0.00/0.03 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.00/0.04 % Command : satallax -s schedule_3_1 -E eprover -P picomus -M modes -p tstp -t %d %s 0.02/0.22 % Computer : n127.star.cs.uiowa.edu 0.02/0.22 % Model : x86_64 x86_64 0.02/0.22 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz 0.02/0.22 % Memory : 32218.625MB 0.02/0.22 % OS : Linux 3.10.0-693.2.2.el7.x86_64 0.02/0.22 % CPULimit : 300 0.02/0.22 % DateTime : Sun Jul 15 13:03:09 CDT 2018 0.02/0.22 % CPUTime : 173.01/172.98 % SZS status Theorem 173.01/172.98 % Mode: mode500 173.01/172.98 % Inferences: 321 173.01/172.98 % SZS output start Proof 173.01/172.98 thf(ty_$i, type, $i : $tType). 173.01/172.98 thf(ty_kfst, type, kfst : ($i>$i)). 173.01/172.98 thf(ty_binintersect, type, binintersect : ($i>$i>$i)). 173.01/172.98 thf(ty_singleton, type, singleton : ($i>$o)). 173.01/172.98 thf(ty_if, type, if : ($i>$o>$i>$i>$i)). 173.01/172.98 thf(ty_subset, type, subset : ($i>$i>$o)). 173.01/172.98 thf(ty_eigen__2, type, eigen__2 : $i). 173.01/172.98 thf(ty_setunion, type, setunion : ($i>$i)). 173.01/172.98 thf(ty_iskpair, type, iskpair : ($i>$o)). 173.01/172.98 thf(ty_breln1invset, type, breln1invset : ($i>$i>$i)). 173.01/172.98 thf(ty_eigen__1, type, eigen__1 : ($i>$i)). 173.01/172.98 thf(ty_eigen__0, type, eigen__0 : $i). 173.01/172.98 thf(ty_cartprod, type, cartprod : ($i>$i>$i)). 173.01/172.98 thf(ty_breln, type, breln : ($i>$i>$i>$o)). 173.01/172.98 thf(ty_emptyset, type, emptyset : $i). 173.01/172.98 thf(ty_breln1, type, breln1 : ($i>$i>$o)). 173.01/172.98 thf(ty_setminus, type, setminus : ($i>$i>$i)). 173.01/172.98 thf(ty_kpair, type, kpair : ($i>$i>$i)). 173.01/172.98 thf(ty_descr, type, descr : (($i>$o)>$i)). 173.01/172.98 thf(ty_reflwellordering, type, reflwellordering : ($i>$i>$o)). 173.01/172.98 thf(ty_breln1compset, type, breln1compset : ($i>$i>$i>$i)). 173.01/172.98 thf(ty_breln1Set, type, breln1Set : ($i>$i)). 173.01/172.98 thf(ty_ap, type, ap : ($i>$i>$i>$i>$i)). 173.01/172.98 thf(ty_funcSet, type, funcSet : ($i>$i>$i)). 173.01/172.98 thf(ty_symdiff, type, symdiff : ($i>$i>$i)). 173.01/172.98 thf(ty_dsetconstr, type, dsetconstr : ($i>($i>$o)>$i)). 173.01/172.98 thf(ty_powerset, type, powerset : ($i>$i)). 173.01/172.98 thf(ty_omega, type, omega : $i). 173.01/172.98 thf(ty_ex1, type, ex1 : ($i>($i>$o)>$o)). 173.01/172.98 thf(ty_ksnd, type, ksnd : ($i>$i)). 173.01/172.98 thf(ty_prop2set, type, prop2set : ($o>$i)). 173.01/172.98 thf(ty_set2prop, type, set2prop : ($i>$o)). 173.01/172.98 thf(ty_lam, type, lam : ($i>$i>($i>$i)>$i)). 173.01/172.98 thf(ty_in, type, in : ($i>$i>$o)). 173.01/172.98 thf(ty_func, type, func : ($i>$i>$i>$o)). 173.01/172.98 thf(ty_binunion, type, binunion : ($i>$i>$i)). 173.01/172.98 thf(ty_dpsetconstr, type, dpsetconstr : ($i>$i>($i>$i>$o)>$i)). 173.01/172.98 thf(ty_image1, type, image1 : ($i>($i>$i)>$i)). 173.01/172.98 thf(ty_setadjoin, type, setadjoin : ($i>$i>$i)). 173.01/172.98 thf(h0, assumption, (![X1:($i>$i)>$o]:(![X2:$i>$i]:((X1 @ X2) => (X1 @ (eps__0 @ X1))))),introduced(assumption,[])). 173.01/172.98 thf(eigendef_eigen__1, definition, (eigen__1 = (eps__0 @ (^[X1:$i>$i]:(~((![X2:$i]:(((in @ X2) @ ((image1 @ eigen__0) @ X1)) => (~((![X3:$i]:(((in @ X3) @ eigen__0) => (~((X2 = (X1 @ X3))))))))))))))), introduced(definition,[new_symbols(definition,[eigen__1]))). 173.01/172.98 thf(h1, assumption, (![X1:$i>$o]:(![X2:$i]:((X1 @ X2) => (X1 @ (eps__1 @ X1))))),introduced(assumption,[])). 173.01/172.98 thf(eigendef_eigen__0, definition, (eigen__0 = (eps__1 @ (^[X1:$i]:(~((![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))))))), introduced(definition,[new_symbols(definition,[eigen__0]))). 173.01/172.98 thf(eigendef_eigen__2, definition, (eigen__2 = (eps__1 @ (^[X1:$i]:(~((((in @ X1) @ ((image1 @ eigen__0) @ eigen__1)) => (~((![X2:$i]:(((in @ X2) @ eigen__0) => (~((X1 = (eigen__1 @ X2)))))))))))))), introduced(definition,[new_symbols(definition,[eigen__2]))). 173.01/172.98 thf(sP1,plain,(sP1 <=> ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))) => (~((![X4:$i]:((![X5:$i]:(((in @ X5) @ X4) = (~((![X6:$i]:(((in @ X6) @ X1) => (~((X5 = (X2 @ X6)))))))))) => (X3 = X4))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => (![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP1])]))). 173.01/172.98 thf(sP2,plain,(sP2 <=> ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3)))))))))))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((~((![X3:$i]:((X1 @ X3) => (~((![X4:$i]:((X1 @ X4) => (X3 = X4))))))))) = (~((![X3:$i]:((X2 @ X3) => (~((![X4:$i]:((X2 @ X4) => (X3 = X4))))))))))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((~((![X3:$i]:((X1 @ X3) => (~((![X4:$i]:((X1 @ X4) => (X3 = X4))))))))) => ((~((![X3:$i]:((X2 @ X3) => (~((![X4:$i]:((X2 @ X4) => (X3 = X4))))))))) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in 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=> (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) 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@ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) 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@ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => 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@ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ 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=> ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => sP22)))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP38])]))). 173.10/173.01 thf(sP39,plain,(sP39 <=> 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => sP25)))),introduced(definition,[new_symbols(definition,[sP40])]))). 173.10/173.01 thf(sP41,plain,(sP41 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => sP22)))))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP41])]))). 173.10/173.01 thf(sP42,plain,(sP42 <=> ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => sP12)),introduced(definition,[new_symbols(definition,[sP42])]))). 173.10/173.01 thf(sP43,plain,(sP43 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => sP5)),introduced(definition,[new_symbols(definition,[sP43])]))). 173.10/173.01 thf(sP44,plain,(sP44 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => sP22)))))))))))),introduced(definition,[new_symbols(definition,[sP44])]))). 173.10/173.01 thf(sP45,plain,(sP45 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => sP35),introduced(definition,[new_symbols(definition,[sP45])]))). 173.10/173.01 thf(sP46,plain,(sP46 <=> (((!!) @ (subset @ emptyset)) => sP21),introduced(definition,[new_symbols(definition,[sP46])]))). 173.10/173.01 thf(sP47,plain,(sP47 <=> ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => sP1)))))),introduced(definition,[new_symbols(definition,[sP47])]))). 173.10/173.01 thf(sP48,plain,(sP48 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => sP31)))))))),introduced(definition,[new_symbols(definition,[sP48])]))). 173.10/173.01 thf(sP49,plain,(sP49 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => sP28))),introduced(definition,[new_symbols(definition,[sP49])]))). 173.10/173.01 thf(sP50,plain,(sP50 <=> ((![X1:$i]:((~((X1 = emptyset))) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (~((X1 = emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ 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(((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => sP33)))))),introduced(definition,[new_symbols(definition,[sP68])]))). 173.10/173.01 thf(sP69,plain,(sP69 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => sP47))))),introduced(definition,[new_symbols(definition,[sP69])]))). 173.10/173.01 thf(sP70,plain,(sP70 <=> ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => sP14)))))))))))),introduced(definition,[new_symbols(definition,[sP70])]))). 173.10/173.01 thf(sP71,plain,(sP71 <=> ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => sP30),introduced(definition,[new_symbols(definition,[sP71])]))). 173.10/173.01 thf(sP72,plain,(sP72 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => sP41)),introduced(definition,[new_symbols(definition,[sP72])]))). 173.10/173.01 thf(sP73,plain,(sP73 <=> (((in @ eigen__2) @ ((image1 @ eigen__0) @ eigen__1)) = (~((![X1:$i]:(((in @ X1) @ eigen__0) => (~((eigen__2 = (eigen__1 @ X1))))))))),introduced(definition,[new_symbols(definition,[sP73])]))). 173.10/173.01 thf(sP74,plain,(sP74 <=> ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => sP72)))))))))))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP74])]))). 173.10/173.01 thf(sP75,plain,(sP75 <=> ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => sP14)))))))),introduced(definition,[new_symbols(definition,[sP75])]))). 173.10/173.01 thf(sP76,plain,(sP76 <=> ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => sP33)))),introduced(definition,[new_symbols(definition,[sP76])]))). 173.10/173.01 thf(sP77,plain,(sP77 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ 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X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => sP72))))))))))))))))))),introduced(definition,[new_symbols(definition,[sP77])]))). 173.10/173.01 thf(sP78,plain,(sP78 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => sP14))))),introduced(definition,[new_symbols(definition,[sP78])]))). 173.10/173.01 thf(sP79,plain,(sP79 <=> ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => sP20),introduced(definition,[new_symbols(definition,[sP79])]))). 173.10/173.01 thf(sP80,plain,(sP80 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => sP54),introduced(definition,[new_symbols(definition,[sP80])]))). 173.10/173.01 thf(sP81,plain,(sP81 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => sP42)))),introduced(definition,[new_symbols(definition,[sP81])]))). 173.10/173.01 thf(sP82,plain,(sP82 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => sP59)),introduced(definition,[new_symbols(definition,[sP82])]))). 173.10/173.01 thf(sP83,plain,(sP83 <=> ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => sP56)),introduced(definition,[new_symbols(definition,[sP83])]))). 173.10/173.01 thf(sP84,plain,(sP84 <=> ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))) => (~((![X4:$i]:((![X5:$i]:(((in @ X5) @ X4) = (~((![X6:$i]:(((in @ X6) @ X1) => (~((X5 = (X2 @ X6)))))))))) => (X3 = X4))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => (![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))))),introduced(definition,[new_symbols(definition,[sP84])]))). 173.10/173.01 thf(sP85,plain,(sP85 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => sP36)))),introduced(definition,[new_symbols(definition,[sP85])]))). 173.10/173.01 thf(sP86,plain,(sP86 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => sP60))))),introduced(definition,[new_symbols(definition,[sP86])]))). 173.10/173.01 thf(sP87,plain,(sP87 <=> (![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ 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((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => sP83))))))))))),introduced(definition,[new_symbols(definition,[sP89])]))). 173.10/173.01 thf(sP90,plain,(sP90 <=> ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => sP58),introduced(definition,[new_symbols(definition,[sP90])]))). 173.10/173.01 thf(sP91,plain,(sP91 <=> ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => sP75)),introduced(definition,[new_symbols(definition,[sP91])]))). 173.10/173.01 thf(sP92,plain,(sP92 <=> (![X1:$o]:((((in @ eigen__2) @ ((image1 @ eigen__0) @ eigen__1)) = X1) => X1)),introduced(definition,[new_symbols(definition,[sP92])]))). 173.10/173.01 thf(sP93,plain,(sP93 <=> (((in @ eigen__2) @ ((image1 @ eigen__0) @ eigen__1)) => sP92),introduced(definition,[new_symbols(definition,[sP93])]))). 173.10/173.01 thf(sP94,plain,(sP94 <=> ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ 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sP46),introduced(definition,[new_symbols(definition,[sP96])]))). 173.10/173.01 thf(sP97,plain,(sP97 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => sP82))))),introduced(definition,[new_symbols(definition,[sP97])]))). 173.10/173.01 thf(sP98,plain,(sP98 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => sP78),introduced(definition,[new_symbols(definition,[sP98])]))). 173.10/173.01 thf(sP99,plain,(sP99 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => sP66)),introduced(definition,[new_symbols(definition,[sP99])]))). 173.10/173.01 thf(sP100,plain,(sP100 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => sP32))))),introduced(definition,[new_symbols(definition,[sP100])]))). 