.97/25.04 (exu__Cong => 166.97/25.04 (emptyset__Cong => 166.97/25.04 ( 166.97/25.04 setadjoin__Cong => 166.97/25.04 ( 166.97/25.04 powerset__Cong => 166.97/25.04 ( 166.97/25.04 setunion__Cong => 166.97/25.04 ( 166.97/25.04 omega__Cong => 166.97/25.04 ( 166.97/25.04 exuEu => 166.97/25.04 ( 166.97/25.04 descr__Cong => 166.97/25.04 ( 166.97/25.04 dsetconstr__Cong => 166.97/25.04 ( 166.97/25.04 subsetI1 => 166.97/25.04 ( 166.97/25.04 eqimpsubset2 => 166.97/25.04 ( 166.97/25.04 eqimpsubset1 => 166.97/25.04 ( 166.97/25.04 subsetI2 => 166.97/25.04 ( 166.97/25.04 emptysetsubset => 166.97/25.04 ( 166.97/25.04 subsetE => 166.97/25.04 ( 166.97/25.04 subsetE2 => 166.97/25.04 ( 166.97/25.04 notsubsetI => 166.97/25.04 ( 166.97/25.04 notequalI1 => 166.97/25.04 ( 166.97/25.04 notequalI2 => 166.97/25.04 ( 166.97/25.04 subsetRefl => 166.97/25.04 ( 166.97/25.04 subsetTrans => 166.97/25.04 ( 166.97/25.04 setadjoinSub => 166.97/25.04 ( 166.97/25.04 setadjoinSub2 => 166.97/25.04 ( 166.97/25.04 subset2powerset => 166.97/25.04 ( 166.97/25.04 setextsub => 166.97/25.04 ( 166.97/25.04 subsetemptysetimpeq => 166.97/25.04 ( 166.97/25.04 powersetI1 => 166.97/25.04 ( 166.97/25.04 powersetE1 => 166.97/25.04 ( 166.97/25.04 inPowerset => 166.97/25.04 ( 166.97/25.04 powersetsubset => 166.97/25.04 ( 166.97/25.04 sepInPowerset => 166.97/25.04 ( 166.97/25.04 sepSubset => 166.97/25.04 ( 166.97/25.04 binunionIL => 166.97/25.04 ( 166.97/25.04 upairset2IR => 166.97/25.04 ( 166.97/25.04 binunionIR => 166.97/25.04 ( 166.97/25.04 binunionEcases => 166.97/25.04 ( 166.97/25.04 binunionE => 166.97/25.04 ( 166.97/25.04 binunionLsub => 166.97/25.04 ( 166.97/25.04 binunionRsub => 166.97/25.04 ( 166.97/25.04 binintersectI => 166.97/25.04 ( 166.97/25.04 binintersectSubset5 => 166.97/25.04 ( 166.97/25.04 binintersectEL => 166.97/25.04 ( 166.97/25.04 binintersectLsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset2 => 166.97/25.04 ( 166.97/25.04 binintersectSubset3 => 166.97/25.04 ( 166.97/25.04 binintersectER => 166.97/25.04 ( 166.97/25.04 disjointsetsI1 => 166.97/25.04 ( 166.97/25.04 binintersectRsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset4 => 166.97/25.04 ( 166.97/25.04 binintersectSubset1 => 166.97/25.04 ( 166.97/25.04 bs114d => 166.97/25.04 ( 166.97/25.04 setminusI => 166.97/25.04 ( 166.97/25.04 setminusEL => 166.97/25.04 ( 166.97/25.04 setminusER => 166.97/25.04 ( 166.97/25.04 setminusSubset2 => 166.97/25.04 ( 166.97/25.04 setminusERneg => 166.97/25.04 ( 166.97/25.04 setminusELneg => 166.97/25.04 ( 166.97/25.04 setminusILneg => 166.97/25.04 ( 166.97/25.04 setminusIRneg => 166.97/25.04 ( 166.97/25.04 setminusLsub => 166.97/25.04 ( 166.97/25.04 setminusSubset1 => 166.97/25.04 ( 166.97/25.04 symdiffE => 166.97/25.04 ( 166.97/25.04 symdiffI1 => 166.97/25.04 ( 166.97/25.04 symdiffI2 => 166.97/25.04 ( 166.97/25.04 symdiffIneg1 => 166.97/25.04 ( 166.97/25.04 symdiffIneg2 => 166.97/25.04 ( 166.97/25.04 secondinupair => 166.97/25.04 ( 166.97/25.04 setukpairIL => 166.97/25.04 ( 166.97/25.04 setukpairIR => 166.97/25.04 ( 166.97/25.04 kpairiskpair => 166.97/25.04 ( 166.97/25.04 kpairp => 166.97/25.04 ( 166.97/25.04 singletonsubset => 166.97/25.04 ( 166.97/25.04 singletoninpowerset => 166.97/25.04 ( 166.97/25.04 singletoninpowunion => 166.97/25.04 ( 166.97/25.04 upairset2E => 166.97/25.04 ( 166.97/25.04 upairsubunion => 166.97/25.04 ( 166.97/25.04 upairinpowunion => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem1 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem2 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem3 => 166.97/25.04 ( 166.97/25.04 cartprodpairin => 166.97/25.04 ( 166.97/25.04 cartprodmempair1 => 166.97/25.04 ( 166.97/25.04 cartprodmempair => 166.97/25.04 ( 166.97/25.04 setunionE2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton1 => 166.97/25.04 ( 166.97/25.04 setunionsingleton2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton => 166.97/25.04 ( 166.97/25.04 singletonprop => 166.97/25.04 ( 166.97/25.04 ex1E1 => 166.97/25.04 ( 166.97/25.04 ex1I => 166.97/25.04 ( 166.97/25.04 ex1I2 => 166.97/25.04 ( 166.97/25.04 singletonsuniq => 166.97/25.04 ( 166.97/25.04 setukpairinjL1 => 166.97/25.04 ( 166.97/25.04 kfstsingleton => 166.97/25.04 ( 166.97/25.04 theprop => 166.97/25.04 ( 166.97/25.04 kfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodfstin => 166.97/25.04 ( 166.97/25.04 setukpairinjL2 => 166.97/25.04 ( 166.97/25.04 setukpairinjL => 166.97/25.04 ( 166.97/25.04 setukpairinjR11 => 166.97/25.04 ( 166.97/25.04 setukpairinjR12 => 166.97/25.04 ( 166.97/25.04 setukpairinjR1 => 166.97/25.04 ( 166.97/25.04 upairequniteq => 166.97/25.04 ( 166.97/25.04 setukpairinjR2 => 166.97/25.04 ( 166.97/25.04 setukpairinjR => 166.97/25.04 ( 166.97/25.04 ksndsingleton => 166.97/25.04 ( 166.97/25.04 ksndpairEq => 166.97/25.04 ( 166.97/25.04 kpairsurjEq => 166.97/25.04 ( 166.97/25.04 cartprodsndin => 166.97/25.04 ( 166.97/25.04 cartprodpairmemEL => 166.97/25.04 ( 166.97/25.04 cartprodpairmemER => 166.97/25.04 ( 166.97/25.04 cartprodmempaircEq => 166.97/25.04 ( 166.97/25.04 cartprodfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodsndpairEq => 166.97/25.04 ( 166.97/25.04 cartprodpairsurjEq => 166.97/25.04 ( 166.97/25.04 dpsetconstrI => 166.97/25.04 ( 166.97/25.04 dpsetconstrSub => 166.97/25.04 ( 166.97/25.04 setOfPairsIsBReln => 166.97/25.04 ( 166.97/25.04 !!(( 166.97/25.04 ^[Y0 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y1 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y2 : ( 166.97/25.04 $i * $i) > $o]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y3 : $i]: 166.97/25.04 (in(Y3, 166.97/25.04 Y0) => 166.97/25.04 (!!(( 166.97/25.04 ^[Y4 : $i]: 166.97/25.04 (in(Y4, 166.97/25.04 Y1) => 166.97/25.04 (in( 166.97/25.04 kpair( 166.97/25.04 Y3, Y4), 166.97/25.04 dpsetconstr( 166.97/25.04 Y0, Y1, 166.97/25.04 Y2)) => 166.97/25.04 Y2(Y3, 166.97/25.04 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.04 inference('lazy_cnf_imply', [status(thm)], ['6'])). 166.97/25.04 tff('8', plain, 166.97/25.04 ~ (omegaIndAx => 166.97/25.04 (replAx => 166.97/25.04 (foundationAx => 166.97/25.04 (wellorderingAx => 166.97/25.04 (descrp => 166.97/25.04 (dsetconstrI => 166.97/25.04 (dsetconstrEL => 166.97/25.04 (dsetconstrER => 166.97/25.04 (exuE1 => 166.97/25.04 (prop2setE => 166.97/25.04 (emptysetE => 166.97/25.04 (emptysetimpfalse => 166.97/25.04 (notinemptyset => 166.97/25.04 (exuE3e => 166.97/25.04 (setext => 166.97/25.04 (emptyI => 166.97/25.04 (noeltsimpempty => 166.97/25.04 (setbeta => 166.97/25.04 (nonemptyE1 => 166.97/25.04 (nonemptyI => 166.97/25.04 (nonemptyI1 => 166.97/25.04 (setadjoinIL => 166.97/25.04 (emptyinunitempty => 166.97/25.04 (setadjoinIR => 166.97/25.04 (setadjoinE => 166.97/25.04 (setadjoinOr => 166.97/25.04 (setoftrueEq => 166.97/25.04 (powersetI => 166.97/25.04 (emptyinPowerset => 166.97/25.04 (emptyInPowerset => 166.97/25.04 (powersetE => 166.97/25.04 (setunionI => 166.97/25.04 (setunionE => 166.97/25.04 (subPowSU => 166.97/25.04 (exuE2 => 166.97/25.04 (nonemptyImpWitness => 166.97/25.04 (uniqinunit => 166.97/25.04 (notinsingleton => 166.97/25.04 (eqinunit => 166.97/25.04 (singletonsswitch => 166.97/25.04 (upairsetE => 166.97/25.04 (upairsetIL => 166.97/25.04 (upairsetIR => 166.97/25.04 (emptyE1 => 166.97/25.04 (vacuousDall => 166.97/25.04 (quantDeMorgan1 => 166.97/25.04 (quantDeMorgan2 => 166.97/25.04 (quantDeMorgan3 => 166.97/25.04 (quantDeMorgan4 => 166.97/25.04 (prop2setI => 166.97/25.04 (prop2set2propI => 166.97/25.04 (notdexE => 166.97/25.04 (notdallE => 166.97/25.04 (exuI1 => 166.97/25.04 (exuI3 => 166.97/25.04 (exuI2 => 166.97/25.04 (inCongP => 166.97/25.04 (in__Cong => 166.97/25.04 (exuE3u => 166.97/25.04 (exu__Cong => 166.97/25.04 (emptyset__Cong => 166.97/25.04 (setadjoin__Cong => 166.97/25.04 ( 166.97/25.04 powerset__Cong => 166.97/25.04 ( 166.97/25.04 setunion__Cong => 166.97/25.04 ( 166.97/25.04 omega__Cong => 166.97/25.04 ( 166.97/25.04 exuEu => 166.97/25.04 ( 166.97/25.04 descr__Cong => 166.97/25.04 ( 166.97/25.04 dsetconstr__Cong => 166.97/25.04 ( 166.97/25.04 subsetI1 => 166.97/25.04 ( 166.97/25.04 eqimpsubset2 => 166.97/25.04 ( 166.97/25.04 eqimpsubset1 => 166.97/25.04 ( 166.97/25.04 subsetI2 => 166.97/25.04 ( 166.97/25.04 emptysetsubset => 166.97/25.04 ( 166.97/25.04 subsetE => 166.97/25.04 ( 166.97/25.04 subsetE2 => 166.97/25.04 ( 166.97/25.04 notsubsetI => 166.97/25.04 ( 166.97/25.04 notequalI1 => 166.97/25.04 ( 166.97/25.04 notequalI2 => 166.97/25.04 ( 166.97/25.04 subsetRefl => 166.97/25.04 ( 166.97/25.04 subsetTrans => 166.97/25.04 ( 166.97/25.04 setadjoinSub => 166.97/25.04 ( 166.97/25.04 setadjoinSub2 => 166.97/25.04 ( 166.97/25.04 subset2powerset => 166.97/25.04 ( 166.97/25.04 setextsub => 166.97/25.04 ( 166.97/25.04 subsetemptysetimpeq => 166.97/25.04 ( 166.97/25.04 powersetI1 => 166.97/25.04 ( 166.97/25.04 powersetE1 => 166.97/25.04 ( 166.97/25.04 inPowerset => 166.97/25.04 ( 166.97/25.04 powersetsubset => 166.97/25.04 ( 166.97/25.04 sepInPowerset => 166.97/25.04 ( 166.97/25.04 sepSubset => 166.97/25.04 ( 166.97/25.04 binunionIL => 166.97/25.04 ( 166.97/25.04 upairset2IR => 166.97/25.04 ( 166.97/25.04 binunionIR => 166.97/25.04 ( 166.97/25.04 binunionEcases => 166.97/25.04 ( 166.97/25.04 binunionE => 166.97/25.04 ( 166.97/25.04 binunionLsub => 166.97/25.04 ( 166.97/25.04 binunionRsub => 166.97/25.04 ( 166.97/25.04 binintersectI => 166.97/25.04 ( 166.97/25.04 binintersectSubset5 => 166.97/25.04 ( 166.97/25.04 binintersectEL => 166.97/25.04 ( 166.97/25.04 binintersectLsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset2 => 166.97/25.04 ( 166.97/25.04 binintersectSubset3 => 166.97/25.04 ( 166.97/25.04 binintersectER => 166.97/25.04 ( 166.97/25.04 disjointsetsI1 => 166.97/25.04 ( 166.97/25.04 binintersectRsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset4 => 166.97/25.04 ( 166.97/25.04 binintersectSubset1 => 166.97/25.04 ( 166.97/25.04 bs114d => 166.97/25.04 ( 166.97/25.04 setminusI => 166.97/25.04 ( 166.97/25.04 setminusEL => 166.97/25.04 ( 166.97/25.04 setminusER => 166.97/25.04 ( 166.97/25.04 setminusSubset2 => 166.97/25.04 ( 166.97/25.04 setminusERneg => 166.97/25.04 ( 166.97/25.04 setminusELneg => 166.97/25.04 ( 166.97/25.04 setminusILneg => 166.97/25.04 ( 166.97/25.04 setminusIRneg => 166.97/25.04 ( 166.97/25.04 setminusLsub => 166.97/25.04 ( 166.97/25.04 setminusSubset1 => 166.97/25.04 ( 166.97/25.04 symdiffE => 166.97/25.04 ( 166.97/25.04 symdiffI1 => 166.97/25.04 ( 166.97/25.04 symdiffI2 => 166.97/25.04 ( 166.97/25.04 symdiffIneg1 => 166.97/25.04 ( 166.97/25.04 symdiffIneg2 => 166.97/25.04 ( 166.97/25.04 secondinupair => 166.97/25.04 ( 166.97/25.04 setukpairIL => 166.97/25.04 ( 166.97/25.04 setukpairIR => 166.97/25.04 ( 166.97/25.04 kpairiskpair => 166.97/25.04 ( 166.97/25.04 kpairp => 166.97/25.04 ( 166.97/25.04 singletonsubset => 166.97/25.04 ( 166.97/25.04 singletoninpowerset => 166.97/25.04 ( 166.97/25.04 singletoninpowunion => 166.97/25.04 ( 166.97/25.04 upairset2E => 166.97/25.04 ( 166.97/25.04 upairsubunion => 166.97/25.04 ( 166.97/25.04 upairinpowunion => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem1 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem2 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem3 => 166.97/25.04 ( 166.97/25.04 cartprodpairin => 166.97/25.04 ( 166.97/25.04 cartprodmempair1 => 166.97/25.04 ( 166.97/25.04 cartprodmempair => 166.97/25.04 ( 166.97/25.04 setunionE2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton1 => 166.97/25.04 ( 166.97/25.04 setunionsingleton2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton => 166.97/25.04 ( 166.97/25.04 singletonprop => 166.97/25.04 ( 166.97/25.04 ex1E1 => 166.97/25.04 ( 166.97/25.04 ex1I => 166.97/25.04 ( 166.97/25.04 ex1I2 => 166.97/25.04 ( 166.97/25.04 singletonsuniq => 166.97/25.04 ( 166.97/25.04 setukpairinjL1 => 166.97/25.04 ( 166.97/25.04 kfstsingleton => 166.97/25.04 ( 166.97/25.04 theprop => 166.97/25.04 ( 166.97/25.04 kfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodfstin => 166.97/25.04 ( 166.97/25.04 setukpairinjL2 => 166.97/25.04 ( 166.97/25.04 setukpairinjL => 166.97/25.04 ( 166.97/25.04 setukpairinjR11 => 166.97/25.04 ( 166.97/25.04 setukpairinjR12 => 166.97/25.04 ( 166.97/25.04 setukpairinjR1 => 166.97/25.04 ( 166.97/25.04 upairequniteq => 166.97/25.04 ( 166.97/25.04 setukpairinjR2 => 166.97/25.04 ( 166.97/25.04 setukpairinjR => 166.97/25.04 ( 166.97/25.04 ksndsingleton => 166.97/25.04 ( 166.97/25.04 ksndpairEq => 166.97/25.04 ( 166.97/25.04 kpairsurjEq => 166.97/25.04 ( 166.97/25.04 cartprodsndin => 166.97/25.04 ( 166.97/25.04 cartprodpairmemEL => 166.97/25.04 ( 166.97/25.04 cartprodpairmemER => 166.97/25.04 ( 166.97/25.04 cartprodmempaircEq => 166.97/25.04 ( 166.97/25.04 cartprodfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodsndpairEq => 166.97/25.04 ( 166.97/25.04 cartprodpairsurjEq => 166.97/25.04 ( 166.97/25.04 dpsetconstrI => 166.97/25.04 ( 166.97/25.04 dpsetconstrSub => 166.97/25.04 ( 166.97/25.04 setOfPairsIsBReln => 166.97/25.04 ( 166.97/25.04 !!(( 166.97/25.04 ^[Y0 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y1 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y2 : ( 166.97/25.04 $i * $i) > $o]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y3 : $i]: 166.97/25.04 (in(Y3, 166.97/25.04 Y0) => 166.97/25.04 (!!(( 166.97/25.04 ^[Y4 : $i]: 166.97/25.04 (in(Y4, 166.97/25.04 Y1) => 166.97/25.04 (in( 166.97/25.04 kpair( 166.97/25.04 Y3, Y4), 166.97/25.04 dpsetconstr( 166.97/25.04 Y0, Y1, 166.97/25.04 Y2)) => 166.97/25.04 Y2(Y3, 166.97/25.04 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.04 inference('lazy_cnf_imply', [status(thm)], ['7'])). 166.97/25.04 tff('9', plain, 166.97/25.04 ~ (replAx => 166.97/25.04 (foundationAx => 166.97/25.04 (wellorderingAx => 166.97/25.04 (descrp => 166.97/25.04 (dsetconstrI => 166.97/25.04 (dsetconstrEL => 166.97/25.04 (dsetconstrER => 166.97/25.04 (exuE1 => 166.97/25.04 (prop2setE => 166.97/25.04 (emptysetE => 166.97/25.04 (emptysetimpfalse => 166.97/25.04 (notinemptyset => 166.97/25.04 (exuE3e => 166.97/25.04 (setext => 166.97/25.04 (emptyI => 166.97/25.04 (noeltsimpempty => 166.97/25.04 (setbeta => 166.97/25.04 (nonemptyE1 => 166.97/25.04 (nonemptyI => 166.97/25.04 (nonemptyI1 => 166.97/25.04 (setadjoinIL => 166.97/25.04 (emptyinunitempty => 166.97/25.04 (setadjoinIR => 166.97/25.04 (setadjoinE => 166.97/25.04 (setadjoinOr => 166.97/25.04 (setoftrueEq => 166.97/25.04 (powersetI => 166.97/25.04 (emptyinPowerset => 166.97/25.04 (emptyInPowerset => 166.97/25.04 (powersetE => 166.97/25.04 (setunionI => 166.97/25.04 (setunionE => 166.97/25.04 (subPowSU => 166.97/25.04 (exuE2 => 166.97/25.04 (nonemptyImpWitness => 166.97/25.04 (uniqinunit => 166.97/25.04 (notinsingleton => 166.97/25.04 (eqinunit => 166.97/25.04 (singletonsswitch => 166.97/25.04 (upairsetE => 166.97/25.04 (upairsetIL => 166.97/25.04 (upairsetIR => 166.97/25.04 (emptyE1 => 166.97/25.04 (vacuousDall => 166.97/25.04 (quantDeMorgan1 => 166.97/25.04 (quantDeMorgan2 => 166.97/25.04 (quantDeMorgan3 => 166.97/25.04 (quantDeMorgan4 => 166.97/25.04 (prop2setI => 166.97/25.04 (prop2set2propI => 166.97/25.04 (notdexE => 166.97/25.04 (notdallE => 166.97/25.04 (exuI1 => 166.97/25.04 (exuI3 => 166.97/25.04 (exuI2 => 166.97/25.04 (inCongP => 166.97/25.04 (in__Cong => 166.97/25.04 (exuE3u => 166.97/25.04 (exu__Cong => 166.97/25.04 (emptyset__Cong => 166.97/25.04 (setadjoin__Cong => 166.97/25.04 (powerset__Cong => 166.97/25.04 ( 166.97/25.04 setunion__Cong => 166.97/25.04 ( 166.97/25.04 omega__Cong => 166.97/25.04 ( 166.97/25.04 exuEu => 166.97/25.04 ( 166.97/25.04 descr__Cong => 166.97/25.04 ( 166.97/25.04 dsetconstr__Cong => 166.97/25.04 ( 166.97/25.04 subsetI1 => 166.97/25.04 ( 166.97/25.04 eqimpsubset2 => 166.97/25.04 ( 166.97/25.04 eqimpsubset1 => 166.97/25.04 ( 166.97/25.04 subsetI2 => 166.97/25.04 ( 166.97/25.04 emptysetsubset => 166.97/25.04 ( 166.97/25.04 subsetE => 166.97/25.04 ( 166.97/25.04 subsetE2 => 166.97/25.04 ( 166.97/25.04 notsubsetI => 166.97/25.04 ( 166.97/25.04 notequalI1 => 166.97/25.04 ( 166.97/25.04 notequalI2 => 166.97/25.04 ( 166.97/25.04 subsetRefl => 166.97/25.04 ( 166.97/25.04 subsetTrans => 166.97/25.04 ( 166.97/25.04 setadjoinSub => 166.97/25.04 ( 166.97/25.04 setadjoinSub2 => 166.97/25.04 ( 166.97/25.04 subset2powerset => 166.97/25.04 ( 166.97/25.04 setextsub => 166.97/25.04 ( 166.97/25.04 subsetemptysetimpeq => 166.97/25.04 ( 166.97/25.04 powersetI1 => 166.97/25.04 ( 166.97/25.04 powersetE1 => 166.97/25.04 ( 166.97/25.04 inPowerset => 166.97/25.04 ( 166.97/25.04 powersetsubset => 166.97/25.04 ( 166.97/25.04 sepInPowerset => 166.97/25.04 ( 166.97/25.04 sepSubset => 166.97/25.04 ( 166.97/25.04 binunionIL => 166.97/25.04 ( 166.97/25.04 upairset2IR => 166.97/25.04 ( 166.97/25.04 binunionIR => 166.97/25.04 ( 166.97/25.04 binunionEcases => 166.97/25.04 ( 166.97/25.04 binunionE => 166.97/25.04 ( 166.97/25.04 binunionLsub => 166.97/25.04 ( 166.97/25.04 binunionRsub => 166.97/25.04 ( 166.97/25.04 binintersectI => 166.97/25.04 ( 166.97/25.04 binintersectSubset5 => 166.97/25.04 ( 166.97/25.04 binintersectEL => 166.97/25.04 ( 166.97/25.04 binintersectLsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset2 => 166.97/25.04 ( 166.97/25.04 binintersectSubset3 => 166.97/25.04 ( 166.97/25.04 binintersectER => 166.97/25.04 ( 166.97/25.04 disjointsetsI1 => 166.97/25.04 ( 166.97/25.04 binintersectRsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset4 => 166.97/25.04 ( 166.97/25.04 binintersectSubset1 => 166.97/25.04 ( 166.97/25.04 bs114d => 166.97/25.04 ( 166.97/25.04 setminusI => 166.97/25.04 ( 166.97/25.04 setminusEL => 166.97/25.04 ( 166.97/25.04 setminusER => 166.97/25.04 ( 166.97/25.04 setminusSubset2 => 166.97/25.04 ( 166.97/25.04 setminusERneg => 166.97/25.04 ( 166.97/25.04 setminusELneg => 166.97/25.04 ( 166.97/25.04 setminusILneg => 166.97/25.04 ( 166.97/25.04 setminusIRneg => 166.97/25.04 ( 166.97/25.04 setminusLsub => 166.97/25.04 ( 166.97/25.04 setminusSubset1 => 166.97/25.04 ( 166.97/25.04 symdiffE => 166.97/25.04 ( 166.97/25.04 symdiffI1 => 166.97/25.04 ( 166.97/25.04 symdiffI2 => 166.97/25.04 ( 166.97/25.04 symdiffIneg1 => 166.97/25.04 ( 166.97/25.04 symdiffIneg2 => 166.97/25.04 ( 166.97/25.04 secondinupair => 166.97/25.04 ( 166.97/25.04 setukpairIL => 166.97/25.04 ( 166.97/25.04 setukpairIR => 166.97/25.04 ( 166.97/25.04 kpairiskpair => 166.97/25.04 ( 166.97/25.04 kpairp => 166.97/25.04 ( 166.97/25.04 singletonsubset => 166.97/25.04 ( 166.97/25.04 singletoninpowerset => 166.97/25.04 ( 166.97/25.04 singletoninpowunion => 166.97/25.04 ( 166.97/25.04 upairset2E => 166.97/25.04 ( 166.97/25.04 upairsubunion => 166.97/25.04 ( 166.97/25.04 upairinpowunion => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem1 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem2 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem3 => 166.97/25.04 ( 166.97/25.04 cartprodpairin => 166.97/25.04 ( 166.97/25.04 cartprodmempair1 => 166.97/25.04 ( 166.97/25.04 cartprodmempair => 166.97/25.04 ( 166.97/25.04 setunionE2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton1 => 166.97/25.04 ( 166.97/25.04 setunionsingleton2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton => 166.97/25.04 ( 166.97/25.04 singletonprop => 166.97/25.04 ( 166.97/25.04 ex1E1 => 166.97/25.04 ( 166.97/25.04 ex1I => 166.97/25.04 ( 166.97/25.04 ex1I2 => 166.97/25.04 ( 166.97/25.04 singletonsuniq => 166.97/25.04 ( 166.97/25.04 setukpairinjL1 => 166.97/25.04 ( 166.97/25.04 kfstsingleton => 166.97/25.04 ( 166.97/25.04 theprop => 166.97/25.04 ( 166.97/25.04 kfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodfstin => 166.97/25.04 ( 166.97/25.04 setukpairinjL2 => 166.97/25.04 ( 166.97/25.04 setukpairinjL => 166.97/25.04 ( 166.97/25.04 setukpairinjR11 => 166.97/25.04 ( 166.97/25.04 setukpairinjR12 => 166.97/25.04 ( 166.97/25.04 setukpairinjR1 => 166.97/25.04 ( 166.97/25.04 upairequniteq => 166.97/25.04 ( 166.97/25.04 setukpairinjR2 => 166.97/25.04 ( 166.97/25.04 setukpairinjR => 166.97/25.04 ( 166.97/25.04 ksndsingleton => 166.97/25.04 ( 166.97/25.04 ksndpairEq => 166.97/25.04 ( 166.97/25.04 kpairsurjEq => 166.97/25.04 ( 166.97/25.04 cartprodsndin => 166.97/25.04 ( 166.97/25.04 cartprodpairmemEL => 166.97/25.04 ( 166.97/25.04 cartprodpairmemER => 166.97/25.04 ( 166.97/25.04 cartprodmempaircEq => 166.97/25.04 ( 166.97/25.04 cartprodfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodsndpairEq => 166.97/25.04 ( 166.97/25.04 cartprodpairsurjEq => 166.97/25.04 ( 166.97/25.04 dpsetconstrI => 166.97/25.04 ( 166.97/25.04 dpsetconstrSub => 166.97/25.04 ( 166.97/25.04 setOfPairsIsBReln => 166.97/25.04 ( 166.97/25.04 !!(( 166.97/25.04 ^[Y0 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y1 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y2 : ( 166.97/25.04 $i * $i) > $o]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y3 : $i]: 166.97/25.04 (in(Y3, 166.97/25.04 Y0) => 166.97/25.04 (!!(( 166.97/25.04 ^[Y4 : $i]: 166.97/25.04 (in(Y4, 166.97/25.04 Y1) => 166.97/25.04 (in( 166.97/25.04 kpair( 166.97/25.04 Y3, Y4), 166.97/25.04 dpsetconstr( 166.97/25.04 Y0, Y1, 166.97/25.04 Y2)) => 166.97/25.04 Y2(Y3, 166.97/25.04 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.04 inference('lazy_cnf_imply', [status(thm)], ['8'])). 166.97/25.04 tff('10', plain, 166.97/25.04 ~ (foundationAx => 166.97/25.04 (wellorderingAx => 166.97/25.04 (descrp => 166.97/25.04 (dsetconstrI => 166.97/25.04 (dsetconstrEL => 166.97/25.04 (dsetconstrER => 166.97/25.04 (exuE1 => 166.97/25.04 (prop2setE => 166.97/25.04 (emptysetE => 166.97/25.04 (emptysetimpfalse => 166.97/25.04 (notinemptyset => 166.97/25.04 (exuE3e => 166.97/25.04 (setext => 166.97/25.04 (emptyI => 166.97/25.04 (noeltsimpempty => 166.97/25.04 (setbeta => 166.97/25.04 (nonemptyE1 => 166.97/25.04 (nonemptyI => 166.97/25.04 (nonemptyI1 => 166.97/25.04 (setadjoinIL => 166.97/25.04 (emptyinunitempty => 166.97/25.04 (setadjoinIR => 166.97/25.04 (setadjoinE => 166.97/25.04 (setadjoinOr => 166.97/25.04 (setoftrueEq => 166.97/25.04 (powersetI => 166.97/25.04 (emptyinPowerset => 166.97/25.04 (emptyInPowerset => 166.97/25.04 (powersetE => 166.97/25.04 (setunionI => 166.97/25.04 (setunionE => 166.97/25.04 (subPowSU => 166.97/25.04 (exuE2 => 166.97/25.04 (nonemptyImpWitness => 166.97/25.04 (uniqinunit => 166.97/25.04 (notinsingleton => 166.97/25.04 (eqinunit => 166.97/25.04 (singletonsswitch => 166.97/25.04 (upairsetE => 166.97/25.04 (upairsetIL => 166.97/25.04 (upairsetIR => 166.97/25.04 (emptyE1 => 166.97/25.04 (vacuousDall => 166.97/25.04 (quantDeMorgan1 => 166.97/25.04 (quantDeMorgan2 => 166.97/25.04 (quantDeMorgan3 => 166.97/25.04 (quantDeMorgan4 => 166.97/25.04 (prop2setI => 166.97/25.04 (prop2set2propI => 166.97/25.04 (notdexE => 166.97/25.04 (notdallE => 166.97/25.04 (exuI1 => 166.97/25.04 (exuI3 => 166.97/25.04 (exuI2 => 166.97/25.04 (inCongP => 166.97/25.04 (in__Cong => 166.97/25.04 (exuE3u => 166.97/25.04 (exu__Cong => 166.97/25.04 (emptyset__Cong => 166.97/25.04 (setadjoin__Cong => 166.97/25.04 (powerset__Cong => 166.97/25.04 (setunion__Cong => 166.97/25.04 ( 166.97/25.04 omega__Cong => 166.97/25.04 ( 166.97/25.04 exuEu => 166.97/25.04 ( 166.97/25.04 descr__Cong => 166.97/25.04 ( 166.97/25.04 dsetconstr__Cong => 166.97/25.04 ( 166.97/25.04 subsetI1 => 166.97/25.04 ( 166.97/25.04 eqimpsubset2 => 166.97/25.04 ( 166.97/25.04 eqimpsubset1 => 166.97/25.04 ( 166.97/25.04 subsetI2 => 166.97/25.04 ( 166.97/25.04 emptysetsubset => 166.97/25.04 ( 166.97/25.04 subsetE => 166.97/25.04 ( 166.97/25.04 subsetE2 => 166.97/25.04 ( 166.97/25.04 notsubsetI => 166.97/25.04 ( 166.97/25.04 notequalI1 => 166.97/25.04 ( 166.97/25.04 notequalI2 => 166.97/25.04 ( 166.97/25.04 subsetRefl => 166.97/25.04 ( 166.97/25.04 subsetTrans => 166.97/25.04 ( 166.97/25.04 setadjoinSub => 166.97/25.04 ( 166.97/25.04 setadjoinSub2 => 166.97/25.04 ( 166.97/25.04 subset2powerset => 166.97/25.04 ( 166.97/25.04 setextsub => 166.97/25.04 ( 166.97/25.04 subsetemptysetimpeq => 166.97/25.04 ( 166.97/25.04 powersetI1 => 166.97/25.04 ( 166.97/25.04 powersetE1 => 166.97/25.04 ( 166.97/25.04 inPowerset => 166.97/25.04 ( 166.97/25.04 powersetsubset => 166.97/25.04 ( 166.97/25.04 sepInPowerset => 166.97/25.04 ( 166.97/25.04 sepSubset => 166.97/25.04 ( 166.97/25.04 binunionIL => 166.97/25.04 ( 166.97/25.04 upairset2IR => 166.97/25.04 ( 166.97/25.04 binunionIR => 166.97/25.04 ( 166.97/25.04 binunionEcases => 166.97/25.04 ( 166.97/25.04 binunionE => 166.97/25.04 ( 166.97/25.04 binunionLsub => 166.97/25.04 ( 166.97/25.04 binunionRsub => 166.97/25.04 ( 166.97/25.04 binintersectI => 166.97/25.04 ( 166.97/25.04 binintersectSubset5 => 166.97/25.04 ( 166.97/25.04 binintersectEL => 166.97/25.04 ( 166.97/25.04 binintersectLsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset2 => 166.97/25.04 ( 166.97/25.04 binintersectSubset3 => 166.97/25.04 ( 166.97/25.04 binintersectER => 166.97/25.04 ( 166.97/25.04 disjointsetsI1 => 166.97/25.04 ( 166.97/25.04 binintersectRsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset4 => 166.97/25.04 ( 166.97/25.04 binintersectSubset1 => 166.97/25.04 ( 166.97/25.04 bs114d => 166.97/25.04 ( 166.97/25.04 setminusI => 166.97/25.04 ( 166.97/25.04 setminusEL => 166.97/25.04 ( 166.97/25.04 setminusER => 166.97/25.04 ( 166.97/25.04 setminusSubset2 => 166.97/25.04 ( 166.97/25.04 setminusERneg => 166.97/25.04 ( 166.97/25.04 setminusELneg => 166.97/25.04 ( 166.97/25.04 setminusILneg => 166.97/25.04 ( 166.97/25.04 setminusIRneg => 166.97/25.04 ( 166.97/25.04 setminusLsub => 166.97/25.04 ( 166.97/25.04 setminusSubset1 => 166.97/25.04 ( 166.97/25.04 symdiffE => 166.97/25.04 ( 166.97/25.04 symdiffI1 => 166.97/25.04 ( 166.97/25.04 symdiffI2 => 166.97/25.04 ( 166.97/25.04 symdiffIneg1 => 166.97/25.04 ( 166.97/25.04 symdiffIneg2 => 166.97/25.04 ( 166.97/25.04 secondinupair => 166.97/25.04 ( 166.97/25.04 setukpairIL => 166.97/25.04 ( 166.97/25.04 setukpairIR => 166.97/25.04 ( 166.97/25.04 kpairiskpair => 166.97/25.04 ( 166.97/25.04 kpairp => 166.97/25.04 ( 166.97/25.04 singletonsubset => 166.97/25.04 ( 166.97/25.04 singletoninpowerset => 166.97/25.04 ( 166.97/25.04 singletoninpowunion => 166.97/25.04 ( 166.97/25.04 upairset2E => 166.97/25.04 ( 166.97/25.04 upairsubunion => 166.97/25.04 ( 166.97/25.04 upairinpowunion => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem1 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem2 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem3 => 166.97/25.04 ( 166.97/25.04 cartprodpairin => 166.97/25.04 ( 166.97/25.04 cartprodmempair1 => 166.97/25.04 ( 166.97/25.04 cartprodmempair => 166.97/25.04 ( 166.97/25.04 setunionE2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton1 => 166.97/25.04 ( 166.97/25.04 setunionsingleton2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton => 166.97/25.04 ( 166.97/25.04 singletonprop => 166.97/25.04 ( 166.97/25.04 ex1E1 => 166.97/25.04 ( 166.97/25.04 ex1I => 166.97/25.04 ( 166.97/25.04 ex1I2 => 166.97/25.04 ( 166.97/25.04 singletonsuniq => 166.97/25.04 ( 166.97/25.04 setukpairinjL1 => 166.97/25.04 ( 166.97/25.04 kfstsingleton => 166.97/25.04 ( 166.97/25.04 theprop => 166.97/25.04 ( 166.97/25.04 kfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodfstin => 166.97/25.04 ( 166.97/25.04 setukpairinjL2 => 166.97/25.04 ( 166.97/25.04 setukpairinjL => 166.97/25.04 ( 166.97/25.04 setukpairinjR11 => 166.97/25.04 ( 166.97/25.04 setukpairinjR12 => 166.97/25.04 ( 166.97/25.04 setukpairinjR1 => 166.97/25.04 ( 166.97/25.04 upairequniteq => 166.97/25.04 ( 166.97/25.04 setukpairinjR2 => 166.97/25.04 ( 166.97/25.04 setukpairinjR => 166.97/25.04 ( 166.97/25.04 ksndsingleton => 166.97/25.04 ( 166.97/25.04 ksndpairEq => 166.97/25.04 ( 166.97/25.04 kpairsurjEq => 166.97/25.04 ( 166.97/25.04 cartprodsndin => 166.97/25.04 ( 166.97/25.04 cartprodpairmemEL => 166.97/25.04 ( 166.97/25.04 cartprodpairmemER => 166.97/25.04 ( 166.97/25.04 cartprodmempaircEq => 166.97/25.04 ( 166.97/25.04 cartprodfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodsndpairEq => 166.97/25.04 ( 166.97/25.04 cartprodpairsurjEq => 166.97/25.04 ( 166.97/25.04 dpsetconstrI => 166.97/25.04 ( 166.97/25.04 dpsetconstrSub => 166.97/25.04 ( 166.97/25.04 setOfPairsIsBReln => 166.97/25.04 ( 166.97/25.04 !!(( 166.97/25.04 ^[Y0 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y1 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y2 : ( 166.97/25.04 $i * $i) > $o]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y3 : $i]: 166.97/25.04 (in(Y3, 166.97/25.04 Y0) => 166.97/25.04 (!!(( 166.97/25.04 ^[Y4 : $i]: 166.97/25.04 (in(Y4, 166.97/25.04 Y1) => 166.97/25.04 (in( 166.97/25.04 kpair( 166.97/25.04 Y3, Y4), 166.97/25.04 dpsetconstr( 166.97/25.04 Y0, Y1, 166.97/25.04 Y2)) => 166.97/25.04 Y2(Y3, 166.97/25.04 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.04 inference('lazy_cnf_imply', [status(thm)], ['9'])). 166.97/25.04 tff('11', plain, 166.97/25.04 ~ (wellorderingAx => 166.97/25.04 (descrp => 166.97/25.04 (dsetconstrI => 166.97/25.04 (dsetconstrEL => 166.97/25.04 (dsetconstrER => 166.97/25.04 (exuE1 => 166.97/25.04 (prop2setE => 166.97/25.04 (emptysetE => 166.97/25.04 (emptysetimpfalse => 166.97/25.04 (notinemptyset => 166.97/25.04 (exuE3e => 166.97/25.04 (setext => 166.97/25.04 (emptyI => 166.97/25.04 (noeltsimpempty => 166.97/25.04 (setbeta => 166.97/25.04 (nonemptyE1 => 166.97/25.04 (nonemptyI => 166.97/25.04 (nonemptyI1 => 166.97/25.04 (setadjoinIL => 166.97/25.04 (emptyinunitempty => 166.97/25.04 (setadjoinIR => 166.97/25.04 (setadjoinE => 166.97/25.04 (setadjoinOr => 166.97/25.04 (setoftrueEq => 166.97/25.04 (powersetI => 166.97/25.04 (emptyinPowerset => 166.97/25.04 (emptyInPowerset => 166.97/25.04 (powersetE => 166.97/25.04 (setunionI => 166.97/25.04 (setunionE => 166.97/25.04 (subPowSU => 166.97/25.04 (exuE2 => 166.97/25.04 (nonemptyImpWitness => 166.97/25.04 (uniqinunit => 166.97/25.04 (notinsingleton => 166.97/25.04 (eqinunit => 166.97/25.04 (singletonsswitch => 166.97/25.04 (upairsetE => 166.97/25.04 (upairsetIL => 166.97/25.04 (upairsetIR => 166.97/25.04 (emptyE1 => 166.97/25.04 (vacuousDall => 166.97/25.04 (quantDeMorgan1 => 166.97/25.04 (quantDeMorgan2 => 166.97/25.04 (quantDeMorgan3 => 166.97/25.04 (quantDeMorgan4 => 166.97/25.04 (prop2setI => 166.97/25.04 (prop2set2propI => 166.97/25.04 (notdexE => 166.97/25.04 (notdallE => 166.97/25.04 (exuI1 => 166.97/25.04 (exuI3 => 166.97/25.04 (exuI2 => 166.97/25.04 (inCongP => 166.97/25.04 (in__Cong => 166.97/25.04 (exuE3u => 166.97/25.04 (exu__Cong => 166.97/25.04 (emptyset__Cong => 166.97/25.04 (setadjoin__Cong => 166.97/25.04 (powerset__Cong => 166.97/25.04 (setunion__Cong => 166.97/25.04 (omega__Cong => 166.97/25.04 ( 166.97/25.04 exuEu => 166.97/25.04 ( 166.97/25.04 descr__Cong => 166.97/25.04 ( 166.97/25.04 dsetconstr__Cong => 166.97/25.04 ( 166.97/25.04 subsetI1 => 166.97/25.04 ( 166.97/25.04 eqimpsubset2 => 166.97/25.04 ( 166.97/25.04 eqimpsubset1 => 166.97/25.04 ( 166.97/25.04 subsetI2 => 166.97/25.04 ( 166.97/25.04 emptysetsubset => 166.97/25.04 ( 166.97/25.04 subsetE => 166.97/25.04 ( 166.97/25.04 subsetE2 => 166.97/25.04 ( 166.97/25.04 notsubsetI => 166.97/25.04 ( 166.97/25.04 notequalI1 => 166.97/25.04 ( 166.97/25.04 notequalI2 => 166.97/25.04 ( 166.97/25.04 subsetRefl => 166.97/25.04 ( 166.97/25.04 subsetTrans => 166.97/25.04 ( 166.97/25.04 setadjoinSub => 166.97/25.04 ( 166.97/25.04 setadjoinSub2 => 166.97/25.04 ( 166.97/25.04 subset2powerset => 166.97/25.04 ( 166.97/25.04 setextsub => 166.97/25.04 ( 166.97/25.04 subsetemptysetimpeq => 166.97/25.04 ( 166.97/25.04 powersetI1 => 166.97/25.04 ( 166.97/25.04 powersetE1 => 166.97/25.04 ( 166.97/25.04 inPowerset => 166.97/25.04 ( 166.97/25.04 powersetsubset => 166.97/25.04 ( 166.97/25.04 sepInPowerset => 166.97/25.04 ( 166.97/25.04 sepSubset => 166.97/25.04 ( 166.97/25.04 binunionIL => 166.97/25.04 ( 166.97/25.04 upairset2IR => 166.97/25.04 ( 166.97/25.04 binunionIR => 166.97/25.04 ( 166.97/25.04 binunionEcases => 166.97/25.04 ( 166.97/25.04 binunionE => 166.97/25.04 ( 166.97/25.04 binunionLsub => 166.97/25.04 ( 166.97/25.04 binunionRsub => 166.97/25.04 ( 166.97/25.04 binintersectI => 166.97/25.04 ( 166.97/25.04 binintersectSubset5 => 166.97/25.04 ( 166.97/25.04 binintersectEL => 166.97/25.04 ( 166.97/25.04 binintersectLsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset2 => 166.97/25.04 ( 166.97/25.04 binintersectSubset3 => 166.97/25.04 ( 166.97/25.04 binintersectER => 166.97/25.04 ( 166.97/25.04 disjointsetsI1 => 166.97/25.04 ( 166.97/25.04 binintersectRsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset4 => 166.97/25.04 ( 166.97/25.04 binintersectSubset1 => 166.97/25.04 ( 166.97/25.04 bs114d => 166.97/25.04 ( 166.97/25.04 setminusI => 166.97/25.04 ( 166.97/25.04 setminusEL => 166.97/25.04 ( 166.97/25.04 setminusER => 166.97/25.04 ( 166.97/25.04 setminusSubset2 => 166.97/25.04 ( 166.97/25.04 setminusERneg => 166.97/25.04 ( 166.97/25.04 setminusELneg => 166.97/25.04 ( 166.97/25.04 setminusILneg => 166.97/25.04 ( 166.97/25.04 setminusIRneg => 166.97/25.04 ( 166.97/25.04 setminusLsub => 166.97/25.04 ( 166.97/25.04 setminusSubset1 => 166.97/25.04 ( 166.97/25.04 symdiffE => 166.97/25.04 ( 166.97/25.04 symdiffI1 => 166.97/25.04 ( 166.97/25.04 symdiffI2 => 166.97/25.04 ( 166.97/25.04 symdiffIneg1 => 166.97/25.04 ( 166.97/25.04 symdiffIneg2 => 166.97/25.04 ( 166.97/25.04 secondinupair => 166.97/25.04 ( 166.97/25.04 setukpairIL => 166.97/25.04 ( 166.97/25.04 setukpairIR => 166.97/25.04 ( 166.97/25.04 kpairiskpair => 166.97/25.04 ( 166.97/25.04 kpairp => 166.97/25.04 ( 166.97/25.04 singletonsubset => 166.97/25.04 ( 166.97/25.04 singletoninpowerset => 166.97/25.04 ( 166.97/25.04 singletoninpowunion => 166.97/25.04 ( 166.97/25.04 upairset2E => 166.97/25.04 ( 166.97/25.04 upairsubunion => 166.97/25.04 ( 166.97/25.04 upairinpowunion => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem1 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem2 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem3 => 166.97/25.04 ( 166.97/25.04 cartprodpairin => 166.97/25.04 ( 166.97/25.04 cartprodmempair1 => 166.97/25.04 ( 166.97/25.04 cartprodmempair => 166.97/25.04 ( 166.97/25.04 setunionE2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton1 => 166.97/25.04 ( 166.97/25.04 setunionsingleton2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton => 166.97/25.04 ( 166.97/25.04 singletonprop => 166.97/25.04 ( 166.97/25.04 ex1E1 => 166.97/25.04 ( 166.97/25.04 ex1I => 166.97/25.04 ( 166.97/25.04 ex1I2 => 166.97/25.04 ( 166.97/25.04 singletonsuniq => 166.97/25.04 ( 166.97/25.04 setukpairinjL1 => 166.97/25.04 ( 166.97/25.04 kfstsingleton => 166.97/25.04 ( 166.97/25.04 theprop => 166.97/25.04 ( 166.97/25.04 kfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodfstin => 166.97/25.04 ( 166.97/25.04 setukpairinjL2 => 166.97/25.04 ( 166.97/25.04 setukpairinjL => 166.97/25.04 ( 166.97/25.04 setukpairinjR11 => 166.97/25.04 ( 166.97/25.04 setukpairinjR12 => 166.97/25.04 ( 166.97/25.04 setukpairinjR1 => 166.97/25.04 ( 166.97/25.04 upairequniteq => 166.97/25.04 ( 166.97/25.04 setukpairinjR2 => 166.97/25.04 ( 166.97/25.04 setukpairinjR => 166.97/25.04 ( 166.97/25.04 ksndsingleton => 166.97/25.04 ( 166.97/25.04 ksndpairEq => 166.97/25.04 ( 166.97/25.04 kpairsurjEq => 166.97/25.04 ( 166.97/25.04 cartprodsndin => 166.97/25.04 ( 166.97/25.04 cartprodpairmemEL => 166.97/25.04 ( 166.97/25.04 cartprodpairmemER => 166.97/25.04 ( 166.97/25.04 cartprodmempaircEq => 166.97/25.04 ( 166.97/25.04 cartprodfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodsndpairEq => 166.97/25.04 ( 166.97/25.04 cartprodpairsurjEq => 166.97/25.04 ( 166.97/25.04 dpsetconstrI => 166.97/25.04 ( 166.97/25.04 dpsetconstrSub => 166.97/25.04 ( 166.97/25.04 setOfPairsIsBReln => 166.97/25.04 ( 166.97/25.04 !!(( 166.97/25.04 ^[Y0 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y1 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y2 : ( 166.97/25.04 $i * $i) > $o]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y3 : $i]: 166.97/25.04 (in(Y3, 166.97/25.04 Y0) => 166.97/25.04 (!!(( 166.97/25.04 ^[Y4 : $i]: 166.97/25.04 (in(Y4, 166.97/25.04 Y1) => 166.97/25.04 (in( 166.97/25.04 kpair( 166.97/25.04 Y3, Y4), 166.97/25.04 dpsetconstr( 166.97/25.04 Y0, Y1, 166.97/25.04 Y2)) => 166.97/25.04 Y2(Y3, 166.97/25.04 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.04 inference('lazy_cnf_imply', [status(thm)], ['10'])). 166.97/25.04 tff('12', plain, 166.97/25.04 ~ (descrp => 166.97/25.04 (dsetconstrI => 166.97/25.04 (dsetconstrEL => 166.97/25.04 (dsetconstrER => 166.97/25.04 (exuE1 => 166.97/25.04 (prop2setE => 166.97/25.04 (emptysetE => 166.97/25.04 (emptysetimpfalse => 166.97/25.04 (notinemptyset => 166.97/25.04 (exuE3e => 166.97/25.04 (setext => 166.97/25.04 (emptyI => 166.97/25.04 (noeltsimpempty => 166.97/25.04 (setbeta => 166.97/25.04 (nonemptyE1 => 166.97/25.04 (nonemptyI => 166.97/25.04 (nonemptyI1 => 166.97/25.04 (setadjoinIL => 166.97/25.04 (emptyinunitempty => 166.97/25.04 (setadjoinIR => 166.97/25.04 (setadjoinE => 166.97/25.04 (setadjoinOr => 166.97/25.04 (setoftrueEq => 166.97/25.04 (powersetI => 166.97/25.04 (emptyinPowerset => 166.97/25.04 (emptyInPowerset => 166.97/25.04 (powersetE => 166.97/25.04 (setunionI => 166.97/25.04 (setunionE => 166.97/25.04 (subPowSU => 166.97/25.04 (exuE2 => 166.97/25.04 (nonemptyImpWitness => 166.97/25.04 (uniqinunit => 166.97/25.04 (notinsingleton => 166.97/25.04 (eqinunit => 166.97/25.04 (singletonsswitch => 166.97/25.04 (upairsetE => 166.97/25.04 (upairsetIL => 166.97/25.04 (upairsetIR => 166.97/25.04 (emptyE1 => 166.97/25.04 (vacuousDall => 166.97/25.04 (quantDeMorgan1 => 166.97/25.04 (quantDeMorgan2 => 166.97/25.04 (quantDeMorgan3 => 166.97/25.04 (quantDeMorgan4 => 166.97/25.04 (prop2setI => 166.97/25.04 (prop2set2propI => 166.97/25.04 (notdexE => 166.97/25.04 (notdallE => 166.97/25.04 (exuI1 => 166.97/25.04 (exuI3 => 166.97/25.04 (exuI2 => 166.97/25.04 (inCongP => 166.97/25.04 (in__Cong => 166.97/25.04 (exuE3u => 166.97/25.04 (exu__Cong => 166.97/25.04 (emptyset__Cong => 166.97/25.04 (setadjoin__Cong => 166.97/25.04 (powerset__Cong => 166.97/25.04 (setunion__Cong => 166.97/25.04 (omega__Cong => 166.97/25.04 (exuEu => 166.97/25.04 ( 166.97/25.04 descr__Cong => 166.97/25.04 ( 166.97/25.04 dsetconstr__Cong => 166.97/25.04 ( 166.97/25.04 subsetI1 => 166.97/25.04 ( 166.97/25.04 eqimpsubset2 => 166.97/25.04 ( 166.97/25.04 eqimpsubset1 => 166.97/25.04 ( 166.97/25.04 subsetI2 => 166.97/25.04 ( 166.97/25.04 emptysetsubset => 166.97/25.04 ( 166.97/25.04 subsetE => 166.97/25.04 ( 166.97/25.04 subsetE2 => 166.97/25.04 ( 166.97/25.04 notsubsetI => 166.97/25.04 ( 166.97/25.04 notequalI1 => 166.97/25.04 ( 166.97/25.04 notequalI2 => 166.97/25.04 ( 166.97/25.04 subsetRefl => 166.97/25.04 ( 166.97/25.04 subsetTrans => 166.97/25.04 ( 166.97/25.04 setadjoinSub => 166.97/25.04 ( 166.97/25.04 setadjoinSub2 => 166.97/25.04 ( 166.97/25.04 subset2powerset => 166.97/25.04 ( 166.97/25.04 setextsub => 166.97/25.04 ( 166.97/25.04 subsetemptysetimpeq => 166.97/25.04 ( 166.97/25.04 powersetI1 => 166.97/25.04 ( 166.97/25.04 powersetE1 => 166.97/25.04 ( 166.97/25.04 inPowerset => 166.97/25.04 ( 166.97/25.04 powersetsubset => 166.97/25.04 ( 166.97/25.04 sepInPowerset => 166.97/25.04 ( 166.97/25.04 sepSubset => 166.97/25.04 ( 166.97/25.04 binunionIL => 166.97/25.04 ( 166.97/25.04 upairset2IR => 166.97/25.04 ( 166.97/25.04 binunionIR => 166.97/25.04 ( 166.97/25.04 binunionEcases => 166.97/25.04 ( 166.97/25.04 binunionE => 166.97/25.04 ( 166.97/25.04 binunionLsub => 166.97/25.04 ( 166.97/25.04 binunionRsub => 166.97/25.04 ( 166.97/25.04 binintersectI => 166.97/25.04 ( 166.97/25.04 binintersectSubset5 => 166.97/25.04 ( 166.97/25.04 binintersectEL => 166.97/25.04 ( 166.97/25.04 binintersectLsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset2 => 166.97/25.04 ( 166.97/25.04 binintersectSubset3 => 166.97/25.04 ( 166.97/25.04 binintersectER => 166.97/25.04 ( 166.97/25.04 disjointsetsI1 => 166.97/25.04 ( 166.97/25.04 binintersectRsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset4 => 166.97/25.04 ( 166.97/25.04 binintersectSubset1 => 166.97/25.04 ( 166.97/25.04 bs114d => 166.97/25.04 ( 166.97/25.04 setminusI => 166.97/25.04 ( 166.97/25.04 setminusEL => 166.97/25.04 ( 166.97/25.04 setminusER => 166.97/25.04 ( 166.97/25.04 setminusSubset2 => 166.97/25.04 ( 166.97/25.04 setminusERneg => 166.97/25.04 ( 166.97/25.04 setminusELneg => 166.97/25.04 ( 166.97/25.04 setminusILneg => 166.97/25.04 ( 166.97/25.04 setminusIRneg => 166.97/25.04 ( 166.97/25.04 setminusLsub => 166.97/25.04 ( 166.97/25.04 setminusSubset1 => 166.97/25.04 ( 166.97/25.04 symdiffE => 166.97/25.04 ( 166.97/25.04 symdiffI1 => 166.97/25.04 ( 166.97/25.04 symdiffI2 => 166.97/25.04 ( 166.97/25.04 symdiffIneg1 => 166.97/25.04 ( 166.97/25.04 symdiffIneg2 => 166.97/25.04 ( 166.97/25.04 secondinupair => 166.97/25.04 ( 166.97/25.04 setukpairIL => 166.97/25.04 ( 166.97/25.04 setukpairIR => 166.97/25.04 ( 166.97/25.04 kpairiskpair => 166.97/25.04 ( 166.97/25.04 kpairp => 166.97/25.04 ( 166.97/25.04 singletonsubset => 166.97/25.04 ( 166.97/25.04 singletoninpowerset => 166.97/25.04 ( 166.97/25.04 singletoninpowunion => 166.97/25.04 ( 166.97/25.04 upairset2E => 166.97/25.04 ( 166.97/25.04 upairsubunion => 166.97/25.04 ( 166.97/25.04 upairinpowunion => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem1 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem2 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem3 => 166.97/25.04 ( 166.97/25.04 cartprodpairin => 166.97/25.04 ( 166.97/25.04 cartprodmempair1 => 166.97/25.04 ( 166.97/25.04 cartprodmempair => 166.97/25.04 ( 166.97/25.04 setunionE2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton1 => 166.97/25.04 ( 166.97/25.04 setunionsingleton2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton => 166.97/25.04 ( 166.97/25.04 singletonprop => 166.97/25.04 ( 166.97/25.04 ex1E1 => 166.97/25.04 ( 166.97/25.04 ex1I => 166.97/25.04 ( 166.97/25.04 ex1I2 => 166.97/25.04 ( 166.97/25.04 singletonsuniq => 166.97/25.04 ( 166.97/25.04 setukpairinjL1 => 166.97/25.04 ( 166.97/25.04 kfstsingleton => 166.97/25.04 ( 166.97/25.04 theprop => 166.97/25.04 ( 166.97/25.04 kfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodfstin => 166.97/25.04 ( 166.97/25.04 setukpairinjL2 => 166.97/25.04 ( 166.97/25.04 setukpairinjL => 166.97/25.04 ( 166.97/25.04 setukpairinjR11 => 166.97/25.04 ( 166.97/25.04 setukpairinjR12 => 166.97/25.04 ( 166.97/25.04 setukpairinjR1 => 166.97/25.04 ( 166.97/25.04 upairequniteq => 166.97/25.04 ( 166.97/25.04 setukpairinjR2 => 166.97/25.04 ( 166.97/25.04 setukpairinjR => 166.97/25.04 ( 166.97/25.04 ksndsingleton => 166.97/25.04 ( 166.97/25.04 ksndpairEq => 166.97/25.04 ( 166.97/25.04 kpairsurjEq => 166.97/25.04 ( 166.97/25.04 cartprodsndin => 166.97/25.04 ( 166.97/25.04 cartprodpairmemEL => 166.97/25.04 ( 166.97/25.04 cartprodpairmemER => 166.97/25.04 ( 166.97/25.04 cartprodmempaircEq => 166.97/25.04 ( 166.97/25.04 cartprodfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodsndpairEq => 166.97/25.04 ( 166.97/25.04 cartprodpairsurjEq => 166.97/25.04 ( 166.97/25.04 dpsetconstrI => 166.97/25.04 ( 166.97/25.04 dpsetconstrSub => 166.97/25.04 ( 166.97/25.04 setOfPairsIsBReln => 166.97/25.04 ( 166.97/25.04 !!(( 166.97/25.04 ^[Y0 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y1 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y2 : ( 166.97/25.04 $i * $i) > $o]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y3 : $i]: 166.97/25.04 (in(Y3, 166.97/25.04 Y0) => 166.97/25.04 (!!(( 166.97/25.04 ^[Y4 : $i]: 166.97/25.04 (in(Y4, 166.97/25.04 Y1) => 166.97/25.04 (in( 166.97/25.04 kpair( 166.97/25.04 Y3, Y4), 166.97/25.04 dpsetconstr( 166.97/25.04 Y0, Y1, 166.97/25.04 Y2)) => 166.97/25.04 Y2(Y3, 166.97/25.04 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.04 inference('lazy_cnf_imply', [status(thm)], ['11'])). 166.97/25.04 tff('13', plain, 166.97/25.04 ~ (dsetconstrI => 166.97/25.04 (dsetconstrEL => 166.97/25.04 (dsetconstrER => 166.97/25.04 (exuE1 => 166.97/25.04 (prop2setE => 166.97/25.04 (emptysetE => 166.97/25.04 (emptysetimpfalse => 166.97/25.04 (notinemptyset => 166.97/25.04 (exuE3e => 166.97/25.04 (setext => 166.97/25.04 (emptyI => 166.97/25.04 (noeltsimpempty => 166.97/25.04 (setbeta => 166.97/25.04 (nonemptyE1 => 166.97/25.04 (nonemptyI => 166.97/25.04 (nonemptyI1 => 166.97/25.04 (setadjoinIL => 166.97/25.04 (emptyinunitempty => 166.97/25.04 (setadjoinIR => 166.97/25.04 (setadjoinE => 166.97/25.04 (setadjoinOr => 166.97/25.04 (setoftrueEq => 166.97/25.04 (powersetI => 166.97/25.04 (emptyinPowerset => 166.97/25.04 (emptyInPowerset => 166.97/25.04 (powersetE => 166.97/25.04 (setunionI => 166.97/25.04 (setunionE => 166.97/25.04 (subPowSU => 166.97/25.04 (exuE2 => 166.97/25.04 (nonemptyImpWitness => 166.97/25.04 (uniqinunit => 166.97/25.04 (notinsingleton => 166.97/25.04 (eqinunit => 166.97/25.04 (singletonsswitch => 166.97/25.04 (upairsetE => 166.97/25.04 (upairsetIL => 166.97/25.04 (upairsetIR => 166.97/25.04 (emptyE1 => 166.97/25.04 (vacuousDall => 166.97/25.04 (quantDeMorgan1 => 166.97/25.04 (quantDeMorgan2 => 166.97/25.04 (quantDeMorgan3 => 166.97/25.04 (quantDeMorgan4 => 166.97/25.04 (prop2setI => 166.97/25.04 (prop2set2propI => 166.97/25.04 (notdexE => 166.97/25.04 (notdallE => 166.97/25.04 (exuI1 => 166.97/25.04 (exuI3 => 166.97/25.04 (exuI2 => 166.97/25.04 (inCongP => 166.97/25.04 (in__Cong => 166.97/25.04 (exuE3u => 166.97/25.04 (exu__Cong => 166.97/25.04 (emptyset__Cong => 166.97/25.04 (setadjoin__Cong => 166.97/25.04 (powerset__Cong => 166.97/25.04 (setunion__Cong => 166.97/25.04 (omega__Cong => 166.97/25.04 (exuEu => 166.97/25.04 (descr__Cong => 166.97/25.04 ( 166.97/25.04 dsetconstr__Cong => 166.97/25.04 ( 166.97/25.04 subsetI1 => 166.97/25.04 ( 166.97/25.04 eqimpsubset2 => 166.97/25.04 ( 166.97/25.04 eqimpsubset1 => 166.97/25.04 ( 166.97/25.04 subsetI2 => 166.97/25.04 ( 166.97/25.04 emptysetsubset => 166.97/25.04 ( 166.97/25.04 subsetE => 166.97/25.04 ( 166.97/25.04 subsetE2 => 166.97/25.04 ( 166.97/25.04 notsubsetI => 166.97/25.04 ( 166.97/25.04 notequalI1 => 166.97/25.04 ( 166.97/25.04 notequalI2 => 166.97/25.04 ( 166.97/25.04 subsetRefl => 166.97/25.04 ( 166.97/25.04 subsetTrans => 166.97/25.04 ( 166.97/25.04 setadjoinSub => 166.97/25.04 ( 166.97/25.04 setadjoinSub2 => 166.97/25.04 ( 166.97/25.04 subset2powerset => 166.97/25.04 ( 166.97/25.04 setextsub => 166.97/25.04 ( 166.97/25.04 subsetemptysetimpeq => 166.97/25.04 ( 166.97/25.04 powersetI1 => 166.97/25.04 ( 166.97/25.04 powersetE1 => 166.97/25.04 ( 166.97/25.04 inPowerset => 166.97/25.04 ( 166.97/25.04 powersetsubset => 166.97/25.04 ( 166.97/25.04 sepInPowerset => 166.97/25.04 ( 166.97/25.04 sepSubset => 166.97/25.04 ( 166.97/25.04 binunionIL => 166.97/25.04 ( 166.97/25.04 upairset2IR => 166.97/25.04 ( 166.97/25.04 binunionIR => 166.97/25.04 ( 166.97/25.04 binunionEcases => 166.97/25.04 ( 166.97/25.04 binunionE => 166.97/25.04 ( 166.97/25.04 binunionLsub => 166.97/25.04 ( 166.97/25.04 binunionRsub => 166.97/25.04 ( 166.97/25.04 binintersectI => 166.97/25.04 ( 166.97/25.04 binintersectSubset5 => 166.97/25.04 ( 166.97/25.04 binintersectEL => 166.97/25.04 ( 166.97/25.04 binintersectLsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset2 => 166.97/25.04 ( 166.97/25.04 binintersectSubset3 => 166.97/25.04 ( 166.97/25.04 binintersectER => 166.97/25.04 ( 166.97/25.04 disjointsetsI1 => 166.97/25.04 ( 166.97/25.04 binintersectRsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset4 => 166.97/25.04 ( 166.97/25.04 binintersectSubset1 => 166.97/25.04 ( 166.97/25.04 bs114d => 166.97/25.04 ( 166.97/25.04 setminusI => 166.97/25.04 ( 166.97/25.04 setminusEL => 166.97/25.04 ( 166.97/25.04 setminusER => 166.97/25.04 ( 166.97/25.04 setminusSubset2 => 166.97/25.04 ( 166.97/25.04 setminusERneg => 166.97/25.04 ( 166.97/25.04 setminusELneg => 166.97/25.04 ( 166.97/25.04 setminusILneg => 166.97/25.04 ( 166.97/25.04 setminusIRneg => 166.97/25.04 ( 166.97/25.04 setminusLsub => 166.97/25.04 ( 166.97/25.04 setminusSubset1 => 166.97/25.04 ( 166.97/25.04 symdiffE => 166.97/25.04 ( 166.97/25.04 symdiffI1 => 166.97/25.04 ( 166.97/25.04 symdiffI2 => 166.97/25.04 ( 166.97/25.04 symdiffIneg1 => 166.97/25.04 ( 166.97/25.04 symdiffIneg2 => 166.97/25.04 ( 166.97/25.04 secondinupair => 166.97/25.04 ( 166.97/25.04 setukpairIL => 166.97/25.04 ( 166.97/25.04 setukpairIR => 166.97/25.04 ( 166.97/25.04 kpairiskpair => 166.97/25.04 ( 166.97/25.04 kpairp => 166.97/25.04 ( 166.97/25.04 singletonsubset => 166.97/25.04 ( 166.97/25.04 singletoninpowerset => 166.97/25.04 ( 166.97/25.04 singletoninpowunion => 166.97/25.04 ( 166.97/25.04 upairset2E => 166.97/25.04 ( 166.97/25.04 upairsubunion => 166.97/25.04 ( 166.97/25.04 upairinpowunion => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem1 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem2 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem3 => 166.97/25.04 ( 166.97/25.04 cartprodpairin => 166.97/25.04 ( 166.97/25.04 cartprodmempair1 => 166.97/25.04 ( 166.97/25.04 cartprodmempair => 166.97/25.04 ( 166.97/25.04 setunionE2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton1 => 166.97/25.04 ( 166.97/25.04 setunionsingleton2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton => 166.97/25.04 ( 166.97/25.04 singletonprop => 166.97/25.04 ( 166.97/25.04 ex1E1 => 166.97/25.04 ( 166.97/25.04 ex1I => 166.97/25.04 ( 166.97/25.04 ex1I2 => 166.97/25.04 ( 166.97/25.04 singletonsuniq => 166.97/25.04 ( 166.97/25.04 setukpairinjL1 => 166.97/25.04 ( 166.97/25.04 kfstsingleton => 166.97/25.04 ( 166.97/25.04 theprop => 166.97/25.04 ( 166.97/25.04 kfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodfstin => 166.97/25.04 ( 166.97/25.04 setukpairinjL2 => 166.97/25.04 ( 166.97/25.04 setukpairinjL => 166.97/25.04 ( 166.97/25.04 setukpairinjR11 => 166.97/25.04 ( 166.97/25.04 setukpairinjR12 => 166.97/25.04 ( 166.97/25.04 setukpairinjR1 => 166.97/25.04 ( 166.97/25.04 upairequniteq => 166.97/25.04 ( 166.97/25.04 setukpairinjR2 => 166.97/25.04 ( 166.97/25.04 setukpairinjR => 166.97/25.04 ( 166.97/25.04 ksndsingleton => 166.97/25.04 ( 166.97/25.04 ksndpairEq => 166.97/25.04 ( 166.97/25.04 kpairsurjEq => 166.97/25.04 ( 166.97/25.04 cartprodsndin => 166.97/25.04 ( 166.97/25.04 cartprodpairmemEL => 166.97/25.04 ( 166.97/25.04 cartprodpairmemER => 166.97/25.04 ( 166.97/25.04 cartprodmempaircEq => 166.97/25.04 ( 166.97/25.04 cartprodfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodsndpairEq => 166.97/25.04 ( 166.97/25.04 cartprodpairsurjEq => 166.97/25.04 ( 166.97/25.04 dpsetconstrI => 166.97/25.04 ( 166.97/25.04 dpsetconstrSub => 166.97/25.04 ( 166.97/25.04 setOfPairsIsBReln => 166.97/25.04 ( 166.97/25.04 !!(( 166.97/25.04 ^[Y0 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y1 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y2 : ( 166.97/25.04 $i * $i) > $o]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y3 : $i]: 166.97/25.04 (in(Y3, 166.97/25.04 Y0) => 166.97/25.04 (!!(( 166.97/25.04 ^[Y4 : $i]: 166.97/25.04 (in(Y4, 166.97/25.04 Y1) => 166.97/25.04 (in( 166.97/25.04 kpair( 166.97/25.04 Y3, Y4), 166.97/25.04 dpsetconstr( 166.97/25.04 Y0, Y1, 166.97/25.04 Y2)) => 166.97/25.04 Y2(Y3, 166.97/25.04 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.04 inference('lazy_cnf_imply', [status(thm)], ['12'])). 166.97/25.04 tff('14', plain, 166.97/25.04 ~ (dsetconstrEL => 166.97/25.04 (dsetconstrER => 166.97/25.04 (exuE1 => 166.97/25.04 (prop2setE => 166.97/25.04 (emptysetE => 166.97/25.04 (emptysetimpfalse => 166.97/25.04 (notinemptyset => 166.97/25.04 (exuE3e => 166.97/25.04 (setext => 166.97/25.04 (emptyI => 166.97/25.04 (noeltsimpempty => 166.97/25.04 (setbeta => 166.97/25.04 (nonemptyE1 => 166.97/25.04 (nonemptyI => 166.97/25.04 (nonemptyI1 => 166.97/25.04 (setadjoinIL => 166.97/25.04 (emptyinunitempty => 166.97/25.04 (setadjoinIR => 166.97/25.04 (setadjoinE => 166.97/25.04 (setadjoinOr => 166.97/25.04 (setoftrueEq => 166.97/25.04 (powersetI => 166.97/25.04 (emptyinPowerset => 166.97/25.04 (emptyInPowerset => 166.97/25.04 (powersetE => 166.97/25.04 (setunionI => 166.97/25.04 (setunionE => 166.97/25.04 (subPowSU => 166.97/25.04 (exuE2 => 166.97/25.04 (nonemptyImpWitness => 166.97/25.04 (uniqinunit => 166.97/25.04 (notinsingleton => 166.97/25.04 (eqinunit => 166.97/25.04 (singletonsswitch => 166.97/25.04 (upairsetE => 166.97/25.04 (upairsetIL => 166.97/25.04 (upairsetIR => 166.97/25.04 (emptyE1 => 166.97/25.04 (vacuousDall => 166.97/25.04 (quantDeMorgan1 => 166.97/25.04 (quantDeMorgan2 => 166.97/25.04 (quantDeMorgan3 => 166.97/25.04 (quantDeMorgan4 => 166.97/25.04 (prop2setI => 166.97/25.04 (prop2set2propI => 166.97/25.04 (notdexE => 166.97/25.04 (notdallE => 166.97/25.04 (exuI1 => 166.97/25.04 (exuI3 => 166.97/25.04 (exuI2 => 166.97/25.04 (inCongP => 166.97/25.04 (in__Cong => 166.97/25.04 (exuE3u => 166.97/25.04 (exu__Cong => 166.97/25.04 (emptyset__Cong => 166.97/25.04 (setadjoin__Cong => 166.97/25.04 (powerset__Cong => 166.97/25.04 (setunion__Cong => 166.97/25.04 (omega__Cong => 166.97/25.04 (exuEu => 166.97/25.04 (descr__Cong => 166.97/25.04 (dsetconstr__Cong => 166.97/25.04 ( 166.97/25.04 subsetI1 => 166.97/25.04 ( 166.97/25.04 eqimpsubset2 => 166.97/25.04 ( 166.97/25.04 eqimpsubset1 => 166.97/25.04 ( 166.97/25.04 subsetI2 => 166.97/25.04 ( 166.97/25.04 emptysetsubset => 166.97/25.04 ( 166.97/25.04 subsetE => 166.97/25.04 ( 166.97/25.04 subsetE2 => 166.97/25.04 ( 166.97/25.04 notsubsetI => 166.97/25.04 ( 166.97/25.04 notequalI1 => 166.97/25.04 ( 166.97/25.04 notequalI2 => 166.97/25.04 ( 166.97/25.04 subsetRefl => 166.97/25.04 ( 166.97/25.04 subsetTrans => 166.97/25.04 ( 166.97/25.04 setadjoinSub => 166.97/25.04 ( 166.97/25.04 setadjoinSub2 => 166.97/25.04 ( 166.97/25.04 subset2powerset => 166.97/25.04 ( 166.97/25.04 setextsub => 166.97/25.04 ( 166.97/25.04 subsetemptysetimpeq => 166.97/25.04 ( 166.97/25.04 powersetI1 => 166.97/25.04 ( 166.97/25.04 powersetE1 => 166.97/25.04 ( 166.97/25.04 inPowerset => 166.97/25.04 ( 166.97/25.04 powersetsubset => 166.97/25.04 ( 166.97/25.04 sepInPowerset => 166.97/25.04 ( 166.97/25.04 sepSubset => 166.97/25.04 ( 166.97/25.04 binunionIL => 166.97/25.04 ( 166.97/25.04 upairset2IR => 166.97/25.04 ( 166.97/25.04 binunionIR => 166.97/25.04 ( 166.97/25.04 binunionEcases => 166.97/25.04 ( 166.97/25.04 binunionE => 166.97/25.04 ( 166.97/25.04 binunionLsub => 166.97/25.04 ( 166.97/25.04 binunionRsub => 166.97/25.04 ( 166.97/25.04 binintersectI => 166.97/25.04 ( 166.97/25.04 binintersectSubset5 => 166.97/25.04 ( 166.97/25.04 binintersectEL => 166.97/25.04 ( 166.97/25.04 binintersectLsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset2 => 166.97/25.04 ( 166.97/25.04 binintersectSubset3 => 166.97/25.04 ( 166.97/25.04 binintersectER => 166.97/25.04 ( 166.97/25.04 disjointsetsI1 => 166.97/25.04 ( 166.97/25.04 binintersectRsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset4 => 166.97/25.04 ( 166.97/25.04 binintersectSubset1 => 166.97/25.04 ( 166.97/25.04 bs114d => 166.97/25.04 ( 166.97/25.04 setminusI => 166.97/25.04 ( 166.97/25.04 setminusEL => 166.97/25.04 ( 166.97/25.04 setminusER => 166.97/25.04 ( 166.97/25.04 setminusSubset2 => 166.97/25.04 ( 166.97/25.04 setminusERneg => 166.97/25.04 ( 166.97/25.04 setminusELneg => 166.97/25.04 ( 166.97/25.04 setminusILneg => 166.97/25.04 ( 166.97/25.04 setminusIRneg => 166.97/25.04 ( 166.97/25.04 setminusLsub => 166.97/25.04 ( 166.97/25.04 setminusSubset1 => 166.97/25.04 ( 166.97/25.04 symdiffE => 166.97/25.04 ( 166.97/25.04 symdiffI1 => 166.97/25.04 ( 166.97/25.04 symdiffI2 => 166.97/25.04 ( 166.97/25.04 symdiffIneg1 => 166.97/25.04 ( 166.97/25.04 symdiffIneg2 => 166.97/25.04 ( 166.97/25.04 secondinupair => 166.97/25.04 ( 166.97/25.04 setukpairIL => 166.97/25.04 ( 166.97/25.04 setukpairIR => 166.97/25.04 ( 166.97/25.04 kpairiskpair => 166.97/25.04 ( 166.97/25.04 kpairp => 166.97/25.04 ( 166.97/25.04 singletonsubset => 166.97/25.04 ( 166.97/25.04 singletoninpowerset => 166.97/25.04 ( 166.97/25.04 singletoninpowunion => 166.97/25.04 ( 166.97/25.04 upairset2E => 166.97/25.04 ( 166.97/25.04 upairsubunion => 166.97/25.04 ( 166.97/25.04 upairinpowunion => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem1 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem2 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem3 => 166.97/25.04 ( 166.97/25.04 cartprodpairin => 166.97/25.04 ( 166.97/25.04 cartprodmempair1 => 166.97/25.04 ( 166.97/25.04 cartprodmempair => 166.97/25.04 ( 166.97/25.04 setunionE2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton1 => 166.97/25.04 ( 166.97/25.04 setunionsingleton2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton => 166.97/25.04 ( 166.97/25.04 singletonprop => 166.97/25.04 ( 166.97/25.04 ex1E1 => 166.97/25.04 ( 166.97/25.04 ex1I => 166.97/25.04 ( 166.97/25.04 ex1I2 => 166.97/25.04 ( 166.97/25.04 singletonsuniq => 166.97/25.04 ( 166.97/25.04 setukpairinjL1 => 166.97/25.04 ( 166.97/25.04 kfstsingleton => 166.97/25.04 ( 166.97/25.04 theprop => 166.97/25.04 ( 166.97/25.04 kfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodfstin => 166.97/25.04 ( 166.97/25.04 setukpairinjL2 => 166.97/25.04 ( 166.97/25.04 setukpairinjL => 166.97/25.04 ( 166.97/25.04 setukpairinjR11 => 166.97/25.04 ( 166.97/25.04 setukpairinjR12 => 166.97/25.04 ( 166.97/25.04 setukpairinjR1 => 166.97/25.04 ( 166.97/25.04 upairequniteq => 166.97/25.04 ( 166.97/25.04 setukpairinjR2 => 166.97/25.04 ( 166.97/25.04 setukpairinjR => 166.97/25.04 ( 166.97/25.04 ksndsingleton => 166.97/25.04 ( 166.97/25.04 ksndpairEq => 166.97/25.04 ( 166.97/25.04 kpairsurjEq => 166.97/25.04 ( 166.97/25.04 cartprodsndin => 166.97/25.04 ( 166.97/25.04 cartprodpairmemEL => 166.97/25.04 ( 166.97/25.04 cartprodpairmemER => 166.97/25.04 ( 166.97/25.04 cartprodmempaircEq => 166.97/25.04 ( 166.97/25.04 cartprodfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodsndpairEq => 166.97/25.04 ( 166.97/25.04 cartprodpairsurjEq => 166.97/25.04 ( 166.97/25.04 dpsetconstrI => 166.97/25.04 ( 166.97/25.04 dpsetconstrSub => 166.97/25.04 ( 166.97/25.04 setOfPairsIsBReln => 166.97/25.04 ( 166.97/25.04 !!(( 166.97/25.04 ^[Y0 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y1 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y2 : ( 166.97/25.04 $i * $i) > $o]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y3 : $i]: 166.97/25.04 (in(Y3, 166.97/25.04 Y0) => 166.97/25.04 (!!(( 166.97/25.04 ^[Y4 : $i]: 166.97/25.04 (in(Y4, 166.97/25.04 Y1) => 166.97/25.04 (in( 166.97/25.04 kpair( 166.97/25.04 Y3, Y4), 166.97/25.04 dpsetconstr( 166.97/25.04 Y0, Y1, 166.97/25.04 Y2)) => 166.97/25.04 Y2(Y3, 166.97/25.04 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.04 inference('lazy_cnf_imply', [status(thm)], ['13'])). 166.97/25.04 tff('15', plain, 166.97/25.04 ~ (dsetconstrER => 166.97/25.04 (exuE1 => 166.97/25.04 (prop2setE => 166.97/25.04 (emptysetE => 166.97/25.04 (emptysetimpfalse => 166.97/25.04 (notinemptyset => 166.97/25.04 (exuE3e => 166.97/25.04 (setext => 166.97/25.04 (emptyI => 166.97/25.04 (noeltsimpempty => 166.97/25.04 (setbeta => 166.97/25.04 (nonemptyE1 => 166.97/25.04 (nonemptyI => 166.97/25.04 (nonemptyI1 => 166.97/25.04 (setadjoinIL => 166.97/25.04 (emptyinunitempty => 166.97/25.04 (setadjoinIR => 166.97/25.04 (setadjoinE => 166.97/25.04 (setadjoinOr => 166.97/25.04 (setoftrueEq => 166.97/25.04 (powersetI => 166.97/25.04 (emptyinPowerset => 166.97/25.04 (emptyInPowerset => 166.97/25.04 (powersetE => 166.97/25.04 (setunionI => 166.97/25.04 (setunionE => 166.97/25.04 (subPowSU => 166.97/25.04 (exuE2 => 166.97/25.04 (nonemptyImpWitness => 166.97/25.04 (uniqinunit => 166.97/25.04 (notinsingleton => 166.97/25.04 (eqinunit => 166.97/25.04 (singletonsswitch => 166.97/25.04 (upairsetE => 166.97/25.04 (upairsetIL => 166.97/25.04 (upairsetIR => 166.97/25.04 (emptyE1 => 166.97/25.04 (vacuousDall => 166.97/25.04 (quantDeMorgan1 => 166.97/25.04 (quantDeMorgan2 => 166.97/25.04 (quantDeMorgan3 => 166.97/25.04 (quantDeMorgan4 => 166.97/25.04 (prop2setI => 166.97/25.04 (prop2set2propI => 166.97/25.04 (notdexE => 166.97/25.04 (notdallE => 166.97/25.04 (exuI1 => 166.97/25.04 (exuI3 => 166.97/25.04 (exuI2 => 166.97/25.04 (inCongP => 166.97/25.04 (in__Cong => 166.97/25.04 (exuE3u => 166.97/25.04 (exu__Cong => 166.97/25.04 (emptyset__Cong => 166.97/25.04 (setadjoin__Cong => 166.97/25.04 (powerset__Cong => 166.97/25.04 (setunion__Cong => 166.97/25.04 (omega__Cong => 166.97/25.04 (exuEu => 166.97/25.04 (descr__Cong => 166.97/25.04 (dsetconstr__Cong => 166.97/25.04 (subsetI1 => 166.97/25.04 ( 166.97/25.04 eqimpsubset2 => 166.97/25.04 ( 166.97/25.04 eqimpsubset1 => 166.97/25.04 ( 166.97/25.04 subsetI2 => 166.97/25.04 ( 166.97/25.04 emptysetsubset => 166.97/25.04 ( 166.97/25.04 subsetE => 166.97/25.04 ( 166.97/25.04 subsetE2 => 166.97/25.04 ( 166.97/25.04 notsubsetI => 166.97/25.04 ( 166.97/25.04 notequalI1 => 166.97/25.04 ( 166.97/25.04 notequalI2 => 166.97/25.04 ( 166.97/25.04 subsetRefl => 166.97/25.04 ( 166.97/25.04 subsetTrans => 166.97/25.04 ( 166.97/25.04 setadjoinSub => 166.97/25.04 ( 166.97/25.04 setadjoinSub2 => 166.97/25.04 ( 166.97/25.04 subset2powerset => 166.97/25.04 ( 166.97/25.04 setextsub => 166.97/25.04 ( 166.97/25.04 subsetemptysetimpeq => 166.97/25.04 ( 166.97/25.04 powersetI1 => 166.97/25.04 ( 166.97/25.04 powersetE1 => 166.97/25.04 ( 166.97/25.04 inPowerset => 166.97/25.04 ( 166.97/25.04 powersetsubset => 166.97/25.04 ( 166.97/25.04 sepInPowerset => 166.97/25.04 ( 166.97/25.04 sepSubset => 166.97/25.04 ( 166.97/25.04 binunionIL => 166.97/25.04 ( 166.97/25.04 upairset2IR => 166.97/25.04 ( 166.97/25.04 binunionIR => 166.97/25.04 ( 166.97/25.04 binunionEcases => 166.97/25.04 ( 166.97/25.04 binunionE => 166.97/25.04 ( 166.97/25.04 binunionLsub => 166.97/25.04 ( 166.97/25.04 binunionRsub => 166.97/25.04 ( 166.97/25.04 binintersectI => 166.97/25.04 ( 166.97/25.04 binintersectSubset5 => 166.97/25.04 ( 166.97/25.04 binintersectEL => 166.97/25.04 ( 166.97/25.04 binintersectLsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset2 => 166.97/25.04 ( 166.97/25.04 binintersectSubset3 => 166.97/25.04 ( 166.97/25.04 binintersectER => 166.97/25.04 ( 166.97/25.04 disjointsetsI1 => 166.97/25.04 ( 166.97/25.04 binintersectRsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset4 => 166.97/25.04 ( 166.97/25.04 binintersectSubset1 => 166.97/25.04 ( 166.97/25.04 bs114d => 166.97/25.04 ( 166.97/25.04 setminusI => 166.97/25.04 ( 166.97/25.04 setminusEL => 166.97/25.04 ( 166.97/25.04 setminusER => 166.97/25.04 ( 166.97/25.04 setminusSubset2 => 166.97/25.04 ( 166.97/25.04 setminusERneg => 166.97/25.04 ( 166.97/25.04 setminusELneg => 166.97/25.04 ( 166.97/25.04 setminusILneg => 166.97/25.04 ( 166.97/25.04 setminusIRneg => 166.97/25.04 ( 166.97/25.04 setminusLsub => 166.97/25.04 ( 166.97/25.04 setminusSubset1 => 166.97/25.04 ( 166.97/25.04 symdiffE => 166.97/25.04 ( 166.97/25.04 symdiffI1 => 166.97/25.04 ( 166.97/25.04 symdiffI2 => 166.97/25.04 ( 166.97/25.04 symdiffIneg1 => 166.97/25.04 ( 166.97/25.04 symdiffIneg2 => 166.97/25.04 ( 166.97/25.04 secondinupair => 166.97/25.04 ( 166.97/25.04 setukpairIL => 166.97/25.04 ( 166.97/25.04 setukpairIR => 166.97/25.04 ( 166.97/25.04 kpairiskpair => 166.97/25.04 ( 166.97/25.04 kpairp => 166.97/25.04 ( 166.97/25.04 singletonsubset => 166.97/25.04 ( 166.97/25.04 singletoninpowerset => 166.97/25.04 ( 166.97/25.04 singletoninpowunion => 166.97/25.04 ( 166.97/25.04 upairset2E => 166.97/25.04 ( 166.97/25.04 upairsubunion => 166.97/25.04 ( 166.97/25.04 upairinpowunion => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem1 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem2 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem3 => 166.97/25.04 ( 166.97/25.04 cartprodpairin => 166.97/25.04 ( 166.97/25.04 cartprodmempair1 => 166.97/25.04 ( 166.97/25.04 cartprodmempair => 166.97/25.04 ( 166.97/25.04 setunionE2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton1 => 166.97/25.04 ( 166.97/25.04 setunionsingleton2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton => 166.97/25.04 ( 166.97/25.04 singletonprop => 166.97/25.04 ( 166.97/25.04 ex1E1 => 166.97/25.04 ( 166.97/25.04 ex1I => 166.97/25.04 ( 166.97/25.04 ex1I2 => 166.97/25.04 ( 166.97/25.04 singletonsuniq => 166.97/25.04 ( 166.97/25.04 setukpairinjL1 => 166.97/25.04 ( 166.97/25.04 kfstsingleton => 166.97/25.04 ( 166.97/25.04 theprop => 166.97/25.04 ( 166.97/25.04 kfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodfstin => 166.97/25.04 ( 166.97/25.04 setukpairinjL2 => 166.97/25.04 ( 166.97/25.04 setukpairinjL => 166.97/25.04 ( 166.97/25.04 setukpairinjR11 => 166.97/25.04 ( 166.97/25.04 setukpairinjR12 => 166.97/25.04 ( 166.97/25.04 setukpairinjR1 => 166.97/25.04 ( 166.97/25.04 upairequniteq => 166.97/25.04 ( 166.97/25.04 setukpairinjR2 => 166.97/25.04 ( 166.97/25.04 setukpairinjR => 166.97/25.04 ( 166.97/25.04 ksndsingleton => 166.97/25.04 ( 166.97/25.04 ksndpairEq => 166.97/25.04 ( 166.97/25.04 kpairsurjEq => 166.97/25.04 ( 166.97/25.04 cartprodsndin => 166.97/25.04 ( 166.97/25.04 cartprodpairmemEL => 166.97/25.04 ( 166.97/25.04 cartprodpairmemER => 166.97/25.04 ( 166.97/25.04 cartprodmempaircEq => 166.97/25.04 ( 166.97/25.04 cartprodfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodsndpairEq => 166.97/25.04 ( 166.97/25.04 cartprodpairsurjEq => 166.97/25.04 ( 166.97/25.04 dpsetconstrI => 166.97/25.04 ( 166.97/25.04 dpsetconstrSub => 166.97/25.04 ( 166.97/25.04 setOfPairsIsBReln => 166.97/25.04 ( 166.97/25.04 !!(( 166.97/25.04 ^[Y0 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y1 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y2 : ( 166.97/25.04 $i * $i) > $o]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y3 : $i]: 166.97/25.04 (in(Y3, 166.97/25.04 Y0) => 166.97/25.04 (!!(( 166.97/25.04 ^[Y4 : $i]: 166.97/25.04 (in(Y4, 166.97/25.04 Y1) => 166.97/25.04 (in( 166.97/25.04 kpair( 166.97/25.04 Y3, Y4), 166.97/25.04 dpsetconstr( 166.97/25.04 Y0, Y1, 166.97/25.04 Y2)) => 166.97/25.04 Y2(Y3, 166.97/25.04 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.04 inference('lazy_cnf_imply', [status(thm)], ['14'])). 166.97/25.04 tff('16', plain, 166.97/25.04 ~ (exuE1 => 166.97/25.04 (prop2setE => 166.97/25.04 (emptysetE => 166.97/25.04 (emptysetimpfalse => 166.97/25.04 (notinemptyset => 166.97/25.04 (exuE3e => 166.97/25.04 (setext => 166.97/25.04 (emptyI => 166.97/25.04 (noeltsimpempty => 166.97/25.04 (setbeta => 166.97/25.04 (nonemptyE1 => 166.97/25.04 (nonemptyI => 166.97/25.04 (nonemptyI1 => 166.97/25.04 (setadjoinIL => 166.97/25.04 (emptyinunitempty => 166.97/25.04 (setadjoinIR => 166.97/25.04 (setadjoinE => 166.97/25.04 (setadjoinOr => 166.97/25.04 (setoftrueEq => 166.97/25.04 (powersetI => 166.97/25.04 (emptyinPowerset => 166.97/25.04 (emptyInPowerset => 166.97/25.04 (powersetE => 166.97/25.04 (setunionI => 166.97/25.04 (setunionE => 166.97/25.04 (subPowSU => 166.97/25.04 (exuE2 => 166.97/25.04 (nonemptyImpWitness => 166.97/25.04 (uniqinunit => 166.97/25.04 (notinsingleton => 166.97/25.04 (eqinunit => 166.97/25.04 (singletonsswitch => 166.97/25.04 (upairsetE => 166.97/25.04 (upairsetIL => 166.97/25.04 (upairsetIR => 166.97/25.04 (emptyE1 => 166.97/25.04 (vacuousDall => 166.97/25.04 (quantDeMorgan1 => 166.97/25.04 (quantDeMorgan2 => 166.97/25.04 (quantDeMorgan3 => 166.97/25.04 (quantDeMorgan4 => 166.97/25.04 (prop2setI => 166.97/25.04 (prop2set2propI => 166.97/25.04 (notdexE => 166.97/25.04 (notdallE => 166.97/25.04 (exuI1 => 166.97/25.04 (exuI3 => 166.97/25.04 (exuI2 => 166.97/25.04 (inCongP => 166.97/25.04 (in__Cong => 166.97/25.04 (exuE3u => 166.97/25.04 (exu__Cong => 166.97/25.04 (emptyset__Cong => 166.97/25.04 (setadjoin__Cong => 166.97/25.04 (powerset__Cong => 166.97/25.04 (setunion__Cong => 166.97/25.04 (omega__Cong => 166.97/25.04 (exuEu => 166.97/25.04 (descr__Cong => 166.97/25.04 (dsetconstr__Cong => 166.97/25.04 (subsetI1 => 166.97/25.04 (eqimpsubset2 => 166.97/25.04 ( 166.97/25.04 eqimpsubset1 => 166.97/25.04 ( 166.97/25.04 subsetI2 => 166.97/25.04 ( 166.97/25.04 emptysetsubset => 166.97/25.04 ( 166.97/25.04 subsetE => 166.97/25.04 ( 166.97/25.04 subsetE2 => 166.97/25.04 ( 166.97/25.04 notsubsetI => 166.97/25.04 ( 166.97/25.04 notequalI1 => 166.97/25.04 ( 166.97/25.04 notequalI2 => 166.97/25.04 ( 166.97/25.04 subsetRefl => 166.97/25.04 ( 166.97/25.04 subsetTrans => 166.97/25.04 ( 166.97/25.04 setadjoinSub => 166.97/25.04 ( 166.97/25.04 setadjoinSub2 => 166.97/25.04 ( 166.97/25.04 subset2powerset => 166.97/25.04 ( 166.97/25.04 setextsub => 166.97/25.04 ( 166.97/25.04 subsetemptysetimpeq => 166.97/25.04 ( 166.97/25.04 powersetI1 => 166.97/25.04 ( 166.97/25.04 powersetE1 => 166.97/25.04 ( 166.97/25.04 inPowerset => 166.97/25.04 ( 166.97/25.04 powersetsubset => 166.97/25.04 ( 166.97/25.04 sepInPowerset => 166.97/25.04 ( 166.97/25.04 sepSubset => 166.97/25.04 ( 166.97/25.04 binunionIL => 166.97/25.04 ( 166.97/25.04 upairset2IR => 166.97/25.04 ( 166.97/25.04 binunionIR => 166.97/25.04 ( 166.97/25.04 binunionEcases => 166.97/25.04 ( 166.97/25.04 binunionE => 166.97/25.04 ( 166.97/25.04 binunionLsub => 166.97/25.04 ( 166.97/25.04 binunionRsub => 166.97/25.04 ( 166.97/25.04 binintersectI => 166.97/25.04 ( 166.97/25.04 binintersectSubset5 => 166.97/25.04 ( 166.97/25.04 binintersectEL => 166.97/25.04 ( 166.97/25.04 binintersectLsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset2 => 166.97/25.04 ( 166.97/25.04 binintersectSubset3 => 166.97/25.04 ( 166.97/25.04 binintersectER => 166.97/25.04 ( 166.97/25.04 disjointsetsI1 => 166.97/25.04 ( 166.97/25.04 binintersectRsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset4 => 166.97/25.04 ( 166.97/25.04 binintersectSubset1 => 166.97/25.04 ( 166.97/25.04 bs114d => 166.97/25.04 ( 166.97/25.04 setminusI => 166.97/25.04 ( 166.97/25.04 setminusEL => 166.97/25.04 ( 166.97/25.04 setminusER => 166.97/25.04 ( 166.97/25.04 setminusSubset2 => 166.97/25.04 ( 166.97/25.04 setminusERneg => 166.97/25.04 ( 166.97/25.04 setminusELneg => 166.97/25.04 ( 166.97/25.04 setminusILneg => 166.97/25.04 ( 166.97/25.04 setminusIRneg => 166.97/25.04 ( 166.97/25.04 setminusLsub => 166.97/25.04 ( 166.97/25.04 setminusSubset1 => 166.97/25.04 ( 166.97/25.04 symdiffE => 166.97/25.04 ( 166.97/25.04 symdiffI1 => 166.97/25.04 ( 166.97/25.04 symdiffI2 => 166.97/25.04 ( 166.97/25.04 symdiffIneg1 => 166.97/25.04 ( 166.97/25.04 symdiffIneg2 => 166.97/25.04 ( 166.97/25.04 secondinupair => 166.97/25.04 ( 166.97/25.04 setukpairIL => 166.97/25.04 ( 166.97/25.04 setukpairIR => 166.97/25.04 ( 166.97/25.04 kpairiskpair => 166.97/25.04 ( 166.97/25.04 kpairp => 166.97/25.04 ( 166.97/25.04 singletonsubset => 166.97/25.04 ( 166.97/25.04 singletoninpowerset => 166.97/25.04 ( 166.97/25.04 singletoninpowunion => 166.97/25.04 ( 166.97/25.04 upairset2E => 166.97/25.04 ( 166.97/25.04 upairsubunion => 166.97/25.04 ( 166.97/25.04 upairinpowunion => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem1 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem2 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem3 => 166.97/25.04 ( 166.97/25.04 cartprodpairin => 166.97/25.04 ( 166.97/25.04 cartprodmempair1 => 166.97/25.04 ( 166.97/25.04 cartprodmempair => 166.97/25.04 ( 166.97/25.04 setunionE2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton1 => 166.97/25.04 ( 166.97/25.04 setunionsingleton2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton => 166.97/25.04 ( 166.97/25.04 singletonprop => 166.97/25.04 ( 166.97/25.04 ex1E1 => 166.97/25.04 ( 166.97/25.04 ex1I => 166.97/25.04 ( 166.97/25.04 ex1I2 => 166.97/25.04 ( 166.97/25.04 singletonsuniq => 166.97/25.04 ( 166.97/25.04 setukpairinjL1 => 166.97/25.04 ( 166.97/25.04 kfstsingleton => 166.97/25.04 ( 166.97/25.04 theprop => 166.97/25.04 ( 166.97/25.04 kfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodfstin => 166.97/25.04 ( 166.97/25.04 setukpairinjL2 => 166.97/25.04 ( 166.97/25.04 setukpairinjL => 166.97/25.04 ( 166.97/25.04 setukpairinjR11 => 166.97/25.04 ( 166.97/25.04 setukpairinjR12 => 166.97/25.04 ( 166.97/25.04 setukpairinjR1 => 166.97/25.04 ( 166.97/25.04 upairequniteq => 166.97/25.04 ( 166.97/25.04 setukpairinjR2 => 166.97/25.04 ( 166.97/25.04 setukpairinjR => 166.97/25.04 ( 166.97/25.04 ksndsingleton => 166.97/25.04 ( 166.97/25.04 ksndpairEq => 166.97/25.04 ( 166.97/25.04 kpairsurjEq => 166.97/25.04 ( 166.97/25.04 cartprodsndin => 166.97/25.04 ( 166.97/25.04 cartprodpairmemEL => 166.97/25.04 ( 166.97/25.04 cartprodpairmemER => 166.97/25.04 ( 166.97/25.04 cartprodmempaircEq => 166.97/25.04 ( 166.97/25.04 cartprodfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodsndpairEq => 166.97/25.04 ( 166.97/25.04 cartprodpairsurjEq => 166.97/25.04 ( 166.97/25.04 dpsetconstrI => 166.97/25.04 ( 166.97/25.04 dpsetconstrSub => 166.97/25.04 ( 166.97/25.04 setOfPairsIsBReln => 166.97/25.04 ( 166.97/25.04 !!(( 166.97/25.04 ^[Y0 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y1 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y2 : ( 166.97/25.04 $i * $i) > $o]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y3 : $i]: 166.97/25.04 (in(Y3, 166.97/25.04 Y0) => 166.97/25.04 (!!(( 166.97/25.04 ^[Y4 : $i]: 166.97/25.04 (in(Y4, 166.97/25.04 Y1) => 166.97/25.04 (in( 166.97/25.04 kpair( 166.97/25.04 Y3, Y4), 166.97/25.04 dpsetconstr( 166.97/25.04 Y0, Y1, 166.97/25.04 Y2)) => 166.97/25.04 Y2(Y3, 166.97/25.04 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.04 inference('lazy_cnf_imply', [status(thm)], ['15'])). 