173.10/173.01 thf(sP101,plain,(sP101 <=> ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:((~((((in @ emptyset) @ X1) => (~((![X2:$i]:((~((((in @ X2) @ omega) => (~(((in @ X2) @ X1)))))) => ((in @ ((setadjoin @ X2) @ X2)) @ X1)))))))) => (![X2:$i]:(((in @ X2) @ omega) => ((in @ X2) @ X1))))) => ((![X1:$i>$i>$o]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => (~((![X4:$i]:(((X1 @ X3) @ X4) => (~((![X5:$i]:(((X1 @ X3) @ X5) => (X4 = X5))))))))))) => (~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X1 @ X5) @ X4))))))))))))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ X1))))))))))))) => ((![X1:$i]:(~((![X2:$i]:((~(((~(((![X3:$i]:(((in @ X3) @ X2) => (![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X1) => (~(((in @ X4) @ X1)))))) => ((![X5:$i]:(((in @ X5) @ X2) => (((in @ X3) @ X5) = ((in @ X4) @ X5)))) => (X3 = X4)))))))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X2) => (~(((in @ X4) @ X2)))))) => ((~((![X5:$i]:(((in @ X5) @ X3) => ((in @ X5) @ X4))))) => (![X5:$i]:(((in @ X5) @ X4) => ((in @ X5) @ X3)))))))))))) => (~((![X3:$i]:((~(((![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))) => (![X4:$i]:(~(((in @ X4) @ X3))))))) => (~((![X4:$i]:(![X5:$i]:((~(((~((((in @ X4) @ X2) => (~(((in @ X5) @ X3)))))) => (~((![X6:$i]:(((in @ X6) @ X4) => (~(((in @ X6) @ X3)))))))))) => (~((![X6:$i]:(((in @ X6) @ X2) => ((~((![X7:$i]:(((in @ X7) @ X6) => ((in @ X7) @ X4))))) => ((in @ X5) @ X6)))))))))))))))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (X1 @ (descr @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((in @ X3) @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => ((in @ X3) @ X1))))) => sP4)))))))),introduced(definition,[new_symbols(definition,[sP101])]))). 173.10/173.01 thf(sP102,plain,(sP102 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => sP44)))))),introduced(definition,[new_symbols(definition,[sP102])]))). 173.10/173.01 thf(sP103,plain,(sP103 <=> ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => sP72))))))))),introduced(definition,[new_symbols(definition,[sP103])]))). 173.10/173.02 thf(sP104,plain,(sP104 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) = (~((![X3:$i]:(((in @ X2) @ X3) => (~(((in @ X3) @ X1)))))))))) => (((in @ emptyset) @ omega) => sP101)),introduced(definition,[new_symbols(definition,[sP104])]))). 173.10/173.02 thf(sP105,plain,(sP105 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => sP85)))),introduced(definition,[new_symbols(definition,[sP105])]))). 173.10/173.02 thf(sP106,plain,(sP106 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => sP24)))))))),introduced(definition,[new_symbols(definition,[sP106])]))). 173.10/173.02 thf(sP107,plain,(sP107 <=> (((in @ emptyset) @ omega) => sP101),introduced(definition,[new_symbols(definition,[sP107])]))). 173.10/173.02 thf(sP108,plain,(sP108 <=> ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => sP36)),introduced(definition,[new_symbols(definition,[sP108])]))). 173.10/173.02 thf(sP109,plain,(sP109 <=> ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => sP83))))))),introduced(definition,[new_symbols(definition,[sP109])]))). 173.10/173.02 thf(sP110,plain,(sP110 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => sP59),introduced(definition,[new_symbols(definition,[sP110])]))). 173.10/173.02 thf(sP111,plain,(sP111 <=> ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => sP1)),introduced(definition,[new_symbols(definition,[sP111])]))). 173.10/173.02 thf(sP112,plain,(sP112 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => sP49),introduced(definition,[new_symbols(definition,[sP112])]))). 173.10/173.02 thf(sP113,plain,(sP113 <=> ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => sP8))),introduced(definition,[new_symbols(definition,[sP113])]))). 173.10/173.02 thf(sP114,plain,(sP114 <=> (sP73 => (~((![X1:$i]:(((in @ X1) @ eigen__0) => (~((eigen__2 = (eigen__1 @ X1))))))))),introduced(definition,[new_symbols(definition,[sP114])]))). 173.10/173.02 thf(sP115,plain,(sP115 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => sP103))))))),introduced(definition,[new_symbols(definition,[sP115])]))). 173.10/173.02 thf(sP116,plain,(sP116 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => sP77)))),introduced(definition,[new_symbols(definition,[sP116])]))). 173.10/173.02 thf(sP117,plain,(sP117 <=> ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => sP42))),introduced(definition,[new_symbols(definition,[sP117])]))). 173.10/173.02 thf(sP118,plain,(sP118 <=> 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X4)))))))) => sP77),introduced(definition,[new_symbols(definition,[sP120])]))). 173.10/173.02 thf(sP121,plain,(sP121 <=> ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => sP45),introduced(definition,[new_symbols(definition,[sP121])]))). 173.10/173.02 thf(sP122,plain,(sP122 <=> ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => sP90))))),introduced(definition,[new_symbols(definition,[sP122])]))). 173.10/173.02 thf(sP123,plain,(sP123 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => sP103))))),introduced(definition,[new_symbols(definition,[sP123])]))). 173.10/173.02 thf(sP124,plain,(sP124 <=> ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (X1 @ (descr @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((in @ X3) @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => ((in @ X3) @ X1))))) => sP4))),introduced(definition,[new_symbols(definition,[sP124])]))). 173.10/173.02 thf(sP125,plain,(sP125 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => sP112)),introduced(definition,[new_symbols(definition,[sP125])]))). 173.10/173.02 thf(sP126,plain,(sP126 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => sP5),introduced(definition,[new_symbols(definition,[sP126])]))). 173.10/173.02 thf(sP127,plain,(sP127 <=> ((![X1:$i]:((~((X1 = emptyset))) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => sP2))),introduced(definition,[new_symbols(definition,[sP127])]))). 173.10/173.02 thf(sP128,plain,(sP128 <=> (![X1:$i>$i]:(![X2:$i]:(((in @ X2) @ ((image1 @ eigen__0) @ X1)) = (~((![X3:$i]:(((in @ X3) @ eigen__0) => (~((X2 = (X1 @ X3))))))))))),introduced(definition,[new_symbols(definition,[sP128])]))). 173.10/173.02 thf(sP129,plain,(sP129 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => sP22))))),introduced(definition,[new_symbols(definition,[sP129])]))). 173.10/173.02 thf(sP130,plain,(sP130 <=> ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => sP47),introduced(definition,[new_symbols(definition,[sP130])]))). 173.10/173.02 thf(sP131,plain,(sP131 <=> ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => sP13),introduced(definition,[new_symbols(definition,[sP131])]))). 173.10/173.02 thf(sP132,plain,(sP132 <=> ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => sP17),introduced(definition,[new_symbols(definition,[sP132])]))). 173.10/173.02 thf(sP133,plain,(sP133 <=> ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => sP109),introduced(definition,[new_symbols(definition,[sP133])]))). 173.10/173.02 thf(sP134,plain,(sP134 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => sP66),introduced(definition,[new_symbols(definition,[sP134])]))). 173.10/173.02 thf(sP135,plain,(sP135 <=> ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => sP71),introduced(definition,[new_symbols(definition,[sP135])]))). 173.10/173.02 thf(sP136,plain,(sP136 <=> ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => sP90)))),introduced(definition,[new_symbols(definition,[sP136])]))). 173.10/173.02 thf(sP137,plain,(sP137 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ 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((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => sP90)),introduced(definition,[new_symbols(definition,[sP144])]))). 173.10/173.02 thf(sP145,plain,(sP145 <=> ((omega = omega) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((~((![X3:$i]:((X1 @ X3) => (~((![X4:$i]:((X1 @ X4) => (X3 = X4))))))))) => ((~((![X3:$i]:((X2 @ X3) => (~((![X4:$i]:((X2 @ X4) => (X3 = X4))))))))) => ((descr @ X1) = (descr @ X2))))))) => sP62))),introduced(definition,[new_symbols(definition,[sP145])]))). 173.10/173.02 thf(sP146,plain,(sP146 <=> ((![X1:$i]:(![X2:$i]:(((in @ X1) 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(((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => sP85)),introduced(definition,[new_symbols(definition,[sP150])]))). 173.10/173.02 thf(sP151,plain,(sP151 <=> ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => sP83)),introduced(definition,[new_symbols(definition,[sP151])]))). 173.10/173.02 thf(sP152,plain,(sP152 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => sP132)),introduced(definition,[new_symbols(definition,[sP152])]))). 173.10/173.02 thf(sP153,plain,(sP153 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => sP118))),introduced(definition,[new_symbols(definition,[sP153])]))). 173.10/173.02 thf(sP154,plain,(sP154 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => sP11)),introduced(definition,[new_symbols(definition,[sP154])]))). 173.10/173.02 thf(sP155,plain,(sP155 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => sP72),introduced(definition,[new_symbols(definition,[sP155])]))). 173.10/173.02 thf(sP156,plain,(sP156 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => sP27),introduced(definition,[new_symbols(definition,[sP156])]))). 173.10/173.02 thf(sP157,plain,(sP157 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => sP105),introduced(definition,[new_symbols(definition,[sP157])]))). 173.10/173.02 thf(sP158,plain,(sP158 <=> ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (~((X1 = emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => sP143)),introduced(definition,[new_symbols(definition,[sP158])]))). 173.10/173.02 thf(sP159,plain,(sP159 <=> ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((in @ X3) @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => ((in @ X3) @ X1))))) => sP4)),introduced(definition,[new_symbols(definition,[sP159])]))). 173.10/173.02 thf(sP160,plain,(sP160 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => sP98),introduced(definition,[new_symbols(definition,[sP160])]))). 173.10/173.02 thf(sP161,plain,(sP161 <=> 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=> (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => sP120),introduced(definition,[new_symbols(definition,[sP164])]))). 173.10/173.02 thf(sP165,plain,(sP165 <=> ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => sP145)),introduced(definition,[new_symbols(definition,[sP165])]))). 173.10/173.02 thf(sP166,plain,(sP166 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => sP48),introduced(definition,[new_symbols(definition,[sP166])]))). 173.10/173.02 thf(sP167,plain,(sP167 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => 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sP60),introduced(definition,[new_symbols(definition,[sP183])]))). 173.10/173.02 thf(sP184,plain,(sP184 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => sP10),introduced(definition,[new_symbols(definition,[sP184])]))). 173.10/173.02 thf(sP185,plain,(sP185 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => sP12),introduced(definition,[new_symbols(definition,[sP185])]))). 173.10/173.02 thf(sP186,plain,(sP186 <=> ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => sP52)),introduced(definition,[new_symbols(definition,[sP186])]))). 173.10/173.02 thf(sP187,plain,(sP187 <=> ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => sP169),introduced(definition,[new_symbols(definition,[sP187])]))). 173.10/173.02 thf(sP188,plain,(sP188 <=> ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => sP111),introduced(definition,[new_symbols(definition,[sP188])]))). 173.10/173.02 thf(sP189,plain,(sP189 <=> ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => sP11),introduced(definition,[new_symbols(definition,[sP189])]))). 173.10/173.02 thf(sP190,plain,(sP190 <=> ((![X1:$o]:(![X2:$i]:(((in @ X2) @ (prop2set @ X1)) => X1))) => sP9),introduced(definition,[new_symbols(definition,[sP190])]))). 173.10/173.02 thf(sP191,plain,(sP191 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => sP24)))))),introduced(definition,[new_symbols(definition,[sP191])]))). 173.10/173.02 thf(sP192,plain,(sP192 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => sP8),introduced(definition,[new_symbols(definition,[sP192])]))). 173.10/173.02 thf(sP193,plain,(sP193 <=> ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => sP155)))))),introduced(definition,[new_symbols(definition,[sP193])]))). 173.10/173.02 thf(sP194,plain,(sP194 <=> ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ 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@ ((dsetconstr @ X1) @ X2)) @ X1))) => sP153),introduced(definition,[new_symbols(definition,[sP218])]))). 173.10/173.02 thf(sP219,plain,(sP219 <=> ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => sP2),introduced(definition,[new_symbols(definition,[sP219])]))). 173.10/173.02 thf(sP220,plain,(sP220 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => sP166),introduced(definition,[new_symbols(definition,[sP220])]))). 173.10/173.02 thf(sP221,plain,(sP221 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => sP44),introduced(definition,[new_symbols(definition,[sP221])]))). 