166.97/25.04 tff('17', plain, 166.97/25.04 ~ (prop2setE => 166.97/25.04 (emptysetE => 166.97/25.04 (emptysetimpfalse => 166.97/25.04 (notinemptyset => 166.97/25.04 (exuE3e => 166.97/25.04 (setext => 166.97/25.04 (emptyI => 166.97/25.04 (noeltsimpempty => 166.97/25.04 (setbeta => 166.97/25.04 (nonemptyE1 => 166.97/25.04 (nonemptyI => 166.97/25.04 (nonemptyI1 => 166.97/25.04 (setadjoinIL => 166.97/25.04 (emptyinunitempty => 166.97/25.04 (setadjoinIR => 166.97/25.04 (setadjoinE => 166.97/25.04 (setadjoinOr => 166.97/25.04 (setoftrueEq => 166.97/25.04 (powersetI => 166.97/25.04 (emptyinPowerset => 166.97/25.04 (emptyInPowerset => 166.97/25.04 (powersetE => 166.97/25.04 (setunionI => 166.97/25.04 (setunionE => 166.97/25.04 (subPowSU => 166.97/25.04 (exuE2 => 166.97/25.04 (nonemptyImpWitness => 166.97/25.04 (uniqinunit => 166.97/25.04 (notinsingleton => 166.97/25.04 (eqinunit => 166.97/25.04 (singletonsswitch => 166.97/25.04 (upairsetE => 166.97/25.04 (upairsetIL => 166.97/25.04 (upairsetIR => 166.97/25.04 (emptyE1 => 166.97/25.04 (vacuousDall => 166.97/25.04 (quantDeMorgan1 => 166.97/25.04 (quantDeMorgan2 => 166.97/25.04 (quantDeMorgan3 => 166.97/25.04 (quantDeMorgan4 => 166.97/25.04 (prop2setI => 166.97/25.04 (prop2set2propI => 166.97/25.04 (notdexE => 166.97/25.04 (notdallE => 166.97/25.04 (exuI1 => 166.97/25.04 (exuI3 => 166.97/25.04 (exuI2 => 166.97/25.04 (inCongP => 166.97/25.04 (in__Cong => 166.97/25.04 (exuE3u => 166.97/25.04 (exu__Cong => 166.97/25.04 (emptyset__Cong => 166.97/25.04 (setadjoin__Cong => 166.97/25.04 (powerset__Cong => 166.97/25.04 (setunion__Cong => 166.97/25.04 (omega__Cong => 166.97/25.04 (exuEu => 166.97/25.04 (descr__Cong => 166.97/25.04 (dsetconstr__Cong => 166.97/25.04 (subsetI1 => 166.97/25.04 (eqimpsubset2 => 166.97/25.04 (eqimpsubset1 => 166.97/25.04 ( 166.97/25.04 subsetI2 => 166.97/25.04 ( 166.97/25.04 emptysetsubset => 166.97/25.04 ( 166.97/25.04 subsetE => 166.97/25.04 ( 166.97/25.04 subsetE2 => 166.97/25.04 ( 166.97/25.04 notsubsetI => 166.97/25.04 ( 166.97/25.04 notequalI1 => 166.97/25.04 ( 166.97/25.04 notequalI2 => 166.97/25.04 ( 166.97/25.04 subsetRefl => 166.97/25.04 ( 166.97/25.04 subsetTrans => 166.97/25.04 ( 166.97/25.04 setadjoinSub => 166.97/25.04 ( 166.97/25.04 setadjoinSub2 => 166.97/25.04 ( 166.97/25.04 subset2powerset => 166.97/25.04 ( 166.97/25.04 setextsub => 166.97/25.04 ( 166.97/25.04 subsetemptysetimpeq => 166.97/25.04 ( 166.97/25.04 powersetI1 => 166.97/25.04 ( 166.97/25.04 powersetE1 => 166.97/25.04 ( 166.97/25.04 inPowerset => 166.97/25.04 ( 166.97/25.04 powersetsubset => 166.97/25.04 ( 166.97/25.04 sepInPowerset => 166.97/25.04 ( 166.97/25.04 sepSubset => 166.97/25.04 ( 166.97/25.04 binunionIL => 166.97/25.04 ( 166.97/25.04 upairset2IR => 166.97/25.04 ( 166.97/25.04 binunionIR => 166.97/25.04 ( 166.97/25.04 binunionEcases => 166.97/25.04 ( 166.97/25.04 binunionE => 166.97/25.04 ( 166.97/25.04 binunionLsub => 166.97/25.04 ( 166.97/25.04 binunionRsub => 166.97/25.04 ( 166.97/25.04 binintersectI => 166.97/25.04 ( 166.97/25.04 binintersectSubset5 => 166.97/25.04 ( 166.97/25.04 binintersectEL => 166.97/25.04 ( 166.97/25.04 binintersectLsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset2 => 166.97/25.04 ( 166.97/25.04 binintersectSubset3 => 166.97/25.04 ( 166.97/25.04 binintersectER => 166.97/25.04 ( 166.97/25.04 disjointsetsI1 => 166.97/25.04 ( 166.97/25.04 binintersectRsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset4 => 166.97/25.04 ( 166.97/25.04 binintersectSubset1 => 166.97/25.04 ( 166.97/25.04 bs114d => 166.97/25.04 ( 166.97/25.04 setminusI => 166.97/25.04 ( 166.97/25.04 setminusEL => 166.97/25.04 ( 166.97/25.04 setminusER => 166.97/25.04 ( 166.97/25.04 setminusSubset2 => 166.97/25.04 ( 166.97/25.04 setminusERneg => 166.97/25.04 ( 166.97/25.04 setminusELneg => 166.97/25.04 ( 166.97/25.04 setminusILneg => 166.97/25.04 ( 166.97/25.04 setminusIRneg => 166.97/25.04 ( 166.97/25.04 setminusLsub => 166.97/25.04 ( 166.97/25.04 setminusSubset1 => 166.97/25.04 ( 166.97/25.04 symdiffE => 166.97/25.04 ( 166.97/25.04 symdiffI1 => 166.97/25.04 ( 166.97/25.04 symdiffI2 => 166.97/25.04 ( 166.97/25.04 symdiffIneg1 => 166.97/25.04 ( 166.97/25.04 symdiffIneg2 => 166.97/25.04 ( 166.97/25.04 secondinupair => 166.97/25.04 ( 166.97/25.04 setukpairIL => 166.97/25.04 ( 166.97/25.04 setukpairIR => 166.97/25.04 ( 166.97/25.04 kpairiskpair => 166.97/25.04 ( 166.97/25.04 kpairp => 166.97/25.04 ( 166.97/25.04 singletonsubset => 166.97/25.04 ( 166.97/25.04 singletoninpowerset => 166.97/25.04 ( 166.97/25.04 singletoninpowunion => 166.97/25.04 ( 166.97/25.04 upairset2E => 166.97/25.04 ( 166.97/25.04 upairsubunion => 166.97/25.04 ( 166.97/25.04 upairinpowunion => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem1 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem2 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem3 => 166.97/25.04 ( 166.97/25.04 cartprodpairin => 166.97/25.04 ( 166.97/25.04 cartprodmempair1 => 166.97/25.04 ( 166.97/25.04 cartprodmempair => 166.97/25.04 ( 166.97/25.04 setunionE2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton1 => 166.97/25.04 ( 166.97/25.04 setunionsingleton2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton => 166.97/25.04 ( 166.97/25.04 singletonprop => 166.97/25.04 ( 166.97/25.04 ex1E1 => 166.97/25.04 ( 166.97/25.04 ex1I => 166.97/25.04 ( 166.97/25.04 ex1I2 => 166.97/25.04 ( 166.97/25.04 singletonsuniq => 166.97/25.04 ( 166.97/25.04 setukpairinjL1 => 166.97/25.04 ( 166.97/25.04 kfstsingleton => 166.97/25.04 ( 166.97/25.04 theprop => 166.97/25.04 ( 166.97/25.04 kfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodfstin => 166.97/25.04 ( 166.97/25.04 setukpairinjL2 => 166.97/25.04 ( 166.97/25.04 setukpairinjL => 166.97/25.04 ( 166.97/25.04 setukpairinjR11 => 166.97/25.04 ( 166.97/25.04 setukpairinjR12 => 166.97/25.04 ( 166.97/25.04 setukpairinjR1 => 166.97/25.04 ( 166.97/25.04 upairequniteq => 166.97/25.04 ( 166.97/25.04 setukpairinjR2 => 166.97/25.04 ( 166.97/25.04 setukpairinjR => 166.97/25.04 ( 166.97/25.04 ksndsingleton => 166.97/25.04 ( 166.97/25.04 ksndpairEq => 166.97/25.04 ( 166.97/25.04 kpairsurjEq => 166.97/25.04 ( 166.97/25.04 cartprodsndin => 166.97/25.04 ( 166.97/25.04 cartprodpairmemEL => 166.97/25.04 ( 166.97/25.04 cartprodpairmemER => 166.97/25.04 ( 166.97/25.04 cartprodmempaircEq => 166.97/25.04 ( 166.97/25.04 cartprodfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodsndpairEq => 166.97/25.04 ( 166.97/25.04 cartprodpairsurjEq => 166.97/25.04 ( 166.97/25.04 dpsetconstrI => 166.97/25.04 ( 166.97/25.04 dpsetconstrSub => 166.97/25.04 ( 166.97/25.04 setOfPairsIsBReln => 166.97/25.04 ( 166.97/25.04 !!(( 166.97/25.04 ^[Y0 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y1 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y2 : ( 166.97/25.04 $i * $i) > $o]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y3 : $i]: 166.97/25.04 (in(Y3, 166.97/25.04 Y0) => 166.97/25.04 (!!(( 166.97/25.04 ^[Y4 : $i]: 166.97/25.04 (in(Y4, 166.97/25.04 Y1) => 166.97/25.04 (in( 166.97/25.04 kpair( 166.97/25.04 Y3, Y4), 166.97/25.04 dpsetconstr( 166.97/25.04 Y0, Y1, 166.97/25.04 Y2)) => 166.97/25.04 Y2(Y3, 166.97/25.04 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.04 inference('lazy_cnf_imply', [status(thm)], ['16'])). 166.97/25.04 tff('18', plain, 166.97/25.04 ~ (emptysetE => 166.97/25.04 (emptysetimpfalse => 166.97/25.04 (notinemptyset => 166.97/25.04 (exuE3e => 166.97/25.04 (setext => 166.97/25.04 (emptyI => 166.97/25.04 (noeltsimpempty => 166.97/25.04 (setbeta => 166.97/25.04 (nonemptyE1 => 166.97/25.04 (nonemptyI => 166.97/25.04 (nonemptyI1 => 166.97/25.04 (setadjoinIL => 166.97/25.04 (emptyinunitempty => 166.97/25.04 (setadjoinIR => 166.97/25.04 (setadjoinE => 166.97/25.04 (setadjoinOr => 166.97/25.04 (setoftrueEq => 166.97/25.04 (powersetI => 166.97/25.04 (emptyinPowerset => 166.97/25.04 (emptyInPowerset => 166.97/25.04 (powersetE => 166.97/25.04 (setunionI => 166.97/25.04 (setunionE => 166.97/25.04 (subPowSU => 166.97/25.04 (exuE2 => 166.97/25.04 (nonemptyImpWitness => 166.97/25.04 (uniqinunit => 166.97/25.04 (notinsingleton => 166.97/25.04 (eqinunit => 166.97/25.04 (singletonsswitch => 166.97/25.04 (upairsetE => 166.97/25.04 (upairsetIL => 166.97/25.04 (upairsetIR => 166.97/25.04 (emptyE1 => 166.97/25.04 (vacuousDall => 166.97/25.04 (quantDeMorgan1 => 166.97/25.04 (quantDeMorgan2 => 166.97/25.04 (quantDeMorgan3 => 166.97/25.04 (quantDeMorgan4 => 166.97/25.04 (prop2setI => 166.97/25.04 (prop2set2propI => 166.97/25.04 (notdexE => 166.97/25.04 (notdallE => 166.97/25.04 (exuI1 => 166.97/25.04 (exuI3 => 166.97/25.04 (exuI2 => 166.97/25.04 (inCongP => 166.97/25.04 (in__Cong => 166.97/25.04 (exuE3u => 166.97/25.04 (exu__Cong => 166.97/25.04 (emptyset__Cong => 166.97/25.04 (setadjoin__Cong => 166.97/25.04 (powerset__Cong => 166.97/25.04 (setunion__Cong => 166.97/25.04 (omega__Cong => 166.97/25.04 (exuEu => 166.97/25.04 (descr__Cong => 166.97/25.04 (dsetconstr__Cong => 166.97/25.04 (subsetI1 => 166.97/25.04 (eqimpsubset2 => 166.97/25.04 (eqimpsubset1 => 166.97/25.04 (subsetI2 => 166.97/25.04 ( 166.97/25.04 emptysetsubset => 166.97/25.04 ( 166.97/25.04 subsetE => 166.97/25.04 ( 166.97/25.04 subsetE2 => 166.97/25.04 ( 166.97/25.04 notsubsetI => 166.97/25.04 ( 166.97/25.04 notequalI1 => 166.97/25.04 ( 166.97/25.04 notequalI2 => 166.97/25.04 ( 166.97/25.04 subsetRefl => 166.97/25.04 ( 166.97/25.04 subsetTrans => 166.97/25.04 ( 166.97/25.04 setadjoinSub => 166.97/25.04 ( 166.97/25.04 setadjoinSub2 => 166.97/25.04 ( 166.97/25.04 subset2powerset => 166.97/25.04 ( 166.97/25.04 setextsub => 166.97/25.04 ( 166.97/25.04 subsetemptysetimpeq => 166.97/25.04 ( 166.97/25.04 powersetI1 => 166.97/25.04 ( 166.97/25.04 powersetE1 => 166.97/25.04 ( 166.97/25.04 inPowerset => 166.97/25.04 ( 166.97/25.04 powersetsubset => 166.97/25.04 ( 166.97/25.04 sepInPowerset => 166.97/25.04 ( 166.97/25.04 sepSubset => 166.97/25.04 ( 166.97/25.04 binunionIL => 166.97/25.04 ( 166.97/25.04 upairset2IR => 166.97/25.04 ( 166.97/25.04 binunionIR => 166.97/25.04 ( 166.97/25.04 binunionEcases => 166.97/25.04 ( 166.97/25.04 binunionE => 166.97/25.04 ( 166.97/25.04 binunionLsub => 166.97/25.04 ( 166.97/25.04 binunionRsub => 166.97/25.04 ( 166.97/25.04 binintersectI => 166.97/25.04 ( 166.97/25.04 binintersectSubset5 => 166.97/25.04 ( 166.97/25.04 binintersectEL => 166.97/25.04 ( 166.97/25.04 binintersectLsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset2 => 166.97/25.04 ( 166.97/25.04 binintersectSubset3 => 166.97/25.04 ( 166.97/25.04 binintersectER => 166.97/25.04 ( 166.97/25.04 disjointsetsI1 => 166.97/25.04 ( 166.97/25.04 binintersectRsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset4 => 166.97/25.04 ( 166.97/25.04 binintersectSubset1 => 166.97/25.04 ( 166.97/25.04 bs114d => 166.97/25.04 ( 166.97/25.04 setminusI => 166.97/25.04 ( 166.97/25.04 setminusEL => 166.97/25.04 ( 166.97/25.04 setminusER => 166.97/25.04 ( 166.97/25.04 setminusSubset2 => 166.97/25.04 ( 166.97/25.04 setminusERneg => 166.97/25.04 ( 166.97/25.04 setminusELneg => 166.97/25.04 ( 166.97/25.04 setminusILneg => 166.97/25.04 ( 166.97/25.04 setminusIRneg => 166.97/25.04 ( 166.97/25.04 setminusLsub => 166.97/25.04 ( 166.97/25.04 setminusSubset1 => 166.97/25.04 ( 166.97/25.04 symdiffE => 166.97/25.04 ( 166.97/25.04 symdiffI1 => 166.97/25.04 ( 166.97/25.04 symdiffI2 => 166.97/25.04 ( 166.97/25.04 symdiffIneg1 => 166.97/25.04 ( 166.97/25.04 symdiffIneg2 => 166.97/25.04 ( 166.97/25.04 secondinupair => 166.97/25.04 ( 166.97/25.04 setukpairIL => 166.97/25.04 ( 166.97/25.04 setukpairIR => 166.97/25.04 ( 166.97/25.04 kpairiskpair => 166.97/25.04 ( 166.97/25.04 kpairp => 166.97/25.04 ( 166.97/25.04 singletonsubset => 166.97/25.04 ( 166.97/25.04 singletoninpowerset => 166.97/25.04 ( 166.97/25.04 singletoninpowunion => 166.97/25.04 ( 166.97/25.04 upairset2E => 166.97/25.04 ( 166.97/25.04 upairsubunion => 166.97/25.04 ( 166.97/25.04 upairinpowunion => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem1 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem2 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem3 => 166.97/25.04 ( 166.97/25.04 cartprodpairin => 166.97/25.04 ( 166.97/25.04 cartprodmempair1 => 166.97/25.04 ( 166.97/25.04 cartprodmempair => 166.97/25.04 ( 166.97/25.04 setunionE2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton1 => 166.97/25.04 ( 166.97/25.04 setunionsingleton2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton => 166.97/25.04 ( 166.97/25.04 singletonprop => 166.97/25.04 ( 166.97/25.04 ex1E1 => 166.97/25.04 ( 166.97/25.04 ex1I => 166.97/25.04 ( 166.97/25.04 ex1I2 => 166.97/25.04 ( 166.97/25.04 singletonsuniq => 166.97/25.04 ( 166.97/25.04 setukpairinjL1 => 166.97/25.04 ( 166.97/25.04 kfstsingleton => 166.97/25.04 ( 166.97/25.04 theprop => 166.97/25.04 ( 166.97/25.04 kfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodfstin => 166.97/25.04 ( 166.97/25.04 setukpairinjL2 => 166.97/25.04 ( 166.97/25.04 setukpairinjL => 166.97/25.04 ( 166.97/25.04 setukpairinjR11 => 166.97/25.04 ( 166.97/25.04 setukpairinjR12 => 166.97/25.04 ( 166.97/25.04 setukpairinjR1 => 166.97/25.04 ( 166.97/25.04 upairequniteq => 166.97/25.04 ( 166.97/25.04 setukpairinjR2 => 166.97/25.04 ( 166.97/25.04 setukpairinjR => 166.97/25.04 ( 166.97/25.04 ksndsingleton => 166.97/25.04 ( 166.97/25.04 ksndpairEq => 166.97/25.04 ( 166.97/25.04 kpairsurjEq => 166.97/25.04 ( 166.97/25.04 cartprodsndin => 166.97/25.04 ( 166.97/25.04 cartprodpairmemEL => 166.97/25.04 ( 166.97/25.04 cartprodpairmemER => 166.97/25.04 ( 166.97/25.04 cartprodmempaircEq => 166.97/25.04 ( 166.97/25.04 cartprodfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodsndpairEq => 166.97/25.04 ( 166.97/25.04 cartprodpairsurjEq => 166.97/25.04 ( 166.97/25.04 dpsetconstrI => 166.97/25.04 ( 166.97/25.04 dpsetconstrSub => 166.97/25.04 ( 166.97/25.04 setOfPairsIsBReln => 166.97/25.04 ( 166.97/25.04 !!(( 166.97/25.04 ^[Y0 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y1 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y2 : ( 166.97/25.04 $i * $i) > $o]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y3 : $i]: 166.97/25.04 (in(Y3, 166.97/25.04 Y0) => 166.97/25.04 (!!(( 166.97/25.04 ^[Y4 : $i]: 166.97/25.04 (in(Y4, 166.97/25.04 Y1) => 166.97/25.04 (in( 166.97/25.04 kpair( 166.97/25.04 Y3, Y4), 166.97/25.04 dpsetconstr( 166.97/25.04 Y0, Y1, 166.97/25.04 Y2)) => 166.97/25.04 Y2(Y3, 166.97/25.04 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.04 inference('lazy_cnf_imply', [status(thm)], ['17'])). 166.97/25.04 tff('19', plain, 166.97/25.04 ~ (emptysetimpfalse => 166.97/25.04 (notinemptyset => 166.97/25.04 (exuE3e => 166.97/25.04 (setext => 166.97/25.04 (emptyI => 166.97/25.04 (noeltsimpempty => 166.97/25.04 (setbeta => 166.97/25.04 (nonemptyE1 => 166.97/25.04 (nonemptyI => 166.97/25.04 (nonemptyI1 => 166.97/25.04 (setadjoinIL => 166.97/25.04 (emptyinunitempty => 166.97/25.04 (setadjoinIR => 166.97/25.04 (setadjoinE => 166.97/25.04 (setadjoinOr => 166.97/25.04 (setoftrueEq => 166.97/25.04 (powersetI => 166.97/25.04 (emptyinPowerset => 166.97/25.04 (emptyInPowerset => 166.97/25.04 (powersetE => 166.97/25.04 (setunionI => 166.97/25.04 (setunionE => 166.97/25.04 (subPowSU => 166.97/25.04 (exuE2 => 166.97/25.04 (nonemptyImpWitness => 166.97/25.04 (uniqinunit => 166.97/25.04 (notinsingleton => 166.97/25.04 (eqinunit => 166.97/25.04 (singletonsswitch => 166.97/25.04 (upairsetE => 166.97/25.04 (upairsetIL => 166.97/25.04 (upairsetIR => 166.97/25.04 (emptyE1 => 166.97/25.04 (vacuousDall => 166.97/25.04 (quantDeMorgan1 => 166.97/25.04 (quantDeMorgan2 => 166.97/25.04 (quantDeMorgan3 => 166.97/25.04 (quantDeMorgan4 => 166.97/25.04 (prop2setI => 166.97/25.04 (prop2set2propI => 166.97/25.04 (notdexE => 166.97/25.04 (notdallE => 166.97/25.04 (exuI1 => 166.97/25.04 (exuI3 => 166.97/25.04 (exuI2 => 166.97/25.04 (inCongP => 166.97/25.04 (in__Cong => 166.97/25.04 (exuE3u => 166.97/25.04 (exu__Cong => 166.97/25.04 (emptyset__Cong => 166.97/25.04 (setadjoin__Cong => 166.97/25.04 (powerset__Cong => 166.97/25.04 (setunion__Cong => 166.97/25.04 (omega__Cong => 166.97/25.04 (exuEu => 166.97/25.04 (descr__Cong => 166.97/25.04 (dsetconstr__Cong => 166.97/25.04 (subsetI1 => 166.97/25.04 (eqimpsubset2 => 166.97/25.04 (eqimpsubset1 => 166.97/25.04 (subsetI2 => 166.97/25.04 (emptysetsubset => 166.97/25.04 ( 166.97/25.04 subsetE => 166.97/25.04 ( 166.97/25.04 subsetE2 => 166.97/25.04 ( 166.97/25.04 notsubsetI => 166.97/25.04 ( 166.97/25.04 notequalI1 => 166.97/25.04 ( 166.97/25.04 notequalI2 => 166.97/25.04 ( 166.97/25.04 subsetRefl => 166.97/25.04 ( 166.97/25.04 subsetTrans => 166.97/25.04 ( 166.97/25.04 setadjoinSub => 166.97/25.04 ( 166.97/25.04 setadjoinSub2 => 166.97/25.04 ( 166.97/25.04 subset2powerset => 166.97/25.04 ( 166.97/25.04 setextsub => 166.97/25.04 ( 166.97/25.04 subsetemptysetimpeq => 166.97/25.04 ( 166.97/25.04 powersetI1 => 166.97/25.04 ( 166.97/25.04 powersetE1 => 166.97/25.04 ( 166.97/25.04 inPowerset => 166.97/25.04 ( 166.97/25.04 powersetsubset => 166.97/25.04 ( 166.97/25.04 sepInPowerset => 166.97/25.04 ( 166.97/25.04 sepSubset => 166.97/25.04 ( 166.97/25.04 binunionIL => 166.97/25.04 ( 166.97/25.04 upairset2IR => 166.97/25.04 ( 166.97/25.04 binunionIR => 166.97/25.04 ( 166.97/25.04 binunionEcases => 166.97/25.04 ( 166.97/25.04 binunionE => 166.97/25.04 ( 166.97/25.04 binunionLsub => 166.97/25.04 ( 166.97/25.04 binunionRsub => 166.97/25.04 ( 166.97/25.04 binintersectI => 166.97/25.04 ( 166.97/25.04 binintersectSubset5 => 166.97/25.04 ( 166.97/25.04 binintersectEL => 166.97/25.04 ( 166.97/25.04 binintersectLsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset2 => 166.97/25.04 ( 166.97/25.04 binintersectSubset3 => 166.97/25.04 ( 166.97/25.04 binintersectER => 166.97/25.04 ( 166.97/25.04 disjointsetsI1 => 166.97/25.04 ( 166.97/25.04 binintersectRsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset4 => 166.97/25.04 ( 166.97/25.04 binintersectSubset1 => 166.97/25.04 ( 166.97/25.04 bs114d => 166.97/25.04 ( 166.97/25.04 setminusI => 166.97/25.04 ( 166.97/25.04 setminusEL => 166.97/25.04 ( 166.97/25.04 setminusER => 166.97/25.04 ( 166.97/25.04 setminusSubset2 => 166.97/25.04 ( 166.97/25.04 setminusERneg => 166.97/25.04 ( 166.97/25.04 setminusELneg => 166.97/25.04 ( 166.97/25.04 setminusILneg => 166.97/25.04 ( 166.97/25.04 setminusIRneg => 166.97/25.04 ( 166.97/25.04 setminusLsub => 166.97/25.04 ( 166.97/25.04 setminusSubset1 => 166.97/25.04 ( 166.97/25.04 symdiffE => 166.97/25.04 ( 166.97/25.04 symdiffI1 => 166.97/25.04 ( 166.97/25.04 symdiffI2 => 166.97/25.04 ( 166.97/25.04 symdiffIneg1 => 166.97/25.04 ( 166.97/25.04 symdiffIneg2 => 166.97/25.04 ( 166.97/25.04 secondinupair => 166.97/25.04 ( 166.97/25.04 setukpairIL => 166.97/25.04 ( 166.97/25.04 setukpairIR => 166.97/25.04 ( 166.97/25.04 kpairiskpair => 166.97/25.04 ( 166.97/25.04 kpairp => 166.97/25.04 ( 166.97/25.04 singletonsubset => 166.97/25.04 ( 166.97/25.04 singletoninpowerset => 166.97/25.04 ( 166.97/25.04 singletoninpowunion => 166.97/25.04 ( 166.97/25.04 upairset2E => 166.97/25.04 ( 166.97/25.04 upairsubunion => 166.97/25.04 ( 166.97/25.04 upairinpowunion => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem1 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem2 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem3 => 166.97/25.04 ( 166.97/25.04 cartprodpairin => 166.97/25.04 ( 166.97/25.04 cartprodmempair1 => 166.97/25.04 ( 166.97/25.04 cartprodmempair => 166.97/25.04 ( 166.97/25.04 setunionE2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton1 => 166.97/25.04 ( 166.97/25.04 setunionsingleton2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton => 166.97/25.04 ( 166.97/25.04 singletonprop => 166.97/25.04 ( 166.97/25.04 ex1E1 => 166.97/25.04 ( 166.97/25.04 ex1I => 166.97/25.04 ( 166.97/25.04 ex1I2 => 166.97/25.04 ( 166.97/25.04 singletonsuniq => 166.97/25.04 ( 166.97/25.04 setukpairinjL1 => 166.97/25.04 ( 166.97/25.04 kfstsingleton => 166.97/25.04 ( 166.97/25.04 theprop => 166.97/25.04 ( 166.97/25.04 kfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodfstin => 166.97/25.04 ( 166.97/25.04 setukpairinjL2 => 166.97/25.04 ( 166.97/25.04 setukpairinjL => 166.97/25.04 ( 166.97/25.04 setukpairinjR11 => 166.97/25.04 ( 166.97/25.04 setukpairinjR12 => 166.97/25.04 ( 166.97/25.04 setukpairinjR1 => 166.97/25.04 ( 166.97/25.04 upairequniteq => 166.97/25.04 ( 166.97/25.04 setukpairinjR2 => 166.97/25.04 ( 166.97/25.04 setukpairinjR => 166.97/25.04 ( 166.97/25.04 ksndsingleton => 166.97/25.04 ( 166.97/25.04 ksndpairEq => 166.97/25.04 ( 166.97/25.04 kpairsurjEq => 166.97/25.04 ( 166.97/25.04 cartprodsndin => 166.97/25.04 ( 166.97/25.04 cartprodpairmemEL => 166.97/25.04 ( 166.97/25.04 cartprodpairmemER => 166.97/25.04 ( 166.97/25.04 cartprodmempaircEq => 166.97/25.04 ( 166.97/25.04 cartprodfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodsndpairEq => 166.97/25.04 ( 166.97/25.04 cartprodpairsurjEq => 166.97/25.04 ( 166.97/25.04 dpsetconstrI => 166.97/25.04 ( 166.97/25.04 dpsetconstrSub => 166.97/25.04 ( 166.97/25.04 setOfPairsIsBReln => 166.97/25.04 ( 166.97/25.04 !!(( 166.97/25.04 ^[Y0 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y1 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y2 : ( 166.97/25.04 $i * $i) > $o]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y3 : $i]: 166.97/25.04 (in(Y3, 166.97/25.04 Y0) => 166.97/25.04 (!!(( 166.97/25.04 ^[Y4 : $i]: 166.97/25.04 (in(Y4, 166.97/25.04 Y1) => 166.97/25.04 (in( 166.97/25.04 kpair( 166.97/25.04 Y3, Y4), 166.97/25.04 dpsetconstr( 166.97/25.04 Y0, Y1, 166.97/25.04 Y2)) => 166.97/25.04 Y2(Y3, 166.97/25.04 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.04 inference('lazy_cnf_imply', [status(thm)], ['18'])). 166.97/25.04 tff('20', plain, 166.97/25.04 ~ (notinemptyset => 166.97/25.04 (exuE3e => 166.97/25.04 (setext => 166.97/25.04 (emptyI => 166.97/25.04 (noeltsimpempty => 166.97/25.04 (setbeta => 166.97/25.04 (nonemptyE1 => 166.97/25.04 (nonemptyI => 166.97/25.04 (nonemptyI1 => 166.97/25.04 (setadjoinIL => 166.97/25.04 (emptyinunitempty => 166.97/25.04 (setadjoinIR => 166.97/25.04 (setadjoinE => 166.97/25.04 (setadjoinOr => 166.97/25.04 (setoftrueEq => 166.97/25.04 (powersetI => 166.97/25.04 (emptyinPowerset => 166.97/25.04 (emptyInPowerset => 166.97/25.04 (powersetE => 166.97/25.04 (setunionI => 166.97/25.04 (setunionE => 166.97/25.04 (subPowSU => 166.97/25.04 (exuE2 => 166.97/25.04 (nonemptyImpWitness => 166.97/25.04 (uniqinunit => 166.97/25.04 (notinsingleton => 166.97/25.04 (eqinunit => 166.97/25.04 (singletonsswitch => 166.97/25.04 (upairsetE => 166.97/25.04 (upairsetIL => 166.97/25.04 (upairsetIR => 166.97/25.04 (emptyE1 => 166.97/25.04 (vacuousDall => 166.97/25.04 (quantDeMorgan1 => 166.97/25.04 (quantDeMorgan2 => 166.97/25.04 (quantDeMorgan3 => 166.97/25.04 (quantDeMorgan4 => 166.97/25.04 (prop2setI => 166.97/25.04 (prop2set2propI => 166.97/25.04 (notdexE => 166.97/25.04 (notdallE => 166.97/25.04 (exuI1 => 166.97/25.04 (exuI3 => 166.97/25.04 (exuI2 => 166.97/25.04 (inCongP => 166.97/25.04 (in__Cong => 166.97/25.04 (exuE3u => 166.97/25.04 (exu__Cong => 166.97/25.04 (emptyset__Cong => 166.97/25.04 (setadjoin__Cong => 166.97/25.04 (powerset__Cong => 166.97/25.04 (setunion__Cong => 166.97/25.04 (omega__Cong => 166.97/25.04 (exuEu => 166.97/25.04 (descr__Cong => 166.97/25.04 (dsetconstr__Cong => 166.97/25.04 (subsetI1 => 166.97/25.04 (eqimpsubset2 => 166.97/25.04 (eqimpsubset1 => 166.97/25.04 (subsetI2 => 166.97/25.04 (emptysetsubset => 166.97/25.04 (subsetE => 166.97/25.04 ( 166.97/25.04 subsetE2 => 166.97/25.04 ( 166.97/25.04 notsubsetI => 166.97/25.04 ( 166.97/25.04 notequalI1 => 166.97/25.04 ( 166.97/25.04 notequalI2 => 166.97/25.04 ( 166.97/25.04 subsetRefl => 166.97/25.04 ( 166.97/25.04 subsetTrans => 166.97/25.04 ( 166.97/25.04 setadjoinSub => 166.97/25.04 ( 166.97/25.04 setadjoinSub2 => 166.97/25.04 ( 166.97/25.04 subset2powerset => 166.97/25.04 ( 166.97/25.04 setextsub => 166.97/25.04 ( 166.97/25.04 subsetemptysetimpeq => 166.97/25.04 ( 166.97/25.04 powersetI1 => 166.97/25.04 ( 166.97/25.04 powersetE1 => 166.97/25.04 ( 166.97/25.04 inPowerset => 166.97/25.04 ( 166.97/25.04 powersetsubset => 166.97/25.04 ( 166.97/25.04 sepInPowerset => 166.97/25.04 ( 166.97/25.04 sepSubset => 166.97/25.04 ( 166.97/25.04 binunionIL => 166.97/25.04 ( 166.97/25.04 upairset2IR => 166.97/25.04 ( 166.97/25.04 binunionIR => 166.97/25.04 ( 166.97/25.04 binunionEcases => 166.97/25.04 ( 166.97/25.04 binunionE => 166.97/25.04 ( 166.97/25.04 binunionLsub => 166.97/25.04 ( 166.97/25.04 binunionRsub => 166.97/25.04 ( 166.97/25.04 binintersectI => 166.97/25.04 ( 166.97/25.04 binintersectSubset5 => 166.97/25.04 ( 166.97/25.04 binintersectEL => 166.97/25.04 ( 166.97/25.04 binintersectLsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset2 => 166.97/25.04 ( 166.97/25.04 binintersectSubset3 => 166.97/25.04 ( 166.97/25.04 binintersectER => 166.97/25.04 ( 166.97/25.04 disjointsetsI1 => 166.97/25.04 ( 166.97/25.04 binintersectRsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset4 => 166.97/25.04 ( 166.97/25.04 binintersectSubset1 => 166.97/25.04 ( 166.97/25.04 bs114d => 166.97/25.04 ( 166.97/25.04 setminusI => 166.97/25.04 ( 166.97/25.04 setminusEL => 166.97/25.04 ( 166.97/25.04 setminusER => 166.97/25.04 ( 166.97/25.04 setminusSubset2 => 166.97/25.04 ( 166.97/25.04 setminusERneg => 166.97/25.04 ( 166.97/25.04 setminusELneg => 166.97/25.04 ( 166.97/25.04 setminusILneg => 166.97/25.04 ( 166.97/25.04 setminusIRneg => 166.97/25.04 ( 166.97/25.04 setminusLsub => 166.97/25.04 ( 166.97/25.04 setminusSubset1 => 166.97/25.04 ( 166.97/25.04 symdiffE => 166.97/25.04 ( 166.97/25.04 symdiffI1 => 166.97/25.04 ( 166.97/25.04 symdiffI2 => 166.97/25.04 ( 166.97/25.04 symdiffIneg1 => 166.97/25.04 ( 166.97/25.04 symdiffIneg2 => 166.97/25.04 ( 166.97/25.04 secondinupair => 166.97/25.04 ( 166.97/25.04 setukpairIL => 166.97/25.04 ( 166.97/25.04 setukpairIR => 166.97/25.04 ( 166.97/25.04 kpairiskpair => 166.97/25.04 ( 166.97/25.04 kpairp => 166.97/25.04 ( 166.97/25.04 singletonsubset => 166.97/25.04 ( 166.97/25.04 singletoninpowerset => 166.97/25.04 ( 166.97/25.04 singletoninpowunion => 166.97/25.04 ( 166.97/25.04 upairset2E => 166.97/25.04 ( 166.97/25.04 upairsubunion => 166.97/25.04 ( 166.97/25.04 upairinpowunion => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem1 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem2 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem3 => 166.97/25.04 ( 166.97/25.04 cartprodpairin => 166.97/25.04 ( 166.97/25.04 cartprodmempair1 => 166.97/25.04 ( 166.97/25.04 cartprodmempair => 166.97/25.04 ( 166.97/25.04 setunionE2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton1 => 166.97/25.04 ( 166.97/25.04 setunionsingleton2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton => 166.97/25.04 ( 166.97/25.04 singletonprop => 166.97/25.04 ( 166.97/25.04 ex1E1 => 166.97/25.04 ( 166.97/25.04 ex1I => 166.97/25.04 ( 166.97/25.04 ex1I2 => 166.97/25.04 ( 166.97/25.04 singletonsuniq => 166.97/25.04 ( 166.97/25.04 setukpairinjL1 => 166.97/25.04 ( 166.97/25.04 kfstsingleton => 166.97/25.04 ( 166.97/25.04 theprop => 166.97/25.04 ( 166.97/25.04 kfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodfstin => 166.97/25.04 ( 166.97/25.04 setukpairinjL2 => 166.97/25.04 ( 166.97/25.04 setukpairinjL => 166.97/25.04 ( 166.97/25.04 setukpairinjR11 => 166.97/25.04 ( 166.97/25.04 setukpairinjR12 => 166.97/25.04 ( 166.97/25.04 setukpairinjR1 => 166.97/25.04 ( 166.97/25.04 upairequniteq => 166.97/25.04 ( 166.97/25.04 setukpairinjR2 => 166.97/25.04 ( 166.97/25.04 setukpairinjR => 166.97/25.04 ( 166.97/25.04 ksndsingleton => 166.97/25.04 ( 166.97/25.04 ksndpairEq => 166.97/25.04 ( 166.97/25.04 kpairsurjEq => 166.97/25.04 ( 166.97/25.04 cartprodsndin => 166.97/25.04 ( 166.97/25.04 cartprodpairmemEL => 166.97/25.04 ( 166.97/25.04 cartprodpairmemER => 166.97/25.04 ( 166.97/25.04 cartprodmempaircEq => 166.97/25.04 ( 166.97/25.04 cartprodfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodsndpairEq => 166.97/25.04 ( 166.97/25.04 cartprodpairsurjEq => 166.97/25.04 ( 166.97/25.04 dpsetconstrI => 166.97/25.04 ( 166.97/25.04 dpsetconstrSub => 166.97/25.04 ( 166.97/25.04 setOfPairsIsBReln => 166.97/25.04 ( 166.97/25.04 !!(( 166.97/25.04 ^[Y0 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y1 : $i]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y2 : ( 166.97/25.04 $i * $i) > $o]: 166.97/25.04 (!!(( 166.97/25.04 ^[Y3 : $i]: 166.97/25.04 (in(Y3, 166.97/25.04 Y0) => 166.97/25.04 (!!(( 166.97/25.04 ^[Y4 : $i]: 166.97/25.04 (in(Y4, 166.97/25.04 Y1) => 166.97/25.04 (in( 166.97/25.04 kpair( 166.97/25.04 Y3, Y4), 166.97/25.04 dpsetconstr( 166.97/25.04 Y0, Y1, 166.97/25.04 Y2)) => 166.97/25.04 Y2(Y3, 166.97/25.04 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.04 inference('lazy_cnf_imply', [status(thm)], ['19'])). 166.97/25.04 tff('21', plain, 166.97/25.04 ~ (exuE3e => 166.97/25.04 (setext => 166.97/25.04 (emptyI => 166.97/25.04 (noeltsimpempty => 166.97/25.04 (setbeta => 166.97/25.04 (nonemptyE1 => 166.97/25.04 (nonemptyI => 166.97/25.04 (nonemptyI1 => 166.97/25.04 (setadjoinIL => 166.97/25.04 (emptyinunitempty => 166.97/25.04 (setadjoinIR => 166.97/25.04 (setadjoinE => 166.97/25.04 (setadjoinOr => 166.97/25.04 (setoftrueEq => 166.97/25.04 (powersetI => 166.97/25.04 (emptyinPowerset => 166.97/25.04 (emptyInPowerset => 166.97/25.04 (powersetE => 166.97/25.04 (setunionI => 166.97/25.04 (setunionE => 166.97/25.04 (subPowSU => 166.97/25.04 (exuE2 => 166.97/25.04 (nonemptyImpWitness => 166.97/25.04 (uniqinunit => 166.97/25.04 (notinsingleton => 166.97/25.04 (eqinunit => 166.97/25.04 (singletonsswitch => 166.97/25.04 (upairsetE => 166.97/25.04 (upairsetIL => 166.97/25.04 (upairsetIR => 166.97/25.04 (emptyE1 => 166.97/25.04 (vacuousDall => 166.97/25.04 (quantDeMorgan1 => 166.97/25.04 (quantDeMorgan2 => 166.97/25.04 (quantDeMorgan3 => 166.97/25.04 (quantDeMorgan4 => 166.97/25.04 (prop2setI => 166.97/25.04 (prop2set2propI => 166.97/25.04 (notdexE => 166.97/25.04 (notdallE => 166.97/25.04 (exuI1 => 166.97/25.04 (exuI3 => 166.97/25.04 (exuI2 => 166.97/25.04 (inCongP => 166.97/25.04 (in__Cong => 166.97/25.04 (exuE3u => 166.97/25.04 (exu__Cong => 166.97/25.04 (emptyset__Cong => 166.97/25.04 (setadjoin__Cong => 166.97/25.04 (powerset__Cong => 166.97/25.04 (setunion__Cong => 166.97/25.04 (omega__Cong => 166.97/25.04 (exuEu => 166.97/25.04 (descr__Cong => 166.97/25.04 (dsetconstr__Cong => 166.97/25.04 (subsetI1 => 166.97/25.04 (eqimpsubset2 => 166.97/25.04 (eqimpsubset1 => 166.97/25.04 (subsetI2 => 166.97/25.04 (emptysetsubset => 166.97/25.04 (subsetE => 166.97/25.04 (subsetE2 => 166.97/25.04 ( 166.97/25.04 notsubsetI => 166.97/25.04 ( 166.97/25.04 notequalI1 => 166.97/25.04 ( 166.97/25.04 notequalI2 => 166.97/25.04 ( 166.97/25.04 subsetRefl => 166.97/25.04 ( 166.97/25.04 subsetTrans => 166.97/25.04 ( 166.97/25.04 setadjoinSub => 166.97/25.04 ( 166.97/25.04 setadjoinSub2 => 166.97/25.04 ( 166.97/25.04 subset2powerset => 166.97/25.04 ( 166.97/25.04 setextsub => 166.97/25.04 ( 166.97/25.04 subsetemptysetimpeq => 166.97/25.04 ( 166.97/25.04 powersetI1 => 166.97/25.04 ( 166.97/25.04 powersetE1 => 166.97/25.04 ( 166.97/25.04 inPowerset => 166.97/25.04 ( 166.97/25.04 powersetsubset => 166.97/25.04 ( 166.97/25.04 sepInPowerset => 166.97/25.04 ( 166.97/25.04 sepSubset => 166.97/25.04 ( 166.97/25.04 binunionIL => 166.97/25.04 ( 166.97/25.04 upairset2IR => 166.97/25.04 ( 166.97/25.04 binunionIR => 166.97/25.04 ( 166.97/25.04 binunionEcases => 166.97/25.04 ( 166.97/25.04 binunionE => 166.97/25.04 ( 166.97/25.04 binunionLsub => 166.97/25.04 ( 166.97/25.04 binunionRsub => 166.97/25.04 ( 166.97/25.04 binintersectI => 166.97/25.04 ( 166.97/25.04 binintersectSubset5 => 166.97/25.04 ( 166.97/25.04 binintersectEL => 166.97/25.04 ( 166.97/25.04 binintersectLsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset2 => 166.97/25.04 ( 166.97/25.04 binintersectSubset3 => 166.97/25.04 ( 166.97/25.04 binintersectER => 166.97/25.04 ( 166.97/25.04 disjointsetsI1 => 166.97/25.04 ( 166.97/25.04 binintersectRsub => 166.97/25.04 ( 166.97/25.04 binintersectSubset4 => 166.97/25.04 ( 166.97/25.04 binintersectSubset1 => 166.97/25.04 ( 166.97/25.04 bs114d => 166.97/25.04 ( 166.97/25.04 setminusI => 166.97/25.04 ( 166.97/25.04 setminusEL => 166.97/25.04 ( 166.97/25.04 setminusER => 166.97/25.04 ( 166.97/25.04 setminusSubset2 => 166.97/25.04 ( 166.97/25.04 setminusERneg => 166.97/25.04 ( 166.97/25.04 setminusELneg => 166.97/25.04 ( 166.97/25.04 setminusILneg => 166.97/25.04 ( 166.97/25.04 setminusIRneg => 166.97/25.04 ( 166.97/25.04 setminusLsub => 166.97/25.04 ( 166.97/25.04 setminusSubset1 => 166.97/25.04 ( 166.97/25.04 symdiffE => 166.97/25.04 ( 166.97/25.04 symdiffI1 => 166.97/25.04 ( 166.97/25.04 symdiffI2 => 166.97/25.04 ( 166.97/25.04 symdiffIneg1 => 166.97/25.04 ( 166.97/25.04 symdiffIneg2 => 166.97/25.04 ( 166.97/25.04 secondinupair => 166.97/25.04 ( 166.97/25.04 setukpairIL => 166.97/25.04 ( 166.97/25.04 setukpairIR => 166.97/25.04 ( 166.97/25.04 kpairiskpair => 166.97/25.04 ( 166.97/25.04 kpairp => 166.97/25.04 ( 166.97/25.04 singletonsubset => 166.97/25.04 ( 166.97/25.04 singletoninpowerset => 166.97/25.04 ( 166.97/25.04 singletoninpowunion => 166.97/25.04 ( 166.97/25.04 upairset2E => 166.97/25.04 ( 166.97/25.04 upairsubunion => 166.97/25.04 ( 166.97/25.04 upairinpowunion => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem1 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem2 => 166.97/25.04 ( 166.97/25.04 ubforcartprodlem3 => 166.97/25.04 ( 166.97/25.04 cartprodpairin => 166.97/25.04 ( 166.97/25.04 cartprodmempair1 => 166.97/25.04 ( 166.97/25.04 cartprodmempair => 166.97/25.04 ( 166.97/25.04 setunionE2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton1 => 166.97/25.04 ( 166.97/25.04 setunionsingleton2 => 166.97/25.04 ( 166.97/25.04 setunionsingleton => 166.97/25.04 ( 166.97/25.04 singletonprop => 166.97/25.04 ( 166.97/25.04 ex1E1 => 166.97/25.04 ( 166.97/25.04 ex1I => 166.97/25.04 ( 166.97/25.04 ex1I2 => 166.97/25.04 ( 166.97/25.04 singletonsuniq => 166.97/25.04 ( 166.97/25.04 setukpairinjL1 => 166.97/25.04 ( 166.97/25.04 kfstsingleton => 166.97/25.04 ( 166.97/25.04 theprop => 166.97/25.04 ( 166.97/25.04 kfstpairEq => 166.97/25.04 ( 166.97/25.04 cartprodfstin => 166.97/25.04 ( 166.97/25.04 setukpairinjL2 => 166.97/25.04 ( 166.97/25.04 setukpairinjL => 166.97/25.04 ( 166.97/25.04 setukpairinjR11 => 166.97/25.04 ( 166.97/25.04 setukpairinjR12 => 166.97/25.04 ( 166.97/25.04 setukpairinjR1 => 166.97/25.04 ( 166.97/25.04 upairequniteq => 166.97/25.04 ( 166.97/25.04 setukpairinjR2 => 166.97/25.04 ( 166.97/25.04 setukpairinjR => 166.97/25.04 ( 166.97/25.04 ksndsingleton => 166.97/25.04 ( 166.97/25.04 ksndpairEq => 166.97/25.04 ( 166.97/25.04 kpairsurjEq => 166.97/25.04 ( 166.97/25.04 cartprodsndin => 166.97/25.04 ( 166.97/25.04 cartprodpairmemEL => 166.97/25.04 ( 166.97/25.04 cartprodpairmemER => 166.97/25.04 ( 166.97/25.04 cartprodmempaircEq => 166.97/25.04 ( 166.97/25.04 cartprodfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodsndpairEq => 166.97/25.05 ( 166.97/25.05 cartprodpairsurjEq => 166.97/25.05 ( 166.97/25.05 dpsetconstrI => 166.97/25.05 ( 166.97/25.05 dpsetconstrSub => 166.97/25.05 ( 166.97/25.05 setOfPairsIsBReln => 166.97/25.05 ( 166.97/25.05 !!(( 166.97/25.05 ^[Y0 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y1 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y2 : ( 166.97/25.05 $i * $i) > $o]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y3 : $i]: 166.97/25.05 (in(Y3, 166.97/25.05 Y0) => 166.97/25.05 (!!(( 166.97/25.05 ^[Y4 : $i]: 166.97/25.05 (in(Y4, 166.97/25.05 Y1) => 166.97/25.05 (in( 166.97/25.05 kpair( 166.97/25.05 Y3, Y4), 166.97/25.05 dpsetconstr( 166.97/25.05 Y0, Y1, 166.97/25.05 Y2)) => 166.97/25.05 Y2(Y3, 166.97/25.05 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.05 inference('lazy_cnf_imply', [status(thm)], ['20'])). 166.97/25.05 tff('22', plain, 166.97/25.05 ~ (setext => 166.97/25.05 (emptyI => 166.97/25.05 (noeltsimpempty => 166.97/25.05 (setbeta => 166.97/25.05 (nonemptyE1 => 166.97/25.05 (nonemptyI => 166.97/25.05 (nonemptyI1 => 166.97/25.05 (setadjoinIL => 166.97/25.05 (emptyinunitempty => 166.97/25.05 (setadjoinIR => 166.97/25.05 (setadjoinE => 166.97/25.05 (setadjoinOr => 166.97/25.05 (setoftrueEq => 166.97/25.05 (powersetI => 166.97/25.05 (emptyinPowerset => 166.97/25.05 (emptyInPowerset => 166.97/25.05 (powersetE => 166.97/25.05 (setunionI => 166.97/25.05 (setunionE => 166.97/25.05 (subPowSU => 166.97/25.05 (exuE2 => 166.97/25.05 (nonemptyImpWitness => 166.97/25.05 (uniqinunit => 166.97/25.05 (notinsingleton => 166.97/25.05 (eqinunit => 166.97/25.05 (singletonsswitch => 166.97/25.05 (upairsetE => 166.97/25.05 (upairsetIL => 166.97/25.05 (upairsetIR => 166.97/25.05 (emptyE1 => 166.97/25.05 (vacuousDall => 166.97/25.05 (quantDeMorgan1 => 166.97/25.05 (quantDeMorgan2 => 166.97/25.05 (quantDeMorgan3 => 166.97/25.05 (quantDeMorgan4 => 166.97/25.05 (prop2setI => 166.97/25.05 (prop2set2propI => 166.97/25.05 (notdexE => 166.97/25.05 (notdallE => 166.97/25.05 (exuI1 => 166.97/25.05 (exuI3 => 166.97/25.05 (exuI2 => 166.97/25.05 (inCongP => 166.97/25.05 (in__Cong => 166.97/25.05 (exuE3u => 166.97/25.05 (exu__Cong => 166.97/25.05 (emptyset__Cong => 166.97/25.05 (setadjoin__Cong => 166.97/25.05 (powerset__Cong => 166.97/25.05 (setunion__Cong => 166.97/25.05 (omega__Cong => 166.97/25.05 (exuEu => 166.97/25.05 (descr__Cong => 166.97/25.05 (dsetconstr__Cong => 166.97/25.05 (subsetI1 => 166.97/25.05 (eqimpsubset2 => 166.97/25.05 (eqimpsubset1 => 166.97/25.05 (subsetI2 => 166.97/25.05 (emptysetsubset => 166.97/25.05 (subsetE => 166.97/25.05 (subsetE2 => 166.97/25.05 (notsubsetI => 166.97/25.05 ( 166.97/25.05 notequalI1 => 166.97/25.05 ( 166.97/25.05 notequalI2 => 166.97/25.05 ( 166.97/25.05 subsetRefl => 166.97/25.05 ( 166.97/25.05 subsetTrans => 166.97/25.05 ( 166.97/25.05 setadjoinSub => 166.97/25.05 ( 166.97/25.05 setadjoinSub2 => 166.97/25.05 ( 166.97/25.05 subset2powerset => 166.97/25.05 ( 166.97/25.05 setextsub => 166.97/25.05 ( 166.97/25.05 subsetemptysetimpeq => 166.97/25.05 ( 166.97/25.05 powersetI1 => 166.97/25.05 ( 166.97/25.05 powersetE1 => 166.97/25.05 ( 166.97/25.05 inPowerset => 166.97/25.05 ( 166.97/25.05 powersetsubset => 166.97/25.05 ( 166.97/25.05 sepInPowerset => 166.97/25.05 ( 166.97/25.05 sepSubset => 166.97/25.05 ( 166.97/25.05 binunionIL => 166.97/25.05 ( 166.97/25.05 upairset2IR => 166.97/25.05 ( 166.97/25.05 binunionIR => 166.97/25.05 ( 166.97/25.05 binunionEcases => 166.97/25.05 ( 166.97/25.05 binunionE => 166.97/25.05 ( 166.97/25.05 binunionLsub => 166.97/25.05 ( 166.97/25.05 binunionRsub => 166.97/25.05 ( 166.97/25.05 binintersectI => 166.97/25.05 ( 166.97/25.05 binintersectSubset5 => 166.97/25.05 ( 166.97/25.05 binintersectEL => 166.97/25.05 ( 166.97/25.05 binintersectLsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset2 => 166.97/25.05 ( 166.97/25.05 binintersectSubset3 => 166.97/25.05 ( 166.97/25.05 binintersectER => 166.97/25.05 ( 166.97/25.05 disjointsetsI1 => 166.97/25.05 ( 166.97/25.05 binintersectRsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset4 => 166.97/25.05 ( 166.97/25.05 binintersectSubset1 => 166.97/25.05 ( 166.97/25.05 bs114d => 166.97/25.05 ( 166.97/25.05 setminusI => 166.97/25.05 ( 166.97/25.05 setminusEL => 166.97/25.05 ( 166.97/25.05 setminusER => 166.97/25.05 ( 166.97/25.05 setminusSubset2 => 166.97/25.05 ( 166.97/25.05 setminusERneg => 166.97/25.05 ( 166.97/25.05 setminusELneg => 166.97/25.05 ( 166.97/25.05 setminusILneg => 166.97/25.05 ( 166.97/25.05 setminusIRneg => 166.97/25.05 ( 166.97/25.05 setminusLsub => 166.97/25.05 ( 166.97/25.05 setminusSubset1 => 166.97/25.05 ( 166.97/25.05 symdiffE => 166.97/25.05 ( 166.97/25.05 symdiffI1 => 166.97/25.05 ( 166.97/25.05 symdiffI2 => 166.97/25.05 ( 166.97/25.05 symdiffIneg1 => 166.97/25.05 ( 166.97/25.05 symdiffIneg2 => 166.97/25.05 ( 166.97/25.05 secondinupair => 166.97/25.05 ( 166.97/25.05 setukpairIL => 166.97/25.05 ( 166.97/25.05 setukpairIR => 166.97/25.05 ( 166.97/25.05 kpairiskpair => 166.97/25.05 ( 166.97/25.05 kpairp => 166.97/25.05 ( 166.97/25.05 singletonsubset => 166.97/25.05 ( 166.97/25.05 singletoninpowerset => 166.97/25.05 ( 166.97/25.05 singletoninpowunion => 166.97/25.05 ( 166.97/25.05 upairset2E => 166.97/25.05 ( 166.97/25.05 upairsubunion => 166.97/25.05 ( 166.97/25.05 upairinpowunion => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem1 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem2 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem3 => 166.97/25.05 ( 166.97/25.05 cartprodpairin => 166.97/25.05 ( 166.97/25.05 cartprodmempair1 => 166.97/25.05 ( 166.97/25.05 cartprodmempair => 166.97/25.05 ( 166.97/25.05 setunionE2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton1 => 166.97/25.05 ( 166.97/25.05 setunionsingleton2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton => 166.97/25.05 ( 166.97/25.05 singletonprop => 166.97/25.05 ( 166.97/25.05 ex1E1 => 166.97/25.05 ( 166.97/25.05 ex1I => 166.97/25.05 ( 166.97/25.05 ex1I2 => 166.97/25.05 ( 166.97/25.05 singletonsuniq => 166.97/25.05 ( 166.97/25.05 setukpairinjL1 => 166.97/25.05 ( 166.97/25.05 kfstsingleton => 166.97/25.05 ( 166.97/25.05 theprop => 166.97/25.05 ( 166.97/25.05 kfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodfstin => 166.97/25.05 ( 166.97/25.05 setukpairinjL2 => 166.97/25.05 ( 166.97/25.05 setukpairinjL => 166.97/25.05 ( 166.97/25.05 setukpairinjR11 => 166.97/25.05 ( 166.97/25.05 setukpairinjR12 => 166.97/25.05 ( 166.97/25.05 setukpairinjR1 => 166.97/25.05 ( 166.97/25.05 upairequniteq => 166.97/25.05 ( 166.97/25.05 setukpairinjR2 => 166.97/25.05 ( 166.97/25.05 setukpairinjR => 166.97/25.05 ( 166.97/25.05 ksndsingleton => 166.97/25.05 ( 166.97/25.05 ksndpairEq => 166.97/25.05 ( 166.97/25.05 kpairsurjEq => 166.97/25.05 ( 166.97/25.05 cartprodsndin => 166.97/25.05 ( 166.97/25.05 cartprodpairmemEL => 166.97/25.05 ( 166.97/25.05 cartprodpairmemER => 166.97/25.05 ( 166.97/25.05 cartprodmempaircEq => 166.97/25.05 ( 166.97/25.05 cartprodfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodsndpairEq => 166.97/25.05 ( 166.97/25.05 cartprodpairsurjEq => 166.97/25.05 ( 166.97/25.05 dpsetconstrI => 166.97/25.05 ( 166.97/25.05 dpsetconstrSub => 166.97/25.05 ( 166.97/25.05 setOfPairsIsBReln => 166.97/25.05 ( 166.97/25.05 !!(( 166.97/25.05 ^[Y0 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y1 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y2 : ( 166.97/25.05 $i * $i) > $o]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y3 : $i]: 166.97/25.05 (in(Y3, 166.97/25.05 Y0) => 166.97/25.05 (!!(( 166.97/25.05 ^[Y4 : $i]: 166.97/25.05 (in(Y4, 166.97/25.05 Y1) => 166.97/25.05 (in( 166.97/25.05 kpair( 166.97/25.05 Y3, Y4), 166.97/25.05 dpsetconstr( 166.97/25.05 Y0, Y1, 166.97/25.05 Y2)) => 166.97/25.05 Y2(Y3, 166.97/25.05 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.05 inference('lazy_cnf_imply', [status(thm)], ['21'])). 166.97/25.05 tff('23', plain, 166.97/25.05 ~ (emptyI => 166.97/25.05 (noeltsimpempty => 166.97/25.05 (setbeta => 166.97/25.05 (nonemptyE1 => 166.97/25.05 (nonemptyI => 166.97/25.05 (nonemptyI1 => 166.97/25.05 (setadjoinIL => 166.97/25.05 (emptyinunitempty => 166.97/25.05 (setadjoinIR => 166.97/25.05 (setadjoinE => 166.97/25.05 (setadjoinOr => 166.97/25.05 (setoftrueEq => 166.97/25.05 (powersetI => 166.97/25.05 (emptyinPowerset => 166.97/25.05 (emptyInPowerset => 166.97/25.05 (powersetE => 166.97/25.05 (setunionI => 166.97/25.05 (setunionE => 166.97/25.05 (subPowSU => 166.97/25.05 (exuE2 => 166.97/25.05 (nonemptyImpWitness => 166.97/25.05 (uniqinunit => 166.97/25.05 (notinsingleton => 166.97/25.05 (eqinunit => 166.97/25.05 (singletonsswitch => 166.97/25.05 (upairsetE => 166.97/25.05 (upairsetIL => 166.97/25.05 (upairsetIR => 166.97/25.05 (emptyE1 => 166.97/25.05 (vacuousDall => 166.97/25.05 (quantDeMorgan1 => 166.97/25.05 (quantDeMorgan2 => 166.97/25.05 (quantDeMorgan3 => 166.97/25.05 (quantDeMorgan4 => 166.97/25.05 (prop2setI => 166.97/25.05 (prop2set2propI => 166.97/25.05 (notdexE => 166.97/25.05 (notdallE => 166.97/25.05 (exuI1 => 166.97/25.05 (exuI3 => 166.97/25.05 (exuI2 => 166.97/25.05 (inCongP => 166.97/25.05 (in__Cong => 166.97/25.05 (exuE3u => 166.97/25.05 (exu__Cong => 166.97/25.05 (emptyset__Cong => 166.97/25.05 (setadjoin__Cong => 166.97/25.05 (powerset__Cong => 166.97/25.05 (setunion__Cong => 166.97/25.05 (omega__Cong => 166.97/25.05 (exuEu => 166.97/25.05 (descr__Cong => 166.97/25.05 (dsetconstr__Cong => 166.97/25.05 (subsetI1 => 166.97/25.05 (eqimpsubset2 => 166.97/25.05 (eqimpsubset1 => 166.97/25.05 (subsetI2 => 166.97/25.05 (emptysetsubset => 166.97/25.05 (subsetE => 166.97/25.05 (subsetE2 => 166.97/25.05 (notsubsetI => 166.97/25.05 (notequalI1 => 166.97/25.05 ( 166.97/25.05 notequalI2 => 166.97/25.05 ( 166.97/25.05 subsetRefl => 166.97/25.05 ( 166.97/25.05 subsetTrans => 166.97/25.05 ( 166.97/25.05 setadjoinSub => 166.97/25.05 ( 166.97/25.05 setadjoinSub2 => 166.97/25.05 ( 166.97/25.05 subset2powerset => 166.97/25.05 ( 166.97/25.05 setextsub => 166.97/25.05 ( 166.97/25.05 subsetemptysetimpeq => 166.97/25.05 ( 166.97/25.05 powersetI1 => 166.97/25.05 ( 166.97/25.05 powersetE1 => 166.97/25.05 ( 166.97/25.05 inPowerset => 166.97/25.05 ( 166.97/25.05 powersetsubset => 166.97/25.05 ( 166.97/25.05 sepInPowerset => 166.97/25.05 ( 166.97/25.05 sepSubset => 166.97/25.05 ( 166.97/25.05 binunionIL => 166.97/25.05 ( 166.97/25.05 upairset2IR => 166.97/25.05 ( 166.97/25.05 binunionIR => 166.97/25.05 ( 166.97/25.05 binunionEcases => 166.97/25.05 ( 166.97/25.05 binunionE => 166.97/25.05 ( 166.97/25.05 binunionLsub => 166.97/25.05 ( 166.97/25.05 binunionRsub => 166.97/25.05 ( 166.97/25.05 binintersectI => 166.97/25.05 ( 166.97/25.05 binintersectSubset5 => 166.97/25.05 ( 166.97/25.05 binintersectEL => 166.97/25.05 ( 166.97/25.05 binintersectLsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset2 => 166.97/25.05 ( 166.97/25.05 binintersectSubset3 => 166.97/25.05 ( 166.97/25.05 binintersectER => 166.97/25.05 ( 166.97/25.05 disjointsetsI1 => 166.97/25.05 ( 166.97/25.05 binintersectRsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset4 => 166.97/25.05 ( 166.97/25.05 binintersectSubset1 => 166.97/25.05 ( 166.97/25.05 bs114d => 166.97/25.05 ( 166.97/25.05 setminusI => 166.97/25.05 ( 166.97/25.05 setminusEL => 166.97/25.05 ( 166.97/25.05 setminusER => 166.97/25.05 ( 166.97/25.05 setminusSubset2 => 166.97/25.05 ( 166.97/25.05 setminusERneg => 166.97/25.05 ( 166.97/25.05 setminusELneg => 166.97/25.05 ( 166.97/25.05 setminusILneg => 166.97/25.05 ( 166.97/25.05 setminusIRneg => 166.97/25.05 ( 166.97/25.05 setminusLsub => 166.97/25.05 ( 166.97/25.05 setminusSubset1 => 166.97/25.05 ( 166.97/25.05 symdiffE => 166.97/25.05 ( 166.97/25.05 symdiffI1 => 166.97/25.05 ( 166.97/25.05 symdiffI2 => 166.97/25.05 ( 166.97/25.05 symdiffIneg1 => 166.97/25.05 ( 166.97/25.05 symdiffIneg2 => 166.97/25.05 ( 166.97/25.05 secondinupair => 166.97/25.05 ( 166.97/25.05 setukpairIL => 166.97/25.05 ( 166.97/25.05 setukpairIR => 166.97/25.05 ( 166.97/25.05 kpairiskpair => 166.97/25.05 ( 166.97/25.05 kpairp => 166.97/25.05 ( 166.97/25.05 singletonsubset => 166.97/25.05 ( 166.97/25.05 singletoninpowerset => 166.97/25.05 ( 166.97/25.05 singletoninpowunion => 166.97/25.05 ( 166.97/25.05 upairset2E => 166.97/25.05 ( 166.97/25.05 upairsubunion => 166.97/25.05 ( 166.97/25.05 upairinpowunion => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem1 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem2 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem3 => 166.97/25.05 ( 166.97/25.05 cartprodpairin => 166.97/25.05 ( 166.97/25.05 cartprodmempair1 => 166.97/25.05 ( 166.97/25.05 cartprodmempair => 166.97/25.05 ( 166.97/25.05 setunionE2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton1 => 166.97/25.05 ( 166.97/25.05 setunionsingleton2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton => 166.97/25.05 ( 166.97/25.05 singletonprop => 166.97/25.05 ( 166.97/25.05 ex1E1 => 166.97/25.05 ( 166.97/25.05 ex1I => 166.97/25.05 ( 166.97/25.05 ex1I2 => 166.97/25.05 ( 166.97/25.05 singletonsuniq => 166.97/25.05 ( 166.97/25.05 setukpairinjL1 => 166.97/25.05 ( 166.97/25.05 kfstsingleton => 166.97/25.05 ( 166.97/25.05 theprop => 166.97/25.05 ( 166.97/25.05 kfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodfstin => 166.97/25.05 ( 166.97/25.05 setukpairinjL2 => 166.97/25.05 ( 166.97/25.05 setukpairinjL => 166.97/25.05 ( 166.97/25.05 setukpairinjR11 => 166.97/25.05 ( 166.97/25.05 setukpairinjR12 => 166.97/25.05 ( 166.97/25.05 setukpairinjR1 => 166.97/25.05 ( 166.97/25.05 upairequniteq => 166.97/25.05 ( 166.97/25.05 setukpairinjR2 => 166.97/25.05 ( 166.97/25.05 setukpairinjR => 166.97/25.05 ( 166.97/25.05 ksndsingleton => 166.97/25.05 ( 166.97/25.05 ksndpairEq => 166.97/25.05 ( 166.97/25.05 kpairsurjEq => 166.97/25.05 ( 166.97/25.05 cartprodsndin => 166.97/25.05 ( 166.97/25.05 cartprodpairmemEL => 166.97/25.05 ( 166.97/25.05 cartprodpairmemER => 166.97/25.05 ( 166.97/25.05 cartprodmempaircEq => 166.97/25.05 ( 166.97/25.05 cartprodfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodsndpairEq => 166.97/25.05 ( 166.97/25.05 cartprodpairsurjEq => 166.97/25.05 ( 166.97/25.05 dpsetconstrI => 166.97/25.05 ( 166.97/25.05 dpsetconstrSub => 166.97/25.05 ( 166.97/25.05 setOfPairsIsBReln => 166.97/25.05 ( 166.97/25.05 !!(( 166.97/25.05 ^[Y0 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y1 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y2 : ( 166.97/25.05 $i * $i) > $o]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y3 : $i]: 166.97/25.05 (in(Y3, 166.97/25.05 Y0) => 166.97/25.05 (!!(( 166.97/25.05 ^[Y4 : $i]: 166.97/25.05 (in(Y4, 166.97/25.05 Y1) => 166.97/25.05 (in( 166.97/25.05 kpair( 166.97/25.05 Y3, Y4), 166.97/25.05 dpsetconstr( 166.97/25.05 Y0, Y1, 166.97/25.05 Y2)) => 166.97/25.05 Y2(Y3, 166.97/25.05 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.05 inference('lazy_cnf_imply', [status(thm)], ['22'])). 166.97/25.05 tff('24', plain, 166.97/25.05 ~ (noeltsimpempty => 166.97/25.05 (setbeta => 166.97/25.05 (nonemptyE1 => 166.97/25.05 (nonemptyI => 166.97/25.05 (nonemptyI1 => 166.97/25.05 (setadjoinIL => 166.97/25.05 (emptyinunitempty => 166.97/25.05 (setadjoinIR => 166.97/25.05 (setadjoinE => 166.97/25.05 (setadjoinOr => 166.97/25.05 (setoftrueEq => 166.97/25.05 (powersetI => 166.97/25.05 (emptyinPowerset => 166.97/25.05 (emptyInPowerset => 166.97/25.05 (powersetE => 166.97/25.05 (setunionI => 166.97/25.05 (setunionE => 166.97/25.05 (subPowSU => 166.97/25.05 (exuE2 => 166.97/25.05 (nonemptyImpWitness => 166.97/25.05 (uniqinunit => 166.97/25.05 (notinsingleton => 166.97/25.05 (eqinunit => 166.97/25.05 (singletonsswitch => 166.97/25.05 (upairsetE => 166.97/25.05 (upairsetIL => 166.97/25.05 (upairsetIR => 166.97/25.05 (emptyE1 => 166.97/25.05 (vacuousDall => 166.97/25.05 (quantDeMorgan1 => 166.97/25.05 (quantDeMorgan2 => 166.97/25.05 (quantDeMorgan3 => 166.97/25.05 (quantDeMorgan4 => 166.97/25.05 (prop2setI => 166.97/25.05 (prop2set2propI => 166.97/25.05 (notdexE => 166.97/25.05 (notdallE => 166.97/25.05 (exuI1 => 166.97/25.05 (exuI3 => 166.97/25.05 (exuI2 => 166.97/25.05 (inCongP => 166.97/25.05 (in__Cong => 166.97/25.05 (exuE3u => 166.97/25.05 (exu__Cong => 166.97/25.05 (emptyset__Cong => 166.97/25.05 (setadjoin__Cong => 166.97/25.05 (powerset__Cong => 166.97/25.05 (setunion__Cong => 166.97/25.05 (omega__Cong => 166.97/25.05 (exuEu => 166.97/25.05 (descr__Cong => 166.97/25.05 (dsetconstr__Cong => 166.97/25.05 (subsetI1 => 166.97/25.05 (eqimpsubset2 => 166.97/25.05 (eqimpsubset1 => 166.97/25.05 (subsetI2 => 166.97/25.05 (emptysetsubset => 166.97/25.05 (subsetE => 166.97/25.05 (subsetE2 => 166.97/25.05 (notsubsetI => 166.97/25.05 (notequalI1 => 166.97/25.05 (notequalI2 => 166.97/25.05 ( 166.97/25.05 subsetRefl => 166.97/25.05 ( 166.97/25.05 subsetTrans => 166.97/25.05 ( 166.97/25.05 setadjoinSub => 166.97/25.05 ( 166.97/25.05 setadjoinSub2 => 166.97/25.05 ( 166.97/25.05 subset2powerset => 166.97/25.05 ( 166.97/25.05 setextsub => 166.97/25.05 ( 166.97/25.05 subsetemptysetimpeq => 166.97/25.05 ( 166.97/25.05 powersetI1 => 166.97/25.05 ( 166.97/25.05 powersetE1 => 166.97/25.05 ( 166.97/25.05 inPowerset => 166.97/25.05 ( 166.97/25.05 powersetsubset => 166.97/25.05 ( 166.97/25.05 sepInPowerset => 166.97/25.05 ( 166.97/25.05 sepSubset => 166.97/25.05 ( 166.97/25.05 binunionIL => 166.97/25.05 ( 166.97/25.05 upairset2IR => 166.97/25.05 ( 166.97/25.05 binunionIR => 166.97/25.05 ( 166.97/25.05 binunionEcases => 166.97/25.05 ( 166.97/25.05 binunionE => 166.97/25.05 ( 166.97/25.05 binunionLsub => 166.97/25.05 ( 166.97/25.05 binunionRsub => 166.97/25.05 ( 166.97/25.05 binintersectI => 166.97/25.05 ( 166.97/25.05 binintersectSubset5 => 166.97/25.05 ( 166.97/25.05 binintersectEL => 166.97/25.05 ( 166.97/25.05 binintersectLsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset2 => 166.97/25.05 ( 166.97/25.05 binintersectSubset3 => 166.97/25.05 ( 166.97/25.05 binintersectER => 166.97/25.05 ( 166.97/25.05 disjointsetsI1 => 166.97/25.05 ( 166.97/25.05 binintersectRsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset4 => 166.97/25.05 ( 166.97/25.05 binintersectSubset1 => 166.97/25.05 ( 166.97/25.05 bs114d => 166.97/25.05 ( 166.97/25.05 setminusI => 166.97/25.05 ( 166.97/25.05 setminusEL => 166.97/25.05 ( 166.97/25.05 setminusER => 166.97/25.05 ( 166.97/25.05 setminusSubset2 => 166.97/25.05 ( 166.97/25.05 setminusERneg => 166.97/25.05 ( 166.97/25.05 setminusELneg => 166.97/25.05 ( 166.97/25.05 setminusILneg => 166.97/25.05 ( 166.97/25.05 setminusIRneg => 166.97/25.05 ( 166.97/25.05 setminusLsub => 166.97/25.05 ( 166.97/25.05 setminusSubset1 => 166.97/25.05 ( 166.97/25.05 symdiffE => 166.97/25.05 ( 166.97/25.05 symdiffI1 => 166.97/25.05 ( 166.97/25.05 symdiffI2 => 166.97/25.05 ( 166.97/25.05 symdiffIneg1 => 166.97/25.05 ( 166.97/25.05 symdiffIneg2 => 166.97/25.05 ( 166.97/25.05 secondinupair => 166.97/25.05 ( 166.97/25.05 setukpairIL => 166.97/25.05 ( 166.97/25.05 setukpairIR => 166.97/25.05 ( 166.97/25.05 kpairiskpair => 166.97/25.05 ( 166.97/25.05 kpairp => 166.97/25.05 ( 166.97/25.05 singletonsubset => 166.97/25.05 ( 166.97/25.05 singletoninpowerset => 166.97/25.05 ( 166.97/25.05 singletoninpowunion => 166.97/25.05 ( 166.97/25.05 upairset2E => 166.97/25.05 ( 166.97/25.05 upairsubunion => 166.97/25.05 ( 166.97/25.05 upairinpowunion => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem1 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem2 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem3 => 166.97/25.05 ( 166.97/25.05 cartprodpairin => 166.97/25.05 ( 166.97/25.05 cartprodmempair1 => 166.97/25.05 ( 166.97/25.05 cartprodmempair => 166.97/25.05 ( 166.97/25.05 setunionE2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton1 => 166.97/25.05 ( 166.97/25.05 setunionsingleton2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton => 166.97/25.05 ( 166.97/25.05 singletonprop => 166.97/25.05 ( 166.97/25.05 ex1E1 => 166.97/25.05 ( 166.97/25.05 ex1I => 166.97/25.05 ( 166.97/25.05 ex1I2 => 166.97/25.05 ( 166.97/25.05 singletonsuniq => 166.97/25.05 ( 166.97/25.05 setukpairinjL1 => 166.97/25.05 ( 166.97/25.05 kfstsingleton => 166.97/25.05 ( 166.97/25.05 theprop => 166.97/25.05 ( 166.97/25.05 kfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodfstin => 166.97/25.05 ( 166.97/25.05 setukpairinjL2 => 166.97/25.05 ( 166.97/25.05 setukpairinjL => 166.97/25.05 ( 166.97/25.05 setukpairinjR11 => 166.97/25.05 ( 166.97/25.05 setukpairinjR12 => 166.97/25.05 ( 166.97/25.05 setukpairinjR1 => 166.97/25.05 ( 166.97/25.05 upairequniteq => 166.97/25.05 ( 166.97/25.05 setukpairinjR2 => 166.97/25.05 ( 166.97/25.05 setukpairinjR => 166.97/25.05 ( 166.97/25.05 ksndsingleton => 166.97/25.05 ( 166.97/25.05 ksndpairEq => 166.97/25.05 ( 166.97/25.05 kpairsurjEq => 166.97/25.05 ( 166.97/25.05 cartprodsndin => 166.97/25.05 ( 166.97/25.05 cartprodpairmemEL => 166.97/25.05 ( 166.97/25.05 cartprodpairmemER => 166.97/25.05 ( 166.97/25.05 cartprodmempaircEq => 166.97/25.05 ( 166.97/25.05 cartprodfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodsndpairEq => 166.97/25.05 ( 166.97/25.05 cartprodpairsurjEq => 166.97/25.05 ( 166.97/25.05 dpsetconstrI => 166.97/25.05 ( 166.97/25.05 dpsetconstrSub => 166.97/25.05 ( 166.97/25.05 setOfPairsIsBReln => 166.97/25.05 ( 166.97/25.05 !!(( 166.97/25.05 ^[Y0 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y1 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y2 : ( 166.97/25.05 $i * $i) > $o]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y3 : $i]: 166.97/25.05 (in(Y3, 166.97/25.05 Y0) => 166.97/25.05 (!!(( 166.97/25.05 ^[Y4 : $i]: 166.97/25.05 (in(Y4, 166.97/25.05 Y1) => 166.97/25.05 (in( 166.97/25.05 kpair( 166.97/25.05 Y3, Y4), 166.97/25.05 dpsetconstr( 166.97/25.05 Y0, Y1, 166.97/25.05 Y2)) => 166.97/25.05 Y2(Y3, 166.97/25.05 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.05 inference('lazy_cnf_imply', [status(thm)], ['23'])). 166.97/25.05 tff('25', plain, 166.97/25.05 ~ (setbeta => 166.97/25.05 (nonemptyE1 => 166.97/25.05 (nonemptyI => 166.97/25.05 (nonemptyI1 => 166.97/25.05 (setadjoinIL => 166.97/25.05 (emptyinunitempty => 166.97/25.05 (setadjoinIR => 166.97/25.05 (setadjoinE => 166.97/25.05 (setadjoinOr => 166.97/25.05 (setoftrueEq => 166.97/25.05 (powersetI => 166.97/25.05 (emptyinPowerset => 166.97/25.05 (emptyInPowerset => 166.97/25.05 (powersetE => 166.97/25.05 (setunionI => 166.97/25.05 (setunionE => 166.97/25.05 (subPowSU => 166.97/25.05 (exuE2 => 166.97/25.05 (nonemptyImpWitness => 166.97/25.05 (uniqinunit => 166.97/25.05 (notinsingleton => 166.97/25.05 (eqinunit => 166.97/25.05 (singletonsswitch => 166.97/25.05 (upairsetE => 166.97/25.05 (upairsetIL => 166.97/25.05 (upairsetIR => 166.97/25.05 (emptyE1 => 166.97/25.05 (vacuousDall => 166.97/25.05 (quantDeMorgan1 => 166.97/25.05 (quantDeMorgan2 => 166.97/25.05 (quantDeMorgan3 => 166.97/25.05 (quantDeMorgan4 => 166.97/25.05 (prop2setI => 166.97/25.05 (prop2set2propI => 166.97/25.05 (notdexE => 166.97/25.05 (notdallE => 166.97/25.05 (exuI1 => 166.97/25.05 (exuI3 => 166.97/25.05 (exuI2 => 166.97/25.05 (inCongP => 166.97/25.05 (in__Cong => 166.97/25.05 (exuE3u => 166.97/25.05 (exu__Cong => 166.97/25.05 (emptyset__Cong => 166.97/25.05 (setadjoin__Cong => 166.97/25.05 (powerset__Cong => 166.97/25.05 (setunion__Cong => 166.97/25.05 (omega__Cong => 166.97/25.05 (exuEu => 166.97/25.05 (descr__Cong => 166.97/25.05 (dsetconstr__Cong => 166.97/25.05 (subsetI1 => 166.97/25.05 (eqimpsubset2 => 166.97/25.05 (eqimpsubset1 => 166.97/25.05 (subsetI2 => 166.97/25.05 (emptysetsubset => 166.97/25.05 (subsetE => 166.97/25.05 (subsetE2 => 166.97/25.05 (notsubsetI => 166.97/25.05 (notequalI1 => 166.97/25.05 (notequalI2 => 166.97/25.05 (subsetRefl => 166.97/25.05 ( 166.97/25.05 subsetTrans => 166.97/25.05 ( 166.97/25.05 setadjoinSub => 166.97/25.05 ( 166.97/25.05 setadjoinSub2 => 166.97/25.05 ( 166.97/25.05 subset2powerset => 166.97/25.05 ( 166.97/25.05 setextsub => 166.97/25.05 ( 166.97/25.05 subsetemptysetimpeq => 166.97/25.05 ( 166.97/25.05 powersetI1 => 166.97/25.05 ( 166.97/25.05 powersetE1 => 166.97/25.05 ( 166.97/25.05 inPowerset => 166.97/25.05 ( 166.97/25.05 powersetsubset => 166.97/25.05 ( 166.97/25.05 sepInPowerset => 166.97/25.05 ( 166.97/25.05 sepSubset => 166.97/25.05 ( 166.97/25.05 binunionIL => 166.97/25.05 ( 166.97/25.05 upairset2IR => 166.97/25.05 ( 166.97/25.05 binunionIR => 166.97/25.05 ( 166.97/25.05 binunionEcases => 166.97/25.05 ( 166.97/25.05 binunionE => 166.97/25.05 ( 166.97/25.05 binunionLsub => 166.97/25.05 ( 166.97/25.05 binunionRsub => 166.97/25.05 ( 166.97/25.05 binintersectI => 166.97/25.05 ( 166.97/25.05 binintersectSubset5 => 166.97/25.05 ( 166.97/25.05 binintersectEL => 166.97/25.05 ( 166.97/25.05 binintersectLsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset2 => 166.97/25.05 ( 166.97/25.05 binintersectSubset3 => 166.97/25.05 ( 166.97/25.05 binintersectER => 166.97/25.05 ( 166.97/25.05 disjointsetsI1 => 166.97/25.05 ( 166.97/25.05 binintersectRsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset4 => 166.97/25.05 ( 166.97/25.05 binintersectSubset1 => 166.97/25.05 ( 166.97/25.05 bs114d => 166.97/25.05 ( 166.97/25.05 setminusI => 166.97/25.05 ( 166.97/25.05 setminusEL => 166.97/25.05 ( 166.97/25.05 setminusER => 166.97/25.05 ( 166.97/25.05 setminusSubset2 => 166.97/25.05 ( 166.97/25.05 setminusERneg => 166.97/25.05 ( 166.97/25.05 setminusELneg => 166.97/25.05 ( 166.97/25.05 setminusILneg => 166.97/25.05 ( 166.97/25.05 setminusIRneg => 166.97/25.05 ( 166.97/25.05 setminusLsub => 166.97/25.05 ( 166.97/25.05 setminusSubset1 => 166.97/25.05 ( 166.97/25.05 symdiffE => 166.97/25.05 ( 166.97/25.05 symdiffI1 => 166.97/25.05 ( 166.97/25.05 symdiffI2 => 166.97/25.05 ( 166.97/25.05 symdiffIneg1 => 166.97/25.05 ( 166.97/25.05 symdiffIneg2 => 166.97/25.05 ( 166.97/25.05 secondinupair => 166.97/25.05 ( 166.97/25.05 setukpairIL => 166.97/25.05 ( 166.97/25.05 setukpairIR => 166.97/25.05 ( 166.97/25.05 kpairiskpair => 166.97/25.05 ( 166.97/25.05 kpairp => 166.97/25.05 ( 166.97/25.05 singletonsubset => 166.97/25.05 ( 166.97/25.05 singletoninpowerset => 166.97/25.05 ( 166.97/25.05 singletoninpowunion => 166.97/25.05 ( 166.97/25.05 upairset2E => 166.97/25.05 ( 166.97/25.05 upairsubunion => 166.97/25.05 ( 166.97/25.05 upairinpowunion => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem1 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem2 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem3 => 166.97/25.05 ( 166.97/25.05 cartprodpairin => 166.97/25.05 ( 166.97/25.05 cartprodmempair1 => 166.97/25.05 ( 166.97/25.05 cartprodmempair => 166.97/25.05 ( 166.97/25.05 setunionE2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton1 => 166.97/25.05 ( 166.97/25.05 setunionsingleton2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton => 166.97/25.05 ( 166.97/25.05 singletonprop => 166.97/25.05 ( 166.97/25.05 ex1E1 => 166.97/25.05 ( 166.97/25.05 ex1I => 166.97/25.05 ( 166.97/25.05 ex1I2 => 166.97/25.05 ( 166.97/25.05 singletonsuniq => 166.97/25.05 ( 166.97/25.05 setukpairinjL1 => 166.97/25.05 ( 166.97/25.05 kfstsingleton => 166.97/25.05 ( 166.97/25.05 theprop => 166.97/25.05 ( 166.97/25.05 kfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodfstin => 166.97/25.05 ( 166.97/25.05 setukpairinjL2 => 166.97/25.05 ( 166.97/25.05 setukpairinjL => 166.97/25.05 ( 166.97/25.05 setukpairinjR11 => 166.97/25.05 ( 166.97/25.05 setukpairinjR12 => 166.97/25.05 ( 166.97/25.05 setukpairinjR1 => 166.97/25.05 ( 166.97/25.05 upairequniteq => 166.97/25.05 ( 166.97/25.05 setukpairinjR2 => 166.97/25.05 ( 166.97/25.05 setukpairinjR => 166.97/25.05 ( 166.97/25.05 ksndsingleton => 166.97/25.05 ( 166.97/25.05 ksndpairEq => 166.97/25.05 ( 166.97/25.05 kpairsurjEq => 166.97/25.05 ( 166.97/25.05 cartprodsndin => 166.97/25.05 ( 166.97/25.05 cartprodpairmemEL => 166.97/25.05 ( 166.97/25.05 cartprodpairmemER => 166.97/25.05 ( 166.97/25.05 cartprodmempaircEq => 166.97/25.05 ( 166.97/25.05 cartprodfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodsndpairEq => 166.97/25.05 ( 166.97/25.05 cartprodpairsurjEq => 166.97/25.05 ( 166.97/25.05 dpsetconstrI => 166.97/25.05 ( 166.97/25.05 dpsetconstrSub => 166.97/25.05 ( 166.97/25.05 setOfPairsIsBReln => 166.97/25.05 ( 166.97/25.05 !!(( 166.97/25.05 ^[Y0 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y1 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y2 : ( 166.97/25.05 $i * $i) > $o]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y3 : $i]: 166.97/25.05 (in(Y3, 166.97/25.05 Y0) => 166.97/25.05 (!!(( 166.97/25.05 ^[Y4 : $i]: 166.97/25.05 (in(Y4, 166.97/25.05 Y1) => 166.97/25.05 (in( 166.97/25.05 kpair( 166.97/25.05 Y3, Y4), 166.97/25.05 dpsetconstr( 166.97/25.05 Y0, Y1, 166.97/25.05 Y2)) => 166.97/25.05 Y2(Y3, 166.97/25.05 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.05 inference('lazy_cnf_imply', [status(thm)], ['24'])). 166.97/25.05 tff('26', plain, 166.97/25.05 ~ (nonemptyE1 => 166.97/25.05 (nonemptyI => 166.97/25.05 (nonemptyI1 => 166.97/25.05 (setadjoinIL => 166.97/25.05 (emptyinunitempty => 166.97/25.05 (setadjoinIR => 166.97/25.05 (setadjoinE => 166.97/25.05 (setadjoinOr => 166.97/25.05 (setoftrueEq => 166.97/25.05 (powersetI => 166.97/25.05 (emptyinPowerset => 166.97/25.05 (emptyInPowerset => 166.97/25.05 (powersetE => 166.97/25.05 (setunionI => 166.97/25.05 (setunionE => 166.97/25.05 (subPowSU => 166.97/25.05 (exuE2 => 166.97/25.05 (nonemptyImpWitness => 166.97/25.05 (uniqinunit => 166.97/25.05 (notinsingleton => 166.97/25.05 (eqinunit => 166.97/25.05 (singletonsswitch => 166.97/25.05 (upairsetE => 166.97/25.05 (upairsetIL => 166.97/25.05 (upairsetIR => 166.97/25.05 (emptyE1 => 166.97/25.05 (vacuousDall => 166.97/25.05 (quantDeMorgan1 => 166.97/25.05 (quantDeMorgan2 => 166.97/25.05 (quantDeMorgan3 => 166.97/25.05 (quantDeMorgan4 => 166.97/25.05 (prop2setI => 166.97/25.05 (prop2set2propI => 166.97/25.05 (notdexE => 166.97/25.05 (notdallE => 166.97/25.05 (exuI1 => 166.97/25.05 (exuI3 => 166.97/25.05 (exuI2 => 166.97/25.05 (inCongP => 166.97/25.05 (in__Cong => 166.97/25.05 (exuE3u => 166.97/25.05 (exu__Cong => 166.97/25.05 (emptyset__Cong => 166.97/25.05 (setadjoin__Cong => 166.97/25.05 (powerset__Cong => 166.97/25.05 (setunion__Cong => 166.97/25.05 (omega__Cong => 166.97/25.05 (exuEu => 166.97/25.05 (descr__Cong => 166.97/25.05 (dsetconstr__Cong => 166.97/25.05 (subsetI1 => 166.97/25.05 (eqimpsubset2 => 166.97/25.05 (eqimpsubset1 => 166.97/25.05 (subsetI2 => 166.97/25.05 (emptysetsubset => 166.97/25.05 (subsetE => 166.97/25.05 (subsetE2 => 166.97/25.05 (notsubsetI => 166.97/25.05 (notequalI1 => 166.97/25.05 (notequalI2 => 166.97/25.05 (subsetRefl => 166.97/25.05 (subsetTrans => 166.97/25.05 ( 166.97/25.05 setadjoinSub => 166.97/25.05 ( 166.97/25.05 setadjoinSub2 => 166.97/25.05 ( 166.97/25.05 subset2powerset => 166.97/25.05 ( 166.97/25.05 setextsub => 166.97/25.05 ( 166.97/25.05 subsetemptysetimpeq => 166.97/25.05 ( 166.97/25.05 powersetI1 => 166.97/25.05 ( 166.97/25.05 powersetE1 => 166.97/25.05 ( 166.97/25.05 inPowerset => 166.97/25.05 ( 166.97/25.05 powersetsubset => 166.97/25.05 ( 166.97/25.05 sepInPowerset => 166.97/25.05 ( 166.97/25.05 sepSubset => 166.97/25.05 ( 166.97/25.05 binunionIL => 166.97/25.05 ( 166.97/25.05 upairset2IR => 166.97/25.05 ( 166.97/25.05 binunionIR => 166.97/25.05 ( 166.97/25.05 binunionEcases => 166.97/25.05 ( 166.97/25.05 binunionE => 166.97/25.05 ( 166.97/25.05 binunionLsub => 166.97/25.05 ( 166.97/25.05 binunionRsub => 166.97/25.05 ( 166.97/25.05 binintersectI => 166.97/25.05 ( 166.97/25.05 binintersectSubset5 => 166.97/25.05 ( 166.97/25.05 binintersectEL => 166.97/25.05 ( 166.97/25.05 binintersectLsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset2 => 166.97/25.05 ( 166.97/25.05 binintersectSubset3 => 166.97/25.05 ( 166.97/25.05 binintersectER => 166.97/25.05 ( 166.97/25.05 disjointsetsI1 => 166.97/25.05 ( 166.97/25.05 binintersectRsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset4 => 166.97/25.05 ( 166.97/25.05 binintersectSubset1 => 166.97/25.05 ( 166.97/25.05 bs114d => 166.97/25.05 ( 166.97/25.05 setminusI => 166.97/25.05 ( 166.97/25.05 setminusEL => 166.97/25.05 ( 166.97/25.05 setminusER => 166.97/25.05 ( 166.97/25.05 setminusSubset2 => 166.97/25.05 ( 166.97/25.05 setminusERneg => 166.97/25.05 ( 166.97/25.05 setminusELneg => 166.97/25.05 ( 166.97/25.05 setminusILneg => 166.97/25.05 ( 166.97/25.05 setminusIRneg => 166.97/25.05 ( 166.97/25.05 setminusLsub => 166.97/25.05 ( 166.97/25.05 setminusSubset1 => 166.97/25.05 ( 166.97/25.05 symdiffE => 166.97/25.05 ( 166.97/25.05 symdiffI1 => 166.97/25.05 ( 166.97/25.05 symdiffI2 => 166.97/25.05 ( 166.97/25.05 symdiffIneg1 => 166.97/25.05 ( 166.97/25.05 symdiffIneg2 => 166.97/25.05 ( 166.97/25.05 secondinupair => 166.97/25.05 ( 166.97/25.05 setukpairIL => 166.97/25.05 ( 166.97/25.05 setukpairIR => 166.97/25.05 ( 166.97/25.05 kpairiskpair => 166.97/25.05 ( 166.97/25.05 kpairp => 166.97/25.05 ( 166.97/25.05 singletonsubset => 166.97/25.05 ( 166.97/25.05 singletoninpowerset => 166.97/25.05 ( 166.97/25.05 singletoninpowunion => 166.97/25.05 ( 166.97/25.05 upairset2E => 166.97/25.05 ( 166.97/25.05 upairsubunion => 166.97/25.05 ( 166.97/25.05 upairinpowunion => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem1 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem2 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem3 => 166.97/25.05 ( 166.97/25.05 cartprodpairin => 166.97/25.05 ( 166.97/25.05 cartprodmempair1 => 166.97/25.05 ( 166.97/25.05 cartprodmempair => 166.97/25.05 ( 166.97/25.05 setunionE2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton1 => 166.97/25.05 ( 166.97/25.05 setunionsingleton2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton => 166.97/25.05 ( 166.97/25.05 singletonprop => 166.97/25.05 ( 166.97/25.05 ex1E1 => 166.97/25.05 ( 166.97/25.05 ex1I => 166.97/25.05 ( 166.97/25.05 ex1I2 => 166.97/25.05 ( 166.97/25.05 singletonsuniq => 166.97/25.05 ( 166.97/25.05 setukpairinjL1 => 166.97/25.05 ( 166.97/25.05 kfstsingleton => 166.97/25.05 ( 166.97/25.05 theprop => 166.97/25.05 ( 166.97/25.05 kfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodfstin => 166.97/25.05 ( 166.97/25.05 setukpairinjL2 => 166.97/25.05 ( 166.97/25.05 setukpairinjL => 166.97/25.05 ( 166.97/25.05 setukpairinjR11 => 166.97/25.05 ( 166.97/25.05 setukpairinjR12 => 166.97/25.05 ( 166.97/25.05 setukpairinjR1 => 166.97/25.05 ( 166.97/25.05 upairequniteq => 166.97/25.05 ( 166.97/25.05 setukpairinjR2 => 166.97/25.05 ( 166.97/25.05 setukpairinjR => 166.97/25.05 ( 166.97/25.05 ksndsingleton => 166.97/25.05 ( 166.97/25.05 ksndpairEq => 166.97/25.05 ( 166.97/25.05 kpairsurjEq => 166.97/25.05 ( 166.97/25.05 cartprodsndin => 166.97/25.05 ( 166.97/25.05 cartprodpairmemEL => 166.97/25.05 ( 166.97/25.05 cartprodpairmemER => 166.97/25.05 ( 166.97/25.05 cartprodmempaircEq => 166.97/25.05 ( 166.97/25.05 cartprodfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodsndpairEq => 166.97/25.05 ( 166.97/25.05 cartprodpairsurjEq => 166.97/25.05 ( 166.97/25.05 dpsetconstrI => 166.97/25.05 ( 166.97/25.05 dpsetconstrSub => 166.97/25.05 ( 166.97/25.05 setOfPairsIsBReln => 166.97/25.05 ( 166.97/25.05 !!(( 166.97/25.05 ^[Y0 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y1 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y2 : ( 166.97/25.05 $i * $i) > $o]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y3 : $i]: 166.97/25.05 (in(Y3, 166.97/25.05 Y0) => 166.97/25.05 (!!(( 166.97/25.05 ^[Y4 : $i]: 166.97/25.05 (in(Y4, 166.97/25.05 Y1) => 166.97/25.05 (in( 166.97/25.05 kpair( 166.97/25.05 Y3, Y4), 166.97/25.05 dpsetconstr( 166.97/25.05 Y0, Y1, 166.97/25.05 Y2)) => 166.97/25.05 Y2(Y3, 166.97/25.05 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.05 inference('lazy_cnf_imply', [status(thm)], ['25'])). 166.97/25.05 tff('27', plain, 166.97/25.05 ~ (nonemptyI => 166.97/25.05 (nonemptyI1 => 166.97/25.05 (setadjoinIL => 166.97/25.05 (emptyinunitempty => 166.97/25.05 (setadjoinIR => 166.97/25.05 (setadjoinE => 166.97/25.05 (setadjoinOr => 166.97/25.05 (setoftrueEq => 166.97/25.05 (powersetI => 166.97/25.05 (emptyinPowerset => 166.97/25.05 (emptyInPowerset => 166.97/25.05 (powersetE => 166.97/25.05 (setunionI => 166.97/25.05 (setunionE => 166.97/25.05 (subPowSU => 166.97/25.05 (exuE2 => 166.97/25.05 (nonemptyImpWitness => 166.97/25.05 (uniqinunit => 166.97/25.05 (notinsingleton => 166.97/25.05 (eqinunit => 166.97/25.05 (singletonsswitch => 166.97/25.05 (upairsetE => 166.97/25.05 (upairsetIL => 166.97/25.05 (upairsetIR => 166.97/25.05 (emptyE1 => 166.97/25.05 (vacuousDall => 166.97/25.05 (quantDeMorgan1 => 166.97/25.05 (quantDeMorgan2 => 166.97/25.05 (quantDeMorgan3 => 166.97/25.05 (quantDeMorgan4 => 166.97/25.05 (prop2setI => 166.97/25.05 (prop2set2propI => 166.97/25.05 (notdexE => 166.97/25.05 (notdallE => 166.97/25.05 (exuI1 => 166.97/25.05 (exuI3 => 166.97/25.05 (exuI2 => 166.97/25.05 (inCongP => 166.97/25.05 (in__Cong => 166.97/25.05 (exuE3u => 166.97/25.05 (exu__Cong => 166.97/25.05 (emptyset__Cong => 166.97/25.05 (setadjoin__Cong => 166.97/25.05 (powerset__Cong => 166.97/25.05 (setunion__Cong => 166.97/25.05 (omega__Cong => 166.97/25.05 (exuEu => 166.97/25.05 (descr__Cong => 166.97/25.05 (dsetconstr__Cong => 166.97/25.05 (subsetI1 => 166.97/25.05 (eqimpsubset2 => 166.97/25.05 (eqimpsubset1 => 166.97/25.05 (subsetI2 => 166.97/25.05 (emptysetsubset => 166.97/25.05 (subsetE => 166.97/25.05 (subsetE2 => 166.97/25.05 (notsubsetI => 166.97/25.05 (notequalI1 => 166.97/25.05 (notequalI2 => 166.97/25.05 (subsetRefl => 166.97/25.05 (subsetTrans => 166.97/25.05 (setadjoinSub => 166.97/25.05 ( 166.97/25.05 setadjoinSub2 => 166.97/25.05 ( 166.97/25.05 subset2powerset => 166.97/25.05 ( 166.97/25.05 setextsub => 166.97/25.05 ( 166.97/25.05 subsetemptysetimpeq => 166.97/25.05 ( 166.97/25.05 powersetI1 => 166.97/25.05 ( 166.97/25.05 powersetE1 => 166.97/25.05 ( 166.97/25.05 inPowerset => 166.97/25.05 ( 166.97/25.05 powersetsubset => 166.97/25.05 ( 166.97/25.05 sepInPowerset => 166.97/25.05 ( 166.97/25.05 sepSubset => 166.97/25.05 ( 166.97/25.05 binunionIL => 166.97/25.05 ( 166.97/25.05 upairset2IR => 166.97/25.05 ( 166.97/25.05 binunionIR => 166.97/25.05 ( 166.97/25.05 binunionEcases => 166.97/25.05 ( 166.97/25.05 binunionE => 166.97/25.05 ( 166.97/25.05 binunionLsub => 166.97/25.05 ( 166.97/25.05 binunionRsub => 166.97/25.05 ( 166.97/25.05 binintersectI => 166.97/25.05 ( 166.97/25.05 binintersectSubset5 => 166.97/25.05 ( 166.97/25.05 binintersectEL => 166.97/25.05 ( 166.97/25.05 binintersectLsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset2 => 166.97/25.05 ( 166.97/25.05 binintersectSubset3 => 166.97/25.05 ( 166.97/25.05 binintersectER => 166.97/25.05 ( 166.97/25.05 disjointsetsI1 => 166.97/25.05 ( 166.97/25.05 binintersectRsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset4 => 166.97/25.05 ( 166.97/25.05 binintersectSubset1 => 166.97/25.05 ( 166.97/25.05 bs114d => 166.97/25.05 ( 166.97/25.05 setminusI => 166.97/25.05 ( 166.97/25.05 setminusEL => 166.97/25.05 ( 166.97/25.05 setminusER => 166.97/25.05 ( 166.97/25.05 setminusSubset2 => 166.97/25.05 ( 166.97/25.05 setminusERneg => 166.97/25.05 ( 166.97/25.05 setminusELneg => 166.97/25.05 ( 166.97/25.05 setminusILneg => 166.97/25.05 ( 166.97/25.05 setminusIRneg => 166.97/25.05 ( 166.97/25.05 setminusLsub => 166.97/25.05 ( 166.97/25.05 setminusSubset1 => 166.97/25.05 ( 166.97/25.05 symdiffE => 166.97/25.05 ( 166.97/25.05 symdiffI1 => 166.97/25.05 ( 166.97/25.05 symdiffI2 => 166.97/25.05 ( 166.97/25.05 symdiffIneg1 => 166.97/25.05 ( 166.97/25.05 symdiffIneg2 => 166.97/25.05 ( 166.97/25.05 secondinupair => 166.97/25.05 ( 166.97/25.05 setukpairIL => 166.97/25.05 ( 166.97/25.05 setukpairIR => 166.97/25.05 ( 166.97/25.05 kpairiskpair => 166.97/25.05 ( 166.97/25.05 kpairp => 166.97/25.05 ( 166.97/25.05 singletonsubset => 166.97/25.05 ( 166.97/25.05 singletoninpowerset => 166.97/25.05 ( 166.97/25.05 singletoninpowunion => 166.97/25.05 ( 166.97/25.05 upairset2E => 166.97/25.05 ( 166.97/25.05 upairsubunion => 166.97/25.05 ( 166.97/25.05 upairinpowunion => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem1 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem2 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem3 => 166.97/25.05 ( 166.97/25.05 cartprodpairin => 166.97/25.05 ( 166.97/25.05 cartprodmempair1 => 166.97/25.05 ( 166.97/25.05 cartprodmempair => 166.97/25.05 ( 166.97/25.05 setunionE2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton1 => 166.97/25.05 ( 166.97/25.05 setunionsingleton2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton => 166.97/25.05 ( 166.97/25.05 singletonprop => 166.97/25.05 ( 166.97/25.05 ex1E1 => 166.97/25.05 ( 166.97/25.05 ex1I => 166.97/25.05 ( 166.97/25.05 ex1I2 => 166.97/25.05 ( 166.97/25.05 singletonsuniq => 166.97/25.05 ( 166.97/25.05 setukpairinjL1 => 166.97/25.05 ( 166.97/25.05 kfstsingleton => 166.97/25.05 ( 166.97/25.05 theprop => 166.97/25.05 ( 166.97/25.05 kfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodfstin => 166.97/25.05 ( 166.97/25.05 setukpairinjL2 => 166.97/25.05 ( 166.97/25.05 setukpairinjL => 166.97/25.05 ( 166.97/25.05 setukpairinjR11 => 166.97/25.05 ( 166.97/25.05 setukpairinjR12 => 166.97/25.05 ( 166.97/25.05 setukpairinjR1 => 166.97/25.05 ( 166.97/25.05 upairequniteq => 166.97/25.05 ( 166.97/25.05 setukpairinjR2 => 166.97/25.05 ( 166.97/25.05 setukpairinjR => 166.97/25.05 ( 166.97/25.05 ksndsingleton => 166.97/25.05 ( 166.97/25.05 ksndpairEq => 166.97/25.05 ( 166.97/25.05 kpairsurjEq => 166.97/25.05 ( 166.97/25.05 cartprodsndin => 166.97/25.05 ( 166.97/25.05 cartprodpairmemEL => 166.97/25.05 ( 166.97/25.05 cartprodpairmemER => 166.97/25.05 ( 166.97/25.05 cartprodmempaircEq => 166.97/25.05 ( 166.97/25.05 cartprodfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodsndpairEq => 166.97/25.05 ( 166.97/25.05 cartprodpairsurjEq => 166.97/25.05 ( 166.97/25.05 dpsetconstrI => 166.97/25.05 ( 166.97/25.05 dpsetconstrSub => 166.97/25.05 ( 166.97/25.05 setOfPairsIsBReln => 166.97/25.05 ( 166.97/25.05 !!