173.10/173.02 thf(sP222,plain,(sP222 <=> 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<=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => sP33))),introduced(definition,[new_symbols(definition,[sP226])]))). 173.10/173.02 thf(sP227,plain,(sP227 <=> ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => ((in @ X3) @ X1))))) => sP4),introduced(definition,[new_symbols(definition,[sP227])]))). 173.10/173.02 thf(sP228,plain,(sP228 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => sP116)),introduced(definition,[new_symbols(definition,[sP228])]))). 173.10/173.02 thf(sP229,plain,(sP229 <=> ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => sP214),introduced(definition,[new_symbols(definition,[sP229])]))). 173.10/173.02 thf(sP230,plain,(sP230 <=> ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => sP96),introduced(definition,[new_symbols(definition,[sP230])]))). 173.10/173.02 thf(sP231,plain,(sP231 <=> ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => sP230),introduced(definition,[new_symbols(definition,[sP231])]))). 173.10/173.02 thf(sP232,plain,(sP232 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => sP115),introduced(definition,[new_symbols(definition,[sP232])]))). 173.10/173.02 thf(sP233,plain,(sP233 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => sP33),introduced(definition,[new_symbols(definition,[sP233])]))). 173.10/173.02 thf(sP234,plain,(sP234 <=> (![X1:$i]:(((in @ X1) @ eigen__0) => (~((eigen__2 = (eigen__1 @ X1)))))),introduced(definition,[new_symbols(definition,[sP234])]))). 173.10/173.02 thf(sP235,plain,(sP235 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => sP25)),introduced(definition,[new_symbols(definition,[sP235])]))). 173.10/173.02 thf(sP236,plain,(sP236 <=> ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => sP118)),introduced(definition,[new_symbols(definition,[sP236])]))). 173.10/173.02 thf(sP237,plain,(sP237 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => sP154),introduced(definition,[new_symbols(definition,[sP237])]))). 173.10/173.02 thf(sP238,plain,(sP238 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => sP24),introduced(definition,[new_symbols(definition,[sP238])]))). 173.10/173.02 thf(sP239,plain,(sP239 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => sP82),introduced(definition,[new_symbols(definition,[sP239])]))). 173.10/173.02 thf(sP240,plain,(sP240 <=> ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => sP213),introduced(definition,[new_symbols(definition,[sP240])]))). 173.10/173.02 thf(sP241,plain,(sP241 <=> ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => sP85),introduced(definition,[new_symbols(definition,[sP241])]))). 173.10/173.02 thf(sP242,plain,(sP242 <=> ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => sP186),introduced(definition,[new_symbols(definition,[sP242])]))). 173.10/173.02 thf(sP243,plain,(sP243 <=> ((in @ eigen__2) @ ((image1 @ eigen__0) @ eigen__1)),introduced(definition,[new_symbols(definition,[sP243])]))). 173.10/173.02 thf(sP244,plain,(sP244 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => sP238),introduced(definition,[new_symbols(definition,[sP244])]))). 173.10/173.02 thf(sP245,plain,(sP245 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => sP32)),introduced(definition,[new_symbols(definition,[sP245])]))). 173.10/173.02 thf(sP246,plain,(sP246 <=> ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => sP155),introduced(definition,[new_symbols(definition,[sP246])]))). 173.10/173.02 thf(sP247,plain,(sP247 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => sP176),introduced(definition,[new_symbols(definition,[sP247])]))). 173.10/173.02 thf(sP248,plain,(sP248 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => sP116),introduced(definition,[new_symbols(definition,[sP248])]))). 173.10/173.02 thf(sP249,plain,(sP249 <=> ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => sP32),introduced(definition,[new_symbols(definition,[sP249])]))). 173.10/173.02 thf(sP250,plain,(sP250 <=> ((![X1:$i>$i>$o]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => (~((![X4:$i]:(((X1 @ X3) @ X4) => (~((![X5:$i]:(((X1 @ X3) @ X5) => (X4 = X5))))))))))) => (~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X1 @ X5) @ X4))))))))))))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ X1))))))))))))) => ((![X1:$i]:(~((![X2:$i]:((~(((~(((![X3:$i]:(((in @ X3) @ X2) => (![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X1) => (~(((in @ X4) @ X1)))))) => ((![X5:$i]:(((in @ X5) @ X2) => (((in @ X3) @ X5) = ((in @ X4) @ X5)))) => (X3 = X4)))))))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X2) => (~(((in @ X4) @ X2)))))) => ((~((![X5:$i]:(((in @ X5) @ X3) => ((in @ X5) @ X4))))) => (![X5:$i]:(((in @ X5) @ X4) => ((in @ X5) @ X3)))))))))))) => (~((![X3:$i]:((~(((![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))) => (![X4:$i]:(~(((in @ X4) @ X3))))))) => (~((![X4:$i]:(![X5:$i]:((~(((~((((in @ X4) @ X2) => (~(((in @ X5) @ X3)))))) => (~((![X6:$i]:(((in @ X6) @ X4) => (~(((in @ X6) @ X3)))))))))) => (~((![X6:$i]:(((in @ X6) @ X2) => ((~((![X7:$i]:(((in @ X7) @ X6) => ((in @ X7) @ X4))))) => ((in @ X5) @ X6)))))))))))))))))))) => sP124))),introduced(definition,[new_symbols(definition,[sP250])]))). 173.10/173.02 thf(sP251,plain,(sP251 <=> ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ X1))))))))))))) => ((![X1:$i]:(~((![X2:$i]:((~(((~(((![X3:$i]:(((in @ X3) @ X2) => (![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X1) => (~(((in @ X4) @ X1)))))) => ((![X5:$i]:(((in @ X5) @ X2) => (((in @ X3) @ X5) = ((in @ X4) @ X5)))) => (X3 = X4)))))))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X2) => (~(((in @ X4) @ X2)))))) => ((~((![X5:$i]:(((in @ X5) @ X3) => ((in @ X5) @ X4))))) => (![X5:$i]:(((in @ X5) @ X4) => ((in @ X5) @ X3)))))))))))) => (~((![X3:$i]:((~(((![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))) => (![X4:$i]:(~(((in @ X4) @ X3))))))) => (~((![X4:$i]:(![X5:$i]:((~(((~((((in @ X4) @ X2) => (~(((in @ X5) @ X3)))))) => (~((![X6:$i]:(((in @ X6) @ X4) => (~(((in @ X6) @ X3)))))))))) => (~((![X6:$i]:(((in @ X6) @ X2) => ((~((![X7:$i]:(((in @ X7) @ X6) => ((in @ X7) @ X4))))) => ((in @ X5) @ X6)))))))))))))))))))) => sP124)),introduced(definition,[new_symbols(definition,[sP251])]))). 173.10/173.02 thf(sP252,plain,(sP252 <=> ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => sP212),introduced(definition,[new_symbols(definition,[sP252])]))). 173.10/173.02 thf(sP253,plain,(sP253 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => sP240),introduced(definition,[new_symbols(definition,[sP253])]))). 173.10/173.02 thf(sP254,plain,(sP254 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => sP244)),introduced(definition,[new_symbols(definition,[sP254])]))). 173.10/173.02 thf(sP255,plain,(sP255 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => sP97),introduced(definition,[new_symbols(definition,[sP255])]))). 173.10/173.02 thf(sP256,plain,(sP256 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => sP88),introduced(definition,[new_symbols(definition,[sP256])]))). 173.10/173.02 thf(sP257,plain,(sP257 <=> ((emptyset = emptyset) => sP208),introduced(definition,[new_symbols(definition,[sP257])]))). 173.10/173.02 thf(sP258,plain,(sP258 <=> ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => sP192),introduced(definition,[new_symbols(definition,[sP258])]))). 173.10/173.02 thf(sP259,plain,(sP259 <=> ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => sP130),introduced(definition,[new_symbols(definition,[sP259])]))). 173.10/173.02 thf(sP260,plain,(sP260 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => sP225),introduced(definition,[new_symbols(definition,[sP260])]))). 173.10/173.02 thf(sP261,plain,(sP261 <=> (![X1:$i]:(((in @ X1) @ ((image1 @ eigen__0) @ eigen__1)) = (~((![X2:$i]:(((in @ X2) @ eigen__0) => (~((X1 = (eigen__1 @ X2)))))))))),introduced(definition,[new_symbols(definition,[sP261])]))). 173.10/173.02 thf(sP262,plain,(sP262 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => sP255),introduced(definition,[new_symbols(definition,[sP262])]))). 173.10/173.02 thf(sP263,plain,(sP263 <=> ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => sP180),introduced(definition,[new_symbols(definition,[sP263])]))). 173.10/173.02 thf(sP264,plain,(sP264 <=> ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => sP203),introduced(definition,[new_symbols(definition,[sP264])]))). 173.10/173.02 thf(sP265,plain,(sP265 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => sP239))),introduced(definition,[new_symbols(definition,[sP265])]))). 173.10/173.02 thf(sP266,plain,(sP266 <=> ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => sP151),introduced(definition,[new_symbols(definition,[sP266])]))). 173.10/173.02 thf(sP267,plain,(sP267 <=> ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => sP266))),introduced(definition,[new_symbols(definition,[sP267])]))). 173.10/173.02 thf(sP268,plain,(sP268 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => sP74),introduced(definition,[new_symbols(definition,[sP268])]))). 173.10/173.02 thf(sP269,plain,(sP269 <=> ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => sP75),introduced(definition,[new_symbols(definition,[sP269])]))). 173.10/173.02 thf(sP270,plain,(sP270 <=> ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => sP218),introduced(definition,[new_symbols(definition,[sP270])]))). 173.10/173.02 thf(sP271,plain,(sP271 <=> ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => sP121),introduced(definition,[new_symbols(definition,[sP271])]))). 173.10/173.03 thf(sP272,plain,(sP272 <=> ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => sP145),introduced(definition,[new_symbols(definition,[sP272])]))). 173.10/173.03 thf(sP273,plain,(sP273 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => sP239)),introduced(definition,[new_symbols(definition,[sP273])]))). 173.10/173.03 thf(sP274,plain,(sP274 <=> (![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))),introduced(definition,[new_symbols(definition,[sP274])]))). 173.10/173.03 thf(sP275,plain,(sP275 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => sP31)),introduced(definition,[new_symbols(definition,[sP275])]))). 173.10/173.03 thf(sP276,plain,(sP276 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => sP22),introduced(definition,[new_symbols(definition,[sP276])]))). 173.10/173.03 thf(sP277,plain,(sP277 <=> ((![X1:$i]:(~((![X2:$i]:((~(((~(((![X3:$i]:(((in @ X3) @ 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X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => sP254),introduced(definition,[new_symbols(definition,[sP278])]))). 173.10/173.03 thf(sP279,plain,(sP279 <=> ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => sP195),introduced(definition,[new_symbols(definition,[sP279])]))). 173.10/173.03 thf(sP280,plain,(sP280 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => sP102),introduced(definition,[new_symbols(definition,[sP280])]))). 173.10/173.03 thf(sP281,plain,(sP281 <=> ((![X1:$i]:((subset @ X1) @ X1)) => sP233),introduced(definition,[new_symbols(definition,[sP281])]))). 173.10/173.03 thf(sP282,plain,(sP282 <=> ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => sP224),introduced(definition,[new_symbols(definition,[sP282])]))). 173.10/173.03 thf(sP283,plain,(sP283 <=> (![X1:$o>$o]:((X1 @ sP243) => (![X2:$o]:((sP243 = X2) => (X1 @ X2))))),introduced(definition,[new_symbols(definition,[sP283])]))). 173.10/173.03 thf(sP284,plain,(sP284 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => sP235),introduced(definition,[new_symbols(definition,[sP284])]))). 173.10/173.03 thf(sP285,plain,(sP285 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => sP142),introduced(definition,[new_symbols(definition,[sP285])]))). 173.10/173.03 thf(sP286,plain,(sP286 <=> ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) = ((in @ X3) @ X2))) => (X1 = X2)))) => sP279),introduced(definition,[new_symbols(definition,[sP286])]))). 173.10/173.03 thf(sP287,plain,(sP287 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => sP232),introduced(definition,[new_symbols(definition,[sP287])]))). 173.10/173.03 thf(sP288,plain,(sP288 <=> ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => sP231),introduced(definition,[new_symbols(definition,[sP288])]))). 173.10/173.03 thf(sP289,plain,(sP289 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => sP129),introduced(definition,[new_symbols(definition,[sP289])]))). 173.10/173.03 thf(sP290,plain,(sP290 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => sP39),introduced(definition,[new_symbols(definition,[sP290])]))). 173.10/173.03 thf(sP291,plain,(sP291 <=> ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => sP143),introduced(definition,[new_symbols(definition,[sP291])]))). 173.10/173.03 thf(sP292,plain,(sP292 <=> ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => sP259),introduced(definition,[new_symbols(definition,[sP292])]))). 