(( 166.97/25.05 ^[Y0 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y1 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y2 : ( 166.97/25.05 $i * $i) > $o]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y3 : $i]: 166.97/25.05 (in(Y3, 166.97/25.05 Y0) => 166.97/25.05 (!!(( 166.97/25.05 ^[Y4 : $i]: 166.97/25.05 (in(Y4, 166.97/25.05 Y1) => 166.97/25.05 (in( 166.97/25.05 kpair( 166.97/25.05 Y3, Y4), 166.97/25.05 dpsetconstr( 166.97/25.05 Y0, Y1, 166.97/25.05 Y2)) => 166.97/25.05 Y2(Y3, 166.97/25.05 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.05 inference('lazy_cnf_imply', [status(thm)], ['26'])). 166.97/25.05 tff('28', plain, 166.97/25.05 ~ (nonemptyI1 => 166.97/25.05 (setadjoinIL => 166.97/25.05 (emptyinunitempty => 166.97/25.05 (setadjoinIR => 166.97/25.05 (setadjoinE => 166.97/25.05 (setadjoinOr => 166.97/25.05 (setoftrueEq => 166.97/25.05 (powersetI => 166.97/25.05 (emptyinPowerset => 166.97/25.05 (emptyInPowerset => 166.97/25.05 (powersetE => 166.97/25.05 (setunionI => 166.97/25.05 (setunionE => 166.97/25.05 (subPowSU => 166.97/25.05 (exuE2 => 166.97/25.05 (nonemptyImpWitness => 166.97/25.05 (uniqinunit => 166.97/25.05 (notinsingleton => 166.97/25.05 (eqinunit => 166.97/25.05 (singletonsswitch => 166.97/25.05 (upairsetE => 166.97/25.05 (upairsetIL => 166.97/25.05 (upairsetIR => 166.97/25.05 (emptyE1 => 166.97/25.05 (vacuousDall => 166.97/25.05 (quantDeMorgan1 => 166.97/25.05 (quantDeMorgan2 => 166.97/25.05 (quantDeMorgan3 => 166.97/25.05 (quantDeMorgan4 => 166.97/25.05 (prop2setI => 166.97/25.05 (prop2set2propI => 166.97/25.05 (notdexE => 166.97/25.05 (notdallE => 166.97/25.05 (exuI1 => 166.97/25.05 (exuI3 => 166.97/25.05 (exuI2 => 166.97/25.05 (inCongP => 166.97/25.05 (in__Cong => 166.97/25.05 (exuE3u => 166.97/25.05 (exu__Cong => 166.97/25.05 (emptyset__Cong => 166.97/25.05 (setadjoin__Cong => 166.97/25.05 (powerset__Cong => 166.97/25.05 (setunion__Cong => 166.97/25.05 (omega__Cong => 166.97/25.05 (exuEu => 166.97/25.05 (descr__Cong => 166.97/25.05 (dsetconstr__Cong => 166.97/25.05 (subsetI1 => 166.97/25.05 (eqimpsubset2 => 166.97/25.05 (eqimpsubset1 => 166.97/25.05 (subsetI2 => 166.97/25.05 (emptysetsubset => 166.97/25.05 (subsetE => 166.97/25.05 (subsetE2 => 166.97/25.05 (notsubsetI => 166.97/25.05 (notequalI1 => 166.97/25.05 (notequalI2 => 166.97/25.05 (subsetRefl => 166.97/25.05 (subsetTrans => 166.97/25.05 (setadjoinSub => 166.97/25.05 (setadjoinSub2 => 166.97/25.05 ( 166.97/25.05 subset2powerset => 166.97/25.05 ( 166.97/25.05 setextsub => 166.97/25.05 ( 166.97/25.05 subsetemptysetimpeq => 166.97/25.05 ( 166.97/25.05 powersetI1 => 166.97/25.05 ( 166.97/25.05 powersetE1 => 166.97/25.05 ( 166.97/25.05 inPowerset => 166.97/25.05 ( 166.97/25.05 powersetsubset => 166.97/25.05 ( 166.97/25.05 sepInPowerset => 166.97/25.05 ( 166.97/25.05 sepSubset => 166.97/25.05 ( 166.97/25.05 binunionIL => 166.97/25.05 ( 166.97/25.05 upairset2IR => 166.97/25.05 ( 166.97/25.05 binunionIR => 166.97/25.05 ( 166.97/25.05 binunionEcases => 166.97/25.05 ( 166.97/25.05 binunionE => 166.97/25.05 ( 166.97/25.05 binunionLsub => 166.97/25.05 ( 166.97/25.05 binunionRsub => 166.97/25.05 ( 166.97/25.05 binintersectI => 166.97/25.05 ( 166.97/25.05 binintersectSubset5 => 166.97/25.05 ( 166.97/25.05 binintersectEL => 166.97/25.05 ( 166.97/25.05 binintersectLsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset2 => 166.97/25.05 ( 166.97/25.05 binintersectSubset3 => 166.97/25.05 ( 166.97/25.05 binintersectER => 166.97/25.05 ( 166.97/25.05 disjointsetsI1 => 166.97/25.05 ( 166.97/25.05 binintersectRsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset4 => 166.97/25.05 ( 166.97/25.05 binintersectSubset1 => 166.97/25.05 ( 166.97/25.05 bs114d => 166.97/25.05 ( 166.97/25.05 setminusI => 166.97/25.05 ( 166.97/25.05 setminusEL => 166.97/25.05 ( 166.97/25.05 setminusER => 166.97/25.05 ( 166.97/25.05 setminusSubset2 => 166.97/25.05 ( 166.97/25.05 setminusERneg => 166.97/25.05 ( 166.97/25.05 setminusELneg => 166.97/25.05 ( 166.97/25.05 setminusILneg => 166.97/25.05 ( 166.97/25.05 setminusIRneg => 166.97/25.05 ( 166.97/25.05 setminusLsub => 166.97/25.05 ( 166.97/25.05 setminusSubset1 => 166.97/25.05 ( 166.97/25.05 symdiffE => 166.97/25.05 ( 166.97/25.05 symdiffI1 => 166.97/25.05 ( 166.97/25.05 symdiffI2 => 166.97/25.05 ( 166.97/25.05 symdiffIneg1 => 166.97/25.05 ( 166.97/25.05 symdiffIneg2 => 166.97/25.05 ( 166.97/25.05 secondinupair => 166.97/25.05 ( 166.97/25.05 setukpairIL => 166.97/25.05 ( 166.97/25.05 setukpairIR => 166.97/25.05 ( 166.97/25.05 kpairiskpair => 166.97/25.05 ( 166.97/25.05 kpairp => 166.97/25.05 ( 166.97/25.05 singletonsubset => 166.97/25.05 ( 166.97/25.05 singletoninpowerset => 166.97/25.05 ( 166.97/25.05 singletoninpowunion => 166.97/25.05 ( 166.97/25.05 upairset2E => 166.97/25.05 ( 166.97/25.05 upairsubunion => 166.97/25.05 ( 166.97/25.05 upairinpowunion => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem1 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem2 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem3 => 166.97/25.05 ( 166.97/25.05 cartprodpairin => 166.97/25.05 ( 166.97/25.05 cartprodmempair1 => 166.97/25.05 ( 166.97/25.05 cartprodmempair => 166.97/25.05 ( 166.97/25.05 setunionE2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton1 => 166.97/25.05 ( 166.97/25.05 setunionsingleton2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton => 166.97/25.05 ( 166.97/25.05 singletonprop => 166.97/25.05 ( 166.97/25.05 ex1E1 => 166.97/25.05 ( 166.97/25.05 ex1I => 166.97/25.05 ( 166.97/25.05 ex1I2 => 166.97/25.05 ( 166.97/25.05 singletonsuniq => 166.97/25.05 ( 166.97/25.05 setukpairinjL1 => 166.97/25.05 ( 166.97/25.05 kfstsingleton => 166.97/25.05 ( 166.97/25.05 theprop => 166.97/25.05 ( 166.97/25.05 kfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodfstin => 166.97/25.05 ( 166.97/25.05 setukpairinjL2 => 166.97/25.05 ( 166.97/25.05 setukpairinjL => 166.97/25.05 ( 166.97/25.05 setukpairinjR11 => 166.97/25.05 ( 166.97/25.05 setukpairinjR12 => 166.97/25.05 ( 166.97/25.05 setukpairinjR1 => 166.97/25.05 ( 166.97/25.05 upairequniteq => 166.97/25.05 ( 166.97/25.05 setukpairinjR2 => 166.97/25.05 ( 166.97/25.05 setukpairinjR => 166.97/25.05 ( 166.97/25.05 ksndsingleton => 166.97/25.05 ( 166.97/25.05 ksndpairEq => 166.97/25.05 ( 166.97/25.05 kpairsurjEq => 166.97/25.05 ( 166.97/25.05 cartprodsndin => 166.97/25.05 ( 166.97/25.05 cartprodpairmemEL => 166.97/25.05 ( 166.97/25.05 cartprodpairmemER => 166.97/25.05 ( 166.97/25.05 cartprodmempaircEq => 166.97/25.05 ( 166.97/25.05 cartprodfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodsndpairEq => 166.97/25.05 ( 166.97/25.05 cartprodpairsurjEq => 166.97/25.05 ( 166.97/25.05 dpsetconstrI => 166.97/25.05 ( 166.97/25.05 dpsetconstrSub => 166.97/25.05 ( 166.97/25.05 setOfPairsIsBReln => 166.97/25.05 ( 166.97/25.05 !!(( 166.97/25.05 ^[Y0 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y1 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y2 : ( 166.97/25.05 $i * $i) > $o]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y3 : $i]: 166.97/25.05 (in(Y3, 166.97/25.05 Y0) => 166.97/25.05 (!!(( 166.97/25.05 ^[Y4 : $i]: 166.97/25.05 (in(Y4, 166.97/25.05 Y1) => 166.97/25.05 (in( 166.97/25.05 kpair( 166.97/25.05 Y3, Y4), 166.97/25.05 dpsetconstr( 166.97/25.05 Y0, Y1, 166.97/25.05 Y2)) => 166.97/25.05 Y2(Y3, 166.97/25.05 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.05 inference('lazy_cnf_imply', [status(thm)], ['27'])). 166.97/25.05 tff('29', plain, 166.97/25.05 ~ (setadjoinIL => 166.97/25.05 (emptyinunitempty => 166.97/25.05 (setadjoinIR => 166.97/25.05 (setadjoinE => 166.97/25.05 (setadjoinOr => 166.97/25.05 (setoftrueEq => 166.97/25.05 (powersetI => 166.97/25.05 (emptyinPowerset => 166.97/25.05 (emptyInPowerset => 166.97/25.05 (powersetE => 166.97/25.05 (setunionI => 166.97/25.05 (setunionE => 166.97/25.05 (subPowSU => 166.97/25.05 (exuE2 => 166.97/25.05 (nonemptyImpWitness => 166.97/25.05 (uniqinunit => 166.97/25.05 (notinsingleton => 166.97/25.05 (eqinunit => 166.97/25.05 (singletonsswitch => 166.97/25.05 (upairsetE => 166.97/25.05 (upairsetIL => 166.97/25.05 (upairsetIR => 166.97/25.05 (emptyE1 => 166.97/25.05 (vacuousDall => 166.97/25.05 (quantDeMorgan1 => 166.97/25.05 (quantDeMorgan2 => 166.97/25.05 (quantDeMorgan3 => 166.97/25.05 (quantDeMorgan4 => 166.97/25.05 (prop2setI => 166.97/25.05 (prop2set2propI => 166.97/25.05 (notdexE => 166.97/25.05 (notdallE => 166.97/25.05 (exuI1 => 166.97/25.05 (exuI3 => 166.97/25.05 (exuI2 => 166.97/25.05 (inCongP => 166.97/25.05 (in__Cong => 166.97/25.05 (exuE3u => 166.97/25.05 (exu__Cong => 166.97/25.05 (emptyset__Cong => 166.97/25.05 (setadjoin__Cong => 166.97/25.05 (powerset__Cong => 166.97/25.05 (setunion__Cong => 166.97/25.05 (omega__Cong => 166.97/25.05 (exuEu => 166.97/25.05 (descr__Cong => 166.97/25.05 (dsetconstr__Cong => 166.97/25.05 (subsetI1 => 166.97/25.05 (eqimpsubset2 => 166.97/25.05 (eqimpsubset1 => 166.97/25.05 (subsetI2 => 166.97/25.05 (emptysetsubset => 166.97/25.05 (subsetE => 166.97/25.05 (subsetE2 => 166.97/25.05 (notsubsetI => 166.97/25.05 (notequalI1 => 166.97/25.05 (notequalI2 => 166.97/25.05 (subsetRefl => 166.97/25.05 (subsetTrans => 166.97/25.05 (setadjoinSub => 166.97/25.05 (setadjoinSub2 => 166.97/25.05 (subset2powerset => 166.97/25.05 ( 166.97/25.05 setextsub => 166.97/25.05 ( 166.97/25.05 subsetemptysetimpeq => 166.97/25.05 ( 166.97/25.05 powersetI1 => 166.97/25.05 ( 166.97/25.05 powersetE1 => 166.97/25.05 ( 166.97/25.05 inPowerset => 166.97/25.05 ( 166.97/25.05 powersetsubset => 166.97/25.05 ( 166.97/25.05 sepInPowerset => 166.97/25.05 ( 166.97/25.05 sepSubset => 166.97/25.05 ( 166.97/25.05 binunionIL => 166.97/25.05 ( 166.97/25.05 upairset2IR => 166.97/25.05 ( 166.97/25.05 binunionIR => 166.97/25.05 ( 166.97/25.05 binunionEcases => 166.97/25.05 ( 166.97/25.05 binunionE => 166.97/25.05 ( 166.97/25.05 binunionLsub => 166.97/25.05 ( 166.97/25.05 binunionRsub => 166.97/25.05 ( 166.97/25.05 binintersectI => 166.97/25.05 ( 166.97/25.05 binintersectSubset5 => 166.97/25.05 ( 166.97/25.05 binintersectEL => 166.97/25.05 ( 166.97/25.05 binintersectLsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset2 => 166.97/25.05 ( 166.97/25.05 binintersectSubset3 => 166.97/25.05 ( 166.97/25.05 binintersectER => 166.97/25.05 ( 166.97/25.05 disjointsetsI1 => 166.97/25.05 ( 166.97/25.05 binintersectRsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset4 => 166.97/25.05 ( 166.97/25.05 binintersectSubset1 => 166.97/25.05 ( 166.97/25.05 bs114d => 166.97/25.05 ( 166.97/25.05 setminusI => 166.97/25.05 ( 166.97/25.05 setminusEL => 166.97/25.05 ( 166.97/25.05 setminusER => 166.97/25.05 ( 166.97/25.05 setminusSubset2 => 166.97/25.05 ( 166.97/25.05 setminusERneg => 166.97/25.05 ( 166.97/25.05 setminusELneg => 166.97/25.05 ( 166.97/25.05 setminusILneg => 166.97/25.05 ( 166.97/25.05 setminusIRneg => 166.97/25.05 ( 166.97/25.05 setminusLsub => 166.97/25.05 ( 166.97/25.05 setminusSubset1 => 166.97/25.05 ( 166.97/25.05 symdiffE => 166.97/25.05 ( 166.97/25.05 symdiffI1 => 166.97/25.05 ( 166.97/25.05 symdiffI2 => 166.97/25.05 ( 166.97/25.05 symdiffIneg1 => 166.97/25.05 ( 166.97/25.05 symdiffIneg2 => 166.97/25.05 ( 166.97/25.05 secondinupair => 166.97/25.05 ( 166.97/25.05 setukpairIL => 166.97/25.05 ( 166.97/25.05 setukpairIR => 166.97/25.05 ( 166.97/25.05 kpairiskpair => 166.97/25.05 ( 166.97/25.05 kpairp => 166.97/25.05 ( 166.97/25.05 singletonsubset => 166.97/25.05 ( 166.97/25.05 singletoninpowerset => 166.97/25.05 ( 166.97/25.05 singletoninpowunion => 166.97/25.05 ( 166.97/25.05 upairset2E => 166.97/25.05 ( 166.97/25.05 upairsubunion => 166.97/25.05 ( 166.97/25.05 upairinpowunion => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem1 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem2 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem3 => 166.97/25.05 ( 166.97/25.05 cartprodpairin => 166.97/25.05 ( 166.97/25.05 cartprodmempair1 => 166.97/25.05 ( 166.97/25.05 cartprodmempair => 166.97/25.05 ( 166.97/25.05 setunionE2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton1 => 166.97/25.05 ( 166.97/25.05 setunionsingleton2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton => 166.97/25.05 ( 166.97/25.05 singletonprop => 166.97/25.05 ( 166.97/25.05 ex1E1 => 166.97/25.05 ( 166.97/25.05 ex1I => 166.97/25.05 ( 166.97/25.05 ex1I2 => 166.97/25.05 ( 166.97/25.05 singletonsuniq => 166.97/25.05 ( 166.97/25.05 setukpairinjL1 => 166.97/25.05 ( 166.97/25.05 kfstsingleton => 166.97/25.05 ( 166.97/25.05 theprop => 166.97/25.05 ( 166.97/25.05 kfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodfstin => 166.97/25.05 ( 166.97/25.05 setukpairinjL2 => 166.97/25.05 ( 166.97/25.05 setukpairinjL => 166.97/25.05 ( 166.97/25.05 setukpairinjR11 => 166.97/25.05 ( 166.97/25.05 setukpairinjR12 => 166.97/25.05 ( 166.97/25.05 setukpairinjR1 => 166.97/25.05 ( 166.97/25.05 upairequniteq => 166.97/25.05 ( 166.97/25.05 setukpairinjR2 => 166.97/25.05 ( 166.97/25.05 setukpairinjR => 166.97/25.05 ( 166.97/25.05 ksndsingleton => 166.97/25.05 ( 166.97/25.05 ksndpairEq => 166.97/25.05 ( 166.97/25.05 kpairsurjEq => 166.97/25.05 ( 166.97/25.05 cartprodsndin => 166.97/25.05 ( 166.97/25.05 cartprodpairmemEL => 166.97/25.05 ( 166.97/25.05 cartprodpairmemER => 166.97/25.05 ( 166.97/25.05 cartprodmempaircEq => 166.97/25.05 ( 166.97/25.05 cartprodfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodsndpairEq => 166.97/25.05 ( 166.97/25.05 cartprodpairsurjEq => 166.97/25.05 ( 166.97/25.05 dpsetconstrI => 166.97/25.05 ( 166.97/25.05 dpsetconstrSub => 166.97/25.05 ( 166.97/25.05 setOfPairsIsBReln => 166.97/25.05 ( 166.97/25.05 !!(( 166.97/25.05 ^[Y0 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y1 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y2 : ( 166.97/25.05 $i * $i) > $o]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y3 : $i]: 166.97/25.05 (in(Y3, 166.97/25.05 Y0) => 166.97/25.05 (!!(( 166.97/25.05 ^[Y4 : $i]: 166.97/25.05 (in(Y4, 166.97/25.05 Y1) => 166.97/25.05 (in( 166.97/25.05 kpair( 166.97/25.05 Y3, Y4), 166.97/25.05 dpsetconstr( 166.97/25.05 Y0, Y1, 166.97/25.05 Y2)) => 166.97/25.05 Y2(Y3, 166.97/25.05 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.05 inference('lazy_cnf_imply', [status(thm)], ['28'])). 166.97/25.05 tff('30', plain, 166.97/25.05 ~ (emptyinunitempty => 166.97/25.05 (setadjoinIR => 166.97/25.05 (setadjoinE => 166.97/25.05 (setadjoinOr => 166.97/25.05 (setoftrueEq => 166.97/25.05 (powersetI => 166.97/25.05 (emptyinPowerset => 166.97/25.05 (emptyInPowerset => 166.97/25.05 (powersetE => 166.97/25.05 (setunionI => 166.97/25.05 (setunionE => 166.97/25.05 (subPowSU => 166.97/25.05 (exuE2 => 166.97/25.05 (nonemptyImpWitness => 166.97/25.05 (uniqinunit => 166.97/25.05 (notinsingleton => 166.97/25.05 (eqinunit => 166.97/25.05 (singletonsswitch => 166.97/25.05 (upairsetE => 166.97/25.05 (upairsetIL => 166.97/25.05 (upairsetIR => 166.97/25.05 (emptyE1 => 166.97/25.05 (vacuousDall => 166.97/25.05 (quantDeMorgan1 => 166.97/25.05 (quantDeMorgan2 => 166.97/25.05 (quantDeMorgan3 => 166.97/25.05 (quantDeMorgan4 => 166.97/25.05 (prop2setI => 166.97/25.05 (prop2set2propI => 166.97/25.05 (notdexE => 166.97/25.05 (notdallE => 166.97/25.05 (exuI1 => 166.97/25.05 (exuI3 => 166.97/25.05 (exuI2 => 166.97/25.05 (inCongP => 166.97/25.05 (in__Cong => 166.97/25.05 (exuE3u => 166.97/25.05 (exu__Cong => 166.97/25.05 (emptyset__Cong => 166.97/25.05 (setadjoin__Cong => 166.97/25.05 (powerset__Cong => 166.97/25.05 (setunion__Cong => 166.97/25.05 (omega__Cong => 166.97/25.05 (exuEu => 166.97/25.05 (descr__Cong => 166.97/25.05 (dsetconstr__Cong => 166.97/25.05 (subsetI1 => 166.97/25.05 (eqimpsubset2 => 166.97/25.05 (eqimpsubset1 => 166.97/25.05 (subsetI2 => 166.97/25.05 (emptysetsubset => 166.97/25.05 (subsetE => 166.97/25.05 (subsetE2 => 166.97/25.05 (notsubsetI => 166.97/25.05 (notequalI1 => 166.97/25.05 (notequalI2 => 166.97/25.05 (subsetRefl => 166.97/25.05 (subsetTrans => 166.97/25.05 (setadjoinSub => 166.97/25.05 (setadjoinSub2 => 166.97/25.05 (subset2powerset => 166.97/25.05 (setextsub => 166.97/25.05 ( 166.97/25.05 subsetemptysetimpeq => 166.97/25.05 ( 166.97/25.05 powersetI1 => 166.97/25.05 ( 166.97/25.05 powersetE1 => 166.97/25.05 ( 166.97/25.05 inPowerset => 166.97/25.05 ( 166.97/25.05 powersetsubset => 166.97/25.05 ( 166.97/25.05 sepInPowerset => 166.97/25.05 ( 166.97/25.05 sepSubset => 166.97/25.05 ( 166.97/25.05 binunionIL => 166.97/25.05 ( 166.97/25.05 upairset2IR => 166.97/25.05 ( 166.97/25.05 binunionIR => 166.97/25.05 ( 166.97/25.05 binunionEcases => 166.97/25.05 ( 166.97/25.05 binunionE => 166.97/25.05 ( 166.97/25.05 binunionLsub => 166.97/25.05 ( 166.97/25.05 binunionRsub => 166.97/25.05 ( 166.97/25.05 binintersectI => 166.97/25.05 ( 166.97/25.05 binintersectSubset5 => 166.97/25.05 ( 166.97/25.05 binintersectEL => 166.97/25.05 ( 166.97/25.05 binintersectLsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset2 => 166.97/25.05 ( 166.97/25.05 binintersectSubset3 => 166.97/25.05 ( 166.97/25.05 binintersectER => 166.97/25.05 ( 166.97/25.05 disjointsetsI1 => 166.97/25.05 ( 166.97/25.05 binintersectRsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset4 => 166.97/25.05 ( 166.97/25.05 binintersectSubset1 => 166.97/25.05 ( 166.97/25.05 bs114d => 166.97/25.05 ( 166.97/25.05 setminusI => 166.97/25.05 ( 166.97/25.05 setminusEL => 166.97/25.05 ( 166.97/25.05 setminusER => 166.97/25.05 ( 166.97/25.05 setminusSubset2 => 166.97/25.05 ( 166.97/25.05 setminusERneg => 166.97/25.05 ( 166.97/25.05 setminusELneg => 166.97/25.05 ( 166.97/25.05 setminusILneg => 166.97/25.05 ( 166.97/25.05 setminusIRneg => 166.97/25.05 ( 166.97/25.05 setminusLsub => 166.97/25.05 ( 166.97/25.05 setminusSubset1 => 166.97/25.05 ( 166.97/25.05 symdiffE => 166.97/25.05 ( 166.97/25.05 symdiffI1 => 166.97/25.05 ( 166.97/25.05 symdiffI2 => 166.97/25.05 ( 166.97/25.05 symdiffIneg1 => 166.97/25.05 ( 166.97/25.05 symdiffIneg2 => 166.97/25.05 ( 166.97/25.05 secondinupair => 166.97/25.05 ( 166.97/25.05 setukpairIL => 166.97/25.05 ( 166.97/25.05 setukpairIR => 166.97/25.05 ( 166.97/25.05 kpairiskpair => 166.97/25.05 ( 166.97/25.05 kpairp => 166.97/25.05 ( 166.97/25.05 singletonsubset => 166.97/25.05 ( 166.97/25.05 singletoninpowerset => 166.97/25.05 ( 166.97/25.05 singletoninpowunion => 166.97/25.05 ( 166.97/25.05 upairset2E => 166.97/25.05 ( 166.97/25.05 upairsubunion => 166.97/25.05 ( 166.97/25.05 upairinpowunion => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem1 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem2 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem3 => 166.97/25.05 ( 166.97/25.05 cartprodpairin => 166.97/25.05 ( 166.97/25.05 cartprodmempair1 => 166.97/25.05 ( 166.97/25.05 cartprodmempair => 166.97/25.05 ( 166.97/25.05 setunionE2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton1 => 166.97/25.05 ( 166.97/25.05 setunionsingleton2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton => 166.97/25.05 ( 166.97/25.05 singletonprop => 166.97/25.05 ( 166.97/25.05 ex1E1 => 166.97/25.05 ( 166.97/25.05 ex1I => 166.97/25.05 ( 166.97/25.05 ex1I2 => 166.97/25.05 ( 166.97/25.05 singletonsuniq => 166.97/25.05 ( 166.97/25.05 setukpairinjL1 => 166.97/25.05 ( 166.97/25.05 kfstsingleton => 166.97/25.05 ( 166.97/25.05 theprop => 166.97/25.05 ( 166.97/25.05 kfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodfstin => 166.97/25.05 ( 166.97/25.05 setukpairinjL2 => 166.97/25.05 ( 166.97/25.05 setukpairinjL => 166.97/25.05 ( 166.97/25.05 setukpairinjR11 => 166.97/25.05 ( 166.97/25.05 setukpairinjR12 => 166.97/25.05 ( 166.97/25.05 setukpairinjR1 => 166.97/25.05 ( 166.97/25.05 upairequniteq => 166.97/25.05 ( 166.97/25.05 setukpairinjR2 => 166.97/25.05 ( 166.97/25.05 setukpairinjR => 166.97/25.05 ( 166.97/25.05 ksndsingleton => 166.97/25.05 ( 166.97/25.05 ksndpairEq => 166.97/25.05 ( 166.97/25.05 kpairsurjEq => 166.97/25.05 ( 166.97/25.05 cartprodsndin => 166.97/25.05 ( 166.97/25.05 cartprodpairmemEL => 166.97/25.05 ( 166.97/25.05 cartprodpairmemER => 166.97/25.05 ( 166.97/25.05 cartprodmempaircEq => 166.97/25.05 ( 166.97/25.05 cartprodfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodsndpairEq => 166.97/25.05 ( 166.97/25.05 cartprodpairsurjEq => 166.97/25.05 ( 166.97/25.05 dpsetconstrI => 166.97/25.05 ( 166.97/25.05 dpsetconstrSub => 166.97/25.05 ( 166.97/25.05 setOfPairsIsBReln => 166.97/25.05 ( 166.97/25.05 !!(( 166.97/25.05 ^[Y0 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y1 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y2 : ( 166.97/25.05 $i * $i) > $o]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y3 : $i]: 166.97/25.05 (in(Y3, 166.97/25.05 Y0) => 166.97/25.05 (!!(( 166.97/25.05 ^[Y4 : $i]: 166.97/25.05 (in(Y4, 166.97/25.05 Y1) => 166.97/25.05 (in( 166.97/25.05 kpair( 166.97/25.05 Y3, Y4), 166.97/25.05 dpsetconstr( 166.97/25.05 Y0, Y1, 166.97/25.05 Y2)) => 166.97/25.05 Y2(Y3, 166.97/25.05 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.05 inference('lazy_cnf_imply', [status(thm)], ['29'])). 166.97/25.05 tff('31', plain, 166.97/25.05 ~ (setadjoinIR => 166.97/25.05 (setadjoinE => 166.97/25.05 (setadjoinOr => 166.97/25.05 (setoftrueEq => 166.97/25.05 (powersetI => 166.97/25.05 (emptyinPowerset => 166.97/25.05 (emptyInPowerset => 166.97/25.05 (powersetE => 166.97/25.05 (setunionI => 166.97/25.05 (setunionE => 166.97/25.05 (subPowSU => 166.97/25.05 (exuE2 => 166.97/25.05 (nonemptyImpWitness => 166.97/25.05 (uniqinunit => 166.97/25.05 (notinsingleton => 166.97/25.05 (eqinunit => 166.97/25.05 (singletonsswitch => 166.97/25.05 (upairsetE => 166.97/25.05 (upairsetIL => 166.97/25.05 (upairsetIR => 166.97/25.05 (emptyE1 => 166.97/25.05 (vacuousDall => 166.97/25.05 (quantDeMorgan1 => 166.97/25.05 (quantDeMorgan2 => 166.97/25.05 (quantDeMorgan3 => 166.97/25.05 (quantDeMorgan4 => 166.97/25.05 (prop2setI => 166.97/25.05 (prop2set2propI => 166.97/25.05 (notdexE => 166.97/25.05 (notdallE => 166.97/25.05 (exuI1 => 166.97/25.05 (exuI3 => 166.97/25.05 (exuI2 => 166.97/25.05 (inCongP => 166.97/25.05 (in__Cong => 166.97/25.05 (exuE3u => 166.97/25.05 (exu__Cong => 166.97/25.05 (emptyset__Cong => 166.97/25.05 (setadjoin__Cong => 166.97/25.05 (powerset__Cong => 166.97/25.05 (setunion__Cong => 166.97/25.05 (omega__Cong => 166.97/25.05 (exuEu => 166.97/25.05 (descr__Cong => 166.97/25.05 (dsetconstr__Cong => 166.97/25.05 (subsetI1 => 166.97/25.05 (eqimpsubset2 => 166.97/25.05 (eqimpsubset1 => 166.97/25.05 (subsetI2 => 166.97/25.05 (emptysetsubset => 166.97/25.05 (subsetE => 166.97/25.05 (subsetE2 => 166.97/25.05 (notsubsetI => 166.97/25.05 (notequalI1 => 166.97/25.05 (notequalI2 => 166.97/25.05 (subsetRefl => 166.97/25.05 (subsetTrans => 166.97/25.05 (setadjoinSub => 166.97/25.05 (setadjoinSub2 => 166.97/25.05 (subset2powerset => 166.97/25.05 (setextsub => 166.97/25.05 (subsetemptysetimpeq => 166.97/25.05 ( 166.97/25.05 powersetI1 => 166.97/25.05 ( 166.97/25.05 powersetE1 => 166.97/25.05 ( 166.97/25.05 inPowerset => 166.97/25.05 ( 166.97/25.05 powersetsubset => 166.97/25.05 ( 166.97/25.05 sepInPowerset => 166.97/25.05 ( 166.97/25.05 sepSubset => 166.97/25.05 ( 166.97/25.05 binunionIL => 166.97/25.05 ( 166.97/25.05 upairset2IR => 166.97/25.05 ( 166.97/25.05 binunionIR => 166.97/25.05 ( 166.97/25.05 binunionEcases => 166.97/25.05 ( 166.97/25.05 binunionE => 166.97/25.05 ( 166.97/25.05 binunionLsub => 166.97/25.05 ( 166.97/25.05 binunionRsub => 166.97/25.05 ( 166.97/25.05 binintersectI => 166.97/25.05 ( 166.97/25.05 binintersectSubset5 => 166.97/25.05 ( 166.97/25.05 binintersectEL => 166.97/25.05 ( 166.97/25.05 binintersectLsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset2 => 166.97/25.05 ( 166.97/25.05 binintersectSubset3 => 166.97/25.05 ( 166.97/25.05 binintersectER => 166.97/25.05 ( 166.97/25.05 disjointsetsI1 => 166.97/25.05 ( 166.97/25.05 binintersectRsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset4 => 166.97/25.05 ( 166.97/25.05 binintersectSubset1 => 166.97/25.05 ( 166.97/25.05 bs114d => 166.97/25.05 ( 166.97/25.05 setminusI => 166.97/25.05 ( 166.97/25.05 setminusEL => 166.97/25.05 ( 166.97/25.05 setminusER => 166.97/25.05 ( 166.97/25.05 setminusSubset2 => 166.97/25.05 ( 166.97/25.05 setminusERneg => 166.97/25.05 ( 166.97/25.05 setminusELneg => 166.97/25.05 ( 166.97/25.05 setminusILneg => 166.97/25.05 ( 166.97/25.05 setminusIRneg => 166.97/25.05 ( 166.97/25.05 setminusLsub => 166.97/25.05 ( 166.97/25.05 setminusSubset1 => 166.97/25.05 ( 166.97/25.05 symdiffE => 166.97/25.05 ( 166.97/25.05 symdiffI1 => 166.97/25.05 ( 166.97/25.05 symdiffI2 => 166.97/25.05 ( 166.97/25.05 symdiffIneg1 => 166.97/25.05 ( 166.97/25.05 symdiffIneg2 => 166.97/25.05 ( 166.97/25.05 secondinupair => 166.97/25.05 ( 166.97/25.05 setukpairIL => 166.97/25.05 ( 166.97/25.05 setukpairIR => 166.97/25.05 ( 166.97/25.05 kpairiskpair => 166.97/25.05 ( 166.97/25.05 kpairp => 166.97/25.05 ( 166.97/25.05 singletonsubset => 166.97/25.05 ( 166.97/25.05 singletoninpowerset => 166.97/25.05 ( 166.97/25.05 singletoninpowunion => 166.97/25.05 ( 166.97/25.05 upairset2E => 166.97/25.05 ( 166.97/25.05 upairsubunion => 166.97/25.05 ( 166.97/25.05 upairinpowunion => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem1 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem2 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem3 => 166.97/25.05 ( 166.97/25.05 cartprodpairin => 166.97/25.05 ( 166.97/25.05 cartprodmempair1 => 166.97/25.05 ( 166.97/25.05 cartprodmempair => 166.97/25.05 ( 166.97/25.05 setunionE2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton1 => 166.97/25.05 ( 166.97/25.05 setunionsingleton2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton => 166.97/25.05 ( 166.97/25.05 singletonprop => 166.97/25.05 ( 166.97/25.05 ex1E1 => 166.97/25.05 ( 166.97/25.05 ex1I => 166.97/25.05 ( 166.97/25.05 ex1I2 => 166.97/25.05 ( 166.97/25.05 singletonsuniq => 166.97/25.05 ( 166.97/25.05 setukpairinjL1 => 166.97/25.05 ( 166.97/25.05 kfstsingleton => 166.97/25.05 ( 166.97/25.05 theprop => 166.97/25.05 ( 166.97/25.05 kfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodfstin => 166.97/25.05 ( 166.97/25.05 setukpairinjL2 => 166.97/25.05 ( 166.97/25.05 setukpairinjL => 166.97/25.05 ( 166.97/25.05 setukpairinjR11 => 166.97/25.05 ( 166.97/25.05 setukpairinjR12 => 166.97/25.05 ( 166.97/25.05 setukpairinjR1 => 166.97/25.05 ( 166.97/25.05 upairequniteq => 166.97/25.05 ( 166.97/25.05 setukpairinjR2 => 166.97/25.05 ( 166.97/25.05 setukpairinjR => 166.97/25.05 ( 166.97/25.05 ksndsingleton => 166.97/25.05 ( 166.97/25.05 ksndpairEq => 166.97/25.05 ( 166.97/25.05 kpairsurjEq => 166.97/25.05 ( 166.97/25.05 cartprodsndin => 166.97/25.05 ( 166.97/25.05 cartprodpairmemEL => 166.97/25.05 ( 166.97/25.05 cartprodpairmemER => 166.97/25.05 ( 166.97/25.05 cartprodmempaircEq => 166.97/25.05 ( 166.97/25.05 cartprodfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodsndpairEq => 166.97/25.05 ( 166.97/25.05 cartprodpairsurjEq => 166.97/25.05 ( 166.97/25.05 dpsetconstrI => 166.97/25.05 ( 166.97/25.05 dpsetconstrSub => 166.97/25.05 ( 166.97/25.05 setOfPairsIsBReln => 166.97/25.05 ( 166.97/25.05 !!(( 166.97/25.05 ^[Y0 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y1 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y2 : ( 166.97/25.05 $i * $i) > $o]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y3 : $i]: 166.97/25.05 (in(Y3, 166.97/25.05 Y0) => 166.97/25.05 (!!(( 166.97/25.05 ^[Y4 : $i]: 166.97/25.05 (in(Y4, 166.97/25.05 Y1) => 166.97/25.05 (in( 166.97/25.05 kpair( 166.97/25.05 Y3, Y4), 166.97/25.05 dpsetconstr( 166.97/25.05 Y0, Y1, 166.97/25.05 Y2)) => 166.97/25.05 Y2(Y3, 166.97/25.05 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.05 inference('lazy_cnf_imply', [status(thm)], ['30'])). 166.97/25.05 tff('32', plain, 166.97/25.05 ~ (setadjoinE => 166.97/25.05 (setadjoinOr => 166.97/25.05 (setoftrueEq => 166.97/25.05 (powersetI => 166.97/25.05 (emptyinPowerset => 166.97/25.05 (emptyInPowerset => 166.97/25.05 (powersetE => 166.97/25.05 (setunionI => 166.97/25.05 (setunionE => 166.97/25.05 (subPowSU => 166.97/25.05 (exuE2 => 166.97/25.05 (nonemptyImpWitness => 166.97/25.05 (uniqinunit => 166.97/25.05 (notinsingleton => 166.97/25.05 (eqinunit => 166.97/25.05 (singletonsswitch => 166.97/25.05 (upairsetE => 166.97/25.05 (upairsetIL => 166.97/25.05 (upairsetIR => 166.97/25.05 (emptyE1 => 166.97/25.05 (vacuousDall => 166.97/25.05 (quantDeMorgan1 => 166.97/25.05 (quantDeMorgan2 => 166.97/25.05 (quantDeMorgan3 => 166.97/25.05 (quantDeMorgan4 => 166.97/25.05 (prop2setI => 166.97/25.05 (prop2set2propI => 166.97/25.05 (notdexE => 166.97/25.05 (notdallE => 166.97/25.05 (exuI1 => 166.97/25.05 (exuI3 => 166.97/25.05 (exuI2 => 166.97/25.05 (inCongP => 166.97/25.05 (in__Cong => 166.97/25.05 (exuE3u => 166.97/25.05 (exu__Cong => 166.97/25.05 (emptyset__Cong => 166.97/25.05 (setadjoin__Cong => 166.97/25.05 (powerset__Cong => 166.97/25.05 (setunion__Cong => 166.97/25.05 (omega__Cong => 166.97/25.05 (exuEu => 166.97/25.05 (descr__Cong => 166.97/25.05 (dsetconstr__Cong => 166.97/25.05 (subsetI1 => 166.97/25.05 (eqimpsubset2 => 166.97/25.05 (eqimpsubset1 => 166.97/25.05 (subsetI2 => 166.97/25.05 (emptysetsubset => 166.97/25.05 (subsetE => 166.97/25.05 (subsetE2 => 166.97/25.05 (notsubsetI => 166.97/25.05 (notequalI1 => 166.97/25.05 (notequalI2 => 166.97/25.05 (subsetRefl => 166.97/25.05 (subsetTrans => 166.97/25.05 (setadjoinSub => 166.97/25.05 (setadjoinSub2 => 166.97/25.05 (subset2powerset => 166.97/25.05 (setextsub => 166.97/25.05 (subsetemptysetimpeq => 166.97/25.05 (powersetI1 => 166.97/25.05 ( 166.97/25.05 powersetE1 => 166.97/25.05 ( 166.97/25.05 inPowerset => 166.97/25.05 ( 166.97/25.05 powersetsubset => 166.97/25.05 ( 166.97/25.05 sepInPowerset => 166.97/25.05 ( 166.97/25.05 sepSubset => 166.97/25.05 ( 166.97/25.05 binunionIL => 166.97/25.05 ( 166.97/25.05 upairset2IR => 166.97/25.05 ( 166.97/25.05 binunionIR => 166.97/25.05 ( 166.97/25.05 binunionEcases => 166.97/25.05 ( 166.97/25.05 binunionE => 166.97/25.05 ( 166.97/25.05 binunionLsub => 166.97/25.05 ( 166.97/25.05 binunionRsub => 166.97/25.05 ( 166.97/25.05 binintersectI => 166.97/25.05 ( 166.97/25.05 binintersectSubset5 => 166.97/25.05 ( 166.97/25.05 binintersectEL => 166.97/25.05 ( 166.97/25.05 binintersectLsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset2 => 166.97/25.05 ( 166.97/25.05 binintersectSubset3 => 166.97/25.05 ( 166.97/25.05 binintersectER => 166.97/25.05 ( 166.97/25.05 disjointsetsI1 => 166.97/25.05 ( 166.97/25.05 binintersectRsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset4 => 166.97/25.05 ( 166.97/25.05 binintersectSubset1 => 166.97/25.05 ( 166.97/25.05 bs114d => 166.97/25.05 ( 166.97/25.05 setminusI => 166.97/25.05 ( 166.97/25.05 setminusEL => 166.97/25.05 ( 166.97/25.05 setminusER => 166.97/25.05 ( 166.97/25.05 setminusSubset2 => 166.97/25.05 ( 166.97/25.05 setminusERneg => 166.97/25.05 ( 166.97/25.05 setminusELneg => 166.97/25.05 ( 166.97/25.05 setminusILneg => 166.97/25.05 ( 166.97/25.05 setminusIRneg => 166.97/25.05 ( 166.97/25.05 setminusLsub => 166.97/25.05 ( 166.97/25.05 setminusSubset1 => 166.97/25.05 ( 166.97/25.05 symdiffE => 166.97/25.05 ( 166.97/25.05 symdiffI1 => 166.97/25.05 ( 166.97/25.05 symdiffI2 => 166.97/25.05 ( 166.97/25.05 symdiffIneg1 => 166.97/25.05 ( 166.97/25.05 symdiffIneg2 => 166.97/25.05 ( 166.97/25.05 secondinupair => 166.97/25.05 ( 166.97/25.05 setukpairIL => 166.97/25.05 ( 166.97/25.05 setukpairIR => 166.97/25.05 ( 166.97/25.05 kpairiskpair => 166.97/25.05 ( 166.97/25.05 kpairp => 166.97/25.05 ( 166.97/25.05 singletonsubset => 166.97/25.05 ( 166.97/25.05 singletoninpowerset => 166.97/25.05 ( 166.97/25.05 singletoninpowunion => 166.97/25.05 ( 166.97/25.05 upairset2E => 166.97/25.05 ( 166.97/25.05 upairsubunion => 166.97/25.05 ( 166.97/25.05 upairinpowunion => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem1 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem2 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem3 => 166.97/25.05 ( 166.97/25.05 cartprodpairin => 166.97/25.05 ( 166.97/25.05 cartprodmempair1 => 166.97/25.05 ( 166.97/25.05 cartprodmempair => 166.97/25.05 ( 166.97/25.05 setunionE2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton1 => 166.97/25.05 ( 166.97/25.05 setunionsingleton2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton => 166.97/25.05 ( 166.97/25.05 singletonprop => 166.97/25.05 ( 166.97/25.05 ex1E1 => 166.97/25.05 ( 166.97/25.05 ex1I => 166.97/25.05 ( 166.97/25.05 ex1I2 => 166.97/25.05 ( 166.97/25.05 singletonsuniq => 166.97/25.05 ( 166.97/25.05 setukpairinjL1 => 166.97/25.05 ( 166.97/25.05 kfstsingleton => 166.97/25.05 ( 166.97/25.05 theprop => 166.97/25.05 ( 166.97/25.05 kfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodfstin => 166.97/25.05 ( 166.97/25.05 setukpairinjL2 => 166.97/25.05 ( 166.97/25.05 setukpairinjL => 166.97/25.05 ( 166.97/25.05 setukpairinjR11 => 166.97/25.05 ( 166.97/25.05 setukpairinjR12 => 166.97/25.05 ( 166.97/25.05 setukpairinjR1 => 166.97/25.05 ( 166.97/25.05 upairequniteq => 166.97/25.05 ( 166.97/25.05 setukpairinjR2 => 166.97/25.05 ( 166.97/25.05 setukpairinjR => 166.97/25.05 ( 166.97/25.05 ksndsingleton => 166.97/25.05 ( 166.97/25.05 ksndpairEq => 166.97/25.05 ( 166.97/25.05 kpairsurjEq => 166.97/25.05 ( 166.97/25.05 cartprodsndin => 166.97/25.05 ( 166.97/25.05 cartprodpairmemEL => 166.97/25.05 ( 166.97/25.05 cartprodpairmemER => 166.97/25.05 ( 166.97/25.05 cartprodmempaircEq => 166.97/25.05 ( 166.97/25.05 cartprodfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodsndpairEq => 166.97/25.05 ( 166.97/25.05 cartprodpairsurjEq => 166.97/25.05 ( 166.97/25.05 dpsetconstrI => 166.97/25.05 ( 166.97/25.05 dpsetconstrSub => 166.97/25.05 ( 166.97/25.05 setOfPairsIsBReln => 166.97/25.05 ( 166.97/25.05 !!(( 166.97/25.05 ^[Y0 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y1 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y2 : ( 166.97/25.05 $i * $i) > $o]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y3 : $i]: 166.97/25.05 (in(Y3, 166.97/25.05 Y0) => 166.97/25.05 (!!(( 166.97/25.05 ^[Y4 : $i]: 166.97/25.05 (in(Y4, 166.97/25.05 Y1) => 166.97/25.05 (in( 166.97/25.05 kpair( 166.97/25.05 Y3, Y4), 166.97/25.05 dpsetconstr( 166.97/25.05 Y0, Y1, 166.97/25.05 Y2)) => 166.97/25.05 Y2(Y3, 166.97/25.05 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.05 inference('lazy_cnf_imply', [status(thm)], ['31'])). 166.97/25.05 tff('33', plain, 166.97/25.05 ~ (setadjoinOr => 166.97/25.05 (setoftrueEq => 166.97/25.05 (powersetI => 166.97/25.05 (emptyinPowerset => 166.97/25.05 (emptyInPowerset => 166.97/25.05 (powersetE => 166.97/25.05 (setunionI => 166.97/25.05 (setunionE => 166.97/25.05 (subPowSU => 166.97/25.05 (exuE2 => 166.97/25.05 (nonemptyImpWitness => 166.97/25.05 (uniqinunit => 166.97/25.05 (notinsingleton => 166.97/25.05 (eqinunit => 166.97/25.05 (singletonsswitch => 166.97/25.05 (upairsetE => 166.97/25.05 (upairsetIL => 166.97/25.05 (upairsetIR => 166.97/25.05 (emptyE1 => 166.97/25.05 (vacuousDall => 166.97/25.05 (quantDeMorgan1 => 166.97/25.05 (quantDeMorgan2 => 166.97/25.05 (quantDeMorgan3 => 166.97/25.05 (quantDeMorgan4 => 166.97/25.05 (prop2setI => 166.97/25.05 (prop2set2propI => 166.97/25.05 (notdexE => 166.97/25.05 (notdallE => 166.97/25.05 (exuI1 => 166.97/25.05 (exuI3 => 166.97/25.05 (exuI2 => 166.97/25.05 (inCongP => 166.97/25.05 (in__Cong => 166.97/25.05 (exuE3u => 166.97/25.05 (exu__Cong => 166.97/25.05 (emptyset__Cong => 166.97/25.05 (setadjoin__Cong => 166.97/25.05 (powerset__Cong => 166.97/25.05 (setunion__Cong => 166.97/25.05 (omega__Cong => 166.97/25.05 (exuEu => 166.97/25.05 (descr__Cong => 166.97/25.05 (dsetconstr__Cong => 166.97/25.05 (subsetI1 => 166.97/25.05 (eqimpsubset2 => 166.97/25.05 (eqimpsubset1 => 166.97/25.05 (subsetI2 => 166.97/25.05 (emptysetsubset => 166.97/25.05 (subsetE => 166.97/25.05 (subsetE2 => 166.97/25.05 (notsubsetI => 166.97/25.05 (notequalI1 => 166.97/25.05 (notequalI2 => 166.97/25.05 (subsetRefl => 166.97/25.05 (subsetTrans => 166.97/25.05 (setadjoinSub => 166.97/25.05 (setadjoinSub2 => 166.97/25.05 (subset2powerset => 166.97/25.05 (setextsub => 166.97/25.05 (subsetemptysetimpeq => 166.97/25.05 (powersetI1 => 166.97/25.05 (powersetE1 => 166.97/25.05 ( 166.97/25.05 inPowerset => 166.97/25.05 ( 166.97/25.05 powersetsubset => 166.97/25.05 ( 166.97/25.05 sepInPowerset => 166.97/25.05 ( 166.97/25.05 sepSubset => 166.97/25.05 ( 166.97/25.05 binunionIL => 166.97/25.05 ( 166.97/25.05 upairset2IR => 166.97/25.05 ( 166.97/25.05 binunionIR => 166.97/25.05 ( 166.97/25.05 binunionEcases => 166.97/25.05 ( 166.97/25.05 binunionE => 166.97/25.05 ( 166.97/25.05 binunionLsub => 166.97/25.05 ( 166.97/25.05 binunionRsub => 166.97/25.05 ( 166.97/25.05 binintersectI => 166.97/25.05 ( 166.97/25.05 binintersectSubset5 => 166.97/25.05 ( 166.97/25.05 binintersectEL => 166.97/25.05 ( 166.97/25.05 binintersectLsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset2 => 166.97/25.05 ( 166.97/25.05 binintersectSubset3 => 166.97/25.05 ( 166.97/25.05 binintersectER => 166.97/25.05 ( 166.97/25.05 disjointsetsI1 => 166.97/25.05 ( 166.97/25.05 binintersectRsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset4 => 166.97/25.05 ( 166.97/25.05 binintersectSubset1 => 166.97/25.05 ( 166.97/25.05 bs114d => 166.97/25.05 ( 166.97/25.05 setminusI => 166.97/25.05 ( 166.97/25.05 setminusEL => 166.97/25.05 ( 166.97/25.05 setminusER => 166.97/25.05 ( 166.97/25.05 setminusSubset2 => 166.97/25.05 ( 166.97/25.05 setminusERneg => 166.97/25.05 ( 166.97/25.05 setminusELneg => 166.97/25.05 ( 166.97/25.05 setminusILneg => 166.97/25.05 ( 166.97/25.05 setminusIRneg => 166.97/25.05 ( 166.97/25.05 setminusLsub => 166.97/25.05 ( 166.97/25.05 setminusSubset1 => 166.97/25.05 ( 166.97/25.05 symdiffE => 166.97/25.05 ( 166.97/25.05 symdiffI1 => 166.97/25.05 ( 166.97/25.05 symdiffI2 => 166.97/25.05 ( 166.97/25.05 symdiffIneg1 => 166.97/25.05 ( 166.97/25.05 symdiffIneg2 => 166.97/25.05 ( 166.97/25.05 secondinupair => 166.97/25.05 ( 166.97/25.05 setukpairIL => 166.97/25.05 ( 166.97/25.05 setukpairIR => 166.97/25.05 ( 166.97/25.05 kpairiskpair => 166.97/25.05 ( 166.97/25.05 kpairp => 166.97/25.05 ( 166.97/25.05 singletonsubset => 166.97/25.05 ( 166.97/25.05 singletoninpowerset => 166.97/25.05 ( 166.97/25.05 singletoninpowunion => 166.97/25.05 ( 166.97/25.05 upairset2E => 166.97/25.05 ( 166.97/25.05 upairsubunion => 166.97/25.05 ( 166.97/25.05 upairinpowunion => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem1 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem2 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem3 => 166.97/25.05 ( 166.97/25.05 cartprodpairin => 166.97/25.05 ( 166.97/25.05 cartprodmempair1 => 166.97/25.05 ( 166.97/25.05 cartprodmempair => 166.97/25.05 ( 166.97/25.05 setunionE2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton1 => 166.97/25.05 ( 166.97/25.05 setunionsingleton2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton => 166.97/25.05 ( 166.97/25.05 singletonprop => 166.97/25.05 ( 166.97/25.05 ex1E1 => 166.97/25.05 ( 166.97/25.05 ex1I => 166.97/25.05 ( 166.97/25.05 ex1I2 => 166.97/25.05 ( 166.97/25.05 singletonsuniq => 166.97/25.05 ( 166.97/25.05 setukpairinjL1 => 166.97/25.05 ( 166.97/25.05 kfstsingleton => 166.97/25.05 ( 166.97/25.05 theprop => 166.97/25.05 ( 166.97/25.05 kfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodfstin => 166.97/25.05 ( 166.97/25.05 setukpairinjL2 => 166.97/25.05 ( 166.97/25.05 setukpairinjL => 166.97/25.05 ( 166.97/25.05 setukpairinjR11 => 166.97/25.05 ( 166.97/25.05 setukpairinjR12 => 166.97/25.05 ( 166.97/25.05 setukpairinjR1 => 166.97/25.05 ( 166.97/25.05 upairequniteq => 166.97/25.05 ( 166.97/25.05 setukpairinjR2 => 166.97/25.05 ( 166.97/25.05 setukpairinjR => 166.97/25.05 ( 166.97/25.05 ksndsingleton => 166.97/25.05 ( 166.97/25.05 ksndpairEq => 166.97/25.05 ( 166.97/25.05 kpairsurjEq => 166.97/25.05 ( 166.97/25.05 cartprodsndin => 166.97/25.05 ( 166.97/25.05 cartprodpairmemEL => 166.97/25.05 ( 166.97/25.05 cartprodpairmemER => 166.97/25.05 ( 166.97/25.05 cartprodmempaircEq => 166.97/25.05 ( 166.97/25.05 cartprodfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodsndpairEq => 166.97/25.05 ( 166.97/25.05 cartprodpairsurjEq => 166.97/25.05 ( 166.97/25.05 dpsetconstrI => 166.97/25.05 ( 166.97/25.05 dpsetconstrSub => 166.97/25.05 ( 166.97/25.05 setOfPairsIsBReln => 166.97/25.05 ( 166.97/25.05 !!(( 166.97/25.05 ^[Y0 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y1 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y2 : ( 166.97/25.05 $i * $i) > $o]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y3 : $i]: 166.97/25.05 (in(Y3, 166.97/25.05 Y0) => 166.97/25.05 (!!(( 166.97/25.05 ^[Y4 : $i]: 166.97/25.05 (in(Y4, 166.97/25.05 Y1) => 166.97/25.05 (in( 166.97/25.05 kpair( 166.97/25.05 Y3, Y4), 166.97/25.05 dpsetconstr( 166.97/25.05 Y0, Y1, 166.97/25.05 Y2)) => 166.97/25.05 Y2(Y3, 166.97/25.05 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.05 inference('lazy_cnf_imply', [status(thm)], ['32'])). 166.97/25.05 tff('34', plain, 166.97/25.05 ~ (setoftrueEq => 166.97/25.05 (powersetI => 166.97/25.05 (emptyinPowerset => 166.97/25.05 (emptyInPowerset => 166.97/25.05 (powersetE => 166.97/25.05 (setunionI => 166.97/25.05 (setunionE => 166.97/25.05 (subPowSU => 166.97/25.05 (exuE2 => 166.97/25.05 (nonemptyImpWitness => 166.97/25.05 (uniqinunit => 166.97/25.05 (notinsingleton => 166.97/25.05 (eqinunit => 166.97/25.05 (singletonsswitch => 166.97/25.05 (upairsetE => 166.97/25.05 (upairsetIL => 166.97/25.05 (upairsetIR => 166.97/25.05 (emptyE1 => 166.97/25.05 (vacuousDall => 166.97/25.05 (quantDeMorgan1 => 166.97/25.05 (quantDeMorgan2 => 166.97/25.05 (quantDeMorgan3 => 166.97/25.05 (quantDeMorgan4 => 166.97/25.05 (prop2setI => 166.97/25.05 (prop2set2propI => 166.97/25.05 (notdexE => 166.97/25.05 (notdallE => 166.97/25.05 (exuI1 => 166.97/25.05 (exuI3 => 166.97/25.05 (exuI2 => 166.97/25.05 (inCongP => 166.97/25.05 (in__Cong => 166.97/25.05 (exuE3u => 166.97/25.05 (exu__Cong => 166.97/25.05 (emptyset__Cong => 166.97/25.05 (setadjoin__Cong => 166.97/25.05 (powerset__Cong => 166.97/25.05 (setunion__Cong => 166.97/25.05 (omega__Cong => 166.97/25.05 (exuEu => 166.97/25.05 (descr__Cong => 166.97/25.05 (dsetconstr__Cong => 166.97/25.05 (subsetI1 => 166.97/25.05 (eqimpsubset2 => 166.97/25.05 (eqimpsubset1 => 166.97/25.05 (subsetI2 => 166.97/25.05 (emptysetsubset => 166.97/25.05 (subsetE => 166.97/25.05 (subsetE2 => 166.97/25.05 (notsubsetI => 166.97/25.05 (notequalI1 => 166.97/25.05 (notequalI2 => 166.97/25.05 (subsetRefl => 166.97/25.05 (subsetTrans => 166.97/25.05 (setadjoinSub => 166.97/25.05 (setadjoinSub2 => 166.97/25.05 (subset2powerset => 166.97/25.05 (setextsub => 166.97/25.05 (subsetemptysetimpeq => 166.97/25.05 (powersetI1 => 166.97/25.05 (powersetE1 => 166.97/25.05 (inPowerset => 166.97/25.05 ( 166.97/25.05 powersetsubset => 166.97/25.05 ( 166.97/25.05 sepInPowerset => 166.97/25.05 ( 166.97/25.05 sepSubset => 166.97/25.05 ( 166.97/25.05 binunionIL => 166.97/25.05 ( 166.97/25.05 upairset2IR => 166.97/25.05 ( 166.97/25.05 binunionIR => 166.97/25.05 ( 166.97/25.05 binunionEcases => 166.97/25.05 ( 166.97/25.05 binunionE => 166.97/25.05 ( 166.97/25.05 binunionLsub => 166.97/25.05 ( 166.97/25.05 binunionRsub => 166.97/25.05 ( 166.97/25.05 binintersectI => 166.97/25.05 ( 166.97/25.05 binintersectSubset5 => 166.97/25.05 ( 166.97/25.05 binintersectEL => 166.97/25.05 ( 166.97/25.05 binintersectLsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset2 => 166.97/25.05 ( 166.97/25.05 binintersectSubset3 => 166.97/25.05 ( 166.97/25.05 binintersectER => 166.97/25.05 ( 166.97/25.05 disjointsetsI1 => 166.97/25.05 ( 166.97/25.05 binintersectRsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset4 => 166.97/25.05 ( 166.97/25.05 binintersectSubset1 => 166.97/25.05 ( 166.97/25.05 bs114d => 166.97/25.05 ( 166.97/25.05 setminusI => 166.97/25.05 ( 166.97/25.05 setminusEL => 166.97/25.05 ( 166.97/25.05 setminusER => 166.97/25.05 ( 166.97/25.05 setminusSubset2 => 166.97/25.05 ( 166.97/25.05 setminusERneg => 166.97/25.05 ( 166.97/25.05 setminusELneg => 166.97/25.05 ( 166.97/25.05 setminusILneg => 166.97/25.05 ( 166.97/25.05 setminusIRneg => 166.97/25.05 ( 166.97/25.05 setminusLsub => 166.97/25.05 ( 166.97/25.05 setminusSubset1 => 166.97/25.05 ( 166.97/25.05 symdiffE => 166.97/25.05 ( 166.97/25.05 symdiffI1 => 166.97/25.05 ( 166.97/25.05 symdiffI2 => 166.97/25.05 ( 166.97/25.05 symdiffIneg1 => 166.97/25.05 ( 166.97/25.05 symdiffIneg2 => 166.97/25.05 ( 166.97/25.05 secondinupair => 166.97/25.05 ( 166.97/25.05 setukpairIL => 166.97/25.05 ( 166.97/25.05 setukpairIR => 166.97/25.05 ( 166.97/25.05 kpairiskpair => 166.97/25.05 ( 166.97/25.05 kpairp => 166.97/25.05 ( 166.97/25.05 singletonsubset => 166.97/25.05 ( 166.97/25.05 singletoninpowerset => 166.97/25.05 ( 166.97/25.05 singletoninpowunion => 166.97/25.05 ( 166.97/25.05 upairset2E => 166.97/25.05 ( 166.97/25.05 upairsubunion => 166.97/25.05 ( 166.97/25.05 upairinpowunion => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem1 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem2 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem3 => 166.97/25.05 ( 166.97/25.05 cartprodpairin => 166.97/25.05 ( 166.97/25.05 cartprodmempair1 => 166.97/25.05 ( 166.97/25.05 cartprodmempair => 166.97/25.05 ( 166.97/25.05 setunionE2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton1 => 166.97/25.05 ( 166.97/25.05 setunionsingleton2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton => 166.97/25.05 ( 166.97/25.05 singletonprop => 166.97/25.05 ( 166.97/25.05 ex1E1 => 166.97/25.05 ( 166.97/25.05 ex1I => 166.97/25.05 ( 166.97/25.05 ex1I2 => 166.97/25.05 ( 166.97/25.05 singletonsuniq => 166.97/25.05 ( 166.97/25.05 setukpairinjL1 => 166.97/25.05 ( 166.97/25.05 kfstsingleton => 166.97/25.05 ( 166.97/25.05 theprop => 166.97/25.05 ( 166.97/25.05 kfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodfstin => 166.97/25.05 ( 166.97/25.05 setukpairinjL2 => 166.97/25.05 ( 166.97/25.05 setukpairinjL => 166.97/25.05 ( 166.97/25.05 setukpairinjR11 => 166.97/25.05 ( 166.97/25.05 setukpairinjR12 => 166.97/25.05 ( 166.97/25.05 setukpairinjR1 => 166.97/25.05 ( 166.97/25.05 upairequniteq => 166.97/25.05 ( 166.97/25.05 setukpairinjR2 => 166.97/25.05 ( 166.97/25.05 setukpairinjR => 166.97/25.05 ( 166.97/25.05 ksndsingleton => 166.97/25.05 ( 166.97/25.05 ksndpairEq => 166.97/25.05 ( 166.97/25.05 kpairsurjEq => 166.97/25.05 ( 166.97/25.05 cartprodsndin => 166.97/25.05 ( 166.97/25.05 cartprodpairmemEL => 166.97/25.05 ( 166.97/25.05 cartprodpairmemER => 166.97/25.05 ( 166.97/25.05 cartprodmempaircEq => 166.97/25.05 ( 166.97/25.05 cartprodfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodsndpairEq => 166.97/25.05 ( 166.97/25.05 cartprodpairsurjEq => 166.97/25.05 ( 166.97/25.05 dpsetconstrI => 166.97/25.05 ( 166.97/25.05 dpsetconstrSub => 166.97/25.05 ( 166.97/25.05 setOfPairsIsBReln => 166.97/25.05 ( 166.97/25.05 !!(( 166.97/25.05 ^[Y0 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y1 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y2 : ( 166.97/25.05 $i * $i) > $o]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y3 : $i]: 166.97/25.05 (in(Y3, 166.97/25.05 Y0) => 166.97/25.05 (!!(( 166.97/25.05 ^[Y4 : $i]: 166.97/25.05 (in(Y4, 166.97/25.05 Y1) => 166.97/25.05 (in( 166.97/25.05 kpair( 166.97/25.05 Y3, Y4), 166.97/25.05 dpsetconstr( 166.97/25.05 Y0, Y1, 166.97/25.05 Y2)) => 166.97/25.05 Y2(Y3, 166.97/25.05 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.05 inference('lazy_cnf_imply', [status(thm)], ['33'])). 166.97/25.05 tff('35', plain, 166.97/25.05 ~ (powersetI => 166.97/25.05 (emptyinPowerset => 166.97/25.05 (emptyInPowerset => 166.97/25.05 (powersetE => 166.97/25.05 (setunionI => 166.97/25.05 (setunionE => 166.97/25.05 (subPowSU => 166.97/25.05 (exuE2 => 166.97/25.05 (nonemptyImpWitness => 166.97/25.05 (uniqinunit => 166.97/25.05 (notinsingleton => 166.97/25.05 (eqinunit => 166.97/25.05 (singletonsswitch => 166.97/25.05 (upairsetE => 166.97/25.05 (upairsetIL => 166.97/25.05 (upairsetIR => 166.97/25.05 (emptyE1 => 166.97/25.05 (vacuousDall => 166.97/25.05 (quantDeMorgan1 => 166.97/25.05 (quantDeMorgan2 => 166.97/25.05 (quantDeMorgan3 => 166.97/25.05 (quantDeMorgan4 => 166.97/25.05 (prop2setI => 166.97/25.05 (prop2set2propI => 166.97/25.05 (notdexE => 166.97/25.05 (notdallE => 166.97/25.05 (exuI1 => 166.97/25.05 (exuI3 => 166.97/25.05 (exuI2 => 166.97/25.05 (inCongP => 166.97/25.05 (in__Cong => 166.97/25.05 (exuE3u => 166.97/25.05 (exu__Cong => 166.97/25.05 (emptyset__Cong => 166.97/25.05 (setadjoin__Cong => 166.97/25.05 (powerset__Cong => 166.97/25.05 (setunion__Cong => 166.97/25.05 (omega__Cong => 166.97/25.05 (exuEu => 166.97/25.05 (descr__Cong => 166.97/25.05 (dsetconstr__Cong => 166.97/25.05 (subsetI1 => 166.97/25.05 (eqimpsubset2 => 166.97/25.05 (eqimpsubset1 => 166.97/25.05 (subsetI2 => 166.97/25.05 (emptysetsubset => 166.97/25.05 (subsetE => 166.97/25.05 (subsetE2 => 166.97/25.05 (notsubsetI => 166.97/25.05 (notequalI1 => 166.97/25.05 (notequalI2 => 166.97/25.05 (subsetRefl => 166.97/25.05 (subsetTrans => 166.97/25.05 (setadjoinSub => 166.97/25.05 (setadjoinSub2 => 166.97/25.05 (subset2powerset => 166.97/25.05 (setextsub => 166.97/25.05 (subsetemptysetimpeq => 166.97/25.05 (powersetI1 => 166.97/25.05 (powersetE1 => 166.97/25.05 (inPowerset => 166.97/25.05 (powersetsubset => 166.97/25.05 ( 166.97/25.05 sepInPowerset => 166.97/25.05 ( 166.97/25.05 sepSubset => 166.97/25.05 ( 166.97/25.05 binunionIL => 166.97/25.05 ( 166.97/25.05 upairset2IR => 166.97/25.05 ( 166.97/25.05 binunionIR => 166.97/25.05 ( 166.97/25.05 binunionEcases => 166.97/25.05 ( 166.97/25.05 binunionE => 166.97/25.05 ( 166.97/25.05 binunionLsub => 166.97/25.05 ( 166.97/25.05 binunionRsub => 166.97/25.05 ( 166.97/25.05 binintersectI => 166.97/25.05 ( 166.97/25.05 binintersectSubset5 => 166.97/25.05 ( 166.97/25.05 binintersectEL => 166.97/25.05 ( 166.97/25.05 binintersectLsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset2 => 166.97/25.05 ( 166.97/25.05 binintersectSubset3 => 166.97/25.05 ( 166.97/25.05 binintersectER => 166.97/25.05 ( 166.97/25.05 disjointsetsI1 => 166.97/25.05 ( 166.97/25.05 binintersectRsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset4 => 166.97/25.05 ( 166.97/25.05 binintersectSubset1 => 166.97/25.05 ( 166.97/25.05 bs114d => 166.97/25.05 ( 166.97/25.05 setminusI => 166.97/25.05 ( 166.97/25.05 setminusEL => 166.97/25.05 ( 166.97/25.05 setminusER => 166.97/25.05 ( 166.97/25.05 setminusSubset2 => 166.97/25.05 ( 166.97/25.05 setminusERneg => 166.97/25.05 ( 166.97/25.05 setminusELneg => 166.97/25.05 ( 166.97/25.05 setminusILneg => 166.97/25.05 ( 166.97/25.05 setminusIRneg => 166.97/25.05 ( 166.97/25.05 setminusLsub => 166.97/25.05 ( 166.97/25.05 setminusSubset1 => 166.97/25.05 ( 166.97/25.05 symdiffE => 166.97/25.05 ( 166.97/25.05 symdiffI1 => 166.97/25.05 ( 166.97/25.05 symdiffI2 => 166.97/25.05 ( 166.97/25.05 symdiffIneg1 => 166.97/25.05 ( 166.97/25.05 symdiffIneg2 => 166.97/25.05 ( 166.97/25.05 secondinupair => 166.97/25.05 ( 166.97/25.05 setukpairIL => 166.97/25.05 ( 166.97/25.05 setukpairIR => 166.97/25.05 ( 166.97/25.05 kpairiskpair => 166.97/25.05 ( 166.97/25.05 kpairp => 166.97/25.05 ( 166.97/25.05 singletonsubset => 166.97/25.05 ( 166.97/25.05 singletoninpowerset => 166.97/25.05 ( 166.97/25.05 singletoninpowunion => 166.97/25.05 ( 166.97/25.05 upairset2E => 166.97/25.05 ( 166.97/25.05 upairsubunion => 166.97/25.05 ( 166.97/25.05 upairinpowunion => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem1 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem2 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem3 => 166.97/25.05 ( 166.97/25.05 cartprodpairin => 166.97/25.05 ( 166.97/25.05 cartprodmempair1 => 166.97/25.05 ( 166.97/25.05 cartprodmempair => 166.97/25.05 ( 166.97/25.05 setunionE2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton1 => 166.97/25.05 ( 166.97/25.05 setunionsingleton2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton => 166.97/25.05 ( 166.97/25.05 singletonprop => 166.97/25.05 ( 166.97/25.05 ex1E1 => 166.97/25.05 ( 166.97/25.05 ex1I => 166.97/25.05 ( 166.97/25.05 ex1I2 => 166.97/25.05 ( 166.97/25.05 singletonsuniq => 166.97/25.05 ( 166.97/25.05 setukpairinjL1 => 166.97/25.05 ( 166.97/25.05 kfstsingleton => 166.97/25.05 ( 166.97/25.05 theprop => 166.97/25.05 ( 166.97/25.05 kfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodfstin => 166.97/25.05 ( 166.97/25.05 setukpairinjL2 => 166.97/25.05 ( 166.97/25.05 setukpairinjL => 166.97/25.05 ( 166.97/25.05 setukpairinjR11 => 166.97/25.05 ( 166.97/25.05 setukpairinjR12 => 166.97/25.05 ( 166.97/25.05 setukpairinjR1 => 166.97/25.05 ( 166.97/25.05 upairequniteq => 166.97/25.05 ( 166.97/25.05 setukpairinjR2 => 166.97/25.05 ( 166.97/25.05 setukpairinjR => 166.97/25.05 ( 166.97/25.05 ksndsingleton => 166.97/25.05 ( 166.97/25.05 ksndpairEq => 166.97/25.05 ( 166.97/25.05 kpairsurjEq => 166.97/25.05 ( 166.97/25.05 cartprodsndin => 166.97/25.05 ( 166.97/25.05 cartprodpairmemEL => 166.97/25.05 ( 166.97/25.05 cartprodpairmemER => 166.97/25.05 ( 166.97/25.05 cartprodmempaircEq => 166.97/25.05 ( 166.97/25.05 cartprodfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodsndpairEq => 166.97/25.05 ( 166.97/25.05 cartprodpairsurjEq => 166.97/25.05 ( 166.97/25.05 dpsetconstrI => 166.97/25.05 ( 166.97/25.05 dpsetconstrSub => 166.97/25.05 ( 166.97/25.05 setOfPairsIsBReln => 166.97/25.05 ( 166.97/25.05 !!(( 166.97/25.05 ^[Y0 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y1 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y2 : ( 166.97/25.05 $i * $i) > $o]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y3 : $i]: 166.97/25.05 (in(Y3, 166.97/25.05 Y0) => 166.97/25.05 (!!(( 166.97/25.05 ^[Y4 : $i]: 166.97/25.05 (in(Y4, 166.97/25.05 Y1) => 166.97/25.05 (in( 166.97/25.05 kpair( 166.97/25.05 Y3, Y4), 166.97/25.05 dpsetconstr( 166.97/25.05 Y0, Y1, 166.97/25.05 Y2)) => 166.97/25.05 Y2(Y3, 166.97/25.05 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.05 inference('lazy_cnf_imply', [status(thm)], ['34'])). 166.97/25.05 tff('36', plain, 166.97/25.05 ~ (emptyinPowerset => 166.97/25.05 (emptyInPowerset => 166.97/25.05 (powersetE => 166.97/25.05 (setunionI => 166.97/25.05 (setunionE => 166.97/25.05 (subPowSU => 166.97/25.05 (exuE2 => 166.97/25.05 (nonemptyImpWitness => 166.97/25.05 (uniqinunit => 166.97/25.05 (notinsingleton => 166.97/25.05 (eqinunit => 166.97/25.05 (singletonsswitch => 166.97/25.05 (upairsetE => 166.97/25.05 (upairsetIL => 166.97/25.05 (upairsetIR => 166.97/25.05 (emptyE1 => 166.97/25.05 (vacuousDall => 166.97/25.05 (quantDeMorgan1 => 166.97/25.05 (quantDeMorgan2 => 166.97/25.05 (quantDeMorgan3 => 166.97/25.05 (quantDeMorgan4 => 166.97/25.05 (prop2setI => 166.97/25.05 (prop2set2propI => 166.97/25.05 (notdexE => 166.97/25.05 (notdallE => 166.97/25.05 (exuI1 => 166.97/25.05 (exuI3 => 166.97/25.05 (exuI2 => 166.97/25.05 (inCongP => 166.97/25.05 (in__Cong => 166.97/25.05 (exuE3u => 166.97/25.05 (exu__Cong => 166.97/25.05 (emptyset__Cong => 166.97/25.05 (setadjoin__Cong => 166.97/25.05 (powerset__Cong => 166.97/25.05 (setunion__Cong => 166.97/25.05 (omega__Cong => 166.97/25.05 (exuEu => 166.97/25.05 (descr__Cong => 166.97/25.05 (dsetconstr__Cong => 166.97/25.05 (subsetI1 => 166.97/25.05 (eqimpsubset2 => 166.97/25.05 (eqimpsubset1 => 166.97/25.05 (subsetI2 => 166.97/25.05 (emptysetsubset => 166.97/25.05 (subsetE => 166.97/25.05 (subsetE2 => 166.97/25.05 (notsubsetI => 166.97/25.05 (notequalI1 => 166.97/25.05 (notequalI2 => 166.97/25.05 (subsetRefl => 166.97/25.05 (subsetTrans => 166.97/25.05 (setadjoinSub => 166.97/25.05 (setadjoinSub2 => 166.97/25.05 (subset2powerset => 166.97/25.05 (setextsub => 166.97/25.05 (subsetemptysetimpeq => 166.97/25.05 (powersetI1 => 166.97/25.05 (powersetE1 => 166.97/25.05 (inPowerset => 166.97/25.05 (powersetsubset => 166.97/25.05 (sepInPowerset => 166.97/25.05 ( 166.97/25.05 sepSubset => 166.97/25.05 ( 166.97/25.05 binunionIL => 166.97/25.05 ( 166.97/25.05 upairset2IR => 166.97/25.05 ( 166.97/25.05 binunionIR => 166.97/25.05 ( 166.97/25.05 binunionEcases => 166.97/25.05 ( 166.97/25.05 binunionE => 166.97/25.05 ( 166.97/25.05 binunionLsub => 166.97/25.05 ( 166.97/25.05 binunionRsub => 166.97/25.05 ( 166.97/25.05 binintersectI => 166.97/25.05 ( 166.97/25.05 binintersectSubset5 => 166.97/25.05 ( 166.97/25.05 binintersectEL => 166.97/25.05 ( 166.97/25.05 binintersectLsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset2 => 166.97/25.05 ( 166.97/25.05 binintersectSubset3 => 166.97/25.05 ( 166.97/25.05 binintersectER => 166.97/25.05 ( 166.97/25.05 disjointsetsI1 => 166.97/25.05 ( 166.97/25.05 binintersectRsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset4 => 166.97/25.05 ( 166.97/25.05 binintersectSubset1 => 166.97/25.05 ( 166.97/25.05 bs114d => 166.97/25.05 ( 166.97/25.05 setminusI => 166.97/25.05 ( 166.97/25.05 setminusEL => 166.97/25.05 ( 166.97/25.05 setminusER => 166.97/25.05 ( 166.97/25.05 setminusSubset2 => 166.97/25.05 ( 166.97/25.05 setminusERneg => 166.97/25.05 ( 166.97/25.05 setminusELneg => 166.97/25.05 ( 166.97/25.05 setminusILneg => 166.97/25.05 ( 166.97/25.05 setminusIRneg => 166.97/25.05 ( 166.97/25.05 setminusLsub => 166.97/25.05 ( 166.97/25.05 setminusSubset1 => 166.97/25.05 ( 166.97/25.05 symdiffE => 166.97/25.05 ( 166.97/25.05 symdiffI1 => 166.97/25.05 ( 166.97/25.05 symdiffI2 => 166.97/25.05 ( 166.97/25.05 symdiffIneg1 => 166.97/25.05 ( 166.97/25.05 symdiffIneg2 => 166.97/25.05 ( 166.97/25.05 secondinupair => 166.97/25.05 ( 166.97/25.05 setukpairIL => 166.97/25.05 ( 166.97/25.05 setukpairIR => 166.97/25.05 ( 166.97/25.05 kpairiskpair => 166.97/25.05 ( 166.97/25.05 kpairp => 166.97/25.05 ( 166.97/25.05 singletonsubset => 166.97/25.05 ( 166.97/25.05 singletoninpowerset => 166.97/25.05 ( 166.97/25.05 singletoninpowunion => 166.97/25.05 ( 166.97/25.05 upairset2E => 166.97/25.05 ( 166.97/25.05 upairsubunion => 166.97/25.05 ( 166.97/25.05 upairinpowunion => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem1 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem2 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem3 => 166.97/25.05 ( 166.97/25.05 cartprodpairin => 166.97/25.05 ( 166.97/25.05 cartprodmempair1 => 166.97/25.05 ( 166.97/25.05 cartprodmempair => 166.97/25.05 ( 166.97/25.05 setunionE2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton1 => 166.97/25.05 ( 166.97/25.05 setunionsingleton2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton => 166.97/25.05 ( 166.97/25.05 singletonprop => 166.97/25.05 ( 166.97/25.05 ex1E1 => 166.97/25.05 ( 166.97/25.05 ex1I => 166.97/25.05 ( 166.97/25.05 ex1I2 => 166.97/25.05 ( 166.97/25.05 singletonsuniq => 166.97/25.05 ( 166.97/25.05 setukpairinjL1 => 166.97/25.05 ( 166.97/25.05 kfstsingleton => 166.97/25.05 ( 166.97/25.05 theprop => 166.97/25.05 ( 166.97/25.05 kfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodfstin => 166.97/25.05 ( 166.97/25.05 setukpairinjL2 => 166.97/25.05 ( 166.97/25.05 setukpairinjL => 166.97/25.05 ( 166.97/25.05 setukpairinjR11 => 166.97/25.05 ( 166.97/25.05 setukpairinjR12 => 166.97/25.05 ( 166.97/25.05 setukpairinjR1 => 166.97/25.05 ( 166.97/25.05 upairequniteq => 166.97/25.05 ( 166.97/25.05 setukpairinjR2 => 166.97/25.05 ( 166.97/25.05 setukpairinjR => 166.97/25.05 ( 166.97/25.05 ksndsingleton => 166.97/25.05 ( 166.97/25.05 ksndpairEq => 166.97/25.05 ( 166.97/25.05 kpairsurjEq => 166.97/25.05 ( 166.97/25.05 cartprodsndin => 166.97/25.05 ( 166.97/25.05 cartprodpairmemEL => 166.97/25.05 ( 166.97/25.05 cartprodpairmemER => 166.97/25.05 ( 166.97/25.05 cartprodmempaircEq => 166.97/25.05 ( 166.97/25.05 cartprodfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodsndpairEq => 166.97/25.05 ( 166.97/25.05 cartprodpairsurjEq => 166.97/25.05 ( 166.97/25.05 dpsetconstrI => 166.97/25.05 ( 166.97/25.05 dpsetconstrSub => 166.97/25.05 ( 166.97/25.05 setOfPairsIsBReln => 166.97/25.05 ( 166.97/25.05 !!(( 166.97/25.05 ^[Y0 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y1 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y2 : ( 166.97/25.05 $i * $i) > $o]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y3 : $i]: 166.97/25.05 (in(Y3, 166.97/25.05 Y0) => 166.97/25.05 (!!(( 166.97/25.05 ^[Y4 : $i]: 166.97/25.05 (in(Y4, 166.97/25.05 Y1) => 166.97/25.05 (in( 166.97/25.05 kpair( 166.97/25.05 Y3, Y4), 166.97/25.05 dpsetconstr( 166.97/25.05 Y0, Y1, 166.97/25.05 Y2)) => 166.97/25.05 Y2(Y3, 166.97/25.05 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.05 inference('lazy_cnf_imply', [status(thm)], ['35'])). 166.97/25.05 tff('37', plain, 166.97/25.05 ~ (emptyInPowerset => 166.97/25.05 (powersetE => 166.97/25.05 (setunionI => 166.97/25.05 (setunionE => 166.97/25.05 (subPowSU => 166.97/25.05 (exuE2 => 166.97/25.05 (nonemptyImpWitness => 166.97/25.05 (uniqinunit => 166.97/25.05 (notinsingleton => 166.97/25.05 (eqinunit => 166.97/25.05 (singletonsswitch => 166.97/25.05 (upairsetE => 166.97/25.05 (upairsetIL => 166.97/25.05 (upairsetIR => 166.97/25.05 (emptyE1 => 166.97/25.05 (vacuousDall => 166.97/25.05 (quantDeMorgan1 => 166.97/25.05 (quantDeMorgan2 => 166.97/25.05 (quantDeMorgan3 => 166.97/25.05 (quantDeMorgan4 => 166.97/25.05 (prop2setI => 166.97/25.05 (prop2set2propI => 166.97/25.05 (notdexE => 166.97/25.05 (notdallE => 166.97/25.05 (exuI1 => 166.97/25.05 (exuI3 => 166.97/25.05 (exuI2 => 166.97/25.05 (inCongP => 166.97/25.05 (in__Cong => 166.97/25.05 (exuE3u => 166.97/25.05 (exu__Cong => 166.97/25.05 (emptyset__Cong => 166.97/25.05 (setadjoin__Cong => 166.97/25.05 (powerset__Cong => 166.97/25.05 (setunion__Cong => 166.97/25.05 (omega__Cong => 166.97/25.05 (exuEu => 166.97/25.05 (descr__Cong => 166.97/25.05 (dsetconstr__Cong => 166.97/25.05 (subsetI1 => 166.97/25.05 (eqimpsubset2 => 166.97/25.05 (eqimpsubset1 => 166.97/25.05 (subsetI2 => 166.97/25.05 (emptysetsubset => 166.97/25.05 (subsetE => 166.97/25.05 (subsetE2 => 166.97/25.05 (notsubsetI => 166.97/25.05 (notequalI1 => 166.97/25.05 (notequalI2 => 166.97/25.05 (subsetRefl => 166.97/25.05 (subsetTrans => 166.97/25.05 (setadjoinSub => 166.97/25.05 (setadjoinSub2 => 166.97/25.05 (subset2powerset => 166.97/25.05 (setextsub => 166.97/25.05 (subsetemptysetimpeq => 166.97/25.05 (powersetI1 => 166.97/25.05 (powersetE1 => 166.97/25.05 (inPowerset => 166.97/25.05 (powersetsubset => 166.97/25.05 (sepInPowerset => 166.97/25.05 (sepSubset => 166.97/25.05 ( 166.97/25.05 binunionIL => 166.97/25.05 ( 166.97/25.05 upairset2IR => 166.97/25.05 ( 166.97/25.05 binunionIR => 166.97/25.05 ( 166.97/25.05 binunionEcases => 166.97/25.05 ( 166.97/25.05 binunionE => 166.97/25.05 ( 166.97/25.05 binunionLsub => 166.97/25.05 ( 166.97/25.05 binunionRsub => 166.97/25.05 ( 166.97/25.05 binintersectI => 166.97/25.05 ( 166.97/25.05 binintersectSubset5 => 166.97/25.05 ( 166.97/25.05 binintersectEL => 166.97/25.05 ( 166.97/25.05 binintersectLsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset2 => 166.97/25.05 ( 166.97/25.05 binintersectSubset3 => 166.97/25.05 ( 166.97/25.05 binintersectER => 166.97/25.05 ( 166.97/25.05 disjointsetsI1 => 166.97/25.05 ( 166.97/25.05 binintersectRsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset4 => 166.97/25.05 ( 166.97/25.05 binintersectSubset1 => 166.97/25.05 ( 166.97/25.05 bs114d => 166.97/25.05 ( 166.97/25.05 setminusI => 166.97/25.05 ( 166.97/25.05 setminusEL => 166.97/25.05 ( 166.97/25.05 setminusER => 166.97/25.05 ( 166.97/25.05 setminusSubset2 => 166.97/25.05 ( 166.97/25.05 setminusERneg => 166.97/25.05 ( 166.97/25.05 setminusELneg => 166.97/25.05 ( 166.97/25.05 setminusILneg => 166.97/25.05 ( 166.97/25.05 setminusIRneg => 166.97/25.05 ( 166.97/25.05 setminusLsub => 166.97/25.05 ( 166.97/25.05 setminusSubset1 => 166.97/25.05 ( 166.97/25.05 symdiffE => 166.97/25.05 ( 166.97/25.05 symdiffI1 => 166.97/25.05 ( 166.97/25.05 symdiffI2 => 166.97/25.05 ( 166.97/25.05 symdiffIneg1 => 166.97/25.05 ( 166.97/25.05 symdiffIneg2 => 166.97/25.05 ( 166.97/25.05 secondinupair => 166.97/25.05 ( 166.97/25.05 setukpairIL => 166.97/25.05 ( 166.97/25.05 setukpairIR => 166.97/25.05 ( 166.97/25.05 kpairiskpair => 166.97/25.05 ( 166.97/25.05 kpairp => 166.97/25.05 ( 166.97/25.05 singletonsubset => 166.97/25.05 ( 166.97/25.05 singletoninpowerset => 166.97/25.05 ( 166.97/25.05 singletoninpowunion => 166.97/25.05 ( 166.97/25.05 upairset2E => 166.97/25.05 ( 166.97/25.05 upairsubunion => 166.97/25.05 ( 166.97/25.05 upairinpowunion => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem1 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem2 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem3 => 166.97/25.05 ( 166.97/25.05 cartprodpairin => 166.97/25.05 ( 166.97/25.05 cartprodmempair1 => 166.97/25.05 ( 166.97/25.05 cartprodmempair => 166.97/25.05 ( 166.97/25.05 setunionE2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton1 => 166.97/25.05 ( 166.97/25.05 setunionsingleton2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton => 166.97/25.05 ( 166.97/25.05 singletonprop => 166.97/25.05 ( 166.97/25.05 ex1E1 => 166.97/25.05 ( 166.97/25.05 ex1I => 166.97/25.05 ( 166.97/25.05 ex1I2 => 166.97/25.05 ( 166.97/25.05 singletonsuniq => 166.97/25.05 ( 166.97/25.05 setukpairinjL1 => 166.97/25.05 ( 166.97/25.05 kfstsingleton => 166.97/25.05 ( 166.97/25.05 theprop => 166.97/25.05 ( 166.97/25.05 kfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodfstin => 166.97/25.05 ( 166.97/25.05 setukpairinjL2 => 166.97/25.05 ( 166.97/25.05 setukpairinjL => 166.97/25.05 ( 166.97/25.05 setukpairinjR11 => 166.97/25.05 ( 166.97/25.05 setukpairinjR12 => 166.97/25.05 ( 166.97/25.05 setukpairinjR1 => 166.97/25.05 ( 166.97/25.05 upairequniteq => 166.97/25.05 ( 166.97/25.05 setukpairinjR2 => 166.97/25.05 ( 166.97/25.05 setukpairinjR => 166.97/25.05 ( 166.97/25.05 ksndsingleton => 166.97/25.05 ( 166.97/25.05 ksndpairEq => 166.97/25.05 ( 166.97/25.05 kpairsurjEq => 166.97/25.05 ( 166.97/25.05 cartprodsndin => 166.97/25.05 ( 166.97/25.05 cartprodpairmemEL => 166.97/25.05 ( 166.97/25.05 cartprodpairmemER => 166.97/25.05 ( 166.97/25.05 cartprodmempaircEq => 166.97/25.05 ( 166.97/25.05 cartprodfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodsndpairEq => 166.97/25.05 ( 166.97/25.05 cartprodpairsurjEq => 166.97/25.05 ( 166.97/25.05 dpsetconstrI => 166.97/25.05 ( 166.97/25.05 dpsetconstrSub => 166.97/25.05 ( 166.97/25.05 setOfPairsIsBReln => 166.97/25.05 ( 166.97/25.05 !!(( 166.97/25.05 ^[Y0 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y1 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y2 : ( 166.97/25.05 $i * $i) > $o]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y3 : $i]: 166.97/25.05 (in(Y3, 166.97/25.05 Y0) => 166.97/25.05 (!!(( 166.97/25.05 ^[Y4 : $i]: 166.97/25.05 (in(Y4, 166.97/25.05 Y1) => 166.97/25.05 (in( 166.97/25.05 kpair( 166.97/25.05 Y3, Y4), 166.97/25.05 dpsetconstr( 166.97/25.05 Y0, Y1, 166.97/25.05 Y2)) => 166.97/25.05 Y2(Y3, 166.97/25.05 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.05 inference('lazy_cnf_imply', [status(thm)], ['36'])). 166.97/25.05 tff('38', plain, 166.97/25.05 ~ (powersetE => 166.97/25.05 (setunionI => 166.97/25.05 (setunionE => 166.97/25.05 (subPowSU => 166.97/25.05 (exuE2 => 166.97/25.05 (nonemptyImpWitness => 166.97/25.05 (uniqinunit => 166.97/25.05 (notinsingleton => 166.97/25.05 (eqinunit => 166.97/25.05 (singletonsswitch => 166.97/25.05 (upairsetE => 166.97/25.05 (upairsetIL => 166.97/25.05 (upairsetIR => 166.97/25.05 (emptyE1 => 166.97/25.05 (vacuousDall => 166.97/25.05 (quantDeMorgan1 => 166.97/25.05 (quantDeMorgan2 => 166.97/25.05 (quantDeMorgan3 => 166.97/25.05 (quantDeMorgan4 => 166.97/25.05 (prop2setI => 166.97/25.05 (prop2set2propI => 166.97/25.05 (notdexE => 166.97/25.05 (notdallE => 166.97/25.05 (exuI1 => 166.97/25.05 (exuI3 => 166.97/25.05 (exuI2 => 166.97/25.05 (inCongP => 166.97/25.05 (in__Cong => 166.97/25.05 (exuE3u => 166.97/25.05 (exu__Cong => 166.97/25.05 (emptyset__Cong => 166.97/25.05 (setadjoin__Cong => 166.97/25.05 (powerset__Cong => 166.97/25.05 (setunion__Cong => 166.97/25.05 (omega__Cong => 166.97/25.05 (exuEu => 166.97/25.05 (descr__Cong => 166.97/25.05 (dsetconstr__Cong => 166.97/25.05 (subsetI1 => 166.97/25.05 (eqimpsubset2 => 166.97/25.05 (eqimpsubset1 => 166.97/25.05 (subsetI2 => 166.97/25.05 (emptysetsubset => 166.97/25.05 (subsetE => 166.97/25.05 (subsetE2 => 166.97/25.05 (notsubsetI => 166.97/25.05 (notequalI1 => 166.97/25.05 (notequalI2 => 166.97/25.05 (subsetRefl => 166.97/25.05 (subsetTrans => 166.97/25.05 (setadjoinSub => 166.97/25.05 (setadjoinSub2 => 166.97/25.05 (subset2powerset => 166.97/25.05 (setextsub => 166.97/25.05 (subsetemptysetimpeq => 166.97/25.05 (powersetI1 => 166.97/25.05 (powersetE1 => 166.97/25.05 (inPowerset => 166.97/25.05 (powersetsubset => 166.97/25.05 (sepInPowerset => 166.97/25.05 (sepSubset => 166.97/25.05 (binunionIL => 166.97/25.05 ( 166.97/25.05 upairset2IR => 166.97/25.05 ( 166.97/25.05 binunionIR => 166.97/25.05 ( 166.97/25.05 binunionEcases => 166.97/25.05 ( 166.97/25.05 binunionE => 166.97/25.05 ( 166.97/25.05 binunionLsub => 166.97/25.05 ( 166.97/25.05 binunionRsub => 166.97/25.05 ( 166.97/25.05 binintersectI => 166.97/25.05 ( 166.97/25.05 binintersectSubset5 => 166.97/25.05 ( 166.97/25.05 binintersectEL => 166.97/25.05 ( 166.97/25.05 binintersectLsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset2 => 166.97/25.05 ( 166.97/25.05 binintersectSubset3 => 166.97/25.05 ( 166.97/25.05 binintersectER => 166.97/25.05 ( 166.97/25.05 disjointsetsI1 => 166.97/25.05 ( 166.97/25.05 binintersectRsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset4 => 166.97/25.05 ( 166.97/25.05 binintersectSubset1 => 166.97/25.05 ( 166.97/25.05 bs114d => 166.97/25.05 ( 166.97/25.05 setminusI => 166.97/25.05 ( 166.97/25.05 setminusEL => 166.97/25.05 ( 166.97/25.05 setminusER => 166.97/25.05 ( 166.97/25.05 setminusSubset2 => 166.97/25.05 ( 166.97/25.05 setminusERneg => 166.97/25.05 ( 166.97/25.05 setminusELneg => 166.97/25.05 ( 166.97/25.05 setminusILneg => 166.97/25.05 ( 166.97/25.05 setminusIRneg => 166.97/25.05 ( 166.97/25.05 setminusLsub => 166.97/25.05 ( 166.97/25.05 setminusSubset1 => 166.97/25.05 ( 166.97/25.05 symdiffE => 166.97/25.05 ( 166.97/25.05 symdiffI1 => 166.97/25.05 ( 166.97/25.05 symdiffI2 => 166.97/25.05 ( 166.97/25.05 symdiffIneg1 => 166.97/25.05 ( 166.97/25.05 symdiffIneg2 => 166.97/25.05 ( 166.97/25.05 secondinupair => 166.97/25.05 ( 166.97/25.05 setukpairIL => 166.97/25.05 ( 166.97/25.05 setukpairIR => 166.97/25.05 ( 166.97/25.05 kpairiskpair => 166.97/25.05 ( 166.97/25.05 kpairp => 166.97/25.05 ( 166.97/25.05 singletonsubset => 166.97/25.05 ( 166.97/25.05 singletoninpowerset => 166.97/25.05 ( 166.97/25.05 singletoninpowunion => 166.97/25.05 ( 166.97/25.05 upairset2E => 166.97/25.05 ( 166.97/25.05 upairsubunion => 166.97/25.05 ( 166.97/25.05 upairinpowunion => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem1 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem2 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem3 => 166.97/25.05 ( 166.97/25.05 cartprodpairin => 166.97/25.05 ( 166.97/25.05 cartprodmempair1 => 166.97/25.05 ( 166.97/25.05 cartprodmempair => 166.97/25.05 ( 166.97/25.05 setunionE2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton1 => 166.97/25.05 ( 166.97/25.05 setunionsingleton2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton => 166.97/25.05 ( 166.97/25.05 singletonprop => 166.97/25.05 ( 166.97/25.05 ex1E1 => 166.97/25.05 ( 166.97/25.05 ex1I => 166.97/25.05 ( 166.97/25.05 ex1I2 => 166.97/25.05 ( 166.97/25.05 singletonsuniq => 166.97/25.05 ( 166.97/25.05 setukpairinjL1 => 166.97/25.05 ( 166.97/25.05 kfstsingleton => 166.97/25.05 ( 166.97/25.05 theprop => 166.97/25.05 ( 166.97/25.05 kfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodfstin => 166.97/25.05 ( 166.97/25.05 setukpairinjL2 => 166.97/25.05 ( 166.97/25.05 setukpairinjL => 166.97/25.05 ( 166.97/25.05 setukpairinjR11 => 166.97/25.05 ( 166.97/25.05 setukpairinjR12 => 166.97/25.05 ( 166.97/25.05 setukpairinjR1 => 166.97/25.05 ( 166.97/25.05 upairequniteq => 166.97/25.05 ( 166.97/25.05 setukpairinjR2 => 166.97/25.05 ( 166.97/25.05 setukpairinjR => 166.97/25.05 ( 166.97/25.05 ksndsingleton => 166.97/25.05 ( 166.97/25.05 ksndpairEq => 166.97/25.05 ( 166.97/25.05 kpairsurjEq => 166.97/25.05 ( 166.97/25.05 cartprodsndin => 166.97/25.05 ( 166.97/25.05 cartprodpairmemEL => 166.97/25.05 ( 166.97/25.05 cartprodpairmemER => 166.97/25.05 ( 166.97/25.05 cartprodmempaircEq => 166.97/25.05 ( 166.97/25.05 cartprodfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodsndpairEq => 166.97/25.05 ( 166.97/25.05 cartprodpairsurjEq => 166.97/25.05 ( 166.97/25.05 dpsetconstrI => 166.97/25.05 ( 166.97/25.05 dpsetconstrSub => 166.97/25.05 ( 166.97/25.05 setOfPairsIsBReln => 166.97/25.05 ( 166.97/25.05 !!