173.10/173.03 thf(sP293,plain,(sP293 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => sP275),introduced(definition,[new_symbols(definition,[sP293])]))). 173.10/173.03 thf(sP294,plain,(sP294 <=> ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => sP209),introduced(definition,[new_symbols(definition,[sP294])]))). 173.10/173.03 thf(sP295,plain,(sP295 <=> ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => sP252)),introduced(definition,[new_symbols(definition,[sP295])]))). 173.10/173.03 thf(sP296,plain,(sP296 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => sP25),introduced(definition,[new_symbols(definition,[sP296])]))). 173.10/173.03 thf(sP297,plain,(sP297 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => sP293),introduced(definition,[new_symbols(definition,[sP297])]))). 173.10/173.03 thf(sP298,plain,(sP298 <=> ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => sP150),introduced(definition,[new_symbols(definition,[sP298])]))). 173.10/173.03 thf(sP299,plain,(sP299 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => sP177),introduced(definition,[new_symbols(definition,[sP299])]))). 173.10/173.03 thf(sP300,plain,(sP300 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => sP229),introduced(definition,[new_symbols(definition,[sP300])]))). 173.10/173.03 thf(sP301,plain,(sP301 <=> ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => sP36),introduced(definition,[new_symbols(definition,[sP301])]))). 173.10/173.03 thf(sP302,plain,(sP302 <=> ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => sP52),introduced(definition,[new_symbols(definition,[sP302])]))). 173.10/173.03 thf(sP303,plain,(sP303 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => sP239),introduced(definition,[new_symbols(definition,[sP303])]))). 173.10/173.03 thf(sP304,plain,(sP304 <=> ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => sP252),introduced(definition,[new_symbols(definition,[sP304])]))). 173.10/173.03 thf(sP305,plain,(sP305 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => sP210),introduced(definition,[new_symbols(definition,[sP305])]))). 173.10/173.03 thf(sP306,plain,(sP306 <=> ((![X1:$i]:((~((((in @ emptyset) @ X1) => (~((![X2:$i]:((~((((in @ X2) @ omega) => (~(((in @ X2) @ X1)))))) => ((in @ ((setadjoin @ X2) @ X2)) @ X1)))))))) => (![X2:$i]:(((in @ X2) @ omega) => ((in @ X2) @ X1))))) => sP250),introduced(definition,[new_symbols(definition,[sP306])]))). 173.10/173.03 thf(sP307,plain,(sP307 <=> (![X1:$i]:(((in @ X1) @ ((image1 @ eigen__0) @ eigen__1)) => (~((![X2:$i]:(((in @ X2) @ eigen__0) => (~((X1 = (eigen__1 @ X2)))))))))),introduced(definition,[new_symbols(definition,[sP307])]))). 173.10/173.03 thf(sP308,plain,(sP308 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => sP244),introduced(definition,[new_symbols(definition,[sP308])]))). 173.10/173.03 thf(sP309,plain,(sP309 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => sP31),introduced(definition,[new_symbols(definition,[sP309])]))). 173.10/173.03 thf(sP310,plain,(sP310 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => sP149),introduced(definition,[new_symbols(definition,[sP310])]))). 173.10/173.03 thf(sP311,plain,(sP311 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => sP38),introduced(definition,[new_symbols(definition,[sP311])]))). 173.10/173.03 thf(sP312,plain,(sP312 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => sP216),introduced(definition,[new_symbols(definition,[sP312])]))). 173.10/173.03 thf(sP313,plain,(sP313 <=> ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => sP266)),introduced(definition,[new_symbols(definition,[sP313])]))). 173.10/173.03 thf(sP314,plain,(sP314 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => sP220),introduced(definition,[new_symbols(definition,[sP314])]))). 173.10/173.03 thf(sP315,plain,(sP315 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => sP191),introduced(definition,[new_symbols(definition,[sP315])]))). 173.10/173.03 thf(sP316,plain,(sP316 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => sP118),introduced(definition,[new_symbols(definition,[sP316])]))). 173.10/173.03 thf(sP317,plain,(sP317 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => sP247),introduced(definition,[new_symbols(definition,[sP317])]))). 173.10/173.03 thf(sP318,plain,(sP318 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => sP125),introduced(definition,[new_symbols(definition,[sP318])]))). 173.10/173.03 thf(sP319,plain,(sP319 <=> ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => sP1),introduced(definition,[new_symbols(definition,[sP319])]))). 173.10/173.03 thf(sP320,plain,(sP320 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => sP262),introduced(definition,[new_symbols(definition,[sP320])]))). 173.10/173.03 thf(sP321,plain,(sP321 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => sP172),introduced(definition,[new_symbols(definition,[sP321])]))). 173.10/173.03 thf(sP322,plain,(sP322 <=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => sP91),introduced(definition,[new_symbols(definition,[sP322])]))). 173.10/173.03 thf(sP323,plain,(sP323 <=> ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => sP266),introduced(definition,[new_symbols(definition,[sP323])]))). 173.10/173.03 thf(sP324,plain,(sP324 <=> ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))))) => sP158),introduced(definition,[new_symbols(definition,[sP324])]))). 173.10/173.03 thf(sP325,plain,(sP325 <=> ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => sP135),introduced(definition,[new_symbols(definition,[sP325])]))). 173.10/173.03 thf(sP326,plain,(sP326 <=> ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => sP80),introduced(definition,[new_symbols(definition,[sP326])]))). 173.10/173.03 thf(sP327,plain,(sP327 <=> ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => sP256),introduced(definition,[new_symbols(definition,[sP327])]))). 173.10/173.03 thf(def_exu,definition,(exu = (^[X1:$i>$o]:(~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3)))))))))))). 173.10/173.03 thf(def_setextAx,definition,(setextAx = (![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) = ((in @ X3) @ X2))) => (X1 = X2)))))). 173.10/173.03 thf(def_emptysetAx,definition,(emptysetAx = (![X1:$i]:(~(((in @ X1) @ emptyset)))))). 173.10/173.03 thf(def_setadjoinAx,definition,(setadjoinAx = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) = ((~((X3 = X1))) => ((in @ X3) @ X2)))))))). 173.10/173.03 thf(def_powersetAx,definition,(powersetAx = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) = (![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1)))))))). 173.10/173.03 thf(def_setunionAx,definition,(setunionAx = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) = (~((![X3:$i]:(((in @ X2) @ X3) => (~(((in @ X3) @ X1)))))))))))). 173.10/173.03 thf(def_omega0Ax,definition,(omega0Ax = ((in @ emptyset) @ omega))). 173.10/173.03 thf(def_omegaSAx,definition,(omegaSAx = (![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))))). 173.10/173.03 thf(def_omegaIndAx,definition,(omegaIndAx = (![X1:$i]:((~((((in @ emptyset) @ X1) => (~((![X2:$i]:((~((((in @ X2) @ omega) => (~(((in @ X2) @ X1)))))) => ((in @ ((setadjoin @ X2) @ X2)) @ X1)))))))) => (![X2:$i]:(((in @ X2) @ omega) => ((in @ X2) @ X1))))))). 173.10/173.03 thf(def_replAx,definition,(replAx = (![X1:$i>$i>$o]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => (exu @ (X1 @ X3)))) => (~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X1 @ X5) @ X4))))))))))))))))))). 173.10/173.03 thf(def_foundationAx,definition,(foundationAx = (![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ X1))))))))))))))). 173.10/173.03 thf(def_wellorderingAx,definition,(wellorderingAx = (![X1:$i]:(~((![X2:$i]:((~(((~(((![X3:$i]:(((in @ X3) @ X2) => (![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X1) => (~(((in @ X4) @ X1)))))) => ((![X5:$i]:(((in @ X5) @ X2) => (((in @ X3) @ X5) = ((in @ X4) @ X5)))) => (X3 = X4)))))))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X2) => (~(((in @ X4) @ X2)))))) => ((~((![X5:$i]:(((in @ X5) @ X3) => ((in @ X5) @ X4))))) => (![X5:$i]:(((in @ X5) @ X4) => ((in @ X5) @ X3)))))))))))) => (~((![X3:$i]:((~(((![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))) => (![X4:$i]:(~(((in @ X4) @ X3))))))) => (~((![X4:$i]:(![X5:$i]:((~(((~((((in @ X4) @ X2) => (~(((in @ X5) @ X3)))))) => (~((![X6:$i]:(((in @ X6) @ X4) => (~(((in @ X6) @ X3)))))))))) => (~((![X6:$i]:(((in @ X6) @ X2) => ((~((![X7:$i]:(((in @ X7) @ X6) => ((in @ X7) @ X4))))) => ((in @ X5) @ X6)))))))))))))))))))))). 173.10/173.03 thf(def_descrp,definition,(descrp = (![X1:$i>$o]:((exu @ X1) => (X1 @ (descr @ X1)))))). 173.10/173.03 thf(def_dsetconstrI,definition,(dsetconstrI = (![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((in @ X3) @ ((dsetconstr @ X1) @ X2))))))))). 173.10/173.03 thf(def_dsetconstrEL,definition,(dsetconstrEL = (![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => ((in @ X3) @ X1))))))). 173.10/173.03 thf(def_dsetconstrER,definition,(dsetconstrER = (![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => (X2 @ X3))))))). 173.10/173.03 thf(def_exuE1,definition,(exuE1 = (![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))))))). 173.10/173.03 thf(def_prop2setE,definition,(prop2setE = (![X1:$o]:(![X2:$i]:(((in @ X2) @ (prop2set @ X1)) => X1))))). 173.10/173.03 thf(def_emptysetE,definition,(emptysetE = (![X1:$i]:(((in @ X1) @ emptyset) => (![X2:$o]:X2))))). 173.10/173.03 thf(def_emptysetimpfalse,definition,(emptysetimpfalse = (![X1:$i]:(~(((in @ X1) @ emptyset)))))). 173.10/173.03 thf(def_notinemptyset,definition,(notinemptyset = (![X1:$i]:(~(((in @ X1) @ emptyset)))))). 173.10/173.03 thf(def_exuE3e,definition,(exuE3e = (![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))))). 173.10/173.03 thf(def_setext,definition,(setext = (![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))))). 173.10/173.03 thf(def_emptyI,definition,(emptyI = (![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))))). 173.10/173.03 thf(def_noeltsimpempty,definition,(noeltsimpempty = (![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))))). 173.10/173.03 thf(def_setbeta,definition,(setbeta = (![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))))). 173.10/173.03 thf(def_nonempty,definition,(nonempty = (^[X1:$i]:(~((X1 = emptyset)))))). 173.10/173.03 thf(def_nonemptyE1,definition,(nonemptyE1 = (![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))))). 173.10/173.03 thf(def_nonemptyI,definition,(nonemptyI = (![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))))). 173.10/173.03 thf(def_nonemptyI1,definition,(nonemptyI1 = (![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))))). 173.10/173.03 thf(def_setadjoinIL,definition,(setadjoinIL = (![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))))). 173.10/173.03 thf(def_emptyinunitempty,definition,(emptyinunitempty = ((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)))). 173.10/173.03 thf(def_setadjoinIR,definition,(setadjoinIR = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))))). 173.10/173.03 thf(def_setadjoinE,definition,(setadjoinE = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))))). 173.10/173.03 thf(def_setadjoinOr,definition,(setadjoinOr = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))))). 173.10/173.03 thf(def_setoftrueEq,definition,(setoftrueEq = (![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)))). 173.10/173.03 thf(def_powersetI,definition,(powersetI = (![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))))). 173.10/173.03 thf(def_emptyinPowerset,definition,(emptyinPowerset = (![X1:$i]:((in @ emptyset) @ (powerset @ X1))))). 173.10/173.03 thf(def_emptyInPowerset,definition,(emptyInPowerset = (![X1:$i]:((in @ emptyset) @ (powerset @ X1))))). 173.10/173.03 thf(def_powersetE,definition,(powersetE = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))))). 173.10/173.03 thf(def_setunionI,definition,(setunionI = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))))). 173.10/173.03 thf(def_setunionE,definition,(setunionE = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))))). 173.10/173.03 thf(def_subPowSU,definition,(subPowSU = (![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))))). 173.10/173.03 thf(def_exuE2,definition,(exuE2 = (![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))))). 173.10/173.03 thf(def_nonemptyImpWitness,definition,(nonemptyImpWitness = (![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))))). 173.10/173.03 thf(def_uniqinunit,definition,(uniqinunit = (![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))))). 173.10/173.03 thf(def_notinsingleton,definition,(notinsingleton = (![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))))). 173.10/173.03 thf(def_eqinunit,definition,(eqinunit = (![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))))). 173.10/173.03 thf(def_singletonsswitch,definition,(singletonsswitch = (![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))). 