(( 166.97/25.05 ^[Y0 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y1 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y2 : ( 166.97/25.05 $i * $i) > $o]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y3 : $i]: 166.97/25.05 (in(Y3, 166.97/25.05 Y0) => 166.97/25.05 (!!(( 166.97/25.05 ^[Y4 : $i]: 166.97/25.05 (in(Y4, 166.97/25.05 Y1) => 166.97/25.05 (in( 166.97/25.05 kpair( 166.97/25.05 Y3, Y4), 166.97/25.05 dpsetconstr( 166.97/25.05 Y0, Y1, 166.97/25.05 Y2)) => 166.97/25.05 Y2(Y3, 166.97/25.05 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.05 inference('lazy_cnf_imply', [status(thm)], ['37'])). 166.97/25.05 tff('39', plain, 166.97/25.05 ~ (setunionI => 166.97/25.05 (setunionE => 166.97/25.05 (subPowSU => 166.97/25.05 (exuE2 => 166.97/25.05 (nonemptyImpWitness => 166.97/25.05 (uniqinunit => 166.97/25.05 (notinsingleton => 166.97/25.05 (eqinunit => 166.97/25.05 (singletonsswitch => 166.97/25.05 (upairsetE => 166.97/25.05 (upairsetIL => 166.97/25.05 (upairsetIR => 166.97/25.05 (emptyE1 => 166.97/25.05 (vacuousDall => 166.97/25.05 (quantDeMorgan1 => 166.97/25.05 (quantDeMorgan2 => 166.97/25.05 (quantDeMorgan3 => 166.97/25.05 (quantDeMorgan4 => 166.97/25.05 (prop2setI => 166.97/25.05 (prop2set2propI => 166.97/25.05 (notdexE => 166.97/25.05 (notdallE => 166.97/25.05 (exuI1 => 166.97/25.05 (exuI3 => 166.97/25.05 (exuI2 => 166.97/25.05 (inCongP => 166.97/25.05 (in__Cong => 166.97/25.05 (exuE3u => 166.97/25.05 (exu__Cong => 166.97/25.05 (emptyset__Cong => 166.97/25.05 (setadjoin__Cong => 166.97/25.05 (powerset__Cong => 166.97/25.05 (setunion__Cong => 166.97/25.05 (omega__Cong => 166.97/25.05 (exuEu => 166.97/25.05 (descr__Cong => 166.97/25.05 (dsetconstr__Cong => 166.97/25.05 (subsetI1 => 166.97/25.05 (eqimpsubset2 => 166.97/25.05 (eqimpsubset1 => 166.97/25.05 (subsetI2 => 166.97/25.05 (emptysetsubset => 166.97/25.05 (subsetE => 166.97/25.05 (subsetE2 => 166.97/25.05 (notsubsetI => 166.97/25.05 (notequalI1 => 166.97/25.05 (notequalI2 => 166.97/25.05 (subsetRefl => 166.97/25.05 (subsetTrans => 166.97/25.05 (setadjoinSub => 166.97/25.05 (setadjoinSub2 => 166.97/25.05 (subset2powerset => 166.97/25.05 (setextsub => 166.97/25.05 (subsetemptysetimpeq => 166.97/25.05 (powersetI1 => 166.97/25.05 (powersetE1 => 166.97/25.05 (inPowerset => 166.97/25.05 (powersetsubset => 166.97/25.05 (sepInPowerset => 166.97/25.05 (sepSubset => 166.97/25.05 (binunionIL => 166.97/25.05 (upairset2IR => 166.97/25.05 ( 166.97/25.05 binunionIR => 166.97/25.05 ( 166.97/25.05 binunionEcases => 166.97/25.05 ( 166.97/25.05 binunionE => 166.97/25.05 ( 166.97/25.05 binunionLsub => 166.97/25.05 ( 166.97/25.05 binunionRsub => 166.97/25.05 ( 166.97/25.05 binintersectI => 166.97/25.05 ( 166.97/25.05 binintersectSubset5 => 166.97/25.05 ( 166.97/25.05 binintersectEL => 166.97/25.05 ( 166.97/25.05 binintersectLsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset2 => 166.97/25.05 ( 166.97/25.05 binintersectSubset3 => 166.97/25.05 ( 166.97/25.05 binintersectER => 166.97/25.05 ( 166.97/25.05 disjointsetsI1 => 166.97/25.05 ( 166.97/25.05 binintersectRsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset4 => 166.97/25.05 ( 166.97/25.05 binintersectSubset1 => 166.97/25.05 ( 166.97/25.05 bs114d => 166.97/25.05 ( 166.97/25.05 setminusI => 166.97/25.05 ( 166.97/25.05 setminusEL => 166.97/25.05 ( 166.97/25.05 setminusER => 166.97/25.05 ( 166.97/25.05 setminusSubset2 => 166.97/25.05 ( 166.97/25.05 setminusERneg => 166.97/25.05 ( 166.97/25.05 setminusELneg => 166.97/25.05 ( 166.97/25.05 setminusILneg => 166.97/25.05 ( 166.97/25.05 setminusIRneg => 166.97/25.05 ( 166.97/25.05 setminusLsub => 166.97/25.05 ( 166.97/25.05 setminusSubset1 => 166.97/25.05 ( 166.97/25.05 symdiffE => 166.97/25.05 ( 166.97/25.05 symdiffI1 => 166.97/25.05 ( 166.97/25.05 symdiffI2 => 166.97/25.05 ( 166.97/25.05 symdiffIneg1 => 166.97/25.05 ( 166.97/25.05 symdiffIneg2 => 166.97/25.05 ( 166.97/25.05 secondinupair => 166.97/25.05 ( 166.97/25.05 setukpairIL => 166.97/25.05 ( 166.97/25.05 setukpairIR => 166.97/25.05 ( 166.97/25.05 kpairiskpair => 166.97/25.05 ( 166.97/25.05 kpairp => 166.97/25.05 ( 166.97/25.05 singletonsubset => 166.97/25.05 ( 166.97/25.05 singletoninpowerset => 166.97/25.05 ( 166.97/25.05 singletoninpowunion => 166.97/25.05 ( 166.97/25.05 upairset2E => 166.97/25.05 ( 166.97/25.05 upairsubunion => 166.97/25.05 ( 166.97/25.05 upairinpowunion => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem1 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem2 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem3 => 166.97/25.05 ( 166.97/25.05 cartprodpairin => 166.97/25.05 ( 166.97/25.05 cartprodmempair1 => 166.97/25.05 ( 166.97/25.05 cartprodmempair => 166.97/25.05 ( 166.97/25.05 setunionE2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton1 => 166.97/25.05 ( 166.97/25.05 setunionsingleton2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton => 166.97/25.05 ( 166.97/25.05 singletonprop => 166.97/25.05 ( 166.97/25.05 ex1E1 => 166.97/25.05 ( 166.97/25.05 ex1I => 166.97/25.05 ( 166.97/25.05 ex1I2 => 166.97/25.05 ( 166.97/25.05 singletonsuniq => 166.97/25.05 ( 166.97/25.05 setukpairinjL1 => 166.97/25.05 ( 166.97/25.05 kfstsingleton => 166.97/25.05 ( 166.97/25.05 theprop => 166.97/25.05 ( 166.97/25.05 kfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodfstin => 166.97/25.05 ( 166.97/25.05 setukpairinjL2 => 166.97/25.05 ( 166.97/25.05 setukpairinjL => 166.97/25.05 ( 166.97/25.05 setukpairinjR11 => 166.97/25.05 ( 166.97/25.05 setukpairinjR12 => 166.97/25.05 ( 166.97/25.05 setukpairinjR1 => 166.97/25.05 ( 166.97/25.05 upairequniteq => 166.97/25.05 ( 166.97/25.05 setukpairinjR2 => 166.97/25.05 ( 166.97/25.05 setukpairinjR => 166.97/25.05 ( 166.97/25.05 ksndsingleton => 166.97/25.05 ( 166.97/25.05 ksndpairEq => 166.97/25.05 ( 166.97/25.05 kpairsurjEq => 166.97/25.05 ( 166.97/25.05 cartprodsndin => 166.97/25.05 ( 166.97/25.05 cartprodpairmemEL => 166.97/25.05 ( 166.97/25.05 cartprodpairmemER => 166.97/25.05 ( 166.97/25.05 cartprodmempaircEq => 166.97/25.05 ( 166.97/25.05 cartprodfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodsndpairEq => 166.97/25.05 ( 166.97/25.05 cartprodpairsurjEq => 166.97/25.05 ( 166.97/25.05 dpsetconstrI => 166.97/25.05 ( 166.97/25.05 dpsetconstrSub => 166.97/25.05 ( 166.97/25.05 setOfPairsIsBReln => 166.97/25.05 ( 166.97/25.05 !!(( 166.97/25.05 ^[Y0 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y1 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y2 : ( 166.97/25.05 $i * $i) > $o]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y3 : $i]: 166.97/25.05 (in(Y3, 166.97/25.05 Y0) => 166.97/25.05 (!!(( 166.97/25.05 ^[Y4 : $i]: 166.97/25.05 (in(Y4, 166.97/25.05 Y1) => 166.97/25.05 (in( 166.97/25.05 kpair( 166.97/25.05 Y3, Y4), 166.97/25.05 dpsetconstr( 166.97/25.05 Y0, Y1, 166.97/25.05 Y2)) => 166.97/25.05 Y2(Y3, 166.97/25.05 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.05 inference('lazy_cnf_imply', [status(thm)], ['38'])). 166.97/25.05 tff('40', plain, 166.97/25.05 ~ (setunionE => 166.97/25.05 (subPowSU => 166.97/25.05 (exuE2 => 166.97/25.05 (nonemptyImpWitness => 166.97/25.05 (uniqinunit => 166.97/25.05 (notinsingleton => 166.97/25.05 (eqinunit => 166.97/25.05 (singletonsswitch => 166.97/25.05 (upairsetE => 166.97/25.05 (upairsetIL => 166.97/25.05 (upairsetIR => 166.97/25.05 (emptyE1 => 166.97/25.05 (vacuousDall => 166.97/25.05 (quantDeMorgan1 => 166.97/25.05 (quantDeMorgan2 => 166.97/25.05 (quantDeMorgan3 => 166.97/25.05 (quantDeMorgan4 => 166.97/25.05 (prop2setI => 166.97/25.05 (prop2set2propI => 166.97/25.05 (notdexE => 166.97/25.05 (notdallE => 166.97/25.05 (exuI1 => 166.97/25.05 (exuI3 => 166.97/25.05 (exuI2 => 166.97/25.05 (inCongP => 166.97/25.05 (in__Cong => 166.97/25.05 (exuE3u => 166.97/25.05 (exu__Cong => 166.97/25.05 (emptyset__Cong => 166.97/25.05 (setadjoin__Cong => 166.97/25.05 (powerset__Cong => 166.97/25.05 (setunion__Cong => 166.97/25.05 (omega__Cong => 166.97/25.05 (exuEu => 166.97/25.05 (descr__Cong => 166.97/25.05 (dsetconstr__Cong => 166.97/25.05 (subsetI1 => 166.97/25.05 (eqimpsubset2 => 166.97/25.05 (eqimpsubset1 => 166.97/25.05 (subsetI2 => 166.97/25.05 (emptysetsubset => 166.97/25.05 (subsetE => 166.97/25.05 (subsetE2 => 166.97/25.05 (notsubsetI => 166.97/25.05 (notequalI1 => 166.97/25.05 (notequalI2 => 166.97/25.05 (subsetRefl => 166.97/25.05 (subsetTrans => 166.97/25.05 (setadjoinSub => 166.97/25.05 (setadjoinSub2 => 166.97/25.05 (subset2powerset => 166.97/25.05 (setextsub => 166.97/25.05 (subsetemptysetimpeq => 166.97/25.05 (powersetI1 => 166.97/25.05 (powersetE1 => 166.97/25.05 (inPowerset => 166.97/25.05 (powersetsubset => 166.97/25.05 (sepInPowerset => 166.97/25.05 (sepSubset => 166.97/25.05 (binunionIL => 166.97/25.05 (upairset2IR => 166.97/25.05 (binunionIR => 166.97/25.05 ( 166.97/25.05 binunionEcases => 166.97/25.05 ( 166.97/25.05 binunionE => 166.97/25.05 ( 166.97/25.05 binunionLsub => 166.97/25.05 ( 166.97/25.05 binunionRsub => 166.97/25.05 ( 166.97/25.05 binintersectI => 166.97/25.05 ( 166.97/25.05 binintersectSubset5 => 166.97/25.05 ( 166.97/25.05 binintersectEL => 166.97/25.05 ( 166.97/25.05 binintersectLsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset2 => 166.97/25.05 ( 166.97/25.05 binintersectSubset3 => 166.97/25.05 ( 166.97/25.05 binintersectER => 166.97/25.05 ( 166.97/25.05 disjointsetsI1 => 166.97/25.05 ( 166.97/25.05 binintersectRsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset4 => 166.97/25.05 ( 166.97/25.05 binintersectSubset1 => 166.97/25.05 ( 166.97/25.05 bs114d => 166.97/25.05 ( 166.97/25.05 setminusI => 166.97/25.05 ( 166.97/25.05 setminusEL => 166.97/25.05 ( 166.97/25.05 setminusER => 166.97/25.05 ( 166.97/25.05 setminusSubset2 => 166.97/25.05 ( 166.97/25.05 setminusERneg => 166.97/25.05 ( 166.97/25.05 setminusELneg => 166.97/25.05 ( 166.97/25.05 setminusILneg => 166.97/25.05 ( 166.97/25.05 setminusIRneg => 166.97/25.05 ( 166.97/25.05 setminusLsub => 166.97/25.05 ( 166.97/25.05 setminusSubset1 => 166.97/25.05 ( 166.97/25.05 symdiffE => 166.97/25.05 ( 166.97/25.05 symdiffI1 => 166.97/25.05 ( 166.97/25.05 symdiffI2 => 166.97/25.05 ( 166.97/25.05 symdiffIneg1 => 166.97/25.05 ( 166.97/25.05 symdiffIneg2 => 166.97/25.05 ( 166.97/25.05 secondinupair => 166.97/25.05 ( 166.97/25.05 setukpairIL => 166.97/25.05 ( 166.97/25.05 setukpairIR => 166.97/25.05 ( 166.97/25.05 kpairiskpair => 166.97/25.05 ( 166.97/25.05 kpairp => 166.97/25.05 ( 166.97/25.05 singletonsubset => 166.97/25.05 ( 166.97/25.05 singletoninpowerset => 166.97/25.05 ( 166.97/25.05 singletoninpowunion => 166.97/25.05 ( 166.97/25.05 upairset2E => 166.97/25.05 ( 166.97/25.05 upairsubunion => 166.97/25.05 ( 166.97/25.05 upairinpowunion => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem1 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem2 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem3 => 166.97/25.05 ( 166.97/25.05 cartprodpairin => 166.97/25.05 ( 166.97/25.05 cartprodmempair1 => 166.97/25.05 ( 166.97/25.05 cartprodmempair => 166.97/25.05 ( 166.97/25.05 setunionE2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton1 => 166.97/25.05 ( 166.97/25.05 setunionsingleton2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton => 166.97/25.05 ( 166.97/25.05 singletonprop => 166.97/25.05 ( 166.97/25.05 ex1E1 => 166.97/25.05 ( 166.97/25.05 ex1I => 166.97/25.05 ( 166.97/25.05 ex1I2 => 166.97/25.05 ( 166.97/25.05 singletonsuniq => 166.97/25.05 ( 166.97/25.05 setukpairinjL1 => 166.97/25.05 ( 166.97/25.05 kfstsingleton => 166.97/25.05 ( 166.97/25.05 theprop => 166.97/25.05 ( 166.97/25.05 kfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodfstin => 166.97/25.05 ( 166.97/25.05 setukpairinjL2 => 166.97/25.05 ( 166.97/25.05 setukpairinjL => 166.97/25.05 ( 166.97/25.05 setukpairinjR11 => 166.97/25.05 ( 166.97/25.05 setukpairinjR12 => 166.97/25.05 ( 166.97/25.05 setukpairinjR1 => 166.97/25.05 ( 166.97/25.05 upairequniteq => 166.97/25.05 ( 166.97/25.05 setukpairinjR2 => 166.97/25.05 ( 166.97/25.05 setukpairinjR => 166.97/25.05 ( 166.97/25.05 ksndsingleton => 166.97/25.05 ( 166.97/25.05 ksndpairEq => 166.97/25.05 ( 166.97/25.05 kpairsurjEq => 166.97/25.05 ( 166.97/25.05 cartprodsndin => 166.97/25.05 ( 166.97/25.05 cartprodpairmemEL => 166.97/25.05 ( 166.97/25.05 cartprodpairmemER => 166.97/25.05 ( 166.97/25.05 cartprodmempaircEq => 166.97/25.05 ( 166.97/25.05 cartprodfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodsndpairEq => 166.97/25.05 ( 166.97/25.05 cartprodpairsurjEq => 166.97/25.05 ( 166.97/25.05 dpsetconstrI => 166.97/25.05 ( 166.97/25.05 dpsetconstrSub => 166.97/25.05 ( 166.97/25.05 setOfPairsIsBReln => 166.97/25.05 ( 166.97/25.05 !!(( 166.97/25.05 ^[Y0 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y1 : $i]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y2 : ( 166.97/25.05 $i * $i) > $o]: 166.97/25.05 (!!(( 166.97/25.05 ^[Y3 : $i]: 166.97/25.05 (in(Y3, 166.97/25.05 Y0) => 166.97/25.05 (!!(( 166.97/25.05 ^[Y4 : $i]: 166.97/25.05 (in(Y4, 166.97/25.05 Y1) => 166.97/25.05 (in( 166.97/25.05 kpair( 166.97/25.05 Y3, Y4), 166.97/25.05 dpsetconstr( 166.97/25.05 Y0, Y1, 166.97/25.05 Y2)) => 166.97/25.05 Y2(Y3, 166.97/25.05 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.05 inference('lazy_cnf_imply', [status(thm)], ['39'])). 166.97/25.05 tff('41', plain, 166.97/25.05 ~ (subPowSU => 166.97/25.05 (exuE2 => 166.97/25.05 (nonemptyImpWitness => 166.97/25.05 (uniqinunit => 166.97/25.05 (notinsingleton => 166.97/25.05 (eqinunit => 166.97/25.05 (singletonsswitch => 166.97/25.05 (upairsetE => 166.97/25.05 (upairsetIL => 166.97/25.05 (upairsetIR => 166.97/25.05 (emptyE1 => 166.97/25.05 (vacuousDall => 166.97/25.05 (quantDeMorgan1 => 166.97/25.05 (quantDeMorgan2 => 166.97/25.05 (quantDeMorgan3 => 166.97/25.05 (quantDeMorgan4 => 166.97/25.05 (prop2setI => 166.97/25.05 (prop2set2propI => 166.97/25.05 (notdexE => 166.97/25.05 (notdallE => 166.97/25.05 (exuI1 => 166.97/25.05 (exuI3 => 166.97/25.05 (exuI2 => 166.97/25.05 (inCongP => 166.97/25.05 (in__Cong => 166.97/25.05 (exuE3u => 166.97/25.05 (exu__Cong => 166.97/25.05 (emptyset__Cong => 166.97/25.05 (setadjoin__Cong => 166.97/25.05 (powerset__Cong => 166.97/25.05 (setunion__Cong => 166.97/25.05 (omega__Cong => 166.97/25.05 (exuEu => 166.97/25.05 (descr__Cong => 166.97/25.05 (dsetconstr__Cong => 166.97/25.05 (subsetI1 => 166.97/25.05 (eqimpsubset2 => 166.97/25.05 (eqimpsubset1 => 166.97/25.05 (subsetI2 => 166.97/25.05 (emptysetsubset => 166.97/25.05 (subsetE => 166.97/25.05 (subsetE2 => 166.97/25.05 (notsubsetI => 166.97/25.05 (notequalI1 => 166.97/25.05 (notequalI2 => 166.97/25.05 (subsetRefl => 166.97/25.05 (subsetTrans => 166.97/25.05 (setadjoinSub => 166.97/25.05 (setadjoinSub2 => 166.97/25.05 (subset2powerset => 166.97/25.05 (setextsub => 166.97/25.05 (subsetemptysetimpeq => 166.97/25.05 (powersetI1 => 166.97/25.05 (powersetE1 => 166.97/25.05 (inPowerset => 166.97/25.05 (powersetsubset => 166.97/25.05 (sepInPowerset => 166.97/25.05 (sepSubset => 166.97/25.05 (binunionIL => 166.97/25.05 (upairset2IR => 166.97/25.05 (binunionIR => 166.97/25.05 (binunionEcases => 166.97/25.05 ( 166.97/25.05 binunionE => 166.97/25.05 ( 166.97/25.05 binunionLsub => 166.97/25.05 ( 166.97/25.05 binunionRsub => 166.97/25.05 ( 166.97/25.05 binintersectI => 166.97/25.05 ( 166.97/25.05 binintersectSubset5 => 166.97/25.05 ( 166.97/25.05 binintersectEL => 166.97/25.05 ( 166.97/25.05 binintersectLsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset2 => 166.97/25.05 ( 166.97/25.05 binintersectSubset3 => 166.97/25.05 ( 166.97/25.05 binintersectER => 166.97/25.05 ( 166.97/25.05 disjointsetsI1 => 166.97/25.05 ( 166.97/25.05 binintersectRsub => 166.97/25.05 ( 166.97/25.05 binintersectSubset4 => 166.97/25.05 ( 166.97/25.05 binintersectSubset1 => 166.97/25.05 ( 166.97/25.05 bs114d => 166.97/25.05 ( 166.97/25.05 setminusI => 166.97/25.05 ( 166.97/25.05 setminusEL => 166.97/25.05 ( 166.97/25.05 setminusER => 166.97/25.05 ( 166.97/25.05 setminusSubset2 => 166.97/25.05 ( 166.97/25.05 setminusERneg => 166.97/25.05 ( 166.97/25.05 setminusELneg => 166.97/25.05 ( 166.97/25.05 setminusILneg => 166.97/25.05 ( 166.97/25.05 setminusIRneg => 166.97/25.05 ( 166.97/25.05 setminusLsub => 166.97/25.05 ( 166.97/25.05 setminusSubset1 => 166.97/25.05 ( 166.97/25.05 symdiffE => 166.97/25.05 ( 166.97/25.05 symdiffI1 => 166.97/25.05 ( 166.97/25.05 symdiffI2 => 166.97/25.05 ( 166.97/25.05 symdiffIneg1 => 166.97/25.05 ( 166.97/25.05 symdiffIneg2 => 166.97/25.05 ( 166.97/25.05 secondinupair => 166.97/25.05 ( 166.97/25.05 setukpairIL => 166.97/25.05 ( 166.97/25.05 setukpairIR => 166.97/25.05 ( 166.97/25.05 kpairiskpair => 166.97/25.05 ( 166.97/25.05 kpairp => 166.97/25.05 ( 166.97/25.05 singletonsubset => 166.97/25.05 ( 166.97/25.05 singletoninpowerset => 166.97/25.05 ( 166.97/25.05 singletoninpowunion => 166.97/25.05 ( 166.97/25.05 upairset2E => 166.97/25.05 ( 166.97/25.05 upairsubunion => 166.97/25.05 ( 166.97/25.05 upairinpowunion => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem1 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem2 => 166.97/25.05 ( 166.97/25.05 ubforcartprodlem3 => 166.97/25.05 ( 166.97/25.05 cartprodpairin => 166.97/25.05 ( 166.97/25.05 cartprodmempair1 => 166.97/25.05 ( 166.97/25.05 cartprodmempair => 166.97/25.05 ( 166.97/25.05 setunionE2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton1 => 166.97/25.05 ( 166.97/25.05 setunionsingleton2 => 166.97/25.05 ( 166.97/25.05 setunionsingleton => 166.97/25.05 ( 166.97/25.05 singletonprop => 166.97/25.05 ( 166.97/25.05 ex1E1 => 166.97/25.05 ( 166.97/25.05 ex1I => 166.97/25.05 ( 166.97/25.05 ex1I2 => 166.97/25.05 ( 166.97/25.05 singletonsuniq => 166.97/25.05 ( 166.97/25.05 setukpairinjL1 => 166.97/25.05 ( 166.97/25.05 kfstsingleton => 166.97/25.05 ( 166.97/25.05 theprop => 166.97/25.05 ( 166.97/25.05 kfstpairEq => 166.97/25.05 ( 166.97/25.05 cartprodfstin => 166.97/25.05 ( 166.97/25.05 setukpairinjL2 => 166.97/25.05 ( 166.97/25.05 setukpairinjL => 166.97/25.05 ( 166.97/25.05 setukpairinjR11 => 166.97/25.05 ( 166.97/25.05 setukpairinjR12 => 166.97/25.05 ( 166.97/25.05 setukpairinjR1 => 166.97/25.05 ( 166.97/25.05 upairequniteq => 166.97/25.05 ( 166.97/25.05 setukpairinjR2 => 166.97/25.05 ( 166.97/25.05 setukpairinjR => 166.97/25.05 ( 166.97/25.06 ksndsingleton => 166.97/25.06 ( 166.97/25.06 ksndpairEq => 166.97/25.06 ( 166.97/25.06 kpairsurjEq => 166.97/25.06 ( 166.97/25.06 cartprodsndin => 166.97/25.06 ( 166.97/25.06 cartprodpairmemEL => 166.97/25.06 ( 166.97/25.06 cartprodpairmemER => 166.97/25.06 ( 166.97/25.06 cartprodmempaircEq => 166.97/25.06 ( 166.97/25.06 cartprodfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodsndpairEq => 166.97/25.06 ( 166.97/25.06 cartprodpairsurjEq => 166.97/25.06 ( 166.97/25.06 dpsetconstrI => 166.97/25.06 ( 166.97/25.06 dpsetconstrSub => 166.97/25.06 ( 166.97/25.06 setOfPairsIsBReln => 166.97/25.06 ( 166.97/25.06 !!(( 166.97/25.06 ^[Y0 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y1 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y2 : ( 166.97/25.06 $i * $i) > $o]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y3 : $i]: 166.97/25.06 (in(Y3, 166.97/25.06 Y0) => 166.97/25.06 (!!(( 166.97/25.06 ^[Y4 : $i]: 166.97/25.06 (in(Y4, 166.97/25.06 Y1) => 166.97/25.06 (in( 166.97/25.06 kpair( 166.97/25.06 Y3, Y4), 166.97/25.06 dpsetconstr( 166.97/25.06 Y0, Y1, 166.97/25.06 Y2)) => 166.97/25.06 Y2(Y3, 166.97/25.06 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.06 inference('lazy_cnf_imply', [status(thm)], ['40'])). 166.97/25.06 tff('42', plain, 166.97/25.06 ~ (exuE2 => 166.97/25.06 (nonemptyImpWitness => 166.97/25.06 (uniqinunit => 166.97/25.06 (notinsingleton => 166.97/25.06 (eqinunit => 166.97/25.06 (singletonsswitch => 166.97/25.06 (upairsetE => 166.97/25.06 (upairsetIL => 166.97/25.06 (upairsetIR => 166.97/25.06 (emptyE1 => 166.97/25.06 (vacuousDall => 166.97/25.06 (quantDeMorgan1 => 166.97/25.06 (quantDeMorgan2 => 166.97/25.06 (quantDeMorgan3 => 166.97/25.06 (quantDeMorgan4 => 166.97/25.06 (prop2setI => 166.97/25.06 (prop2set2propI => 166.97/25.06 (notdexE => 166.97/25.06 (notdallE => 166.97/25.06 (exuI1 => 166.97/25.06 (exuI3 => 166.97/25.06 (exuI2 => 166.97/25.06 (inCongP => 166.97/25.06 (in__Cong => 166.97/25.06 (exuE3u => 166.97/25.06 (exu__Cong => 166.97/25.06 (emptyset__Cong => 166.97/25.06 (setadjoin__Cong => 166.97/25.06 (powerset__Cong => 166.97/25.06 (setunion__Cong => 166.97/25.06 (omega__Cong => 166.97/25.06 (exuEu => 166.97/25.06 (descr__Cong => 166.97/25.06 (dsetconstr__Cong => 166.97/25.06 (subsetI1 => 166.97/25.06 (eqimpsubset2 => 166.97/25.06 (eqimpsubset1 => 166.97/25.06 (subsetI2 => 166.97/25.06 (emptysetsubset => 166.97/25.06 (subsetE => 166.97/25.06 (subsetE2 => 166.97/25.06 (notsubsetI => 166.97/25.06 (notequalI1 => 166.97/25.06 (notequalI2 => 166.97/25.06 (subsetRefl => 166.97/25.06 (subsetTrans => 166.97/25.06 (setadjoinSub => 166.97/25.06 (setadjoinSub2 => 166.97/25.06 (subset2powerset => 166.97/25.06 (setextsub => 166.97/25.06 (subsetemptysetimpeq => 166.97/25.06 (powersetI1 => 166.97/25.06 (powersetE1 => 166.97/25.06 (inPowerset => 166.97/25.06 (powersetsubset => 166.97/25.06 (sepInPowerset => 166.97/25.06 (sepSubset => 166.97/25.06 (binunionIL => 166.97/25.06 (upairset2IR => 166.97/25.06 (binunionIR => 166.97/25.06 (binunionEcases => 166.97/25.06 (binunionE => 166.97/25.06 ( 166.97/25.06 binunionLsub => 166.97/25.06 ( 166.97/25.06 binunionRsub => 166.97/25.06 ( 166.97/25.06 binintersectI => 166.97/25.06 ( 166.97/25.06 binintersectSubset5 => 166.97/25.06 ( 166.97/25.06 binintersectEL => 166.97/25.06 ( 166.97/25.06 binintersectLsub => 166.97/25.06 ( 166.97/25.06 binintersectSubset2 => 166.97/25.06 ( 166.97/25.06 binintersectSubset3 => 166.97/25.06 ( 166.97/25.06 binintersectER => 166.97/25.06 ( 166.97/25.06 disjointsetsI1 => 166.97/25.06 ( 166.97/25.06 binintersectRsub => 166.97/25.06 ( 166.97/25.06 binintersectSubset4 => 166.97/25.06 ( 166.97/25.06 binintersectSubset1 => 166.97/25.06 ( 166.97/25.06 bs114d => 166.97/25.06 ( 166.97/25.06 setminusI => 166.97/25.06 ( 166.97/25.06 setminusEL => 166.97/25.06 ( 166.97/25.06 setminusER => 166.97/25.06 ( 166.97/25.06 setminusSubset2 => 166.97/25.06 ( 166.97/25.06 setminusERneg => 166.97/25.06 ( 166.97/25.06 setminusELneg => 166.97/25.06 ( 166.97/25.06 setminusILneg => 166.97/25.06 ( 166.97/25.06 setminusIRneg => 166.97/25.06 ( 166.97/25.06 setminusLsub => 166.97/25.06 ( 166.97/25.06 setminusSubset1 => 166.97/25.06 ( 166.97/25.06 symdiffE => 166.97/25.06 ( 166.97/25.06 symdiffI1 => 166.97/25.06 ( 166.97/25.06 symdiffI2 => 166.97/25.06 ( 166.97/25.06 symdiffIneg1 => 166.97/25.06 ( 166.97/25.06 symdiffIneg2 => 166.97/25.06 ( 166.97/25.06 secondinupair => 166.97/25.06 ( 166.97/25.06 setukpairIL => 166.97/25.06 ( 166.97/25.06 setukpairIR => 166.97/25.06 ( 166.97/25.06 kpairiskpair => 166.97/25.06 ( 166.97/25.06 kpairp => 166.97/25.06 ( 166.97/25.06 singletonsubset => 166.97/25.06 ( 166.97/25.06 singletoninpowerset => 166.97/25.06 ( 166.97/25.06 singletoninpowunion => 166.97/25.06 ( 166.97/25.06 upairset2E => 166.97/25.06 ( 166.97/25.06 upairsubunion => 166.97/25.06 ( 166.97/25.06 upairinpowunion => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem1 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem2 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem3 => 166.97/25.06 ( 166.97/25.06 cartprodpairin => 166.97/25.06 ( 166.97/25.06 cartprodmempair1 => 166.97/25.06 ( 166.97/25.06 cartprodmempair => 166.97/25.06 ( 166.97/25.06 setunionE2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton1 => 166.97/25.06 ( 166.97/25.06 setunionsingleton2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton => 166.97/25.06 ( 166.97/25.06 singletonprop => 166.97/25.06 ( 166.97/25.06 ex1E1 => 166.97/25.06 ( 166.97/25.06 ex1I => 166.97/25.06 ( 166.97/25.06 ex1I2 => 166.97/25.06 ( 166.97/25.06 singletonsuniq => 166.97/25.06 ( 166.97/25.06 setukpairinjL1 => 166.97/25.06 ( 166.97/25.06 kfstsingleton => 166.97/25.06 ( 166.97/25.06 theprop => 166.97/25.06 ( 166.97/25.06 kfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodfstin => 166.97/25.06 ( 166.97/25.06 setukpairinjL2 => 166.97/25.06 ( 166.97/25.06 setukpairinjL => 166.97/25.06 ( 166.97/25.06 setukpairinjR11 => 166.97/25.06 ( 166.97/25.06 setukpairinjR12 => 166.97/25.06 ( 166.97/25.06 setukpairinjR1 => 166.97/25.06 ( 166.97/25.06 upairequniteq => 166.97/25.06 ( 166.97/25.06 setukpairinjR2 => 166.97/25.06 ( 166.97/25.06 setukpairinjR => 166.97/25.06 ( 166.97/25.06 ksndsingleton => 166.97/25.06 ( 166.97/25.06 ksndpairEq => 166.97/25.06 ( 166.97/25.06 kpairsurjEq => 166.97/25.06 ( 166.97/25.06 cartprodsndin => 166.97/25.06 ( 166.97/25.06 cartprodpairmemEL => 166.97/25.06 ( 166.97/25.06 cartprodpairmemER => 166.97/25.06 ( 166.97/25.06 cartprodmempaircEq => 166.97/25.06 ( 166.97/25.06 cartprodfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodsndpairEq => 166.97/25.06 ( 166.97/25.06 cartprodpairsurjEq => 166.97/25.06 ( 166.97/25.06 dpsetconstrI => 166.97/25.06 ( 166.97/25.06 dpsetconstrSub => 166.97/25.06 ( 166.97/25.06 setOfPairsIsBReln => 166.97/25.06 ( 166.97/25.06 !!(( 166.97/25.06 ^[Y0 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y1 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y2 : ( 166.97/25.06 $i * $i) > $o]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y3 : $i]: 166.97/25.06 (in(Y3, 166.97/25.06 Y0) => 166.97/25.06 (!!(( 166.97/25.06 ^[Y4 : $i]: 166.97/25.06 (in(Y4, 166.97/25.06 Y1) => 166.97/25.06 (in( 166.97/25.06 kpair( 166.97/25.06 Y3, Y4), 166.97/25.06 dpsetconstr( 166.97/25.06 Y0, Y1, 166.97/25.06 Y2)) => 166.97/25.06 Y2(Y3, 166.97/25.06 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.06 inference('lazy_cnf_imply', [status(thm)], ['41'])). 166.97/25.06 tff('43', plain, 166.97/25.06 ~ (nonemptyImpWitness => 166.97/25.06 (uniqinunit => 166.97/25.06 (notinsingleton => 166.97/25.06 (eqinunit => 166.97/25.06 (singletonsswitch => 166.97/25.06 (upairsetE => 166.97/25.06 (upairsetIL => 166.97/25.06 (upairsetIR => 166.97/25.06 (emptyE1 => 166.97/25.06 (vacuousDall => 166.97/25.06 (quantDeMorgan1 => 166.97/25.06 (quantDeMorgan2 => 166.97/25.06 (quantDeMorgan3 => 166.97/25.06 (quantDeMorgan4 => 166.97/25.06 (prop2setI => 166.97/25.06 (prop2set2propI => 166.97/25.06 (notdexE => 166.97/25.06 (notdallE => 166.97/25.06 (exuI1 => 166.97/25.06 (exuI3 => 166.97/25.06 (exuI2 => 166.97/25.06 (inCongP => 166.97/25.06 (in__Cong => 166.97/25.06 (exuE3u => 166.97/25.06 (exu__Cong => 166.97/25.06 (emptyset__Cong => 166.97/25.06 (setadjoin__Cong => 166.97/25.06 (powerset__Cong => 166.97/25.06 (setunion__Cong => 166.97/25.06 (omega__Cong => 166.97/25.06 (exuEu => 166.97/25.06 (descr__Cong => 166.97/25.06 (dsetconstr__Cong => 166.97/25.06 (subsetI1 => 166.97/25.06 (eqimpsubset2 => 166.97/25.06 (eqimpsubset1 => 166.97/25.06 (subsetI2 => 166.97/25.06 (emptysetsubset => 166.97/25.06 (subsetE => 166.97/25.06 (subsetE2 => 166.97/25.06 (notsubsetI => 166.97/25.06 (notequalI1 => 166.97/25.06 (notequalI2 => 166.97/25.06 (subsetRefl => 166.97/25.06 (subsetTrans => 166.97/25.06 (setadjoinSub => 166.97/25.06 (setadjoinSub2 => 166.97/25.06 (subset2powerset => 166.97/25.06 (setextsub => 166.97/25.06 (subsetemptysetimpeq => 166.97/25.06 (powersetI1 => 166.97/25.06 (powersetE1 => 166.97/25.06 (inPowerset => 166.97/25.06 (powersetsubset => 166.97/25.06 (sepInPowerset => 166.97/25.06 (sepSubset => 166.97/25.06 (binunionIL => 166.97/25.06 (upairset2IR => 166.97/25.06 (binunionIR => 166.97/25.06 (binunionEcases => 166.97/25.06 (binunionE => 166.97/25.06 (binunionLsub => 166.97/25.06 ( 166.97/25.06 binunionRsub => 166.97/25.06 ( 166.97/25.06 binintersectI => 166.97/25.06 ( 166.97/25.06 binintersectSubset5 => 166.97/25.06 ( 166.97/25.06 binintersectEL => 166.97/25.06 ( 166.97/25.06 binintersectLsub => 166.97/25.06 ( 166.97/25.06 binintersectSubset2 => 166.97/25.06 ( 166.97/25.06 binintersectSubset3 => 166.97/25.06 ( 166.97/25.06 binintersectER => 166.97/25.06 ( 166.97/25.06 disjointsetsI1 => 166.97/25.06 ( 166.97/25.06 binintersectRsub => 166.97/25.06 ( 166.97/25.06 binintersectSubset4 => 166.97/25.06 ( 166.97/25.06 binintersectSubset1 => 166.97/25.06 ( 166.97/25.06 bs114d => 166.97/25.06 ( 166.97/25.06 setminusI => 166.97/25.06 ( 166.97/25.06 setminusEL => 166.97/25.06 ( 166.97/25.06 setminusER => 166.97/25.06 ( 166.97/25.06 setminusSubset2 => 166.97/25.06 ( 166.97/25.06 setminusERneg => 166.97/25.06 ( 166.97/25.06 setminusELneg => 166.97/25.06 ( 166.97/25.06 setminusILneg => 166.97/25.06 ( 166.97/25.06 setminusIRneg => 166.97/25.06 ( 166.97/25.06 setminusLsub => 166.97/25.06 ( 166.97/25.06 setminusSubset1 => 166.97/25.06 ( 166.97/25.06 symdiffE => 166.97/25.06 ( 166.97/25.06 symdiffI1 => 166.97/25.06 ( 166.97/25.06 symdiffI2 => 166.97/25.06 ( 166.97/25.06 symdiffIneg1 => 166.97/25.06 ( 166.97/25.06 symdiffIneg2 => 166.97/25.06 ( 166.97/25.06 secondinupair => 166.97/25.06 ( 166.97/25.06 setukpairIL => 166.97/25.06 ( 166.97/25.06 setukpairIR => 166.97/25.06 ( 166.97/25.06 kpairiskpair => 166.97/25.06 ( 166.97/25.06 kpairp => 166.97/25.06 ( 166.97/25.06 singletonsubset => 166.97/25.06 ( 166.97/25.06 singletoninpowerset => 166.97/25.06 ( 166.97/25.06 singletoninpowunion => 166.97/25.06 ( 166.97/25.06 upairset2E => 166.97/25.06 ( 166.97/25.06 upairsubunion => 166.97/25.06 ( 166.97/25.06 upairinpowunion => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem1 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem2 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem3 => 166.97/25.06 ( 166.97/25.06 cartprodpairin => 166.97/25.06 ( 166.97/25.06 cartprodmempair1 => 166.97/25.06 ( 166.97/25.06 cartprodmempair => 166.97/25.06 ( 166.97/25.06 setunionE2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton1 => 166.97/25.06 ( 166.97/25.06 setunionsingleton2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton => 166.97/25.06 ( 166.97/25.06 singletonprop => 166.97/25.06 ( 166.97/25.06 ex1E1 => 166.97/25.06 ( 166.97/25.06 ex1I => 166.97/25.06 ( 166.97/25.06 ex1I2 => 166.97/25.06 ( 166.97/25.06 singletonsuniq => 166.97/25.06 ( 166.97/25.06 setukpairinjL1 => 166.97/25.06 ( 166.97/25.06 kfstsingleton => 166.97/25.06 ( 166.97/25.06 theprop => 166.97/25.06 ( 166.97/25.06 kfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodfstin => 166.97/25.06 ( 166.97/25.06 setukpairinjL2 => 166.97/25.06 ( 166.97/25.06 setukpairinjL => 166.97/25.06 ( 166.97/25.06 setukpairinjR11 => 166.97/25.06 ( 166.97/25.06 setukpairinjR12 => 166.97/25.06 ( 166.97/25.06 setukpairinjR1 => 166.97/25.06 ( 166.97/25.06 upairequniteq => 166.97/25.06 ( 166.97/25.06 setukpairinjR2 => 166.97/25.06 ( 166.97/25.06 setukpairinjR => 166.97/25.06 ( 166.97/25.06 ksndsingleton => 166.97/25.06 ( 166.97/25.06 ksndpairEq => 166.97/25.06 ( 166.97/25.06 kpairsurjEq => 166.97/25.06 ( 166.97/25.06 cartprodsndin => 166.97/25.06 ( 166.97/25.06 cartprodpairmemEL => 166.97/25.06 ( 166.97/25.06 cartprodpairmemER => 166.97/25.06 ( 166.97/25.06 cartprodmempaircEq => 166.97/25.06 ( 166.97/25.06 cartprodfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodsndpairEq => 166.97/25.06 ( 166.97/25.06 cartprodpairsurjEq => 166.97/25.06 ( 166.97/25.06 dpsetconstrI => 166.97/25.06 ( 166.97/25.06 dpsetconstrSub => 166.97/25.06 ( 166.97/25.06 setOfPairsIsBReln => 166.97/25.06 ( 166.97/25.06 !!(( 166.97/25.06 ^[Y0 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y1 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y2 : ( 166.97/25.06 $i * $i) > $o]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y3 : $i]: 166.97/25.06 (in(Y3, 166.97/25.06 Y0) => 166.97/25.06 (!!(( 166.97/25.06 ^[Y4 : $i]: 166.97/25.06 (in(Y4, 166.97/25.06 Y1) => 166.97/25.06 (in( 166.97/25.06 kpair( 166.97/25.06 Y3, Y4), 166.97/25.06 dpsetconstr( 166.97/25.06 Y0, Y1, 166.97/25.06 Y2)) => 166.97/25.06 Y2(Y3, 166.97/25.06 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.06 inference('lazy_cnf_imply', [status(thm)], ['42'])). 166.97/25.06 tff('44', plain, 166.97/25.06 ~ (uniqinunit => 166.97/25.06 (notinsingleton => 166.97/25.06 (eqinunit => 166.97/25.06 (singletonsswitch => 166.97/25.06 (upairsetE => 166.97/25.06 (upairsetIL => 166.97/25.06 (upairsetIR => 166.97/25.06 (emptyE1 => 166.97/25.06 (vacuousDall => 166.97/25.06 (quantDeMorgan1 => 166.97/25.06 (quantDeMorgan2 => 166.97/25.06 (quantDeMorgan3 => 166.97/25.06 (quantDeMorgan4 => 166.97/25.06 (prop2setI => 166.97/25.06 (prop2set2propI => 166.97/25.06 (notdexE => 166.97/25.06 (notdallE => 166.97/25.06 (exuI1 => 166.97/25.06 (exuI3 => 166.97/25.06 (exuI2 => 166.97/25.06 (inCongP => 166.97/25.06 (in__Cong => 166.97/25.06 (exuE3u => 166.97/25.06 (exu__Cong => 166.97/25.06 (emptyset__Cong => 166.97/25.06 (setadjoin__Cong => 166.97/25.06 (powerset__Cong => 166.97/25.06 (setunion__Cong => 166.97/25.06 (omega__Cong => 166.97/25.06 (exuEu => 166.97/25.06 (descr__Cong => 166.97/25.06 (dsetconstr__Cong => 166.97/25.06 (subsetI1 => 166.97/25.06 (eqimpsubset2 => 166.97/25.06 (eqimpsubset1 => 166.97/25.06 (subsetI2 => 166.97/25.06 (emptysetsubset => 166.97/25.06 (subsetE => 166.97/25.06 (subsetE2 => 166.97/25.06 (notsubsetI => 166.97/25.06 (notequalI1 => 166.97/25.06 (notequalI2 => 166.97/25.06 (subsetRefl => 166.97/25.06 (subsetTrans => 166.97/25.06 (setadjoinSub => 166.97/25.06 (setadjoinSub2 => 166.97/25.06 (subset2powerset => 166.97/25.06 (setextsub => 166.97/25.06 (subsetemptysetimpeq => 166.97/25.06 (powersetI1 => 166.97/25.06 (powersetE1 => 166.97/25.06 (inPowerset => 166.97/25.06 (powersetsubset => 166.97/25.06 (sepInPowerset => 166.97/25.06 (sepSubset => 166.97/25.06 (binunionIL => 166.97/25.06 (upairset2IR => 166.97/25.06 (binunionIR => 166.97/25.06 (binunionEcases => 166.97/25.06 (binunionE => 166.97/25.06 (binunionLsub => 166.97/25.06 (binunionRsub => 166.97/25.06 ( 166.97/25.06 binintersectI => 166.97/25.06 ( 166.97/25.06 binintersectSubset5 => 166.97/25.06 ( 166.97/25.06 binintersectEL => 166.97/25.06 ( 166.97/25.06 binintersectLsub => 166.97/25.06 ( 166.97/25.06 binintersectSubset2 => 166.97/25.06 ( 166.97/25.06 binintersectSubset3 => 166.97/25.06 ( 166.97/25.06 binintersectER => 166.97/25.06 ( 166.97/25.06 disjointsetsI1 => 166.97/25.06 ( 166.97/25.06 binintersectRsub => 166.97/25.06 ( 166.97/25.06 binintersectSubset4 => 166.97/25.06 ( 166.97/25.06 binintersectSubset1 => 166.97/25.06 ( 166.97/25.06 bs114d => 166.97/25.06 ( 166.97/25.06 setminusI => 166.97/25.06 ( 166.97/25.06 setminusEL => 166.97/25.06 ( 166.97/25.06 setminusER => 166.97/25.06 ( 166.97/25.06 setminusSubset2 => 166.97/25.06 ( 166.97/25.06 setminusERneg => 166.97/25.06 ( 166.97/25.06 setminusELneg => 166.97/25.06 ( 166.97/25.06 setminusILneg => 166.97/25.06 ( 166.97/25.06 setminusIRneg => 166.97/25.06 ( 166.97/25.06 setminusLsub => 166.97/25.06 ( 166.97/25.06 setminusSubset1 => 166.97/25.06 ( 166.97/25.06 symdiffE => 166.97/25.06 ( 166.97/25.06 symdiffI1 => 166.97/25.06 ( 166.97/25.06 symdiffI2 => 166.97/25.06 ( 166.97/25.06 symdiffIneg1 => 166.97/25.06 ( 166.97/25.06 symdiffIneg2 => 166.97/25.06 ( 166.97/25.06 secondinupair => 166.97/25.06 ( 166.97/25.06 setukpairIL => 166.97/25.06 ( 166.97/25.06 setukpairIR => 166.97/25.06 ( 166.97/25.06 kpairiskpair => 166.97/25.06 ( 166.97/25.06 kpairp => 166.97/25.06 ( 166.97/25.06 singletonsubset => 166.97/25.06 ( 166.97/25.06 singletoninpowerset => 166.97/25.06 ( 166.97/25.06 singletoninpowunion => 166.97/25.06 ( 166.97/25.06 upairset2E => 166.97/25.06 ( 166.97/25.06 upairsubunion => 166.97/25.06 ( 166.97/25.06 upairinpowunion => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem1 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem2 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem3 => 166.97/25.06 ( 166.97/25.06 cartprodpairin => 166.97/25.06 ( 166.97/25.06 cartprodmempair1 => 166.97/25.06 ( 166.97/25.06 cartprodmempair => 166.97/25.06 ( 166.97/25.06 setunionE2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton1 => 166.97/25.06 ( 166.97/25.06 setunionsingleton2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton => 166.97/25.06 ( 166.97/25.06 singletonprop => 166.97/25.06 ( 166.97/25.06 ex1E1 => 166.97/25.06 ( 166.97/25.06 ex1I => 166.97/25.06 ( 166.97/25.06 ex1I2 => 166.97/25.06 ( 166.97/25.06 singletonsuniq => 166.97/25.06 ( 166.97/25.06 setukpairinjL1 => 166.97/25.06 ( 166.97/25.06 kfstsingleton => 166.97/25.06 ( 166.97/25.06 theprop => 166.97/25.06 ( 166.97/25.06 kfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodfstin => 166.97/25.06 ( 166.97/25.06 setukpairinjL2 => 166.97/25.06 ( 166.97/25.06 setukpairinjL => 166.97/25.06 ( 166.97/25.06 setukpairinjR11 => 166.97/25.06 ( 166.97/25.06 setukpairinjR12 => 166.97/25.06 ( 166.97/25.06 setukpairinjR1 => 166.97/25.06 ( 166.97/25.06 upairequniteq => 166.97/25.06 ( 166.97/25.06 setukpairinjR2 => 166.97/25.06 ( 166.97/25.06 setukpairinjR => 166.97/25.06 ( 166.97/25.06 ksndsingleton => 166.97/25.06 ( 166.97/25.06 ksndpairEq => 166.97/25.06 ( 166.97/25.06 kpairsurjEq => 166.97/25.06 ( 166.97/25.06 cartprodsndin => 166.97/25.06 ( 166.97/25.06 cartprodpairmemEL => 166.97/25.06 ( 166.97/25.06 cartprodpairmemER => 166.97/25.06 ( 166.97/25.06 cartprodmempaircEq => 166.97/25.06 ( 166.97/25.06 cartprodfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodsndpairEq => 166.97/25.06 ( 166.97/25.06 cartprodpairsurjEq => 166.97/25.06 ( 166.97/25.06 dpsetconstrI => 166.97/25.06 ( 166.97/25.06 dpsetconstrSub => 166.97/25.06 ( 166.97/25.06 setOfPairsIsBReln => 166.97/25.06 ( 166.97/25.06 !!(( 166.97/25.06 ^[Y0 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y1 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y2 : ( 166.97/25.06 $i * $i) > $o]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y3 : $i]: 166.97/25.06 (in(Y3, 166.97/25.06 Y0) => 166.97/25.06 (!!(( 166.97/25.06 ^[Y4 : $i]: 166.97/25.06 (in(Y4, 166.97/25.06 Y1) => 166.97/25.06 (in( 166.97/25.06 kpair( 166.97/25.06 Y3, Y4), 166.97/25.06 dpsetconstr( 166.97/25.06 Y0, Y1, 166.97/25.06 Y2)) => 166.97/25.06 Y2(Y3, 166.97/25.06 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.06 inference('lazy_cnf_imply', [status(thm)], ['43'])). 166.97/25.06 tff('45', plain, 166.97/25.06 ~ (notinsingleton => 166.97/25.06 (eqinunit => 166.97/25.06 (singletonsswitch => 166.97/25.06 (upairsetE => 166.97/25.06 (upairsetIL => 166.97/25.06 (upairsetIR => 166.97/25.06 (emptyE1 => 166.97/25.06 (vacuousDall => 166.97/25.06 (quantDeMorgan1 => 166.97/25.06 (quantDeMorgan2 => 166.97/25.06 (quantDeMorgan3 => 166.97/25.06 (quantDeMorgan4 => 166.97/25.06 (prop2setI => 166.97/25.06 (prop2set2propI => 166.97/25.06 (notdexE => 166.97/25.06 (notdallE => 166.97/25.06 (exuI1 => 166.97/25.06 (exuI3 => 166.97/25.06 (exuI2 => 166.97/25.06 (inCongP => 166.97/25.06 (in__Cong => 166.97/25.06 (exuE3u => 166.97/25.06 (exu__Cong => 166.97/25.06 (emptyset__Cong => 166.97/25.06 (setadjoin__Cong => 166.97/25.06 (powerset__Cong => 166.97/25.06 (setunion__Cong => 166.97/25.06 (omega__Cong => 166.97/25.06 (exuEu => 166.97/25.06 (descr__Cong => 166.97/25.06 (dsetconstr__Cong => 166.97/25.06 (subsetI1 => 166.97/25.06 (eqimpsubset2 => 166.97/25.06 (eqimpsubset1 => 166.97/25.06 (subsetI2 => 166.97/25.06 (emptysetsubset => 166.97/25.06 (subsetE => 166.97/25.06 (subsetE2 => 166.97/25.06 (notsubsetI => 166.97/25.06 (notequalI1 => 166.97/25.06 (notequalI2 => 166.97/25.06 (subsetRefl => 166.97/25.06 (subsetTrans => 166.97/25.06 (setadjoinSub => 166.97/25.06 (setadjoinSub2 => 166.97/25.06 (subset2powerset => 166.97/25.06 (setextsub => 166.97/25.06 (subsetemptysetimpeq => 166.97/25.06 (powersetI1 => 166.97/25.06 (powersetE1 => 166.97/25.06 (inPowerset => 166.97/25.06 (powersetsubset => 166.97/25.06 (sepInPowerset => 166.97/25.06 (sepSubset => 166.97/25.06 (binunionIL => 166.97/25.06 (upairset2IR => 166.97/25.06 (binunionIR => 166.97/25.06 (binunionEcases => 166.97/25.06 (binunionE => 166.97/25.06 (binunionLsub => 166.97/25.06 (binunionRsub => 166.97/25.06 (binintersectI => 166.97/25.06 ( 166.97/25.06 binintersectSubset5 => 166.97/25.06 ( 166.97/25.06 binintersectEL => 166.97/25.06 ( 166.97/25.06 binintersectLsub => 166.97/25.06 ( 166.97/25.06 binintersectSubset2 => 166.97/25.06 ( 166.97/25.06 binintersectSubset3 => 166.97/25.06 ( 166.97/25.06 binintersectER => 166.97/25.06 ( 166.97/25.06 disjointsetsI1 => 166.97/25.06 ( 166.97/25.06 binintersectRsub => 166.97/25.06 ( 166.97/25.06 binintersectSubset4 => 166.97/25.06 ( 166.97/25.06 binintersectSubset1 => 166.97/25.06 ( 166.97/25.06 bs114d => 166.97/25.06 ( 166.97/25.06 setminusI => 166.97/25.06 ( 166.97/25.06 setminusEL => 166.97/25.06 ( 166.97/25.06 setminusER => 166.97/25.06 ( 166.97/25.06 setminusSubset2 => 166.97/25.06 ( 166.97/25.06 setminusERneg => 166.97/25.06 ( 166.97/25.06 setminusELneg => 166.97/25.06 ( 166.97/25.06 setminusILneg => 166.97/25.06 ( 166.97/25.06 setminusIRneg => 166.97/25.06 ( 166.97/25.06 setminusLsub => 166.97/25.06 ( 166.97/25.06 setminusSubset1 => 166.97/25.06 ( 166.97/25.06 symdiffE => 166.97/25.06 ( 166.97/25.06 symdiffI1 => 166.97/25.06 ( 166.97/25.06 symdiffI2 => 166.97/25.06 ( 166.97/25.06 symdiffIneg1 => 166.97/25.06 ( 166.97/25.06 symdiffIneg2 => 166.97/25.06 ( 166.97/25.06 secondinupair => 166.97/25.06 ( 166.97/25.06 setukpairIL => 166.97/25.06 ( 166.97/25.06 setukpairIR => 166.97/25.06 ( 166.97/25.06 kpairiskpair => 166.97/25.06 ( 166.97/25.06 kpairp => 166.97/25.06 ( 166.97/25.06 singletonsubset => 166.97/25.06 ( 166.97/25.06 singletoninpowerset => 166.97/25.06 ( 166.97/25.06 singletoninpowunion => 166.97/25.06 ( 166.97/25.06 upairset2E => 166.97/25.06 ( 166.97/25.06 upairsubunion => 166.97/25.06 ( 166.97/25.06 upairinpowunion => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem1 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem2 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem3 => 166.97/25.06 ( 166.97/25.06 cartprodpairin => 166.97/25.06 ( 166.97/25.06 cartprodmempair1 => 166.97/25.06 ( 166.97/25.06 cartprodmempair => 166.97/25.06 ( 166.97/25.06 setunionE2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton1 => 166.97/25.06 ( 166.97/25.06 setunionsingleton2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton => 166.97/25.06 ( 166.97/25.06 singletonprop => 166.97/25.06 ( 166.97/25.06 ex1E1 => 166.97/25.06 ( 166.97/25.06 ex1I => 166.97/25.06 ( 166.97/25.06 ex1I2 => 166.97/25.06 ( 166.97/25.06 singletonsuniq => 166.97/25.06 ( 166.97/25.06 setukpairinjL1 => 166.97/25.06 ( 166.97/25.06 kfstsingleton => 166.97/25.06 ( 166.97/25.06 theprop => 166.97/25.06 ( 166.97/25.06 kfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodfstin => 166.97/25.06 ( 166.97/25.06 setukpairinjL2 => 166.97/25.06 ( 166.97/25.06 setukpairinjL => 166.97/25.06 ( 166.97/25.06 setukpairinjR11 => 166.97/25.06 ( 166.97/25.06 setukpairinjR12 => 166.97/25.06 ( 166.97/25.06 setukpairinjR1 => 166.97/25.06 ( 166.97/25.06 upairequniteq => 166.97/25.06 ( 166.97/25.06 setukpairinjR2 => 166.97/25.06 ( 166.97/25.06 setukpairinjR => 166.97/25.06 ( 166.97/25.06 ksndsingleton => 166.97/25.06 ( 166.97/25.06 ksndpairEq => 166.97/25.06 ( 166.97/25.06 kpairsurjEq => 166.97/25.06 ( 166.97/25.06 cartprodsndin => 166.97/25.06 ( 166.97/25.06 cartprodpairmemEL => 166.97/25.06 ( 166.97/25.06 cartprodpairmemER => 166.97/25.06 ( 166.97/25.06 cartprodmempaircEq => 166.97/25.06 ( 166.97/25.06 cartprodfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodsndpairEq => 166.97/25.06 ( 166.97/25.06 cartprodpairsurjEq => 166.97/25.06 ( 166.97/25.06 dpsetconstrI => 166.97/25.06 ( 166.97/25.06 dpsetconstrSub => 166.97/25.06 ( 166.97/25.06 setOfPairsIsBReln => 166.97/25.06 ( 166.97/25.06 !!(( 166.97/25.06 ^[Y0 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y1 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y2 : ( 166.97/25.06 $i * $i) > $o]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y3 : $i]: 166.97/25.06 (in(Y3, 166.97/25.06 Y0) => 166.97/25.06 (!!(( 166.97/25.06 ^[Y4 : $i]: 166.97/25.06 (in(Y4, 166.97/25.06 Y1) => 166.97/25.06 (in( 166.97/25.06 kpair( 166.97/25.06 Y3, Y4), 166.97/25.06 dpsetconstr( 166.97/25.06 Y0, Y1, 166.97/25.06 Y2)) => 166.97/25.06 Y2(Y3, 166.97/25.06 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.06 inference('lazy_cnf_imply', [status(thm)], ['44'])). 166.97/25.06 tff('46', plain, 166.97/25.06 ~ (eqinunit => 166.97/25.06 (singletonsswitch => 166.97/25.06 (upairsetE => 166.97/25.06 (upairsetIL => 166.97/25.06 (upairsetIR => 166.97/25.06 (emptyE1 => 166.97/25.06 (vacuousDall => 166.97/25.06 (quantDeMorgan1 => 166.97/25.06 (quantDeMorgan2 => 166.97/25.06 (quantDeMorgan3 => 166.97/25.06 (quantDeMorgan4 => 166.97/25.06 (prop2setI => 166.97/25.06 (prop2set2propI => 166.97/25.06 (notdexE => 166.97/25.06 (notdallE => 166.97/25.06 (exuI1 => 166.97/25.06 (exuI3 => 166.97/25.06 (exuI2 => 166.97/25.06 (inCongP => 166.97/25.06 (in__Cong => 166.97/25.06 (exuE3u => 166.97/25.06 (exu__Cong => 166.97/25.06 (emptyset__Cong => 166.97/25.06 (setadjoin__Cong => 166.97/25.06 (powerset__Cong => 166.97/25.06 (setunion__Cong => 166.97/25.06 (omega__Cong => 166.97/25.06 (exuEu => 166.97/25.06 (descr__Cong => 166.97/25.06 (dsetconstr__Cong => 166.97/25.06 (subsetI1 => 166.97/25.06 (eqimpsubset2 => 166.97/25.06 (eqimpsubset1 => 166.97/25.06 (subsetI2 => 166.97/25.06 (emptysetsubset => 166.97/25.06 (subsetE => 166.97/25.06 (subsetE2 => 166.97/25.06 (notsubsetI => 166.97/25.06 (notequalI1 => 166.97/25.06 (notequalI2 => 166.97/25.06 (subsetRefl => 166.97/25.06 (subsetTrans => 166.97/25.06 (setadjoinSub => 166.97/25.06 (setadjoinSub2 => 166.97/25.06 (subset2powerset => 166.97/25.06 (setextsub => 166.97/25.06 (subsetemptysetimpeq => 166.97/25.06 (powersetI1 => 166.97/25.06 (powersetE1 => 166.97/25.06 (inPowerset => 166.97/25.06 (powersetsubset => 166.97/25.06 (sepInPowerset => 166.97/25.06 (sepSubset => 166.97/25.06 (binunionIL => 166.97/25.06 (upairset2IR => 166.97/25.06 (binunionIR => 166.97/25.06 (binunionEcases => 166.97/25.06 (binunionE => 166.97/25.06 (binunionLsub => 166.97/25.06 (binunionRsub => 166.97/25.06 (binintersectI => 166.97/25.06 (binintersectSubset5 => 166.97/25.06 ( 166.97/25.06 binintersectEL => 166.97/25.06 ( 166.97/25.06 binintersectLsub => 166.97/25.06 ( 166.97/25.06 binintersectSubset2 => 166.97/25.06 ( 166.97/25.06 binintersectSubset3 => 166.97/25.06 ( 166.97/25.06 binintersectER => 166.97/25.06 ( 166.97/25.06 disjointsetsI1 => 166.97/25.06 ( 166.97/25.06 binintersectRsub => 166.97/25.06 ( 166.97/25.06 binintersectSubset4 => 166.97/25.06 ( 166.97/25.06 binintersectSubset1 => 166.97/25.06 ( 166.97/25.06 bs114d => 166.97/25.06 ( 166.97/25.06 setminusI => 166.97/25.06 ( 166.97/25.06 setminusEL => 166.97/25.06 ( 166.97/25.06 setminusER => 166.97/25.06 ( 166.97/25.06 setminusSubset2 => 166.97/25.06 ( 166.97/25.06 setminusERneg => 166.97/25.06 ( 166.97/25.06 setminusELneg => 166.97/25.06 ( 166.97/25.06 setminusILneg => 166.97/25.06 ( 166.97/25.06 setminusIRneg => 166.97/25.06 ( 166.97/25.06 setminusLsub => 166.97/25.06 ( 166.97/25.06 setminusSubset1 => 166.97/25.06 ( 166.97/25.06 symdiffE => 166.97/25.06 ( 166.97/25.06 symdiffI1 => 166.97/25.06 ( 166.97/25.06 symdiffI2 => 166.97/25.06 ( 166.97/25.06 symdiffIneg1 => 166.97/25.06 ( 166.97/25.06 symdiffIneg2 => 166.97/25.06 ( 166.97/25.06 secondinupair => 166.97/25.06 ( 166.97/25.06 setukpairIL => 166.97/25.06 ( 166.97/25.06 setukpairIR => 166.97/25.06 ( 166.97/25.06 kpairiskpair => 166.97/25.06 ( 166.97/25.06 kpairp => 166.97/25.06 ( 166.97/25.06 singletonsubset => 166.97/25.06 ( 166.97/25.06 singletoninpowerset => 166.97/25.06 ( 166.97/25.06 singletoninpowunion => 166.97/25.06 ( 166.97/25.06 upairset2E => 166.97/25.06 ( 166.97/25.06 upairsubunion => 166.97/25.06 ( 166.97/25.06 upairinpowunion => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem1 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem2 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem3 => 166.97/25.06 ( 166.97/25.06 cartprodpairin => 166.97/25.06 ( 166.97/25.06 cartprodmempair1 => 166.97/25.06 ( 166.97/25.06 cartprodmempair => 166.97/25.06 ( 166.97/25.06 setunionE2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton1 => 166.97/25.06 ( 166.97/25.06 setunionsingleton2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton => 166.97/25.06 ( 166.97/25.06 singletonprop => 166.97/25.06 ( 166.97/25.06 ex1E1 => 166.97/25.06 ( 166.97/25.06 ex1I => 166.97/25.06 ( 166.97/25.06 ex1I2 => 166.97/25.06 ( 166.97/25.06 singletonsuniq => 166.97/25.06 ( 166.97/25.06 setukpairinjL1 => 166.97/25.06 ( 166.97/25.06 kfstsingleton => 166.97/25.06 ( 166.97/25.06 theprop => 166.97/25.06 ( 166.97/25.06 kfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodfstin => 166.97/25.06 ( 166.97/25.06 setukpairinjL2 => 166.97/25.06 ( 166.97/25.06 setukpairinjL => 166.97/25.06 ( 166.97/25.06 setukpairinjR11 => 166.97/25.06 ( 166.97/25.06 setukpairinjR12 => 166.97/25.06 ( 166.97/25.06 setukpairinjR1 => 166.97/25.06 ( 166.97/25.06 upairequniteq => 166.97/25.06 ( 166.97/25.06 setukpairinjR2 => 166.97/25.06 ( 166.97/25.06 setukpairinjR => 166.97/25.06 ( 166.97/25.06 ksndsingleton => 166.97/25.06 ( 166.97/25.06 ksndpairEq => 166.97/25.06 ( 166.97/25.06 kpairsurjEq => 166.97/25.06 ( 166.97/25.06 cartprodsndin => 166.97/25.06 ( 166.97/25.06 cartprodpairmemEL => 166.97/25.06 ( 166.97/25.06 cartprodpairmemER => 166.97/25.06 ( 166.97/25.06 cartprodmempaircEq => 166.97/25.06 ( 166.97/25.06 cartprodfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodsndpairEq => 166.97/25.06 ( 166.97/25.06 cartprodpairsurjEq => 166.97/25.06 ( 166.97/25.06 dpsetconstrI => 166.97/25.06 ( 166.97/25.06 dpsetconstrSub => 166.97/25.06 ( 166.97/25.06 setOfPairsIsBReln => 166.97/25.06 ( 166.97/25.06 !!(( 166.97/25.06 ^[Y0 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y1 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y2 : ( 166.97/25.06 $i * $i) > $o]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y3 : $i]: 166.97/25.06 (in(Y3, 166.97/25.06 Y0) => 166.97/25.06 (!!(( 166.97/25.06 ^[Y4 : $i]: 166.97/25.06 (in(Y4, 166.97/25.06 Y1) => 166.97/25.06 (in( 166.97/25.06 kpair( 166.97/25.06 Y3, Y4), 166.97/25.06 dpsetconstr( 166.97/25.06 Y0, Y1, 166.97/25.06 Y2)) => 166.97/25.06 Y2(Y3, 166.97/25.06 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.06 inference('lazy_cnf_imply', [status(thm)], ['45'])). 166.97/25.06 tff('47', plain, 166.97/25.06 ~ (singletonsswitch => 166.97/25.06 (upairsetE => 166.97/25.06 (upairsetIL => 166.97/25.06 (upairsetIR => 166.97/25.06 (emptyE1 => 166.97/25.06 (vacuousDall => 166.97/25.06 (quantDeMorgan1 => 166.97/25.06 (quantDeMorgan2 => 166.97/25.06 (quantDeMorgan3 => 166.97/25.06 (quantDeMorgan4 => 166.97/25.06 (prop2setI => 166.97/25.06 (prop2set2propI => 166.97/25.06 (notdexE => 166.97/25.06 (notdallE => 166.97/25.06 (exuI1 => 166.97/25.06 (exuI3 => 166.97/25.06 (exuI2 => 166.97/25.06 (inCongP => 166.97/25.06 (in__Cong => 166.97/25.06 (exuE3u => 166.97/25.06 (exu__Cong => 166.97/25.06 (emptyset__Cong => 166.97/25.06 (setadjoin__Cong => 166.97/25.06 (powerset__Cong => 166.97/25.06 (setunion__Cong => 166.97/25.06 (omega__Cong => 166.97/25.06 (exuEu => 166.97/25.06 (descr__Cong => 166.97/25.06 (dsetconstr__Cong => 166.97/25.06 (subsetI1 => 166.97/25.06 (eqimpsubset2 => 166.97/25.06 (eqimpsubset1 => 166.97/25.06 (subsetI2 => 166.97/25.06 (emptysetsubset => 166.97/25.06 (subsetE => 166.97/25.06 (subsetE2 => 166.97/25.06 (notsubsetI => 166.97/25.06 (notequalI1 => 166.97/25.06 (notequalI2 => 166.97/25.06 (subsetRefl => 166.97/25.06 (subsetTrans => 166.97/25.06 (setadjoinSub => 166.97/25.06 (setadjoinSub2 => 166.97/25.06 (subset2powerset => 166.97/25.06 (setextsub => 166.97/25.06 (subsetemptysetimpeq => 166.97/25.06 (powersetI1 => 166.97/25.06 (powersetE1 => 166.97/25.06 (inPowerset => 166.97/25.06 (powersetsubset => 166.97/25.06 (sepInPowerset => 166.97/25.06 (sepSubset => 166.97/25.06 (binunionIL => 166.97/25.06 (upairset2IR => 166.97/25.06 (binunionIR => 166.97/25.06 (binunionEcases => 166.97/25.06 (binunionE => 166.97/25.06 (binunionLsub => 166.97/25.06 (binunionRsub => 166.97/25.06 (binintersectI => 166.97/25.06 (binintersectSubset5 => 166.97/25.06 (binintersectEL => 166.97/25.06 ( 166.97/25.06 binintersectLsub => 166.97/25.06 ( 166.97/25.06 binintersectSubset2 => 166.97/25.06 ( 166.97/25.06 binintersectSubset3 => 166.97/25.06 ( 166.97/25.06 binintersectER => 166.97/25.06 ( 166.97/25.06 disjointsetsI1 => 166.97/25.06 ( 166.97/25.06 binintersectRsub => 166.97/25.06 ( 166.97/25.06 binintersectSubset4 => 166.97/25.06 ( 166.97/25.06 binintersectSubset1 => 166.97/25.06 ( 166.97/25.06 bs114d => 166.97/25.06 ( 166.97/25.06 setminusI => 166.97/25.06 ( 166.97/25.06 setminusEL => 166.97/25.06 ( 166.97/25.06 setminusER => 166.97/25.06 ( 166.97/25.06 setminusSubset2 => 166.97/25.06 ( 166.97/25.06 setminusERneg => 166.97/25.06 ( 166.97/25.06 setminusELneg => 166.97/25.06 ( 166.97/25.06 setminusILneg => 166.97/25.06 ( 166.97/25.06 setminusIRneg => 166.97/25.06 ( 166.97/25.06 setminusLsub => 166.97/25.06 ( 166.97/25.06 setminusSubset1 => 166.97/25.06 ( 166.97/25.06 symdiffE => 166.97/25.06 ( 166.97/25.06 symdiffI1 => 166.97/25.06 ( 166.97/25.06 symdiffI2 => 166.97/25.06 ( 166.97/25.06 symdiffIneg1 => 166.97/25.06 ( 166.97/25.06 symdiffIneg2 => 166.97/25.06 ( 166.97/25.06 secondinupair => 166.97/25.06 ( 166.97/25.06 setukpairIL => 166.97/25.06 ( 166.97/25.06 setukpairIR => 166.97/25.06 ( 166.97/25.06 kpairiskpair => 166.97/25.06 ( 166.97/25.06 kpairp => 166.97/25.06 ( 166.97/25.06 singletonsubset => 166.97/25.06 ( 166.97/25.06 singletoninpowerset => 166.97/25.06 ( 166.97/25.06 singletoninpowunion => 166.97/25.06 ( 166.97/25.06 upairset2E => 166.97/25.06 ( 166.97/25.06 upairsubunion => 166.97/25.06 ( 166.97/25.06 upairinpowunion => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem1 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem2 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem3 => 166.97/25.06 ( 166.97/25.06 cartprodpairin => 166.97/25.06 ( 166.97/25.06 cartprodmempair1 => 166.97/25.06 ( 166.97/25.06 cartprodmempair => 166.97/25.06 ( 166.97/25.06 setunionE2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton1 => 166.97/25.06 ( 166.97/25.06 setunionsingleton2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton => 166.97/25.06 ( 166.97/25.06 singletonprop => 166.97/25.06 ( 166.97/25.06 ex1E1 => 166.97/25.06 ( 166.97/25.06 ex1I => 166.97/25.06 ( 166.97/25.06 ex1I2 => 166.97/25.06 ( 166.97/25.06 singletonsuniq => 166.97/25.06 ( 166.97/25.06 setukpairinjL1 => 166.97/25.06 ( 166.97/25.06 kfstsingleton => 166.97/25.06 ( 166.97/25.06 theprop => 166.97/25.06 ( 166.97/25.06 kfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodfstin => 166.97/25.06 ( 166.97/25.06 setukpairinjL2 => 166.97/25.06 ( 166.97/25.06 setukpairinjL => 166.97/25.06 ( 166.97/25.06 setukpairinjR11 => 166.97/25.06 ( 166.97/25.06 setukpairinjR12 => 166.97/25.06 ( 166.97/25.06 setukpairinjR1 => 166.97/25.06 ( 166.97/25.06 upairequniteq => 166.97/25.06 ( 166.97/25.06 setukpairinjR2 => 166.97/25.06 ( 166.97/25.06 setukpairinjR => 166.97/25.06 ( 166.97/25.06 ksndsingleton => 166.97/25.06 ( 166.97/25.06 ksndpairEq => 166.97/25.06 ( 166.97/25.06 kpairsurjEq => 166.97/25.06 ( 166.97/25.06 cartprodsndin => 166.97/25.06 ( 166.97/25.06 cartprodpairmemEL => 166.97/25.06 ( 166.97/25.06 cartprodpairmemER => 166.97/25.06 ( 166.97/25.06 cartprodmempaircEq => 166.97/25.06 ( 166.97/25.06 cartprodfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodsndpairEq => 166.97/25.06 ( 166.97/25.06 cartprodpairsurjEq => 166.97/25.06 ( 166.97/25.06 dpsetconstrI => 166.97/25.06 ( 166.97/25.06 dpsetconstrSub => 166.97/25.06 ( 166.97/25.06 setOfPairsIsBReln => 166.97/25.06 ( 166.97/25.06 !!(( 166.97/25.06 ^[Y0 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y1 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y2 : ( 166.97/25.06 $i * $i) > $o]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y3 : $i]: 166.97/25.06 (in(Y3, 166.97/25.06 Y0) => 166.97/25.06 (!!(( 166.97/25.06 ^[Y4 : $i]: 166.97/25.06 (in(Y4, 166.97/25.06 Y1) => 166.97/25.06 (in( 166.97/25.06 kpair( 166.97/25.06 Y3, Y4), 166.97/25.06 dpsetconstr( 166.97/25.06 Y0, Y1, 166.97/25.06 Y2)) => 166.97/25.06 Y2(Y3, 166.97/25.06 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.06 inference('lazy_cnf_imply', [status(thm)], ['46'])). 166.97/25.06 tff('48', plain, 166.97/25.06 ~ (upairsetE => 166.97/25.06 (upairsetIL => 166.97/25.06 (upairsetIR => 166.97/25.06 (emptyE1 => 166.97/25.06 (vacuousDall => 166.97/25.06 (quantDeMorgan1 => 166.97/25.06 (quantDeMorgan2 => 166.97/25.06 (quantDeMorgan3 => 166.97/25.06 (quantDeMorgan4 => 166.97/25.06 (prop2setI => 166.97/25.06 (prop2set2propI => 166.97/25.06 (notdexE => 166.97/25.06 (notdallE => 166.97/25.06 (exuI1 => 166.97/25.06 (exuI3 => 166.97/25.06 (exuI2 => 166.97/25.06 (inCongP => 166.97/25.06 (in__Cong => 166.97/25.06 (exuE3u => 166.97/25.06 (exu__Cong => 166.97/25.06 (emptyset__Cong => 166.97/25.06 (setadjoin__Cong => 166.97/25.06 (powerset__Cong => 166.97/25.06 (setunion__Cong => 166.97/25.06 (omega__Cong => 166.97/25.06 (exuEu => 166.97/25.06 (descr__Cong => 166.97/25.06 (dsetconstr__Cong => 166.97/25.06 (subsetI1 => 166.97/25.06 (eqimpsubset2 => 166.97/25.06 (eqimpsubset1 => 166.97/25.06 (subsetI2 => 166.97/25.06 (emptysetsubset => 166.97/25.06 (subsetE => 166.97/25.06 (subsetE2 => 166.97/25.06 (notsubsetI => 166.97/25.06 (notequalI1 => 166.97/25.06 (notequalI2 => 166.97/25.06 (subsetRefl => 166.97/25.06 (subsetTrans => 166.97/25.06 (setadjoinSub => 166.97/25.06 (setadjoinSub2 => 166.97/25.06 (subset2powerset => 166.97/25.06 (setextsub => 166.97/25.06 (subsetemptysetimpeq => 166.97/25.06 (powersetI1 => 166.97/25.06 (powersetE1 => 166.97/25.06 (inPowerset => 166.97/25.06 (powersetsubset => 166.97/25.06 (sepInPowerset => 166.97/25.06 (sepSubset => 166.97/25.06 (binunionIL => 166.97/25.06 (upairset2IR => 166.97/25.06 (binunionIR => 166.97/25.06 (binunionEcases => 166.97/25.06 (binunionE => 166.97/25.06 (binunionLsub => 166.97/25.06 (binunionRsub => 166.97/25.06 (binintersectI => 166.97/25.06 (binintersectSubset5 => 166.97/25.06 (binintersectEL => 166.97/25.06 (binintersectLsub => 166.97/25.06 ( 166.97/25.06 binintersectSubset2 => 166.97/25.06 ( 166.97/25.06 binintersectSubset3 => 166.97/25.06 ( 166.97/25.06 binintersectER => 166.97/25.06 ( 166.97/25.06 disjointsetsI1 => 166.97/25.06 ( 166.97/25.06 binintersectRsub => 166.97/25.06 ( 166.97/25.06 binintersectSubset4 => 166.97/25.06 ( 166.97/25.06 binintersectSubset1 => 166.97/25.06 ( 166.97/25.06 bs114d => 166.97/25.06 ( 166.97/25.06 setminusI => 166.97/25.06 ( 166.97/25.06 setminusEL => 166.97/25.06 ( 166.97/25.06 setminusER => 166.97/25.06 ( 166.97/25.06 setminusSubset2 => 166.97/25.06 ( 166.97/25.06 setminusERneg => 166.97/25.06 ( 166.97/25.06 setminusELneg => 166.97/25.06 ( 166.97/25.06 setminusILneg => 166.97/25.06 ( 166.97/25.06 setminusIRneg => 166.97/25.06 ( 166.97/25.06 setminusLsub => 166.97/25.06 ( 166.97/25.06 setminusSubset1 => 166.97/25.06 ( 166.97/25.06 symdiffE => 166.97/25.06 ( 166.97/25.06 symdiffI1 => 166.97/25.06 ( 166.97/25.06 symdiffI2 => 166.97/25.06 ( 166.97/25.06 symdiffIneg1 => 166.97/25.06 ( 166.97/25.06 symdiffIneg2 => 166.97/25.06 ( 166.97/25.06 secondinupair => 166.97/25.06 ( 166.97/25.06 setukpairIL => 166.97/25.06 ( 166.97/25.06 setukpairIR => 166.97/25.06 ( 166.97/25.06 kpairiskpair => 166.97/25.06 ( 166.97/25.06 kpairp => 166.97/25.06 ( 166.97/25.06 singletonsubset => 166.97/25.06 ( 166.97/25.06 singletoninpowerset => 166.97/25.06 ( 166.97/25.06 singletoninpowunion => 166.97/25.06 ( 166.97/25.06 upairset2E => 166.97/25.06 ( 166.97/25.06 upairsubunion => 166.97/25.06 ( 166.97/25.06 upairinpowunion => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem1 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem2 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem3 => 166.97/25.06 ( 166.97/25.06 cartprodpairin => 166.97/25.06 ( 166.97/25.06 cartprodmempair1 => 166.97/25.06 ( 166.97/25.06 cartprodmempair => 166.97/25.06 ( 166.97/25.06 setunionE2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton1 => 166.97/25.06 ( 166.97/25.06 setunionsingleton2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton => 166.97/25.06 ( 166.97/25.06 singletonprop => 166.97/25.06 ( 166.97/25.06 ex1E1 => 166.97/25.06 ( 166.97/25.06 ex1I => 166.97/25.06 ( 166.97/25.06 ex1I2 => 166.97/25.06 ( 166.97/25.06 singletonsuniq => 166.97/25.06 ( 166.97/25.06 setukpairinjL1 => 166.97/25.06 ( 166.97/25.06 kfstsingleton => 166.97/25.06 ( 166.97/25.06 theprop => 166.97/25.06 ( 166.97/25.06 kfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodfstin => 166.97/25.06 ( 166.97/25.06 setukpairinjL2 => 166.97/25.06 ( 166.97/25.06 setukpairinjL => 166.97/25.06 ( 166.97/25.06 setukpairinjR11 => 166.97/25.06 ( 166.97/25.06 setukpairinjR12 => 166.97/25.06 ( 166.97/25.06 setukpairinjR1 => 166.97/25.06 ( 166.97/25.06 upairequniteq => 166.97/25.06 ( 166.97/25.06 setukpairinjR2 => 166.97/25.06 ( 166.97/25.06 setukpairinjR => 166.97/25.06 ( 166.97/25.06 ksndsingleton => 166.97/25.06 ( 166.97/25.06 ksndpairEq => 166.97/25.06 ( 166.97/25.06 kpairsurjEq => 166.97/25.06 ( 166.97/25.06 cartprodsndin => 166.97/25.06 ( 166.97/25.06 cartprodpairmemEL => 166.97/25.06 ( 166.97/25.06 cartprodpairmemER => 166.97/25.06 ( 166.97/25.06 cartprodmempaircEq => 166.97/25.06 ( 166.97/25.06 cartprodfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodsndpairEq => 166.97/25.06 ( 166.97/25.06 cartprodpairsurjEq => 166.97/25.06 ( 166.97/25.06 dpsetconstrI => 166.97/25.06 ( 166.97/25.06 dpsetconstrSub => 166.97/25.06 ( 166.97/25.06 setOfPairsIsBReln => 166.97/25.06 ( 166.97/25.06 !!(( 166.97/25.06 ^[Y0 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y1 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y2 : ( 166.97/25.06 $i * $i) > $o]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y3 : $i]: 166.97/25.06 (in(Y3, 166.97/25.06 Y0) => 166.97/25.06 (!!(( 166.97/25.06 ^[Y4 : $i]: 166.97/25.06 (in(Y4, 166.97/25.06 Y1) => 166.97/25.06 (in( 166.97/25.06 kpair( 166.97/25.06 Y3, Y4), 166.97/25.06 dpsetconstr( 166.97/25.06 Y0, Y1, 166.97/25.06 Y2)) => 166.97/25.06 Y2(Y3, 166.97/25.06 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.06 inference('lazy_cnf_imply', [status(thm)], ['47'])). 166.97/25.06 tff('49', plain, 166.97/25.06 ~ (upairsetIL => 166.97/25.06 (upairsetIR => 166.97/25.06 (emptyE1 => 166.97/25.06 (vacuousDall => 166.97/25.06 (quantDeMorgan1 => 166.97/25.06 (quantDeMorgan2 => 166.97/25.06 (quantDeMorgan3 => 166.97/25.06 (quantDeMorgan4 => 166.97/25.06 (prop2setI => 166.97/25.06 (prop2set2propI => 166.97/25.06 (notdexE => 166.97/25.06 (notdallE => 166.97/25.06 (exuI1 => 166.97/25.06 (exuI3 => 166.97/25.06 (exuI2 => 166.97/25.06 (inCongP => 166.97/25.06 (in__Cong => 166.97/25.06 (exuE3u => 166.97/25.06 (exu__Cong => 166.97/25.06 (emptyset__Cong => 166.97/25.06 (setadjoin__Cong => 166.97/25.06 (powerset__Cong => 166.97/25.06 (setunion__Cong => 166.97/25.06 (omega__Cong => 166.97/25.06 (exuEu => 166.97/25.06 (descr__Cong => 166.97/25.06 (dsetconstr__Cong => 166.97/25.06 (subsetI1 => 166.97/25.06 (eqimpsubset2 => 166.97/25.06 (eqimpsubset1 => 166.97/25.06 (subsetI2 => 166.97/25.06 (emptysetsubset => 166.97/25.06 (subsetE => 166.97/25.06 (subsetE2 => 166.97/25.06 (notsubsetI => 166.97/25.06 (notequalI1 => 166.97/25.06 (notequalI2 => 166.97/25.06 (subsetRefl => 166.97/25.06 (subsetTrans => 166.97/25.06 (setadjoinSub => 166.97/25.06 (setadjoinSub2 => 166.97/25.06 (subset2powerset => 166.97/25.06 (setextsub => 166.97/25.06 (subsetemptysetimpeq => 166.97/25.06 (powersetI1 => 166.97/25.06 (powersetE1 => 166.97/25.06 (inPowerset => 166.97/25.06 (powersetsubset => 166.97/25.06 (sepInPowerset => 166.97/25.06 (sepSubset => 166.97/25.06 (binunionIL => 166.97/25.06 (upairset2IR => 166.97/25.06 (binunionIR => 166.97/25.06 (binunionEcases => 166.97/25.06 (binunionE => 166.97/25.06 (binunionLsub => 166.97/25.06 (binunionRsub => 166.97/25.06 (binintersectI => 166.97/25.06 (binintersectSubset5 => 166.97/25.06 (binintersectEL => 166.97/25.06 (binintersectLsub => 166.97/25.06 (binintersectSubset2 => 166.97/25.06 ( 166.97/25.06 binintersectSubset3 => 166.97/25.06 ( 166.97/25.06 binintersectER => 166.97/25.06 ( 166.97/25.06 disjointsetsI1 => 166.97/25.06 ( 166.97/25.06 binintersectRsub => 166.97/25.06 ( 166.97/25.06 binintersectSubset4 => 166.97/25.06 ( 166.97/25.06 binintersectSubset1 => 166.97/25.06 ( 166.97/25.06 bs114d => 166.97/25.06 ( 166.97/25.06 setminusI => 166.97/25.06 ( 166.97/25.06 setminusEL => 166.97/25.06 ( 166.97/25.06 setminusER => 166.97/25.06 ( 166.97/25.06 setminusSubset2 => 166.97/25.06 ( 166.97/25.06 setminusERneg => 166.97/25.06 ( 166.97/25.06 setminusELneg => 166.97/25.06 ( 166.97/25.06 setminusILneg => 166.97/25.06 ( 166.97/25.06 setminusIRneg => 166.97/25.06 ( 166.97/25.06 setminusLsub => 166.97/25.06 ( 166.97/25.06 setminusSubset1 => 166.97/25.06 ( 166.97/25.06 symdiffE => 166.97/25.06 ( 166.97/25.06 symdiffI1 => 166.97/25.06 ( 166.97/25.06 symdiffI2 => 166.97/25.06 ( 166.97/25.06 symdiffIneg1 => 166.97/25.06 ( 166.97/25.06 symdiffIneg2 => 166.97/25.06 ( 166.97/25.06 secondinupair => 166.97/25.06 ( 166.97/25.06 setukpairIL => 166.97/25.06 ( 166.97/25.06 setukpairIR => 166.97/25.06 ( 166.97/25.06 kpairiskpair => 166.97/25.06 ( 166.97/25.06 kpairp => 166.97/25.06 ( 166.97/25.06 singletonsubset => 166.97/25.06 ( 166.97/25.06 singletoninpowerset => 166.97/25.06 ( 166.97/25.06 singletoninpowunion => 166.97/25.06 ( 166.97/25.06 upairset2E => 166.97/25.06 ( 166.97/25.06 upairsubunion => 166.97/25.06 ( 166.97/25.06 upairinpowunion => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem1 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem2 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem3 => 166.97/25.06 ( 166.97/25.06 cartprodpairin => 166.97/25.06 ( 166.97/25.06 cartprodmempair1 => 166.97/25.06 ( 166.97/25.06 cartprodmempair => 166.97/25.06 ( 166.97/25.06 setunionE2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton1 => 166.97/25.06 ( 166.97/25.06 setunionsingleton2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton => 166.97/25.06 ( 166.97/25.06 singletonprop => 166.97/25.06 ( 166.97/25.06 ex1E1 => 166.97/25.06 ( 166.97/25.06 ex1I => 166.97/25.06 ( 166.97/25.06 ex1I2 => 166.97/25.06 ( 166.97/25.06 singletonsuniq => 166.97/25.06 ( 166.97/25.06 setukpairinjL1 => 166.97/25.06 ( 166.97/25.06 kfstsingleton => 166.97/25.06 ( 166.97/25.06 theprop => 166.97/25.06 ( 166.97/25.06 kfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodfstin => 166.97/25.06 ( 166.97/25.06 setukpairinjL2 => 166.97/25.06 ( 166.97/25.06 setukpairinjL => 166.97/25.06 ( 166.97/25.06 setukpairinjR11 => 166.97/25.06 ( 166.97/25.06 setukpairinjR12 => 166.97/25.06 ( 166.97/25.06 setukpairinjR1 => 166.97/25.06 ( 166.97/25.06 upairequniteq => 166.97/25.06 ( 166.97/25.06 setukpairinjR2 => 166.97/25.06 ( 166.97/25.06 setukpairinjR => 166.97/25.06 ( 166.97/25.06 ksndsingleton => 166.97/25.06 ( 166.97/25.06 ksndpairEq => 166.97/25.06 ( 166.97/25.06 kpairsurjEq => 166.97/25.06 ( 166.97/25.06 cartprodsndin => 166.97/25.06 ( 166.97/25.06 cartprodpairmemEL => 166.97/25.06 ( 166.97/25.06 cartprodpairmemER => 166.97/25.06 ( 166.97/25.06 cartprodmempaircEq => 166.97/25.06 ( 166.97/25.06 cartprodfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodsndpairEq => 166.97/25.06 ( 166.97/25.06 cartprodpairsurjEq => 166.97/25.06 ( 166.97/25.06 dpsetconstrI => 166.97/25.06 ( 166.97/25.06 dpsetconstrSub => 166.97/25.06 ( 166.97/25.06 setOfPairsIsBReln => 166.97/25.06 ( 166.97/25.06 !!(( 166.97/25.06 ^[Y0 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y1 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y2 : ( 166.97/25.06 $i * $i) > $o]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y3 : $i]: 166.97/25.06 (in(Y3, 166.97/25.06 Y0) => 166.97/25.06 (!!(( 166.97/25.06 ^[Y4 : $i]: 166.97/25.06 (in(Y4, 166.97/25.06 Y1) => 166.97/25.06 (in( 166.97/25.06 kpair( 166.97/25.06 Y3, Y4), 166.97/25.06 dpsetconstr( 166.97/25.06 Y0, Y1, 166.97/25.06 Y2)) => 166.97/25.06 Y2(Y3, 166.97/25.06 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.06 inference('lazy_cnf_imply', [status(thm)], ['48'])). 166.97/25.06 tff('50', plain, 166.97/25.06 ~ (upairsetIR => 166.97/25.06 (emptyE1 => 166.97/25.06 (vacuousDall => 166.97/25.06 (quantDeMorgan1 => 166.97/25.06 (quantDeMorgan2 => 166.97/25.06 (quantDeMorgan3 => 166.97/25.06 (quantDeMorgan4 => 166.97/25.06 (prop2setI => 166.97/25.06 (prop2set2propI => 166.97/25.06 (notdexE => 166.97/25.06 (notdallE => 166.97/25.06 (exuI1 => 166.97/25.06 (exuI3 => 166.97/25.06 (exuI2 => 166.97/25.06 (inCongP => 166.97/25.06 (in__Cong => 166.97/25.06 (exuE3u => 166.97/25.06 (exu__Cong => 166.97/25.06 (emptyset__Cong => 166.97/25.06 (setadjoin__Cong => 166.97/25.06 (powerset__Cong => 166.97/25.06 (setunion__Cong => 166.97/25.06 (omega__Cong => 166.97/25.06 (exuEu => 166.97/25.06 (descr__Cong => 166.97/25.06 (dsetconstr__Cong => 166.97/25.06 (subsetI1 => 166.97/25.06 (eqimpsubset2 => 166.97/25.06 (eqimpsubset1 => 166.97/25.06 (subsetI2 => 166.97/25.06 (emptysetsubset => 166.97/25.06 (subsetE => 166.97/25.06 (subsetE2 => 166.97/25.06 (notsubsetI => 166.97/25.06 (notequalI1 => 166.97/25.06 (notequalI2 => 166.97/25.06 (subsetRefl => 166.97/25.06 (subsetTrans => 166.97/25.06 (setadjoinSub => 166.97/25.06 (setadjoinSub2 => 166.97/25.06 (subset2powerset => 166.97/25.06 (setextsub => 166.97/25.06 (subsetemptysetimpeq => 166.97/25.06 (powersetI1 => 166.97/25.06 (powersetE1 => 166.97/25.06 (inPowerset => 166.97/25.06 (powersetsubset => 166.97/25.06 (sepInPowerset => 166.97/25.06 (sepSubset => 166.97/25.06 (binunionIL => 166.97/25.06 (upairset2IR => 166.97/25.06 (binunionIR => 166.97/25.06 (binunionEcases => 166.97/25.06 (binunionE => 166.97/25.06 (binunionLsub => 166.97/25.06 (binunionRsub => 166.97/25.06 (binintersectI => 166.97/25.06 (binintersectSubset5 => 166.97/25.06 (binintersectEL => 166.97/25.06 (binintersectLsub => 166.97/25.06 (binintersectSubset2 => 166.97/25.06 (binintersectSubset3 => 166.97/25.06 ( 166.97/25.06 binintersectER => 166.97/25.06 ( 166.97/25.06 disjointsetsI1 => 166.97/25.06 ( 166.97/25.06 binintersectRsub => 166.97/25.06 ( 166.97/25.06 binintersectSubset4 => 166.97/25.06 ( 166.97/25.06 binintersectSubset1 => 166.97/25.06 ( 166.97/25.06 bs114d => 166.97/25.06 ( 166.97/25.06 setminusI => 166.97/25.06 ( 166.97/25.06 setminusEL => 166.97/25.06 ( 166.97/25.06 setminusER => 166.97/25.06 ( 166.97/25.06 setminusSubset2 => 166.97/25.06 ( 166.97/25.06 setminusERneg => 166.97/25.06 ( 166.97/25.06 setminusELneg => 166.97/25.06 ( 166.97/25.06 setminusILneg => 166.97/25.06 ( 166.97/25.06 setminusIRneg => 166.97/25.06 ( 166.97/25.06 setminusLsub => 166.97/25.06 ( 166.97/25.06 setminusSubset1 => 166.97/25.06 ( 166.97/25.06 symdiffE => 166.97/25.06 ( 166.97/25.06 symdiffI1 => 166.97/25.06 ( 166.97/25.06 symdiffI2 => 166.97/25.06 ( 166.97/25.06 symdiffIneg1 => 166.97/25.06 ( 166.97/25.06 symdiffIneg2 => 166.97/25.06 ( 166.97/25.06 secondinupair => 166.97/25.06 ( 166.97/25.06 setukpairIL => 166.97/25.06 ( 166.97/25.06 setukpairIR => 166.97/25.06 ( 166.97/25.06 kpairiskpair => 166.97/25.06 ( 166.97/25.06 kpairp => 166.97/25.06 ( 166.97/25.06 singletonsubset => 166.97/25.06 ( 166.97/25.06 singletoninpowerset => 166.97/25.06 ( 166.97/25.06 singletoninpowunion => 166.97/25.06 ( 166.97/25.06 upairset2E => 166.97/25.06 ( 166.97/25.06 upairsubunion => 166.97/25.06 ( 166.97/25.06 upairinpowunion => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem1 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem2 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem3 => 166.97/25.06 ( 166.97/25.06 cartprodpairin => 166.97/25.06 ( 166.97/25.06 cartprodmempair1 => 166.97/25.06 ( 166.97/25.06 cartprodmempair => 166.97/25.06 ( 166.97/25.06 setunionE2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton1 => 166.97/25.06 ( 166.97/25.06 setunionsingleton2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton => 166.97/25.06 ( 166.97/25.06 singletonprop => 166.97/25.06 ( 166.97/25.06 ex1E1 => 166.97/25.06 ( 166.97/25.06 ex1I => 166.97/25.06 ( 166.97/25.06 ex1I2 => 166.97/25.06 ( 166.97/25.06 singletonsuniq => 166.97/25.06 ( 166.97/25.06 setukpairinjL1 => 166.97/25.06 ( 166.97/25.06 kfstsingleton => 166.97/25.06 ( 166.97/25.06 theprop => 166.97/25.06 ( 166.97/25.06 kfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodfstin => 166.97/25.06 ( 166.97/25.06 setukpairinjL2 => 166.97/25.06 ( 166.97/25.06 setukpairinjL => 166.97/25.06 ( 166.97/25.06 setukpairinjR11 => 166.97/25.06 ( 166.97/25.06 setukpairinjR12 => 166.97/25.06 ( 166.97/25.06 setukpairinjR1 => 166.97/25.06 ( 166.97/25.06 upairequniteq => 166.97/25.06 ( 166.97/25.06 setukpairinjR2 => 166.97/25.06 ( 166.97/25.06 setukpairinjR => 166.97/25.06 ( 166.97/25.06 ksndsingleton => 166.97/25.06 ( 166.97/25.06 ksndpairEq => 166.97/25.06 ( 166.97/25.06 kpairsurjEq => 166.97/25.06 ( 166.97/25.06 cartprodsndin => 166.97/25.06 ( 166.97/25.06 cartprodpairmemEL => 166.97/25.06 ( 166.97/25.06 cartprodpairmemER => 166.97/25.06 ( 166.97/25.06 cartprodmempaircEq => 166.97/25.06 ( 166.97/25.06 cartprodfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodsndpairEq => 166.97/25.06 ( 166.97/25.06 cartprodpairsurjEq => 166.97/25.06 ( 166.97/25.06 dpsetconstrI => 166.97/25.06 ( 166.97/25.06 dpsetconstrSub => 166.97/25.06 ( 166.97/25.06 setOfPairsIsBReln => 166.97/25.06 ( 166.97/25.06 !!(( 166.97/25.06 ^[Y0 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y1 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y2 : ( 166.97/25.06 $i * $i) > $o]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y3 : $i]: 166.97/25.06 (in(Y3, 166.97/25.06 Y0) => 166.97/25.06 (!!(( 166.97/25.06 ^[Y4 : $i]: 166.97/25.06 (in(Y4, 166.97/25.06 Y1) => 166.97/25.06 (in( 166.97/25.06 kpair( 166.97/25.06 Y3, Y4), 166.97/25.06 dpsetconstr( 166.97/25.06 Y0, Y1, 166.97/25.06 Y2)) => 166.97/25.06 Y2(Y3, 166.97/25.06 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.06 inference('lazy_cnf_imply', [status(thm)], ['49'])). 166.97/25.06 tff('51', plain, 166.97/25.06 ~ (emptyE1 => 166.97/25.06 (vacuousDall => 166.97/25.06 (quantDeMorgan1 => 166.97/25.06 (quantDeMorgan2 => 166.97/25.06 (quantDeMorgan3 => 166.97/25.06 (quantDeMorgan4 => 166.97/25.06 (prop2setI => 166.97/25.06 (prop2set2propI => 166.97/25.06 (notdexE => 166.97/25.06 (notdallE => 166.97/25.06 (exuI1 => 166.97/25.06 (exuI3 => 166.97/25.06 (exuI2 => 166.97/25.06 (inCongP => 166.97/25.06 (in__Cong => 166.97/25.06 (exuE3u => 166.97/25.06 (exu__Cong => 166.97/25.06 (emptyset__Cong => 166.97/25.06 (setadjoin__Cong => 166.97/25.06 (powerset__Cong => 166.97/25.06 (setunion__Cong => 166.97/25.06 (omega__Cong => 166.97/25.06 (exuEu => 166.97/25.06 (descr__Cong => 166.97/25.06 (dsetconstr__Cong => 166.97/25.06 (subsetI1 => 166.97/25.06 (eqimpsubset2 => 166.97/25.06 (eqimpsubset1 => 166.97/25.06 (subsetI2 => 166.97/25.06 (emptysetsubset => 166.97/25.06 (subsetE => 166.97/25.06 (subsetE2 => 166.97/25.06 (notsubsetI => 166.97/25.06 (notequalI1 => 166.97/25.06 (notequalI2 => 166.97/25.06 (subsetRefl => 166.97/25.06 (subsetTrans => 166.97/25.06 (setadjoinSub => 166.97/25.06 (setadjoinSub2 => 166.97/25.06 (subset2powerset => 166.97/25.06 (setextsub => 166.97/25.06 (subsetemptysetimpeq => 166.97/25.06 (powersetI1 => 166.97/25.06 (powersetE1 => 166.97/25.06 (inPowerset => 166.97/25.06 (powersetsubset => 166.97/25.06 (sepInPowerset => 166.97/25.06 (sepSubset => 166.97/25.06 (binunionIL => 166.97/25.06 (upairset2IR => 166.97/25.06 (binunionIR => 166.97/25.06 (binunionEcases => 166.97/25.06 (binunionE => 166.97/25.06 (binunionLsub => 166.97/25.06 (binunionRsub => 166.97/25.06 (binintersectI => 166.97/25.06 (binintersectSubset5 => 166.97/25.06 (binintersectEL => 166.97/25.06 (binintersectLsub => 166.97/25.06 (binintersectSubset2 => 166.97/25.06 (binintersectSubset3 => 166.97/25.06 (binintersectER => 166.97/25.06 ( 166.97/25.06 disjointsetsI1 => 166.97/25.06 ( 166.97/25.06 binintersectRsub => 166.97/25.06 ( 166.97/25.06 binintersectSubset4 => 166.97/25.06 ( 166.97/25.06 binintersectSubset1 => 166.97/25.06 ( 166.97/25.06 bs114d => 166.97/25.06 ( 166.97/25.06 setminusI => 166.97/25.06 ( 166.97/25.06 setminusEL => 166.97/25.06 ( 166.97/25.06 setminusER => 166.97/25.06 ( 166.97/25.06 setminusSubset2 => 166.97/25.06 ( 166.97/25.06 setminusERneg => 166.97/25.06 ( 166.97/25.06 setminusELneg => 166.97/25.06 ( 166.97/25.06 setminusILneg => 166.97/25.06 ( 166.97/25.06 setminusIRneg => 166.97/25.06 ( 166.97/25.06 setminusLsub => 166.97/25.06 ( 166.97/25.06 setminusSubset1 => 166.97/25.06 ( 166.97/25.06 symdiffE => 166.97/25.06 ( 166.97/25.06 symdiffI1 => 166.97/25.06 ( 166.97/25.06 symdiffI2 => 166.97/25.06 ( 166.97/25.06 symdiffIneg1 => 166.97/25.06 ( 166.97/25.06 symdiffIneg2 => 166.97/25.06 ( 166.97/25.06 secondinupair => 166.97/25.06 ( 166.97/25.06 setukpairIL => 166.97/25.06 ( 166.97/25.06 setukpairIR => 166.97/25.06 ( 166.97/25.06 kpairiskpair => 166.97/25.06 ( 166.97/25.06 kpairp => 166.97/25.06 ( 166.97/25.06 singletonsubset => 166.97/25.06 ( 166.97/25.06 singletoninpowerset => 166.97/25.06 ( 166.97/25.06 singletoninpowunion => 166.97/25.06 ( 166.97/25.06 upairset2E => 166.97/25.06 ( 166.97/25.06 upairsubunion => 166.97/25.06 ( 166.97/25.06 upairinpowunion => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem1 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem2 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem3 => 166.97/25.06 ( 166.97/25.06 cartprodpairin => 166.97/25.06 ( 166.97/25.06 cartprodmempair1 => 166.97/25.06 ( 166.97/25.06 cartprodmempair => 166.97/25.06 ( 166.97/25.06 setunionE2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton1 => 166.97/25.06 ( 166.97/25.06 setunionsingleton2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton => 166.97/25.06 ( 166.97/25.06 singletonprop => 166.97/25.06 ( 166.97/25.06 ex1E1 => 166.97/25.06 ( 166.97/25.06 ex1I => 166.97/25.06 ( 166.97/25.06 ex1I2 => 166.97/25.06 ( 166.97/25.06 singletonsuniq => 166.97/25.06 ( 166.97/25.06 setukpairinjL1 => 166.97/25.06 ( 166.97/25.06 kfstsingleton => 166.97/25.06 ( 166.97/25.06 theprop => 166.97/25.06 ( 166.97/25.06 kfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodfstin => 166.97/25.06 ( 166.97/25.06 setukpairinjL2 => 166.97/25.06 ( 166.97/25.06 setukpairinjL => 166.97/25.06 ( 166.97/25.06 setukpairinjR11 => 166.97/25.06 ( 166.97/25.06 setukpairinjR12 => 166.97/25.06 ( 166.97/25.06 setukpairinjR1 => 166.97/25.06 ( 166.97/25.06 upairequniteq => 166.97/25.06 ( 166.97/25.06 setukpairinjR2 => 166.97/25.06 ( 166.97/25.06 setukpairinjR => 166.97/25.06 ( 166.97/25.06 ksndsingleton => 166.97/25.06 ( 166.97/25.06 ksndpairEq => 166.97/25.06 ( 166.97/25.06 kpairsurjEq => 166.97/25.06 ( 166.97/25.06 cartprodsndin => 166.97/25.06 ( 166.97/25.06 cartprodpairmemEL => 166.97/25.06 ( 166.97/25.06 cartprodpairmemER => 166.97/25.06 ( 166.97/25.06 cartprodmempaircEq => 166.97/25.06 ( 166.97/25.06 cartprodfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodsndpairEq => 166.97/25.06 ( 166.97/25.06 cartprodpairsurjEq => 166.97/25.06 ( 166.97/25.06 dpsetconstrI => 166.97/25.06 ( 166.97/25.06 dpsetconstrSub => 166.97/25.06 ( 166.97/25.06 setOfPairsIsBReln => 166.97/25.06 ( 166.97/25.06 !!(( 166.97/25.06 ^[Y0 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y1 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y2 : ( 166.97/25.06 $i * $i) > $o]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y3 : $i]: 166.97/25.06 (in(Y3, 166.97/25.06 Y0) => 166.97/25.06 (!!(( 166.97/25.06 ^[Y4 : $i]: 166.97/25.06 (in(Y4, 166.97/25.06 Y1) => 166.97/25.06 (in( 166.97/25.06 kpair( 166.97/25.06 Y3, Y4), 166.97/25.06 dpsetconstr( 166.97/25.06 Y0, Y1, 166.97/25.06 Y2)) => 166.97/25.06 Y2(Y3, 166.97/25.06 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.06 inference('lazy_cnf_imply', [status(thm)], ['50'])). 166.97/25.06 tff('52', plain, 166.97/25.06 ~ (vacuousDall => 166.97/25.06 (quantDeMorgan1 => 166.97/25.06 (quantDeMorgan2 => 166.97/25.06 (quantDeMorgan3 => 166.97/25.06 (quantDeMorgan4 => 166.97/25.06 (prop2setI => 166.97/25.06 (prop2set2propI => 166.97/25.06 (notdexE => 166.97/25.06 (notdallE => 166.97/25.06 (exuI1 => 166.97/25.06 (exuI3 => 166.97/25.06 (exuI2 => 166.97/25.06 (inCongP => 166.97/25.06 (in__Cong => 166.97/25.06 (exuE3u => 166.97/25.06 (exu__Cong => 166.97/25.06 (emptyset__Cong => 166.97/25.06 (setadjoin__Cong => 166.97/25.06 (powerset__Cong => 166.97/25.06 (setunion__Cong => 166.97/25.06 (omega__Cong => 166.97/25.06 (exuEu => 166.97/25.06 (descr__Cong => 166.97/25.06 (dsetconstr__Cong => 166.97/25.06 (subsetI1 => 166.97/25.06 (eqimpsubset2 => 166.97/25.06 (eqimpsubset1 => 166.97/25.06 (subsetI2 => 166.97/25.06 (emptysetsubset => 166.97/25.06 (subsetE => 166.97/25.06 (subsetE2 => 166.97/25.06 (notsubsetI => 166.97/25.06 (notequalI1 => 166.97/25.06 (notequalI2 => 166.97/25.06 (subsetRefl => 166.97/25.06 (subsetTrans => 166.97/25.06 (setadjoinSub => 166.97/25.06 (setadjoinSub2 => 166.97/25.06 (subset2powerset => 166.97/25.06 (setextsub => 166.97/25.06 (subsetemptysetimpeq => 166.97/25.06 (powersetI1 => 166.97/25.06 (powersetE1 => 166.97/25.06 (inPowerset => 166.97/25.06 (powersetsubset => 166.97/25.06 (sepInPowerset => 166.97/25.06 (sepSubset => 166.97/25.06 (binunionIL => 166.97/25.06 (upairset2IR => 166.97/25.06 (binunionIR => 166.97/25.06 (binunionEcases => 166.97/25.06 (binunionE => 166.97/25.06 (binunionLsub => 166.97/25.06 (binunionRsub => 166.97/25.06 (binintersectI => 166.97/25.06 (binintersectSubset5 => 166.97/25.06 (binintersectEL => 166.97/25.06 (binintersectLsub => 166.97/25.06 (binintersectSubset2 => 166.97/25.06 (binintersectSubset3 => 166.97/25.06 (binintersectER => 166.97/25.06 (disjointsetsI1 => 166.97/25.06 ( 166.97/25.06 binintersectRsub => 166.97/25.06 ( 166.97/25.06 binintersectSubset4 => 166.97/25.06 ( 166.97/25.06 binintersectSubset1 => 166.97/25.06 ( 166.97/25.06 bs114d => 166.97/25.06 ( 166.97/25.06 setminusI => 166.97/25.06 ( 166.97/25.06 setminusEL => 166.97/25.06 ( 166.97/25.06 setminusER => 166.97/25.06 ( 166.97/25.06 setminusSubset2 => 166.97/25.06 ( 166.97/25.06 setminusERneg => 166.97/25.06 ( 166.97/25.06 setminusELneg => 166.97/25.06 ( 166.97/25.06 setminusILneg => 166.97/25.06 ( 166.97/25.06 setminusIRneg => 166.97/25.06 ( 166.97/25.06 setminusLsub => 166.97/25.06 ( 166.97/25.06 setminusSubset1 => 166.97/25.06 ( 166.97/25.06 symdiffE => 166.97/25.06 ( 166.97/25.06 symdiffI1 => 166.97/25.06 ( 166.97/25.06 symdiffI2 => 166.97/25.06 ( 166.97/25.06 symdiffIneg1 => 166.97/25.06 ( 166.97/25.06 symdiffIneg2 => 166.97/25.06 ( 166.97/25.06 secondinupair => 166.97/25.06 ( 166.97/25.06 setukpairIL => 166.97/25.06 ( 166.97/25.06 setukpairIR => 166.97/25.06 ( 166.97/25.06 kpairiskpair => 166.97/25.06 ( 166.97/25.06 kpairp => 166.97/25.06 ( 166.97/25.06 singletonsubset => 166.97/25.06 ( 166.97/25.06 singletoninpowerset => 166.97/25.06 ( 166.97/25.06 singletoninpowunion => 166.97/25.06 ( 166.97/25.06 upairset2E => 166.97/25.06 ( 166.97/25.06 upairsubunion => 166.97/25.06 ( 166.97/25.06 upairinpowunion => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem1 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem2 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem3 => 166.97/25.06 ( 166.97/25.06 cartprodpairin => 166.97/25.06 ( 166.97/25.06 cartprodmempair1 => 166.97/25.06 ( 166.97/25.06 cartprodmempair => 166.97/25.06 ( 166.97/25.06 setunionE2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton1 => 166.97/25.06 ( 166.97/25.06 setunionsingleton2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton => 166.97/25.06 ( 166.97/25.06 singletonprop => 166.97/25.06 ( 166.97/25.06 ex1E1 => 166.97/25.06 ( 166.97/25.06 ex1I => 166.97/25.06 ( 166.97/25.06 ex1I2 => 166.97/25.06 ( 166.97/25.06 singletonsuniq => 166.97/25.06 ( 166.97/25.06 setukpairinjL1 => 166.97/25.06 ( 166.97/25.06 kfstsingleton => 166.97/25.06 ( 166.97/25.06 theprop => 166.97/25.06 ( 166.97/25.06 kfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodfstin => 166.97/25.06 ( 166.97/25.06 setukpairinjL2 => 166.97/25.06 ( 166.97/25.06 setukpairinjL => 166.97/25.06 ( 166.97/25.06 setukpairinjR11 => 166.97/25.06 ( 166.97/25.06 setukpairinjR12 => 166.97/25.06 ( 166.97/25.06 setukpairinjR1 => 166.97/25.06 ( 166.97/25.06 upairequniteq => 166.97/25.06 ( 166.97/25.06 setukpairinjR2 => 166.97/25.06 ( 166.97/25.06 setukpairinjR => 166.97/25.06 ( 166.97/25.06 ksndsingleton => 166.97/25.06 ( 166.97/25.06 ksndpairEq => 166.97/25.06 ( 166.97/25.06 kpairsurjEq => 166.97/25.06 ( 166.97/25.06 cartprodsndin => 166.97/25.06 ( 166.97/25.06 cartprodpairmemEL => 166.97/25.06 ( 166.97/25.06 cartprodpairmemER => 166.97/25.06 ( 166.97/25.06 cartprodmempaircEq => 166.97/25.06 ( 166.97/25.06 cartprodfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodsndpairEq => 166.97/25.06 ( 166.97/25.06 cartprodpairsurjEq => 166.97/25.06 ( 166.97/25.06 dpsetconstrI => 166.97/25.06 ( 166.97/25.06 dpsetconstrSub => 166.97/25.06 ( 166.97/25.06 setOfPairsIsBReln => 166.97/25.06 ( 166.97/25.06 !!(( 166.97/25.06 ^[Y0 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y1 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y2 : ( 166.97/25.06 $i * $i) > $o]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y3 : $i]: 166.97/25.06 (in(Y3, 166.97/25.06 Y0) => 166.97/25.06 (!!(( 166.97/25.06 ^[Y4 : $i]: 166.97/25.06 (in(Y4, 166.97/25.06 Y1) => 166.97/25.06 (in( 166.97/25.06 kpair( 166.97/25.06 Y3, Y4), 166.97/25.06 dpsetconstr( 166.97/25.06 Y0, Y1, 166.97/25.06 Y2)) => 166.97/25.06 Y2(Y3, 166.97/25.06 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.06 inference('lazy_cnf_imply', [status(thm)], ['51'])). 166.97/25.06 tff('53', plain, 166.97/25.06 ~ (quantDeMorgan1 => 166.97/25.06 (quantDeMorgan2 => 166.97/25.06 (quantDeMorgan3 => 166.97/25.06 (quantDeMorgan4 => 166.97/25.06 (prop2setI => 166.97/25.06 (prop2set2propI => 166.97/25.06 (notdexE => 166.97/25.06 (notdallE => 166.97/25.06 (exuI1 => 166.97/25.06 (exuI3 => 166.97/25.06 (exuI2 => 166.97/25.06 (inCongP => 166.97/25.06 (in__Cong => 166.97/25.06 (exuE3u => 166.97/25.06 (exu__Cong => 166.97/25.06 (emptyset__Cong => 166.97/25.06 (setadjoin__Cong => 166.97/25.06 (powerset__Cong => 166.97/25.06 (setunion__Cong => 166.97/25.06 (omega__Cong => 166.97/25.06 (exuEu => 166.97/25.06 (descr__Cong => 166.97/25.06 (dsetconstr__Cong => 166.97/25.06 (subsetI1 => 166.97/25.06 (eqimpsubset2 => 166.97/25.06 (eqimpsubset1 => 166.97/25.06 (subsetI2 => 166.97/25.06 (emptysetsubset => 166.97/25.06 (subsetE => 166.97/25.06 (subsetE2 => 166.97/25.06 (notsubsetI => 166.97/25.06 (notequalI1 => 166.97/25.06 (notequalI2 => 166.97/25.06 (subsetRefl => 166.97/25.06 (subsetTrans => 166.97/25.06 (setadjoinSub => 166.97/25.06 (setadjoinSub2 => 166.97/25.06 (subset2powerset => 166.97/25.06 (setextsub => 166.97/25.06 (subsetemptysetimpeq => 166.97/25.06 (powersetI1 => 166.97/25.06 (powersetE1 => 166.97/25.06 (inPowerset => 166.97/25.06 (powersetsubset => 166.97/25.06 (sepInPowerset => 166.97/25.06 (sepSubset => 166.97/25.06 (binunionIL => 166.97/25.06 (upairset2IR => 166.97/25.06 (binunionIR => 166.97/25.06 (binunionEcases => 166.97/25.06 (binunionE => 166.97/25.06 (binunionLsub => 166.97/25.06 (binunionRsub => 166.97/25.06 (binintersectI => 166.97/25.06 (binintersectSubset5 => 166.97/25.06 (binintersectEL => 166.97/25.06 (binintersectLsub => 166.97/25.06 (binintersectSubset2 => 166.97/25.06 (binintersectSubset3 => 166.97/25.06 (binintersectER => 166.97/25.06 (disjointsetsI1 => 166.97/25.06 (binintersectRsub => 166.97/25.06 ( 166.97/25.06 binintersectSubset4 => 166.97/25.06 ( 166.97/25.06 binintersectSubset1 => 166.97/25.06 ( 166.97/25.06 bs114d => 166.97/25.06 ( 166.97/25.06 setminusI => 166.97/25.06 ( 166.97/25.06 setminusEL => 166.97/25.06 ( 166.97/25.06 setminusER => 166.97/25.06 ( 166.97/25.06 setminusSubset2 => 166.97/25.06 ( 166.97/25.06 setminusERneg => 166.97/25.06 ( 166.97/25.06 setminusELneg => 166.97/25.06 ( 166.97/25.06 setminusILneg => 166.97/25.06 ( 166.97/25.06 setminusIRneg => 166.97/25.06 ( 166.97/25.06 setminusLsub => 166.97/25.06 ( 166.97/25.06 setminusSubset1 => 166.97/25.06 ( 166.97/25.06 symdiffE => 166.97/25.06 ( 166.97/25.06 symdiffI1 => 166.97/25.06 ( 166.97/25.06 symdiffI2 => 166.97/25.06 ( 166.97/25.06 symdiffIneg1 => 166.97/25.06 ( 166.97/25.06 symdiffIneg2 => 166.97/25.06 ( 166.97/25.06 secondinupair => 166.97/25.06 ( 166.97/25.06 setukpairIL => 166.97/25.06 ( 166.97/25.06 setukpairIR => 166.97/25.06 ( 166.97/25.06 kpairiskpair => 166.97/25.06 ( 166.97/25.06 kpairp => 166.97/25.06 ( 166.97/25.06 singletonsubset => 166.97/25.06 ( 166.97/25.06 singletoninpowerset => 166.97/25.06 ( 166.97/25.06 singletoninpowunion => 166.97/25.06 ( 166.97/25.06 upairset2E => 166.97/25.06 ( 166.97/25.06 upairsubunion => 166.97/25.06 ( 166.97/25.06 upairinpowunion => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem1 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem2 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem3 => 166.97/25.06 ( 166.97/25.06 cartprodpairin => 166.97/25.06 ( 166.97/25.06 cartprodmempair1 => 166.97/25.06 ( 166.97/25.06 cartprodmempair => 166.97/25.06 ( 166.97/25.06 setunionE2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton1 => 166.97/25.06 ( 166.97/25.06 setunionsingleton2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton => 166.97/25.06 ( 166.97/25.06 singletonprop => 166.97/25.06 ( 166.97/25.06 ex1E1 => 166.97/25.06 ( 166.97/25.06 ex1I => 166.97/25.06 ( 166.97/25.06 ex1I2 => 166.97/25.06 ( 166.97/25.06 singletonsuniq => 166.97/25.06 ( 166.97/25.06 setukpairinjL1 => 166.97/25.06 ( 166.97/25.06 kfstsingleton => 166.97/25.06 ( 166.97/25.06 theprop => 166.97/25.06 ( 166.97/25.06 kfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodfstin => 166.97/25.06 ( 166.97/25.06 setukpairinjL2 => 166.97/25.06 ( 166.97/25.06 setukpairinjL => 166.97/25.06 ( 166.97/25.06 setukpairinjR11 => 166.97/25.06 ( 166.97/25.06 setukpairinjR12 => 166.97/25.06 ( 166.97/25.06 setukpairinjR1 => 166.97/25.06 ( 166.97/25.06 upairequniteq => 166.97/25.06 ( 166.97/25.06 setukpairinjR2 => 166.97/25.06 ( 166.97/25.06 setukpairinjR => 166.97/25.06 ( 166.97/25.06 ksndsingleton => 166.97/25.06 ( 166.97/25.06 ksndpairEq => 166.97/25.06 ( 166.97/25.06 kpairsurjEq => 166.97/25.06 ( 166.97/25.06 cartprodsndin => 166.97/25.06 ( 166.97/25.06 cartprodpairmemEL => 166.97/25.06 ( 166.97/25.06 cartprodpairmemER => 166.97/25.06 ( 166.97/25.06 cartprodmempaircEq => 166.97/25.06 ( 166.97/25.06 cartprodfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodsndpairEq => 166.97/25.06 ( 166.97/25.06 cartprodpairsurjEq => 166.97/25.06 ( 166.97/25.06 dpsetconstrI => 166.97/25.06 ( 166.97/25.06 dpsetconstrSub => 166.97/25.06 ( 166.97/25.06 setOfPairsIsBReln => 166.97/25.06 ( 166.97/25.06 !!(( 166.97/25.06 ^[Y0 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y1 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y2 : ( 166.97/25.06 $i * $i) > $o]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y3 : $i]: 166.97/25.06 (in(Y3, 166.97/25.06 Y0) => 166.97/25.06 (!!(( 166.97/25.06 ^[Y4 : $i]: 166.97/25.06 (in(Y4, 166.97/25.06 Y1) => 166.97/25.06 (in( 166.97/25.06 kpair( 166.97/25.06 Y3, Y4), 166.97/25.06 dpsetconstr( 166.97/25.06 Y0, Y1, 166.97/25.06 Y2)) => 166.97/25.06 Y2(Y3, 166.97/25.06 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.06 inference('lazy_cnf_imply', [status(thm)], ['52'])). 166.97/25.06 tff('54', plain, 166.97/25.06 ~ (quantDeMorgan2 => 166.97/25.06 (quantDeMorgan3 => 166.97/25.06 (quantDeMorgan4 => 166.97/25.06 (prop2setI => 166.97/25.06 (prop2set2propI => 166.97/25.06 (notdexE => 166.97/25.06 (notdallE => 166.97/25.06 (exuI1 => 166.97/25.06 (exuI3 => 166.97/25.06 (exuI2 => 166.97/25.06 (inCongP => 166.97/25.06 (in__Cong => 166.97/25.06 (exuE3u => 166.97/25.06 (exu__Cong => 166.97/25.06 (emptyset__Cong => 166.97/25.06 (setadjoin__Cong => 166.97/25.06 (powerset__Cong => 166.97/25.06 (setunion__Cong => 166.97/25.06 (omega__Cong => 166.97/25.06 (exuEu => 166.97/25.06 (descr__Cong => 166.97/25.06 (dsetconstr__Cong => 166.97/25.06 (subsetI1 => 166.97/25.06 (eqimpsubset2 => 166.97/25.06 (eqimpsubset1 => 166.97/25.06 (subsetI2 => 166.97/25.06 (emptysetsubset => 166.97/25.06 (subsetE => 166.97/25.06 (subsetE2 => 166.97/25.06 (notsubsetI => 166.97/25.06 (notequalI1 => 166.97/25.06 (notequalI2 => 166.97/25.06 (subsetRefl => 166.97/25.06 (subsetTrans => 166.97/25.06 (setadjoinSub => 166.97/25.06 (setadjoinSub2 => 166.97/25.06 (subset2powerset => 166.97/25.06 (setextsub => 166.97/25.06 (subsetemptysetimpeq => 166.97/25.06 (powersetI1 => 166.97/25.06 (powersetE1 => 166.97/25.06 (inPowerset => 166.97/25.06 (powersetsubset => 166.97/25.06 (sepInPowerset => 166.97/25.06 (sepSubset => 166.97/25.06 (binunionIL => 166.97/25.06 (upairset2IR => 166.97/25.06 (binunionIR => 166.97/25.06 (binunionEcases => 166.97/25.06 (binunionE => 166.97/25.06 (binunionLsub => 166.97/25.06 (binunionRsub => 166.97/25.06 (binintersectI => 166.97/25.06 (binintersectSubset5 => 166.97/25.06 (binintersectEL => 166.97/25.06 (binintersectLsub => 166.97/25.06 (binintersectSubset2 => 166.97/25.06 (binintersectSubset3 => 166.97/25.06 (binintersectER => 166.97/25.06 (disjointsetsI1 => 166.97/25.06 (binintersectRsub => 166.97/25.06 (binintersectSubset4 => 166.97/25.06 ( 166.97/25.06 binintersectSubset1 => 166.97/25.06 ( 166.97/25.06 bs114d => 166.97/25.06 ( 166.97/25.06 setminusI => 166.97/25.06 ( 166.97/25.06 setminusEL => 166.97/25.06 ( 166.97/25.06 setminusER => 166.97/25.06 ( 166.97/25.06 setminusSubset2 => 166.97/25.06 ( 166.97/25.06 setminusERneg => 166.97/25.06 ( 166.97/25.06 setminusELneg => 166.97/25.06 ( 166.97/25.06 setminusILneg => 166.97/25.06 ( 166.97/25.06 setminusIRneg => 166.97/25.06 ( 166.97/25.06 setminusLsub => 166.97/25.06 ( 166.97/25.06 setminusSubset1 => 166.97/25.06 ( 166.97/25.06 symdiffE => 166.97/25.06 ( 166.97/25.06 symdiffI1 => 166.97/25.06 ( 166.97/25.06 symdiffI2 => 166.97/25.06 ( 166.97/25.06 symdiffIneg1 => 166.97/25.06 ( 166.97/25.06 symdiffIneg2 => 166.97/25.06 ( 166.97/25.06 secondinupair => 166.97/25.06 ( 166.97/25.06 setukpairIL => 166.97/25.06 ( 166.97/25.06 setukpairIR => 166.97/25.06 ( 166.97/25.06 kpairiskpair => 166.97/25.06 ( 166.97/25.06 kpairp => 166.97/25.06 ( 166.97/25.06 singletonsubset => 166.97/25.06 ( 166.97/25.06 singletoninpowerset => 166.97/25.06 ( 166.97/25.06 singletoninpowunion => 166.97/25.06 ( 166.97/25.06 upairset2E => 166.97/25.06 ( 166.97/25.06 upairsubunion => 166.97/25.06 ( 166.97/25.06 upairinpowunion => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem1 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem2 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem3 => 166.97/25.06 ( 166.97/25.06 cartprodpairin => 166.97/25.06 ( 166.97/25.06 cartprodmempair1 => 166.97/25.06 ( 166.97/25.06 cartprodmempair => 166.97/25.06 ( 166.97/25.06 setunionE2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton1 => 166.97/25.06 ( 166.97/25.06 setunionsingleton2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton => 166.97/25.06 ( 166.97/25.06 singletonprop => 166.97/25.06 ( 166.97/25.06 ex1E1 => 166.97/25.06 ( 166.97/25.06 ex1I => 166.97/25.06 ( 166.97/25.06 ex1I2 => 166.97/25.06 ( 166.97/25.06 singletonsuniq => 166.97/25.06 ( 166.97/25.06 setukpairinjL1 => 166.97/25.06 ( 166.97/25.06 kfstsingleton => 166.97/25.06 ( 166.97/25.06 theprop => 166.97/25.06 ( 166.97/25.06 kfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodfstin => 166.97/25.06 ( 166.97/25.06 setukpairinjL2 => 166.97/25.06 ( 166.97/25.06 setukpairinjL => 166.97/25.06 ( 166.97/25.06 setukpairinjR11 => 166.97/25.06 ( 166.97/25.06 setukpairinjR12 => 166.97/25.06 ( 166.97/25.06 setukpairinjR1 => 166.97/25.06 ( 166.97/25.06 upairequniteq => 166.97/25.06 ( 166.97/25.06 setukpairinjR2 => 166.97/25.06 ( 166.97/25.06 setukpairinjR => 166.97/25.06 ( 166.97/25.06 ksndsingleton => 166.97/25.06 ( 166.97/25.06 ksndpairEq => 166.97/25.06 ( 166.97/25.06 kpairsurjEq => 166.97/25.06 ( 166.97/25.06 cartprodsndin => 166.97/25.06 ( 166.97/25.06 cartprodpairmemEL => 166.97/25.06 ( 166.97/25.06 cartprodpairmemER => 166.97/25.06 ( 166.97/25.06 cartprodmempaircEq => 166.97/25.06 ( 166.97/25.06 cartprodfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodsndpairEq => 166.97/25.06 ( 166.97/25.06 cartprodpairsurjEq => 166.97/25.06 ( 166.97/25.06 dpsetconstrI => 166.97/25.06 ( 166.97/25.06 dpsetconstrSub => 166.97/25.06 ( 166.97/25.06 setOfPairsIsBReln => 166.97/25.06 ( 166.97/25.06 !!(( 166.97/25.06 ^[Y0 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y1 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y2 : ( 166.97/25.06 $i * $i) > $o]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y3 : $i]: 166.97/25.06 (in(Y3, 166.97/25.06 Y0) => 166.97/25.06 (!!(( 166.97/25.06 ^[Y4 : $i]: 166.97/25.06 (in(Y4, 166.97/25.06 Y1) => 166.97/25.06 (in( 166.97/25.06 kpair( 166.97/25.06 Y3, Y4), 166.97/25.06 dpsetconstr( 166.97/25.06 Y0, Y1, 166.97/25.06 Y2)) => 166.97/25.06 Y2(Y3, 166.97/25.06 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.06 inference('lazy_cnf_imply', [status(thm)], ['53'])). 166.97/25.06 tff('55', plain, 166.97/25.06 ~ (quantDeMorgan3 => 166.97/25.06 (quantDeMorgan4 => 166.97/25.06 (prop2setI => 166.97/25.06 (prop2set2propI => 166.97/25.06 (notdexE => 166.97/25.06 (notdallE => 166.97/25.06 (exuI1 => 166.97/25.06 (exuI3 => 166.97/25.06 (exuI2 => 166.97/25.06 (inCongP => 166.97/25.06 (in__Cong => 166.97/25.06 (exuE3u => 166.97/25.06 (exu__Cong => 166.97/25.06 (emptyset__Cong => 166.97/25.06 (setadjoin__Cong => 166.97/25.06 (powerset__Cong => 166.97/25.06 (setunion__Cong => 166.97/25.06 (omega__Cong => 166.97/25.06 (exuEu => 166.97/25.06 (descr__Cong => 166.97/25.06 (dsetconstr__Cong => 166.97/25.06 (subsetI1 => 166.97/25.06 (eqimpsubset2 => 166.97/25.06 (eqimpsubset1 => 166.97/25.06 (subsetI2 => 166.97/25.06 (emptysetsubset => 166.97/25.06 (subsetE => 166.97/25.06 (subsetE2 => 166.97/25.06 (notsubsetI => 166.97/25.06 (notequalI1 => 166.97/25.06 (notequalI2 => 166.97/25.06 (subsetRefl => 166.97/25.06 (subsetTrans => 166.97/25.06 (setadjoinSub => 166.97/25.06 (setadjoinSub2 => 166.97/25.06 (subset2powerset => 166.97/25.06 (setextsub => 166.97/25.06 (subsetemptysetimpeq => 166.97/25.06 (powersetI1 => 166.97/25.06 (powersetE1 => 166.97/25.06 (inPowerset => 166.97/25.06 (powersetsubset => 166.97/25.06 (sepInPowerset => 166.97/25.06 (sepSubset => 166.97/25.06 (binunionIL => 166.97/25.06 (upairset2IR => 166.97/25.06 (binunionIR => 166.97/25.06 (binunionEcases => 166.97/25.06 (binunionE => 166.97/25.06 (binunionLsub => 166.97/25.06 (binunionRsub => 166.97/25.06 (binintersectI => 166.97/25.06 (binintersectSubset5 => 166.97/25.06 (binintersectEL => 166.97/25.06 (binintersectLsub => 166.97/25.06 (binintersectSubset2 => 166.97/25.06 (binintersectSubset3 => 166.97/25.06 (binintersectER => 166.97/25.06 (disjointsetsI1 => 166.97/25.06 (binintersectRsub => 166.97/25.06 (binintersectSubset4 => 166.97/25.06 (binintersectSubset1 => 166.97/25.06 ( 166.97/25.06 bs114d => 166.97/25.06 ( 166.97/25.06 setminusI => 166.97/25.06 ( 166.97/25.06 setminusEL => 166.97/25.06 ( 166.97/25.06 setminusER => 166.97/25.06 ( 166.97/25.06 setminusSubset2 => 166.97/25.06 ( 166.97/25.06 setminusERneg => 166.97/25.06 ( 166.97/25.06 setminusELneg => 166.97/25.06 ( 166.97/25.06 setminusILneg => 166.97/25.06 ( 166.97/25.06 setminusIRneg => 166.97/25.06 ( 166.97/25.06 setminusLsub => 166.97/25.06 ( 166.97/25.06 setminusSubset1 => 166.97/25.06 ( 166.97/25.06 symdiffE => 166.97/25.06 ( 166.97/25.06 symdiffI1 => 166.97/25.06 ( 166.97/25.06 symdiffI2 => 166.97/25.06 ( 166.97/25.06 symdiffIneg1 => 166.97/25.06 ( 166.97/25.06 symdiffIneg2 => 166.97/25.06 ( 166.97/25.06 secondinupair => 166.97/25.06 ( 166.97/25.06 setukpairIL => 166.97/25.06 ( 166.97/25.06 setukpairIR => 166.97/25.06 ( 166.97/25.06 kpairiskpair => 166.97/25.06 ( 166.97/25.06 kpairp => 166.97/25.06 ( 166.97/25.06 singletonsubset => 166.97/25.06 ( 166.97/25.06 singletoninpowerset => 166.97/25.06 ( 166.97/25.06 singletoninpowunion => 166.97/25.06 ( 166.97/25.06 upairset2E => 166.97/25.06 ( 166.97/25.06 upairsubunion => 166.97/25.06 ( 166.97/25.06 upairinpowunion => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem1 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem2 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem3 => 166.97/25.06 ( 166.97/25.06 cartprodpairin => 166.97/25.06 ( 166.97/25.06 cartprodmempair1 => 166.97/25.06 ( 166.97/25.06 cartprodmempair => 166.97/25.06 ( 166.97/25.06 setunionE2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton1 => 166.97/25.06 ( 166.97/25.06 setunionsingleton2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton => 166.97/25.06 ( 166.97/25.06 singletonprop => 166.97/25.06 ( 166.97/25.06 ex1E1 => 166.97/25.06 ( 166.97/25.06 ex1I => 166.97/25.06 ( 166.97/25.06 ex1I2 => 166.97/25.06 ( 166.97/25.06 singletonsuniq => 166.97/25.06 ( 166.97/25.06 setukpairinjL1 => 166.97/25.06 ( 166.97/25.06 kfstsingleton => 166.97/25.06 ( 166.97/25.06 theprop => 166.97/25.06 ( 166.97/25.06 kfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodfstin => 166.97/25.06 ( 166.97/25.06 setukpairinjL2 => 166.97/25.06 ( 166.97/25.06 setukpairinjL => 166.97/25.06 ( 166.97/25.06 setukpairinjR11 => 166.97/25.06 ( 166.97/25.06 setukpairinjR12 => 166.97/25.06 ( 166.97/25.06 setukpairinjR1 => 166.97/25.06 ( 166.97/25.06 upairequniteq => 166.97/25.06 ( 166.97/25.06 setukpairinjR2 => 166.97/25.06 ( 166.97/25.06 setukpairinjR => 166.97/25.06 ( 166.97/25.06 ksndsingleton => 166.97/25.06 ( 166.97/25.06 ksndpairEq => 166.97/25.06 ( 166.97/25.06 kpairsurjEq => 166.97/25.06 ( 166.97/25.06 cartprodsndin => 166.97/25.06 ( 166.97/25.06 cartprodpairmemEL => 166.97/25.06 ( 166.97/25.06 cartprodpairmemER => 166.97/25.06 ( 166.97/25.06 cartprodmempaircEq => 166.97/25.06 ( 166.97/25.06 cartprodfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodsndpairEq => 166.97/25.06 ( 166.97/25.06 cartprodpairsurjEq => 166.97/25.06 ( 166.97/25.06 dpsetconstrI => 166.97/25.06 ( 166.97/25.06 dpsetconstrSub => 166.97/25.06 ( 166.97/25.06 setOfPairsIsBReln => 166.97/25.06 ( 166.97/25.06 !!(( 166.97/25.06 ^[Y0 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y1 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y2 : ( 166.97/25.06 $i * $i) > $o]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y3 : $i]: 166.97/25.06 (in(Y3, 166.97/25.06 Y0) => 166.97/25.06 (!!(( 166.97/25.06 ^[Y4 : $i]: 166.97/25.06 (in(Y4, 166.97/25.06 Y1) => 166.97/25.06 (in( 166.97/25.06 kpair( 166.97/25.06 Y3, Y4), 166.97/25.06 dpsetconstr( 166.97/25.06 Y0, Y1, 166.97/25.06 Y2)) => 166.97/25.06 Y2(Y3, 166.97/25.06 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.06 inference('lazy_cnf_imply', [status(thm)], ['54'])). 166.97/25.06 tff('56', plain, 166.97/25.06 ~ (quantDeMorgan4 => 166.97/25.06 (prop2setI => 166.97/25.06 (prop2set2propI => 166.97/25.06 (notdexE => 166.97/25.06 (notdallE => 166.97/25.06 (exuI1 => 166.97/25.06 (exuI3 => 166.97/25.06 (exuI2 => 166.97/25.06 (inCongP => 166.97/25.06 (in__Cong => 166.97/25.06 (exuE3u => 166.97/25.06 (exu__Cong => 166.97/25.06 (emptyset__Cong => 166.97/25.06 (setadjoin__Cong => 166.97/25.06 (powerset__Cong => 166.97/25.06 (setunion__Cong => 166.97/25.06 (omega__Cong => 166.97/25.06 (exuEu => 166.97/25.06 (descr__Cong => 166.97/25.06 (dsetconstr__Cong => 166.97/25.06 (subsetI1 => 166.97/25.06 (eqimpsubset2 => 166.97/25.06 (eqimpsubset1 => 166.97/25.06 (subsetI2 => 166.97/25.06 (emptysetsubset => 166.97/25.06 (subsetE => 166.97/25.06 (subsetE2 => 166.97/25.06 (notsubsetI => 166.97/25.06 (notequalI1 => 166.97/25.06 (notequalI2 => 166.97/25.06 (subsetRefl => 166.97/25.06 (subsetTrans => 166.97/25.06 (setadjoinSub => 166.97/25.06 (setadjoinSub2 => 166.97/25.06 (subset2powerset => 166.97/25.06 (setextsub => 166.97/25.06 (subsetemptysetimpeq => 166.97/25.06 (powersetI1 => 166.97/25.06 (powersetE1 => 166.97/25.06 (inPowerset => 166.97/25.06 (powersetsubset => 166.97/25.06 (sepInPowerset => 166.97/25.06 (sepSubset => 166.97/25.06 (binunionIL => 166.97/25.06 (upairset2IR => 166.97/25.06 (binunionIR => 166.97/25.06 (binunionEcases => 166.97/25.06 (binunionE => 166.97/25.06 (binunionLsub => 166.97/25.06 (binunionRsub => 166.97/25.06 (binintersectI => 166.97/25.06 (binintersectSubset5 => 166.97/25.06 (binintersectEL => 166.97/25.06 (binintersectLsub => 166.97/25.06 (binintersectSubset2 => 166.97/25.06 (binintersectSubset3 => 166.97/25.06 (binintersectER => 166.97/25.06 (disjointsetsI1 => 166.97/25.06 (binintersectRsub => 166.97/25.06 (binintersectSubset4 => 166.97/25.06 (binintersectSubset1 => 166.97/25.06 (bs114d => 166.97/25.06 ( 166.97/25.06 setminusI => 166.97/25.06 ( 166.97/25.06 setminusEL => 166.97/25.06 ( 166.97/25.06 setminusER => 166.97/25.06 ( 166.97/25.06 setminusSubset2 => 166.97/25.06 ( 166.97/25.06 setminusERneg => 166.97/25.06 ( 166.97/25.06 setminusELneg => 166.97/25.06 ( 166.97/25.06 setminusILneg => 166.97/25.06 ( 166.97/25.06 setminusIRneg => 166.97/25.06 ( 166.97/25.06 setminusLsub => 166.97/25.06 ( 166.97/25.06 setminusSubset1 => 166.97/25.06 ( 166.97/25.06 symdiffE => 166.97/25.06 ( 166.97/25.06 symdiffI1 => 166.97/25.06 ( 166.97/25.06 symdiffI2 => 166.97/25.06 ( 166.97/25.06 symdiffIneg1 => 166.97/25.06 ( 166.97/25.06 symdiffIneg2 => 166.97/25.06 ( 166.97/25.06 secondinupair => 166.97/25.06 ( 166.97/25.06 setukpairIL => 166.97/25.06 ( 166.97/25.06 setukpairIR => 166.97/25.06 ( 166.97/25.06 kpairiskpair => 166.97/25.06 ( 166.97/25.06 kpairp => 166.97/25.06 ( 166.97/25.06 singletonsubset => 166.97/25.06 ( 166.97/25.06 singletoninpowerset => 166.97/25.06 ( 166.97/25.06 singletoninpowunion => 166.97/25.06 ( 166.97/25.06 upairset2E => 166.97/25.06 ( 166.97/25.06 upairsubunion => 166.97/25.06 ( 166.97/25.06 upairinpowunion => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem1 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem2 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem3 => 166.97/25.06 ( 166.97/25.06 cartprodpairin => 166.97/25.06 ( 166.97/25.06 cartprodmempair1 => 166.97/25.06 ( 166.97/25.06 cartprodmempair => 166.97/25.06 ( 166.97/25.06 setunionE2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton1 => 166.97/25.06 ( 166.97/25.06 setunionsingleton2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton => 166.97/25.06 ( 166.97/25.06 singletonprop => 166.97/25.06 ( 166.97/25.06 ex1E1 => 166.97/25.06 ( 166.97/25.06 ex1I => 166.97/25.06 ( 166.97/25.06 ex1I2 => 166.97/25.06 ( 166.97/25.06 singletonsuniq => 166.97/25.06 ( 166.97/25.06 setukpairinjL1 => 166.97/25.06 ( 166.97/25.06 kfstsingleton => 166.97/25.06 ( 166.97/25.06 theprop => 166.97/25.06 ( 166.97/25.06 kfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodfstin => 166.97/25.06 ( 166.97/25.06 setukpairinjL2 => 166.97/25.06 ( 166.97/25.06 setukpairinjL => 166.97/25.06 ( 166.97/25.06 setukpairinjR11 => 166.97/25.06 ( 166.97/25.06 setukpairinjR12 => 166.97/25.06 ( 166.97/25.06 setukpairinjR1 => 166.97/25.06 ( 166.97/25.06 upairequniteq => 166.97/25.06 ( 166.97/25.06 setukpairinjR2 => 166.97/25.06 ( 166.97/25.06 setukpairinjR => 166.97/25.06 ( 166.97/25.06 ksndsingleton => 166.97/25.06 ( 166.97/25.06 ksndpairEq => 166.97/25.06 ( 166.97/25.06 kpairsurjEq => 166.97/25.06 ( 166.97/25.06 cartprodsndin => 166.97/25.06 ( 166.97/25.06 cartprodpairmemEL => 166.97/25.06 ( 166.97/25.06 cartprodpairmemER => 166.97/25.06 ( 166.97/25.06 cartprodmempaircEq => 166.97/25.06 ( 166.97/25.06 cartprodfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodsndpairEq => 166.97/25.06 ( 166.97/25.06 cartprodpairsurjEq => 166.97/25.06 ( 166.97/25.06 dpsetconstrI => 166.97/25.06 ( 166.97/25.06 dpsetconstrSub => 166.97/25.06 ( 166.97/25.06 setOfPairsIsBReln => 166.97/25.06 ( 166.97/25.06 !!(( 166.97/25.06 ^[Y0 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y1 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y2 : ( 166.97/25.06 $i * $i) > $o]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y3 : $i]: 166.97/25.06 (in(Y3, 166.97/25.06 Y0) => 166.97/25.06 (!!(( 166.97/25.06 ^[Y4 : $i]: 166.97/25.06 (in(Y4, 166.97/25.06 Y1) => 166.97/25.06 (in( 166.97/25.06 kpair( 166.97/25.06 Y3, Y4), 166.97/25.06 dpsetconstr( 166.97/25.06 Y0, Y1, 166.97/25.06 Y2)) => 166.97/25.06 Y2(Y3, 166.97/25.06 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.06 inference('lazy_cnf_imply', [status(thm)], ['55'])). 166.97/25.06 tff('57', plain, 166.97/25.06 ~ (prop2setI => 166.97/25.06 (prop2set2propI => 166.97/25.06 (notdexE => 166.97/25.06 (notdallE => 166.97/25.06 (exuI1 => 166.97/25.06 (exuI3 => 166.97/25.06 (exuI2 => 166.97/25.06 (inCongP => 166.97/25.06 (in__Cong => 166.97/25.06 (exuE3u => 166.97/25.06 (exu__Cong => 166.97/25.06 (emptyset__Cong => 166.97/25.06 (setadjoin__Cong => 166.97/25.06 (powerset__Cong => 166.97/25.06 (setunion__Cong => 166.97/25.06 (omega__Cong => 166.97/25.06 (exuEu => 166.97/25.06 (descr__Cong => 166.97/25.06 (dsetconstr__Cong => 166.97/25.06 (subsetI1 => 166.97/25.06 (eqimpsubset2 => 166.97/25.06 (eqimpsubset1 => 166.97/25.06 (subsetI2 => 166.97/25.06 (emptysetsubset => 166.97/25.06 (subsetE => 166.97/25.06 (subsetE2 => 166.97/25.06 (notsubsetI => 166.97/25.06 (notequalI1 => 166.97/25.06 (notequalI2 => 166.97/25.06 (subsetRefl => 166.97/25.06 (subsetTrans => 166.97/25.06 (setadjoinSub => 166.97/25.06 (setadjoinSub2 => 166.97/25.06 (subset2powerset => 166.97/25.06 (setextsub => 166.97/25.06 (subsetemptysetimpeq => 166.97/25.06 (powersetI1 => 166.97/25.06 (powersetE1 => 166.97/25.06 (inPowerset => 166.97/25.06 (powersetsubset => 166.97/25.06 (sepInPowerset => 166.97/25.06 (sepSubset => 166.97/25.06 (binunionIL => 166.97/25.06 (upairset2IR => 166.97/25.06 (binunionIR => 166.97/25.06 (binunionEcases => 166.97/25.06 (binunionE => 166.97/25.06 (binunionLsub => 166.97/25.06 (binunionRsub => 166.97/25.06 (binintersectI => 166.97/25.06 (binintersectSubset5 => 166.97/25.06 (binintersectEL => 166.97/25.06 (binintersectLsub => 166.97/25.06 (binintersectSubset2 => 166.97/25.06 (binintersectSubset3 => 166.97/25.06 (binintersectER => 166.97/25.06 (disjointsetsI1 => 166.97/25.06 (binintersectRsub => 166.97/25.06 (binintersectSubset4 => 166.97/25.06 (binintersectSubset1 => 166.97/25.06 (bs114d => 166.97/25.06 (setminusI => 166.97/25.06 ( 166.97/25.06 setminusEL => 166.97/25.06 ( 166.97/25.06 setminusER => 166.97/25.06 ( 166.97/25.06 setminusSubset2 => 166.97/25.06 ( 166.97/25.06 setminusERneg => 166.97/25.06 ( 166.97/25.06 setminusELneg => 166.97/25.06 ( 166.97/25.06 setminusILneg => 166.97/25.06 ( 166.97/25.06 setminusIRneg => 166.97/25.06 ( 166.97/25.06 setminusLsub => 166.97/25.06 ( 166.97/25.06 setminusSubset1 => 166.97/25.06 ( 166.97/25.06 symdiffE => 166.97/25.06 ( 166.97/25.06 symdiffI1 => 166.97/25.06 ( 166.97/25.06 symdiffI2 => 166.97/25.06 ( 166.97/25.06 symdiffIneg1 => 166.97/25.06 ( 166.97/25.06 symdiffIneg2 => 166.97/25.06 ( 166.97/25.06 secondinupair => 166.97/25.06 ( 166.97/25.06 setukpairIL => 166.97/25.06 ( 166.97/25.06 setukpairIR => 166.97/25.06 ( 166.97/25.06 kpairiskpair => 166.97/25.06 ( 166.97/25.06 kpairp => 166.97/25.06 ( 166.97/25.06 singletonsubset => 166.97/25.06 ( 166.97/25.06 singletoninpowerset => 166.97/25.06 ( 166.97/25.06 singletoninpowunion => 166.97/25.06 ( 166.97/25.06 upairset2E => 166.97/25.06 ( 166.97/25.06 upairsubunion => 166.97/25.06 ( 166.97/25.06 upairinpowunion => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem1 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem2 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem3 => 166.97/25.06 ( 166.97/25.06 cartprodpairin => 166.97/25.06 ( 166.97/25.06 cartprodmempair1 => 166.97/25.06 ( 166.97/25.06 cartprodmempair => 166.97/25.06 ( 166.97/25.06 setunionE2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton1 => 166.97/25.06 ( 166.97/25.06 setunionsingleton2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton => 166.97/25.06 ( 166.97/25.06 singletonprop => 166.97/25.06 ( 166.97/25.06 ex1E1 => 166.97/25.06 ( 166.97/25.06 ex1I => 166.97/25.06 ( 166.97/25.06 ex1I2 => 166.97/25.06 ( 166.97/25.06 singletonsuniq => 166.97/25.06 ( 166.97/25.06 setukpairinjL1 => 166.97/25.06 ( 166.97/25.06 kfstsingleton => 166.97/25.06 ( 166.97/25.06 theprop => 166.97/25.06 ( 166.97/25.06 kfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodfstin => 166.97/25.06 ( 166.97/25.06 setukpairinjL2 => 166.97/25.06 ( 166.97/25.06 setukpairinjL => 166.97/25.06 ( 166.97/25.06 setukpairinjR11 => 166.97/25.06 ( 166.97/25.06 setukpairinjR12 => 166.97/25.06 ( 166.97/25.06 setukpairinjR1 => 166.97/25.06 ( 166.97/25.06 upairequniteq => 166.97/25.06 ( 166.97/25.06 setukpairinjR2 => 166.97/25.06 ( 166.97/25.06 setukpairinjR => 166.97/25.06 ( 166.97/25.06 ksndsingleton => 166.97/25.06 ( 166.97/25.06 ksndpairEq => 166.97/25.06 ( 166.97/25.06 kpairsurjEq => 166.97/25.06 ( 166.97/25.06 cartprodsndin => 166.97/25.06 ( 166.97/25.06 cartprodpairmemEL => 166.97/25.06 ( 166.97/25.06 cartprodpairmemER => 166.97/25.06 ( 166.97/25.06 cartprodmempaircEq => 166.97/25.06 ( 166.97/25.06 cartprodfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodsndpairEq => 166.97/25.06 ( 166.97/25.06 cartprodpairsurjEq => 166.97/25.06 ( 166.97/25.06 dpsetconstrI => 166.97/25.06 ( 166.97/25.06 dpsetconstrSub => 166.97/25.06 ( 166.97/25.06 setOfPairsIsBReln => 166.97/25.06 ( 166.97/25.06 !!(( 166.97/25.06 ^[Y0 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y1 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y2 : ( 166.97/25.06 $i * $i) > $o]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y3 : $i]: 166.97/25.06 (in(Y3, 166.97/25.06 Y0) => 166.97/25.06 (!!(( 166.97/25.06 ^[Y4 : $i]: 166.97/25.06 (in(Y4, 166.97/25.06 Y1) => 166.97/25.06 (in( 166.97/25.06 kpair( 166.97/25.06 Y3, Y4), 166.97/25.06 dpsetconstr( 166.97/25.06 Y0, Y1, 166.97/25.06 Y2)) => 166.97/25.06 Y2(Y3, 166.97/25.06 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.06 inference('lazy_cnf_imply', [status(thm)], ['56'])). 166.97/25.06 tff('58', plain, 166.97/25.06 ~ (prop2set2propI => 166.97/25.06 (notdexE => 166.97/25.06 (notdallE => 166.97/25.06 (exuI1 => 166.97/25.06 (exuI3 => 166.97/25.06 (exuI2 => 166.97/25.06 (inCongP => 166.97/25.06 (in__Cong => 166.97/25.06 (exuE3u => 166.97/25.06 (exu__Cong => 166.97/25.06 (emptyset__Cong => 166.97/25.06 (setadjoin__Cong => 166.97/25.06 (powerset__Cong => 166.97/25.06 (setunion__Cong => 166.97/25.06 (omega__Cong => 166.97/25.06 (exuEu => 166.97/25.06 (descr__Cong => 166.97/25.06 (dsetconstr__Cong => 166.97/25.06 (subsetI1 => 166.97/25.06 (eqimpsubset2 => 166.97/25.06 (eqimpsubset1 => 166.97/25.06 (subsetI2 => 166.97/25.06 (emptysetsubset => 166.97/25.06 (subsetE => 166.97/25.06 (subsetE2 => 166.97/25.06 (notsubsetI => 166.97/25.06 (notequalI1 => 166.97/25.06 (notequalI2 => 166.97/25.06 (subsetRefl => 166.97/25.06 (subsetTrans => 166.97/25.06 (setadjoinSub => 166.97/25.06 (setadjoinSub2 => 166.97/25.06 (subset2powerset => 166.97/25.06 (setextsub => 166.97/25.06 (subsetemptysetimpeq => 166.97/25.06 (powersetI1 => 166.97/25.06 (powersetE1 => 166.97/25.06 (inPowerset => 166.97/25.06 (powersetsubset => 166.97/25.06 (sepInPowerset => 166.97/25.06 (sepSubset => 166.97/25.06 (binunionIL => 166.97/25.06 (upairset2IR => 166.97/25.06 (binunionIR => 166.97/25.06 (binunionEcases => 166.97/25.06 (binunionE => 166.97/25.06 (binunionLsub => 166.97/25.06 (binunionRsub => 166.97/25.06 (binintersectI => 166.97/25.06 (binintersectSubset5 => 166.97/25.06 (binintersectEL => 166.97/25.06 (binintersectLsub => 166.97/25.06 (binintersectSubset2 => 166.97/25.06 (binintersectSubset3 => 166.97/25.06 (binintersectER => 166.97/25.06 (disjointsetsI1 => 166.97/25.06 (binintersectRsub => 166.97/25.06 (binintersectSubset4 => 166.97/25.06 (binintersectSubset1 => 166.97/25.06 (bs114d => 166.97/25.06 (setminusI => 166.97/25.06 (setminusEL => 166.97/25.06 ( 166.97/25.06 setminusER => 166.97/25.06 ( 166.97/25.06 setminusSubset2 => 166.97/25.06 ( 166.97/25.06 setminusERneg => 166.97/25.06 ( 166.97/25.06 setminusELneg => 166.97/25.06 ( 166.97/25.06 setminusILneg => 166.97/25.06 ( 166.97/25.06 setminusIRneg => 166.97/25.06 ( 166.97/25.06 setminusLsub => 166.97/25.06 ( 166.97/25.06 setminusSubset1 => 166.97/25.06 ( 166.97/25.06 symdiffE => 166.97/25.06 ( 166.97/25.06 symdiffI1 => 166.97/25.06 ( 166.97/25.06 symdiffI2 => 166.97/25.06 ( 166.97/25.06 symdiffIneg1 => 166.97/25.06 ( 166.97/25.06 symdiffIneg2 => 166.97/25.06 ( 166.97/25.06 secondinupair => 166.97/25.06 ( 166.97/25.06 setukpairIL => 166.97/25.06 ( 166.97/25.06 setukpairIR => 166.97/25.06 ( 166.97/25.06 kpairiskpair => 166.97/25.06 ( 166.97/25.06 kpairp => 166.97/25.06 ( 166.97/25.06 singletonsubset => 166.97/25.06 ( 166.97/25.06 singletoninpowerset => 166.97/25.06 ( 166.97/25.06 singletoninpowunion => 166.97/25.06 ( 166.97/25.06 upairset2E => 166.97/25.06 ( 166.97/25.06 upairsubunion => 166.97/25.06 ( 166.97/25.06 upairinpowunion => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem1 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem2 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem3 => 166.97/25.06 ( 166.97/25.06 cartprodpairin => 166.97/25.06 ( 166.97/25.06 cartprodmempair1 => 166.97/25.06 ( 166.97/25.06 cartprodmempair => 166.97/25.06 ( 166.97/25.06 setunionE2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton1 => 166.97/25.06 ( 166.97/25.06 setunionsingleton2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton => 166.97/25.06 ( 166.97/25.06 singletonprop => 166.97/25.06 ( 166.97/25.06 ex1E1 => 166.97/25.06 ( 166.97/25.06 ex1I => 166.97/25.06 ( 166.97/25.06 ex1I2 => 166.97/25.06 ( 166.97/25.06 singletonsuniq => 166.97/25.06 ( 166.97/25.06 setukpairinjL1 => 166.97/25.06 ( 166.97/25.06 kfstsingleton => 166.97/25.06 ( 166.97/25.06 theprop => 166.97/25.06 ( 166.97/25.06 kfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodfstin => 166.97/25.06 ( 166.97/25.06 setukpairinjL2 => 166.97/25.06 ( 166.97/25.06 setukpairinjL => 166.97/25.06 ( 166.97/25.06 setukpairinjR11 => 166.97/25.06 ( 166.97/25.06 setukpairinjR12 => 166.97/25.06 ( 166.97/25.06 setukpairinjR1 => 166.97/25.06 ( 166.97/25.06 upairequniteq => 166.97/25.06 ( 166.97/25.06 setukpairinjR2 => 166.97/25.06 ( 166.97/25.06 setukpairinjR => 166.97/25.06 ( 166.97/25.06 ksndsingleton => 166.97/25.06 ( 166.97/25.06 ksndpairEq => 166.97/25.06 ( 166.97/25.06 kpairsurjEq => 166.97/25.06 ( 166.97/25.06 cartprodsndin => 166.97/25.06 ( 166.97/25.06 cartprodpairmemEL => 166.97/25.06 ( 166.97/25.06 cartprodpairmemER => 166.97/25.06 ( 166.97/25.06 cartprodmempaircEq => 166.97/25.06 ( 166.97/25.06 cartprodfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodsndpairEq => 166.97/25.06 ( 166.97/25.06 cartprodpairsurjEq => 166.97/25.06 ( 166.97/25.06 dpsetconstrI => 166.97/25.06 ( 166.97/25.06 dpsetconstrSub => 166.97/25.06 ( 166.97/25.06 setOfPairsIsBReln => 166.97/25.06 ( 166.97/25.06 !!(( 166.97/25.06 ^[Y0 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y1 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y2 : ( 166.97/25.06 $i * $i) > $o]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y3 : $i]: 166.97/25.06 (in(Y3, 166.97/25.06 Y0) => 166.97/25.06 (!!(( 166.97/25.06 ^[Y4 : $i]: 166.97/25.06 (in(Y4, 166.97/25.06 Y1) => 166.97/25.06 (in( 166.97/25.06 kpair( 166.97/25.06 Y3, Y4), 166.97/25.06 dpsetconstr( 166.97/25.06 Y0, Y1, 166.97/25.06 Y2)) => 166.97/25.06 Y2(Y3, 166.97/25.06 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.06 inference('lazy_cnf_imply', [status(thm)], ['57'])). 166.97/25.06 tff('59', plain, 166.97/25.06 ~ (notdexE => 166.97/25.06 (notdallE => 166.97/25.06 (exuI1 => 166.97/25.06 (exuI3 => 166.97/25.06 (exuI2 => 166.97/25.06 (inCongP => 166.97/25.06 (in__Cong => 166.97/25.06 (exuE3u => 166.97/25.06 (exu__Cong => 166.97/25.06 (emptyset__Cong => 166.97/25.06 (setadjoin__Cong => 166.97/25.06 (powerset__Cong => 166.97/25.06 (setunion__Cong => 166.97/25.06 (omega__Cong => 166.97/25.06 (exuEu => 166.97/25.06 (descr__Cong => 166.97/25.06 (dsetconstr__Cong => 166.97/25.06 (subsetI1 => 166.97/25.06 (eqimpsubset2 => 166.97/25.06 (eqimpsubset1 => 166.97/25.06 (subsetI2 => 166.97/25.06 (emptysetsubset => 166.97/25.06 (subsetE => 166.97/25.06 (subsetE2 => 166.97/25.06 (notsubsetI => 166.97/25.06 (notequalI1 => 166.97/25.06 (notequalI2 => 166.97/25.06 (subsetRefl => 166.97/25.06 (subsetTrans => 166.97/25.06 (setadjoinSub => 166.97/25.06 (setadjoinSub2 => 166.97/25.06 (subset2powerset => 166.97/25.06 (setextsub => 166.97/25.06 (subsetemptysetimpeq => 166.97/25.06 (powersetI1 => 166.97/25.06 (powersetE1 => 166.97/25.06 (inPowerset => 166.97/25.06 (powersetsubset => 166.97/25.06 (sepInPowerset => 166.97/25.06 (sepSubset => 166.97/25.06 (binunionIL => 166.97/25.06 (upairset2IR => 166.97/25.06 (binunionIR => 166.97/25.06 (binunionEcases => 166.97/25.06 (binunionE => 166.97/25.06 (binunionLsub => 166.97/25.06 (binunionRsub => 166.97/25.06 (binintersectI => 166.97/25.06 (binintersectSubset5 => 166.97/25.06 (binintersectEL => 166.97/25.06 (binintersectLsub => 166.97/25.06 (binintersectSubset2 => 166.97/25.06 (binintersectSubset3 => 166.97/25.06 (binintersectER => 166.97/25.06 (disjointsetsI1 => 166.97/25.06 (binintersectRsub => 166.97/25.06 (binintersectSubset4 => 166.97/25.06 (binintersectSubset1 => 166.97/25.06 (bs114d => 166.97/25.06 (setminusI => 166.97/25.06 (setminusEL => 166.97/25.06 (setminusER => 166.97/25.06 ( 166.97/25.06 setminusSubset2 => 166.97/25.06 ( 166.97/25.06 setminusERneg => 166.97/25.06 ( 166.97/25.06 setminusELneg => 166.97/25.06 ( 166.97/25.06 setminusILneg => 166.97/25.06 ( 166.97/25.06 setminusIRneg => 166.97/25.06 ( 166.97/25.06 setminusLsub => 166.97/25.06 ( 166.97/25.06 setminusSubset1 => 166.97/25.06 ( 166.97/25.06 symdiffE => 166.97/25.06 ( 166.97/25.06 symdiffI1 => 166.97/25.06 ( 166.97/25.06 symdiffI2 => 166.97/25.06 ( 166.97/25.06 symdiffIneg1 => 166.97/25.06 ( 166.97/25.06 symdiffIneg2 => 166.97/25.06 ( 166.97/25.06 secondinupair => 166.97/25.06 ( 166.97/25.06 setukpairIL => 166.97/25.06 ( 166.97/25.06 setukpairIR => 166.97/25.06 ( 166.97/25.06 kpairiskpair => 166.97/25.06 ( 166.97/25.06 kpairp => 166.97/25.06 ( 166.97/25.06 singletonsubset => 166.97/25.06 ( 166.97/25.06 singletoninpowerset => 166.97/25.06 ( 166.97/25.06 singletoninpowunion => 166.97/25.06 ( 166.97/25.06 upairset2E => 166.97/25.06 ( 166.97/25.06 upairsubunion => 166.97/25.06 ( 166.97/25.06 upairinpowunion => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem1 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem2 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem3 => 166.97/25.06 ( 166.97/25.06 cartprodpairin => 166.97/25.06 ( 166.97/25.06 cartprodmempair1 => 166.97/25.06 ( 166.97/25.06 cartprodmempair => 166.97/25.06 ( 166.97/25.06 setunionE2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton1 => 166.97/25.06 ( 166.97/25.06 setunionsingleton2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton => 166.97/25.06 ( 166.97/25.06 singletonprop => 166.97/25.06 ( 166.97/25.06 ex1E1 => 166.97/25.06 ( 166.97/25.06 ex1I => 166.97/25.06 ( 166.97/25.06 ex1I2 => 166.97/25.06 ( 166.97/25.06 singletonsuniq => 166.97/25.06 ( 166.97/25.06 setukpairinjL1 => 166.97/25.06 ( 166.97/25.06 kfstsingleton => 166.97/25.06 ( 166.97/25.06 theprop => 166.97/25.06 ( 166.97/25.06 kfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodfstin => 166.97/25.06 ( 166.97/25.06 setukpairinjL2 => 166.97/25.06 ( 166.97/25.06 setukpairinjL => 166.97/25.06 ( 166.97/25.06 setukpairinjR11 => 166.97/25.06 ( 166.97/25.06 setukpairinjR12 => 166.97/25.06 ( 166.97/25.06 setukpairinjR1 => 166.97/25.06 ( 166.97/25.06 upairequniteq => 166.97/25.06 ( 166.97/25.06 setukpairinjR2 => 166.97/25.06 ( 166.97/25.06 setukpairinjR => 166.97/25.06 ( 166.97/25.06 ksndsingleton => 166.97/25.06 ( 166.97/25.06 ksndpairEq => 166.97/25.06 ( 166.97/25.06 kpairsurjEq => 166.97/25.06 ( 166.97/25.06 cartprodsndin => 166.97/25.06 ( 166.97/25.06 cartprodpairmemEL => 166.97/25.06 ( 166.97/25.06 cartprodpairmemER => 166.97/25.06 ( 166.97/25.06 cartprodmempaircEq => 166.97/25.06 ( 166.97/25.06 cartprodfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodsndpairEq => 166.97/25.06 ( 166.97/25.06 cartprodpairsurjEq => 166.97/25.06 ( 166.97/25.06 dpsetconstrI => 166.97/25.06 ( 166.97/25.06 dpsetconstrSub => 166.97/25.06 ( 166.97/25.06 setOfPairsIsBReln => 166.97/25.06 ( 166.97/25.06 !!(( 166.97/25.06 ^[Y0 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y1 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y2 : ( 166.97/25.06 $i * $i) > $o]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y3 : $i]: 166.97/25.06 (in(Y3, 166.97/25.06 Y0) => 166.97/25.06 (!!(( 166.97/25.06 ^[Y4 : $i]: 166.97/25.06 (in(Y4, 166.97/25.06 Y1) => 166.97/25.06 (in( 166.97/25.06 kpair( 166.97/25.06 Y3, Y4), 166.97/25.06 dpsetconstr( 166.97/25.06 Y0, Y1, 166.97/25.06 Y2)) => 166.97/25.06 Y2(Y3, 166.97/25.06 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.06 inference('lazy_cnf_imply', [status(thm)], ['58'])). 166.97/25.06 tff('60', plain, 166.97/25.06 ~ (notdallE => 166.97/25.06 (exuI1 => 166.97/25.06 (exuI3 => 166.97/25.06 (exuI2 => 166.97/25.06 (inCongP => 166.97/25.06 (in__Cong => 166.97/25.06 (exuE3u => 166.97/25.06 (exu__Cong => 166.97/25.06 (emptyset__Cong => 166.97/25.06 (setadjoin__Cong => 166.97/25.06 (powerset__Cong => 166.97/25.06 (setunion__Cong => 166.97/25.06 (omega__Cong => 166.97/25.06 (exuEu => 166.97/25.06 (descr__Cong => 166.97/25.06 (dsetconstr__Cong => 166.97/25.06 (subsetI1 => 166.97/25.06 (eqimpsubset2 => 166.97/25.06 (eqimpsubset1 => 166.97/25.06 (subsetI2 => 166.97/25.06 (emptysetsubset => 166.97/25.06 (subsetE => 166.97/25.06 (subsetE2 => 166.97/25.06 (notsubsetI => 166.97/25.06 (notequalI1 => 166.97/25.06 (notequalI2 => 166.97/25.06 (subsetRefl => 166.97/25.06 (subsetTrans => 166.97/25.06 (setadjoinSub => 166.97/25.06 (setadjoinSub2 => 166.97/25.06 (subset2powerset => 166.97/25.06 (setextsub => 166.97/25.06 (subsetemptysetimpeq => 166.97/25.06 (powersetI1 => 166.97/25.06 (powersetE1 => 166.97/25.06 (inPowerset => 166.97/25.06 (powersetsubset => 166.97/25.06 (sepInPowerset => 166.97/25.06 (sepSubset => 166.97/25.06 (binunionIL => 166.97/25.06 (upairset2IR => 166.97/25.06 (binunionIR => 166.97/25.06 (binunionEcases => 166.97/25.06 (binunionE => 166.97/25.06 (binunionLsub => 166.97/25.06 (binunionRsub => 166.97/25.06 (binintersectI => 166.97/25.06 (binintersectSubset5 => 166.97/25.06 (binintersectEL => 166.97/25.06 (binintersectLsub => 166.97/25.06 (binintersectSubset2 => 166.97/25.06 (binintersectSubset3 => 166.97/25.06 (binintersectER => 166.97/25.06 (disjointsetsI1 => 166.97/25.06 (binintersectRsub => 166.97/25.06 (binintersectSubset4 => 166.97/25.06 (binintersectSubset1 => 166.97/25.06 (bs114d => 166.97/25.06 (setminusI => 166.97/25.06 (setminusEL => 166.97/25.06 (setminusER => 166.97/25.06 (setminusSubset2 => 166.97/25.06 ( 166.97/25.06 setminusERneg => 166.97/25.06 ( 166.97/25.06 setminusELneg => 166.97/25.06 ( 166.97/25.06 setminusILneg => 166.97/25.06 ( 166.97/25.06 setminusIRneg => 166.97/25.06 ( 166.97/25.06 setminusLsub => 166.97/25.06 ( 166.97/25.06 setminusSubset1 => 166.97/25.06 ( 166.97/25.06 symdiffE => 166.97/25.06 ( 166.97/25.06 symdiffI1 => 166.97/25.06 ( 166.97/25.06 symdiffI2 => 166.97/25.06 ( 166.97/25.06 symdiffIneg1 => 166.97/25.06 ( 166.97/25.06 symdiffIneg2 => 166.97/25.06 ( 166.97/25.06 secondinupair => 166.97/25.06 ( 166.97/25.06 setukpairIL => 166.97/25.06 ( 166.97/25.06 setukpairIR => 166.97/25.06 ( 166.97/25.06 kpairiskpair => 166.97/25.06 ( 166.97/25.06 kpairp => 166.97/25.06 ( 166.97/25.06 singletonsubset => 166.97/25.06 ( 166.97/25.06 singletoninpowerset => 166.97/25.06 ( 166.97/25.06 singletoninpowunion => 166.97/25.06 ( 166.97/25.06 upairset2E => 166.97/25.06 ( 166.97/25.06 upairsubunion => 166.97/25.06 ( 166.97/25.06 upairinpowunion => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem1 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem2 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem3 => 166.97/25.06 ( 166.97/25.06 cartprodpairin => 166.97/25.06 ( 166.97/25.06 cartprodmempair1 => 166.97/25.06 ( 166.97/25.06 cartprodmempair => 166.97/25.06 ( 166.97/25.06 setunionE2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton1 => 166.97/25.06 ( 166.97/25.06 setunionsingleton2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton => 166.97/25.06 ( 166.97/25.06 singletonprop => 166.97/25.06 ( 166.97/25.06 ex1E1 => 166.97/25.06 ( 166.97/25.06 ex1I => 166.97/25.06 ( 166.97/25.06 ex1I2 => 166.97/25.06 ( 166.97/25.06 singletonsuniq => 166.97/25.06 ( 166.97/25.06 setukpairinjL1 => 166.97/25.06 ( 166.97/25.06 kfstsingleton => 166.97/25.06 ( 166.97/25.06 theprop => 166.97/25.06 ( 166.97/25.06 kfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodfstin => 166.97/25.06 ( 166.97/25.06 setukpairinjL2 => 166.97/25.06 ( 166.97/25.06 setukpairinjL => 166.97/25.06 ( 166.97/25.06 setukpairinjR11 => 166.97/25.06 ( 166.97/25.06 setukpairinjR12 => 166.97/25.06 ( 166.97/25.06 setukpairinjR1 => 166.97/25.06 ( 166.97/25.06 upairequniteq => 166.97/25.06 ( 166.97/25.06 setukpairinjR2 => 166.97/25.06 ( 166.97/25.06 setukpairinjR => 166.97/25.06 ( 166.97/25.06 ksndsingleton => 166.97/25.06 ( 166.97/25.06 ksndpairEq => 166.97/25.06 ( 166.97/25.06 kpairsurjEq => 166.97/25.06 ( 166.97/25.06 cartprodsndin => 166.97/25.06 ( 166.97/25.06 cartprodpairmemEL => 166.97/25.06 ( 166.97/25.06 cartprodpairmemER => 166.97/25.06 ( 166.97/25.06 cartprodmempaircEq => 166.97/25.06 ( 166.97/25.06 cartprodfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodsndpairEq => 166.97/25.06 ( 166.97/25.06 cartprodpairsurjEq => 166.97/25.06 ( 166.97/25.06 dpsetconstrI => 166.97/25.06 ( 166.97/25.06 dpsetconstrSub => 166.97/25.06 ( 166.97/25.06 setOfPairsIsBReln => 166.97/25.06 ( 166.97/25.06 !!(( 166.97/25.06 ^[Y0 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y1 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y2 : ( 166.97/25.06 $i * $i) > $o]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y3 : $i]: 166.97/25.06 (in(Y3, 166.97/25.06 Y0) => 166.97/25.06 (!!(( 166.97/25.06 ^[Y4 : $i]: 166.97/25.06 (in(Y4, 166.97/25.06 Y1) => 166.97/25.06 (in( 166.97/25.06 kpair( 166.97/25.06 Y3, Y4), 166.97/25.06 dpsetconstr( 166.97/25.06 Y0, Y1, 166.97/25.06 Y2)) => 166.97/25.06 Y2(Y3, 166.97/25.06 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.06 inference('lazy_cnf_imply', [status(thm)], ['59'])). 166.97/25.06 tff('61', plain, 166.97/25.06 ~ (exuI1 => 166.97/25.06 (exuI3 => 166.97/25.06 (exuI2 => 166.97/25.06 (inCongP => 166.97/25.06 (in__Cong => 166.97/25.06 (exuE3u => 166.97/25.06 (exu__Cong => 166.97/25.06 (emptyset__Cong => 166.97/25.06 (setadjoin__Cong => 166.97/25.06 (powerset__Cong => 166.97/25.06 (setunion__Cong => 166.97/25.06 (omega__Cong => 166.97/25.06 (exuEu => 166.97/25.06 (descr__Cong => 166.97/25.06 (dsetconstr__Cong => 166.97/25.06 (subsetI1 => 166.97/25.06 (eqimpsubset2 => 166.97/25.06 (eqimpsubset1 => 166.97/25.06 (subsetI2 => 166.97/25.06 (emptysetsubset => 166.97/25.06 (subsetE => 166.97/25.06 (subsetE2 => 166.97/25.06 (notsubsetI => 166.97/25.06 (notequalI1 => 166.97/25.06 (notequalI2 => 166.97/25.06 (subsetRefl => 166.97/25.06 (subsetTrans => 166.97/25.06 (setadjoinSub => 166.97/25.06 (setadjoinSub2 => 166.97/25.06 (subset2powerset => 166.97/25.06 (setextsub => 166.97/25.06 (subsetemptysetimpeq => 166.97/25.06 (powersetI1 => 166.97/25.06 (powersetE1 => 166.97/25.06 (inPowerset => 166.97/25.06 (powersetsubset => 166.97/25.06 (sepInPowerset => 166.97/25.06 (sepSubset => 166.97/25.06 (binunionIL => 166.97/25.06 (upairset2IR => 166.97/25.06 (binunionIR => 166.97/25.06 (binunionEcases => 166.97/25.06 (binunionE => 166.97/25.06 (binunionLsub => 166.97/25.06 (binunionRsub => 166.97/25.06 (binintersectI => 166.97/25.06 (binintersectSubset5 => 166.97/25.06 (binintersectEL => 166.97/25.06 (binintersectLsub => 166.97/25.06 (binintersectSubset2 => 166.97/25.06 (binintersectSubset3 => 166.97/25.06 (binintersectER => 166.97/25.06 (disjointsetsI1 => 166.97/25.06 (binintersectRsub => 166.97/25.06 (binintersectSubset4 => 166.97/25.06 (binintersectSubset1 => 166.97/25.06 (bs114d => 166.97/25.06 (setminusI => 166.97/25.06 (setminusEL => 166.97/25.06 (setminusER => 166.97/25.06 (setminusSubset2 => 166.97/25.06 (setminusERneg => 166.97/25.06 ( 166.97/25.06 setminusELneg => 166.97/25.06 ( 166.97/25.06 setminusILneg => 166.97/25.06 ( 166.97/25.06 setminusIRneg => 166.97/25.06 ( 166.97/25.06 setminusLsub => 166.97/25.06 ( 166.97/25.06 setminusSubset1 => 166.97/25.06 ( 166.97/25.06 symdiffE => 166.97/25.06 ( 166.97/25.06 symdiffI1 => 166.97/25.06 ( 166.97/25.06 symdiffI2 => 166.97/25.06 ( 166.97/25.06 symdiffIneg1 => 166.97/25.06 ( 166.97/25.06 symdiffIneg2 => 166.97/25.06 ( 166.97/25.06 secondinupair => 166.97/25.06 ( 166.97/25.06 setukpairIL => 166.97/25.06 ( 166.97/25.06 setukpairIR => 166.97/25.06 ( 166.97/25.06 kpairiskpair => 166.97/25.06 ( 166.97/25.06 kpairp => 166.97/25.06 ( 166.97/25.06 singletonsubset => 166.97/25.06 ( 166.97/25.06 singletoninpowerset => 166.97/25.06 ( 166.97/25.06 singletoninpowunion => 166.97/25.06 ( 166.97/25.06 upairset2E => 166.97/25.06 ( 166.97/25.06 upairsubunion => 166.97/25.06 ( 166.97/25.06 upairinpowunion => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem1 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem2 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem3 => 166.97/25.06 ( 166.97/25.06 cartprodpairin => 166.97/25.06 ( 166.97/25.06 cartprodmempair1 => 166.97/25.06 ( 166.97/25.06 cartprodmempair => 166.97/25.06 ( 166.97/25.06 setunionE2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton1 => 166.97/25.06 ( 166.97/25.06 setunionsingleton2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton => 166.97/25.06 ( 166.97/25.06 singletonprop => 166.97/25.06 ( 166.97/25.06 ex1E1 => 166.97/25.06 ( 166.97/25.06 ex1I => 166.97/25.06 ( 166.97/25.06 ex1I2 => 166.97/25.06 ( 166.97/25.06 singletonsuniq => 166.97/25.06 ( 166.97/25.06 setukpairinjL1 => 166.97/25.06 ( 166.97/25.06 kfstsingleton => 166.97/25.06 ( 166.97/25.06 theprop => 166.97/25.06 ( 166.97/25.06 kfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodfstin => 166.97/25.06 ( 166.97/25.06 setukpairinjL2 => 166.97/25.06 ( 166.97/25.06 setukpairinjL => 166.97/25.06 ( 166.97/25.06 setukpairinjR11 => 166.97/25.06 ( 166.97/25.06 setukpairinjR12 => 166.97/25.06 ( 166.97/25.06 setukpairinjR1 => 166.97/25.06 ( 166.97/25.06 upairequniteq => 166.97/25.06 ( 166.97/25.06 setukpairinjR2 => 166.97/25.06 ( 166.97/25.06 setukpairinjR => 166.97/25.06 ( 166.97/25.06 ksndsingleton => 166.97/25.06 ( 166.97/25.06 ksndpairEq => 166.97/25.06 ( 166.97/25.06 kpairsurjEq => 166.97/25.06 ( 166.97/25.06 cartprodsndin => 166.97/25.06 ( 166.97/25.06 cartprodpairmemEL => 166.97/25.06 ( 166.97/25.06 cartprodpairmemER => 166.97/25.06 ( 166.97/25.06 cartprodmempaircEq => 166.97/25.06 ( 166.97/25.06 cartprodfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodsndpairEq => 166.97/25.06 ( 166.97/25.06 cartprodpairsurjEq => 166.97/25.06 ( 166.97/25.06 dpsetconstrI => 166.97/25.06 ( 166.97/25.06 dpsetconstrSub => 166.97/25.06 ( 166.97/25.06 setOfPairsIsBReln => 166.97/25.06 ( 166.97/25.06 !!(( 166.97/25.06 ^[Y0 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y1 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y2 : ( 166.97/25.06 $i * $i) > $o]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y3 : $i]: 166.97/25.06 (in(Y3, 166.97/25.06 Y0) => 166.97/25.06 (!!(( 166.97/25.06 ^[Y4 : $i]: 166.97/25.06 (in(Y4, 166.97/25.06 Y1) => 166.97/25.06 (in( 166.97/25.06 kpair( 166.97/25.06 Y3, Y4), 166.97/25.06 dpsetconstr( 166.97/25.06 Y0, Y1, 166.97/25.06 Y2)) => 166.97/25.06 Y2(Y3, 166.97/25.06 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.06 inference('lazy_cnf_imply', [status(thm)], ['60'])). 166.97/25.06 tff('62', plain, 166.97/25.06 ~ (exuI3 => 166.97/25.06 (exuI2 => 166.97/25.06 (inCongP => 166.97/25.06 (in__Cong => 166.97/25.06 (exuE3u => 166.97/25.06 (exu__Cong => 166.97/25.06 (emptyset__Cong => 166.97/25.06 (setadjoin__Cong => 166.97/25.06 (powerset__Cong => 166.97/25.06 (setunion__Cong => 166.97/25.06 (omega__Cong => 166.97/25.06 (exuEu => 166.97/25.06 (descr__Cong => 166.97/25.06 (dsetconstr__Cong => 166.97/25.06 (subsetI1 => 166.97/25.06 (eqimpsubset2 => 166.97/25.06 (eqimpsubset1 => 166.97/25.06 (subsetI2 => 166.97/25.06 (emptysetsubset => 166.97/25.06 (subsetE => 166.97/25.06 (subsetE2 => 166.97/25.06 (notsubsetI => 166.97/25.06 (notequalI1 => 166.97/25.06 (notequalI2 => 166.97/25.06 (subsetRefl => 166.97/25.06 (subsetTrans => 166.97/25.06 (setadjoinSub => 166.97/25.06 (setadjoinSub2 => 166.97/25.06 (subset2powerset => 166.97/25.06 (setextsub => 166.97/25.06 (subsetemptysetimpeq => 166.97/25.06 (powersetI1 => 166.97/25.06 (powersetE1 => 166.97/25.06 (inPowerset => 166.97/25.06 (powersetsubset => 166.97/25.06 (sepInPowerset => 166.97/25.06 (sepSubset => 166.97/25.06 (binunionIL => 166.97/25.06 (upairset2IR => 166.97/25.06 (binunionIR => 166.97/25.06 (binunionEcases => 166.97/25.06 (binunionE => 166.97/25.06 (binunionLsub => 166.97/25.06 (binunionRsub => 166.97/25.06 (binintersectI => 166.97/25.06 (binintersectSubset5 => 166.97/25.06 (binintersectEL => 166.97/25.06 (binintersectLsub => 166.97/25.06 (binintersectSubset2 => 166.97/25.06 (binintersectSubset3 => 166.97/25.06 (binintersectER => 166.97/25.06 (disjointsetsI1 => 166.97/25.06 (binintersectRsub => 166.97/25.06 (binintersectSubset4 => 166.97/25.06 (binintersectSubset1 => 166.97/25.06 (bs114d => 166.97/25.06 (setminusI => 166.97/25.06 (setminusEL => 166.97/25.06 (setminusER => 166.97/25.06 (setminusSubset2 => 166.97/25.06 (setminusERneg => 166.97/25.06 (setminusELneg => 166.97/25.06 ( 166.97/25.06 setminusILneg => 166.97/25.06 ( 166.97/25.06 setminusIRneg => 166.97/25.06 ( 166.97/25.06 setminusLsub => 166.97/25.06 ( 166.97/25.06 setminusSubset1 => 166.97/25.06 ( 166.97/25.06 symdiffE => 166.97/25.06 ( 166.97/25.06 symdiffI1 => 166.97/25.06 ( 166.97/25.06 symdiffI2 => 166.97/25.06 ( 166.97/25.06 symdiffIneg1 => 166.97/25.06 ( 166.97/25.06 symdiffIneg2 => 166.97/25.06 ( 166.97/25.06 secondinupair => 166.97/25.06 ( 166.97/25.06 setukpairIL => 166.97/25.06 ( 166.97/25.06 setukpairIR => 166.97/25.06 ( 166.97/25.06 kpairiskpair => 166.97/25.06 ( 166.97/25.06 kpairp => 166.97/25.06 ( 166.97/25.06 singletonsubset => 166.97/25.06 ( 166.97/25.06 singletoninpowerset => 166.97/25.06 ( 166.97/25.06 singletoninpowunion => 166.97/25.06 ( 166.97/25.06 upairset2E => 166.97/25.06 ( 166.97/25.06 upairsubunion => 166.97/25.06 ( 166.97/25.06 upairinpowunion => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem1 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem2 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem3 => 166.97/25.06 ( 166.97/25.06 cartprodpairin => 166.97/25.06 ( 166.97/25.06 cartprodmempair1 => 166.97/25.06 ( 166.97/25.06 cartprodmempair => 166.97/25.06 ( 166.97/25.06 setunionE2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton1 => 166.97/25.06 ( 166.97/25.06 setunionsingleton2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton => 166.97/25.06 ( 166.97/25.06 singletonprop => 166.97/25.06 ( 166.97/25.06 ex1E1 => 166.97/25.06 ( 166.97/25.06 ex1I => 166.97/25.06 ( 166.97/25.06 ex1I2 => 166.97/25.06 ( 166.97/25.06 singletonsuniq => 166.97/25.06 ( 166.97/25.06 setukpairinjL1 => 166.97/25.06 ( 166.97/25.06 kfstsingleton => 166.97/25.06 ( 166.97/25.06 theprop => 166.97/25.06 ( 166.97/25.06 kfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodfstin => 166.97/25.06 ( 166.97/25.06 setukpairinjL2 => 166.97/25.06 ( 166.97/25.06 setukpairinjL => 166.97/25.06 ( 166.97/25.06 setukpairinjR11 => 166.97/25.06 ( 166.97/25.06 setukpairinjR12 => 166.97/25.06 ( 166.97/25.06 setukpairinjR1 => 166.97/25.06 ( 166.97/25.06 upairequniteq => 166.97/25.06 ( 166.97/25.06 setukpairinjR2 => 166.97/25.06 ( 166.97/25.06 setukpairinjR => 166.97/25.06 ( 166.97/25.06 ksndsingleton => 166.97/25.06 ( 166.97/25.06 ksndpairEq => 166.97/25.06 ( 166.97/25.06 kpairsurjEq => 166.97/25.06 ( 166.97/25.06 cartprodsndin => 166.97/25.06 ( 166.97/25.06 cartprodpairmemEL => 166.97/25.06 ( 166.97/25.06 cartprodpairmemER => 166.97/25.06 ( 166.97/25.06 cartprodmempaircEq => 166.97/25.06 ( 166.97/25.06 cartprodfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodsndpairEq => 166.97/25.06 ( 166.97/25.06 cartprodpairsurjEq => 166.97/25.06 ( 166.97/25.06 dpsetconstrI => 166.97/25.06 ( 166.97/25.06 dpsetconstrSub => 166.97/25.06 ( 166.97/25.06 setOfPairsIsBReln => 166.97/25.06 ( 166.97/25.06 !!(( 166.97/25.06 ^[Y0 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y1 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y2 : ( 166.97/25.06 $i * $i) > $o]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y3 : $i]: 166.97/25.06 (in(Y3, 166.97/25.06 Y0) => 166.97/25.06 (!!(( 166.97/25.06 ^[Y4 : $i]: 166.97/25.06 (in(Y4, 166.97/25.06 Y1) => 166.97/25.06 (in( 166.97/25.06 kpair( 166.97/25.06 Y3, Y4), 166.97/25.06 dpsetconstr( 166.97/25.06 Y0, Y1, 166.97/25.06 Y2)) => 166.97/25.06 Y2(Y3, 166.97/25.06 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.06 inference('lazy_cnf_imply', [status(thm)], ['61'])). 166.97/25.06 tff('63', plain, 166.97/25.06 ~ (exuI2 => 166.97/25.06 (inCongP => 166.97/25.06 (in__Cong => 166.97/25.06 (exuE3u => 166.97/25.06 (exu__Cong => 166.97/25.06 (emptyset__Cong => 166.97/25.06 (setadjoin__Cong => 166.97/25.06 (powerset__Cong => 166.97/25.06 (setunion__Cong => 166.97/25.06 (omega__Cong => 166.97/25.06 (exuEu => 166.97/25.06 (descr__Cong => 166.97/25.06 (dsetconstr__Cong => 166.97/25.06 (subsetI1 => 166.97/25.06 (eqimpsubset2 => 166.97/25.06 (eqimpsubset1 => 166.97/25.06 (subsetI2 => 166.97/25.06 (emptysetsubset => 166.97/25.06 (subsetE => 166.97/25.06 (subsetE2 => 166.97/25.06 (notsubsetI => 166.97/25.06 (notequalI1 => 166.97/25.06 (notequalI2 => 166.97/25.06 (subsetRefl => 166.97/25.06 (subsetTrans => 166.97/25.06 (setadjoinSub => 166.97/25.06 (setadjoinSub2 => 166.97/25.06 (subset2powerset => 166.97/25.06 (setextsub => 166.97/25.06 (subsetemptysetimpeq => 166.97/25.06 (powersetI1 => 166.97/25.06 (powersetE1 => 166.97/25.06 (inPowerset => 166.97/25.06 (powersetsubset => 166.97/25.06 (sepInPowerset => 166.97/25.06 (sepSubset => 166.97/25.06 (binunionIL => 166.97/25.06 (upairset2IR => 166.97/25.06 (binunionIR => 166.97/25.06 (binunionEcases => 166.97/25.06 (binunionE => 166.97/25.06 (binunionLsub => 166.97/25.06 (binunionRsub => 166.97/25.06 (binintersectI => 166.97/25.06 (binintersectSubset5 => 166.97/25.06 (binintersectEL => 166.97/25.06 (binintersectLsub => 166.97/25.06 (binintersectSubset2 => 166.97/25.06 (binintersectSubset3 => 166.97/25.06 (binintersectER => 166.97/25.06 (disjointsetsI1 => 166.97/25.06 (binintersectRsub => 166.97/25.06 (binintersectSubset4 => 166.97/25.06 (binintersectSubset1 => 166.97/25.06 (bs114d => 166.97/25.06 (setminusI => 166.97/25.06 (setminusEL => 166.97/25.06 (setminusER => 166.97/25.06 (setminusSubset2 => 166.97/25.06 (setminusERneg => 166.97/25.06 (setminusELneg => 166.97/25.06 (setminusILneg => 166.97/25.06 ( 166.97/25.06 setminusIRneg => 166.97/25.06 ( 166.97/25.06 setminusLsub => 166.97/25.06 ( 166.97/25.06 setminusSubset1 => 166.97/25.06 ( 166.97/25.06 symdiffE => 166.97/25.06 ( 166.97/25.06 symdiffI1 => 166.97/25.06 ( 166.97/25.06 symdiffI2 => 166.97/25.06 ( 166.97/25.06 symdiffIneg1 => 166.97/25.06 ( 166.97/25.06 symdiffIneg2 => 166.97/25.06 ( 166.97/25.06 secondinupair => 166.97/25.06 ( 166.97/25.06 setukpairIL => 166.97/25.06 ( 166.97/25.06 setukpairIR => 166.97/25.06 ( 166.97/25.06 kpairiskpair => 166.97/25.06 ( 166.97/25.06 kpairp => 166.97/25.06 ( 166.97/25.06 singletonsubset => 166.97/25.06 ( 166.97/25.06 singletoninpowerset => 166.97/25.06 ( 166.97/25.06 singletoninpowunion => 166.97/25.06 ( 166.97/25.06 upairset2E => 166.97/25.06 ( 166.97/25.06 upairsubunion => 166.97/25.06 ( 166.97/25.06 upairinpowunion => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem1 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem2 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem3 => 166.97/25.06 ( 166.97/25.06 cartprodpairin => 166.97/25.06 ( 166.97/25.06 cartprodmempair1 => 166.97/25.06 ( 166.97/25.06 cartprodmempair => 166.97/25.06 ( 166.97/25.06 setunionE2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton1 => 166.97/25.06 ( 166.97/25.06 setunionsingleton2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton => 166.97/25.06 ( 166.97/25.06 singletonprop => 166.97/25.06 ( 166.97/25.06 ex1E1 => 166.97/25.06 ( 166.97/25.06 ex1I => 166.97/25.06 ( 166.97/25.06 ex1I2 => 166.97/25.06 ( 166.97/25.06 singletonsuniq => 166.97/25.06 ( 166.97/25.06 setukpairinjL1 => 166.97/25.06 ( 166.97/25.06 kfstsingleton => 166.97/25.06 ( 166.97/25.06 theprop => 166.97/25.06 ( 166.97/25.06 kfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodfstin => 166.97/25.06 ( 166.97/25.06 setukpairinjL2 => 166.97/25.06 ( 166.97/25.06 setukpairinjL => 166.97/25.06 ( 166.97/25.06 setukpairinjR11 => 166.97/25.06 ( 166.97/25.06 setukpairinjR12 => 166.97/25.06 ( 166.97/25.06 setukpairinjR1 => 166.97/25.06 ( 166.97/25.06 upairequniteq => 166.97/25.06 ( 166.97/25.06 setukpairinjR2 => 166.97/25.06 ( 166.97/25.06 setukpairinjR => 166.97/25.06 ( 166.97/25.06 ksndsingleton => 166.97/25.06 ( 166.97/25.06 ksndpairEq => 166.97/25.06 ( 166.97/25.06 kpairsurjEq => 166.97/25.06 ( 166.97/25.06 cartprodsndin => 166.97/25.06 ( 166.97/25.06 cartprodpairmemEL => 166.97/25.06 ( 166.97/25.06 cartprodpairmemER => 166.97/25.06 ( 166.97/25.06 cartprodmempaircEq => 166.97/25.06 ( 166.97/25.06 cartprodfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodsndpairEq => 166.97/25.06 ( 166.97/25.06 cartprodpairsurjEq => 166.97/25.06 ( 166.97/25.06 dpsetconstrI => 166.97/25.06 ( 166.97/25.06 dpsetconstrSub => 166.97/25.06 ( 166.97/25.06 setOfPairsIsBReln => 166.97/25.06 ( 166.97/25.06 !!(( 166.97/25.06 ^[Y0 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y1 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y2 : ( 166.97/25.06 $i * $i) > $o]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y3 : $i]: 166.97/25.06 (in(Y3, 166.97/25.06 Y0) => 166.97/25.06 (!!(( 166.97/25.06 ^[Y4 : $i]: 166.97/25.06 (in(Y4, 166.97/25.06 Y1) => 166.97/25.06 (in( 166.97/25.06 kpair( 166.97/25.06 Y3, Y4), 166.97/25.06 dpsetconstr( 166.97/25.06 Y0, Y1, 166.97/25.06 Y2)) => 166.97/25.06 Y2(Y3, 166.97/25.06 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.06 inference('lazy_cnf_imply', [status(thm)], ['62'])). 