173.10/173.03 thf(def_upairsetE,definition,(upairsetE = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))))). 173.10/173.03 thf(def_upairsetIL,definition,(upairsetIL = (![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))))). 173.10/173.03 thf(def_upairsetIR,definition,(upairsetIR = (![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))))). 173.10/173.03 thf(def_emptyE1,definition,(emptyE1 = (![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))))). 173.10/173.03 thf(def_vacuousDall,definition,(vacuousDall = (![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))))). 173.10/173.03 thf(def_quantDeMorgan1,definition,(quantDeMorgan1 = (![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))))). 173.10/173.03 thf(def_quantDeMorgan2,definition,(quantDeMorgan2 = (![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))). 173.10/173.03 thf(def_quantDeMorgan3,definition,(quantDeMorgan3 = (![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))). 173.10/173.03 thf(def_quantDeMorgan4,definition,(quantDeMorgan4 = (![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))))). 173.10/173.03 thf(def_prop2setI,definition,(prop2setI = (![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))))). 173.10/173.03 thf(def_prop2set2propI,definition,(prop2set2propI = (![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))))). 173.10/173.03 thf(def_notdexE,definition,(notdexE = (![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))). 173.10/173.03 thf(def_notdallE,definition,(notdallE = (![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))))). 173.10/173.03 thf(def_exuI1,definition,(exuI1 = (![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))))). 173.10/173.03 thf(def_exuI3,definition,(exuI3 = (![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))))). 173.10/173.03 thf(def_exuI2,definition,(exuI2 = (![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))))). 173.10/173.03 thf(def_inCongP,definition,(inCongP = (![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))))). 173.10/173.03 thf(def_in__Cong,definition,(in__Cong = (![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))))). 173.10/173.03 thf(def_exuE3u,definition,(exuE3u = (![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))))). 173.10/173.03 thf(def_exu__Cong,definition,(exu__Cong = (![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))))). 173.10/173.03 thf(def_emptyset__Cong,definition,(emptyset__Cong = (emptyset = emptyset))). 173.10/173.03 thf(def_setadjoin__Cong,definition,(setadjoin__Cong = (![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))))). 173.10/173.03 thf(def_powerset__Cong,definition,(powerset__Cong = (![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))))). 173.10/173.03 thf(def_setunion__Cong,definition,(setunion__Cong = (![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))))). 173.10/173.03 thf(def_omega__Cong,definition,(omega__Cong = (omega = omega))). 173.10/173.03 thf(def_exuEu,definition,(exuEu = (![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))))). 173.10/173.03 thf(def_descr__Cong,definition,(descr__Cong = (![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))))). 173.10/173.03 thf(def_dsetconstr__Cong,definition,(dsetconstr__Cong = (![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))))). 173.10/173.03 thf(def_subsetI1,definition,(subsetI1 = (![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))))). 173.10/173.03 thf(def_eqimpsubset2,definition,(eqimpsubset2 = (![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))))). 173.10/173.03 thf(def_eqimpsubset1,definition,(eqimpsubset1 = (![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))))). 173.10/173.03 thf(def_subsetI2,definition,(subsetI2 = (![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))))). 173.10/173.03 thf(def_emptysetsubset,definition,(emptysetsubset = ((!!) @ (subset @ emptyset)))). 173.10/173.03 thf(def_subsetE,definition,(subsetE = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))))). 173.10/173.03 thf(def_subsetE2,definition,(subsetE2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))))). 173.10/173.03 thf(def_notsubsetI,definition,(notsubsetI = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))))). 173.10/173.03 thf(def_notequalI1,definition,(notequalI1 = (![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))))). 173.10/173.03 thf(def_notequalI2,definition,(notequalI2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))))). 173.10/173.03 thf(def_subsetRefl,definition,(subsetRefl = (![X1:$i]:((subset @ X1) @ X1)))). 173.10/173.03 thf(def_subsetTrans,definition,(subsetTrans = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))))). 173.10/173.03 thf(def_setadjoinSub,definition,(setadjoinSub = (![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))))). 173.10/173.03 thf(def_setadjoinSub2,definition,(setadjoinSub2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))))). 173.10/173.03 thf(def_subset2powerset,definition,(subset2powerset = (![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))))). 173.10/173.03 thf(def_setextsub,definition,(setextsub = (![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))))). 173.10/173.03 thf(def_subsetemptysetimpeq,definition,(subsetemptysetimpeq = (![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))))). 173.10/173.03 thf(def_powersetI1,definition,(powersetI1 = (![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))))). 173.10/173.03 thf(def_powersetE1,definition,(powersetE1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))))). 173.10/173.03 thf(def_inPowerset,definition,(inPowerset = (![X1:$i]:((in @ X1) @ (powerset @ X1))))). 173.10/173.03 thf(def_powersetsubset,definition,(powersetsubset = (![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))))). 173.10/173.03 thf(def_sepInPowerset,definition,(sepInPowerset = (![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))))). 173.10/173.03 thf(def_sepSubset,definition,(sepSubset = (![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))))). 173.10/173.03 thf(def_binunionIL,definition,(binunionIL = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))))). 173.10/173.03 thf(def_upairset2IR,definition,(upairset2IR = (![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))))). 173.10/173.03 thf(def_binunionIR,definition,(binunionIR = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))))). 173.10/173.03 thf(def_binunionEcases,definition,(binunionEcases = (![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))))). 173.10/173.03 thf(def_binunionE,definition,(binunionE = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))))). 173.10/173.03 thf(def_binunionLsub,definition,(binunionLsub = (![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))))). 173.10/173.03 thf(def_binunionRsub,definition,(binunionRsub = (![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))))). 173.10/173.03 thf(def_binintersectI,definition,(binintersectI = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))))). 173.10/173.03 thf(def_binintersectSubset5,definition,(binintersectSubset5 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))). 173.10/173.03 thf(def_binintersectEL,definition,(binintersectEL = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))))). 173.10/173.03 thf(def_binintersectLsub,definition,(binintersectLsub = (![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))))). 173.10/173.03 thf(def_binintersectSubset2,definition,(binintersectSubset2 = (![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))))). 173.10/173.03 thf(def_binintersectSubset3,definition,(binintersectSubset3 = (![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))))). 173.10/173.03 thf(def_binintersectER,definition,(binintersectER = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))))). 173.10/173.03 thf(def_disjointsetsI1,definition,(disjointsetsI1 = (![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))))). 173.10/173.03 thf(def_binintersectRsub,definition,(binintersectRsub = (![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))))). 173.10/173.03 thf(def_binintersectSubset4,definition,(binintersectSubset4 = (![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))))). 173.10/173.03 thf(def_binintersectSubset1,definition,(binintersectSubset1 = (![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))))). 173.10/173.03 thf(def_bs114d,definition,(bs114d = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))))). 173.10/173.03 thf(def_setminusI,definition,(setminusI = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))))). 173.10/173.03 thf(def_setminusEL,definition,(setminusEL = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))))). 173.10/173.03 thf(def_setminusER,definition,(setminusER = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))))). 173.10/173.03 thf(def_setminusSubset2,definition,(setminusSubset2 = (![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))))). 173.10/173.03 thf(def_setminusERneg,definition,(setminusERneg = (![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))))). 173.10/173.03 thf(def_setminusELneg,definition,(setminusELneg = (![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))))). 173.10/173.03 thf(def_setminusILneg,definition,(setminusILneg = (![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))). 173.10/173.03 thf(def_setminusIRneg,definition,(setminusIRneg = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))). 173.10/173.03 thf(def_setminusLsub,definition,(setminusLsub = (![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))))). 173.10/173.03 thf(def_setminusSubset1,definition,(setminusSubset1 = (![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))))). 173.10/173.03 thf(def_symdiffE,definition,(symdiffE = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))))). 173.10/173.03 thf(def_symdiffI1,definition,(symdiffI1 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))))). 173.10/173.03 thf(def_symdiffI2,definition,(symdiffI2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))))). 173.10/173.03 thf(def_symdiffIneg1,definition,(symdiffIneg1 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))))). 173.10/173.03 thf(def_symdiffIneg2,definition,(symdiffIneg2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))))). 173.10/173.03 thf(def_secondinupair,definition,(secondinupair = (![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))))). 173.10/173.03 thf(def_setukpairIL,definition,(setukpairIL = (![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))))). 173.10/173.03 thf(def_setukpairIR,definition,(setukpairIR = (![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))))). 173.10/173.03 thf(def_kpairiskpair,definition,(kpairiskpair = (![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))))). 173.10/173.03 thf(def_kpairp,definition,(kpairp = (![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))))). 173.10/173.03 thf(def_singletonsubset,definition,(singletonsubset = (![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))))). 173.10/173.03 thf(def_singletoninpowerset,definition,(singletoninpowerset = (![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))))). 173.10/173.03 thf(def_singletoninpowunion,definition,(singletoninpowunion = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))))). 173.10/173.03 thf(def_upairset2E,definition,(upairset2E = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))))). 173.10/173.03 thf(def_upairsubunion,definition,(upairsubunion = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))))). 173.10/173.03 thf(def_upairinpowunion,definition,(upairinpowunion = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))))). 173.10/173.03 thf(def_ubforcartprodlem1,definition,(ubforcartprodlem1 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))))). 173.10/173.03 thf(def_ubforcartprodlem2,definition,(ubforcartprodlem2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))))). 173.10/173.03 thf(def_ubforcartprodlem3,definition,(ubforcartprodlem3 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))))). 173.10/173.03 thf(def_cartprodpairin,definition,(cartprodpairin = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))))). 173.10/173.03 thf(def_cartprodmempair1,definition,(cartprodmempair1 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))))). 173.10/173.03 thf(def_cartprodmempair,definition,(cartprodmempair = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))))). 173.10/173.03 thf(def_setunionE2,definition,(setunionE2 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))))). 173.10/173.03 thf(def_setunionsingleton1,definition,(setunionsingleton1 = (![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)))). 173.10/173.03 thf(def_setunionsingleton2,definition,(setunionsingleton2 = (![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))))). 173.10/173.03 thf(def_setunionsingleton,definition,(setunionsingleton = (![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)))). 