166.97/25.06 tff('64', plain, 166.97/25.06 ~ (inCongP => 166.97/25.06 (in__Cong => 166.97/25.06 (exuE3u => 166.97/25.06 (exu__Cong => 166.97/25.06 (emptyset__Cong => 166.97/25.06 (setadjoin__Cong => 166.97/25.06 (powerset__Cong => 166.97/25.06 (setunion__Cong => 166.97/25.06 (omega__Cong => 166.97/25.06 (exuEu => 166.97/25.06 (descr__Cong => 166.97/25.06 (dsetconstr__Cong => 166.97/25.06 (subsetI1 => 166.97/25.06 (eqimpsubset2 => 166.97/25.06 (eqimpsubset1 => 166.97/25.06 (subsetI2 => 166.97/25.06 (emptysetsubset => 166.97/25.06 (subsetE => 166.97/25.06 (subsetE2 => 166.97/25.06 (notsubsetI => 166.97/25.06 (notequalI1 => 166.97/25.06 (notequalI2 => 166.97/25.06 (subsetRefl => 166.97/25.06 (subsetTrans => 166.97/25.06 (setadjoinSub => 166.97/25.06 (setadjoinSub2 => 166.97/25.06 (subset2powerset => 166.97/25.06 (setextsub => 166.97/25.06 (subsetemptysetimpeq => 166.97/25.06 (powersetI1 => 166.97/25.06 (powersetE1 => 166.97/25.06 (inPowerset => 166.97/25.06 (powersetsubset => 166.97/25.06 (sepInPowerset => 166.97/25.06 (sepSubset => 166.97/25.06 (binunionIL => 166.97/25.06 (upairset2IR => 166.97/25.06 (binunionIR => 166.97/25.06 (binunionEcases => 166.97/25.06 (binunionE => 166.97/25.06 (binunionLsub => 166.97/25.06 (binunionRsub => 166.97/25.06 (binintersectI => 166.97/25.06 (binintersectSubset5 => 166.97/25.06 (binintersectEL => 166.97/25.06 (binintersectLsub => 166.97/25.06 (binintersectSubset2 => 166.97/25.06 (binintersectSubset3 => 166.97/25.06 (binintersectER => 166.97/25.06 (disjointsetsI1 => 166.97/25.06 (binintersectRsub => 166.97/25.06 (binintersectSubset4 => 166.97/25.06 (binintersectSubset1 => 166.97/25.06 (bs114d => 166.97/25.06 (setminusI => 166.97/25.06 (setminusEL => 166.97/25.06 (setminusER => 166.97/25.06 (setminusSubset2 => 166.97/25.06 (setminusERneg => 166.97/25.06 (setminusELneg => 166.97/25.06 (setminusILneg => 166.97/25.06 (setminusIRneg => 166.97/25.06 ( 166.97/25.06 setminusLsub => 166.97/25.06 ( 166.97/25.06 setminusSubset1 => 166.97/25.06 ( 166.97/25.06 symdiffE => 166.97/25.06 ( 166.97/25.06 symdiffI1 => 166.97/25.06 ( 166.97/25.06 symdiffI2 => 166.97/25.06 ( 166.97/25.06 symdiffIneg1 => 166.97/25.06 ( 166.97/25.06 symdiffIneg2 => 166.97/25.06 ( 166.97/25.06 secondinupair => 166.97/25.06 ( 166.97/25.06 setukpairIL => 166.97/25.06 ( 166.97/25.06 setukpairIR => 166.97/25.06 ( 166.97/25.06 kpairiskpair => 166.97/25.06 ( 166.97/25.06 kpairp => 166.97/25.06 ( 166.97/25.06 singletonsubset => 166.97/25.06 ( 166.97/25.06 singletoninpowerset => 166.97/25.06 ( 166.97/25.06 singletoninpowunion => 166.97/25.06 ( 166.97/25.06 upairset2E => 166.97/25.06 ( 166.97/25.06 upairsubunion => 166.97/25.06 ( 166.97/25.06 upairinpowunion => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem1 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem2 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem3 => 166.97/25.06 ( 166.97/25.06 cartprodpairin => 166.97/25.06 ( 166.97/25.06 cartprodmempair1 => 166.97/25.06 ( 166.97/25.06 cartprodmempair => 166.97/25.06 ( 166.97/25.06 setunionE2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton1 => 166.97/25.06 ( 166.97/25.06 setunionsingleton2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton => 166.97/25.06 ( 166.97/25.06 singletonprop => 166.97/25.06 ( 166.97/25.06 ex1E1 => 166.97/25.06 ( 166.97/25.06 ex1I => 166.97/25.06 ( 166.97/25.06 ex1I2 => 166.97/25.06 ( 166.97/25.06 singletonsuniq => 166.97/25.06 ( 166.97/25.06 setukpairinjL1 => 166.97/25.06 ( 166.97/25.06 kfstsingleton => 166.97/25.06 ( 166.97/25.06 theprop => 166.97/25.06 ( 166.97/25.06 kfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodfstin => 166.97/25.06 ( 166.97/25.06 setukpairinjL2 => 166.97/25.06 ( 166.97/25.06 setukpairinjL => 166.97/25.06 ( 166.97/25.06 setukpairinjR11 => 166.97/25.06 ( 166.97/25.06 setukpairinjR12 => 166.97/25.06 ( 166.97/25.06 setukpairinjR1 => 166.97/25.06 ( 166.97/25.06 upairequniteq => 166.97/25.06 ( 166.97/25.06 setukpairinjR2 => 166.97/25.06 ( 166.97/25.06 setukpairinjR => 166.97/25.06 ( 166.97/25.06 ksndsingleton => 166.97/25.06 ( 166.97/25.06 ksndpairEq => 166.97/25.06 ( 166.97/25.06 kpairsurjEq => 166.97/25.06 ( 166.97/25.06 cartprodsndin => 166.97/25.06 ( 166.97/25.06 cartprodpairmemEL => 166.97/25.06 ( 166.97/25.06 cartprodpairmemER => 166.97/25.06 ( 166.97/25.06 cartprodmempaircEq => 166.97/25.06 ( 166.97/25.06 cartprodfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodsndpairEq => 166.97/25.06 ( 166.97/25.06 cartprodpairsurjEq => 166.97/25.06 ( 166.97/25.06 dpsetconstrI => 166.97/25.06 ( 166.97/25.06 dpsetconstrSub => 166.97/25.06 ( 166.97/25.06 setOfPairsIsBReln => 166.97/25.06 ( 166.97/25.06 !!(( 166.97/25.06 ^[Y0 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y1 : $i]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y2 : ( 166.97/25.06 $i * $i) > $o]: 166.97/25.06 (!!(( 166.97/25.06 ^[Y3 : $i]: 166.97/25.06 (in(Y3, 166.97/25.06 Y0) => 166.97/25.06 (!!(( 166.97/25.06 ^[Y4 : $i]: 166.97/25.06 (in(Y4, 166.97/25.06 Y1) => 166.97/25.06 (in( 166.97/25.06 kpair( 166.97/25.06 Y3, Y4), 166.97/25.06 dpsetconstr( 166.97/25.06 Y0, Y1, 166.97/25.06 Y2)) => 166.97/25.06 Y2(Y3, 166.97/25.06 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.06 inference('lazy_cnf_imply', [status(thm)], ['63'])). 166.97/25.06 tff('65', plain, 166.97/25.06 ~ (in__Cong => 166.97/25.06 (exuE3u => 166.97/25.06 (exu__Cong => 166.97/25.06 (emptyset__Cong => 166.97/25.06 (setadjoin__Cong => 166.97/25.06 (powerset__Cong => 166.97/25.06 (setunion__Cong => 166.97/25.06 (omega__Cong => 166.97/25.06 (exuEu => 166.97/25.06 (descr__Cong => 166.97/25.06 (dsetconstr__Cong => 166.97/25.06 (subsetI1 => 166.97/25.06 (eqimpsubset2 => 166.97/25.06 (eqimpsubset1 => 166.97/25.06 (subsetI2 => 166.97/25.06 (emptysetsubset => 166.97/25.06 (subsetE => 166.97/25.06 (subsetE2 => 166.97/25.06 (notsubsetI => 166.97/25.06 (notequalI1 => 166.97/25.06 (notequalI2 => 166.97/25.06 (subsetRefl => 166.97/25.06 (subsetTrans => 166.97/25.06 (setadjoinSub => 166.97/25.06 (setadjoinSub2 => 166.97/25.06 (subset2powerset => 166.97/25.06 (setextsub => 166.97/25.06 (subsetemptysetimpeq => 166.97/25.06 (powersetI1 => 166.97/25.06 (powersetE1 => 166.97/25.06 (inPowerset => 166.97/25.06 (powersetsubset => 166.97/25.06 (sepInPowerset => 166.97/25.06 (sepSubset => 166.97/25.06 (binunionIL => 166.97/25.06 (upairset2IR => 166.97/25.06 (binunionIR => 166.97/25.06 (binunionEcases => 166.97/25.06 (binunionE => 166.97/25.06 (binunionLsub => 166.97/25.06 (binunionRsub => 166.97/25.06 (binintersectI => 166.97/25.06 (binintersectSubset5 => 166.97/25.06 (binintersectEL => 166.97/25.06 (binintersectLsub => 166.97/25.06 (binintersectSubset2 => 166.97/25.06 (binintersectSubset3 => 166.97/25.06 (binintersectER => 166.97/25.06 (disjointsetsI1 => 166.97/25.06 (binintersectRsub => 166.97/25.06 (binintersectSubset4 => 166.97/25.06 (binintersectSubset1 => 166.97/25.06 (bs114d => 166.97/25.06 (setminusI => 166.97/25.06 (setminusEL => 166.97/25.06 (setminusER => 166.97/25.06 (setminusSubset2 => 166.97/25.06 (setminusERneg => 166.97/25.06 (setminusELneg => 166.97/25.06 (setminusILneg => 166.97/25.06 (setminusIRneg => 166.97/25.06 (setminusLsub => 166.97/25.06 ( 166.97/25.06 setminusSubset1 => 166.97/25.06 ( 166.97/25.06 symdiffE => 166.97/25.06 ( 166.97/25.06 symdiffI1 => 166.97/25.06 ( 166.97/25.06 symdiffI2 => 166.97/25.06 ( 166.97/25.06 symdiffIneg1 => 166.97/25.06 ( 166.97/25.06 symdiffIneg2 => 166.97/25.06 ( 166.97/25.06 secondinupair => 166.97/25.06 ( 166.97/25.06 setukpairIL => 166.97/25.06 ( 166.97/25.06 setukpairIR => 166.97/25.06 ( 166.97/25.06 kpairiskpair => 166.97/25.06 ( 166.97/25.06 kpairp => 166.97/25.06 ( 166.97/25.06 singletonsubset => 166.97/25.06 ( 166.97/25.06 singletoninpowerset => 166.97/25.06 ( 166.97/25.06 singletoninpowunion => 166.97/25.06 ( 166.97/25.06 upairset2E => 166.97/25.06 ( 166.97/25.06 upairsubunion => 166.97/25.06 ( 166.97/25.06 upairinpowunion => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem1 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem2 => 166.97/25.06 ( 166.97/25.06 ubforcartprodlem3 => 166.97/25.06 ( 166.97/25.06 cartprodpairin => 166.97/25.06 ( 166.97/25.06 cartprodmempair1 => 166.97/25.06 ( 166.97/25.06 cartprodmempair => 166.97/25.06 ( 166.97/25.06 setunionE2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton1 => 166.97/25.06 ( 166.97/25.06 setunionsingleton2 => 166.97/25.06 ( 166.97/25.06 setunionsingleton => 166.97/25.06 ( 166.97/25.06 singletonprop => 166.97/25.06 ( 166.97/25.06 ex1E1 => 166.97/25.06 ( 166.97/25.06 ex1I => 166.97/25.06 ( 166.97/25.06 ex1I2 => 166.97/25.06 ( 166.97/25.06 singletonsuniq => 166.97/25.06 ( 166.97/25.06 setukpairinjL1 => 166.97/25.06 ( 166.97/25.06 kfstsingleton => 166.97/25.06 ( 166.97/25.06 theprop => 166.97/25.06 ( 166.97/25.06 kfstpairEq => 166.97/25.06 ( 166.97/25.06 cartprodfstin => 166.97/25.06 ( 166.97/25.06 setukpairinjL2 => 166.97/25.06 ( 166.97/25.06 setukpairinjL => 166.97/25.07 ( 166.97/25.07 setukpairinjR11 => 166.97/25.07 ( 166.97/25.07 setukpairinjR12 => 166.97/25.07 ( 166.97/25.07 setukpairinjR1 => 166.97/25.07 ( 166.97/25.07 upairequniteq => 166.97/25.07 ( 166.97/25.07 setukpairinjR2 => 166.97/25.07 ( 166.97/25.07 setukpairinjR => 166.97/25.07 ( 166.97/25.07 ksndsingleton => 166.97/25.07 ( 166.97/25.07 ksndpairEq => 166.97/25.07 ( 166.97/25.07 kpairsurjEq => 166.97/25.07 ( 166.97/25.07 cartprodsndin => 166.97/25.07 ( 166.97/25.07 cartprodpairmemEL => 166.97/25.07 ( 166.97/25.07 cartprodpairmemER => 166.97/25.07 ( 166.97/25.07 cartprodmempaircEq => 166.97/25.07 ( 166.97/25.07 cartprodfstpairEq => 166.97/25.07 ( 166.97/25.07 cartprodsndpairEq => 166.97/25.07 ( 166.97/25.07 cartprodpairsurjEq => 166.97/25.07 ( 166.97/25.07 dpsetconstrI => 166.97/25.07 ( 166.97/25.07 dpsetconstrSub => 166.97/25.07 ( 166.97/25.07 setOfPairsIsBReln => 166.97/25.07 ( 166.97/25.07 !!(( 166.97/25.07 ^[Y0 : $i]: 166.97/25.07 (!!(( 166.97/25.07 ^[Y1 : $i]: 166.97/25.07 (!!(( 166.97/25.07 ^[Y2 : ( 166.97/25.07 $i * $i) > $o]: 166.97/25.07 (!!(( 166.97/25.07 ^[Y3 : $i]: 166.97/25.07 (in(Y3, 166.97/25.07 Y0) => 166.97/25.07 (!!(( 166.97/25.07 ^[Y4 : $i]: 166.97/25.07 (in(Y4, 166.97/25.07 Y1) => 166.97/25.07 (in( 166.97/25.07 kpair( 166.97/25.07 Y3, Y4), 166.97/25.07 dpsetconstr( 166.97/25.07 Y0, Y1, 166.97/25.07 Y2)) => 166.97/25.07 Y2(Y3, 166.97/25.07 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.07 inference('lazy_cnf_imply', [status(thm)], ['64'])). 166.97/25.07 tff('66', plain, 166.97/25.07 ~ (exuE3u => 166.97/25.07 (exu__Cong => 166.97/25.07 (emptyset__Cong => 166.97/25.07 (setadjoin__Cong => 166.97/25.07 (powerset__Cong => 166.97/25.07 (setunion__Cong => 166.97/25.07 (omega__Cong => 166.97/25.07 (exuEu => 166.97/25.07 (descr__Cong => 166.97/25.07 (dsetconstr__Cong => 166.97/25.07 (subsetI1 => 166.97/25.07 (eqimpsubset2 => 166.97/25.07 (eqimpsubset1 => 166.97/25.07 (subsetI2 => 166.97/25.07 (emptysetsubset => 166.97/25.07 (subsetE => 166.97/25.07 (subsetE2 => 166.97/25.07 (notsubsetI => 166.97/25.07 (notequalI1 => 166.97/25.07 (notequalI2 => 166.97/25.07 (subsetRefl => 166.97/25.07 (subsetTrans => 166.97/25.07 (setadjoinSub => 166.97/25.07 (setadjoinSub2 => 166.97/25.07 (subset2powerset => 166.97/25.07 (setextsub => 166.97/25.07 (subsetemptysetimpeq => 166.97/25.07 (powersetI1 => 166.97/25.07 (powersetE1 => 166.97/25.07 (inPowerset => 166.97/25.07 (powersetsubset => 166.97/25.07 (sepInPowerset => 166.97/25.07 (sepSubset => 166.97/25.07 (binunionIL => 166.97/25.07 (upairset2IR => 166.97/25.07 (binunionIR => 166.97/25.07 (binunionEcases => 166.97/25.07 (binunionE => 166.97/25.07 (binunionLsub => 166.97/25.07 (binunionRsub => 166.97/25.07 (binintersectI => 166.97/25.07 (binintersectSubset5 => 166.97/25.07 (binintersectEL => 166.97/25.07 (binintersectLsub => 166.97/25.07 (binintersectSubset2 => 166.97/25.07 (binintersectSubset3 => 166.97/25.07 (binintersectER => 166.97/25.07 (disjointsetsI1 => 166.97/25.07 (binintersectRsub => 166.97/25.07 (binintersectSubset4 => 166.97/25.07 (binintersectSubset1 => 166.97/25.07 (bs114d => 166.97/25.07 (setminusI => 166.97/25.07 (setminusEL => 166.97/25.07 (setminusER => 166.97/25.07 (setminusSubset2 => 166.97/25.07 (setminusERneg => 166.97/25.07 (setminusELneg => 166.97/25.07 (setminusILneg => 166.97/25.07 (setminusIRneg => 166.97/25.07 (setminusLsub => 166.97/25.07 (setminusSubset1 => 166.97/25.07 ( 166.97/25.07 symdiffE => 166.97/25.07 ( 166.97/25.07 symdiffI1 => 166.97/25.07 ( 166.97/25.07 symdiffI2 => 166.97/25.07 ( 166.97/25.07 symdiffIneg1 => 166.97/25.07 ( 166.97/25.07 symdiffIneg2 => 166.97/25.07 ( 166.97/25.07 secondinupair => 166.97/25.07 ( 166.97/25.07 setukpairIL => 166.97/25.07 ( 166.97/25.07 setukpairIR => 166.97/25.07 ( 166.97/25.07 kpairiskpair => 166.97/25.07 ( 166.97/25.07 kpairp => 166.97/25.07 ( 166.97/25.07 singletonsubset => 166.97/25.07 ( 166.97/25.07 singletoninpowerset => 166.97/25.07 ( 166.97/25.07 singletoninpowunion => 166.97/25.07 ( 166.97/25.07 upairset2E => 166.97/25.07 ( 166.97/25.07 upairsubunion => 166.97/25.07 ( 166.97/25.07 upairinpowunion => 166.97/25.07 ( 166.97/25.07 ubforcartprodlem1 => 166.97/25.07 ( 166.97/25.07 ubforcartprodlem2 => 166.97/25.07 ( 166.97/25.07 ubforcartprodlem3 => 166.97/25.07 ( 166.97/25.07 cartprodpairin => 166.97/25.07 ( 166.97/25.07 cartprodmempair1 => 166.97/25.07 ( 166.97/25.07 cartprodmempair => 166.97/25.07 ( 166.97/25.07 setunionE2 => 166.97/25.07 ( 166.97/25.07 setunionsingleton1 => 166.97/25.07 ( 166.97/25.07 setunionsingleton2 => 166.97/25.07 ( 166.97/25.07 setunionsingleton => 166.97/25.07 ( 166.97/25.07 singletonprop => 166.97/25.07 ( 166.97/25.07 ex1E1 => 166.97/25.07 ( 166.97/25.07 ex1I => 166.97/25.07 ( 166.97/25.07 ex1I2 => 166.97/25.07 ( 166.97/25.07 singletonsuniq => 166.97/25.07 ( 166.97/25.07 setukpairinjL1 => 166.97/25.07 ( 166.97/25.07 kfstsingleton => 166.97/25.07 ( 166.97/25.07 theprop => 166.97/25.07 ( 166.97/25.07 kfstpairEq => 166.97/25.07 ( 166.97/25.07 cartprodfstin => 166.97/25.07 ( 166.97/25.07 setukpairinjL2 => 166.97/25.07 ( 166.97/25.07 setukpairinjL => 166.97/25.07 ( 166.97/25.07 setukpairinjR11 => 166.97/25.07 ( 166.97/25.07 setukpairinjR12 => 166.97/25.07 ( 166.97/25.07 setukpairinjR1 => 166.97/25.07 ( 166.97/25.07 upairequniteq => 166.97/25.07 ( 166.97/25.07 setukpairinjR2 => 166.97/25.07 ( 166.97/25.07 setukpairinjR => 166.97/25.07 ( 166.97/25.07 ksndsingleton => 166.97/25.07 ( 166.97/25.07 ksndpairEq => 166.97/25.07 ( 166.97/25.07 kpairsurjEq => 166.97/25.07 ( 166.97/25.07 cartprodsndin => 166.97/25.07 ( 166.97/25.07 cartprodpairmemEL => 166.97/25.07 ( 166.97/25.07 cartprodpairmemER => 166.97/25.07 ( 166.97/25.07 cartprodmempaircEq => 166.97/25.07 ( 166.97/25.07 cartprodfstpairEq => 166.97/25.07 ( 166.97/25.07 cartprodsndpairEq => 166.97/25.07 ( 166.97/25.07 cartprodpairsurjEq => 166.97/25.07 ( 166.97/25.07 dpsetconstrI => 166.97/25.07 ( 166.97/25.07 dpsetconstrSub => 166.97/25.07 ( 166.97/25.07 setOfPairsIsBReln => 166.97/25.07 ( 166.97/25.07 !!(( 166.97/25.07 ^[Y0 : $i]: 166.97/25.07 (!!(( 166.97/25.07 ^[Y1 : $i]: 166.97/25.07 (!!(( 166.97/25.07 ^[Y2 : ( 166.97/25.07 $i * $i) > $o]: 166.97/25.07 (!!(( 166.97/25.07 ^[Y3 : $i]: 166.97/25.07 (in(Y3, 166.97/25.07 Y0) => 166.97/25.07 (!!(( 166.97/25.07 ^[Y4 : $i]: 166.97/25.07 (in(Y4, 166.97/25.07 Y1) => 166.97/25.07 (in( 166.97/25.07 kpair( 166.97/25.07 Y3, Y4), 166.97/25.07 dpsetconstr( 166.97/25.07 Y0, Y1, 166.97/25.07 Y2)) => 166.97/25.07 Y2(Y3, 166.97/25.07 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.07 inference('lazy_cnf_imply', [status(thm)], ['65'])). 166.97/25.07 tff('67', plain, 166.97/25.07 ~ (exu__Cong => 166.97/25.07 (emptyset__Cong => 166.97/25.07 (setadjoin__Cong => 166.97/25.07 (powerset__Cong => 166.97/25.07 (setunion__Cong => 166.97/25.07 (omega__Cong => 166.97/25.07 (exuEu => 166.97/25.07 (descr__Cong => 166.97/25.07 (dsetconstr__Cong => 166.97/25.07 (subsetI1 => 166.97/25.07 (eqimpsubset2 => 166.97/25.07 (eqimpsubset1 => 166.97/25.07 (subsetI2 => 166.97/25.07 (emptysetsubset => 166.97/25.07 (subsetE => 166.97/25.07 (subsetE2 => 166.97/25.07 (notsubsetI => 166.97/25.07 (notequalI1 => 166.97/25.07 (notequalI2 => 166.97/25.07 (subsetRefl => 166.97/25.07 (subsetTrans => 166.97/25.07 (setadjoinSub => 166.97/25.07 (setadjoinSub2 => 166.97/25.07 (subset2powerset => 166.97/25.07 (setextsub => 166.97/25.07 (subsetemptysetimpeq => 166.97/25.07 (powersetI1 => 166.97/25.07 (powersetE1 => 166.97/25.07 (inPowerset => 166.97/25.07 (powersetsubset => 166.97/25.07 (sepInPowerset => 166.97/25.07 (sepSubset => 166.97/25.07 (binunionIL => 166.97/25.07 (upairset2IR => 166.97/25.07 (binunionIR => 166.97/25.07 (binunionEcases => 166.97/25.07 (binunionE => 166.97/25.07 (binunionLsub => 166.97/25.07 (binunionRsub => 166.97/25.07 (binintersectI => 166.97/25.07 (binintersectSubset5 => 166.97/25.07 (binintersectEL => 166.97/25.07 (binintersectLsub => 166.97/25.07 (binintersectSubset2 => 166.97/25.07 (binintersectSubset3 => 166.97/25.07 (binintersectER => 166.97/25.07 (disjointsetsI1 => 166.97/25.07 (binintersectRsub => 166.97/25.07 (binintersectSubset4 => 166.97/25.07 (binintersectSubset1 => 166.97/25.07 (bs114d => 166.97/25.07 (setminusI => 166.97/25.07 (setminusEL => 166.97/25.07 (setminusER => 166.97/25.07 (setminusSubset2 => 166.97/25.07 (setminusERneg => 166.97/25.07 (setminusELneg => 166.97/25.07 (setminusILneg => 166.97/25.07 (setminusIRneg => 166.97/25.07 (setminusLsub => 166.97/25.07 (setminusSubset1 => 166.97/25.07 (symdiffE => 166.97/25.07 ( 166.97/25.07 symdiffI1 => 166.97/25.07 ( 166.97/25.07 symdiffI2 => 166.97/25.07 ( 166.97/25.07 symdiffIneg1 => 166.97/25.07 ( 166.97/25.07 symdiffIneg2 => 166.97/25.07 ( 166.97/25.07 secondinupair => 166.97/25.07 ( 166.97/25.07 setukpairIL => 166.97/25.07 ( 166.97/25.07 setukpairIR => 166.97/25.07 ( 166.97/25.07 kpairiskpair => 166.97/25.07 ( 166.97/25.07 kpairp => 166.97/25.07 ( 166.97/25.07 singletonsubset => 166.97/25.07 ( 166.97/25.07 singletoninpowerset => 166.97/25.07 ( 166.97/25.07 singletoninpowunion => 166.97/25.07 ( 166.97/25.07 upairset2E => 166.97/25.07 ( 166.97/25.07 upairsubunion => 166.97/25.07 ( 166.97/25.07 upairinpowunion => 166.97/25.07 ( 166.97/25.07 ubforcartprodlem1 => 166.97/25.07 ( 166.97/25.07 ubforcartprodlem2 => 166.97/25.07 ( 166.97/25.07 ubforcartprodlem3 => 166.97/25.07 ( 166.97/25.07 cartprodpairin => 166.97/25.07 ( 166.97/25.07 cartprodmempair1 => 166.97/25.07 ( 166.97/25.07 cartprodmempair => 166.97/25.07 ( 166.97/25.07 setunionE2 => 166.97/25.07 ( 166.97/25.07 setunionsingleton1 => 166.97/25.07 ( 166.97/25.07 setunionsingleton2 => 166.97/25.07 ( 166.97/25.07 setunionsingleton => 166.97/25.07 ( 166.97/25.07 singletonprop => 166.97/25.07 ( 166.97/25.07 ex1E1 => 166.97/25.07 ( 166.97/25.07 ex1I => 166.97/25.07 ( 166.97/25.07 ex1I2 => 166.97/25.07 ( 166.97/25.07 singletonsuniq => 166.97/25.07 ( 166.97/25.07 setukpairinjL1 => 166.97/25.07 ( 166.97/25.07 kfstsingleton => 166.97/25.07 ( 166.97/25.07 theprop => 166.97/25.07 ( 166.97/25.07 kfstpairEq => 166.97/25.07 ( 166.97/25.07 cartprodfstin => 166.97/25.07 ( 166.97/25.07 setukpairinjL2 => 166.97/25.07 ( 166.97/25.07 setukpairinjL => 166.97/25.07 ( 166.97/25.07 setukpairinjR11 => 166.97/25.07 ( 166.97/25.07 setukpairinjR12 => 166.97/25.07 ( 166.97/25.07 setukpairinjR1 => 166.97/25.07 ( 166.97/25.07 upairequniteq => 166.97/25.07 ( 166.97/25.07 setukpairinjR2 => 166.97/25.07 ( 166.97/25.07 setukpairinjR => 166.97/25.07 ( 166.97/25.07 ksndsingleton => 166.97/25.07 ( 166.97/25.07 ksndpairEq => 166.97/25.07 ( 166.97/25.07 kpairsurjEq => 166.97/25.07 ( 166.97/25.07 cartprodsndin => 166.97/25.07 ( 166.97/25.07 cartprodpairmemEL => 166.97/25.07 ( 166.97/25.07 cartprodpairmemER => 166.97/25.07 ( 166.97/25.07 cartprodmempaircEq => 166.97/25.07 ( 166.97/25.07 cartprodfstpairEq => 166.97/25.07 ( 166.97/25.07 cartprodsndpairEq => 166.97/25.07 ( 166.97/25.07 cartprodpairsurjEq => 166.97/25.07 ( 166.97/25.07 dpsetconstrI => 166.97/25.07 ( 166.97/25.07 dpsetconstrSub => 166.97/25.07 ( 166.97/25.07 setOfPairsIsBReln => 166.97/25.07 ( 166.97/25.07 !!(( 166.97/25.07 ^[Y0 : $i]: 166.97/25.07 (!!(( 166.97/25.07 ^[Y1 : $i]: 166.97/25.07 (!!(( 166.97/25.07 ^[Y2 : ( 166.97/25.07 $i * $i) > $o]: 166.97/25.07 (!!(( 166.97/25.07 ^[Y3 : $i]: 166.97/25.07 (in(Y3, 166.97/25.07 Y0) => 166.97/25.07 (!!(( 166.97/25.07 ^[Y4 : $i]: 166.97/25.07 (in(Y4, 166.97/25.07 Y1) => 166.97/25.07 (in( 166.97/25.07 kpair( 166.97/25.07 Y3, Y4), 166.97/25.07 dpsetconstr( 166.97/25.07 Y0, Y1, 166.97/25.07 Y2)) => 166.97/25.07 Y2(Y3, 166.97/25.07 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.07 inference('lazy_cnf_imply', [status(thm)], ['66'])). 166.97/25.07 tff('68', plain, 166.97/25.07 ~ (emptyset__Cong => 166.97/25.07 (setadjoin__Cong => 166.97/25.07 (powerset__Cong => 166.97/25.07 (setunion__Cong => 166.97/25.07 (omega__Cong => 166.97/25.07 (exuEu => 166.97/25.07 (descr__Cong => 166.97/25.07 (dsetconstr__Cong => 166.97/25.07 (subsetI1 => 166.97/25.07 (eqimpsubset2 => 166.97/25.07 (eqimpsubset1 => 166.97/25.07 (subsetI2 => 166.97/25.07 (emptysetsubset => 166.97/25.07 (subsetE => 166.97/25.07 (subsetE2 => 166.97/25.07 (notsubsetI => 166.97/25.07 (notequalI1 => 166.97/25.07 (notequalI2 => 166.97/25.07 (subsetRefl => 166.97/25.07 (subsetTrans => 166.97/25.07 (setadjoinSub => 166.97/25.07 (setadjoinSub2 => 166.97/25.07 (subset2powerset => 166.97/25.07 (setextsub => 166.97/25.07 (subsetemptysetimpeq => 166.97/25.07 (powersetI1 => 166.97/25.07 (powersetE1 => 166.97/25.07 (inPowerset => 166.97/25.07 (powersetsubset => 166.97/25.07 (sepInPowerset => 166.97/25.07 (sepSubset => 166.97/25.07 (binunionIL => 166.97/25.07 (upairset2IR => 166.97/25.07 (binunionIR => 166.97/25.07 (binunionEcases => 166.97/25.07 (binunionE => 166.97/25.07 (binunionLsub => 166.97/25.07 (binunionRsub => 166.97/25.07 (binintersectI => 166.97/25.07 (binintersectSubset5 => 166.97/25.07 (binintersectEL => 166.97/25.07 (binintersectLsub => 166.97/25.07 (binintersectSubset2 => 166.97/25.07 (binintersectSubset3 => 166.97/25.07 (binintersectER => 166.97/25.07 (disjointsetsI1 => 166.97/25.07 (binintersectRsub => 166.97/25.07 (binintersectSubset4 => 166.97/25.07 (binintersectSubset1 => 166.97/25.07 (bs114d => 166.97/25.07 (setminusI => 166.97/25.07 (setminusEL => 166.97/25.07 (setminusER => 166.97/25.07 (setminusSubset2 => 166.97/25.07 (setminusERneg => 166.97/25.07 (setminusELneg => 166.97/25.07 (setminusILneg => 166.97/25.07 (setminusIRneg => 166.97/25.07 (setminusLsub => 166.97/25.07 (setminusSubset1 => 166.97/25.07 (symdiffE => 166.97/25.07 (symdiffI1 => 166.97/25.07 ( 166.97/25.07 symdiffI2 => 166.97/25.07 ( 166.97/25.07 symdiffIneg1 => 166.97/25.07 ( 166.97/25.07 symdiffIneg2 => 166.97/25.07 ( 166.97/25.07 secondinupair => 166.97/25.07 ( 166.97/25.07 setukpairIL => 166.97/25.07 ( 166.97/25.07 setukpairIR => 166.97/25.07 ( 166.97/25.07 kpairiskpair => 166.97/25.07 ( 166.97/25.07 kpairp => 166.97/25.07 ( 166.97/25.07 singletonsubset => 166.97/25.07 ( 166.97/25.07 singletoninpowerset => 166.97/25.07 ( 166.97/25.07 singletoninpowunion => 166.97/25.07 ( 166.97/25.07 upairset2E => 166.97/25.07 ( 166.97/25.07 upairsubunion => 166.97/25.07 ( 166.97/25.07 upairinpowunion => 166.97/25.07 ( 166.97/25.07 ubforcartprodlem1 => 166.97/25.07 ( 166.97/25.07 ubforcartprodlem2 => 166.97/25.07 ( 166.97/25.07 ubforcartprodlem3 => 166.97/25.07 ( 166.97/25.07 cartprodpairin => 166.97/25.07 ( 166.97/25.07 cartprodmempair1 => 166.97/25.07 ( 166.97/25.07 cartprodmempair => 166.97/25.07 ( 166.97/25.07 setunionE2 => 166.97/25.07 ( 166.97/25.07 setunionsingleton1 => 166.97/25.07 ( 166.97/25.07 setunionsingleton2 => 166.97/25.07 ( 166.97/25.07 setunionsingleton => 166.97/25.07 ( 166.97/25.07 singletonprop => 166.97/25.07 ( 166.97/25.07 ex1E1 => 166.97/25.07 ( 166.97/25.07 ex1I => 166.97/25.07 ( 166.97/25.07 ex1I2 => 166.97/25.07 ( 166.97/25.07 singletonsuniq => 166.97/25.07 ( 166.97/25.07 setukpairinjL1 => 166.97/25.07 ( 166.97/25.07 kfstsingleton => 166.97/25.07 ( 166.97/25.07 theprop => 166.97/25.07 ( 166.97/25.07 kfstpairEq => 166.97/25.07 ( 166.97/25.07 cartprodfstin => 166.97/25.07 ( 166.97/25.07 setukpairinjL2 => 166.97/25.07 ( 166.97/25.07 setukpairinjL => 166.97/25.07 ( 166.97/25.07 setukpairinjR11 => 166.97/25.07 ( 166.97/25.07 setukpairinjR12 => 166.97/25.07 ( 166.97/25.07 setukpairinjR1 => 166.97/25.07 ( 166.97/25.07 upairequniteq => 166.97/25.07 ( 166.97/25.07 setukpairinjR2 => 166.97/25.07 ( 166.97/25.07 setukpairinjR => 166.97/25.07 ( 166.97/25.07 ksndsingleton => 166.97/25.07 ( 166.97/25.07 ksndpairEq => 166.97/25.07 ( 166.97/25.07 kpairsurjEq => 166.97/25.07 ( 166.97/25.07 cartprodsndin => 166.97/25.07 ( 166.97/25.07 cartprodpairmemEL => 166.97/25.07 ( 166.97/25.07 cartprodpairmemER => 166.97/25.07 ( 166.97/25.07 cartprodmempaircEq => 166.97/25.07 ( 166.97/25.07 cartprodfstpairEq => 166.97/25.07 ( 166.97/25.07 cartprodsndpairEq => 166.97/25.07 ( 166.97/25.07 cartprodpairsurjEq => 166.97/25.07 ( 166.97/25.07 dpsetconstrI => 166.97/25.07 ( 166.97/25.07 dpsetconstrSub => 166.97/25.07 ( 166.97/25.07 setOfPairsIsBReln => 166.97/25.07 ( 166.97/25.07 !!(( 166.97/25.07 ^[Y0 : $i]: 166.97/25.07 (!!(( 166.97/25.07 ^[Y1 : $i]: 166.97/25.07 (!!(( 166.97/25.07 ^[Y2 : ( 166.97/25.07 $i * $i) > $o]: 166.97/25.07 (!!(( 166.97/25.07 ^[Y3 : $i]: 166.97/25.07 (in(Y3, 166.97/25.07 Y0) => 166.97/25.07 (!!(( 166.97/25.07 ^[Y4 : $i]: 166.97/25.07 (in(Y4, 166.97/25.07 Y1) => 166.97/25.07 (in( 166.97/25.07 kpair( 166.97/25.07 Y3, Y4), 166.97/25.07 dpsetconstr( 166.97/25.07 Y0, Y1, 166.97/25.07 Y2)) => 166.97/25.07 Y2(Y3, 166.97/25.07 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.07 inference('lazy_cnf_imply', [status(thm)], ['67'])). 166.97/25.07 tff('69', plain, 166.97/25.07 ~ (setadjoin__Cong => 166.97/25.07 (powerset__Cong => 166.97/25.07 (setunion__Cong => 166.97/25.07 (omega__Cong => 166.97/25.07 (exuEu => 166.97/25.07 (descr__Cong => 166.97/25.07 (dsetconstr__Cong => 166.97/25.07 (subsetI1 => 166.97/25.07 (eqimpsubset2 => 166.97/25.07 (eqimpsubset1 => 166.97/25.07 (subsetI2 => 166.97/25.07 (emptysetsubset => 166.97/25.07 (subsetE => 166.97/25.07 (subsetE2 => 166.97/25.07 (notsubsetI => 166.97/25.07 (notequalI1 => 166.97/25.07 (notequalI2 => 166.97/25.07 (subsetRefl => 166.97/25.07 (subsetTrans => 166.97/25.07 (setadjoinSub => 166.97/25.07 (setadjoinSub2 => 166.97/25.07 (subset2powerset => 166.97/25.07 (setextsub => 166.97/25.07 (subsetemptysetimpeq => 166.97/25.07 (powersetI1 => 166.97/25.07 (powersetE1 => 166.97/25.07 (inPowerset => 166.97/25.07 (powersetsubset => 166.97/25.07 (sepInPowerset => 166.97/25.07 (sepSubset => 166.97/25.07 (binunionIL => 166.97/25.07 (upairset2IR => 166.97/25.07 (binunionIR => 166.97/25.07 (binunionEcases => 166.97/25.07 (binunionE => 166.97/25.07 (binunionLsub => 166.97/25.07 (binunionRsub => 166.97/25.07 (binintersectI => 166.97/25.07 (binintersectSubset5 => 166.97/25.07 (binintersectEL => 166.97/25.07 (binintersectLsub => 166.97/25.07 (binintersectSubset2 => 166.97/25.07 (binintersectSubset3 => 166.97/25.07 (binintersectER => 166.97/25.07 (disjointsetsI1 => 166.97/25.07 (binintersectRsub => 166.97/25.07 (binintersectSubset4 => 166.97/25.07 (binintersectSubset1 => 166.97/25.07 (bs114d => 166.97/25.07 (setminusI => 166.97/25.07 (setminusEL => 166.97/25.07 (setminusER => 166.97/25.07 (setminusSubset2 => 166.97/25.07 (setminusERneg => 166.97/25.07 (setminusELneg => 166.97/25.07 (setminusILneg => 166.97/25.07 (setminusIRneg => 166.97/25.07 (setminusLsub => 166.97/25.07 (setminusSubset1 => 166.97/25.07 (symdiffE => 166.97/25.07 (symdiffI1 => 166.97/25.07 (symdiffI2 => 166.97/25.07 ( 166.97/25.07 symdiffIneg1 => 166.97/25.07 ( 166.97/25.07 symdiffIneg2 => 166.97/25.07 ( 166.97/25.07 secondinupair => 166.97/25.07 ( 166.97/25.07 setukpairIL => 166.97/25.07 ( 166.97/25.07 setukpairIR => 166.97/25.07 ( 166.97/25.07 kpairiskpair => 166.97/25.07 ( 166.97/25.07 kpairp => 166.97/25.07 ( 166.97/25.07 singletonsubset => 166.97/25.07 ( 166.97/25.07 singletoninpowerset => 166.97/25.07 ( 166.97/25.07 singletoninpowunion => 166.97/25.07 ( 166.97/25.07 upairset2E => 166.97/25.07 ( 166.97/25.07 upairsubunion => 166.97/25.07 ( 166.97/25.07 upairinpowunion => 166.97/25.07 ( 166.97/25.07 ubforcartprodlem1 => 166.97/25.07 ( 166.97/25.07 ubforcartprodlem2 => 166.97/25.07 ( 166.97/25.07 ubforcartprodlem3 => 166.97/25.07 ( 166.97/25.07 cartprodpairin => 166.97/25.07 ( 166.97/25.07 cartprodmempair1 => 166.97/25.07 ( 166.97/25.07 cartprodmempair => 166.97/25.07 ( 166.97/25.07 setunionE2 => 166.97/25.07 ( 166.97/25.07 setunionsingleton1 => 166.97/25.07 ( 166.97/25.07 setunionsingleton2 => 166.97/25.07 ( 166.97/25.07 setunionsingleton => 166.97/25.07 ( 166.97/25.07 singletonprop => 166.97/25.07 ( 166.97/25.07 ex1E1 => 166.97/25.07 ( 166.97/25.07 ex1I => 166.97/25.07 ( 166.97/25.07 ex1I2 => 166.97/25.07 ( 166.97/25.07 singletonsuniq => 166.97/25.07 ( 166.97/25.07 setukpairinjL1 => 166.97/25.07 ( 166.97/25.07 kfstsingleton => 166.97/25.07 ( 166.97/25.07 theprop => 166.97/25.07 ( 166.97/25.07 kfstpairEq => 166.97/25.07 ( 166.97/25.07 cartprodfstin => 166.97/25.07 ( 166.97/25.07 setukpairinjL2 => 166.97/25.07 ( 166.97/25.07 setukpairinjL => 166.97/25.07 ( 166.97/25.07 setukpairinjR11 => 166.97/25.07 ( 166.97/25.07 setukpairinjR12 => 166.97/25.07 ( 166.97/25.07 setukpairinjR1 => 166.97/25.07 ( 166.97/25.07 upairequniteq => 166.97/25.07 ( 166.97/25.07 setukpairinjR2 => 166.97/25.07 ( 166.97/25.07 setukpairinjR => 166.97/25.07 ( 166.97/25.07 ksndsingleton => 166.97/25.07 ( 166.97/25.07 ksndpairEq => 166.97/25.07 ( 166.97/25.07 kpairsurjEq => 166.97/25.07 ( 166.97/25.07 cartprodsndin => 166.97/25.07 ( 166.97/25.07 cartprodpairmemEL => 166.97/25.07 ( 166.97/25.07 cartprodpairmemER => 166.97/25.07 ( 166.97/25.07 cartprodmempaircEq => 166.97/25.07 ( 166.97/25.07 cartprodfstpairEq => 166.97/25.07 ( 166.97/25.07 cartprodsndpairEq => 166.97/25.07 ( 166.97/25.07 cartprodpairsurjEq => 166.97/25.07 ( 166.97/25.07 dpsetconstrI => 166.97/25.07 ( 166.97/25.07 dpsetconstrSub => 166.97/25.07 ( 166.97/25.07 setOfPairsIsBReln => 166.97/25.07 ( 166.97/25.07 !!(( 166.97/25.07 ^[Y0 : $i]: 166.97/25.07 (!!(( 166.97/25.07 ^[Y1 : $i]: 166.97/25.07 (!!(( 166.97/25.07 ^[Y2 : ( 166.97/25.07 $i * $i) > $o]: 166.97/25.07 (!!(( 166.97/25.07 ^[Y3 : $i]: 166.97/25.07 (in(Y3, 166.97/25.07 Y0) => 166.97/25.07 (!!(( 166.97/25.07 ^[Y4 : $i]: 166.97/25.07 (in(Y4, 166.97/25.07 Y1) => 166.97/25.07 (in( 166.97/25.07 kpair( 166.97/25.07 Y3, Y4), 166.97/25.07 dpsetconstr( 166.97/25.07 Y0, Y1, 166.97/25.07 Y2)) => 166.97/25.07 Y2(Y3, 166.97/25.07 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.07 inference('lazy_cnf_imply', [status(thm)], ['68'])). 166.97/25.07 tff('70', plain, 166.97/25.07 ~ (powerset__Cong => 166.97/25.07 (setunion__Cong => 166.97/25.07 (omega__Cong => 166.97/25.07 (exuEu => 166.97/25.07 (descr__Cong => 166.97/25.07 (dsetconstr__Cong => 166.97/25.07 (subsetI1 => 166.97/25.07 (eqimpsubset2 => 166.97/25.07 (eqimpsubset1 => 166.97/25.07 (subsetI2 => 166.97/25.07 (emptysetsubset => 166.97/25.07 (subsetE => 166.97/25.07 (subsetE2 => 166.97/25.07 (notsubsetI => 166.97/25.07 (notequalI1 => 166.97/25.07 (notequalI2 => 166.97/25.07 (subsetRefl => 166.97/25.07 (subsetTrans => 166.97/25.07 (setadjoinSub => 166.97/25.07 (setadjoinSub2 => 166.97/25.07 (subset2powerset => 166.97/25.07 (setextsub => 166.97/25.07 (subsetemptysetimpeq => 166.97/25.07 (powersetI1 => 166.97/25.07 (powersetE1 => 166.97/25.07 (inPowerset => 166.97/25.07 (powersetsubset => 166.97/25.07 (sepInPowerset => 166.97/25.07 (sepSubset => 166.97/25.07 (binunionIL => 166.97/25.07 (upairset2IR => 166.97/25.07 (binunionIR => 166.97/25.07 (binunionEcases => 166.97/25.07 (binunionE => 166.97/25.07 (binunionLsub => 166.97/25.07 (binunionRsub => 166.97/25.07 (binintersectI => 166.97/25.07 (binintersectSubset5 => 166.97/25.07 (binintersectEL => 166.97/25.07 (binintersectLsub => 166.97/25.07 (binintersectSubset2 => 166.97/25.07 (binintersectSubset3 => 166.97/25.07 (binintersectER => 166.97/25.07 (disjointsetsI1 => 166.97/25.07 (binintersectRsub => 166.97/25.07 (binintersectSubset4 => 166.97/25.07 (binintersectSubset1 => 166.97/25.07 (bs114d => 166.97/25.07 (setminusI => 166.97/25.07 (setminusEL => 166.97/25.07 (setminusER => 166.97/25.07 (setminusSubset2 => 166.97/25.07 (setminusERneg => 166.97/25.07 (setminusELneg => 166.97/25.07 (setminusILneg => 166.97/25.07 (setminusIRneg => 166.97/25.07 (setminusLsub => 166.97/25.07 (setminusSubset1 => 166.97/25.07 (symdiffE => 166.97/25.07 (symdiffI1 => 166.97/25.07 (symdiffI2 => 166.97/25.07 (symdiffIneg1 => 166.97/25.07 ( 166.97/25.07 symdiffIneg2 => 166.97/25.07 ( 166.97/25.07 secondinupair => 166.97/25.07 ( 166.97/25.07 setukpairIL => 166.97/25.07 ( 166.97/25.07 setukpairIR => 166.97/25.07 ( 166.97/25.07 kpairiskpair => 166.97/25.07 ( 166.97/25.07 kpairp => 166.97/25.07 ( 166.97/25.07 singletonsubset => 166.97/25.07 ( 166.97/25.07 singletoninpowerset => 166.97/25.07 ( 166.97/25.07 singletoninpowunion => 166.97/25.07 ( 166.97/25.07 upairset2E => 166.97/25.07 ( 166.97/25.07 upairsubunion => 166.97/25.07 ( 166.97/25.07 upairinpowunion => 166.97/25.07 ( 166.97/25.07 ubforcartprodlem1 => 166.97/25.07 ( 166.97/25.07 ubforcartprodlem2 => 166.97/25.07 ( 166.97/25.07 ubforcartprodlem3 => 166.97/25.07 ( 166.97/25.07 cartprodpairin => 166.97/25.07 ( 166.97/25.07 cartprodmempair1 => 166.97/25.07 ( 166.97/25.07 cartprodmempair => 166.97/25.07 ( 166.97/25.07 setunionE2 => 166.97/25.07 ( 166.97/25.07 setunionsingleton1 => 166.97/25.07 ( 166.97/25.07 setunionsingleton2 => 166.97/25.07 ( 166.97/25.07 setunionsingleton => 166.97/25.07 ( 166.97/25.07 singletonprop => 166.97/25.07 ( 166.97/25.07 ex1E1 => 166.97/25.07 ( 166.97/25.07 ex1I => 166.97/25.07 ( 166.97/25.07 ex1I2 => 166.97/25.07 ( 166.97/25.07 singletonsuniq => 166.97/25.07 ( 166.97/25.07 setukpairinjL1 => 166.97/25.07 ( 166.97/25.07 kfstsingleton => 166.97/25.07 ( 166.97/25.07 theprop => 166.97/25.07 ( 166.97/25.07 kfstpairEq => 166.97/25.07 ( 166.97/25.07 cartprodfstin => 166.97/25.07 ( 166.97/25.07 setukpairinjL2 => 166.97/25.07 ( 166.97/25.07 setukpairinjL => 166.97/25.07 ( 166.97/25.07 setukpairinjR11 => 166.97/25.07 ( 166.97/25.07 setukpairinjR12 => 166.97/25.07 ( 166.97/25.07 setukpairinjR1 => 166.97/25.07 ( 166.97/25.07 upairequniteq => 166.97/25.07 ( 166.97/25.07 setukpairinjR2 => 166.97/25.07 ( 166.97/25.07 setukpairinjR => 166.97/25.07 ( 166.97/25.07 ksndsingleton => 166.97/25.07 ( 166.97/25.07 ksndpairEq => 166.97/25.07 ( 166.97/25.07 kpairsurjEq => 166.97/25.07 ( 166.97/25.07 cartprodsndin => 166.97/25.07 ( 166.97/25.07 cartprodpairmemEL => 166.97/25.07 ( 166.97/25.07 cartprodpairmemER => 166.97/25.07 ( 166.97/25.07 cartprodmempaircEq => 166.97/25.07 ( 166.97/25.07 cartprodfstpairEq => 166.97/25.07 ( 166.97/25.07 cartprodsndpairEq => 166.97/25.07 ( 166.97/25.07 cartprodpairsurjEq => 166.97/25.07 ( 166.97/25.07 dpsetconstrI => 166.97/25.07 ( 166.97/25.07 dpsetconstrSub => 166.97/25.07 ( 166.97/25.07 setOfPairsIsBReln => 166.97/25.07 ( 166.97/25.07 !!(( 166.97/25.07 ^[Y0 : $i]: 166.97/25.07 (!!(( 166.97/25.07 ^[Y1 : $i]: 166.97/25.07 (!!(( 166.97/25.07 ^[Y2 : ( 166.97/25.07 $i * $i) > $o]: 166.97/25.07 (!!(( 166.97/25.07 ^[Y3 : $i]: 166.97/25.07 (in(Y3, 166.97/25.07 Y0) => 166.97/25.07 (!!(( 166.97/25.07 ^[Y4 : $i]: 166.97/25.07 (in(Y4, 166.97/25.07 Y1) => 166.97/25.07 (in( 166.97/25.07 kpair( 166.97/25.07 Y3, Y4), 166.97/25.07 dpsetconstr( 166.97/25.07 Y0, Y1, 166.97/25.07 Y2)) => 166.97/25.07 Y2(Y3, 166.97/25.07 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.07 inference('lazy_cnf_imply', [status(thm)], ['69'])). 166.97/25.07 tff('71', plain, 166.97/25.07 ~ (setunion__Cong => 166.97/25.07 (omega__Cong => 166.97/25.07 (exuEu => 166.97/25.07 (descr__Cong => 166.97/25.07 (dsetconstr__Cong => 166.97/25.07 (subsetI1 => 166.97/25.07 (eqimpsubset2 => 166.97/25.07 (eqimpsubset1 => 166.97/25.07 (subsetI2 => 166.97/25.07 (emptysetsubset => 166.97/25.07 (subsetE => 166.97/25.07 (subsetE2 => 166.97/25.07 (notsubsetI => 166.97/25.07 (notequalI1 => 166.97/25.07 (notequalI2 => 166.97/25.07 (subsetRefl => 166.97/25.07 (subsetTrans => 166.97/25.07 (setadjoinSub => 166.97/25.07 (setadjoinSub2 => 166.97/25.07 (subset2powerset => 166.97/25.07 (setextsub => 166.97/25.07 (subsetemptysetimpeq => 166.97/25.07 (powersetI1 => 166.97/25.07 (powersetE1 => 166.97/25.07 (inPowerset => 166.97/25.07 (powersetsubset => 166.97/25.07 (sepInPowerset => 166.97/25.07 (sepSubset => 166.97/25.07 (binunionIL => 166.97/25.07 (upairset2IR => 166.97/25.07 (binunionIR => 166.97/25.07 (binunionEcases => 166.97/25.07 (binunionE => 166.97/25.07 (binunionLsub => 166.97/25.07 (binunionRsub => 166.97/25.07 (binintersectI => 166.97/25.07 (binintersectSubset5 => 166.97/25.07 (binintersectEL => 166.97/25.07 (binintersectLsub => 166.97/25.07 (binintersectSubset2 => 166.97/25.07 (binintersectSubset3 => 166.97/25.07 (binintersectER => 166.97/25.07 (disjointsetsI1 => 166.97/25.07 (binintersectRsub => 166.97/25.07 (binintersectSubset4 => 166.97/25.07 (binintersectSubset1 => 166.97/25.07 (bs114d => 166.97/25.07 (setminusI => 166.97/25.07 (setminusEL => 166.97/25.07 (setminusER => 166.97/25.07 (setminusSubset2 => 166.97/25.07 (setminusERneg => 166.97/25.07 (setminusELneg => 166.97/25.07 (setminusILneg => 166.97/25.07 (setminusIRneg => 166.97/25.07 (setminusLsub => 166.97/25.07 (setminusSubset1 => 166.97/25.07 (symdiffE => 166.97/25.07 (symdiffI1 => 166.97/25.07 (symdiffI2 => 166.97/25.07 (symdiffIneg1 => 166.97/25.07 (symdiffIneg2 => 166.97/25.07 ( 166.97/25.07 secondinupair => 166.97/25.07 ( 166.97/25.07 setukpairIL => 166.97/25.07 ( 166.97/25.07 setukpairIR => 166.97/25.07 ( 166.97/25.07 kpairiskpair => 166.97/25.07 ( 166.97/25.07 kpairp => 166.97/25.07 ( 166.97/25.07 singletonsubset => 166.97/25.07 ( 166.97/25.07 singletoninpowerset => 166.97/25.07 ( 166.97/25.07 singletoninpowunion => 166.97/25.07 ( 166.97/25.07 upairset2E => 166.97/25.07 ( 166.97/25.07 upairsubunion => 166.97/25.07 ( 166.97/25.07 upairinpowunion => 166.97/25.07 ( 166.97/25.07 ubforcartprodlem1 => 166.97/25.07 ( 166.97/25.07 ubforcartprodlem2 => 166.97/25.07 ( 166.97/25.07 ubforcartprodlem3 => 166.97/25.07 ( 166.97/25.07 cartprodpairin => 166.97/25.07 ( 166.97/25.07 cartprodmempair1 => 166.97/25.07 ( 166.97/25.07 cartprodmempair => 166.97/25.07 ( 166.97/25.07 setunionE2 => 166.97/25.07 ( 166.97/25.07 setunionsingleton1 => 166.97/25.07 ( 166.97/25.07 setunionsingleton2 => 166.97/25.07 ( 166.97/25.07 setunionsingleton => 166.97/25.07 ( 166.97/25.07 singletonprop => 166.97/25.07 ( 166.97/25.07 ex1E1 => 166.97/25.07 ( 166.97/25.07 ex1I => 166.97/25.07 ( 166.97/25.07 ex1I2 => 166.97/25.07 ( 166.97/25.07 singletonsuniq => 166.97/25.07 ( 166.97/25.07 setukpairinjL1 => 166.97/25.07 ( 166.97/25.07 kfstsingleton => 166.97/25.07 ( 166.97/25.07 theprop => 166.97/25.07 ( 166.97/25.07 kfstpairEq => 166.97/25.07 ( 166.97/25.07 cartprodfstin => 166.97/25.07 ( 166.97/25.07 setukpairinjL2 => 166.97/25.07 ( 166.97/25.07 setukpairinjL => 166.97/25.07 ( 166.97/25.07 setukpairinjR11 => 166.97/25.07 ( 166.97/25.07 setukpairinjR12 => 166.97/25.07 ( 166.97/25.07 setukpairinjR1 => 166.97/25.07 ( 166.97/25.07 upairequniteq => 166.97/25.07 ( 166.97/25.07 setukpairinjR2 => 166.97/25.07 ( 166.97/25.07 setukpairinjR => 166.97/25.07 ( 166.97/25.07 ksndsingleton => 166.97/25.07 ( 166.97/25.07 ksndpairEq => 166.97/25.07 ( 166.97/25.07 kpairsurjEq => 166.97/25.07 ( 166.97/25.07 cartprodsndin => 166.97/25.07 ( 166.97/25.07 cartprodpairmemEL => 166.97/25.07 ( 166.97/25.07 cartprodpairmemER => 166.97/25.07 ( 166.97/25.07 cartprodmempaircEq => 166.97/25.07 ( 166.97/25.07 cartprodfstpairEq => 166.97/25.07 ( 166.97/25.07 cartprodsndpairEq => 166.97/25.07 ( 166.97/25.07 cartprodpairsurjEq => 166.97/25.07 ( 166.97/25.07 dpsetconstrI => 166.97/25.07 ( 166.97/25.07 dpsetconstrSub => 166.97/25.07 ( 166.97/25.07 setOfPairsIsBReln => 166.97/25.07 ( 166.97/25.07 !!(( 166.97/25.07 ^[Y0 : $i]: 166.97/25.07 (!!(( 166.97/25.07 ^[Y1 : $i]: 166.97/25.07 (!!(( 166.97/25.07 ^[Y2 : ( 166.97/25.07 $i * $i) > $o]: 166.97/25.07 (!!(( 166.97/25.07 ^[Y3 : $i]: 166.97/25.07 (in(Y3, 166.97/25.07 Y0) => 166.97/25.07 (!!(( 166.97/25.07 ^[Y4 : $i]: 166.97/25.07 (in(Y4, 166.97/25.07 Y1) => 166.97/25.07 (in( 166.97/25.07 kpair( 166.97/25.07 Y3, Y4), 166.97/25.07 dpsetconstr( 166.97/25.07 Y0, Y1, 166.97/25.07 Y2)) => 166.97/25.07 Y2(Y3, 166.97/25.07 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 166.97/25.07 inference('lazy_cnf_imply', [status(thm)], ['70'])). 166.97/25.07 tff('72', plain, 166.97/25.07 ~ (omega__Cong => 166.97/25.07 (exuEu => 166.97/25.07 (descr__Cong => 166.97/25.07 (dsetconstr__Cong => 166.97/25.07 (subsetI1 => 166.97/25.07 (eqimpsubset2 => 166.97/25.07 (eqimpsubset1 => 166.97/25.07 (subsetI2 => 166.97/25.07 (emptysetsubset => 166.97/25.07 (subsetE => 166.97/25.07 (subsetE2 => 166.97/25.07 (notsubsetI => 166.97/25.07 (notequalI1 => 166.97/25.07 (notequalI2 => 166.97/25.07 (subsetRefl => 166.97/25.07 (subsetTrans => 166.97/25.07 (setadjoinSub => 166.97/25.07 (setadjoinSub2 => 166.97/25.07 (subset2powerset => 166.97/25.07 (setextsub => 166.97/25.07 (subsetemptysetimpeq => 166.97/25.07 (powersetI1 => 166.97/25.07 (powersetE1 => 166.97/25.07 (inPowerset => 166.97/25.07 (powersetsubset => 166.97/25.07 (sepInPowerset => 166.97/25.07 (sepSubset => 166.97/25.07 (binunionIL => 166.97/25.07 (upairset2IR => 166.97/25.07 (binunionIR => 166.97/25.07 (binunionEcases => 166.97/25.07 (binunionE => 166.97/25.07 (binunionLsub => 166.97/25.07 (binunionRsub => 166.97/25.07 (binintersectI => 166.97/25.07 (binintersectSubset5 => 166.97/25.07 (binintersectEL => 166.97/25.07 (binintersectLsub => 166.97/25.07 (binintersectSubset2 => 166.97/25.07 (binintersectSubset3 => 166.97/25.07 (binintersectER => 166.97/25.07 (disjointsetsI1 => 166.97/25.07 (binintersectRsub => 166.97/25.07 (binintersectSubset4 => 166.97/25.07 (binintersectSubset1 => 166.97/25.07 (bs114d => 166.97/25.07 (setminusI => 166.97/25.07 (setminusEL => 166.97/25.07 (setminusER => 166.97/25.07 (setminusSubset2 => 166.97/25.07 (setminusERneg => 166.97/25.07 (setminusELneg => 166.97/25.07 (setminusILneg => 166.97/25.07 (setminusIRneg => 166.97/25.07 (setminusLsub => 166.97/25.07 (setminusSubset1 => 166.97/25.07 (symdiffE => 166.97/25.07 (symdiffI1 => 166.97/25.07 (symdiffI2 => 166.97/25.07 (symdiffIneg1 => 166.97/25.07 (symdiffIneg2 => 166.97/25.07 (secondinupair => 166.97/25.07 ( 166.97/25.07 setukpairIL => 166.97/25.07 ( 166.97/25.07 setukpairIR => 166.97/25.07 ( 166.97/25.07 kpairiskpair => 166.97/25.07 ( 166.97/25.07 kpairp => 166.97/25.07 ( 166.97/25.07 singletonsubset => 166.97/25.07 ( 166.97/25.07 singletoninpowerset => 166.97/25.07 ( 166.97/25.07 singletoninpowunion => 166.97/25.07 ( 166.97/25.07 upairset2E => 166.97/25.07 ( 166.97/25.07 upairsubunion => 166.97/25.07 ( 166.97/25.07 upairinpowunion => 166.97/25.07 ( 166.97/25.07 ubforcartprodlem1 => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem2 => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem3 => 167.19/25.07 ( 167.19/25.07 cartprodpairin => 167.19/25.07 ( 167.19/25.07 cartprodmempair1 => 167.19/25.07 ( 167.19/25.07 cartprodmempair => 167.19/25.07 ( 167.19/25.07 setunionE2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton1 => 167.19/25.07 ( 167.19/25.07 setunionsingleton2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton => 167.19/25.07 ( 167.19/25.07 singletonprop => 167.19/25.07 ( 167.19/25.07 ex1E1 => 167.19/25.07 ( 167.19/25.07 ex1I => 167.19/25.07 ( 167.19/25.07 ex1I2 => 167.19/25.07 ( 167.19/25.07 singletonsuniq => 167.19/25.07 ( 167.19/25.07 setukpairinjL1 => 167.19/25.07 ( 167.19/25.07 kfstsingleton => 167.19/25.07 ( 167.19/25.07 theprop => 167.19/25.07 ( 167.19/25.07 kfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodfstin => 167.19/25.07 ( 167.19/25.07 setukpairinjL2 => 167.19/25.07 ( 167.19/25.07 setukpairinjL => 167.19/25.07 ( 167.19/25.07 setukpairinjR11 => 167.19/25.07 ( 167.19/25.07 setukpairinjR12 => 167.19/25.07 ( 167.19/25.07 setukpairinjR1 => 167.19/25.07 ( 167.19/25.07 upairequniteq => 167.19/25.07 ( 167.19/25.07 setukpairinjR2 => 167.19/25.07 ( 167.19/25.07 setukpairinjR => 167.19/25.07 ( 167.19/25.07 ksndsingleton => 167.19/25.07 ( 167.19/25.07 ksndpairEq => 167.19/25.07 ( 167.19/25.07 kpairsurjEq => 167.19/25.07 ( 167.19/25.07 cartprodsndin => 167.19/25.07 ( 167.19/25.07 cartprodpairmemEL => 167.19/25.07 ( 167.19/25.07 cartprodpairmemER => 167.19/25.07 ( 167.19/25.07 cartprodmempaircEq => 167.19/25.07 ( 167.19/25.07 cartprodfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodsndpairEq => 167.19/25.07 ( 167.19/25.07 cartprodpairsurjEq => 167.19/25.07 ( 167.19/25.07 dpsetconstrI => 167.19/25.07 ( 167.19/25.07 dpsetconstrSub => 167.19/25.07 ( 167.19/25.07 setOfPairsIsBReln => 167.19/25.07 ( 167.19/25.07 !!(( 167.19/25.07 ^[Y0 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y1 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y2 : ( 167.19/25.07 $i * $i) > $o]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y3 : $i]: 167.19/25.07 (in(Y3, 167.19/25.07 Y0) => 167.19/25.07 (!!(( 167.19/25.07 ^[Y4 : $i]: 167.19/25.07 (in(Y4, 167.19/25.07 Y1) => 167.19/25.07 (in( 167.19/25.07 kpair( 167.19/25.07 Y3, Y4), 167.19/25.07 dpsetconstr( 167.19/25.07 Y0, Y1, 167.19/25.07 Y2)) => 167.19/25.07 Y2(Y3, 167.19/25.07 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.07 inference('lazy_cnf_imply', [status(thm)], ['71'])). 167.19/25.07 tff('73', plain, 167.19/25.07 ~ (exuEu => 167.19/25.07 (descr__Cong => 167.19/25.07 (dsetconstr__Cong => 167.19/25.07 (subsetI1 => 167.19/25.07 (eqimpsubset2 => 167.19/25.07 (eqimpsubset1 => 167.19/25.07 (subsetI2 => 167.19/25.07 (emptysetsubset => 167.19/25.07 (subsetE => 167.19/25.07 (subsetE2 => 167.19/25.07 (notsubsetI => 167.19/25.07 (notequalI1 => 167.19/25.07 (notequalI2 => 167.19/25.07 (subsetRefl => 167.19/25.07 (subsetTrans => 167.19/25.07 (setadjoinSub => 167.19/25.07 (setadjoinSub2 => 167.19/25.07 (subset2powerset => 167.19/25.07 (setextsub => 167.19/25.07 (subsetemptysetimpeq => 167.19/25.07 (powersetI1 => 167.19/25.07 (powersetE1 => 167.19/25.07 (inPowerset => 167.19/25.07 (powersetsubset => 167.19/25.07 (sepInPowerset => 167.19/25.07 (sepSubset => 167.19/25.07 (binunionIL => 167.19/25.07 (upairset2IR => 167.19/25.07 (binunionIR => 167.19/25.07 (binunionEcases => 167.19/25.07 (binunionE => 167.19/25.07 (binunionLsub => 167.19/25.07 (binunionRsub => 167.19/25.07 (binintersectI => 167.19/25.07 (binintersectSubset5 => 167.19/25.07 (binintersectEL => 167.19/25.07 (binintersectLsub => 167.19/25.07 (binintersectSubset2 => 167.19/25.07 (binintersectSubset3 => 167.19/25.07 (binintersectER => 167.19/25.07 (disjointsetsI1 => 167.19/25.07 (binintersectRsub => 167.19/25.07 (binintersectSubset4 => 167.19/25.07 (binintersectSubset1 => 167.19/25.07 (bs114d => 167.19/25.07 (setminusI => 167.19/25.07 (setminusEL => 167.19/25.07 (setminusER => 167.19/25.07 (setminusSubset2 => 167.19/25.07 (setminusERneg => 167.19/25.07 (setminusELneg => 167.19/25.07 (setminusILneg => 167.19/25.07 (setminusIRneg => 167.19/25.07 (setminusLsub => 167.19/25.07 (setminusSubset1 => 167.19/25.07 (symdiffE => 167.19/25.07 (symdiffI1 => 167.19/25.07 (symdiffI2 => 167.19/25.07 (symdiffIneg1 => 167.19/25.07 (symdiffIneg2 => 167.19/25.07 (secondinupair => 167.19/25.07 (setukpairIL => 167.19/25.07 ( 167.19/25.07 setukpairIR => 167.19/25.07 ( 167.19/25.07 kpairiskpair => 167.19/25.07 ( 167.19/25.07 kpairp => 167.19/25.07 ( 167.19/25.07 singletonsubset => 167.19/25.07 ( 167.19/25.07 singletoninpowerset => 167.19/25.07 ( 167.19/25.07 singletoninpowunion => 167.19/25.07 ( 167.19/25.07 upairset2E => 167.19/25.07 ( 167.19/25.07 upairsubunion => 167.19/25.07 ( 167.19/25.07 upairinpowunion => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem1 => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem2 => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem3 => 167.19/25.07 ( 167.19/25.07 cartprodpairin => 167.19/25.07 ( 167.19/25.07 cartprodmempair1 => 167.19/25.07 ( 167.19/25.07 cartprodmempair => 167.19/25.07 ( 167.19/25.07 setunionE2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton1 => 167.19/25.07 ( 167.19/25.07 setunionsingleton2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton => 167.19/25.07 ( 167.19/25.07 singletonprop => 167.19/25.07 ( 167.19/25.07 ex1E1 => 167.19/25.07 ( 167.19/25.07 ex1I => 167.19/25.07 ( 167.19/25.07 ex1I2 => 167.19/25.07 ( 167.19/25.07 singletonsuniq => 167.19/25.07 ( 167.19/25.07 setukpairinjL1 => 167.19/25.07 ( 167.19/25.07 kfstsingleton => 167.19/25.07 ( 167.19/25.07 theprop => 167.19/25.07 ( 167.19/25.07 kfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodfstin => 167.19/25.07 ( 167.19/25.07 setukpairinjL2 => 167.19/25.07 ( 167.19/25.07 setukpairinjL => 167.19/25.07 ( 167.19/25.07 setukpairinjR11 => 167.19/25.07 ( 167.19/25.07 setukpairinjR12 => 167.19/25.07 ( 167.19/25.07 setukpairinjR1 => 167.19/25.07 ( 167.19/25.07 upairequniteq => 167.19/25.07 ( 167.19/25.07 setukpairinjR2 => 167.19/25.07 ( 167.19/25.07 setukpairinjR => 167.19/25.07 ( 167.19/25.07 ksndsingleton => 167.19/25.07 ( 167.19/25.07 ksndpairEq => 167.19/25.07 ( 167.19/25.07 kpairsurjEq => 167.19/25.07 ( 167.19/25.07 cartprodsndin => 167.19/25.07 ( 167.19/25.07 cartprodpairmemEL => 167.19/25.07 ( 167.19/25.07 cartprodpairmemER => 167.19/25.07 ( 167.19/25.07 cartprodmempaircEq => 167.19/25.07 ( 167.19/25.07 cartprodfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodsndpairEq => 167.19/25.07 ( 167.19/25.07 cartprodpairsurjEq => 167.19/25.07 ( 167.19/25.07 dpsetconstrI => 167.19/25.07 ( 167.19/25.07 dpsetconstrSub => 167.19/25.07 ( 167.19/25.07 setOfPairsIsBReln => 167.19/25.07 ( 167.19/25.07 !!(( 167.19/25.07 ^[Y0 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y1 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y2 : ( 167.19/25.07 $i * $i) > $o]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y3 : $i]: 167.19/25.07 (in(Y3, 167.19/25.07 Y0) => 167.19/25.07 (!!(( 167.19/25.07 ^[Y4 : $i]: 167.19/25.07 (in(Y4, 167.19/25.07 Y1) => 167.19/25.07 (in( 167.19/25.07 kpair( 167.19/25.07 Y3, Y4), 167.19/25.07 dpsetconstr( 167.19/25.07 Y0, Y1, 167.19/25.07 Y2)) => 167.19/25.07 Y2(Y3, 167.19/25.07 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.07 inference('lazy_cnf_imply', [status(thm)], ['72'])). 167.19/25.07 tff('74', plain, 167.19/25.07 ~ (descr__Cong => 167.19/25.07 (dsetconstr__Cong => 167.19/25.07 (subsetI1 => 167.19/25.07 (eqimpsubset2 => 167.19/25.07 (eqimpsubset1 => 167.19/25.07 (subsetI2 => 167.19/25.07 (emptysetsubset => 167.19/25.07 (subsetE => 167.19/25.07 (subsetE2 => 167.19/25.07 (notsubsetI => 167.19/25.07 (notequalI1 => 167.19/25.07 (notequalI2 => 167.19/25.07 (subsetRefl => 167.19/25.07 (subsetTrans => 167.19/25.07 (setadjoinSub => 167.19/25.07 (setadjoinSub2 => 167.19/25.07 (subset2powerset => 167.19/25.07 (setextsub => 167.19/25.07 (subsetemptysetimpeq => 167.19/25.07 (powersetI1 => 167.19/25.07 (powersetE1 => 167.19/25.07 (inPowerset => 167.19/25.07 (powersetsubset => 167.19/25.07 (sepInPowerset => 167.19/25.07 (sepSubset => 167.19/25.07 (binunionIL => 167.19/25.07 (upairset2IR => 167.19/25.07 (binunionIR => 167.19/25.07 (binunionEcases => 167.19/25.07 (binunionE => 167.19/25.07 (binunionLsub => 167.19/25.07 (binunionRsub => 167.19/25.07 (binintersectI => 167.19/25.07 (binintersectSubset5 => 167.19/25.07 (binintersectEL => 167.19/25.07 (binintersectLsub => 167.19/25.07 (binintersectSubset2 => 167.19/25.07 (binintersectSubset3 => 167.19/25.07 (binintersectER => 167.19/25.07 (disjointsetsI1 => 167.19/25.07 (binintersectRsub => 167.19/25.07 (binintersectSubset4 => 167.19/25.07 (binintersectSubset1 => 167.19/25.07 (bs114d => 167.19/25.07 (setminusI => 167.19/25.07 (setminusEL => 167.19/25.07 (setminusER => 167.19/25.07 (setminusSubset2 => 167.19/25.07 (setminusERneg => 167.19/25.07 (setminusELneg => 167.19/25.07 (setminusILneg => 167.19/25.07 (setminusIRneg => 167.19/25.07 (setminusLsub => 167.19/25.07 (setminusSubset1 => 167.19/25.07 (symdiffE => 167.19/25.07 (symdiffI1 => 167.19/25.07 (symdiffI2 => 167.19/25.07 (symdiffIneg1 => 167.19/25.07 (symdiffIneg2 => 167.19/25.07 (secondinupair => 167.19/25.07 (setukpairIL => 167.19/25.07 (setukpairIR => 167.19/25.07 ( 167.19/25.07 kpairiskpair => 167.19/25.07 ( 167.19/25.07 kpairp => 167.19/25.07 ( 167.19/25.07 singletonsubset => 167.19/25.07 ( 167.19/25.07 singletoninpowerset => 167.19/25.07 ( 167.19/25.07 singletoninpowunion => 167.19/25.07 ( 167.19/25.07 upairset2E => 167.19/25.07 ( 167.19/25.07 upairsubunion => 167.19/25.07 ( 167.19/25.07 upairinpowunion => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem1 => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem2 => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem3 => 167.19/25.07 ( 167.19/25.07 cartprodpairin => 167.19/25.07 ( 167.19/25.07 cartprodmempair1 => 167.19/25.07 ( 167.19/25.07 cartprodmempair => 167.19/25.07 ( 167.19/25.07 setunionE2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton1 => 167.19/25.07 ( 167.19/25.07 setunionsingleton2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton => 167.19/25.07 ( 167.19/25.07 singletonprop => 167.19/25.07 ( 167.19/25.07 ex1E1 => 167.19/25.07 ( 167.19/25.07 ex1I => 167.19/25.07 ( 167.19/25.07 ex1I2 => 167.19/25.07 ( 167.19/25.07 singletonsuniq => 167.19/25.07 ( 167.19/25.07 setukpairinjL1 => 167.19/25.07 ( 167.19/25.07 kfstsingleton => 167.19/25.07 ( 167.19/25.07 theprop => 167.19/25.07 ( 167.19/25.07 kfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodfstin => 167.19/25.07 ( 167.19/25.07 setukpairinjL2 => 167.19/25.07 ( 167.19/25.07 setukpairinjL => 167.19/25.07 ( 167.19/25.07 setukpairinjR11 => 167.19/25.07 ( 167.19/25.07 setukpairinjR12 => 167.19/25.07 ( 167.19/25.07 setukpairinjR1 => 167.19/25.07 ( 167.19/25.07 upairequniteq => 167.19/25.07 ( 167.19/25.07 setukpairinjR2 => 167.19/25.07 ( 167.19/25.07 setukpairinjR => 167.19/25.07 ( 167.19/25.07 ksndsingleton => 167.19/25.07 ( 167.19/25.07 ksndpairEq => 167.19/25.07 ( 167.19/25.07 kpairsurjEq => 167.19/25.07 ( 167.19/25.07 cartprodsndin => 167.19/25.07 ( 167.19/25.07 cartprodpairmemEL => 167.19/25.07 ( 167.19/25.07 cartprodpairmemER => 167.19/25.07 ( 167.19/25.07 cartprodmempaircEq => 167.19/25.07 ( 167.19/25.07 cartprodfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodsndpairEq => 167.19/25.07 ( 167.19/25.07 cartprodpairsurjEq => 167.19/25.07 ( 167.19/25.07 dpsetconstrI => 167.19/25.07 ( 167.19/25.07 dpsetconstrSub => 167.19/25.07 ( 167.19/25.07 setOfPairsIsBReln => 167.19/25.07 ( 167.19/25.07 !!(( 167.19/25.07 ^[Y0 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y1 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y2 : ( 167.19/25.07 $i * $i) > $o]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y3 : $i]: 167.19/25.07 (in(Y3, 167.19/25.07 Y0) => 167.19/25.07 (!!(( 167.19/25.07 ^[Y4 : $i]: 167.19/25.07 (in(Y4, 167.19/25.07 Y1) => 167.19/25.07 (in( 167.19/25.07 kpair( 167.19/25.07 Y3, Y4), 167.19/25.07 dpsetconstr( 167.19/25.07 Y0, Y1, 167.19/25.07 Y2)) => 167.19/25.07 Y2(Y3, 167.19/25.07 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.07 inference('lazy_cnf_imply', [status(thm)], ['73'])). 167.19/25.07 tff('75', plain, 167.19/25.07 ~ (dsetconstr__Cong => 167.19/25.07 (subsetI1 => 167.19/25.07 (eqimpsubset2 => 167.19/25.07 (eqimpsubset1 => 167.19/25.07 (subsetI2 => 167.19/25.07 (emptysetsubset => 167.19/25.07 (subsetE => 167.19/25.07 (subsetE2 => 167.19/25.07 (notsubsetI => 167.19/25.07 (notequalI1 => 167.19/25.07 (notequalI2 => 167.19/25.07 (subsetRefl => 167.19/25.07 (subsetTrans => 167.19/25.07 (setadjoinSub => 167.19/25.07 (setadjoinSub2 => 167.19/25.07 (subset2powerset => 167.19/25.07 (setextsub => 167.19/25.07 (subsetemptysetimpeq => 167.19/25.07 (powersetI1 => 167.19/25.07 (powersetE1 => 167.19/25.07 (inPowerset => 167.19/25.07 (powersetsubset => 167.19/25.07 (sepInPowerset => 167.19/25.07 (sepSubset => 167.19/25.07 (binunionIL => 167.19/25.07 (upairset2IR => 167.19/25.07 (binunionIR => 167.19/25.07 (binunionEcases => 167.19/25.07 (binunionE => 167.19/25.07 (binunionLsub => 167.19/25.07 (binunionRsub => 167.19/25.07 (binintersectI => 167.19/25.07 (binintersectSubset5 => 167.19/25.07 (binintersectEL => 167.19/25.07 (binintersectLsub => 167.19/25.07 (binintersectSubset2 => 167.19/25.07 (binintersectSubset3 => 167.19/25.07 (binintersectER => 167.19/25.07 (disjointsetsI1 => 167.19/25.07 (binintersectRsub => 167.19/25.07 (binintersectSubset4 => 167.19/25.07 (binintersectSubset1 => 167.19/25.07 (bs114d => 167.19/25.07 (setminusI => 167.19/25.07 (setminusEL => 167.19/25.07 (setminusER => 167.19/25.07 (setminusSubset2 => 167.19/25.07 (setminusERneg => 167.19/25.07 (setminusELneg => 167.19/25.07 (setminusILneg => 167.19/25.07 (setminusIRneg => 167.19/25.07 (setminusLsub => 167.19/25.07 (setminusSubset1 => 167.19/25.07 (symdiffE => 167.19/25.07 (symdiffI1 => 167.19/25.07 (symdiffI2 => 167.19/25.07 (symdiffIneg1 => 167.19/25.07 (symdiffIneg2 => 167.19/25.07 (secondinupair => 167.19/25.07 (setukpairIL => 167.19/25.07 (setukpairIR => 167.19/25.07 (kpairiskpair => 167.19/25.07 ( 167.19/25.07 kpairp => 167.19/25.07 ( 167.19/25.07 singletonsubset => 167.19/25.07 ( 167.19/25.07 singletoninpowerset => 167.19/25.07 ( 167.19/25.07 singletoninpowunion => 167.19/25.07 ( 167.19/25.07 upairset2E => 167.19/25.07 ( 167.19/25.07 upairsubunion => 167.19/25.07 ( 167.19/25.07 upairinpowunion => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem1 => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem2 => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem3 => 167.19/25.07 ( 167.19/25.07 cartprodpairin => 167.19/25.07 ( 167.19/25.07 cartprodmempair1 => 167.19/25.07 ( 167.19/25.07 cartprodmempair => 167.19/25.07 ( 167.19/25.07 setunionE2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton1 => 167.19/25.07 ( 167.19/25.07 setunionsingleton2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton => 167.19/25.07 ( 167.19/25.07 singletonprop => 167.19/25.07 ( 167.19/25.07 ex1E1 => 167.19/25.07 ( 167.19/25.07 ex1I => 167.19/25.07 ( 167.19/25.07 ex1I2 => 167.19/25.07 ( 167.19/25.07 singletonsuniq => 167.19/25.07 ( 167.19/25.07 setukpairinjL1 => 167.19/25.07 ( 167.19/25.07 kfstsingleton => 167.19/25.07 ( 167.19/25.07 theprop => 167.19/25.07 ( 167.19/25.07 kfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodfstin => 167.19/25.07 ( 167.19/25.07 setukpairinjL2 => 167.19/25.07 ( 167.19/25.07 setukpairinjL => 167.19/25.07 ( 167.19/25.07 setukpairinjR11 => 167.19/25.07 ( 167.19/25.07 setukpairinjR12 => 167.19/25.07 ( 167.19/25.07 setukpairinjR1 => 167.19/25.07 ( 167.19/25.07 upairequniteq => 167.19/25.07 ( 167.19/25.07 setukpairinjR2 => 167.19/25.07 ( 167.19/25.07 setukpairinjR => 167.19/25.07 ( 167.19/25.07 ksndsingleton => 167.19/25.07 ( 167.19/25.07 ksndpairEq => 167.19/25.07 ( 167.19/25.07 kpairsurjEq => 167.19/25.07 ( 167.19/25.07 cartprodsndin => 167.19/25.07 ( 167.19/25.07 cartprodpairmemEL => 167.19/25.07 ( 167.19/25.07 cartprodpairmemER => 167.19/25.07 ( 167.19/25.07 cartprodmempaircEq => 167.19/25.07 ( 167.19/25.07 cartprodfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodsndpairEq => 167.19/25.07 ( 167.19/25.07 cartprodpairsurjEq => 167.19/25.07 ( 167.19/25.07 dpsetconstrI => 167.19/25.07 ( 167.19/25.07 dpsetconstrSub => 167.19/25.07 ( 167.19/25.07 setOfPairsIsBReln => 167.19/25.07 ( 167.19/25.07 !!(( 167.19/25.07 ^[Y0 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y1 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y2 : ( 167.19/25.07 $i * $i) > $o]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y3 : $i]: 167.19/25.07 (in(Y3, 167.19/25.07 Y0) => 167.19/25.07 (!!(( 167.19/25.07 ^[Y4 : $i]: 167.19/25.07 (in(Y4, 167.19/25.07 Y1) => 167.19/25.07 (in( 167.19/25.07 kpair( 167.19/25.07 Y3, Y4), 167.19/25.07 dpsetconstr( 167.19/25.07 Y0, Y1, 167.19/25.07 Y2)) => 167.19/25.07 Y2(Y3, 167.19/25.07 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.07 inference('lazy_cnf_imply', [status(thm)], ['74'])). 167.19/25.07 tff('76', plain, 167.19/25.07 ~ (subsetI1 => 167.19/25.07 (eqimpsubset2 => 167.19/25.07 (eqimpsubset1 => 167.19/25.07 (subsetI2 => 167.19/25.07 (emptysetsubset => 167.19/25.07 (subsetE => 167.19/25.07 (subsetE2 => 167.19/25.07 (notsubsetI => 167.19/25.07 (notequalI1 => 167.19/25.07 (notequalI2 => 167.19/25.07 (subsetRefl => 167.19/25.07 (subsetTrans => 167.19/25.07 (setadjoinSub => 167.19/25.07 (setadjoinSub2 => 167.19/25.07 (subset2powerset => 167.19/25.07 (setextsub => 167.19/25.07 (subsetemptysetimpeq => 167.19/25.07 (powersetI1 => 167.19/25.07 (powersetE1 => 167.19/25.07 (inPowerset => 167.19/25.07 (powersetsubset => 167.19/25.07 (sepInPowerset => 167.19/25.07 (sepSubset => 167.19/25.07 (binunionIL => 167.19/25.07 (upairset2IR => 167.19/25.07 (binunionIR => 167.19/25.07 (binunionEcases => 167.19/25.07 (binunionE => 167.19/25.07 (binunionLsub => 167.19/25.07 (binunionRsub => 167.19/25.07 (binintersectI => 167.19/25.07 (binintersectSubset5 => 167.19/25.07 (binintersectEL => 167.19/25.07 (binintersectLsub => 167.19/25.07 (binintersectSubset2 => 167.19/25.07 (binintersectSubset3 => 167.19/25.07 (binintersectER => 167.19/25.07 (disjointsetsI1 => 167.19/25.07 (binintersectRsub => 167.19/25.07 (binintersectSubset4 => 167.19/25.07 (binintersectSubset1 => 167.19/25.07 (bs114d => 167.19/25.07 (setminusI => 167.19/25.07 (setminusEL => 167.19/25.07 (setminusER => 167.19/25.07 (setminusSubset2 => 167.19/25.07 (setminusERneg => 167.19/25.07 (setminusELneg => 167.19/25.07 (setminusILneg => 167.19/25.07 (setminusIRneg => 167.19/25.07 (setminusLsub => 167.19/25.07 (setminusSubset1 => 167.19/25.07 (symdiffE => 167.19/25.07 (symdiffI1 => 167.19/25.07 (symdiffI2 => 167.19/25.07 (symdiffIneg1 => 167.19/25.07 (symdiffIneg2 => 167.19/25.07 (secondinupair => 167.19/25.07 (setukpairIL => 167.19/25.07 (setukpairIR => 167.19/25.07 (kpairiskpair => 167.19/25.07 (kpairp => 167.19/25.07 ( 167.19/25.07 singletonsubset => 167.19/25.07 ( 167.19/25.07 singletoninpowerset => 167.19/25.07 ( 167.19/25.07 singletoninpowunion => 167.19/25.07 ( 167.19/25.07 upairset2E => 167.19/25.07 ( 167.19/25.07 upairsubunion => 167.19/25.07 ( 167.19/25.07 upairinpowunion => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem1 => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem2 => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem3 => 167.19/25.07 ( 167.19/25.07 cartprodpairin => 167.19/25.07 ( 167.19/25.07 cartprodmempair1 => 167.19/25.07 ( 167.19/25.07 cartprodmempair => 167.19/25.07 ( 167.19/25.07 setunionE2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton1 => 167.19/25.07 ( 167.19/25.07 setunionsingleton2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton => 167.19/25.07 ( 167.19/25.07 singletonprop => 167.19/25.07 ( 167.19/25.07 ex1E1 => 167.19/25.07 ( 167.19/25.07 ex1I => 167.19/25.07 ( 167.19/25.07 ex1I2 => 167.19/25.07 ( 167.19/25.07 singletonsuniq => 167.19/25.07 ( 167.19/25.07 setukpairinjL1 => 167.19/25.07 ( 167.19/25.07 kfstsingleton => 167.19/25.07 ( 167.19/25.07 theprop => 167.19/25.07 ( 167.19/25.07 kfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodfstin => 167.19/25.07 ( 167.19/25.07 setukpairinjL2 => 167.19/25.07 ( 167.19/25.07 setukpairinjL => 167.19/25.07 ( 167.19/25.07 setukpairinjR11 => 167.19/25.07 ( 167.19/25.07 setukpairinjR12 => 167.19/25.07 ( 167.19/25.07 setukpairinjR1 => 167.19/25.07 ( 167.19/25.07 upairequniteq => 167.19/25.07 ( 167.19/25.07 setukpairinjR2 => 167.19/25.07 ( 167.19/25.07 setukpairinjR => 167.19/25.07 ( 167.19/25.07 ksndsingleton => 167.19/25.07 ( 167.19/25.07 ksndpairEq => 167.19/25.07 ( 167.19/25.07 kpairsurjEq => 167.19/25.07 ( 167.19/25.07 cartprodsndin => 167.19/25.07 ( 167.19/25.07 cartprodpairmemEL => 167.19/25.07 ( 167.19/25.07 cartprodpairmemER => 167.19/25.07 ( 167.19/25.07 cartprodmempaircEq => 167.19/25.07 ( 167.19/25.07 cartprodfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodsndpairEq => 167.19/25.07 ( 167.19/25.07 cartprodpairsurjEq => 167.19/25.07 ( 167.19/25.07 dpsetconstrI => 167.19/25.07 ( 167.19/25.07 dpsetconstrSub => 167.19/25.07 ( 167.19/25.07 setOfPairsIsBReln => 167.19/25.07 ( 167.19/25.07 !!(( 167.19/25.07 ^[Y0 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y1 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y2 : ( 167.19/25.07 $i * $i) > $o]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y3 : $i]: 167.19/25.07 (in(Y3, 167.19/25.07 Y0) => 167.19/25.07 (!!(( 167.19/25.07 ^[Y4 : $i]: 167.19/25.07 (in(Y4, 167.19/25.07 Y1) => 167.19/25.07 (in( 167.19/25.07 kpair( 167.19/25.07 Y3, Y4), 167.19/25.07 dpsetconstr( 167.19/25.07 Y0, Y1, 167.19/25.07 Y2)) => 167.19/25.07 Y2(Y3, 167.19/25.07 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.07 inference('lazy_cnf_imply', [status(thm)], ['75'])). 167.19/25.07 tff('77', plain, 167.19/25.07 ~ (eqimpsubset2 => 167.19/25.07 (eqimpsubset1 => 167.19/25.07 (subsetI2 => 167.19/25.07 (emptysetsubset => 167.19/25.07 (subsetE => 167.19/25.07 (subsetE2 => 167.19/25.07 (notsubsetI => 167.19/25.07 (notequalI1 => 167.19/25.07 (notequalI2 => 167.19/25.07 (subsetRefl => 167.19/25.07 (subsetTrans => 167.19/25.07 (setadjoinSub => 167.19/25.07 (setadjoinSub2 => 167.19/25.07 (subset2powerset => 167.19/25.07 (setextsub => 167.19/25.07 (subsetemptysetimpeq => 167.19/25.07 (powersetI1 => 167.19/25.07 (powersetE1 => 167.19/25.07 (inPowerset => 167.19/25.07 (powersetsubset => 167.19/25.07 (sepInPowerset => 167.19/25.07 (sepSubset => 167.19/25.07 (binunionIL => 167.19/25.07 (upairset2IR => 167.19/25.07 (binunionIR => 167.19/25.07 (binunionEcases => 167.19/25.07 (binunionE => 167.19/25.07 (binunionLsub => 167.19/25.07 (binunionRsub => 167.19/25.07 (binintersectI => 167.19/25.07 (binintersectSubset5 => 167.19/25.07 (binintersectEL => 167.19/25.07 (binintersectLsub => 167.19/25.07 (binintersectSubset2 => 167.19/25.07 (binintersectSubset3 => 167.19/25.07 (binintersectER => 167.19/25.07 (disjointsetsI1 => 167.19/25.07 (binintersectRsub => 167.19/25.07 (binintersectSubset4 => 167.19/25.07 (binintersectSubset1 => 167.19/25.07 (bs114d => 167.19/25.07 (setminusI => 167.19/25.07 (setminusEL => 167.19/25.07 (setminusER => 167.19/25.07 (setminusSubset2 => 167.19/25.07 (setminusERneg => 167.19/25.07 (setminusELneg => 167.19/25.07 (setminusILneg => 167.19/25.07 (setminusIRneg => 167.19/25.07 (setminusLsub => 167.19/25.07 (setminusSubset1 => 167.19/25.07 (symdiffE => 167.19/25.07 (symdiffI1 => 167.19/25.07 (symdiffI2 => 167.19/25.07 (symdiffIneg1 => 167.19/25.07 (symdiffIneg2 => 167.19/25.07 (secondinupair => 167.19/25.07 (setukpairIL => 167.19/25.07 (setukpairIR => 167.19/25.07 (kpairiskpair => 167.19/25.07 (kpairp => 167.19/25.07 (singletonsubset => 167.19/25.07 ( 167.19/25.07 singletoninpowerset => 167.19/25.07 ( 167.19/25.07 singletoninpowunion => 167.19/25.07 ( 167.19/25.07 upairset2E => 167.19/25.07 ( 167.19/25.07 upairsubunion => 167.19/25.07 ( 167.19/25.07 upairinpowunion => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem1 => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem2 => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem3 => 167.19/25.07 ( 167.19/25.07 cartprodpairin => 167.19/25.07 ( 167.19/25.07 cartprodmempair1 => 167.19/25.07 ( 167.19/25.07 cartprodmempair => 167.19/25.07 ( 167.19/25.07 setunionE2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton1 => 167.19/25.07 ( 167.19/25.07 setunionsingleton2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton => 167.19/25.07 ( 167.19/25.07 singletonprop => 167.19/25.07 ( 167.19/25.07 ex1E1 => 167.19/25.07 ( 167.19/25.07 ex1I => 167.19/25.07 ( 167.19/25.07 ex1I2 => 167.19/25.07 ( 167.19/25.07 singletonsuniq => 167.19/25.07 ( 167.19/25.07 setukpairinjL1 => 167.19/25.07 ( 167.19/25.07 kfstsingleton => 167.19/25.07 ( 167.19/25.07 theprop => 167.19/25.07 ( 167.19/25.07 kfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodfstin => 167.19/25.07 ( 167.19/25.07 setukpairinjL2 => 167.19/25.07 ( 167.19/25.07 setukpairinjL => 167.19/25.07 ( 167.19/25.07 setukpairinjR11 => 167.19/25.07 ( 167.19/25.07 setukpairinjR12 => 167.19/25.07 ( 167.19/25.07 setukpairinjR1 => 167.19/25.07 ( 167.19/25.07 upairequniteq => 167.19/25.07 ( 167.19/25.07 setukpairinjR2 => 167.19/25.07 ( 167.19/25.07 setukpairinjR => 167.19/25.07 ( 167.19/25.07 ksndsingleton => 167.19/25.07 ( 167.19/25.07 ksndpairEq => 167.19/25.07 ( 167.19/25.07 kpairsurjEq => 167.19/25.07 ( 167.19/25.07 cartprodsndin => 167.19/25.07 ( 167.19/25.07 cartprodpairmemEL => 167.19/25.07 ( 167.19/25.07 cartprodpairmemER => 167.19/25.07 ( 167.19/25.07 cartprodmempaircEq => 167.19/25.07 ( 167.19/25.07 cartprodfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodsndpairEq => 167.19/25.07 ( 167.19/25.07 cartprodpairsurjEq => 167.19/25.07 ( 167.19/25.07 dpsetconstrI => 167.19/25.07 ( 167.19/25.07 dpsetconstrSub => 167.19/25.07 ( 167.19/25.07 setOfPairsIsBReln => 167.19/25.07 ( 167.19/25.07 !!(( 167.19/25.07 ^[Y0 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y1 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y2 : ( 167.19/25.07 $i * $i) > $o]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y3 : $i]: 167.19/25.07 (in(Y3, 167.19/25.07 Y0) => 167.19/25.07 (!!(( 167.19/25.07 ^[Y4 : $i]: 167.19/25.07 (in(Y4, 167.19/25.07 Y1) => 167.19/25.07 (in( 167.19/25.07 kpair( 167.19/25.07 Y3, Y4), 167.19/25.07 dpsetconstr( 167.19/25.07 Y0, Y1, 167.19/25.07 Y2)) => 167.19/25.07 Y2(Y3, 167.19/25.07 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.07 inference('lazy_cnf_imply', [status(thm)], ['76'])). 167.19/25.07 tff('78', plain, 167.19/25.07 ~ (eqimpsubset1 => 167.19/25.07 (subsetI2 => 167.19/25.07 (emptysetsubset => 167.19/25.07 (subsetE => 167.19/25.07 (subsetE2 => 167.19/25.07 (notsubsetI => 167.19/25.07 (notequalI1 => 167.19/25.07 (notequalI2 => 167.19/25.07 (subsetRefl => 167.19/25.07 (subsetTrans => 167.19/25.07 (setadjoinSub => 167.19/25.07 (setadjoinSub2 => 167.19/25.07 (subset2powerset => 167.19/25.07 (setextsub => 167.19/25.07 (subsetemptysetimpeq => 167.19/25.07 (powersetI1 => 167.19/25.07 (powersetE1 => 167.19/25.07 (inPowerset => 167.19/25.07 (powersetsubset => 167.19/25.07 (sepInPowerset => 167.19/25.07 (sepSubset => 167.19/25.07 (binunionIL => 167.19/25.07 (upairset2IR => 167.19/25.07 (binunionIR => 167.19/25.07 (binunionEcases => 167.19/25.07 (binunionE => 167.19/25.07 (binunionLsub => 167.19/25.07 (binunionRsub => 167.19/25.07 (binintersectI => 167.19/25.07 (binintersectSubset5 => 167.19/25.07 (binintersectEL => 167.19/25.07 (binintersectLsub => 167.19/25.07 (binintersectSubset2 => 167.19/25.07 (binintersectSubset3 => 167.19/25.07 (binintersectER => 167.19/25.07 (disjointsetsI1 => 167.19/25.07 (binintersectRsub => 167.19/25.07 (binintersectSubset4 => 167.19/25.07 (binintersectSubset1 => 167.19/25.07 (bs114d => 167.19/25.07 (setminusI => 167.19/25.07 (setminusEL => 167.19/25.07 (setminusER => 167.19/25.07 (setminusSubset2 => 167.19/25.07 (setminusERneg => 167.19/25.07 (setminusELneg => 167.19/25.07 (setminusILneg => 167.19/25.07 (setminusIRneg => 167.19/25.07 (setminusLsub => 167.19/25.07 (setminusSubset1 => 167.19/25.07 (symdiffE => 167.19/25.07 (symdiffI1 => 167.19/25.07 (symdiffI2 => 167.19/25.07 (symdiffIneg1 => 167.19/25.07 (symdiffIneg2 => 167.19/25.07 (secondinupair => 167.19/25.07 (setukpairIL => 167.19/25.07 (setukpairIR => 167.19/25.07 (kpairiskpair => 167.19/25.07 (kpairp => 167.19/25.07 (singletonsubset => 167.19/25.07 (singletoninpowerset => 167.19/25.07 ( 167.19/25.07 singletoninpowunion => 167.19/25.07 ( 167.19/25.07 upairset2E => 167.19/25.07 ( 167.19/25.07 upairsubunion => 167.19/25.07 ( 167.19/25.07 upairinpowunion => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem1 => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem2 => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem3 => 167.19/25.07 ( 167.19/25.07 cartprodpairin => 167.19/25.07 ( 167.19/25.07 cartprodmempair1 => 167.19/25.07 ( 167.19/25.07 cartprodmempair => 167.19/25.07 ( 167.19/25.07 setunionE2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton1 => 167.19/25.07 ( 167.19/25.07 setunionsingleton2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton => 167.19/25.07 ( 167.19/25.07 singletonprop => 167.19/25.07 ( 167.19/25.07 ex1E1 => 167.19/25.07 ( 167.19/25.07 ex1I => 167.19/25.07 ( 167.19/25.07 ex1I2 => 167.19/25.07 ( 167.19/25.07 singletonsuniq => 167.19/25.07 ( 167.19/25.07 setukpairinjL1 => 167.19/25.07 ( 167.19/25.07 kfstsingleton => 167.19/25.07 ( 167.19/25.07 theprop => 167.19/25.07 ( 167.19/25.07 kfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodfstin => 167.19/25.07 ( 167.19/25.07 setukpairinjL2 => 167.19/25.07 ( 167.19/25.07 setukpairinjL => 167.19/25.07 ( 167.19/25.07 setukpairinjR11 => 167.19/25.07 ( 167.19/25.07 setukpairinjR12 => 167.19/25.07 ( 167.19/25.07 setukpairinjR1 => 167.19/25.07 ( 167.19/25.07 upairequniteq => 167.19/25.07 ( 167.19/25.07 setukpairinjR2 => 167.19/25.07 ( 167.19/25.07 setukpairinjR => 167.19/25.07 ( 167.19/25.07 ksndsingleton => 167.19/25.07 ( 167.19/25.07 ksndpairEq => 167.19/25.07 ( 167.19/25.07 kpairsurjEq => 167.19/25.07 ( 167.19/25.07 cartprodsndin => 167.19/25.07 ( 167.19/25.07 cartprodpairmemEL => 167.19/25.07 ( 167.19/25.07 cartprodpairmemER => 167.19/25.07 ( 167.19/25.07 cartprodmempaircEq => 167.19/25.07 ( 167.19/25.07 cartprodfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodsndpairEq => 167.19/25.07 ( 167.19/25.07 cartprodpairsurjEq => 167.19/25.07 ( 167.19/25.07 dpsetconstrI => 167.19/25.07 ( 167.19/25.07 dpsetconstrSub => 167.19/25.07 ( 167.19/25.07 setOfPairsIsBReln => 167.19/25.07 ( 167.19/25.07 !!(( 167.19/25.07 ^[Y0 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y1 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y2 : ( 167.19/25.07 $i * $i) > $o]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y3 : $i]: 167.19/25.07 (in(Y3, 167.19/25.07 Y0) => 167.19/25.07 (!!(( 167.19/25.07 ^[Y4 : $i]: 167.19/25.07 (in(Y4, 167.19/25.07 Y1) => 167.19/25.07 (in( 167.19/25.07 kpair( 167.19/25.07 Y3, Y4), 167.19/25.07 dpsetconstr( 167.19/25.07 Y0, Y1, 167.19/25.07 Y2)) => 167.19/25.07 Y2(Y3, 167.19/25.07 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.07 inference('lazy_cnf_imply', [status(thm)], ['77'])). 167.19/25.07 tff('79', plain, 167.19/25.07 ~ (subsetI2 => 167.19/25.07 (emptysetsubset => 167.19/25.07 (subsetE => 167.19/25.07 (subsetE2 => 167.19/25.07 (notsubsetI => 167.19/25.07 (notequalI1 => 167.19/25.07 (notequalI2 => 167.19/25.07 (subsetRefl => 167.19/25.07 (subsetTrans => 167.19/25.07 (setadjoinSub => 167.19/25.07 (setadjoinSub2 => 167.19/25.07 (subset2powerset => 167.19/25.07 (setextsub => 167.19/25.07 (subsetemptysetimpeq => 167.19/25.07 (powersetI1 => 167.19/25.07 (powersetE1 => 167.19/25.07 (inPowerset => 167.19/25.07 (powersetsubset => 167.19/25.07 (sepInPowerset => 167.19/25.07 (sepSubset => 167.19/25.07 (binunionIL => 167.19/25.07 (upairset2IR => 167.19/25.07 (binunionIR => 167.19/25.07 (binunionEcases => 167.19/25.07 (binunionE => 167.19/25.07 (binunionLsub => 167.19/25.07 (binunionRsub => 167.19/25.07 (binintersectI => 167.19/25.07 (binintersectSubset5 => 167.19/25.07 (binintersectEL => 167.19/25.07 (binintersectLsub => 167.19/25.07 (binintersectSubset2 => 167.19/25.07 (binintersectSubset3 => 167.19/25.07 (binintersectER => 167.19/25.07 (disjointsetsI1 => 167.19/25.07 (binintersectRsub => 167.19/25.07 (binintersectSubset4 => 167.19/25.07 (binintersectSubset1 => 167.19/25.07 (bs114d => 167.19/25.07 (setminusI => 167.19/25.07 (setminusEL => 167.19/25.07 (setminusER => 167.19/25.07 (setminusSubset2 => 167.19/25.07 (setminusERneg => 167.19/25.07 (setminusELneg => 167.19/25.07 (setminusILneg => 167.19/25.07 (setminusIRneg => 167.19/25.07 (setminusLsub => 167.19/25.07 (setminusSubset1 => 167.19/25.07 (symdiffE => 167.19/25.07 (symdiffI1 => 167.19/25.07 (symdiffI2 => 167.19/25.07 (symdiffIneg1 => 167.19/25.07 (symdiffIneg2 => 167.19/25.07 (secondinupair => 167.19/25.07 (setukpairIL => 167.19/25.07 (setukpairIR => 167.19/25.07 (kpairiskpair => 167.19/25.07 (kpairp => 167.19/25.07 (singletonsubset => 167.19/25.07 (singletoninpowerset => 167.19/25.07 (singletoninpowunion => 167.19/25.07 ( 167.19/25.07 upairset2E => 167.19/25.07 ( 167.19/25.07 upairsubunion => 167.19/25.07 ( 167.19/25.07 upairinpowunion => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem1 => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem2 => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem3 => 167.19/25.07 ( 167.19/25.07 cartprodpairin => 167.19/25.07 ( 167.19/25.07 cartprodmempair1 => 167.19/25.07 ( 167.19/25.07 cartprodmempair => 167.19/25.07 ( 167.19/25.07 setunionE2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton1 => 167.19/25.07 ( 167.19/25.07 setunionsingleton2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton => 167.19/25.07 ( 167.19/25.07 singletonprop => 167.19/25.07 ( 167.19/25.07 ex1E1 => 167.19/25.07 ( 167.19/25.07 ex1I => 167.19/25.07 ( 167.19/25.07 ex1I2 => 167.19/25.07 ( 167.19/25.07 singletonsuniq => 167.19/25.07 ( 167.19/25.07 setukpairinjL1 => 167.19/25.07 ( 167.19/25.07 kfstsingleton => 167.19/25.07 ( 167.19/25.07 theprop => 167.19/25.07 ( 167.19/25.07 kfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodfstin => 167.19/25.07 ( 167.19/25.07 setukpairinjL2 => 167.19/25.07 ( 167.19/25.07 setukpairinjL => 167.19/25.07 ( 167.19/25.07 setukpairinjR11 => 167.19/25.07 ( 167.19/25.07 setukpairinjR12 => 167.19/25.07 ( 167.19/25.07 setukpairinjR1 => 167.19/25.07 ( 167.19/25.07 upairequniteq => 167.19/25.07 ( 167.19/25.07 setukpairinjR2 => 167.19/25.07 ( 167.19/25.07 setukpairinjR => 167.19/25.07 ( 167.19/25.07 ksndsingleton => 167.19/25.07 ( 167.19/25.07 ksndpairEq => 167.19/25.07 ( 167.19/25.07 kpairsurjEq => 167.19/25.07 ( 167.19/25.07 cartprodsndin => 167.19/25.07 ( 167.19/25.07 cartprodpairmemEL => 167.19/25.07 ( 167.19/25.07 cartprodpairmemER => 167.19/25.07 ( 167.19/25.07 cartprodmempaircEq => 167.19/25.07 ( 167.19/25.07 cartprodfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodsndpairEq => 167.19/25.07 ( 167.19/25.07 cartprodpairsurjEq => 167.19/25.07 ( 167.19/25.07 dpsetconstrI => 167.19/25.07 ( 167.19/25.07 dpsetconstrSub => 167.19/25.07 ( 167.19/25.07 setOfPairsIsBReln => 167.19/25.07 ( 167.19/25.07 !!(( 167.19/25.07 ^[Y0 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y1 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y2 : ( 167.19/25.07 $i * $i) > $o]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y3 : $i]: 167.19/25.07 (in(Y3, 167.19/25.07 Y0) => 167.19/25.07 (!!(( 167.19/25.07 ^[Y4 : $i]: 167.19/25.07 (in(Y4, 167.19/25.07 Y1) => 167.19/25.07 (in( 167.19/25.07 kpair( 167.19/25.07 Y3, Y4), 167.19/25.07 dpsetconstr( 167.19/25.07 Y0, Y1, 167.19/25.07 Y2)) => 167.19/25.07 Y2(Y3, 167.19/25.07 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.07 inference('lazy_cnf_imply', [status(thm)], ['78'])). 167.19/25.07 tff('80', plain, 167.19/25.07 ~ (emptysetsubset => 167.19/25.07 (subsetE => 167.19/25.07 (subsetE2 => 167.19/25.07 (notsubsetI => 167.19/25.07 (notequalI1 => 167.19/25.07 (notequalI2 => 167.19/25.07 (subsetRefl => 167.19/25.07 (subsetTrans => 167.19/25.07 (setadjoinSub => 167.19/25.07 (setadjoinSub2 => 167.19/25.07 (subset2powerset => 167.19/25.07 (setextsub => 167.19/25.07 (subsetemptysetimpeq => 167.19/25.07 (powersetI1 => 167.19/25.07 (powersetE1 => 167.19/25.07 (inPowerset => 167.19/25.07 (powersetsubset => 167.19/25.07 (sepInPowerset => 167.19/25.07 (sepSubset => 167.19/25.07 (binunionIL => 167.19/25.07 (upairset2IR => 167.19/25.07 (binunionIR => 167.19/25.07 (binunionEcases => 167.19/25.07 (binunionE => 167.19/25.07 (binunionLsub => 167.19/25.07 (binunionRsub => 167.19/25.07 (binintersectI => 167.19/25.07 (binintersectSubset5 => 167.19/25.07 (binintersectEL => 167.19/25.07 (binintersectLsub => 167.19/25.07 (binintersectSubset2 => 167.19/25.07 (binintersectSubset3 => 167.19/25.07 (binintersectER => 167.19/25.07 (disjointsetsI1 => 167.19/25.07 (binintersectRsub => 167.19/25.07 (binintersectSubset4 => 167.19/25.07 (binintersectSubset1 => 167.19/25.07 (bs114d => 167.19/25.07 (setminusI => 167.19/25.07 (setminusEL => 167.19/25.07 (setminusER => 167.19/25.07 (setminusSubset2 => 167.19/25.07 (setminusERneg => 167.19/25.07 (setminusELneg => 167.19/25.07 (setminusILneg => 167.19/25.07 (setminusIRneg => 167.19/25.07 (setminusLsub => 167.19/25.07 (setminusSubset1 => 167.19/25.07 (symdiffE => 167.19/25.07 (symdiffI1 => 167.19/25.07 (symdiffI2 => 167.19/25.07 (symdiffIneg1 => 167.19/25.07 (symdiffIneg2 => 167.19/25.07 (secondinupair => 167.19/25.07 (setukpairIL => 167.19/25.07 (setukpairIR => 167.19/25.07 (kpairiskpair => 167.19/25.07 (kpairp => 167.19/25.07 (singletonsubset => 167.19/25.07 (singletoninpowerset => 167.19/25.07 (singletoninpowunion => 167.19/25.07 (upairset2E => 167.19/25.07 ( 167.19/25.07 upairsubunion => 167.19/25.07 ( 167.19/25.07 upairinpowunion => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem1 => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem2 => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem3 => 167.19/25.07 ( 167.19/25.07 cartprodpairin => 167.19/25.07 ( 167.19/25.07 cartprodmempair1 => 167.19/25.07 ( 167.19/25.07 cartprodmempair => 167.19/25.07 ( 167.19/25.07 setunionE2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton1 => 167.19/25.07 ( 167.19/25.07 setunionsingleton2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton => 167.19/25.07 ( 167.19/25.07 singletonprop => 167.19/25.07 ( 167.19/25.07 ex1E1 => 167.19/25.07 ( 167.19/25.07 ex1I => 167.19/25.07 ( 167.19/25.07 ex1I2 => 167.19/25.07 ( 167.19/25.07 singletonsuniq => 167.19/25.07 ( 167.19/25.07 setukpairinjL1 => 167.19/25.07 ( 167.19/25.07 kfstsingleton => 167.19/25.07 ( 167.19/25.07 theprop => 167.19/25.07 ( 167.19/25.07 kfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodfstin => 167.19/25.07 ( 167.19/25.07 setukpairinjL2 => 167.19/25.07 ( 167.19/25.07 setukpairinjL => 167.19/25.07 ( 167.19/25.07 setukpairinjR11 => 167.19/25.07 ( 167.19/25.07 setukpairinjR12 => 167.19/25.07 ( 167.19/25.07 setukpairinjR1 => 167.19/25.07 ( 167.19/25.07 upairequniteq => 167.19/25.07 ( 167.19/25.07 setukpairinjR2 => 167.19/25.07 ( 167.19/25.07 setukpairinjR => 167.19/25.07 ( 167.19/25.07 ksndsingleton => 167.19/25.07 ( 167.19/25.07 ksndpairEq => 167.19/25.07 ( 167.19/25.07 kpairsurjEq => 167.19/25.07 ( 167.19/25.07 cartprodsndin => 167.19/25.07 ( 167.19/25.07 cartprodpairmemEL => 167.19/25.07 ( 167.19/25.07 cartprodpairmemER => 167.19/25.07 ( 167.19/25.07 cartprodmempaircEq => 167.19/25.07 ( 167.19/25.07 cartprodfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodsndpairEq => 167.19/25.07 ( 167.19/25.07 cartprodpairsurjEq => 167.19/25.07 ( 167.19/25.07 dpsetconstrI => 167.19/25.07 ( 167.19/25.07 dpsetconstrSub => 167.19/25.07 ( 167.19/25.07 setOfPairsIsBReln => 167.19/25.07 ( 167.19/25.07 !!(( 167.19/25.07 ^[Y0 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y1 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y2 : ( 167.19/25.07 $i * $i) > $o]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y3 : $i]: 167.19/25.07 (in(Y3, 167.19/25.07 Y0) => 167.19/25.07 (!!(( 167.19/25.07 ^[Y4 : $i]: 167.19/25.07 (in(Y4, 167.19/25.07 Y1) => 167.19/25.07 (in( 167.19/25.07 kpair( 167.19/25.07 Y3, Y4), 167.19/25.07 dpsetconstr( 167.19/25.07 Y0, Y1, 167.19/25.07 Y2)) => 167.19/25.07 Y2(Y3, 167.19/25.07 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.07 inference('lazy_cnf_imply', [status(thm)], ['79'])). 167.19/25.07 tff('81', plain, 167.19/25.07 ~ (subsetE => 167.19/25.07 (subsetE2 => 167.19/25.07 (notsubsetI => 167.19/25.07 (notequalI1 => 167.19/25.07 (notequalI2 => 167.19/25.07 (subsetRefl => 167.19/25.07 (subsetTrans => 167.19/25.07 (setadjoinSub => 167.19/25.07 (setadjoinSub2 => 167.19/25.07 (subset2powerset => 167.19/25.07 (setextsub => 167.19/25.07 (subsetemptysetimpeq => 167.19/25.07 (powersetI1 => 167.19/25.07 (powersetE1 => 167.19/25.07 (inPowerset => 167.19/25.07 (powersetsubset => 167.19/25.07 (sepInPowerset => 167.19/25.07 (sepSubset => 167.19/25.07 (binunionIL => 167.19/25.07 (upairset2IR => 167.19/25.07 (binunionIR => 167.19/25.07 (binunionEcases => 167.19/25.07 (binunionE => 167.19/25.07 (binunionLsub => 167.19/25.07 (binunionRsub => 167.19/25.07 (binintersectI => 167.19/25.07 (binintersectSubset5 => 167.19/25.07 (binintersectEL => 167.19/25.07 (binintersectLsub => 167.19/25.07 (binintersectSubset2 => 167.19/25.07 (binintersectSubset3 => 167.19/25.07 (binintersectER => 167.19/25.07 (disjointsetsI1 => 167.19/25.07 (binintersectRsub => 167.19/25.07 (binintersectSubset4 => 167.19/25.07 (binintersectSubset1 => 167.19/25.07 (bs114d => 167.19/25.07 (setminusI => 167.19/25.07 (setminusEL => 167.19/25.07 (setminusER => 167.19/25.07 (setminusSubset2 => 167.19/25.07 (setminusERneg => 167.19/25.07 (setminusELneg => 167.19/25.07 (setminusILneg => 167.19/25.07 (setminusIRneg => 167.19/25.07 (setminusLsub => 167.19/25.07 (setminusSubset1 => 167.19/25.07 (symdiffE => 167.19/25.07 (symdiffI1 => 167.19/25.07 (symdiffI2 => 167.19/25.07 (symdiffIneg1 => 167.19/25.07 (symdiffIneg2 => 167.19/25.07 (secondinupair => 167.19/25.07 (setukpairIL => 167.19/25.07 (setukpairIR => 167.19/25.07 (kpairiskpair => 167.19/25.07 (kpairp => 167.19/25.07 (singletonsubset => 167.19/25.07 (singletoninpowerset => 167.19/25.07 (singletoninpowunion => 167.19/25.07 (upairset2E => 167.19/25.07 (upairsubunion => 167.19/25.07 ( 167.19/25.07 upairinpowunion => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem1 => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem2 => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem3 => 167.19/25.07 ( 167.19/25.07 cartprodpairin => 167.19/25.07 ( 167.19/25.07 cartprodmempair1 => 167.19/25.07 ( 167.19/25.07 cartprodmempair => 167.19/25.07 ( 167.19/25.07 setunionE2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton1 => 167.19/25.07 ( 167.19/25.07 setunionsingleton2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton => 167.19/25.07 ( 167.19/25.07 singletonprop => 167.19/25.07 ( 167.19/25.07 ex1E1 => 167.19/25.07 ( 167.19/25.07 ex1I => 167.19/25.07 ( 167.19/25.07 ex1I2 => 167.19/25.07 ( 167.19/25.07 singletonsuniq => 167.19/25.07 ( 167.19/25.07 setukpairinjL1 => 167.19/25.07 ( 167.19/25.07 kfstsingleton => 167.19/25.07 ( 167.19/25.07 theprop => 167.19/25.07 ( 167.19/25.07 kfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodfstin => 167.19/25.07 ( 167.19/25.07 setukpairinjL2 => 167.19/25.07 ( 167.19/25.07 setukpairinjL => 167.19/25.07 ( 167.19/25.07 setukpairinjR11 => 167.19/25.07 ( 167.19/25.07 setukpairinjR12 => 167.19/25.07 ( 167.19/25.07 setukpairinjR1 => 167.19/25.07 ( 167.19/25.07 upairequniteq => 167.19/25.07 ( 167.19/25.07 setukpairinjR2 => 167.19/25.07 ( 167.19/25.07 setukpairinjR => 167.19/25.07 ( 167.19/25.07 ksndsingleton => 167.19/25.07 ( 167.19/25.07 ksndpairEq => 167.19/25.07 ( 167.19/25.07 kpairsurjEq => 167.19/25.07 ( 167.19/25.07 cartprodsndin => 167.19/25.07 ( 167.19/25.07 cartprodpairmemEL => 167.19/25.07 ( 167.19/25.07 cartprodpairmemER => 167.19/25.07 ( 167.19/25.07 cartprodmempaircEq => 167.19/25.07 ( 167.19/25.07 cartprodfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodsndpairEq => 167.19/25.07 ( 167.19/25.07 cartprodpairsurjEq => 167.19/25.07 ( 167.19/25.07 dpsetconstrI => 167.19/25.07 ( 167.19/25.07 dpsetconstrSub => 167.19/25.07 ( 167.19/25.07 setOfPairsIsBReln => 167.19/25.07 ( 167.19/25.07 !!(( 167.19/25.07 ^[Y0 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y1 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y2 : ( 167.19/25.07 $i * $i) > $o]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y3 : $i]: 167.19/25.07 (in(Y3, 167.19/25.07 Y0) => 167.19/25.07 (!!(( 167.19/25.07 ^[Y4 : $i]: 167.19/25.07 (in(Y4, 167.19/25.07 Y1) => 167.19/25.07 (in( 167.19/25.07 kpair( 167.19/25.07 Y3, Y4), 167.19/25.07 dpsetconstr( 167.19/25.07 Y0, Y1, 167.19/25.07 Y2)) => 167.19/25.07 Y2(Y3, 167.19/25.07 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.07 inference('lazy_cnf_imply', [status(thm)], ['80'])). 167.19/25.07 tff('82', plain, 167.19/25.07 ~ (subsetE2 => 167.19/25.07 (notsubsetI => 167.19/25.07 (notequalI1 => 167.19/25.07 (notequalI2 => 167.19/25.07 (subsetRefl => 167.19/25.07 (subsetTrans => 167.19/25.07 (setadjoinSub => 167.19/25.07 (setadjoinSub2 => 167.19/25.07 (subset2powerset => 167.19/25.07 (setextsub => 167.19/25.07 (subsetemptysetimpeq => 167.19/25.07 (powersetI1 => 167.19/25.07 (powersetE1 => 167.19/25.07 (inPowerset => 167.19/25.07 (powersetsubset => 167.19/25.07 (sepInPowerset => 167.19/25.07 (sepSubset => 167.19/25.07 (binunionIL => 167.19/25.07 (upairset2IR => 167.19/25.07 (binunionIR => 167.19/25.07 (binunionEcases => 167.19/25.07 (binunionE => 167.19/25.07 (binunionLsub => 167.19/25.07 (binunionRsub => 167.19/25.07 (binintersectI => 167.19/25.07 (binintersectSubset5 => 167.19/25.07 (binintersectEL => 167.19/25.07 (binintersectLsub => 167.19/25.07 (binintersectSubset2 => 167.19/25.07 (binintersectSubset3 => 167.19/25.07 (binintersectER => 167.19/25.07 (disjointsetsI1 => 167.19/25.07 (binintersectRsub => 167.19/25.07 (binintersectSubset4 => 167.19/25.07 (binintersectSubset1 => 167.19/25.07 (bs114d => 167.19/25.07 (setminusI => 167.19/25.07 (setminusEL => 167.19/25.07 (setminusER => 167.19/25.07 (setminusSubset2 => 167.19/25.07 (setminusERneg => 167.19/25.07 (setminusELneg => 167.19/25.07 (setminusILneg => 167.19/25.07 (setminusIRneg => 167.19/25.07 (setminusLsub => 167.19/25.07 (setminusSubset1 => 167.19/25.07 (symdiffE => 167.19/25.07 (symdiffI1 => 167.19/25.07 (symdiffI2 => 167.19/25.07 (symdiffIneg1 => 167.19/25.07 (symdiffIneg2 => 167.19/25.07 (secondinupair => 167.19/25.07 (setukpairIL => 167.19/25.07 (setukpairIR => 167.19/25.07 (kpairiskpair => 167.19/25.07 (kpairp => 167.19/25.07 (singletonsubset => 167.19/25.07 (singletoninpowerset => 167.19/25.07 (singletoninpowunion => 167.19/25.07 (upairset2E => 167.19/25.07 (upairsubunion => 167.19/25.07 (upairinpowunion => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem1 => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem2 => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem3 => 167.19/25.07 ( 167.19/25.07 cartprodpairin => 167.19/25.07 ( 167.19/25.07 cartprodmempair1 => 167.19/25.07 ( 167.19/25.07 cartprodmempair => 167.19/25.07 ( 167.19/25.07 setunionE2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton1 => 167.19/25.07 ( 167.19/25.07 setunionsingleton2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton => 167.19/25.07 ( 167.19/25.07 singletonprop => 167.19/25.07 ( 167.19/25.07 ex1E1 => 167.19/25.07 ( 167.19/25.07 ex1I => 167.19/25.07 ( 167.19/25.07 ex1I2 => 167.19/25.07 ( 167.19/25.07 singletonsuniq => 167.19/25.07 ( 167.19/25.07 setukpairinjL1 => 167.19/25.07 ( 167.19/25.07 kfstsingleton => 167.19/25.07 ( 167.19/25.07 theprop => 167.19/25.07 ( 167.19/25.07 kfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodfstin => 167.19/25.07 ( 167.19/25.07 setukpairinjL2 => 167.19/25.07 ( 167.19/25.07 setukpairinjL => 167.19/25.07 ( 167.19/25.07 setukpairinjR11 => 167.19/25.07 ( 167.19/25.07 setukpairinjR12 => 167.19/25.07 ( 167.19/25.07 setukpairinjR1 => 167.19/25.07 ( 167.19/25.07 upairequniteq => 167.19/25.07 ( 167.19/25.07 setukpairinjR2 => 167.19/25.07 ( 167.19/25.07 setukpairinjR => 167.19/25.07 ( 167.19/25.07 ksndsingleton => 167.19/25.07 ( 167.19/25.07 ksndpairEq => 167.19/25.07 ( 167.19/25.07 kpairsurjEq => 167.19/25.07 ( 167.19/25.07 cartprodsndin => 167.19/25.07 ( 167.19/25.07 cartprodpairmemEL => 167.19/25.07 ( 167.19/25.07 cartprodpairmemER => 167.19/25.07 ( 167.19/25.07 cartprodmempaircEq => 167.19/25.07 ( 167.19/25.07 cartprodfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodsndpairEq => 167.19/25.07 ( 167.19/25.07 cartprodpairsurjEq => 167.19/25.07 ( 167.19/25.07 dpsetconstrI => 167.19/25.07 ( 167.19/25.07 dpsetconstrSub => 167.19/25.07 ( 167.19/25.07 setOfPairsIsBReln => 167.19/25.07 ( 167.19/25.07 !!(( 167.19/25.07 ^[Y0 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y1 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y2 : ( 167.19/25.07 $i * $i) > $o]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y3 : $i]: 167.19/25.07 (in(Y3, 167.19/25.07 Y0) => 167.19/25.07 (!!(( 167.19/25.07 ^[Y4 : $i]: 167.19/25.07 (in(Y4, 167.19/25.07 Y1) => 167.19/25.07 (in( 167.19/25.07 kpair( 167.19/25.07 Y3, Y4), 167.19/25.07 dpsetconstr( 167.19/25.07 Y0, Y1, 167.19/25.07 Y2)) => 167.19/25.07 Y2(Y3, 167.19/25.07 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.07 inference('lazy_cnf_imply', [status(thm)], ['81'])). 167.19/25.07 tff('83', plain, 167.19/25.07 ~ (notsubsetI => 167.19/25.07 (notequalI1 => 167.19/25.07 (notequalI2 => 167.19/25.07 (subsetRefl => 167.19/25.07 (subsetTrans => 167.19/25.07 (setadjoinSub => 167.19/25.07 (setadjoinSub2 => 167.19/25.07 (subset2powerset => 167.19/25.07 (setextsub => 167.19/25.07 (subsetemptysetimpeq => 167.19/25.07 (powersetI1 => 167.19/25.07 (powersetE1 => 167.19/25.07 (inPowerset => 167.19/25.07 (powersetsubset => 167.19/25.07 (sepInPowerset => 167.19/25.07 (sepSubset => 167.19/25.07 (binunionIL => 167.19/25.07 (upairset2IR => 167.19/25.07 (binunionIR => 167.19/25.07 (binunionEcases => 167.19/25.07 (binunionE => 167.19/25.07 (binunionLsub => 167.19/25.07 (binunionRsub => 167.19/25.07 (binintersectI => 167.19/25.07 (binintersectSubset5 => 167.19/25.07 (binintersectEL => 167.19/25.07 (binintersectLsub => 167.19/25.07 (binintersectSubset2 => 167.19/25.07 (binintersectSubset3 => 167.19/25.07 (binintersectER => 167.19/25.07 (disjointsetsI1 => 167.19/25.07 (binintersectRsub => 167.19/25.07 (binintersectSubset4 => 167.19/25.07 (binintersectSubset1 => 167.19/25.07 (bs114d => 167.19/25.07 (setminusI => 167.19/25.07 (setminusEL => 167.19/25.07 (setminusER => 167.19/25.07 (setminusSubset2 => 167.19/25.07 (setminusERneg => 167.19/25.07 (setminusELneg => 167.19/25.07 (setminusILneg => 167.19/25.07 (setminusIRneg => 167.19/25.07 (setminusLsub => 167.19/25.07 (setminusSubset1 => 167.19/25.07 (symdiffE => 167.19/25.07 (symdiffI1 => 167.19/25.07 (symdiffI2 => 167.19/25.07 (symdiffIneg1 => 167.19/25.07 (symdiffIneg2 => 167.19/25.07 (secondinupair => 167.19/25.07 (setukpairIL => 167.19/25.07 (setukpairIR => 167.19/25.07 (kpairiskpair => 167.19/25.07 (kpairp => 167.19/25.07 (singletonsubset => 167.19/25.07 (singletoninpowerset => 167.19/25.07 (singletoninpowunion => 167.19/25.07 (upairset2E => 167.19/25.07 (upairsubunion => 167.19/25.07 (upairinpowunion => 167.19/25.07 (ubforcartprodlem1 => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem2 => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem3 => 167.19/25.07 ( 167.19/25.07 cartprodpairin => 167.19/25.07 ( 167.19/25.07 cartprodmempair1 => 167.19/25.07 ( 167.19/25.07 cartprodmempair => 167.19/25.07 ( 167.19/25.07 setunionE2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton1 => 167.19/25.07 ( 167.19/25.07 setunionsingleton2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton => 167.19/25.07 ( 167.19/25.07 singletonprop => 167.19/25.07 ( 167.19/25.07 ex1E1 => 167.19/25.07 ( 167.19/25.07 ex1I => 167.19/25.07 ( 167.19/25.07 ex1I2 => 167.19/25.07 ( 167.19/25.07 singletonsuniq => 167.19/25.07 ( 167.19/25.07 setukpairinjL1 => 167.19/25.07 ( 167.19/25.07 kfstsingleton => 167.19/25.07 ( 167.19/25.07 theprop => 167.19/25.07 ( 167.19/25.07 kfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodfstin => 167.19/25.07 ( 167.19/25.07 setukpairinjL2 => 167.19/25.07 ( 167.19/25.07 setukpairinjL => 167.19/25.07 ( 167.19/25.07 setukpairinjR11 => 167.19/25.07 ( 167.19/25.07 setukpairinjR12 => 167.19/25.07 ( 167.19/25.07 setukpairinjR1 => 167.19/25.07 ( 167.19/25.07 upairequniteq => 167.19/25.07 ( 167.19/25.07 setukpairinjR2 => 167.19/25.07 ( 167.19/25.07 setukpairinjR => 167.19/25.07 ( 167.19/25.07 ksndsingleton => 167.19/25.07 ( 167.19/25.07 ksndpairEq => 167.19/25.07 ( 167.19/25.07 kpairsurjEq => 167.19/25.07 ( 167.19/25.07 cartprodsndin => 167.19/25.07 ( 167.19/25.07 cartprodpairmemEL => 167.19/25.07 ( 167.19/25.07 cartprodpairmemER => 167.19/25.07 ( 167.19/25.07 cartprodmempaircEq => 167.19/25.07 ( 167.19/25.07 cartprodfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodsndpairEq => 167.19/25.07 ( 167.19/25.07 cartprodpairsurjEq => 167.19/25.07 ( 167.19/25.07 dpsetconstrI => 167.19/25.07 ( 167.19/25.07 dpsetconstrSub => 167.19/25.07 ( 167.19/25.07 setOfPairsIsBReln => 167.19/25.07 ( 167.19/25.07 !!(( 167.19/25.07 ^[Y0 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y1 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y2 : ( 167.19/25.07 $i * $i) > $o]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y3 : $i]: 167.19/25.07 (in(Y3, 167.19/25.07 Y0) => 167.19/25.07 (!!(( 167.19/25.07 ^[Y4 : $i]: 167.19/25.07 (in(Y4, 167.19/25.07 Y1) => 167.19/25.07 (in( 167.19/25.07 kpair( 167.19/25.07 Y3, Y4), 167.19/25.07 dpsetconstr( 167.19/25.07 Y0, Y1, 167.19/25.07 Y2)) => 167.19/25.07 Y2(Y3, 167.19/25.07 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.07 inference('lazy_cnf_imply', [status(thm)], ['82'])). 167.19/25.07 tff('84', plain, 167.19/25.07 ~ (notequalI1 => 167.19/25.07 (notequalI2 => 167.19/25.07 (subsetRefl => 167.19/25.07 (subsetTrans => 167.19/25.07 (setadjoinSub => 167.19/25.07 (setadjoinSub2 => 167.19/25.07 (subset2powerset => 167.19/25.07 (setextsub => 167.19/25.07 (subsetemptysetimpeq => 167.19/25.07 (powersetI1 => 167.19/25.07 (powersetE1 => 167.19/25.07 (inPowerset => 167.19/25.07 (powersetsubset => 167.19/25.07 (sepInPowerset => 167.19/25.07 (sepSubset => 167.19/25.07 (binunionIL => 167.19/25.07 (upairset2IR => 167.19/25.07 (binunionIR => 167.19/25.07 (binunionEcases => 167.19/25.07 (binunionE => 167.19/25.07 (binunionLsub => 167.19/25.07 (binunionRsub => 167.19/25.07 (binintersectI => 167.19/25.07 (binintersectSubset5 => 167.19/25.07 (binintersectEL => 167.19/25.07 (binintersectLsub => 167.19/25.07 (binintersectSubset2 => 167.19/25.07 (binintersectSubset3 => 167.19/25.07 (binintersectER => 167.19/25.07 (disjointsetsI1 => 167.19/25.07 (binintersectRsub => 167.19/25.07 (binintersectSubset4 => 167.19/25.07 (binintersectSubset1 => 167.19/25.07 (bs114d => 167.19/25.07 (setminusI => 167.19/25.07 (setminusEL => 167.19/25.07 (setminusER => 167.19/25.07 (setminusSubset2 => 167.19/25.07 (setminusERneg => 167.19/25.07 (setminusELneg => 167.19/25.07 (setminusILneg => 167.19/25.07 (setminusIRneg => 167.19/25.07 (setminusLsub => 167.19/25.07 (setminusSubset1 => 167.19/25.07 (symdiffE => 167.19/25.07 (symdiffI1 => 167.19/25.07 (symdiffI2 => 167.19/25.07 (symdiffIneg1 => 167.19/25.07 (symdiffIneg2 => 167.19/25.07 (secondinupair => 167.19/25.07 (setukpairIL => 167.19/25.07 (setukpairIR => 167.19/25.07 (kpairiskpair => 167.19/25.07 (kpairp => 167.19/25.07 (singletonsubset => 167.19/25.07 (singletoninpowerset => 167.19/25.07 (singletoninpowunion => 167.19/25.07 (upairset2E => 167.19/25.07 (upairsubunion => 167.19/25.07 (upairinpowunion => 167.19/25.07 (ubforcartprodlem1 => 167.19/25.07 (ubforcartprodlem2 => 167.19/25.07 ( 167.19/25.07 ubforcartprodlem3 => 167.19/25.07 ( 167.19/25.07 cartprodpairin => 167.19/25.07 ( 167.19/25.07 cartprodmempair1 => 167.19/25.07 ( 167.19/25.07 cartprodmempair => 167.19/25.07 ( 167.19/25.07 setunionE2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton1 => 167.19/25.07 ( 167.19/25.07 setunionsingleton2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton => 167.19/25.07 ( 167.19/25.07 singletonprop => 167.19/25.07 ( 167.19/25.07 ex1E1 => 167.19/25.07 ( 167.19/25.07 ex1I => 167.19/25.07 ( 167.19/25.07 ex1I2 => 167.19/25.07 ( 167.19/25.07 singletonsuniq => 167.19/25.07 ( 167.19/25.07 setukpairinjL1 => 167.19/25.07 ( 167.19/25.07 kfstsingleton => 167.19/25.07 ( 167.19/25.07 theprop => 167.19/25.07 ( 167.19/25.07 kfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodfstin => 167.19/25.07 ( 167.19/25.07 setukpairinjL2 => 167.19/25.07 ( 167.19/25.07 setukpairinjL => 167.19/25.07 ( 167.19/25.07 setukpairinjR11 => 167.19/25.07 ( 167.19/25.07 setukpairinjR12 => 167.19/25.07 ( 167.19/25.07 setukpairinjR1 => 167.19/25.07 ( 167.19/25.07 upairequniteq => 167.19/25.07 ( 167.19/25.07 setukpairinjR2 => 167.19/25.07 ( 167.19/25.07 setukpairinjR => 167.19/25.07 ( 167.19/25.07 ksndsingleton => 167.19/25.07 ( 167.19/25.07 ksndpairEq => 167.19/25.07 ( 167.19/25.07 kpairsurjEq => 167.19/25.07 ( 167.19/25.07 cartprodsndin => 167.19/25.07 ( 167.19/25.07 cartprodpairmemEL => 167.19/25.07 ( 167.19/25.07 cartprodpairmemER => 167.19/25.07 ( 167.19/25.07 cartprodmempaircEq => 167.19/25.07 ( 167.19/25.07 cartprodfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodsndpairEq => 167.19/25.07 ( 167.19/25.07 cartprodpairsurjEq => 167.19/25.07 ( 167.19/25.07 dpsetconstrI => 167.19/25.07 ( 167.19/25.07 dpsetconstrSub => 167.19/25.07 ( 167.19/25.07 setOfPairsIsBReln => 167.19/25.07 ( 167.19/25.07 !!(( 167.19/25.07 ^[Y0 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y1 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y2 : ( 167.19/25.07 $i * $i) > $o]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y3 : $i]: 167.19/25.07 (in(Y3, 167.19/25.07 Y0) => 167.19/25.07 (!!(( 167.19/25.07 ^[Y4 : $i]: 167.19/25.07 (in(Y4, 167.19/25.07 Y1) => 167.19/25.07 (in( 167.19/25.07 kpair( 167.19/25.07 Y3, Y4), 167.19/25.07 dpsetconstr( 167.19/25.07 Y0, Y1, 167.19/25.07 Y2)) => 167.19/25.07 Y2(Y3, 167.19/25.07 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.07 inference('lazy_cnf_imply', [status(thm)], ['83'])). 167.19/25.07 tff('85', plain, 167.19/25.07 ~ (notequalI2 => 167.19/25.07 (subsetRefl => 167.19/25.07 (subsetTrans => 167.19/25.07 (setadjoinSub => 167.19/25.07 (setadjoinSub2 => 167.19/25.07 (subset2powerset => 167.19/25.07 (setextsub => 167.19/25.07 (subsetemptysetimpeq => 167.19/25.07 (powersetI1 => 167.19/25.07 (powersetE1 => 167.19/25.07 (inPowerset => 167.19/25.07 (powersetsubset => 167.19/25.07 (sepInPowerset => 167.19/25.07 (sepSubset => 167.19/25.07 (binunionIL => 167.19/25.07 (upairset2IR => 167.19/25.07 (binunionIR => 167.19/25.07 (binunionEcases => 167.19/25.07 (binunionE => 167.19/25.07 (binunionLsub => 167.19/25.07 (binunionRsub => 167.19/25.07 (binintersectI => 167.19/25.07 (binintersectSubset5 => 167.19/25.07 (binintersectEL => 167.19/25.07 (binintersectLsub => 167.19/25.07 (binintersectSubset2 => 167.19/25.07 (binintersectSubset3 => 167.19/25.07 (binintersectER => 167.19/25.07 (disjointsetsI1 => 167.19/25.07 (binintersectRsub => 167.19/25.07 (binintersectSubset4 => 167.19/25.07 (binintersectSubset1 => 167.19/25.07 (bs114d => 167.19/25.07 (setminusI => 167.19/25.07 (setminusEL => 167.19/25.07 (setminusER => 167.19/25.07 (setminusSubset2 => 167.19/25.07 (setminusERneg => 167.19/25.07 (setminusELneg => 167.19/25.07 (setminusILneg => 167.19/25.07 (setminusIRneg => 167.19/25.07 (setminusLsub => 167.19/25.07 (setminusSubset1 => 167.19/25.07 (symdiffE => 167.19/25.07 (symdiffI1 => 167.19/25.07 (symdiffI2 => 167.19/25.07 (symdiffIneg1 => 167.19/25.07 (symdiffIneg2 => 167.19/25.07 (secondinupair => 167.19/25.07 (setukpairIL => 167.19/25.07 (setukpairIR => 167.19/25.07 (kpairiskpair => 167.19/25.07 (kpairp => 167.19/25.07 (singletonsubset => 167.19/25.07 (singletoninpowerset => 167.19/25.07 (singletoninpowunion => 167.19/25.07 (upairset2E => 167.19/25.07 (upairsubunion => 167.19/25.07 (upairinpowunion => 167.19/25.07 (ubforcartprodlem1 => 167.19/25.07 (ubforcartprodlem2 => 167.19/25.07 (ubforcartprodlem3 => 167.19/25.07 ( 167.19/25.07 cartprodpairin => 167.19/25.07 ( 167.19/25.07 cartprodmempair1 => 167.19/25.07 ( 167.19/25.07 cartprodmempair => 167.19/25.07 ( 167.19/25.07 setunionE2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton1 => 167.19/25.07 ( 167.19/25.07 setunionsingleton2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton => 167.19/25.07 ( 167.19/25.07 singletonprop => 167.19/25.07 ( 167.19/25.07 ex1E1 => 167.19/25.07 ( 167.19/25.07 ex1I => 167.19/25.07 ( 167.19/25.07 ex1I2 => 167.19/25.07 ( 167.19/25.07 singletonsuniq => 167.19/25.07 ( 167.19/25.07 setukpairinjL1 => 167.19/25.07 ( 167.19/25.07 kfstsingleton => 167.19/25.07 ( 167.19/25.07 theprop => 167.19/25.07 ( 167.19/25.07 kfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodfstin => 167.19/25.07 ( 167.19/25.07 setukpairinjL2 => 167.19/25.07 ( 167.19/25.07 setukpairinjL => 167.19/25.07 ( 167.19/25.07 setukpairinjR11 => 167.19/25.07 ( 167.19/25.07 setukpairinjR12 => 167.19/25.07 ( 167.19/25.07 setukpairinjR1 => 167.19/25.07 ( 167.19/25.07 upairequniteq => 167.19/25.07 ( 167.19/25.07 setukpairinjR2 => 167.19/25.07 ( 167.19/25.07 setukpairinjR => 167.19/25.07 ( 167.19/25.07 ksndsingleton => 167.19/25.07 ( 167.19/25.07 ksndpairEq => 167.19/25.07 ( 167.19/25.07 kpairsurjEq => 167.19/25.07 ( 167.19/25.07 cartprodsndin => 167.19/25.07 ( 167.19/25.07 cartprodpairmemEL => 167.19/25.07 ( 167.19/25.07 cartprodpairmemER => 167.19/25.07 ( 167.19/25.07 cartprodmempaircEq => 167.19/25.07 ( 167.19/25.07 cartprodfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodsndpairEq => 167.19/25.07 ( 167.19/25.07 cartprodpairsurjEq => 167.19/25.07 ( 167.19/25.07 dpsetconstrI => 167.19/25.07 ( 167.19/25.07 dpsetconstrSub => 167.19/25.07 ( 167.19/25.07 setOfPairsIsBReln => 167.19/25.07 ( 167.19/25.07 !!(( 167.19/25.07 ^[Y0 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y1 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y2 : ( 167.19/25.07 $i * $i) > $o]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y3 : $i]: 167.19/25.07 (in(Y3, 167.19/25.07 Y0) => 167.19/25.07 (!!(( 167.19/25.07 ^[Y4 : $i]: 167.19/25.07 (in(Y4, 167.19/25.07 Y1) => 167.19/25.07 (in( 167.19/25.07 kpair( 167.19/25.07 Y3, Y4), 167.19/25.07 dpsetconstr( 167.19/25.07 Y0, Y1, 167.19/25.07 Y2)) => 167.19/25.07 Y2(Y3, 167.19/25.07 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.07 inference('lazy_cnf_imply', [status(thm)], ['84'])). 167.19/25.07 tff('86', plain, 167.19/25.07 ~ (subsetRefl => 167.19/25.07 (subsetTrans => 167.19/25.07 (setadjoinSub => 167.19/25.07 (setadjoinSub2 => 167.19/25.07 (subset2powerset => 167.19/25.07 (setextsub => 167.19/25.07 (subsetemptysetimpeq => 167.19/25.07 (powersetI1 => 167.19/25.07 (powersetE1 => 167.19/25.07 (inPowerset => 167.19/25.07 (powersetsubset => 167.19/25.07 (sepInPowerset => 167.19/25.07 (sepSubset => 167.19/25.07 (binunionIL => 167.19/25.07 (upairset2IR => 167.19/25.07 (binunionIR => 167.19/25.07 (binunionEcases => 167.19/25.07 (binunionE => 167.19/25.07 (binunionLsub => 167.19/25.07 (binunionRsub => 167.19/25.07 (binintersectI => 167.19/25.07 (binintersectSubset5 => 167.19/25.07 (binintersectEL => 167.19/25.07 (binintersectLsub => 167.19/25.07 (binintersectSubset2 => 167.19/25.07 (binintersectSubset3 => 167.19/25.07 (binintersectER => 167.19/25.07 (disjointsetsI1 => 167.19/25.07 (binintersectRsub => 167.19/25.07 (binintersectSubset4 => 167.19/25.07 (binintersectSubset1 => 167.19/25.07 (bs114d => 167.19/25.07 (setminusI => 167.19/25.07 (setminusEL => 167.19/25.07 (setminusER => 167.19/25.07 (setminusSubset2 => 167.19/25.07 (setminusERneg => 167.19/25.07 (setminusELneg => 167.19/25.07 (setminusILneg => 167.19/25.07 (setminusIRneg => 167.19/25.07 (setminusLsub => 167.19/25.07 (setminusSubset1 => 167.19/25.07 (symdiffE => 167.19/25.07 (symdiffI1 => 167.19/25.07 (symdiffI2 => 167.19/25.07 (symdiffIneg1 => 167.19/25.07 (symdiffIneg2 => 167.19/25.07 (secondinupair => 167.19/25.07 (setukpairIL => 167.19/25.07 (setukpairIR => 167.19/25.07 (kpairiskpair => 167.19/25.07 (kpairp => 167.19/25.07 (singletonsubset => 167.19/25.07 (singletoninpowerset => 167.19/25.07 (singletoninpowunion => 167.19/25.07 (upairset2E => 167.19/25.07 (upairsubunion => 167.19/25.07 (upairinpowunion => 167.19/25.07 (ubforcartprodlem1 => 167.19/25.07 (ubforcartprodlem2 => 167.19/25.07 (ubforcartprodlem3 => 167.19/25.07 (cartprodpairin => 167.19/25.07 ( 167.19/25.07 cartprodmempair1 => 167.19/25.07 ( 167.19/25.07 cartprodmempair => 167.19/25.07 ( 167.19/25.07 setunionE2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton1 => 167.19/25.07 ( 167.19/25.07 setunionsingleton2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton => 167.19/25.07 ( 167.19/25.07 singletonprop => 167.19/25.07 ( 167.19/25.07 ex1E1 => 167.19/25.07 ( 167.19/25.07 ex1I => 167.19/25.07 ( 167.19/25.07 ex1I2 => 167.19/25.07 ( 167.19/25.07 singletonsuniq => 167.19/25.07 ( 167.19/25.07 setukpairinjL1 => 167.19/25.07 ( 167.19/25.07 kfstsingleton => 167.19/25.07 ( 167.19/25.07 theprop => 167.19/25.07 ( 167.19/25.07 kfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodfstin => 167.19/25.07 ( 167.19/25.07 setukpairinjL2 => 167.19/25.07 ( 167.19/25.07 setukpairinjL => 167.19/25.07 ( 167.19/25.07 setukpairinjR11 => 167.19/25.07 ( 167.19/25.07 setukpairinjR12 => 167.19/25.07 ( 167.19/25.07 setukpairinjR1 => 167.19/25.07 ( 167.19/25.07 upairequniteq => 167.19/25.07 ( 167.19/25.07 setukpairinjR2 => 167.19/25.07 ( 167.19/25.07 setukpairinjR => 167.19/25.07 ( 167.19/25.07 ksndsingleton => 167.19/25.07 ( 167.19/25.07 ksndpairEq => 167.19/25.07 ( 167.19/25.07 kpairsurjEq => 167.19/25.07 ( 167.19/25.07 cartprodsndin => 167.19/25.07 ( 167.19/25.07 cartprodpairmemEL => 167.19/25.07 ( 167.19/25.07 cartprodpairmemER => 167.19/25.07 ( 167.19/25.07 cartprodmempaircEq => 167.19/25.07 ( 167.19/25.07 cartprodfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodsndpairEq => 167.19/25.07 ( 167.19/25.07 cartprodpairsurjEq => 167.19/25.07 ( 167.19/25.07 dpsetconstrI => 167.19/25.07 ( 167.19/25.07 dpsetconstrSub => 167.19/25.07 ( 167.19/25.07 setOfPairsIsBReln => 167.19/25.07 ( 167.19/25.07 !!(( 167.19/25.07 ^[Y0 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y1 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y2 : ( 167.19/25.07 $i * $i) > $o]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y3 : $i]: 167.19/25.07 (in(Y3, 167.19/25.07 Y0) => 167.19/25.07 (!!(( 167.19/25.07 ^[Y4 : $i]: 167.19/25.07 (in(Y4, 167.19/25.07 Y1) => 167.19/25.07 (in( 167.19/25.07 kpair( 167.19/25.07 Y3, Y4), 167.19/25.07 dpsetconstr( 167.19/25.07 Y0, Y1, 167.19/25.07 Y2)) => 167.19/25.07 Y2(Y3, 167.19/25.07 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.07 inference('lazy_cnf_imply', [status(thm)], ['85'])). 167.19/25.07 tff('87', plain, 167.19/25.07 ~ (subsetTrans => 167.19/25.07 (setadjoinSub => 167.19/25.07 (setadjoinSub2 => 167.19/25.07 (subset2powerset => 167.19/25.07 (setextsub => 167.19/25.07 (subsetemptysetimpeq => 167.19/25.07 (powersetI1 => 167.19/25.07 (powersetE1 => 167.19/25.07 (inPowerset => 167.19/25.07 (powersetsubset => 167.19/25.07 (sepInPowerset => 167.19/25.07 (sepSubset => 167.19/25.07 (binunionIL => 167.19/25.07 (upairset2IR => 167.19/25.07 (binunionIR => 167.19/25.07 (binunionEcases => 167.19/25.07 (binunionE => 167.19/25.07 (binunionLsub => 167.19/25.07 (binunionRsub => 167.19/25.07 (binintersectI => 167.19/25.07 (binintersectSubset5 => 167.19/25.07 (binintersectEL => 167.19/25.07 (binintersectLsub => 167.19/25.07 (binintersectSubset2 => 167.19/25.07 (binintersectSubset3 => 167.19/25.07 (binintersectER => 167.19/25.07 (disjointsetsI1 => 167.19/25.07 (binintersectRsub => 167.19/25.07 (binintersectSubset4 => 167.19/25.07 (binintersectSubset1 => 167.19/25.07 (bs114d => 167.19/25.07 (setminusI => 167.19/25.07 (setminusEL => 167.19/25.07 (setminusER => 167.19/25.07 (setminusSubset2 => 167.19/25.07 (setminusERneg => 167.19/25.07 (setminusELneg => 167.19/25.07 (setminusILneg => 167.19/25.07 (setminusIRneg => 167.19/25.07 (setminusLsub => 167.19/25.07 (setminusSubset1 => 167.19/25.07 (symdiffE => 167.19/25.07 (symdiffI1 => 167.19/25.07 (symdiffI2 => 167.19/25.07 (symdiffIneg1 => 167.19/25.07 (symdiffIneg2 => 167.19/25.07 (secondinupair => 167.19/25.07 (setukpairIL => 167.19/25.07 (setukpairIR => 167.19/25.07 (kpairiskpair => 167.19/25.07 (kpairp => 167.19/25.07 (singletonsubset => 167.19/25.07 (singletoninpowerset => 167.19/25.07 (singletoninpowunion => 167.19/25.07 (upairset2E => 167.19/25.07 (upairsubunion => 167.19/25.07 (upairinpowunion => 167.19/25.07 (ubforcartprodlem1 => 167.19/25.07 (ubforcartprodlem2 => 167.19/25.07 (ubforcartprodlem3 => 167.19/25.07 (cartprodpairin => 167.19/25.07 (cartprodmempair1 => 167.19/25.07 ( 167.19/25.07 cartprodmempair => 167.19/25.07 ( 167.19/25.07 setunionE2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton1 => 167.19/25.07 ( 167.19/25.07 setunionsingleton2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton => 167.19/25.07 ( 167.19/25.07 singletonprop => 167.19/25.07 ( 167.19/25.07 ex1E1 => 167.19/25.07 ( 167.19/25.07 ex1I => 167.19/25.07 ( 167.19/25.07 ex1I2 => 167.19/25.07 ( 167.19/25.07 singletonsuniq => 167.19/25.07 ( 167.19/25.07 setukpairinjL1 => 167.19/25.07 ( 167.19/25.07 kfstsingleton => 167.19/25.07 ( 167.19/25.07 theprop => 167.19/25.07 ( 167.19/25.07 kfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodfstin => 167.19/25.07 ( 167.19/25.07 setukpairinjL2 => 167.19/25.07 ( 167.19/25.07 setukpairinjL => 167.19/25.07 ( 167.19/25.07 setukpairinjR11 => 167.19/25.07 ( 167.19/25.07 setukpairinjR12 => 167.19/25.07 ( 167.19/25.07 setukpairinjR1 => 167.19/25.07 ( 167.19/25.07 upairequniteq => 167.19/25.07 ( 167.19/25.07 setukpairinjR2 => 167.19/25.07 ( 167.19/25.07 setukpairinjR => 167.19/25.07 ( 167.19/25.07 ksndsingleton => 167.19/25.07 ( 167.19/25.07 ksndpairEq => 167.19/25.07 ( 167.19/25.07 kpairsurjEq => 167.19/25.07 ( 167.19/25.07 cartprodsndin => 167.19/25.07 ( 167.19/25.07 cartprodpairmemEL => 167.19/25.07 ( 167.19/25.07 cartprodpairmemER => 167.19/25.07 ( 167.19/25.07 cartprodmempaircEq => 167.19/25.07 ( 167.19/25.07 cartprodfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodsndpairEq => 167.19/25.07 ( 167.19/25.07 cartprodpairsurjEq => 167.19/25.07 ( 167.19/25.07 dpsetconstrI => 167.19/25.07 ( 167.19/25.07 dpsetconstrSub => 167.19/25.07 ( 167.19/25.07 setOfPairsIsBReln => 167.19/25.07 ( 167.19/25.07 !!(( 167.19/25.07 ^[Y0 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y1 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y2 : ( 167.19/25.07 $i * $i) > $o]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y3 : $i]: 167.19/25.07 (in(Y3, 167.19/25.07 Y0) => 167.19/25.07 (!!(( 167.19/25.07 ^[Y4 : $i]: 167.19/25.07 (in(Y4, 167.19/25.07 Y1) => 167.19/25.07 (in( 167.19/25.07 kpair( 167.19/25.07 Y3, Y4), 167.19/25.07 dpsetconstr( 167.19/25.07 Y0, Y1, 167.19/25.07 Y2)) => 167.19/25.07 Y2(Y3, 167.19/25.07 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.07 inference('lazy_cnf_imply', [status(thm)], ['86'])). 167.19/25.07 tff('88', plain, 167.19/25.07 ~ (setadjoinSub => 167.19/25.07 (setadjoinSub2 => 167.19/25.07 (subset2powerset => 167.19/25.07 (setextsub => 167.19/25.07 (subsetemptysetimpeq => 167.19/25.07 (powersetI1 => 167.19/25.07 (powersetE1 => 167.19/25.07 (inPowerset => 167.19/25.07 (powersetsubset => 167.19/25.07 (sepInPowerset => 167.19/25.07 (sepSubset => 167.19/25.07 (binunionIL => 167.19/25.07 (upairset2IR => 167.19/25.07 (binunionIR => 167.19/25.07 (binunionEcases => 167.19/25.07 (binunionE => 167.19/25.07 (binunionLsub => 167.19/25.07 (binunionRsub => 167.19/25.07 (binintersectI => 167.19/25.07 (binintersectSubset5 => 167.19/25.07 (binintersectEL => 167.19/25.07 (binintersectLsub => 167.19/25.07 (binintersectSubset2 => 167.19/25.07 (binintersectSubset3 => 167.19/25.07 (binintersectER => 167.19/25.07 (disjointsetsI1 => 167.19/25.07 (binintersectRsub => 167.19/25.07 (binintersectSubset4 => 167.19/25.07 (binintersectSubset1 => 167.19/25.07 (bs114d => 167.19/25.07 (setminusI => 167.19/25.07 (setminusEL => 167.19/25.07 (setminusER => 167.19/25.07 (setminusSubset2 => 167.19/25.07 (setminusERneg => 167.19/25.07 (setminusELneg => 167.19/25.07 (setminusILneg => 167.19/25.07 (setminusIRneg => 167.19/25.07 (setminusLsub => 167.19/25.07 (setminusSubset1 => 167.19/25.07 (symdiffE => 167.19/25.07 (symdiffI1 => 167.19/25.07 (symdiffI2 => 167.19/25.07 (symdiffIneg1 => 167.19/25.07 (symdiffIneg2 => 167.19/25.07 (secondinupair => 167.19/25.07 (setukpairIL => 167.19/25.07 (setukpairIR => 167.19/25.07 (kpairiskpair => 167.19/25.07 (kpairp => 167.19/25.07 (singletonsubset => 167.19/25.07 (singletoninpowerset => 167.19/25.07 (singletoninpowunion => 167.19/25.07 (upairset2E => 167.19/25.07 (upairsubunion => 167.19/25.07 (upairinpowunion => 167.19/25.07 (ubforcartprodlem1 => 167.19/25.07 (ubforcartprodlem2 => 167.19/25.07 (ubforcartprodlem3 => 167.19/25.07 (cartprodpairin => 167.19/25.07 (cartprodmempair1 => 167.19/25.07 (cartprodmempair => 167.19/25.07 ( 167.19/25.07 setunionE2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton1 => 167.19/25.07 ( 167.19/25.07 setunionsingleton2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton => 167.19/25.07 ( 167.19/25.07 singletonprop => 167.19/25.07 ( 167.19/25.07 ex1E1 => 167.19/25.07 ( 167.19/25.07 ex1I => 167.19/25.07 ( 167.19/25.07 ex1I2 => 167.19/25.07 ( 167.19/25.07 singletonsuniq => 167.19/25.07 ( 167.19/25.07 setukpairinjL1 => 167.19/25.07 ( 167.19/25.07 kfstsingleton => 167.19/25.07 ( 167.19/25.07 theprop => 167.19/25.07 ( 167.19/25.07 kfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodfstin => 167.19/25.07 ( 167.19/25.07 setukpairinjL2 => 167.19/25.07 ( 167.19/25.07 setukpairinjL => 167.19/25.07 ( 167.19/25.07 setukpairinjR11 => 167.19/25.07 ( 167.19/25.07 setukpairinjR12 => 167.19/25.07 ( 167.19/25.07 setukpairinjR1 => 167.19/25.07 ( 167.19/25.07 upairequniteq => 167.19/25.07 ( 167.19/25.07 setukpairinjR2 => 167.19/25.07 ( 167.19/25.07 setukpairinjR => 167.19/25.07 ( 167.19/25.07 ksndsingleton => 167.19/25.07 ( 167.19/25.07 ksndpairEq => 167.19/25.07 ( 167.19/25.07 kpairsurjEq => 167.19/25.07 ( 167.19/25.07 cartprodsndin => 167.19/25.07 ( 167.19/25.07 cartprodpairmemEL => 167.19/25.07 ( 167.19/25.07 cartprodpairmemER => 167.19/25.07 ( 167.19/25.07 cartprodmempaircEq => 167.19/25.07 ( 167.19/25.07 cartprodfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodsndpairEq => 167.19/25.07 ( 167.19/25.07 cartprodpairsurjEq => 167.19/25.07 ( 167.19/25.07 dpsetconstrI => 167.19/25.07 ( 167.19/25.07 dpsetconstrSub => 167.19/25.07 ( 167.19/25.07 setOfPairsIsBReln => 167.19/25.07 ( 167.19/25.07 !!(( 167.19/25.07 ^[Y0 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y1 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y2 : ( 167.19/25.07 $i * $i) > $o]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y3 : $i]: 167.19/25.07 (in(Y3, 167.19/25.07 Y0) => 167.19/25.07 (!!(( 167.19/25.07 ^[Y4 : $i]: 167.19/25.07 (in(Y4, 167.19/25.07 Y1) => 167.19/25.07 (in( 167.19/25.07 kpair( 167.19/25.07 Y3, Y4), 167.19/25.07 dpsetconstr( 167.19/25.07 Y0, Y1, 167.19/25.07 Y2)) => 167.19/25.07 Y2(Y3, 167.19/25.07 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.07 inference('lazy_cnf_imply', [status(thm)], ['87'])). 167.19/25.07 tff('89', plain, 167.19/25.07 ~ (setadjoinSub2 => 167.19/25.07 (subset2powerset => 167.19/25.07 (setextsub => 167.19/25.07 (subsetemptysetimpeq => 167.19/25.07 (powersetI1 => 167.19/25.07 (powersetE1 => 167.19/25.07 (inPowerset => 167.19/25.07 (powersetsubset => 167.19/25.07 (sepInPowerset => 167.19/25.07 (sepSubset => 167.19/25.07 (binunionIL => 167.19/25.07 (upairset2IR => 167.19/25.07 (binunionIR => 167.19/25.07 (binunionEcases => 167.19/25.07 (binunionE => 167.19/25.07 (binunionLsub => 167.19/25.07 (binunionRsub => 167.19/25.07 (binintersectI => 167.19/25.07 (binintersectSubset5 => 167.19/25.07 (binintersectEL => 167.19/25.07 (binintersectLsub => 167.19/25.07 (binintersectSubset2 => 167.19/25.07 (binintersectSubset3 => 167.19/25.07 (binintersectER => 167.19/25.07 (disjointsetsI1 => 167.19/25.07 (binintersectRsub => 167.19/25.07 (binintersectSubset4 => 167.19/25.07 (binintersectSubset1 => 167.19/25.07 (bs114d => 167.19/25.07 (setminusI => 167.19/25.07 (setminusEL => 167.19/25.07 (setminusER => 167.19/25.07 (setminusSubset2 => 167.19/25.07 (setminusERneg => 167.19/25.07 (setminusELneg => 167.19/25.07 (setminusILneg => 167.19/25.07 (setminusIRneg => 167.19/25.07 (setminusLsub => 167.19/25.07 (setminusSubset1 => 167.19/25.07 (symdiffE => 167.19/25.07 (symdiffI1 => 167.19/25.07 (symdiffI2 => 167.19/25.07 (symdiffIneg1 => 167.19/25.07 (symdiffIneg2 => 167.19/25.07 (secondinupair => 167.19/25.07 (setukpairIL => 167.19/25.07 (setukpairIR => 167.19/25.07 (kpairiskpair => 167.19/25.07 (kpairp => 167.19/25.07 (singletonsubset => 167.19/25.07 (singletoninpowerset => 167.19/25.07 (singletoninpowunion => 167.19/25.07 (upairset2E => 167.19/25.07 (upairsubunion => 167.19/25.07 (upairinpowunion => 167.19/25.07 (ubforcartprodlem1 => 167.19/25.07 (ubforcartprodlem2 => 167.19/25.07 (ubforcartprodlem3 => 167.19/25.07 (cartprodpairin => 167.19/25.07 (cartprodmempair1 => 167.19/25.07 (cartprodmempair => 167.19/25.07 (setunionE2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton1 => 167.19/25.07 ( 167.19/25.07 setunionsingleton2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton => 167.19/25.07 ( 167.19/25.07 singletonprop => 167.19/25.07 ( 167.19/25.07 ex1E1 => 167.19/25.07 ( 167.19/25.07 ex1I => 167.19/25.07 ( 167.19/25.07 ex1I2 => 167.19/25.07 ( 167.19/25.07 singletonsuniq => 167.19/25.07 ( 167.19/25.07 setukpairinjL1 => 167.19/25.07 ( 167.19/25.07 kfstsingleton => 167.19/25.07 ( 167.19/25.07 theprop => 167.19/25.07 ( 167.19/25.07 kfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodfstin => 167.19/25.07 ( 167.19/25.07 setukpairinjL2 => 167.19/25.07 ( 167.19/25.07 setukpairinjL => 167.19/25.07 ( 167.19/25.07 setukpairinjR11 => 167.19/25.07 ( 167.19/25.07 setukpairinjR12 => 167.19/25.07 ( 167.19/25.07 setukpairinjR1 => 167.19/25.07 ( 167.19/25.07 upairequniteq => 167.19/25.07 ( 167.19/25.07 setukpairinjR2 => 167.19/25.07 ( 167.19/25.07 setukpairinjR => 167.19/25.07 ( 167.19/25.07 ksndsingleton => 167.19/25.07 ( 167.19/25.07 ksndpairEq => 167.19/25.07 ( 167.19/25.07 kpairsurjEq => 167.19/25.07 ( 167.19/25.07 cartprodsndin => 167.19/25.07 ( 167.19/25.07 cartprodpairmemEL => 167.19/25.07 ( 167.19/25.07 cartprodpairmemER => 167.19/25.07 ( 167.19/25.07 cartprodmempaircEq => 167.19/25.07 ( 167.19/25.07 cartprodfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodsndpairEq => 167.19/25.07 ( 167.19/25.07 cartprodpairsurjEq => 167.19/25.07 ( 167.19/25.07 dpsetconstrI => 167.19/25.07 ( 167.19/25.07 dpsetconstrSub => 167.19/25.07 ( 167.19/25.07 setOfPairsIsBReln => 167.19/25.07 ( 167.19/25.07 !!(( 167.19/25.07 ^[Y0 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y1 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y2 : ( 167.19/25.07 $i * $i) > $o]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y3 : $i]: 167.19/25.07 (in(Y3, 167.19/25.07 Y0) => 167.19/25.07 (!!(( 167.19/25.07 ^[Y4 : $i]: 167.19/25.07 (in(Y4, 167.19/25.07 Y1) => 167.19/25.07 (in( 167.19/25.07 kpair( 167.19/25.07 Y3, Y4), 167.19/25.07 dpsetconstr( 167.19/25.07 Y0, Y1, 167.19/25.07 Y2)) => 167.19/25.07 Y2(Y3, 167.19/25.07 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.07 inference('lazy_cnf_imply', [status(thm)], ['88'])). 167.19/25.07 tff('90', plain, 167.19/25.07 ~ (subset2powerset => 167.19/25.07 (setextsub => 167.19/25.07 (subsetemptysetimpeq => 167.19/25.07 (powersetI1 => 167.19/25.07 (powersetE1 => 167.19/25.07 (inPowerset => 167.19/25.07 (powersetsubset => 167.19/25.07 (sepInPowerset => 167.19/25.07 (sepSubset => 167.19/25.07 (binunionIL => 167.19/25.07 (upairset2IR => 167.19/25.07 (binunionIR => 167.19/25.07 (binunionEcases => 167.19/25.07 (binunionE => 167.19/25.07 (binunionLsub => 167.19/25.07 (binunionRsub => 167.19/25.07 (binintersectI => 167.19/25.07 (binintersectSubset5 => 167.19/25.07 (binintersectEL => 167.19/25.07 (binintersectLsub => 167.19/25.07 (binintersectSubset2 => 167.19/25.07 (binintersectSubset3 => 167.19/25.07 (binintersectER => 167.19/25.07 (disjointsetsI1 => 167.19/25.07 (binintersectRsub => 167.19/25.07 (binintersectSubset4 => 167.19/25.07 (binintersectSubset1 => 167.19/25.07 (bs114d => 167.19/25.07 (setminusI => 167.19/25.07 (setminusEL => 167.19/25.07 (setminusER => 167.19/25.07 (setminusSubset2 => 167.19/25.07 (setminusERneg => 167.19/25.07 (setminusELneg => 167.19/25.07 (setminusILneg => 167.19/25.07 (setminusIRneg => 167.19/25.07 (setminusLsub => 167.19/25.07 (setminusSubset1 => 167.19/25.07 (symdiffE => 167.19/25.07 (symdiffI1 => 167.19/25.07 (symdiffI2 => 167.19/25.07 (symdiffIneg1 => 167.19/25.07 (symdiffIneg2 => 167.19/25.07 (secondinupair => 167.19/25.07 (setukpairIL => 167.19/25.07 (setukpairIR => 167.19/25.07 (kpairiskpair => 167.19/25.07 (kpairp => 167.19/25.07 (singletonsubset => 167.19/25.07 (singletoninpowerset => 167.19/25.07 (singletoninpowunion => 167.19/25.07 (upairset2E => 167.19/25.07 (upairsubunion => 167.19/25.07 (upairinpowunion => 167.19/25.07 (ubforcartprodlem1 => 167.19/25.07 (ubforcartprodlem2 => 167.19/25.07 (ubforcartprodlem3 => 167.19/25.07 (cartprodpairin => 167.19/25.07 (cartprodmempair1 => 167.19/25.07 (cartprodmempair => 167.19/25.07 (setunionE2 => 167.19/25.07 (setunionsingleton1 => 167.19/25.07 ( 167.19/25.07 setunionsingleton2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton => 167.19/25.07 ( 167.19/25.07 singletonprop => 167.19/25.07 ( 167.19/25.07 ex1E1 => 167.19/25.07 ( 167.19/25.07 ex1I => 167.19/25.07 ( 167.19/25.07 ex1I2 => 167.19/25.07 ( 167.19/25.07 singletonsuniq => 167.19/25.07 ( 167.19/25.07 setukpairinjL1 => 167.19/25.07 ( 167.19/25.07 kfstsingleton => 167.19/25.07 ( 167.19/25.07 theprop => 167.19/25.07 ( 167.19/25.07 kfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodfstin => 167.19/25.07 ( 167.19/25.07 setukpairinjL2 => 167.19/25.07 ( 167.19/25.07 setukpairinjL => 167.19/25.07 ( 167.19/25.07 setukpairinjR11 => 167.19/25.07 ( 167.19/25.07 setukpairinjR12 => 167.19/25.07 ( 167.19/25.07 setukpairinjR1 => 167.19/25.07 ( 167.19/25.07 upairequniteq => 167.19/25.07 ( 167.19/25.07 setukpairinjR2 => 167.19/25.07 ( 167.19/25.07 setukpairinjR => 167.19/25.07 ( 167.19/25.07 ksndsingleton => 167.19/25.07 ( 167.19/25.07 ksndpairEq => 167.19/25.07 ( 167.19/25.07 kpairsurjEq => 167.19/25.07 ( 167.19/25.07 cartprodsndin => 167.19/25.07 ( 167.19/25.07 cartprodpairmemEL => 167.19/25.07 ( 167.19/25.07 cartprodpairmemER => 167.19/25.07 ( 167.19/25.07 cartprodmempaircEq => 167.19/25.07 ( 167.19/25.07 cartprodfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodsndpairEq => 167.19/25.07 ( 167.19/25.07 cartprodpairsurjEq => 167.19/25.07 ( 167.19/25.07 dpsetconstrI => 167.19/25.07 ( 167.19/25.07 dpsetconstrSub => 167.19/25.07 ( 167.19/25.07 setOfPairsIsBReln => 167.19/25.07 ( 167.19/25.07 !!(( 167.19/25.07 ^[Y0 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y1 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y2 : ( 167.19/25.07 $i * $i) > $o]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y3 : $i]: 167.19/25.07 (in(Y3, 167.19/25.07 Y0) => 167.19/25.07 (!!(( 167.19/25.07 ^[Y4 : $i]: 167.19/25.07 (in(Y4, 167.19/25.07 Y1) => 167.19/25.07 (in( 167.19/25.07 kpair( 167.19/25.07 Y3, Y4), 167.19/25.07 dpsetconstr( 167.19/25.07 Y0, Y1, 167.19/25.07 Y2)) => 167.19/25.07 Y2(Y3, 167.19/25.07 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.07 inference('lazy_cnf_imply', [status(thm)], ['89'])). 167.19/25.07 tff('91', plain, 167.19/25.07 ~ (setextsub => 167.19/25.07 (subsetemptysetimpeq => 167.19/25.07 (powersetI1 => 167.19/25.07 (powersetE1 => 167.19/25.07 (inPowerset => 167.19/25.07 (powersetsubset => 167.19/25.07 (sepInPowerset => 167.19/25.07 (sepSubset => 167.19/25.07 (binunionIL => 167.19/25.07 (upairset2IR => 167.19/25.07 (binunionIR => 167.19/25.07 (binunionEcases => 167.19/25.07 (binunionE => 167.19/25.07 (binunionLsub => 167.19/25.07 (binunionRsub => 167.19/25.07 (binintersectI => 167.19/25.07 (binintersectSubset5 => 167.19/25.07 (binintersectEL => 167.19/25.07 (binintersectLsub => 167.19/25.07 (binintersectSubset2 => 167.19/25.07 (binintersectSubset3 => 167.19/25.07 (binintersectER => 167.19/25.07 (disjointsetsI1 => 167.19/25.07 (binintersectRsub => 167.19/25.07 (binintersectSubset4 => 167.19/25.07 (binintersectSubset1 => 167.19/25.07 (bs114d => 167.19/25.07 (setminusI => 167.19/25.07 (setminusEL => 167.19/25.07 (setminusER => 167.19/25.07 (setminusSubset2 => 167.19/25.07 (setminusERneg => 167.19/25.07 (setminusELneg => 167.19/25.07 (setminusILneg => 167.19/25.07 (setminusIRneg => 167.19/25.07 (setminusLsub => 167.19/25.07 (setminusSubset1 => 167.19/25.07 (symdiffE => 167.19/25.07 (symdiffI1 => 167.19/25.07 (symdiffI2 => 167.19/25.07 (symdiffIneg1 => 167.19/25.07 (symdiffIneg2 => 167.19/25.07 (secondinupair => 167.19/25.07 (setukpairIL => 167.19/25.07 (setukpairIR => 167.19/25.07 (kpairiskpair => 167.19/25.07 (kpairp => 167.19/25.07 (singletonsubset => 167.19/25.07 (singletoninpowerset => 167.19/25.07 (singletoninpowunion => 167.19/25.07 (upairset2E => 167.19/25.07 (upairsubunion => 167.19/25.07 (upairinpowunion => 167.19/25.07 (ubforcartprodlem1 => 167.19/25.07 (ubforcartprodlem2 => 167.19/25.07 (ubforcartprodlem3 => 167.19/25.07 (cartprodpairin => 167.19/25.07 (cartprodmempair1 => 167.19/25.07 (cartprodmempair => 167.19/25.07 (setunionE2 => 167.19/25.07 (setunionsingleton1 => 167.19/25.07 (setunionsingleton2 => 167.19/25.07 ( 167.19/25.07 setunionsingleton => 167.19/25.07 ( 167.19/25.07 singletonprop => 167.19/25.07 ( 167.19/25.07 ex1E1 => 167.19/25.07 ( 167.19/25.07 ex1I => 167.19/25.07 ( 167.19/25.07 ex1I2 => 167.19/25.07 ( 167.19/25.07 singletonsuniq => 167.19/25.07 ( 167.19/25.07 setukpairinjL1 => 167.19/25.07 ( 167.19/25.07 kfstsingleton => 167.19/25.07 ( 167.19/25.07 theprop => 167.19/25.07 ( 167.19/25.07 kfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodfstin => 167.19/25.07 ( 167.19/25.07 setukpairinjL2 => 167.19/25.07 ( 167.19/25.07 setukpairinjL => 167.19/25.07 ( 167.19/25.07 setukpairinjR11 => 167.19/25.07 ( 167.19/25.07 setukpairinjR12 => 167.19/25.07 ( 167.19/25.07 setukpairinjR1 => 167.19/25.07 ( 167.19/25.07 upairequniteq => 167.19/25.07 ( 167.19/25.07 setukpairinjR2 => 167.19/25.07 ( 167.19/25.07 setukpairinjR => 167.19/25.07 ( 167.19/25.07 ksndsingleton => 167.19/25.07 ( 167.19/25.07 ksndpairEq => 167.19/25.07 ( 167.19/25.07 kpairsurjEq => 167.19/25.07 ( 167.19/25.07 cartprodsndin => 167.19/25.07 ( 167.19/25.07 cartprodpairmemEL => 167.19/25.07 ( 167.19/25.07 cartprodpairmemER => 167.19/25.07 ( 167.19/25.07 cartprodmempaircEq => 167.19/25.07 ( 167.19/25.07 cartprodfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodsndpairEq => 167.19/25.07 ( 167.19/25.07 cartprodpairsurjEq => 167.19/25.07 ( 167.19/25.07 dpsetconstrI => 167.19/25.07 ( 167.19/25.07 dpsetconstrSub => 167.19/25.07 ( 167.19/25.07 setOfPairsIsBReln => 167.19/25.07 ( 167.19/25.07 !!(( 167.19/25.07 ^[Y0 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y1 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y2 : ( 167.19/25.07 $i * $i) > $o]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y3 : $i]: 167.19/25.07 (in(Y3, 167.19/25.07 Y0) => 167.19/25.07 (!!(( 167.19/25.07 ^[Y4 : $i]: 167.19/25.07 (in(Y4, 167.19/25.07 Y1) => 167.19/25.07 (in( 167.19/25.07 kpair( 167.19/25.07 Y3, Y4), 167.19/25.07 dpsetconstr( 167.19/25.07 Y0, Y1, 167.19/25.07 Y2)) => 167.19/25.07 Y2(Y3, 167.19/25.07 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.07 inference('lazy_cnf_imply', [status(thm)], ['90'])). 167.19/25.07 tff('92', plain, 167.19/25.07 ~ (subsetemptysetimpeq => 167.19/25.07 (powersetI1 => 167.19/25.07 (powersetE1 => 167.19/25.07 (inPowerset => 167.19/25.07 (powersetsubset => 167.19/25.07 (sepInPowerset => 167.19/25.07 (sepSubset => 167.19/25.07 (binunionIL => 167.19/25.07 (upairset2IR => 167.19/25.07 (binunionIR => 167.19/25.07 (binunionEcases => 167.19/25.07 (binunionE => 167.19/25.07 (binunionLsub => 167.19/25.07 (binunionRsub => 167.19/25.07 (binintersectI => 167.19/25.07 (binintersectSubset5 => 167.19/25.07 (binintersectEL => 167.19/25.07 (binintersectLsub => 167.19/25.07 (binintersectSubset2 => 167.19/25.07 (binintersectSubset3 => 167.19/25.07 (binintersectER => 167.19/25.07 (disjointsetsI1 => 167.19/25.07 (binintersectRsub => 167.19/25.07 (binintersectSubset4 => 167.19/25.07 (binintersectSubset1 => 167.19/25.07 (bs114d => 167.19/25.07 (setminusI => 167.19/25.07 (setminusEL => 167.19/25.07 (setminusER => 167.19/25.07 (setminusSubset2 => 167.19/25.07 (setminusERneg => 167.19/25.07 (setminusELneg => 167.19/25.07 (setminusILneg => 167.19/25.07 (setminusIRneg => 167.19/25.07 (setminusLsub => 167.19/25.07 (setminusSubset1 => 167.19/25.07 (symdiffE => 167.19/25.07 (symdiffI1 => 167.19/25.07 (symdiffI2 => 167.19/25.07 (symdiffIneg1 => 167.19/25.07 (symdiffIneg2 => 167.19/25.07 (secondinupair => 167.19/25.07 (setukpairIL => 167.19/25.07 (setukpairIR => 167.19/25.07 (kpairiskpair => 167.19/25.07 (kpairp => 167.19/25.07 (singletonsubset => 167.19/25.07 (singletoninpowerset => 167.19/25.07 (singletoninpowunion => 167.19/25.07 (upairset2E => 167.19/25.07 (upairsubunion => 167.19/25.07 (upairinpowunion => 167.19/25.07 (ubforcartprodlem1 => 167.19/25.07 (ubforcartprodlem2 => 167.19/25.07 (ubforcartprodlem3 => 167.19/25.07 (cartprodpairin => 167.19/25.07 (cartprodmempair1 => 167.19/25.07 (cartprodmempair => 167.19/25.07 (setunionE2 => 167.19/25.07 (setunionsingleton1 => 167.19/25.07 (setunionsingleton2 => 167.19/25.07 (setunionsingleton => 167.19/25.07 ( 167.19/25.07 singletonprop => 167.19/25.07 ( 167.19/25.07 ex1E1 => 167.19/25.07 ( 167.19/25.07 ex1I => 167.19/25.07 ( 167.19/25.07 ex1I2 => 167.19/25.07 ( 167.19/25.07 singletonsuniq => 167.19/25.07 ( 167.19/25.07 setukpairinjL1 => 167.19/25.07 ( 167.19/25.07 kfstsingleton => 167.19/25.07 ( 167.19/25.07 theprop => 167.19/25.07 ( 167.19/25.07 kfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodfstin => 167.19/25.07 ( 167.19/25.07 setukpairinjL2 => 167.19/25.07 ( 167.19/25.07 setukpairinjL => 167.19/25.07 ( 167.19/25.07 setukpairinjR11 => 167.19/25.07 ( 167.19/25.07 setukpairinjR12 => 167.19/25.07 ( 167.19/25.07 setukpairinjR1 => 167.19/25.07 ( 167.19/25.07 upairequniteq => 167.19/25.07 ( 167.19/25.07 setukpairinjR2 => 167.19/25.07 ( 167.19/25.07 setukpairinjR => 167.19/25.07 ( 167.19/25.07 ksndsingleton => 167.19/25.07 ( 167.19/25.07 ksndpairEq => 167.19/25.07 ( 167.19/25.07 kpairsurjEq => 167.19/25.07 ( 167.19/25.07 cartprodsndin => 167.19/25.07 ( 167.19/25.07 cartprodpairmemEL => 167.19/25.07 ( 167.19/25.07 cartprodpairmemER => 167.19/25.07 ( 167.19/25.07 cartprodmempaircEq => 167.19/25.07 ( 167.19/25.07 cartprodfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodsndpairEq => 167.19/25.07 ( 167.19/25.07 cartprodpairsurjEq => 167.19/25.07 ( 167.19/25.07 dpsetconstrI => 167.19/25.07 ( 167.19/25.07 dpsetconstrSub => 167.19/25.07 ( 167.19/25.07 setOfPairsIsBReln => 167.19/25.07 ( 167.19/25.07 !!(( 167.19/25.07 ^[Y0 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y1 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y2 : ( 167.19/25.07 $i * $i) > $o]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y3 : $i]: 167.19/25.07 (in(Y3, 167.19/25.07 Y0) => 167.19/25.07 (!!(( 167.19/25.07 ^[Y4 : $i]: 167.19/25.07 (in(Y4, 167.19/25.07 Y1) => 167.19/25.07 (in( 167.19/25.07 kpair( 167.19/25.07 Y3, Y4), 167.19/25.07 dpsetconstr( 167.19/25.07 Y0, Y1, 167.19/25.07 Y2)) => 167.19/25.07 Y2(Y3, 167.19/25.07 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.07 inference('lazy_cnf_imply', [status(thm)], ['91'])). 167.19/25.07 tff('93', plain, 167.19/25.07 ~ (powersetI1 => 167.19/25.07 (powersetE1 => 167.19/25.07 (inPowerset => 167.19/25.07 (powersetsubset => 167.19/25.07 (sepInPowerset => 167.19/25.07 (sepSubset => 167.19/25.07 (binunionIL => 167.19/25.07 (upairset2IR => 167.19/25.07 (binunionIR => 167.19/25.07 (binunionEcases => 167.19/25.07 (binunionE => 167.19/25.07 (binunionLsub => 167.19/25.07 (binunionRsub => 167.19/25.07 (binintersectI => 167.19/25.07 (binintersectSubset5 => 167.19/25.07 (binintersectEL => 167.19/25.07 (binintersectLsub => 167.19/25.07 (binintersectSubset2 => 167.19/25.07 (binintersectSubset3 => 167.19/25.07 (binintersectER => 167.19/25.07 (disjointsetsI1 => 167.19/25.07 (binintersectRsub => 167.19/25.07 (binintersectSubset4 => 167.19/25.07 (binintersectSubset1 => 167.19/25.07 (bs114d => 167.19/25.07 (setminusI => 167.19/25.07 (setminusEL => 167.19/25.07 (setminusER => 167.19/25.07 (setminusSubset2 => 167.19/25.07 (setminusERneg => 167.19/25.07 (setminusELneg => 167.19/25.07 (setminusILneg => 167.19/25.07 (setminusIRneg => 167.19/25.07 (setminusLsub => 167.19/25.07 (setminusSubset1 => 167.19/25.07 (symdiffE => 167.19/25.07 (symdiffI1 => 167.19/25.07 (symdiffI2 => 167.19/25.07 (symdiffIneg1 => 167.19/25.07 (symdiffIneg2 => 167.19/25.07 (secondinupair => 167.19/25.07 (setukpairIL => 167.19/25.07 (setukpairIR => 167.19/25.07 (kpairiskpair => 167.19/25.07 (kpairp => 167.19/25.07 (singletonsubset => 167.19/25.07 (singletoninpowerset => 167.19/25.07 (singletoninpowunion => 167.19/25.07 (upairset2E => 167.19/25.07 (upairsubunion => 167.19/25.07 (upairinpowunion => 167.19/25.07 (ubforcartprodlem1 => 167.19/25.07 (ubforcartprodlem2 => 167.19/25.07 (ubforcartprodlem3 => 167.19/25.07 (cartprodpairin => 167.19/25.07 (cartprodmempair1 => 167.19/25.07 (cartprodmempair => 167.19/25.07 (setunionE2 => 167.19/25.07 (setunionsingleton1 => 167.19/25.07 (setunionsingleton2 => 167.19/25.07 (setunionsingleton => 167.19/25.07 (singletonprop => 167.19/25.07 ( 167.19/25.07 ex1E1 => 167.19/25.07 ( 167.19/25.07 ex1I => 167.19/25.07 ( 167.19/25.07 ex1I2 => 167.19/25.07 ( 167.19/25.07 singletonsuniq => 167.19/25.07 ( 167.19/25.07 setukpairinjL1 => 167.19/25.07 ( 167.19/25.07 kfstsingleton => 167.19/25.07 ( 167.19/25.07 theprop => 167.19/25.07 ( 167.19/25.07 kfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodfstin => 167.19/25.07 ( 167.19/25.07 setukpairinjL2 => 167.19/25.07 ( 167.19/25.07 setukpairinjL => 167.19/25.07 ( 167.19/25.07 setukpairinjR11 => 167.19/25.07 ( 167.19/25.07 setukpairinjR12 => 167.19/25.07 ( 167.19/25.07 setukpairinjR1 => 167.19/25.07 ( 167.19/25.07 upairequniteq => 167.19/25.07 ( 167.19/25.07 setukpairinjR2 => 167.19/25.07 ( 167.19/25.07 setukpairinjR => 167.19/25.07 ( 167.19/25.07 ksndsingleton => 167.19/25.07 ( 167.19/25.07 ksndpairEq => 167.19/25.07 ( 167.19/25.07 kpairsurjEq => 167.19/25.07 ( 167.19/25.07 cartprodsndin => 167.19/25.07 ( 167.19/25.07 cartprodpairmemEL => 167.19/25.07 ( 167.19/25.07 cartprodpairmemER => 167.19/25.07 ( 167.19/25.07 cartprodmempaircEq => 167.19/25.07 ( 167.19/25.07 cartprodfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodsndpairEq => 167.19/25.07 ( 167.19/25.07 cartprodpairsurjEq => 167.19/25.07 ( 167.19/25.07 dpsetconstrI => 167.19/25.07 ( 167.19/25.07 dpsetconstrSub => 167.19/25.07 ( 167.19/25.07 setOfPairsIsBReln => 167.19/25.07 ( 167.19/25.07 !!(( 167.19/25.07 ^[Y0 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y1 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y2 : ( 167.19/25.07 $i * $i) > $o]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y3 : $i]: 167.19/25.07 (in(Y3, 167.19/25.07 Y0) => 167.19/25.07 (!!(( 167.19/25.07 ^[Y4 : $i]: 167.19/25.07 (in(Y4, 167.19/25.07 Y1) => 167.19/25.07 (in( 167.19/25.07 kpair( 167.19/25.07 Y3, Y4), 167.19/25.07 dpsetconstr( 167.19/25.07 Y0, Y1, 167.19/25.07 Y2)) => 167.19/25.07 Y2(Y3, 167.19/25.07 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.07 inference('lazy_cnf_imply', [status(thm)], ['92'])). 167.19/25.07 tff('94', plain, 167.19/25.07 ~ (powersetE1 => 167.19/25.07 (inPowerset => 167.19/25.07 (powersetsubset => 167.19/25.07 (sepInPowerset => 167.19/25.07 (sepSubset => 167.19/25.07 (binunionIL => 167.19/25.07 (upairset2IR => 167.19/25.07 (binunionIR => 167.19/25.07 (binunionEcases => 167.19/25.07 (binunionE => 167.19/25.07 (binunionLsub => 167.19/25.07 (binunionRsub => 167.19/25.07 (binintersectI => 167.19/25.07 (binintersectSubset5 => 167.19/25.07 (binintersectEL => 167.19/25.07 (binintersectLsub => 167.19/25.07 (binintersectSubset2 => 167.19/25.07 (binintersectSubset3 => 167.19/25.07 (binintersectER => 167.19/25.07 (disjointsetsI1 => 167.19/25.07 (binintersectRsub => 167.19/25.07 (binintersectSubset4 => 167.19/25.07 (binintersectSubset1 => 167.19/25.07 (bs114d => 167.19/25.07 (setminusI => 167.19/25.07 (setminusEL => 167.19/25.07 (setminusER => 167.19/25.07 (setminusSubset2 => 167.19/25.07 (setminusERneg => 167.19/25.07 (setminusELneg => 167.19/25.07 (setminusILneg => 167.19/25.07 (setminusIRneg => 167.19/25.07 (setminusLsub => 167.19/25.07 (setminusSubset1 => 167.19/25.07 (symdiffE => 167.19/25.07 (symdiffI1 => 167.19/25.07 (symdiffI2 => 167.19/25.07 (symdiffIneg1 => 167.19/25.07 (symdiffIneg2 => 167.19/25.07 (secondinupair => 167.19/25.07 (setukpairIL => 167.19/25.07 (setukpairIR => 167.19/25.07 (kpairiskpair => 167.19/25.07 (kpairp => 167.19/25.07 (singletonsubset => 167.19/25.07 (singletoninpowerset => 167.19/25.07 (singletoninpowunion => 167.19/25.07 (upairset2E => 167.19/25.07 (upairsubunion => 167.19/25.07 (upairinpowunion => 167.19/25.07 (ubforcartprodlem1 => 167.19/25.07 (ubforcartprodlem2 => 167.19/25.07 (ubforcartprodlem3 => 167.19/25.07 (cartprodpairin => 167.19/25.07 (cartprodmempair1 => 167.19/25.07 (cartprodmempair => 167.19/25.07 (setunionE2 => 167.19/25.07 (setunionsingleton1 => 167.19/25.07 (setunionsingleton2 => 167.19/25.07 (setunionsingleton => 167.19/25.07 (singletonprop => 167.19/25.07 (ex1E1 => 167.19/25.07 ( 167.19/25.07 ex1I => 167.19/25.07 ( 167.19/25.07 ex1I2 => 167.19/25.07 ( 167.19/25.07 singletonsuniq => 167.19/25.07 ( 167.19/25.07 setukpairinjL1 => 167.19/25.07 ( 167.19/25.07 kfstsingleton => 167.19/25.07 ( 167.19/25.07 theprop => 167.19/25.07 ( 167.19/25.07 kfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodfstin => 167.19/25.07 ( 167.19/25.07 setukpairinjL2 => 167.19/25.07 ( 167.19/25.07 setukpairinjL => 167.19/25.07 ( 167.19/25.07 setukpairinjR11 => 167.19/25.07 ( 167.19/25.07 setukpairinjR12 => 167.19/25.07 ( 167.19/25.07 setukpairinjR1 => 167.19/25.07 ( 167.19/25.07 upairequniteq => 167.19/25.07 ( 167.19/25.07 setukpairinjR2 => 167.19/25.07 ( 167.19/25.07 setukpairinjR => 167.19/25.07 ( 167.19/25.07 ksndsingleton => 167.19/25.07 ( 167.19/25.07 ksndpairEq => 167.19/25.07 ( 167.19/25.07 kpairsurjEq => 167.19/25.07 ( 167.19/25.07 cartprodsndin => 167.19/25.07 ( 167.19/25.07 cartprodpairmemEL => 167.19/25.07 ( 167.19/25.07 cartprodpairmemER => 167.19/25.07 ( 167.19/25.07 cartprodmempaircEq => 167.19/25.07 ( 167.19/25.07 cartprodfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodsndpairEq => 167.19/25.07 ( 167.19/25.07 cartprodpairsurjEq => 167.19/25.07 ( 167.19/25.07 dpsetconstrI => 167.19/25.07 ( 167.19/25.07 dpsetconstrSub => 167.19/25.07 ( 167.19/25.07 setOfPairsIsBReln => 167.19/25.07 ( 167.19/25.07 !!(( 167.19/25.07 ^[Y0 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y1 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y2 : ( 167.19/25.07 $i * $i) > $o]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y3 : $i]: 167.19/25.07 (in(Y3, 167.19/25.07 Y0) => 167.19/25.07 (!!(( 167.19/25.07 ^[Y4 : $i]: 167.19/25.07 (in(Y4, 167.19/25.07 Y1) => 167.19/25.07 (in( 167.19/25.07 kpair( 167.19/25.07 Y3, Y4), 167.19/25.07 dpsetconstr( 167.19/25.07 Y0, Y1, 167.19/25.07 Y2)) => 167.19/25.07 Y2(Y3, 167.19/25.07 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.07 inference('lazy_cnf_imply', [status(thm)], ['93'])). 167.19/25.07 tff('95', plain, 167.19/25.07 ~ (inPowerset => 167.19/25.07 (powersetsubset => 167.19/25.07 (sepInPowerset => 167.19/25.07 (sepSubset => 167.19/25.07 (binunionIL => 167.19/25.07 (upairset2IR => 167.19/25.07 (binunionIR => 167.19/25.07 (binunionEcases => 167.19/25.07 (binunionE => 167.19/25.07 (binunionLsub => 167.19/25.07 (binunionRsub => 167.19/25.07 (binintersectI => 167.19/25.07 (binintersectSubset5 => 167.19/25.07 (binintersectEL => 167.19/25.07 (binintersectLsub => 167.19/25.07 (binintersectSubset2 => 167.19/25.07 (binintersectSubset3 => 167.19/25.07 (binintersectER => 167.19/25.07 (disjointsetsI1 => 167.19/25.07 (binintersectRsub => 167.19/25.07 (binintersectSubset4 => 167.19/25.07 (binintersectSubset1 => 167.19/25.07 (bs114d => 167.19/25.07 (setminusI => 167.19/25.07 (setminusEL => 167.19/25.07 (setminusER => 167.19/25.07 (setminusSubset2 => 167.19/25.07 (setminusERneg => 167.19/25.07 (setminusELneg => 167.19/25.07 (setminusILneg => 167.19/25.07 (setminusIRneg => 167.19/25.07 (setminusLsub => 167.19/25.07 (setminusSubset1 => 167.19/25.07 (symdiffE => 167.19/25.07 (symdiffI1 => 167.19/25.07 (symdiffI2 => 167.19/25.07 (symdiffIneg1 => 167.19/25.07 (symdiffIneg2 => 167.19/25.07 (secondinupair => 167.19/25.07 (setukpairIL => 167.19/25.07 (setukpairIR => 167.19/25.07 (kpairiskpair => 167.19/25.07 (kpairp => 167.19/25.07 (singletonsubset => 167.19/25.07 (singletoninpowerset => 167.19/25.07 (singletoninpowunion => 167.19/25.07 (upairset2E => 167.19/25.07 (upairsubunion => 167.19/25.07 (upairinpowunion => 167.19/25.07 (ubforcartprodlem1 => 167.19/25.07 (ubforcartprodlem2 => 167.19/25.07 (ubforcartprodlem3 => 167.19/25.07 (cartprodpairin => 167.19/25.07 (cartprodmempair1 => 167.19/25.07 (cartprodmempair => 167.19/25.07 (setunionE2 => 167.19/25.07 (setunionsingleton1 => 167.19/25.07 (setunionsingleton2 => 167.19/25.07 (setunionsingleton => 167.19/25.07 (singletonprop => 167.19/25.07 (ex1E1 => 167.19/25.07 (ex1I => 167.19/25.07 ( 167.19/25.07 ex1I2 => 167.19/25.07 ( 167.19/25.07 singletonsuniq => 167.19/25.07 ( 167.19/25.07 setukpairinjL1 => 167.19/25.07 ( 167.19/25.07 kfstsingleton => 167.19/25.07 ( 167.19/25.07 theprop => 167.19/25.07 ( 167.19/25.07 kfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodfstin => 167.19/25.07 ( 167.19/25.07 setukpairinjL2 => 167.19/25.07 ( 167.19/25.07 setukpairinjL => 167.19/25.07 ( 167.19/25.07 setukpairinjR11 => 167.19/25.07 ( 167.19/25.07 setukpairinjR12 => 167.19/25.07 ( 167.19/25.07 setukpairinjR1 => 167.19/25.07 ( 167.19/25.07 upairequniteq => 167.19/25.07 ( 167.19/25.07 setukpairinjR2 => 167.19/25.07 ( 167.19/25.07 setukpairinjR => 167.19/25.07 ( 167.19/25.07 ksndsingleton => 167.19/25.07 ( 167.19/25.07 ksndpairEq => 167.19/25.07 ( 167.19/25.07 kpairsurjEq => 167.19/25.07 ( 167.19/25.07 cartprodsndin => 167.19/25.07 ( 167.19/25.07 cartprodpairmemEL => 167.19/25.07 ( 167.19/25.07 cartprodpairmemER => 167.19/25.07 ( 167.19/25.07 cartprodmempaircEq => 167.19/25.07 ( 167.19/25.07 cartprodfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodsndpairEq => 167.19/25.07 ( 167.19/25.07 cartprodpairsurjEq => 167.19/25.07 ( 167.19/25.07 dpsetconstrI => 167.19/25.07 ( 167.19/25.07 dpsetconstrSub => 167.19/25.07 ( 167.19/25.07 setOfPairsIsBReln => 167.19/25.07 ( 167.19/25.07 !!(( 167.19/25.07 ^[Y0 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y1 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y2 : ( 167.19/25.07 $i * $i) > $o]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y3 : $i]: 167.19/25.07 (in(Y3, 167.19/25.07 Y0) => 167.19/25.07 (!!(( 167.19/25.07 ^[Y4 : $i]: 167.19/25.07 (in(Y4, 167.19/25.07 Y1) => 167.19/25.07 (in( 167.19/25.07 kpair( 167.19/25.07 Y3, Y4), 167.19/25.07 dpsetconstr( 167.19/25.07 Y0, Y1, 167.19/25.07 Y2)) => 167.19/25.07 Y2(Y3, 167.19/25.07 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.07 inference('lazy_cnf_imply', [status(thm)], ['94'])). 167.19/25.07 tff('96', plain, 167.19/25.07 ~ (powersetsubset => 167.19/25.07 (sepInPowerset => 167.19/25.07 (sepSubset => 167.19/25.07 (binunionIL => 167.19/25.07 (upairset2IR => 167.19/25.07 (binunionIR => 167.19/25.07 (binunionEcases => 167.19/25.07 (binunionE => 167.19/25.07 (binunionLsub => 167.19/25.07 (binunionRsub => 167.19/25.07 (binintersectI => 167.19/25.07 (binintersectSubset5 => 167.19/25.07 (binintersectEL => 167.19/25.07 (binintersectLsub => 167.19/25.07 (binintersectSubset2 => 167.19/25.07 (binintersectSubset3 => 167.19/25.07 (binintersectER => 167.19/25.07 (disjointsetsI1 => 167.19/25.07 (binintersectRsub => 167.19/25.07 (binintersectSubset4 => 167.19/25.07 (binintersectSubset1 => 167.19/25.07 (bs114d => 167.19/25.07 (setminusI => 167.19/25.07 (setminusEL => 167.19/25.07 (setminusER => 167.19/25.07 (setminusSubset2 => 167.19/25.07 (setminusERneg => 167.19/25.07 (setminusELneg => 167.19/25.07 (setminusILneg => 167.19/25.07 (setminusIRneg => 167.19/25.07 (setminusLsub => 167.19/25.07 (setminusSubset1 => 167.19/25.07 (symdiffE => 167.19/25.07 (symdiffI1 => 167.19/25.07 (symdiffI2 => 167.19/25.07 (symdiffIneg1 => 167.19/25.07 (symdiffIneg2 => 167.19/25.07 (secondinupair => 167.19/25.07 (setukpairIL => 167.19/25.07 (setukpairIR => 167.19/25.07 (kpairiskpair => 167.19/25.07 (kpairp => 167.19/25.07 (singletonsubset => 167.19/25.07 (singletoninpowerset => 167.19/25.07 (singletoninpowunion => 167.19/25.07 (upairset2E => 167.19/25.07 (upairsubunion => 167.19/25.07 (upairinpowunion => 167.19/25.07 (ubforcartprodlem1 => 167.19/25.07 (ubforcartprodlem2 => 167.19/25.07 (ubforcartprodlem3 => 167.19/25.07 (cartprodpairin => 167.19/25.07 (cartprodmempair1 => 167.19/25.07 (cartprodmempair => 167.19/25.07 (setunionE2 => 167.19/25.07 (setunionsingleton1 => 167.19/25.07 (setunionsingleton2 => 167.19/25.07 (setunionsingleton => 167.19/25.07 (singletonprop => 167.19/25.07 (ex1E1 => 167.19/25.07 (ex1I => 167.19/25.07 (ex1I2 => 167.19/25.07 ( 167.19/25.07 singletonsuniq => 167.19/25.07 ( 167.19/25.07 setukpairinjL1 => 167.19/25.07 ( 167.19/25.07 kfstsingleton => 167.19/25.07 ( 167.19/25.07 theprop => 167.19/25.07 ( 167.19/25.07 kfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodfstin => 167.19/25.07 ( 167.19/25.07 setukpairinjL2 => 167.19/25.07 ( 167.19/25.07 setukpairinjL => 167.19/25.07 ( 167.19/25.07 setukpairinjR11 => 167.19/25.07 ( 167.19/25.07 setukpairinjR12 => 167.19/25.07 ( 167.19/25.07 setukpairinjR1 => 167.19/25.07 ( 167.19/25.07 upairequniteq => 167.19/25.07 ( 167.19/25.07 setukpairinjR2 => 167.19/25.07 ( 167.19/25.07 setukpairinjR => 167.19/25.07 ( 167.19/25.07 ksndsingleton => 167.19/25.07 ( 167.19/25.07 ksndpairEq => 167.19/25.07 ( 167.19/25.07 kpairsurjEq => 167.19/25.07 ( 167.19/25.07 cartprodsndin => 167.19/25.07 ( 167.19/25.07 cartprodpairmemEL => 167.19/25.07 ( 167.19/25.07 cartprodpairmemER => 167.19/25.07 ( 167.19/25.07 cartprodmempaircEq => 167.19/25.07 ( 167.19/25.07 cartprodfstpairEq => 167.19/25.07 ( 167.19/25.07 cartprodsndpairEq => 167.19/25.07 ( 167.19/25.07 cartprodpairsurjEq => 167.19/25.07 ( 167.19/25.07 dpsetconstrI => 167.19/25.07 ( 167.19/25.07 dpsetconstrSub => 167.19/25.07 ( 167.19/25.07 setOfPairsIsBReln => 167.19/25.07 ( 167.19/25.07 !!