173.10/173.03 thf(def_singletonprop,definition,(singletonprop = (![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))))). 173.10/173.03 thf(def_ex1E1,definition,(ex1E1 = (![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))))). 173.10/173.03 thf(def_ex1I,definition,(ex1I = (![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))))). 173.10/173.03 thf(def_ex1I2,definition,(ex1I2 = (![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))))). 173.10/173.03 thf(def_singletonsuniq,definition,(singletonsuniq = (![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))))). 173.10/173.03 thf(def_setukpairinjL1,definition,(setukpairinjL1 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))))). 173.10/173.03 thf(def_kfstsingleton,definition,(kfstsingleton = (![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))))). 173.10/173.03 thf(def_theprop,definition,(theprop = (![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))))). 173.10/173.03 thf(def_kfstpairEq,definition,(kfstpairEq = (![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))))). 173.10/173.03 thf(def_cartprodfstin,definition,(cartprodfstin = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))))). 173.10/173.03 thf(def_setukpairinjL2,definition,(setukpairinjL2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))))). 173.10/173.03 thf(def_setukpairinjL,definition,(setukpairinjL = (![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))))). 173.10/173.03 thf(def_setukpairinjR11,definition,(setukpairinjR11 = (![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))))). 173.10/173.03 thf(def_setukpairinjR12,definition,(setukpairinjR12 = (![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))))). 173.10/173.03 thf(def_setukpairinjR1,definition,(setukpairinjR1 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))))). 173.10/173.03 thf(def_upairequniteq,definition,(upairequniteq = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))))). 173.10/173.03 thf(def_setukpairinjR2,definition,(setukpairinjR2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))))). 173.10/173.03 thf(def_setukpairinjR,definition,(setukpairinjR = (![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))))). 173.10/173.03 thf(def_ksndsingleton,definition,(ksndsingleton = (![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))))). 173.10/173.03 thf(def_ksndpairEq,definition,(ksndpairEq = (![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))))). 173.10/173.03 thf(def_kpairsurjEq,definition,(kpairsurjEq = (![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))))). 173.10/173.03 thf(def_cartprodsndin,definition,(cartprodsndin = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))))). 173.10/173.03 thf(def_cartprodpairmemEL,definition,(cartprodpairmemEL = (![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))))). 173.10/173.03 thf(def_cartprodpairmemER,definition,(cartprodpairmemER = (![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))))). 173.10/173.03 thf(def_cartprodmempaircEq,definition,(cartprodmempaircEq = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))))). 173.10/173.03 thf(def_cartprodfstpairEq,definition,(cartprodfstpairEq = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))))). 173.10/173.03 thf(def_cartprodsndpairEq,definition,(cartprodsndpairEq = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))))). 173.10/173.03 thf(def_cartprodpairsurjEq,definition,(cartprodpairsurjEq = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))))). 173.10/173.03 thf(def_dpsetconstrI,definition,(dpsetconstrI = (![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))))). 173.10/173.03 thf(def_dpsetconstrSub,definition,(dpsetconstrSub = (![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))))). 173.10/173.03 thf(def_setOfPairsIsBReln,definition,(setOfPairsIsBReln = (![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))))). 173.10/173.03 thf(def_dpsetconstrERa,definition,(dpsetconstrERa = (![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))))). 173.10/173.03 thf(def_dpsetconstrEL1,definition,(dpsetconstrEL1 = (![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))))). 173.10/173.03 thf(def_dpsetconstrEL2,definition,(dpsetconstrEL2 = (![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))))). 173.10/173.03 thf(def_dpsetconstrER,definition,(dpsetconstrER = (![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))). 173.10/173.03 thf(def_funcImageSingleton,definition,(funcImageSingleton = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))). 173.10/173.03 thf(def_apProp,definition,(apProp = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))))). 173.10/173.03 thf(def_app,definition,(app = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))))). 173.10/173.03 thf(def_infuncsetfunc,definition,(infuncsetfunc = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))))). 173.10/173.03 thf(def_ap2p,definition,(ap2p = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))))). 173.10/173.03 thf(def_funcinfuncset,definition,(funcinfuncset = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))))). 173.10/173.03 thf(def_lamProp,definition,(lamProp = (![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))))). 173.10/173.03 thf(def_lamp,definition,(lamp = (![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))))). 173.10/173.03 thf(def_lam2p,definition,(lam2p = (![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))))). 173.10/173.03 thf(def_brelnall1,definition,(brelnall1 = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))))). 173.10/173.03 thf(def_brelnall2,definition,(brelnall2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))))). 173.10/173.03 thf(def_ex1E2,definition,(ex1E2 = (![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))))). 173.10/173.03 thf(def_funcGraphProp1,definition,(funcGraphProp1 = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))))). 173.10/173.03 thf(def_funcGraphProp3,definition,(funcGraphProp3 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))))). 173.10/173.03 thf(def_funcGraphProp2,definition,(funcGraphProp2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))))). 173.10/173.03 thf(def_funcextLem,definition,(funcextLem = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))))). 173.10/173.03 thf(def_funcGraphProp4,definition,(funcGraphProp4 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))))). 173.10/173.03 thf(def_subbreln,definition,(subbreln = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))))). 173.10/173.03 thf(def_eqbreln,definition,(eqbreln = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))))). 173.10/173.03 thf(def_funcext,definition,(funcext = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))))). 173.10/173.03 thf(def_funcext2,definition,(funcext2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))))). 173.10/173.03 thf(def_ap2apEq1,definition,(ap2apEq1 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))))). 173.10/173.03 thf(def_ap2apEq2,definition,(ap2apEq2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))))). 173.10/173.03 thf(def_beta1,definition,(beta1 = (![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))))). 173.10/173.03 thf(def_eta1,definition,(eta1 = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))))). 173.10/173.03 thf(def_lam2lamEq,definition,(lam2lamEq = (![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))))). 173.10/173.03 thf(def_beta2,definition,(beta2 = (![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))))). 173.10/173.03 thf(def_eta2,definition,(eta2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))))). 173.10/173.03 thf(def_iffalseProp1,definition,(iffalseProp1 = (![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))))). 173.10/173.03 thf(def_iffalseProp2,definition,(iffalseProp2 = (![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))))). 173.10/173.03 thf(def_iftrueProp1,definition,(iftrueProp1 = (![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))))). 173.10/173.03 thf(def_iftrueProp2,definition,(iftrueProp2 = (![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))))). 173.10/173.03 thf(def_ifSingleton,definition,(ifSingleton = (![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))))). 173.10/173.03 thf(def_ifp,definition,(ifp = (![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))))). 173.10/173.03 thf(def_theeq,definition,(theeq = (![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))))). 173.10/173.03 thf(def_iftrue,definition,(iftrue = (![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))))). 173.10/173.03 thf(def_iffalse,definition,(iffalse = (![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))))). 173.10/173.03 thf(def_iftrueorfalse,definition,(iftrueorfalse = (![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))))). 173.10/173.03 thf(def_binintersectT_lem,definition,(binintersectT_lem = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))))). 173.10/173.03 thf(def_binunionT_lem,definition,(binunionT_lem = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))))). 173.10/173.03 thf(def_powersetT_lem,definition,(powersetT_lem = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))))). 173.10/173.03 thf(def_setminusT_lem,definition,(setminusT_lem = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))))). 173.10/173.03 thf(def_complementT_lem,definition,(complementT_lem = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))))). 173.10/173.03 thf(def_setextT,definition,(setextT = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))))). 173.10/173.03 thf(def_subsetTI,definition,(subsetTI = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))))). 173.10/173.03 thf(def_powersetTI1,definition,(powersetTI1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))))). 173.10/173.03 thf(def_powersetTE1,definition,(powersetTE1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))))). 173.10/173.03 thf(def_complementTI1,definition,(complementTI1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))))). 173.10/173.03 thf(def_complementTE1,definition,(complementTE1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))))). 173.10/173.03 thf(def_binintersectTELcontra,definition,(binintersectTELcontra = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))))). 173.10/173.03 thf(def_binintersectTERcontra,definition,(binintersectTERcontra = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))))). 173.10/173.03 thf(def_contrasubsetT,definition,(contrasubsetT = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))))). 173.10/173.03 thf(def_contrasubsetT1,definition,(contrasubsetT1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))))). 173.10/173.03 thf(def_contrasubsetT2,definition,(contrasubsetT2 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))))). 173.10/173.03 thf(def_contrasubsetT3,definition,(contrasubsetT3 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))))). 173.10/173.03 thf(def_doubleComplementI1,definition,(doubleComplementI1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))))). 173.10/173.03 thf(def_doubleComplementE1,definition,(doubleComplementE1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))))). 173.10/173.03 thf(def_doubleComplementSub1,definition,(doubleComplementSub1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))))). 173.10/173.03 thf(def_doubleComplementSub2,definition,(doubleComplementSub2 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))))). 173.10/173.03 thf(def_doubleComplementEq,definition,(doubleComplementEq = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))))). 173.10/173.03 thf(def_complementTnotintersectT,definition,(complementTnotintersectT = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))))). 173.10/173.03 thf(def_complementImpComplementIntersect,definition,(complementImpComplementIntersect = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))))). 173.10/173.03 thf(def_complementSubsetComplementIntersect,definition,(complementSubsetComplementIntersect = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))))). 173.10/173.03 thf(def_complementInPowersetComplementIntersect,definition,(complementInPowersetComplementIntersect = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))). 173.10/173.03 thf(def_contraSubsetComplement,definition,(contraSubsetComplement = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))))). 173.10/173.03 thf(def_complementTcontraSubset,definition,(complementTcontraSubset = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))))). 173.10/173.03 thf(def_binunionTILcontra,definition,(binunionTILcontra = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))))). 173.10/173.03 thf(def_binunionTIRcontra,definition,(binunionTIRcontra = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))))). 173.10/173.03 thf(def_inIntersectImpInUnion,definition,(inIntersectImpInUnion = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))))). 