(( 167.19/25.07 ^[Y0 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y1 : $i]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y2 : ( 167.19/25.07 $i * $i) > $o]: 167.19/25.07 (!!(( 167.19/25.07 ^[Y3 : $i]: 167.19/25.07 (in(Y3, 167.19/25.07 Y0) => 167.19/25.07 (!!(( 167.19/25.07 ^[Y4 : $i]: 167.19/25.07 (in(Y4, 167.19/25.07 Y1) => 167.19/25.07 (in( 167.19/25.07 kpair( 167.19/25.07 Y3, Y4), 167.19/25.07 dpsetconstr( 167.19/25.07 Y0, Y1, 167.19/25.07 Y2)) => 167.19/25.07 Y2(Y3, 167.19/25.07 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.07 inference('lazy_cnf_imply', [status(thm)], ['95'])). 167.19/25.07 tff('97', plain, 167.19/25.07 ~ (sepInPowerset => 167.19/25.07 (sepSubset => 167.19/25.07 (binunionIL => 167.19/25.07 (upairset2IR => 167.19/25.07 (binunionIR => 167.19/25.07 (binunionEcases => 167.19/25.07 (binunionE => 167.19/25.07 (binunionLsub => 167.19/25.07 (binunionRsub => 167.19/25.07 (binintersectI => 167.19/25.07 (binintersectSubset5 => 167.19/25.07 (binintersectEL => 167.19/25.08 (binintersectLsub => 167.19/25.08 (binintersectSubset2 => 167.19/25.08 (binintersectSubset3 => 167.19/25.08 (binintersectER => 167.19/25.08 (disjointsetsI1 => 167.19/25.08 (binintersectRsub => 167.19/25.08 (binintersectSubset4 => 167.19/25.08 (binintersectSubset1 => 167.19/25.08 (bs114d => 167.19/25.08 (setminusI => 167.19/25.08 (setminusEL => 167.19/25.08 (setminusER => 167.19/25.08 (setminusSubset2 => 167.19/25.08 (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 ( 167.19/25.08 setukpairinjL1 => 167.19/25.08 ( 167.19/25.08 kfstsingleton => 167.19/25.08 ( 167.19/25.08 theprop => 167.19/25.08 ( 167.19/25.08 kfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodfstin => 167.19/25.08 ( 167.19/25.08 setukpairinjL2 => 167.19/25.08 ( 167.19/25.08 setukpairinjL => 167.19/25.08 ( 167.19/25.08 setukpairinjR11 => 167.19/25.08 ( 167.19/25.08 setukpairinjR12 => 167.19/25.08 ( 167.19/25.08 setukpairinjR1 => 167.19/25.08 ( 167.19/25.08 upairequniteq => 167.19/25.08 ( 167.19/25.08 setukpairinjR2 => 167.19/25.08 ( 167.19/25.08 setukpairinjR => 167.19/25.08 ( 167.19/25.08 ksndsingleton => 167.19/25.08 ( 167.19/25.08 ksndpairEq => 167.19/25.08 ( 167.19/25.08 kpairsurjEq => 167.19/25.08 ( 167.19/25.08 cartprodsndin => 167.19/25.08 ( 167.19/25.08 cartprodpairmemEL => 167.19/25.08 ( 167.19/25.08 cartprodpairmemER => 167.19/25.08 ( 167.19/25.08 cartprodmempaircEq => 167.19/25.08 ( 167.19/25.08 cartprodfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodsndpairEq => 167.19/25.08 ( 167.19/25.08 cartprodpairsurjEq => 167.19/25.08 ( 167.19/25.08 dpsetconstrI => 167.19/25.08 ( 167.19/25.08 dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['96'])). 167.19/25.08 tff('98', plain, 167.19/25.08 ~ (sepSubset => 167.19/25.08 (binunionIL => 167.19/25.08 (upairset2IR => 167.19/25.08 (binunionIR => 167.19/25.08 (binunionEcases => 167.19/25.08 (binunionE => 167.19/25.08 (binunionLsub => 167.19/25.08 (binunionRsub => 167.19/25.08 (binintersectI => 167.19/25.08 (binintersectSubset5 => 167.19/25.08 (binintersectEL => 167.19/25.08 (binintersectLsub => 167.19/25.08 (binintersectSubset2 => 167.19/25.08 (binintersectSubset3 => 167.19/25.08 (binintersectER => 167.19/25.08 (disjointsetsI1 => 167.19/25.08 (binintersectRsub => 167.19/25.08 (binintersectSubset4 => 167.19/25.08 (binintersectSubset1 => 167.19/25.08 (bs114d => 167.19/25.08 (setminusI => 167.19/25.08 (setminusEL => 167.19/25.08 (setminusER => 167.19/25.08 (setminusSubset2 => 167.19/25.08 (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 ( 167.19/25.08 kfstsingleton => 167.19/25.08 ( 167.19/25.08 theprop => 167.19/25.08 ( 167.19/25.08 kfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodfstin => 167.19/25.08 ( 167.19/25.08 setukpairinjL2 => 167.19/25.08 ( 167.19/25.08 setukpairinjL => 167.19/25.08 ( 167.19/25.08 setukpairinjR11 => 167.19/25.08 ( 167.19/25.08 setukpairinjR12 => 167.19/25.08 ( 167.19/25.08 setukpairinjR1 => 167.19/25.08 ( 167.19/25.08 upairequniteq => 167.19/25.08 ( 167.19/25.08 setukpairinjR2 => 167.19/25.08 ( 167.19/25.08 setukpairinjR => 167.19/25.08 ( 167.19/25.08 ksndsingleton => 167.19/25.08 ( 167.19/25.08 ksndpairEq => 167.19/25.08 ( 167.19/25.08 kpairsurjEq => 167.19/25.08 ( 167.19/25.08 cartprodsndin => 167.19/25.08 ( 167.19/25.08 cartprodpairmemEL => 167.19/25.08 ( 167.19/25.08 cartprodpairmemER => 167.19/25.08 ( 167.19/25.08 cartprodmempaircEq => 167.19/25.08 ( 167.19/25.08 cartprodfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodsndpairEq => 167.19/25.08 ( 167.19/25.08 cartprodpairsurjEq => 167.19/25.08 ( 167.19/25.08 dpsetconstrI => 167.19/25.08 ( 167.19/25.08 dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['97'])). 167.19/25.08 tff('99', plain, 167.19/25.08 ~ (binunionIL => 167.19/25.08 (upairset2IR => 167.19/25.08 (binunionIR => 167.19/25.08 (binunionEcases => 167.19/25.08 (binunionE => 167.19/25.08 (binunionLsub => 167.19/25.08 (binunionRsub => 167.19/25.08 (binintersectI => 167.19/25.08 (binintersectSubset5 => 167.19/25.08 (binintersectEL => 167.19/25.08 (binintersectLsub => 167.19/25.08 (binintersectSubset2 => 167.19/25.08 (binintersectSubset3 => 167.19/25.08 (binintersectER => 167.19/25.08 (disjointsetsI1 => 167.19/25.08 (binintersectRsub => 167.19/25.08 (binintersectSubset4 => 167.19/25.08 (binintersectSubset1 => 167.19/25.08 (bs114d => 167.19/25.08 (setminusI => 167.19/25.08 (setminusEL => 167.19/25.08 (setminusER => 167.19/25.08 (setminusSubset2 => 167.19/25.08 (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 ( 167.19/25.08 theprop => 167.19/25.08 ( 167.19/25.08 kfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodfstin => 167.19/25.08 ( 167.19/25.08 setukpairinjL2 => 167.19/25.08 ( 167.19/25.08 setukpairinjL => 167.19/25.08 ( 167.19/25.08 setukpairinjR11 => 167.19/25.08 ( 167.19/25.08 setukpairinjR12 => 167.19/25.08 ( 167.19/25.08 setukpairinjR1 => 167.19/25.08 ( 167.19/25.08 upairequniteq => 167.19/25.08 ( 167.19/25.08 setukpairinjR2 => 167.19/25.08 ( 167.19/25.08 setukpairinjR => 167.19/25.08 ( 167.19/25.08 ksndsingleton => 167.19/25.08 ( 167.19/25.08 ksndpairEq => 167.19/25.08 ( 167.19/25.08 kpairsurjEq => 167.19/25.08 ( 167.19/25.08 cartprodsndin => 167.19/25.08 ( 167.19/25.08 cartprodpairmemEL => 167.19/25.08 ( 167.19/25.08 cartprodpairmemER => 167.19/25.08 ( 167.19/25.08 cartprodmempaircEq => 167.19/25.08 ( 167.19/25.08 cartprodfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodsndpairEq => 167.19/25.08 ( 167.19/25.08 cartprodpairsurjEq => 167.19/25.08 ( 167.19/25.08 dpsetconstrI => 167.19/25.08 ( 167.19/25.08 dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['98'])). 167.19/25.08 tff('100', plain, 167.19/25.08 ~ (upairset2IR => 167.19/25.08 (binunionIR => 167.19/25.08 (binunionEcases => 167.19/25.08 (binunionE => 167.19/25.08 (binunionLsub => 167.19/25.08 (binunionRsub => 167.19/25.08 (binintersectI => 167.19/25.08 (binintersectSubset5 => 167.19/25.08 (binintersectEL => 167.19/25.08 (binintersectLsub => 167.19/25.08 (binintersectSubset2 => 167.19/25.08 (binintersectSubset3 => 167.19/25.08 (binintersectER => 167.19/25.08 (disjointsetsI1 => 167.19/25.08 (binintersectRsub => 167.19/25.08 (binintersectSubset4 => 167.19/25.08 (binintersectSubset1 => 167.19/25.08 (bs114d => 167.19/25.08 (setminusI => 167.19/25.08 (setminusEL => 167.19/25.08 (setminusER => 167.19/25.08 (setminusSubset2 => 167.19/25.08 (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 ( 167.19/25.08 kfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodfstin => 167.19/25.08 ( 167.19/25.08 setukpairinjL2 => 167.19/25.08 ( 167.19/25.08 setukpairinjL => 167.19/25.08 ( 167.19/25.08 setukpairinjR11 => 167.19/25.08 ( 167.19/25.08 setukpairinjR12 => 167.19/25.08 ( 167.19/25.08 setukpairinjR1 => 167.19/25.08 ( 167.19/25.08 upairequniteq => 167.19/25.08 ( 167.19/25.08 setukpairinjR2 => 167.19/25.08 ( 167.19/25.08 setukpairinjR => 167.19/25.08 ( 167.19/25.08 ksndsingleton => 167.19/25.08 ( 167.19/25.08 ksndpairEq => 167.19/25.08 ( 167.19/25.08 kpairsurjEq => 167.19/25.08 ( 167.19/25.08 cartprodsndin => 167.19/25.08 ( 167.19/25.08 cartprodpairmemEL => 167.19/25.08 ( 167.19/25.08 cartprodpairmemER => 167.19/25.08 ( 167.19/25.08 cartprodmempaircEq => 167.19/25.08 ( 167.19/25.08 cartprodfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodsndpairEq => 167.19/25.08 ( 167.19/25.08 cartprodpairsurjEq => 167.19/25.08 ( 167.19/25.08 dpsetconstrI => 167.19/25.08 ( 167.19/25.08 dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['99'])). 167.19/25.08 tff('101', plain, 167.19/25.08 ~ (binunionIR => 167.19/25.08 (binunionEcases => 167.19/25.08 (binunionE => 167.19/25.08 (binunionLsub => 167.19/25.08 (binunionRsub => 167.19/25.08 (binintersectI => 167.19/25.08 (binintersectSubset5 => 167.19/25.08 (binintersectEL => 167.19/25.08 (binintersectLsub => 167.19/25.08 (binintersectSubset2 => 167.19/25.08 (binintersectSubset3 => 167.19/25.08 (binintersectER => 167.19/25.08 (disjointsetsI1 => 167.19/25.08 (binintersectRsub => 167.19/25.08 (binintersectSubset4 => 167.19/25.08 (binintersectSubset1 => 167.19/25.08 (bs114d => 167.19/25.08 (setminusI => 167.19/25.08 (setminusEL => 167.19/25.08 (setminusER => 167.19/25.08 (setminusSubset2 => 167.19/25.08 (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodfstin => 167.19/25.08 ( 167.19/25.08 setukpairinjL2 => 167.19/25.08 ( 167.19/25.08 setukpairinjL => 167.19/25.08 ( 167.19/25.08 setukpairinjR11 => 167.19/25.08 ( 167.19/25.08 setukpairinjR12 => 167.19/25.08 ( 167.19/25.08 setukpairinjR1 => 167.19/25.08 ( 167.19/25.08 upairequniteq => 167.19/25.08 ( 167.19/25.08 setukpairinjR2 => 167.19/25.08 ( 167.19/25.08 setukpairinjR => 167.19/25.08 ( 167.19/25.08 ksndsingleton => 167.19/25.08 ( 167.19/25.08 ksndpairEq => 167.19/25.08 ( 167.19/25.08 kpairsurjEq => 167.19/25.08 ( 167.19/25.08 cartprodsndin => 167.19/25.08 ( 167.19/25.08 cartprodpairmemEL => 167.19/25.08 ( 167.19/25.08 cartprodpairmemER => 167.19/25.08 ( 167.19/25.08 cartprodmempaircEq => 167.19/25.08 ( 167.19/25.08 cartprodfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodsndpairEq => 167.19/25.08 ( 167.19/25.08 cartprodpairsurjEq => 167.19/25.08 ( 167.19/25.08 dpsetconstrI => 167.19/25.08 ( 167.19/25.08 dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['100'])). 167.19/25.08 tff('102', plain, 167.19/25.08 ~ (binunionEcases => 167.19/25.08 (binunionE => 167.19/25.08 (binunionLsub => 167.19/25.08 (binunionRsub => 167.19/25.08 (binintersectI => 167.19/25.08 (binintersectSubset5 => 167.19/25.08 (binintersectEL => 167.19/25.08 (binintersectLsub => 167.19/25.08 (binintersectSubset2 => 167.19/25.08 (binintersectSubset3 => 167.19/25.08 (binintersectER => 167.19/25.08 (disjointsetsI1 => 167.19/25.08 (binintersectRsub => 167.19/25.08 (binintersectSubset4 => 167.19/25.08 (binintersectSubset1 => 167.19/25.08 (bs114d => 167.19/25.08 (setminusI => 167.19/25.08 (setminusEL => 167.19/25.08 (setminusER => 167.19/25.08 (setminusSubset2 => 167.19/25.08 (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 ( 167.19/25.08 setukpairinjL2 => 167.19/25.08 ( 167.19/25.08 setukpairinjL => 167.19/25.08 ( 167.19/25.08 setukpairinjR11 => 167.19/25.08 ( 167.19/25.08 setukpairinjR12 => 167.19/25.08 ( 167.19/25.08 setukpairinjR1 => 167.19/25.08 ( 167.19/25.08 upairequniteq => 167.19/25.08 ( 167.19/25.08 setukpairinjR2 => 167.19/25.08 ( 167.19/25.08 setukpairinjR => 167.19/25.08 ( 167.19/25.08 ksndsingleton => 167.19/25.08 ( 167.19/25.08 ksndpairEq => 167.19/25.08 ( 167.19/25.08 kpairsurjEq => 167.19/25.08 ( 167.19/25.08 cartprodsndin => 167.19/25.08 ( 167.19/25.08 cartprodpairmemEL => 167.19/25.08 ( 167.19/25.08 cartprodpairmemER => 167.19/25.08 ( 167.19/25.08 cartprodmempaircEq => 167.19/25.08 ( 167.19/25.08 cartprodfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodsndpairEq => 167.19/25.08 ( 167.19/25.08 cartprodpairsurjEq => 167.19/25.08 ( 167.19/25.08 dpsetconstrI => 167.19/25.08 ( 167.19/25.08 dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['101'])). 167.19/25.08 tff('103', plain, 167.19/25.08 ~ (binunionE => 167.19/25.08 (binunionLsub => 167.19/25.08 (binunionRsub => 167.19/25.08 (binintersectI => 167.19/25.08 (binintersectSubset5 => 167.19/25.08 (binintersectEL => 167.19/25.08 (binintersectLsub => 167.19/25.08 (binintersectSubset2 => 167.19/25.08 (binintersectSubset3 => 167.19/25.08 (binintersectER => 167.19/25.08 (disjointsetsI1 => 167.19/25.08 (binintersectRsub => 167.19/25.08 (binintersectSubset4 => 167.19/25.08 (binintersectSubset1 => 167.19/25.08 (bs114d => 167.19/25.08 (setminusI => 167.19/25.08 (setminusEL => 167.19/25.08 (setminusER => 167.19/25.08 (setminusSubset2 => 167.19/25.08 (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 ( 167.19/25.08 setukpairinjL => 167.19/25.08 ( 167.19/25.08 setukpairinjR11 => 167.19/25.08 ( 167.19/25.08 setukpairinjR12 => 167.19/25.08 ( 167.19/25.08 setukpairinjR1 => 167.19/25.08 ( 167.19/25.08 upairequniteq => 167.19/25.08 ( 167.19/25.08 setukpairinjR2 => 167.19/25.08 ( 167.19/25.08 setukpairinjR => 167.19/25.08 ( 167.19/25.08 ksndsingleton => 167.19/25.08 ( 167.19/25.08 ksndpairEq => 167.19/25.08 ( 167.19/25.08 kpairsurjEq => 167.19/25.08 ( 167.19/25.08 cartprodsndin => 167.19/25.08 ( 167.19/25.08 cartprodpairmemEL => 167.19/25.08 ( 167.19/25.08 cartprodpairmemER => 167.19/25.08 ( 167.19/25.08 cartprodmempaircEq => 167.19/25.08 ( 167.19/25.08 cartprodfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodsndpairEq => 167.19/25.08 ( 167.19/25.08 cartprodpairsurjEq => 167.19/25.08 ( 167.19/25.08 dpsetconstrI => 167.19/25.08 ( 167.19/25.08 dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['102'])). 167.19/25.08 tff('104', plain, 167.19/25.08 ~ (binunionLsub => 167.19/25.08 (binunionRsub => 167.19/25.08 (binintersectI => 167.19/25.08 (binintersectSubset5 => 167.19/25.08 (binintersectEL => 167.19/25.08 (binintersectLsub => 167.19/25.08 (binintersectSubset2 => 167.19/25.08 (binintersectSubset3 => 167.19/25.08 (binintersectER => 167.19/25.08 (disjointsetsI1 => 167.19/25.08 (binintersectRsub => 167.19/25.08 (binintersectSubset4 => 167.19/25.08 (binintersectSubset1 => 167.19/25.08 (bs114d => 167.19/25.08 (setminusI => 167.19/25.08 (setminusEL => 167.19/25.08 (setminusER => 167.19/25.08 (setminusSubset2 => 167.19/25.08 (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 ( 167.19/25.08 setukpairinjR11 => 167.19/25.08 ( 167.19/25.08 setukpairinjR12 => 167.19/25.08 ( 167.19/25.08 setukpairinjR1 => 167.19/25.08 ( 167.19/25.08 upairequniteq => 167.19/25.08 ( 167.19/25.08 setukpairinjR2 => 167.19/25.08 ( 167.19/25.08 setukpairinjR => 167.19/25.08 ( 167.19/25.08 ksndsingleton => 167.19/25.08 ( 167.19/25.08 ksndpairEq => 167.19/25.08 ( 167.19/25.08 kpairsurjEq => 167.19/25.08 ( 167.19/25.08 cartprodsndin => 167.19/25.08 ( 167.19/25.08 cartprodpairmemEL => 167.19/25.08 ( 167.19/25.08 cartprodpairmemER => 167.19/25.08 ( 167.19/25.08 cartprodmempaircEq => 167.19/25.08 ( 167.19/25.08 cartprodfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodsndpairEq => 167.19/25.08 ( 167.19/25.08 cartprodpairsurjEq => 167.19/25.08 ( 167.19/25.08 dpsetconstrI => 167.19/25.08 ( 167.19/25.08 dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['103'])). 167.19/25.08 tff('105', plain, 167.19/25.08 ~ (binunionRsub => 167.19/25.08 (binintersectI => 167.19/25.08 (binintersectSubset5 => 167.19/25.08 (binintersectEL => 167.19/25.08 (binintersectLsub => 167.19/25.08 (binintersectSubset2 => 167.19/25.08 (binintersectSubset3 => 167.19/25.08 (binintersectER => 167.19/25.08 (disjointsetsI1 => 167.19/25.08 (binintersectRsub => 167.19/25.08 (binintersectSubset4 => 167.19/25.08 (binintersectSubset1 => 167.19/25.08 (bs114d => 167.19/25.08 (setminusI => 167.19/25.08 (setminusEL => 167.19/25.08 (setminusER => 167.19/25.08 (setminusSubset2 => 167.19/25.08 (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 ( 167.19/25.08 setukpairinjR12 => 167.19/25.08 ( 167.19/25.08 setukpairinjR1 => 167.19/25.08 ( 167.19/25.08 upairequniteq => 167.19/25.08 ( 167.19/25.08 setukpairinjR2 => 167.19/25.08 ( 167.19/25.08 setukpairinjR => 167.19/25.08 ( 167.19/25.08 ksndsingleton => 167.19/25.08 ( 167.19/25.08 ksndpairEq => 167.19/25.08 ( 167.19/25.08 kpairsurjEq => 167.19/25.08 ( 167.19/25.08 cartprodsndin => 167.19/25.08 ( 167.19/25.08 cartprodpairmemEL => 167.19/25.08 ( 167.19/25.08 cartprodpairmemER => 167.19/25.08 ( 167.19/25.08 cartprodmempaircEq => 167.19/25.08 ( 167.19/25.08 cartprodfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodsndpairEq => 167.19/25.08 ( 167.19/25.08 cartprodpairsurjEq => 167.19/25.08 ( 167.19/25.08 dpsetconstrI => 167.19/25.08 ( 167.19/25.08 dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['104'])). 167.19/25.08 tff('106', plain, 167.19/25.08 ~ (binintersectI => 167.19/25.08 (binintersectSubset5 => 167.19/25.08 (binintersectEL => 167.19/25.08 (binintersectLsub => 167.19/25.08 (binintersectSubset2 => 167.19/25.08 (binintersectSubset3 => 167.19/25.08 (binintersectER => 167.19/25.08 (disjointsetsI1 => 167.19/25.08 (binintersectRsub => 167.19/25.08 (binintersectSubset4 => 167.19/25.08 (binintersectSubset1 => 167.19/25.08 (bs114d => 167.19/25.08 (setminusI => 167.19/25.08 (setminusEL => 167.19/25.08 (setminusER => 167.19/25.08 (setminusSubset2 => 167.19/25.08 (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 ( 167.19/25.08 setukpairinjR1 => 167.19/25.08 ( 167.19/25.08 upairequniteq => 167.19/25.08 ( 167.19/25.08 setukpairinjR2 => 167.19/25.08 ( 167.19/25.08 setukpairinjR => 167.19/25.08 ( 167.19/25.08 ksndsingleton => 167.19/25.08 ( 167.19/25.08 ksndpairEq => 167.19/25.08 ( 167.19/25.08 kpairsurjEq => 167.19/25.08 ( 167.19/25.08 cartprodsndin => 167.19/25.08 ( 167.19/25.08 cartprodpairmemEL => 167.19/25.08 ( 167.19/25.08 cartprodpairmemER => 167.19/25.08 ( 167.19/25.08 cartprodmempaircEq => 167.19/25.08 ( 167.19/25.08 cartprodfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodsndpairEq => 167.19/25.08 ( 167.19/25.08 cartprodpairsurjEq => 167.19/25.08 ( 167.19/25.08 dpsetconstrI => 167.19/25.08 ( 167.19/25.08 dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['105'])). 167.19/25.08 tff('107', plain, 167.19/25.08 ~ (binintersectSubset5 => 167.19/25.08 (binintersectEL => 167.19/25.08 (binintersectLsub => 167.19/25.08 (binintersectSubset2 => 167.19/25.08 (binintersectSubset3 => 167.19/25.08 (binintersectER => 167.19/25.08 (disjointsetsI1 => 167.19/25.08 (binintersectRsub => 167.19/25.08 (binintersectSubset4 => 167.19/25.08 (binintersectSubset1 => 167.19/25.08 (bs114d => 167.19/25.08 (setminusI => 167.19/25.08 (setminusEL => 167.19/25.08 (setminusER => 167.19/25.08 (setminusSubset2 => 167.19/25.08 (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 ( 167.19/25.08 upairequniteq => 167.19/25.08 ( 167.19/25.08 setukpairinjR2 => 167.19/25.08 ( 167.19/25.08 setukpairinjR => 167.19/25.08 ( 167.19/25.08 ksndsingleton => 167.19/25.08 ( 167.19/25.08 ksndpairEq => 167.19/25.08 ( 167.19/25.08 kpairsurjEq => 167.19/25.08 ( 167.19/25.08 cartprodsndin => 167.19/25.08 ( 167.19/25.08 cartprodpairmemEL => 167.19/25.08 ( 167.19/25.08 cartprodpairmemER => 167.19/25.08 ( 167.19/25.08 cartprodmempaircEq => 167.19/25.08 ( 167.19/25.08 cartprodfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodsndpairEq => 167.19/25.08 ( 167.19/25.08 cartprodpairsurjEq => 167.19/25.08 ( 167.19/25.08 dpsetconstrI => 167.19/25.08 ( 167.19/25.08 dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['106'])). 167.19/25.08 tff('108', plain, 167.19/25.08 ~ (binintersectEL => 167.19/25.08 (binintersectLsub => 167.19/25.08 (binintersectSubset2 => 167.19/25.08 (binintersectSubset3 => 167.19/25.08 (binintersectER => 167.19/25.08 (disjointsetsI1 => 167.19/25.08 (binintersectRsub => 167.19/25.08 (binintersectSubset4 => 167.19/25.08 (binintersectSubset1 => 167.19/25.08 (bs114d => 167.19/25.08 (setminusI => 167.19/25.08 (setminusEL => 167.19/25.08 (setminusER => 167.19/25.08 (setminusSubset2 => 167.19/25.08 (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 ( 167.19/25.08 setukpairinjR2 => 167.19/25.08 ( 167.19/25.08 setukpairinjR => 167.19/25.08 ( 167.19/25.08 ksndsingleton => 167.19/25.08 ( 167.19/25.08 ksndpairEq => 167.19/25.08 ( 167.19/25.08 kpairsurjEq => 167.19/25.08 ( 167.19/25.08 cartprodsndin => 167.19/25.08 ( 167.19/25.08 cartprodpairmemEL => 167.19/25.08 ( 167.19/25.08 cartprodpairmemER => 167.19/25.08 ( 167.19/25.08 cartprodmempaircEq => 167.19/25.08 ( 167.19/25.08 cartprodfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodsndpairEq => 167.19/25.08 ( 167.19/25.08 cartprodpairsurjEq => 167.19/25.08 ( 167.19/25.08 dpsetconstrI => 167.19/25.08 ( 167.19/25.08 dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['107'])). 167.19/25.08 tff('109', plain, 167.19/25.08 ~ (binintersectLsub => 167.19/25.08 (binintersectSubset2 => 167.19/25.08 (binintersectSubset3 => 167.19/25.08 (binintersectER => 167.19/25.08 (disjointsetsI1 => 167.19/25.08 (binintersectRsub => 167.19/25.08 (binintersectSubset4 => 167.19/25.08 (binintersectSubset1 => 167.19/25.08 (bs114d => 167.19/25.08 (setminusI => 167.19/25.08 (setminusEL => 167.19/25.08 (setminusER => 167.19/25.08 (setminusSubset2 => 167.19/25.08 (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 ( 167.19/25.08 setukpairinjR => 167.19/25.08 ( 167.19/25.08 ksndsingleton => 167.19/25.08 ( 167.19/25.08 ksndpairEq => 167.19/25.08 ( 167.19/25.08 kpairsurjEq => 167.19/25.08 ( 167.19/25.08 cartprodsndin => 167.19/25.08 ( 167.19/25.08 cartprodpairmemEL => 167.19/25.08 ( 167.19/25.08 cartprodpairmemER => 167.19/25.08 ( 167.19/25.08 cartprodmempaircEq => 167.19/25.08 ( 167.19/25.08 cartprodfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodsndpairEq => 167.19/25.08 ( 167.19/25.08 cartprodpairsurjEq => 167.19/25.08 ( 167.19/25.08 dpsetconstrI => 167.19/25.08 ( 167.19/25.08 dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['108'])). 167.19/25.08 tff('110', plain, 167.19/25.08 ~ (binintersectSubset2 => 167.19/25.08 (binintersectSubset3 => 167.19/25.08 (binintersectER => 167.19/25.08 (disjointsetsI1 => 167.19/25.08 (binintersectRsub => 167.19/25.08 (binintersectSubset4 => 167.19/25.08 (binintersectSubset1 => 167.19/25.08 (bs114d => 167.19/25.08 (setminusI => 167.19/25.08 (setminusEL => 167.19/25.08 (setminusER => 167.19/25.08 (setminusSubset2 => 167.19/25.08 (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 ( 167.19/25.08 ksndsingleton => 167.19/25.08 ( 167.19/25.08 ksndpairEq => 167.19/25.08 ( 167.19/25.08 kpairsurjEq => 167.19/25.08 ( 167.19/25.08 cartprodsndin => 167.19/25.08 ( 167.19/25.08 cartprodpairmemEL => 167.19/25.08 ( 167.19/25.08 cartprodpairmemER => 167.19/25.08 ( 167.19/25.08 cartprodmempaircEq => 167.19/25.08 ( 167.19/25.08 cartprodfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodsndpairEq => 167.19/25.08 ( 167.19/25.08 cartprodpairsurjEq => 167.19/25.08 ( 167.19/25.08 dpsetconstrI => 167.19/25.08 ( 167.19/25.08 dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['109'])). 167.19/25.08 tff('111', plain, 167.19/25.08 ~ (binintersectSubset3 => 167.19/25.08 (binintersectER => 167.19/25.08 (disjointsetsI1 => 167.19/25.08 (binintersectRsub => 167.19/25.08 (binintersectSubset4 => 167.19/25.08 (binintersectSubset1 => 167.19/25.08 (bs114d => 167.19/25.08 (setminusI => 167.19/25.08 (setminusEL => 167.19/25.08 (setminusER => 167.19/25.08 (setminusSubset2 => 167.19/25.08 (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 ( 167.19/25.08 ksndpairEq => 167.19/25.08 ( 167.19/25.08 kpairsurjEq => 167.19/25.08 ( 167.19/25.08 cartprodsndin => 167.19/25.08 ( 167.19/25.08 cartprodpairmemEL => 167.19/25.08 ( 167.19/25.08 cartprodpairmemER => 167.19/25.08 ( 167.19/25.08 cartprodmempaircEq => 167.19/25.08 ( 167.19/25.08 cartprodfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodsndpairEq => 167.19/25.08 ( 167.19/25.08 cartprodpairsurjEq => 167.19/25.08 ( 167.19/25.08 dpsetconstrI => 167.19/25.08 ( 167.19/25.08 dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['110'])). 167.19/25.08 tff('112', plain, 167.19/25.08 ~ (binintersectER => 167.19/25.08 (disjointsetsI1 => 167.19/25.08 (binintersectRsub => 167.19/25.08 (binintersectSubset4 => 167.19/25.08 (binintersectSubset1 => 167.19/25.08 (bs114d => 167.19/25.08 (setminusI => 167.19/25.08 (setminusEL => 167.19/25.08 (setminusER => 167.19/25.08 (setminusSubset2 => 167.19/25.08 (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 ( 167.19/25.08 kpairsurjEq => 167.19/25.08 ( 167.19/25.08 cartprodsndin => 167.19/25.08 ( 167.19/25.08 cartprodpairmemEL => 167.19/25.08 ( 167.19/25.08 cartprodpairmemER => 167.19/25.08 ( 167.19/25.08 cartprodmempaircEq => 167.19/25.08 ( 167.19/25.08 cartprodfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodsndpairEq => 167.19/25.08 ( 167.19/25.08 cartprodpairsurjEq => 167.19/25.08 ( 167.19/25.08 dpsetconstrI => 167.19/25.08 ( 167.19/25.08 dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['111'])). 167.19/25.08 tff('113', plain, 167.19/25.08 ~ (disjointsetsI1 => 167.19/25.08 (binintersectRsub => 167.19/25.08 (binintersectSubset4 => 167.19/25.08 (binintersectSubset1 => 167.19/25.08 (bs114d => 167.19/25.08 (setminusI => 167.19/25.08 (setminusEL => 167.19/25.08 (setminusER => 167.19/25.08 (setminusSubset2 => 167.19/25.08 (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 ( 167.19/25.08 cartprodsndin => 167.19/25.08 ( 167.19/25.08 cartprodpairmemEL => 167.19/25.08 ( 167.19/25.08 cartprodpairmemER => 167.19/25.08 ( 167.19/25.08 cartprodmempaircEq => 167.19/25.08 ( 167.19/25.08 cartprodfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodsndpairEq => 167.19/25.08 ( 167.19/25.08 cartprodpairsurjEq => 167.19/25.08 ( 167.19/25.08 dpsetconstrI => 167.19/25.08 ( 167.19/25.08 dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['112'])). 167.19/25.08 tff('114', plain, 167.19/25.08 ~ (binintersectRsub => 167.19/25.08 (binintersectSubset4 => 167.19/25.08 (binintersectSubset1 => 167.19/25.08 (bs114d => 167.19/25.08 (setminusI => 167.19/25.08 (setminusEL => 167.19/25.08 (setminusER => 167.19/25.08 (setminusSubset2 => 167.19/25.08 (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 ( 167.19/25.08 cartprodpairmemEL => 167.19/25.08 ( 167.19/25.08 cartprodpairmemER => 167.19/25.08 ( 167.19/25.08 cartprodmempaircEq => 167.19/25.08 ( 167.19/25.08 cartprodfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodsndpairEq => 167.19/25.08 ( 167.19/25.08 cartprodpairsurjEq => 167.19/25.08 ( 167.19/25.08 dpsetconstrI => 167.19/25.08 ( 167.19/25.08 dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['113'])). 167.19/25.08 tff('115', plain, 167.19/25.08 ~ (binintersectSubset4 => 167.19/25.08 (binintersectSubset1 => 167.19/25.08 (bs114d => 167.19/25.08 (setminusI => 167.19/25.08 (setminusEL => 167.19/25.08 (setminusER => 167.19/25.08 (setminusSubset2 => 167.19/25.08 (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 ( 167.19/25.08 cartprodpairmemER => 167.19/25.08 ( 167.19/25.08 cartprodmempaircEq => 167.19/25.08 ( 167.19/25.08 cartprodfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodsndpairEq => 167.19/25.08 ( 167.19/25.08 cartprodpairsurjEq => 167.19/25.08 ( 167.19/25.08 dpsetconstrI => 167.19/25.08 ( 167.19/25.08 dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['114'])). 167.19/25.08 tff('116', plain, 167.19/25.08 ~ (binintersectSubset1 => 167.19/25.08 (bs114d => 167.19/25.08 (setminusI => 167.19/25.08 (setminusEL => 167.19/25.08 (setminusER => 167.19/25.08 (setminusSubset2 => 167.19/25.08 (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 ( 167.19/25.08 cartprodmempaircEq => 167.19/25.08 ( 167.19/25.08 cartprodfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodsndpairEq => 167.19/25.08 ( 167.19/25.08 cartprodpairsurjEq => 167.19/25.08 ( 167.19/25.08 dpsetconstrI => 167.19/25.08 ( 167.19/25.08 dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['115'])). 167.19/25.08 tff('117', plain, 167.19/25.08 ~ (bs114d => 167.19/25.08 (setminusI => 167.19/25.08 (setminusEL => 167.19/25.08 (setminusER => 167.19/25.08 (setminusSubset2 => 167.19/25.08 (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 ( 167.19/25.08 cartprodfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodsndpairEq => 167.19/25.08 ( 167.19/25.08 cartprodpairsurjEq => 167.19/25.08 ( 167.19/25.08 dpsetconstrI => 167.19/25.08 ( 167.19/25.08 dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['116'])). 167.19/25.08 tff('118', plain, 167.19/25.08 ~ (setminusI => 167.19/25.08 (setminusEL => 167.19/25.08 (setminusER => 167.19/25.08 (setminusSubset2 => 167.19/25.08 (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 ( 167.19/25.08 cartprodsndpairEq => 167.19/25.08 ( 167.19/25.08 cartprodpairsurjEq => 167.19/25.08 ( 167.19/25.08 dpsetconstrI => 167.19/25.08 ( 167.19/25.08 dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['117'])). 167.19/25.08 tff('119', plain, 167.19/25.08 ~ (setminusEL => 167.19/25.08 (setminusER => 167.19/25.08 (setminusSubset2 => 167.19/25.08 (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 ( 167.19/25.08 cartprodpairsurjEq => 167.19/25.08 ( 167.19/25.08 dpsetconstrI => 167.19/25.08 ( 167.19/25.08 dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['118'])). 167.19/25.08 tff('120', plain, 167.19/25.08 ~ (setminusER => 167.19/25.08 (setminusSubset2 => 167.19/25.08 (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 ( 167.19/25.08 dpsetconstrI => 167.19/25.08 ( 167.19/25.08 dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['119'])). 167.19/25.08 tff('121', plain, 167.19/25.08 ~ (setminusSubset2 => 167.19/25.08 (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 ( 167.19/25.08 dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['120'])). 167.19/25.08 tff('122', plain, 167.19/25.08 ~ (setminusERneg => 167.19/25.08 (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 ( 167.19/25.08 setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['121'])). 167.19/25.08 tff('123', plain, 167.19/25.08 ~ (setminusELneg => 167.19/25.08 (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['122'])). 167.19/25.08 tff('124', plain, 167.19/25.08 ~ (setminusILneg => 167.19/25.08 (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['123'])). 167.19/25.08 tff('125', plain, 167.19/25.08 ~ (setminusIRneg => 167.19/25.08 (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['124'])). 167.19/25.08 tff('126', plain, 167.19/25.08 ~ (setminusLsub => 167.19/25.08 (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['125'])). 167.19/25.08 tff('127', plain, 167.19/25.08 ~ (setminusSubset1 => 167.19/25.08 (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!(( 167.19/25.08 ^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['126'])). 167.19/25.08 tff('128', plain, 167.19/25.08 ~ (symdiffE => 167.19/25.08 (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!((^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['127'])). 167.19/25.08 tff('129', plain, 167.19/25.08 ~ (symdiffI1 => 167.19/25.08 (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!((^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['128'])). 167.19/25.08 tff('130', plain, 167.19/25.08 ~ (symdiffI2 => 167.19/25.08 (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!((^[Y0 : $i]: 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['129'])). 167.19/25.08 tff('131', plain, 167.19/25.08 ~ (symdiffIneg1 => 167.19/25.08 (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!((^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['130'])). 167.19/25.08 tff('132', plain, 167.19/25.08 ~ (symdiffIneg2 => 167.19/25.08 (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!((^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['131'])). 167.19/25.08 tff('133', plain, 167.19/25.08 ~ (secondinupair => 167.19/25.08 (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!((^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['132'])). 167.19/25.08 tff('134', plain, 167.19/25.08 ~ (setukpairIL => 167.19/25.08 (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!((^[Y0 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['133'])). 167.19/25.08 tff('135', plain, 167.19/25.08 ~ (setukpairIR => 167.19/25.08 (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!((^[Y0 : $i]: 167.19/25.08 (!!((^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['134'])). 167.19/25.08 tff('136', plain, 167.19/25.08 ~ (kpairiskpair => 167.19/25.08 (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!((^[Y0 : $i]: 167.19/25.08 (!!((^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['135'])). 167.19/25.08 tff('137', plain, 167.19/25.08 ~ (kpairp => 167.19/25.08 (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!((^[Y0 : $i]: 167.19/25.08 (!!((^[Y1 : $i]: 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['136'])). 167.19/25.08 tff('138', plain, 167.19/25.08 ~ (singletonsubset => 167.19/25.08 (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!((^[Y0 : $i]: 167.19/25.08 (!!((^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['137'])). 167.19/25.08 tff('139', plain, 167.19/25.08 ~ (singletoninpowerset => 167.19/25.08 (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!((^[Y0 : $i]: 167.19/25.08 (!!((^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['138'])). 167.19/25.08 tff('140', plain, 167.19/25.08 ~ (singletoninpowunion => 167.19/25.08 (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!((^[Y0 : $i]: 167.19/25.08 (!!((^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['139'])). 167.19/25.08 tff('141', plain, 167.19/25.08 ~ (upairset2E => 167.19/25.08 (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!((^[Y0 : $i]: 167.19/25.08 (!!((^[Y1 : $i]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['140'])). 167.19/25.08 tff('142', plain, 167.19/25.08 ~ (upairsubunion => 167.19/25.08 (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!((^[Y0 : $i]: 167.19/25.08 (!!((^[Y1 : $i]: 167.19/25.08 (!!((^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['141'])). 167.19/25.08 tff('143', plain, 167.19/25.08 ~ (upairinpowunion => 167.19/25.08 (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!((^[Y0 : $i]: 167.19/25.08 (!!((^[Y1 : $i]: 167.19/25.08 (!!((^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['142'])). 167.19/25.08 tff('144', plain, 167.19/25.08 ~ (ubforcartprodlem1 => 167.19/25.08 (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!((^[Y0 : $i]: 167.19/25.08 (!!((^[Y1 : $i]: 167.19/25.08 (!!((^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['143'])). 167.19/25.08 tff('145', plain, 167.19/25.08 ~ (ubforcartprodlem2 => 167.19/25.08 (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!((^[Y0 : $i]: 167.19/25.08 (!!((^[Y1 : $i]: 167.19/25.08 (!!((^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['144'])). 167.19/25.08 tff('146', plain, 167.19/25.08 ~ (ubforcartprodlem3 => 167.19/25.08 (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!((^[Y0 : $i]: 167.19/25.08 (!!((^[Y1 : $i]: 167.19/25.08 (!!((^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['145'])). 167.19/25.08 tff('147', plain, 167.19/25.08 ~ (cartprodpairin => 167.19/25.08 (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!((^[Y0 : $i]: 167.19/25.08 (!!((^[Y1 : $i]: 167.19/25.08 (!!((^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 ( 167.19/25.08 in(Y3, 167.19/25.08 Y0) => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['146'])). 167.19/25.08 tff('148', plain, 167.19/25.08 ~ (cartprodmempair1 => 167.19/25.08 (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!((^[Y0 : $i]: 167.19/25.08 (!!((^[Y1 : $i]: 167.19/25.08 (!!((^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!(( 167.19/25.08 ^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['147'])). 167.19/25.08 tff('149', plain, 167.19/25.08 ~ (cartprodmempair => 167.19/25.08 (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!((^[Y0 : $i]: 167.19/25.08 (!!((^[Y1 : $i]: 167.19/25.08 (!!((^[Y2 : ( 167.19/25.08 $i * $i) > $o]: 167.19/25.08 (!!((^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['148'])). 167.19/25.08 tff('150', plain, 167.19/25.08 ~ (setunionE2 => 167.19/25.08 (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!((^[Y0 : $i]: 167.19/25.08 (!!((^[Y1 : $i]: 167.19/25.08 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.08 (!!((^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 (!!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['149'])). 167.19/25.08 tff('151', plain, 167.19/25.08 ~ (setunionsingleton1 => 167.19/25.08 (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!((^[Y0 : $i]: 167.19/25.08 (!!((^[Y1 : $i]: 167.19/25.08 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.08 (!!((^[Y3 : $i]: 167.19/25.08 ( 167.19/25.08 in(Y3, 167.19/25.08 Y0) => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.08 Y2)) => 167.19/25.08 Y2(Y3, 167.19/25.08 Y4))))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.08 inference('lazy_cnf_imply', [status(thm)], ['150'])). 167.19/25.08 tff('152', plain, 167.19/25.08 ~ (setunionsingleton2 => 167.19/25.08 (setunionsingleton => 167.19/25.08 (singletonprop => 167.19/25.08 (ex1E1 => 167.19/25.08 (ex1I => 167.19/25.08 (ex1I2 => 167.19/25.08 (singletonsuniq => 167.19/25.08 (setukpairinjL1 => 167.19/25.08 (kfstsingleton => 167.19/25.08 (theprop => 167.19/25.08 (kfstpairEq => 167.19/25.08 (cartprodfstin => 167.19/25.08 (setukpairinjL2 => 167.19/25.08 (setukpairinjL => 167.19/25.08 (setukpairinjR11 => 167.19/25.08 (setukpairinjR12 => 167.19/25.08 (setukpairinjR1 => 167.19/25.08 (upairequniteq => 167.19/25.08 (setukpairinjR2 => 167.19/25.08 (setukpairinjR => 167.19/25.08 (ksndsingleton => 167.19/25.08 (ksndpairEq => 167.19/25.08 (kpairsurjEq => 167.19/25.08 (cartprodsndin => 167.19/25.08 (cartprodpairmemEL => 167.19/25.08 (cartprodpairmemER => 167.19/25.08 (cartprodmempaircEq => 167.19/25.08 (cartprodfstpairEq => 167.19/25.08 (cartprodsndpairEq => 167.19/25.08 (cartprodpairsurjEq => 167.19/25.08 (dpsetconstrI => 167.19/25.08 (dpsetconstrSub => 167.19/25.08 (setOfPairsIsBReln => 167.19/25.08 (!!((^[Y0 : $i]: 167.19/25.08 (!!((^[Y1 : $i]: 167.19/25.08 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.08 (!!((^[Y3 : $i]: 167.19/25.08 (in(Y3, 167.19/25.08 Y0) => 167.19/25.08 ( 167.19/25.08 !!(( 167.19/25.08 ^[Y4 : $i]: 167.19/25.08 (in(Y4, 167.19/25.08 Y1) => 167.19/25.08 (in( 167.19/25.08 kpair( 167.19/25.08 Y3, Y4), 167.19/25.08 dpsetconstr( 167.19/25.08 Y0, Y1, 167.19/25.09 Y2)) => 167.19/25.09 Y2(Y3, 167.19/25.09 Y4)))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['151'])). 167.19/25.09 tff('153', plain, 167.19/25.09 ~ (setunionsingleton => 167.19/25.09 (singletonprop => 167.19/25.09 (ex1E1 => 167.19/25.09 (ex1I => 167.19/25.09 (ex1I2 => 167.19/25.09 (singletonsuniq => 167.19/25.09 (setukpairinjL1 => 167.19/25.09 (kfstsingleton => 167.19/25.09 (theprop => 167.19/25.09 (kfstpairEq => 167.19/25.09 (cartprodfstin => 167.19/25.09 (setukpairinjL2 => 167.19/25.09 (setukpairinjL => 167.19/25.09 (setukpairinjR11 => 167.19/25.09 (setukpairinjR12 => 167.19/25.09 (setukpairinjR1 => 167.19/25.09 (upairequniteq => 167.19/25.09 (setukpairinjR2 => 167.19/25.09 (setukpairinjR => 167.19/25.09 (ksndsingleton => 167.19/25.09 (ksndpairEq => 167.19/25.09 (kpairsurjEq => 167.19/25.09 (cartprodsndin => 167.19/25.09 (cartprodpairmemEL => 167.19/25.09 (cartprodpairmemER => 167.19/25.09 (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!(( 167.19/25.09 ^[Y4 : $i]: 167.19/25.09 (in(Y4, 167.19/25.09 Y1) => 167.19/25.09 (in( 167.19/25.09 kpair( 167.19/25.09 Y3, Y4), 167.19/25.09 dpsetconstr( 167.19/25.09 Y0, Y1, 167.19/25.09 Y2)) => 167.19/25.09 Y2(Y3, 167.19/25.09 Y4))))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['152'])). 167.19/25.09 tff('154', plain, 167.19/25.09 ~ (singletonprop => 167.19/25.09 (ex1E1 => 167.19/25.09 (ex1I => 167.19/25.09 (ex1I2 => 167.19/25.09 (singletonsuniq => 167.19/25.09 (setukpairinjL1 => 167.19/25.09 (kfstsingleton => 167.19/25.09 (theprop => 167.19/25.09 (kfstpairEq => 167.19/25.09 (cartprodfstin => 167.19/25.09 (setukpairinjL2 => 167.19/25.09 (setukpairinjL => 167.19/25.09 (setukpairinjR11 => 167.19/25.09 (setukpairinjR12 => 167.19/25.09 (setukpairinjR1 => 167.19/25.09 (upairequniteq => 167.19/25.09 (setukpairinjR2 => 167.19/25.09 (setukpairinjR => 167.19/25.09 (ksndsingleton => 167.19/25.09 (ksndpairEq => 167.19/25.09 (kpairsurjEq => 167.19/25.09 (cartprodsndin => 167.19/25.09 (cartprodpairmemEL => 167.19/25.09 (cartprodpairmemER => 167.19/25.09 (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!(( 167.19/25.09 ^[Y4 : $i]: 167.19/25.09 (in(Y4, 167.19/25.09 Y1) => 167.19/25.09 (in( 167.19/25.09 kpair( 167.19/25.09 Y3, Y4), 167.19/25.09 dpsetconstr( 167.19/25.09 Y0, Y1, 167.19/25.09 Y2)) => 167.19/25.09 Y2(Y3, 167.19/25.09 Y4)))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['153'])). 167.19/25.09 tff('155', plain, 167.19/25.09 ~ (ex1E1 => 167.19/25.09 (ex1I => 167.19/25.09 (ex1I2 => 167.19/25.09 (singletonsuniq => 167.19/25.09 (setukpairinjL1 => 167.19/25.09 (kfstsingleton => 167.19/25.09 (theprop => 167.19/25.09 (kfstpairEq => 167.19/25.09 (cartprodfstin => 167.19/25.09 (setukpairinjL2 => 167.19/25.09 (setukpairinjL => 167.19/25.09 (setukpairinjR11 => 167.19/25.09 (setukpairinjR12 => 167.19/25.09 (setukpairinjR1 => 167.19/25.09 (upairequniteq => 167.19/25.09 (setukpairinjR2 => 167.19/25.09 (setukpairinjR => 167.19/25.09 (ksndsingleton => 167.19/25.09 (ksndpairEq => 167.19/25.09 (kpairsurjEq => 167.19/25.09 (cartprodsndin => 167.19/25.09 (cartprodpairmemEL => 167.19/25.09 (cartprodpairmemER => 167.19/25.09 (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!(( 167.19/25.09 ^[Y4 : $i]: 167.19/25.09 ( 167.19/25.09 in(Y4, 167.19/25.09 Y1) => 167.19/25.09 ( 167.19/25.09 in( 167.19/25.09 kpair( 167.19/25.09 Y3, Y4), 167.19/25.09 dpsetconstr( 167.19/25.09 Y0, Y1, 167.19/25.09 Y2)) => 167.19/25.09 Y2(Y3, 167.19/25.09 Y4))))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['154'])). 167.19/25.09 tff('156', plain, 167.19/25.09 ~ (ex1I => 167.19/25.09 (ex1I2 => 167.19/25.09 (singletonsuniq => 167.19/25.09 (setukpairinjL1 => 167.19/25.09 (kfstsingleton => 167.19/25.09 (theprop => 167.19/25.09 (kfstpairEq => 167.19/25.09 (cartprodfstin => 167.19/25.09 (setukpairinjL2 => 167.19/25.09 (setukpairinjL => 167.19/25.09 (setukpairinjR11 => 167.19/25.09 (setukpairinjR12 => 167.19/25.09 (setukpairinjR1 => 167.19/25.09 (upairequniteq => 167.19/25.09 (setukpairinjR2 => 167.19/25.09 (setukpairinjR => 167.19/25.09 (ksndsingleton => 167.19/25.09 (ksndpairEq => 167.19/25.09 (kpairsurjEq => 167.19/25.09 (cartprodsndin => 167.19/25.09 (cartprodpairmemEL => 167.19/25.09 (cartprodpairmemER => 167.19/25.09 (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!(( 167.19/25.09 ^[Y4 : $i]: 167.19/25.09 (in(Y4, 167.19/25.09 Y1) => 167.19/25.09 ( 167.19/25.09 in( 167.19/25.09 kpair( 167.19/25.09 Y3, Y4), 167.19/25.09 dpsetconstr( 167.19/25.09 Y0, Y1, 167.19/25.09 Y2)) => 167.19/25.09 Y2(Y3, 167.19/25.09 Y4)))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['155'])). 167.19/25.09 tff('157', plain, 167.19/25.09 ~ (ex1I2 => 167.19/25.09 (singletonsuniq => 167.19/25.09 (setukpairinjL1 => 167.19/25.09 (kfstsingleton => 167.19/25.09 (theprop => 167.19/25.09 (kfstpairEq => 167.19/25.09 (cartprodfstin => 167.19/25.09 (setukpairinjL2 => 167.19/25.09 (setukpairinjL => 167.19/25.09 (setukpairinjR11 => 167.19/25.09 (setukpairinjR12 => 167.19/25.09 (setukpairinjR1 => 167.19/25.09 (upairequniteq => 167.19/25.09 (setukpairinjR2 => 167.19/25.09 (setukpairinjR => 167.19/25.09 (ksndsingleton => 167.19/25.09 (ksndpairEq => 167.19/25.09 (kpairsurjEq => 167.19/25.09 (cartprodsndin => 167.19/25.09 (cartprodpairmemEL => 167.19/25.09 (cartprodpairmemER => 167.19/25.09 (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, 167.19/25.09 Y1) => 167.19/25.09 (in( 167.19/25.09 kpair( 167.19/25.09 Y3, Y4), 167.19/25.09 dpsetconstr( 167.19/25.09 Y0, Y1, 167.19/25.09 Y2)) => 167.19/25.09 Y2(Y3, 167.19/25.09 Y4))))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['156'])). 167.19/25.09 tff('158', plain, 167.19/25.09 ~ (singletonsuniq => 167.19/25.09 (setukpairinjL1 => 167.19/25.09 (kfstsingleton => 167.19/25.09 (theprop => 167.19/25.09 (kfstpairEq => 167.19/25.09 (cartprodfstin => 167.19/25.09 (setukpairinjL2 => 167.19/25.09 (setukpairinjL => 167.19/25.09 (setukpairinjR11 => 167.19/25.09 (setukpairinjR12 => 167.19/25.09 (setukpairinjR1 => 167.19/25.09 (upairequniteq => 167.19/25.09 (setukpairinjR2 => 167.19/25.09 (setukpairinjR => 167.19/25.09 (ksndsingleton => 167.19/25.09 (ksndpairEq => 167.19/25.09 (kpairsurjEq => 167.19/25.09 (cartprodsndin => 167.19/25.09 (cartprodpairmemEL => 167.19/25.09 (cartprodpairmemER => 167.19/25.09 (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, 167.19/25.09 Y1) => 167.19/25.09 (in( 167.19/25.09 kpair( 167.19/25.09 Y3, Y4), 167.19/25.09 dpsetconstr( 167.19/25.09 Y0, Y1, 167.19/25.09 Y2)) => 167.19/25.09 Y2(Y3, 167.19/25.09 Y4)))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['157'])). 167.19/25.09 tff('159', plain, 167.19/25.09 ~ (setukpairinjL1 => 167.19/25.09 (kfstsingleton => 167.19/25.09 (theprop => 167.19/25.09 (kfstpairEq => 167.19/25.09 (cartprodfstin => 167.19/25.09 (setukpairinjL2 => 167.19/25.09 (setukpairinjL => 167.19/25.09 (setukpairinjR11 => 167.19/25.09 (setukpairinjR12 => 167.19/25.09 (setukpairinjR1 => 167.19/25.09 (upairequniteq => 167.19/25.09 (setukpairinjR2 => 167.19/25.09 (setukpairinjR => 167.19/25.09 (ksndsingleton => 167.19/25.09 (ksndpairEq => 167.19/25.09 (kpairsurjEq => 167.19/25.09 (cartprodsndin => 167.19/25.09 (cartprodpairmemEL => 167.19/25.09 (cartprodpairmemER => 167.19/25.09 (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 ( 167.19/25.09 in(Y4, 167.19/25.09 Y1) => 167.19/25.09 ( 167.19/25.09 in( 167.19/25.09 kpair( 167.19/25.09 Y3, Y4), 167.19/25.09 dpsetconstr( 167.19/25.09 Y0, Y1, 167.19/25.09 Y2)) => 167.19/25.09 Y2(Y3, 167.19/25.09 Y4))))))))))))))))))))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['158'])). 167.19/25.09 tff('160', plain, 167.19/25.09 ~ (kfstsingleton => 167.19/25.09 (theprop => 167.19/25.09 (kfstpairEq => 167.19/25.09 (cartprodfstin => 167.19/25.09 (setukpairinjL2 => 167.19/25.09 (setukpairinjL => 167.19/25.09 (setukpairinjR11 => 167.19/25.09 (setukpairinjR12 => 167.19/25.09 (setukpairinjR1 => 167.19/25.09 (upairequniteq => 167.19/25.09 (setukpairinjR2 => 167.19/25.09 (setukpairinjR => 167.19/25.09 (ksndsingleton => 167.19/25.09 (ksndpairEq => 167.19/25.09 (kpairsurjEq => 167.19/25.09 (cartprodsndin => 167.19/25.09 (cartprodpairmemEL => 167.19/25.09 (cartprodpairmemER => 167.19/25.09 (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, 167.19/25.09 Y1) => 167.19/25.09 ( 167.19/25.09 in( 167.19/25.09 kpair( 167.19/25.09 Y3, Y4), 167.19/25.09 dpsetconstr( 167.19/25.09 Y0, Y1, 167.19/25.09 Y2)) => 167.19/25.09 Y2(Y3, 167.19/25.09 Y4)))))))))))))))))))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['159'])). 167.19/25.09 tff('161', plain, 167.19/25.09 ~ (theprop => 167.19/25.09 (kfstpairEq => 167.19/25.09 (cartprodfstin => 167.19/25.09 (setukpairinjL2 => 167.19/25.09 (setukpairinjL => 167.19/25.09 (setukpairinjR11 => 167.19/25.09 (setukpairinjR12 => 167.19/25.09 (setukpairinjR1 => 167.19/25.09 (upairequniteq => 167.19/25.09 (setukpairinjR2 => 167.19/25.09 (setukpairinjR => 167.19/25.09 (ksndsingleton => 167.19/25.09 (ksndpairEq => 167.19/25.09 (kpairsurjEq => 167.19/25.09 (cartprodsndin => 167.19/25.09 (cartprodpairmemEL => 167.19/25.09 (cartprodpairmemER => 167.19/25.09 (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, Y1) => 167.19/25.09 (in( 167.19/25.09 kpair( 167.19/25.09 Y3, Y4), 167.19/25.09 dpsetconstr( 167.19/25.09 Y0, Y1, 167.19/25.09 Y2)) => 167.19/25.09 Y2(Y3, 167.19/25.09 Y4))))))))))))))))))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['160'])). 167.19/25.09 tff('162', plain, 167.19/25.09 ~ (kfstpairEq => 167.19/25.09 (cartprodfstin => 167.19/25.09 (setukpairinjL2 => 167.19/25.09 (setukpairinjL => 167.19/25.09 (setukpairinjR11 => 167.19/25.09 (setukpairinjR12 => 167.19/25.09 (setukpairinjR1 => 167.19/25.09 (upairequniteq => 167.19/25.09 (setukpairinjR2 => 167.19/25.09 (setukpairinjR => 167.19/25.09 (ksndsingleton => 167.19/25.09 (ksndpairEq => 167.19/25.09 (kpairsurjEq => 167.19/25.09 (cartprodsndin => 167.19/25.09 (cartprodpairmemEL => 167.19/25.09 (cartprodpairmemER => 167.19/25.09 (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, Y1) => 167.19/25.09 (in( 167.19/25.09 kpair( 167.19/25.09 Y3, Y4), 167.19/25.09 dpsetconstr( 167.19/25.09 Y0, Y1, 167.19/25.09 Y2)) => 167.19/25.09 Y2(Y3, Y4)))))))))))))))))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['161'])). 167.19/25.09 tff('163', plain, 167.19/25.09 ~ (cartprodfstin => 167.19/25.09 (setukpairinjL2 => 167.19/25.09 (setukpairinjL => 167.19/25.09 (setukpairinjR11 => 167.19/25.09 (setukpairinjR12 => 167.19/25.09 (setukpairinjR1 => 167.19/25.09 (upairequniteq => 167.19/25.09 (setukpairinjR2 => 167.19/25.09 (setukpairinjR => 167.19/25.09 (ksndsingleton => 167.19/25.09 (ksndpairEq => 167.19/25.09 (kpairsurjEq => 167.19/25.09 (cartprodsndin => 167.19/25.09 (cartprodpairmemEL => 167.19/25.09 (cartprodpairmemER => 167.19/25.09 (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, Y1) => 167.19/25.09 (in( 167.19/25.09 kpair( 167.19/25.09 Y3, Y4), 167.19/25.09 dpsetconstr( 167.19/25.09 Y0, Y1, 167.19/25.09 Y2)) => 167.19/25.09 Y2(Y3, Y4))))))))))))))))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['162'])). 167.19/25.09 tff('164', plain, 167.19/25.09 ~ (setukpairinjL2 => 167.19/25.09 (setukpairinjL => 167.19/25.09 (setukpairinjR11 => 167.19/25.09 (setukpairinjR12 => 167.19/25.09 (setukpairinjR1 => 167.19/25.09 (upairequniteq => 167.19/25.09 (setukpairinjR2 => 167.19/25.09 (setukpairinjR => 167.19/25.09 (ksndsingleton => 167.19/25.09 (ksndpairEq => 167.19/25.09 (kpairsurjEq => 167.19/25.09 (cartprodsndin => 167.19/25.09 (cartprodpairmemEL => 167.19/25.09 (cartprodpairmemER => 167.19/25.09 (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, Y1) => 167.19/25.09 (in( 167.19/25.09 kpair(Y3, 167.19/25.09 Y4), 167.19/25.09 dpsetconstr( 167.19/25.09 Y0, Y1, 167.19/25.09 Y2)) => 167.19/25.09 Y2(Y3, Y4)))))))))))))))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['163'])). 167.19/25.09 tff('165', plain, 167.19/25.09 ~ (setukpairinjL => 167.19/25.09 (setukpairinjR11 => 167.19/25.09 (setukpairinjR12 => 167.19/25.09 (setukpairinjR1 => 167.19/25.09 (upairequniteq => 167.19/25.09 (setukpairinjR2 => 167.19/25.09 (setukpairinjR => 167.19/25.09 (ksndsingleton => 167.19/25.09 (ksndpairEq => 167.19/25.09 (kpairsurjEq => 167.19/25.09 (cartprodsndin => 167.19/25.09 (cartprodpairmemEL => 167.19/25.09 (cartprodpairmemER => 167.19/25.09 (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, Y1) => 167.19/25.09 (in( 167.19/25.09 kpair(Y3, 167.19/25.09 Y4), 167.19/25.09 dpsetconstr( 167.19/25.09 Y0, Y1, 167.19/25.09 Y2)) => 167.19/25.09 Y2(Y3, Y4))))))))))))))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['164'])). 167.19/25.09 tff('166', plain, 167.19/25.09 ~ (setukpairinjR11 => 167.19/25.09 (setukpairinjR12 => 167.19/25.09 (setukpairinjR1 => 167.19/25.09 (upairequniteq => 167.19/25.09 (setukpairinjR2 => 167.19/25.09 (setukpairinjR => 167.19/25.09 (ksndsingleton => 167.19/25.09 (ksndpairEq => 167.19/25.09 (kpairsurjEq => 167.19/25.09 (cartprodsndin => 167.19/25.09 (cartprodpairmemEL => 167.19/25.09 (cartprodpairmemER => 167.19/25.09 (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, Y1) => 167.19/25.09 (in( 167.19/25.09 kpair(Y3, 167.19/25.09 Y4), 167.19/25.09 dpsetconstr( 167.19/25.09 Y0, Y1, 167.19/25.09 Y2)) => 167.19/25.09 Y2(Y3, Y4)))))))))))))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['165'])). 167.19/25.09 tff('167', plain, 167.19/25.09 ~ (setukpairinjR12 => 167.19/25.09 (setukpairinjR1 => 167.19/25.09 (upairequniteq => 167.19/25.09 (setukpairinjR2 => 167.19/25.09 (setukpairinjR => 167.19/25.09 (ksndsingleton => 167.19/25.09 (ksndpairEq => 167.19/25.09 (kpairsurjEq => 167.19/25.09 (cartprodsndin => 167.19/25.09 (cartprodpairmemEL => 167.19/25.09 (cartprodpairmemER => 167.19/25.09 (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, Y1) => 167.19/25.09 (in( 167.19/25.09 kpair(Y3, Y4), 167.19/25.09 dpsetconstr( 167.19/25.09 Y0, Y1, Y2)) => 167.19/25.09 Y2(Y3, Y4))))))))))))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['166'])). 167.19/25.09 tff('168', plain, 167.19/25.09 ~ (setukpairinjR1 => 167.19/25.09 (upairequniteq => 167.19/25.09 (setukpairinjR2 => 167.19/25.09 (setukpairinjR => 167.19/25.09 (ksndsingleton => 167.19/25.09 (ksndpairEq => 167.19/25.09 (kpairsurjEq => 167.19/25.09 (cartprodsndin => 167.19/25.09 (cartprodpairmemEL => 167.19/25.09 (cartprodpairmemER => 167.19/25.09 (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, Y1) => 167.19/25.09 (in( 167.19/25.09 kpair(Y3, Y4), 167.19/25.09 dpsetconstr( 167.19/25.09 Y0, Y1, Y2)) => 167.19/25.09 Y2(Y3, Y4)))))))))))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['167'])). 167.19/25.09 tff('169', plain, 167.19/25.09 ~ (upairequniteq => 167.19/25.09 (setukpairinjR2 => 167.19/25.09 (setukpairinjR => 167.19/25.09 (ksndsingleton => 167.19/25.09 (ksndpairEq => 167.19/25.09 (kpairsurjEq => 167.19/25.09 (cartprodsndin => 167.19/25.09 (cartprodpairmemEL => 167.19/25.09 (cartprodpairmemER => 167.19/25.09 (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, Y1) => 167.19/25.09 (in( 167.19/25.09 kpair(Y3, Y4), 167.19/25.09 dpsetconstr( 167.19/25.09 Y0, Y1, Y2)) => 167.19/25.09 Y2(Y3, Y4))))))))))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['168'])). 167.19/25.09 tff('170', plain, 167.19/25.09 ~ (setukpairinjR2 => 167.19/25.09 (setukpairinjR => 167.19/25.09 (ksndsingleton => 167.19/25.09 (ksndpairEq => 167.19/25.09 (kpairsurjEq => 167.19/25.09 (cartprodsndin => 167.19/25.09 (cartprodpairmemEL => 167.19/25.09 (cartprodpairmemER => 167.19/25.09 (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, Y1) => 167.19/25.09 (in(kpair(Y3, Y4), 167.19/25.09 dpsetconstr(Y0, 167.19/25.09 Y1, Y2)) => 167.19/25.09 Y2(Y3, Y4)))))))))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['169'])). 167.19/25.09 tff('171', plain, 167.19/25.09 ~ (setukpairinjR => 167.19/25.09 (ksndsingleton => 167.19/25.09 (ksndpairEq => 167.19/25.09 (kpairsurjEq => 167.19/25.09 (cartprodsndin => 167.19/25.09 (cartprodpairmemEL => 167.19/25.09 (cartprodpairmemER => 167.19/25.09 (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, Y1) => 167.19/25.09 (in(kpair(Y3, Y4), 167.19/25.09 dpsetconstr(Y0, 167.19/25.09 Y1, Y2)) => 167.19/25.09 Y2(Y3, Y4))))))))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['170'])). 167.19/25.09 tff('172', plain, 167.19/25.09 ~ (ksndsingleton => 167.19/25.09 (ksndpairEq => 167.19/25.09 (kpairsurjEq => 167.19/25.09 (cartprodsndin => 167.19/25.09 (cartprodpairmemEL => 167.19/25.09 (cartprodpairmemER => 167.19/25.09 (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, Y1) => 167.19/25.09 (in(kpair(Y3, Y4), 167.19/25.09 dpsetconstr(Y0, 167.19/25.09 Y1, Y2)) => 167.19/25.09 Y2(Y3, Y4)))))))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['171'])). 167.19/25.09 tff('173', plain, 167.19/25.09 ~ (ksndpairEq => 167.19/25.09 (kpairsurjEq => 167.19/25.09 (cartprodsndin => 167.19/25.09 (cartprodpairmemEL => 167.19/25.09 (cartprodpairmemER => 167.19/25.09 (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, Y1) => 167.19/25.09 (in(kpair(Y3, Y4), 167.19/25.09 dpsetconstr(Y0, 167.19/25.09 Y1, Y2)) => 167.19/25.09 Y2(Y3, Y4))))))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['172'])). 167.19/25.09 tff('174', plain, 167.19/25.09 ~ (kpairsurjEq => 167.19/25.09 (cartprodsndin => 167.19/25.09 (cartprodpairmemEL => 167.19/25.09 (cartprodpairmemER => 167.19/25.09 (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, Y1) => 167.19/25.09 (in(kpair(Y3, Y4), 167.19/25.09 dpsetconstr(Y0, Y1, 167.19/25.09 Y2)) => 167.19/25.09 Y2(Y3, Y4)))))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['173'])). 167.19/25.09 tff('175', plain, 167.19/25.09 ~ (cartprodsndin => 167.19/25.09 (cartprodpairmemEL => 167.19/25.09 (cartprodpairmemER => 167.19/25.09 (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, Y1) => 167.19/25.09 (in(kpair(Y3, Y4), 167.19/25.09 dpsetconstr(Y0, Y1, 167.19/25.09 Y2)) => 167.19/25.09 Y2(Y3, Y4))))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['174'])). 167.19/25.09 tff('176', plain, 167.19/25.09 ~ (cartprodpairmemEL => 167.19/25.09 (cartprodpairmemER => 167.19/25.09 (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, Y1) => 167.19/25.09 (in(kpair(Y3, Y4), 167.19/25.09 dpsetconstr(Y0, Y1, 167.19/25.09 Y2)) => 167.19/25.09 Y2(Y3, Y4)))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['175'])). 167.19/25.09 tff('177', plain, 167.19/25.09 ~ (cartprodpairmemER => 167.19/25.09 (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, Y1) => 167.19/25.09 (in(kpair(Y3, Y4), 167.19/25.09 dpsetconstr(Y0, Y1, Y2)) => 167.19/25.09 Y2(Y3, Y4))))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['176'])). 167.19/25.09 tff('178', plain, 167.19/25.09 ~ (cartprodmempaircEq => 167.19/25.09 (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, Y1) => 167.19/25.09 (in(kpair(Y3, Y4), 167.19/25.09 dpsetconstr(Y0, Y1, Y2)) => 167.19/25.09 Y2(Y3, Y4)))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['177'])). 167.19/25.09 tff('179', plain, 167.19/25.09 ~ (cartprodfstpairEq => 167.19/25.09 (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, Y1) => 167.19/25.09 (in(kpair(Y3, Y4), 167.19/25.09 dpsetconstr(Y0, Y1, Y2)) => 167.19/25.09 Y2(Y3, Y4))))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['178'])). 167.19/25.09 tff('180', plain, 167.19/25.09 ~ (cartprodsndpairEq => 167.19/25.09 (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, Y1) => 167.19/25.09 (in(kpair(Y3, Y4), 167.19/25.09 dpsetconstr(Y0, Y1, Y2)) => 167.19/25.09 Y2(Y3, Y4)))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['179'])). 167.19/25.09 tff('181', plain, 167.19/25.09 ~ (cartprodpairsurjEq => 167.19/25.09 (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, Y1) => 167.19/25.09 (in(kpair(Y3, Y4), 167.19/25.09 dpsetconstr(Y0, Y1, Y2)) => 167.19/25.09 Y2(Y3, Y4))))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['180'])). 167.19/25.09 tff('182', plain, 167.19/25.09 ~ (dpsetconstrI => 167.19/25.09 (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, Y1) => 167.19/25.09 (in(kpair(Y3, Y4), 167.19/25.09 dpsetconstr(Y0, Y1, Y2)) => 167.19/25.09 Y2(Y3, Y4)))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['181'])). 167.19/25.09 tff('183', plain, 167.19/25.09 ~ (dpsetconstrSub => 167.19/25.09 (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, Y1) => 167.19/25.09 (in(kpair(Y3, Y4), 167.19/25.09 dpsetconstr(Y0, Y1, Y2)) => 167.19/25.09 Y2(Y3, Y4))))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['182'])). 167.19/25.09 tff('184', plain, 167.19/25.09 ~ (setOfPairsIsBReln => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, Y1) => 167.19/25.09 (in(kpair(Y3, Y4), 167.19/25.09 dpsetconstr(Y0, Y1, Y2)) => 167.19/25.09 Y2(Y3, Y4)))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['183'])). 167.19/25.09 tff('185', plain, 167.19/25.09 ~ (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (!!((^[Y4 : $i]: 167.19/25.09 (in(Y4, Y1) => 167.19/25.09 (in(kpair(Y3, Y4), 167.19/25.09 dpsetconstr(Y0, Y1, Y2)) => 167.19/25.09 Y2(Y3, Y4))))))))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['184'])). 167.19/25.09 tff('186', plain, 167.19/25.09 ~ (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y2 : $i]: 167.19/25.09 (in(Y2, '#sk211') => 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 (in(Y3, Y0) => 167.19/25.09 (in(kpair(Y2, Y3), 167.19/25.09 dpsetconstr('#sk211', Y0, Y1)) => 167.19/25.09 Y1(Y2, Y3)))))))))))))))), 167.19/25.09 inference('lazy_cnf_exists', [status(thm)], ['185'])). 167.19/25.09 tff('187', plain, 167.19/25.09 ~ (!!((^[Y0 : ($i * $i) > $o]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (in(Y1, '#sk211') => 167.19/25.09 (!!((^[Y2 : $i]: 167.19/25.09 (in(Y2, '#sk212') => 167.19/25.09 (in(kpair(Y1, Y2), 167.19/25.09 dpsetconstr('#sk211', '#sk212', Y0)) => 167.19/25.09 Y0(Y1, Y2))))))))))))), 167.19/25.09 inference('lazy_cnf_exists', [status(thm)], ['186'])). 167.19/25.09 tff('188', plain, 167.19/25.09 ~ (!!((^[Y0 : $i]: 167.19/25.09 (in(Y0, '#sk211') => 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (in(Y1, '#sk212') => 167.19/25.09 (in(kpair(Y0, Y1), 167.19/25.09 dpsetconstr('#sk211', '#sk212', '#sk221')) => 167.19/25.09 '#sk221'(Y0, Y1)))))))))), 167.19/25.09 inference('lazy_cnf_exists', [status(thm)], ['187'])). 167.19/25.09 tff(dpsetconstr, axiom, dpsetconstr(X0, X1, X2) = 167.19/25.09 dsetconstr(cartprod(X0, X1), 167.19/25.09 (^[Y0 : $i]: 167.19/25.09 (??((^[Y1 : $i]: 167.19/25.09 (in(Y1, X0) & 167.19/25.09 (??((^[Y2 : $i]: (in(Y2, X1)&X2(Y1, Y2)&(Y0 = kpair(Y1, Y2))))))))))))). 167.19/25.09 tff('189', plain, 167.19/25.09 ![X2:(($i * $i) > $o),X1,X0]: 167.19/25.09 (dpsetconstr(X0,X1,X2) = 167.19/25.09 dsetconstr(cartprod(X0,X1), 167.19/25.09 ^[V_1]: 167.19/25.09 (?[X6]: 167.19/25.09 (in(X6,X0) & 167.19/25.09 (?[X8]: 167.19/25.09 (in(X8,X1) & X2(X6,X8) & (V_1 = kpair(X6,X8)))))))), 167.19/25.09 define([status(thm)])). 167.19/25.09 tff('190', plain, 167.19/25.09 ~ (!!((^[Y0 : $i]: 167.19/25.09 (in(Y0, '#sk211') => 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (in(Y1, '#sk212') => 167.19/25.09 (in(kpair(Y0, Y1), 167.19/25.09 dsetconstr(cartprod('#sk211', '#sk212'), 167.19/25.09 (^[Y2 : $i]: 167.19/25.09 (??((^[Y3 : $i]: 167.19/25.09 (in(Y3, '#sk211') & 167.19/25.09 (??((^[Y4 : $i]: 167.19/25.09 (in(Y4, '#sk212')& 167.19/25.09 '#sk221'(Y3, Y4)& 167.19/25.09 (Y2 = kpair(Y3, Y4))))))))))))) => 167.19/25.09 '#sk221'(Y0, Y1)))))))))), 167.19/25.09 inference('rw', [status(thm)], ['188', '189'])). 167.19/25.09 tff('191', plain, 167.19/25.09 ~ (in('#sk229', '#sk211') => 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (in(Y0, '#sk212') => 167.19/25.09 (in(kpair('#sk229', Y0), 167.19/25.09 dsetconstr(cartprod('#sk211', '#sk212'), 167.19/25.09 (^[Y1 : $i]: 167.19/25.09 (??((^[Y2 : $i]: 167.19/25.09 (in(Y2, '#sk211') & 167.19/25.09 (??((^[Y3 : $i]: 167.19/25.09 (in(Y3, '#sk212')&'#sk221'(Y2, Y3)& 167.19/25.09 (Y1 = kpair(Y2, Y3))))))))))))) => 167.19/25.09 '#sk221'('#sk229', Y0))))))), 167.19/25.09 inference('lazy_cnf_exists', [status(thm)], ['190'])). 167.19/25.09 tff('192', plain, 167.19/25.09 ~ (!!((^[Y0 : $i]: 167.19/25.09 (in(Y0, '#sk212') => 167.19/25.09 (in(kpair('#sk229', Y0), 167.19/25.09 dsetconstr(cartprod('#sk211', '#sk212'), 167.19/25.09 (^[Y1 : $i]: 167.19/25.09 (??((^[Y2 : $i]: 167.19/25.09 (in(Y2, '#sk211') & 167.19/25.09 (??((^[Y3 : $i]: 167.19/25.09 (in(Y3, '#sk212')&'#sk221'(Y2, Y3)& 167.19/25.09 (Y1 = kpair(Y2, Y3))))))))))))) => 167.19/25.09 '#sk221'('#sk229', Y0)))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['191'])). 167.19/25.09 tff('193', plain, 167.19/25.09 ~ (in('#sk235', '#sk212') => 167.19/25.09 (in(kpair('#sk229', '#sk235'), 167.19/25.09 dsetconstr(cartprod('#sk211', '#sk212'), 167.19/25.09 (^[Y0 : $i]: 167.19/25.09 (??((^[Y1 : $i]: 167.19/25.09 (in(Y1, '#sk211') & 167.19/25.09 (??((^[Y2 : $i]: 167.19/25.09 (in(Y2, '#sk212')&'#sk221'(Y1, Y2)& 167.19/25.09 (Y0 = kpair(Y1, Y2))))))))))))) => 167.19/25.09 '#sk221'('#sk229', '#sk235'))), 167.19/25.09 inference('lazy_cnf_exists', [status(thm)], ['192'])). 167.19/25.09 tff('194', plain, 167.19/25.09 ~ (in(kpair('#sk229', '#sk235'), 167.19/25.09 dsetconstr(cartprod('#sk211', '#sk212'), 167.19/25.09 (^[Y0 : $i]: 167.19/25.09 (??((^[Y1 : $i]: 167.19/25.09 (in(Y1, '#sk211') & 167.19/25.09 (??((^[Y2 : $i]: 167.19/25.09 (in(Y2, '#sk212')&'#sk221'(Y1, Y2)& 167.19/25.09 (Y0 = kpair(Y1, Y2))))))))))))) => 167.19/25.09 '#sk221'('#sk229', '#sk235')), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['193'])). 167.19/25.09 tff('195', plain, ~ '#sk221'('#sk229', '#sk235'), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['194'])). 167.19/25.09 tff('196', plain, 167.19/25.09 in(kpair('#sk229', '#sk235'), 167.19/25.09 dsetconstr(cartprod('#sk211', '#sk212'), 167.19/25.09 (^[Y0 : $i]: 167.19/25.09 (??((^[Y1 : $i]: 167.19/25.09 (in(Y1, '#sk211') & 167.19/25.09 (??((^[Y2 : $i]: 167.19/25.09 (in(Y2, '#sk212')&'#sk221'(Y1, Y2)& 167.19/25.09 (Y0 = kpair(Y1, Y2))))))))))))), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['194'])). 167.19/25.09 tff('197', plain, dsetconstrER, 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['15'])). 167.19/25.09 tff(dsetconstrER, axiom, dsetconstrER => 167.19/25.09 ((!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i > $o]: 167.19/25.09 (!!((^[Y2 : $i]: 167.19/25.09 (in(Y2, dsetconstr(Y0, (^[Y3 : $i]: Y1(Y3)))) => 167.19/25.09 Y1(Y2))))))))))) 167.19/25.09 = $true)). 167.19/25.09 tff('198', plain, 167.19/25.09 dsetconstrER <=> 167.19/25.09 (![X5,X7:(($i > $o)),X9]: 167.19/25.09 (in(X9,dsetconstr(X5,^[V_1]: X7(V_1))) => X7(X9))), 167.19/25.09 inference('rw.lit', [status(esa)], [dsetconstrER])). 167.19/25.09 tff('199', plain, 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i > $o]: 167.19/25.09 (!!((^[Y2 : $i]: 167.19/25.09 (in(Y2, dsetconstr(Y0, (^[Y3 : $i]: Y1(Y3)))) => 167.19/25.09 Y1(Y2))))))))))), 167.19/25.09 inference('rw_clause', [status(thm)], ['197', '198'])). 167.19/25.09 tff('200', plain, 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i > $o]: 167.19/25.09 (!!((^[Y2 : $i]: (in(Y2, dsetconstr(Y0, Y1)) => Y1(Y2))))))))))), 167.19/25.09 inference('ho_norm', [status(thm)], ['199'])). 167.19/25.09 tff('201', plain, 167.19/25.09 ![X1]: 167.19/25.09 (!!((^[Y0 : $i > $o]: 167.19/25.09 (!!((^[Y1 : $i]: (in(Y1, dsetconstr(X1, Y0)) => Y0(Y1)))))))), 167.19/25.09 inference('lazy_cnf_forall', [status(thm)], ['200'])). 167.19/25.09 tff('202', plain, 167.19/25.09 ![X1, X2 : $i > $o]: 167.19/25.09 (!!((^[Y0 : $i]: (in(Y0, dsetconstr(X1, X2)) => X2(Y0))))), 167.19/25.09 inference('lazy_cnf_forall', [status(thm)], ['201'])). 167.19/25.09 tff('203', plain, 167.19/25.09 ![X1, X2 : $i > $o, X3]: (in(X3, dsetconstr(X1, X2)) => X2(X3)), 167.19/25.09 inference('lazy_cnf_forall', [status(thm)], ['202'])). 167.19/25.09 tff('204', plain, 167.19/25.09 ![X1, X2 : $i > $o, X3]: (~ in(X3, dsetconstr(X1, X2)) | X2(X3)), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['203'])). 167.19/25.09 tff('205', plain, 167.19/25.09 (^[Y0 : $i]: 167.19/25.09 (??((^[Y1 : $i]: 167.19/25.09 (in(Y1, '#sk211') & 167.19/25.09 (??((^[Y2 : $i]: 167.19/25.09 (in(Y2, '#sk212')&'#sk221'(Y1, Y2)& 167.19/25.09 (Y0 = kpair(Y1, Y2)))))))))))( 167.19/25.09 kpair('#sk229', '#sk235')), 167.19/25.09 inference('sup-', [status(thm)], ['196', '204'])). 167.19/25.09 tff('206', plain, 167.19/25.09 (??((^[Y0 : $i]: 167.19/25.09 (in(Y0, '#sk211') & 167.19/25.09 (??((^[Y1 : $i]: 167.19/25.09 (in(Y1, '#sk212')&'#sk221'(Y0, Y1)& 167.19/25.09 (kpair('#sk229', '#sk235') = kpair(Y0, Y1)))))))))), 167.19/25.09 inference('ho_norm', [status(thm)], ['205'])). 167.19/25.09 tff('207', plain, 167.19/25.09 (in('#sk3797', '#sk211') & 167.19/25.09 (??((^[Y0 : $i]: 167.19/25.09 (in(Y0, '#sk212')&'#sk221'('#sk3797', Y0)& 167.19/25.09 (kpair('#sk229', '#sk235') = kpair('#sk3797', Y0))))))), 167.19/25.09 inference('lazy_cnf_exists', [status(thm)], ['206'])). 167.19/25.09 tff('208', plain, 167.19/25.09 (??((^[Y0 : $i]: 167.19/25.09 (in(Y0, '#sk212')&'#sk221'('#sk3797', Y0)& 167.19/25.09 (kpair('#sk229', '#sk235') = kpair('#sk3797', Y0)))))), 167.19/25.09 inference('lazy_cnf_and', [status(thm)], ['207'])). 167.19/25.09 tff('209', plain, 167.19/25.09 (in('#sk3798', '#sk212')&'#sk221'('#sk3797', '#sk3798')& 167.19/25.09 (kpair('#sk229', '#sk235') = kpair('#sk3797', '#sk3798'))), 167.19/25.09 inference('lazy_cnf_exists', [status(thm)], ['208'])). 167.19/25.09 tff('210', plain, '#sk221'('#sk3797', '#sk3798'), 167.19/25.09 inference('lazy_cnf_and', [status(thm)], ['209'])). 167.19/25.09 tff('211', plain, kpair('#sk229', '#sk235') = kpair('#sk3797', '#sk3798'), 167.19/25.09 inference('lazy_cnf_and', [status(thm)], ['209'])). 167.19/25.09 tff('212', plain, setukpairinjL, 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['165'])). 167.19/25.09 tff(setukpairinjL, axiom, setukpairinjL => 167.19/25.09 ((!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : $i]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 ((kpair(Y0, Y1) = kpair(Y2, Y3)) => (Y0 = Y2)))))))))))))) 167.19/25.09 = $true)). 167.19/25.09 tff('213', plain, 167.19/25.09 setukpairinjL <=> 167.19/25.09 (![X5,X7,X9,X11]: ((kpair(X5,X7) = kpair(X9,X11)) => (X5 = X9))), 167.19/25.09 inference('rw.lit', [status(esa)], [setukpairinjL])). 167.19/25.09 tff('214', plain, 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : $i]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 ((kpair(Y0, Y1) = kpair(Y2, Y3)) => 167.19/25.09 (Y0 = Y2)))))))))))))), 167.19/25.09 inference('rw_clause', [status(thm)], ['212', '213'])). 167.19/25.09 tff('215', plain, 167.19/25.09 ![X1]: 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : $i]: 167.19/25.09 ((kpair(X1, Y0) = kpair(Y1, Y2)) => (X1 = Y1))))))))))), 167.19/25.09 inference('lazy_cnf_forall', [status(thm)], ['214'])). 167.19/25.09 tff('216', plain, 167.19/25.09 ![X1, X2]: 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: ((kpair(X1, X2) = kpair(Y0, Y1)) => (X1 = Y0)))))))), 167.19/25.09 inference('lazy_cnf_forall', [status(thm)], ['215'])). 167.19/25.09 tff('217', plain, 167.19/25.09 ![X1, X2, X3]: 167.19/25.09 (!!((^[Y0 : $i]: ((kpair(X1, X2) = kpair(X3, Y0)) => (X1 = X3))))), 167.19/25.09 inference('lazy_cnf_forall', [status(thm)], ['216'])). 167.19/25.09 tff('218', plain, 167.19/25.09 ![X1, X2, X3, X4]: ((kpair(X1, X2) = kpair(X3, X4)) => (X1 = X3)), 167.19/25.09 inference('lazy_cnf_forall', [status(thm)], ['217'])). 167.19/25.09 tff('219', plain, 167.19/25.09 ![X1, X2, X3, X4]: (kpair(X1, X2) != kpair(X3, X4) | X1 = X3), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['218'])). 167.19/25.09 tff('220', plain, 167.19/25.09 ![X0, X1]: (kpair(X1, X0) != kpair('#sk229', '#sk235') | X1 = '#sk3797'), 167.19/25.09 inference('sup-', [status(thm)], ['211', '219'])). 167.19/25.09 tff('221', plain, '#sk229' = '#sk3797', 167.19/25.09 inference('eq_res', [status(thm)], ['220'])). 167.19/25.09 tff('222', plain, '#sk221'('#sk229', '#sk3798'), 167.19/25.09 inference('demod', [status(thm)], ['210', '221'])). 167.19/25.09 tff('223', plain, kpair('#sk229', '#sk235') = kpair('#sk3797', '#sk3798'), 167.19/25.09 inference('lazy_cnf_and', [status(thm)], ['209'])). 167.19/25.09 tff('224', plain, '#sk229' = '#sk3797', 167.19/25.09 inference('eq_res', [status(thm)], ['220'])). 167.19/25.09 tff('225', plain, kpair('#sk229', '#sk235') = kpair('#sk229', '#sk3798'), 167.19/25.09 inference('demod', [status(thm)], ['223', '224'])). 167.19/25.09 tff('226', plain, setukpairinjR, 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['171'])). 167.19/25.09 tff(setukpairinjR, axiom, setukpairinjR => 167.19/25.09 ((!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : $i]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 ((kpair(Y0, Y1) = kpair(Y2, Y3)) => (Y1 = Y3)))))))))))))) 167.19/25.09 = $true)). 167.19/25.09 tff('227', plain, 167.19/25.09 setukpairinjR <=> 167.19/25.09 (![X5,X7,X9,X11]: ((kpair(X5,X7) = kpair(X9,X11)) => (X7 = X11))), 167.19/25.09 inference('rw.lit', [status(esa)], [setukpairinjR])). 167.19/25.09 tff('228', plain, 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : $i]: 167.19/25.09 (!!((^[Y3 : $i]: 167.19/25.09 ((kpair(Y0, Y1) = kpair(Y2, Y3)) => 167.19/25.09 (Y1 = Y3)))))))))))))), 167.19/25.09 inference('rw_clause', [status(thm)], ['226', '227'])). 167.19/25.09 tff('229', plain, 167.19/25.09 ![X1]: 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: 167.19/25.09 (!!((^[Y2 : $i]: 167.19/25.09 ((kpair(X1, Y0) = kpair(Y1, Y2)) => (Y0 = Y2))))))))))), 167.19/25.09 inference('lazy_cnf_forall', [status(thm)], ['228'])). 167.19/25.09 tff('230', plain, 167.19/25.09 ![X1, X2]: 167.19/25.09 (!!((^[Y0 : $i]: 167.19/25.09 (!!((^[Y1 : $i]: ((kpair(X1, X2) = kpair(Y0, Y1)) => (X2 = Y1)))))))), 167.19/25.09 inference('lazy_cnf_forall', [status(thm)], ['229'])). 167.19/25.09 tff('231', plain, 167.19/25.09 ![X1, X2, X3]: 167.19/25.09 (!!((^[Y0 : $i]: ((kpair(X1, X2) = kpair(X3, Y0)) => (X2 = Y0))))), 167.19/25.09 inference('lazy_cnf_forall', [status(thm)], ['230'])). 167.19/25.09 tff('232', plain, 167.19/25.09 ![X1, X2, X3, X4]: ((kpair(X1, X2) = kpair(X3, X4)) => (X2 = X4)), 167.19/25.09 inference('lazy_cnf_forall', [status(thm)], ['231'])). 167.19/25.09 tff('233', plain, 167.19/25.09 ![X1, X2, X3, X4]: (kpair(X1, X2) != kpair(X3, X4) | X2 = X4), 167.19/25.09 inference('lazy_cnf_imply', [status(thm)], ['232'])). 167.19/25.09 tff('234', plain, 167.19/25.09 ![X0, X1]: (kpair(X1, X0) != kpair('#sk229', '#sk235') | X0 = '#sk3798'), 167.19/25.09 inference('sup-', [status(thm)], ['225', '233'])). 167.19/25.09 tff('235', plain, '#sk235' = '#sk3798', 167.19/25.09 inference('eq_res', [status(thm)], ['234'])). 167.19/25.09 tff('236', plain, '#sk221'('#sk229', '#sk235'), 167.19/25.09 inference('demod', [status(thm)], ['222', '235'])). 167.19/25.09 tff('237', plain, $false, inference('demod', [status(thm)], ['195', '236'])). 167.19/25.09 167.19/25.09 % SZS output end Refutation 167.19/25.09 167.19/25.09 167.19/25.09 % /export/starexec/sandbox/solver/bin/lams/s4.sh running for 15s 167.19/25.09 % Terminating... 167.19/25.14 % Runner terminated. 167.19/25.15 % Zipperpin 1.5 exiting 167.19/25.17 EOF