173.10/173.03 thf(def_inIntersectImpInUnion2,definition,(inIntersectImpInUnion2 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))))). 173.10/173.03 thf(def_inIntersectImpInIntersectUnions,definition,(inIntersectImpInIntersectUnions = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))))). 173.10/173.03 thf(def_intersectInPowersetIntersectUnions,definition,(intersectInPowersetIntersectUnions = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))). 173.10/173.03 thf(def_inComplementUnionImpNotIn1,definition,(inComplementUnionImpNotIn1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))))). 173.10/173.03 thf(def_inComplementUnionImpInComplement1,definition,(inComplementUnionImpInComplement1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))))). 173.10/173.03 thf(def_binunionTE,definition,(binunionTE = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))))). 173.10/173.03 thf(def_binunionTEcontra,definition,(binunionTEcontra = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))))). 173.10/173.04 thf(def_demorgan2a1,definition,(demorgan2a1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))))). 173.10/173.04 thf(def_complementUnionInPowersetComplement,definition,(complementUnionInPowersetComplement = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))))). 173.10/173.04 thf(def_demorgan2a2,definition,(demorgan2a2 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))))). 173.10/173.04 thf(def_demorgan1a,definition,(demorgan1a = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))))). 173.10/173.04 thf(def_demorgan1b,definition,(demorgan1b = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))))). 173.10/173.04 thf(def_demorgan1,definition,(demorgan1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))))). 173.10/173.04 thf(def_demorgan2a,definition,(demorgan2a = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))))). 173.10/173.04 thf(def_demorgan2b2,definition,(demorgan2b2 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))))). 173.10/173.04 thf(def_demorgan2b,definition,(demorgan2b = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))))). 173.10/173.04 thf(def_demorgan2,definition,(demorgan2 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))))). 173.10/173.04 thf(def_woz13rule0,definition,(woz13rule0 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))))). 173.10/173.04 thf(def_woz13rule1,definition,(woz13rule1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))))). 173.10/173.04 thf(def_woz13rule2,definition,(woz13rule2 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))))). 173.10/173.04 thf(def_woz13rule3,definition,(woz13rule3 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))))). 173.10/173.04 thf(def_woz13rule4,definition,(woz13rule4 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))))). 173.10/173.04 thf(def_woz1_1,definition,(woz1_1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))). 173.10/173.04 thf(def_woz1_2,definition,(woz1_2 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))))). 173.10/173.04 thf(def_woz1_3,definition,(woz1_3 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))). 173.10/173.04 thf(def_woz1_4,definition,(woz1_4 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))))). 173.10/173.04 thf(def_woz1_5,definition,(woz1_5 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))))). 173.10/173.04 thf(def_breln1all2,definition,(breln1all2 = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))))). 173.10/173.04 thf(def_breln1SetBreln1,definition,(breln1SetBreln1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))))). 173.10/173.04 thf(def_choice2fnsingleton,definition,(choice2fnsingleton = (![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))))). 173.10/173.04 thf(def_setOfPairsIsBReln1,definition,(setOfPairsIsBReln1 = (![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))))). 173.10/173.04 thf(def_breln1all1,definition,(breln1all1 = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))))). 173.10/173.04 thf(def_subbreln1,definition,(subbreln1 = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))))). 173.10/173.04 thf(def_eqbreln1,definition,(eqbreln1 = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))))). 173.10/173.04 thf(def_breln1invprop,definition,(breln1invprop = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))))). 173.10/173.04 thf(def_breln1invI,definition,(breln1invI = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))))). 173.10/173.04 thf(def_breln1invE,definition,(breln1invE = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))))). 173.10/173.04 thf(def_breln1compprop,definition,(breln1compprop = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))))). 173.10/173.04 thf(def_breln1compI,definition,(breln1compI = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))))). 173.10/173.04 thf(def_breln1compE,definition,(breln1compE = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))))). 173.10/173.04 thf(def_breln1compEex,definition,(breln1compEex = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))))). 173.10/173.04 thf(def_breln1unionprop,definition,(breln1unionprop = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))))). 173.10/173.04 thf(def_breln1unionIL,definition,(breln1unionIL = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))))). 173.10/173.04 thf(def_breln1unionIR,definition,(breln1unionIR = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))))). 173.10/173.04 thf(def_breln1unionI,definition,(breln1unionI = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))))). 173.10/173.04 thf(def_breln1unionE,definition,(breln1unionE = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))))). 173.10/173.04 thf(def_breln1unionEcases,definition,(breln1unionEcases = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))))). 173.10/173.04 thf(def_breln1unionCommutes,definition,(breln1unionCommutes = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))))). 173.10/173.04 thf(def_woz2Ex,definition,(woz2Ex = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))))). 173.10/173.04 thf(def_woz2W,definition,(woz2W = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))). 173.10/173.04 thf(def_woz2A,definition,(woz2A = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))))). 173.10/173.04 thf(def_woz2B,definition,(woz2B = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))))). 173.10/173.04 thf(def_image1Ex,definition,(image1Ex = (![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))))). 173.10/173.04 thf(def_image1Ex1,definition,(image1Ex1 = (![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))))). 173.10/173.04 thf(def_image1Equiv,definition,(image1Equiv = sP87)). 173.10/173.04 thf(image1E,conjecture,(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => sP274))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))). 173.10/173.04 thf(h2,negated_conjecture,(~(sP286)),inference(assume_negation,[status(cth)],[image1E])). 173.10/173.04 thf(1,plain,(sP286 | ~(sP279)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(2,plain,(sP279 | ~(sP195)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(3,plain,(sP195 | ~(sP175)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(4,plain,(sP175 | ~(sP104)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(5,plain,(sP104 | ~(sP107)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(6,plain,(sP107 | ~(sP101)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(7,plain,(sP101 | ~(sP306)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(8,plain,(sP306 | ~(sP250)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(9,plain,(sP250 | ~(sP251)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(10,plain,(sP251 | ~(sP277)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(11,plain,(sP277 | ~(sP124)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(12,plain,(sP124 | ~(sP159)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(13,plain,(sP159 | ~(sP227)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(14,plain,(sP227 | ~(sP4)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(15,plain,(sP4 | ~(sP138)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(16,plain,(sP138 | ~(sP190)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(17,plain,(sP190 | ~(sP9)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(18,plain,(sP9 | ~(sP29)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(19,plain,(sP29 | ~(sP65)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(20,plain,(sP65 | ~(sP178)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(21,plain,(sP178 | ~(sP23)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(22,plain,(sP23 | ~(sP3)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(23,plain,(sP3 | ~(sP131)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(24,plain,(sP131 | ~(sP13)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(25,plain,(sP13 | ~(sP50)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(26,plain,(sP50 | ~(sP324)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(27,plain,(sP324 | ~(sP158)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(28,plain,(sP158 | ~(sP291)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(29,plain,(sP291 | ~(sP143)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(30,plain,(sP143 | ~(sP57)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(31,plain,(sP57 | ~(sP237)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(32,plain,(sP237 | ~(sP154)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(33,plain,(sP154 | ~(sP189)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(34,plain,(sP189 | ~(sP11)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(35,plain,(sP11 | ~(sP271)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(36,plain,(sP271 | ~(sP121)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(37,plain,(sP121 | ~(sP45)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(38,plain,(sP45 | ~(sP35)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(39,plain,(sP35 | ~(sP53)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(40,plain,(sP53 | ~(sP285)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(41,plain,(sP285 | ~(sP142)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(42,plain,(sP142 | ~(sP127)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(43,plain,(sP127 | ~(sP146)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(44,plain,(sP146 | ~(sP219)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(45,plain,(sP219 | ~(sP2)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(46,plain,(sP2 | ~(sP174)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(47,plain,(sP174 | ~(sP89)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(48,plain,(sP89 | ~(sP182)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(49,plain,(sP182 | ~(sP119)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(50,plain,(sP119 | ~(sP133)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(51,plain,(sP133 | ~(sP109)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(52,plain,(sP109 | ~(sP267)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(53,plain,(sP267 | ~(sP313)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(54,plain,(sP313 | ~(sP323)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(55,plain,(sP323 | ~(sP266)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(56,plain,(sP266 | ~(sP151)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(57,plain,(sP151 | ~(sP163)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(58,plain,(sP163 | ~(sP83)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(59,plain,(sP83 | ~(sP148)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(60,plain,(sP148 | ~(sP56)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(61,plain,(sP56 | ~(sP63)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(62,plain,(sP63 | ~(sP18)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(63,plain,(sP18 | ~(sP242)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(64,plain,(sP242 | ~(sP186)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(65,plain,(sP186 | ~(sP302)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(66,plain,(sP302 | ~(sP52)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(67,plain,(sP52 | ~(sP257)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(68,plain,(sP257 | ~(sP208)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(69,plain,(sP208 | ~(sP165)),inference(prop_rule,[status(thm)],[])). 173.10/173.04 thf(70,plain,(sP165 | ~(sP272)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(71,plain,(sP272 | ~(sP145)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(72,plain,(sP145 | ~(sP263)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(73,plain,(sP263 | ~(sP180)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(74,plain,(sP180 | ~(sP62)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(75,plain,(sP62 | ~(sP288)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(76,plain,(sP288 | ~(sP231)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(77,plain,(sP231 | ~(sP230)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(78,plain,(sP230 | ~(sP96)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(79,plain,(sP96 | ~(sP46)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(80,plain,(sP46 | ~(sP21)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(81,plain,(sP21 | ~(sP68)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(82,plain,(sP68 | ~(sP179)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(83,plain,(sP179 | ~(sP76)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(84,plain,(sP76 | ~(sP226)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(85,plain,(sP226 | ~(sP281)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(86,plain,(sP281 | ~(sP233)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(87,plain,(sP233 | ~(sP33)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(88,plain,(sP33 | ~(sP81)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(89,plain,(sP81 | ~(sP117)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(90,plain,(sP117 | ~(sP211)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(91,plain,(sP211 | ~(sP217)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(92,plain,(sP217 | ~(sP42)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(93,plain,(sP42 | ~(sP185)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(94,plain,(sP185 | ~(sP12)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(95,plain,(sP12 | ~(sP171)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(96,plain,(sP171 | ~(sP270)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(97,plain,(sP270 | ~(sP218)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(98,plain,(sP218 | ~(sP153)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(99,plain,(sP153 | ~(sP236)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(100,plain,(sP236 | ~(sP316)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(101,plain,(sP316 | ~(sP118)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(102,plain,(sP118 | ~(sP260)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(103,plain,(sP260 | ~(sP225)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(104,plain,(sP225 | ~(sP79)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(105,plain,(sP79 | ~(sP20)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(106,plain,(sP20 | ~(sP157)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(107,plain,(sP157 | ~(sP105)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(108,plain,(sP105 | ~(sP298)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(109,plain,(sP298 | ~(sP150)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(110,plain,(sP150 | ~(sP241)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(111,plain,(sP241 | ~(sP85)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(112,plain,(sP85 | ~(sP223)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(113,plain,(sP223 | ~(sP108)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(114,plain,(sP108 | ~(sP301)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(115,plain,(sP301 | ~(sP36)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(116,plain,(sP36 | ~(sP100)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(117,plain,(sP100 | ~(sP205)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(118,plain,(sP205 | ~(sP181)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(119,plain,(sP181 | ~(sP245)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(120,plain,(sP245 | ~(sP249)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(121,plain,(sP249 | ~(sP32)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(122,plain,(sP32 | ~(sP19)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(123,plain,(sP19 | ~(sP43)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(124,plain,(sP43 | ~(sP126)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(125,plain,(sP126 | ~(sP5)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(126,plain,(sP5 | ~(sP40)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(127,plain,(sP40 | ~(sP284)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(128,plain,(sP284 | ~(sP235)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(129,plain,(sP235 | ~(sP296)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(130,plain,(sP296 | ~(sP25)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(131,plain,(sP25 | ~(sP156)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(132,plain,(sP156 | ~(sP27)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(133,plain,(sP27 | ~(sP16)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(134,plain,(sP16 | ~(sP55)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(135,plain,(sP55 | ~(sP113)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(136,plain,(sP113 | ~(sP258)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(137,plain,(sP258 | ~(sP192)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(138,plain,(sP192 | ~(sP8)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(139,plain,(sP8 | ~(sP26)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(140,plain,(sP26 | ~(sP51)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(141,plain,(sP51 | ~(sP86)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(142,plain,(sP86 | ~(sP140)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(143,plain,(sP140 | ~(sP312)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(144,plain,(sP312 | ~(sP216)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(145,plain,(sP216 | ~(sP183)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(146,plain,(sP183 | ~(sP60)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(147,plain,(sP60 | ~(sP198)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(148,plain,(sP198 | ~(sP34)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(149,plain,(sP34 | ~(sP152)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(150,plain,(sP152 | ~(sP204)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(151,plain,(sP204 | ~(sP132)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(152,plain,(sP132 | ~(sP17)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(153,plain,(sP17 | ~(sP187)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(154,plain,(sP187 | ~(sP169)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(155,plain,(sP169 | ~(sP264)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(156,plain,(sP264 | ~(sP203)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(157,plain,(sP203 | ~(sP70)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(158,plain,(sP70 | ~(sP322)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(159,plain,(sP322 | ~(sP91)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(160,plain,(sP91 | ~(sP269)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(161,plain,(sP269 | ~(sP75)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(162,plain,(sP75 | ~(sP160)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(163,plain,(sP160 | ~(sP98)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(164,plain,(sP98 | ~(sP78)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(165,plain,(sP78 | ~(sP94)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(166,plain,(sP94 | ~(sP327)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(167,plain,(sP327 | ~(sP256)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(168,plain,(sP256 | ~(sP88)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(169,plain,(sP88 | ~(sP14)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(170,plain,(sP14 | ~(sP253)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(171,plain,(sP253 | ~(sP240)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(172,plain,(sP240 | ~(sP213)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(173,plain,(sP213 | ~(sP282)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(174,plain,(sP282 | ~(sP224)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(175,plain,(sP224 | ~(sP7)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(176,plain,(sP7 | ~(sP184)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(177,plain,(sP184 | ~(sP10)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(178,plain,(sP10 | ~(sP67)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(179,plain,(sP67 | ~(sP99)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(180,plain,(sP99 | ~(sP134)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(181,plain,(sP134 | ~(sP66)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(182,plain,(sP66 | ~(sP170)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(183,plain,(sP170 | ~(sP168)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(184,plain,(sP168 | ~(sP318)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(185,plain,(sP318 | ~(sP125)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(186,plain,(sP125 | ~(sP162)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(187,plain,(sP162 | ~(sP112)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(188,plain,(sP112 | ~(sP49)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(189,plain,(sP49 | ~(sP215)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(190,plain,(sP215 | ~(sP137)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(191,plain,(sP137 | ~(sP28)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(192,plain,(sP28 | ~(sP290)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(193,plain,(sP290 | ~(sP39)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(194,plain,(sP39 | ~(sP61)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(195,plain,(sP61 | ~(sP161)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(196,plain,(sP161 | ~(sP311)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(197,plain,(sP311 | ~(sP38)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(198,plain,(sP38 | ~(sP268)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(199,plain,(sP268 | ~(sP74)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(200,plain,(sP74 | ~(sP299)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(201,plain,(sP299 | ~(sP177)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(202,plain,(sP177 | ~(sP228)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(203,plain,(sP228 | ~(sP248)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(204,plain,(sP248 | ~(sP116)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(205,plain,(sP116 | ~(sP222)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(206,plain,(sP222 | ~(sP164)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(207,plain,(sP164 | ~(sP120)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(208,plain,(sP120 | ~(sP77)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(209,plain,(sP77 | ~(sP287)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(210,plain,(sP287 | ~(sP232)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(211,plain,(sP232 | ~(sP115)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(212,plain,(sP115 | ~(sP167)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(213,plain,(sP167 | ~(sP123)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(214,plain,(sP123 | ~(sP206)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(215,plain,(sP206 | ~(sP300)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(216,plain,(sP300 | ~(sP229)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(217,plain,(sP229 | ~(sP214)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(218,plain,(sP214 | ~(sP103)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(219,plain,(sP103 | ~(sP139)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(220,plain,(sP139 | ~(sP193)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(221,plain,(sP193 | ~(sP295)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(222,plain,(sP295 | ~(sP304)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(223,plain,(sP304 | ~(sP252)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(224,plain,(sP252 | ~(sP212)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(225,plain,(sP212 | ~(sP246)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(226,plain,(sP246 | ~(sP155)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(227,plain,(sP155 | ~(sP72)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(228,plain,(sP72 | ~(sP196)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(229,plain,(sP196 | ~(sP41)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(230,plain,(sP41 | ~(sP310)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(231,plain,(sP310 | ~(sP149)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(232,plain,(sP149 | ~(sP280)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(233,plain,(sP280 | ~(sP102)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(234,plain,(sP102 | ~(sP317)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(235,plain,(sP317 | ~(sP247)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(236,plain,(sP247 | ~(sP176)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(237,plain,(sP176 | ~(sP200)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(238,plain,(sP200 | ~(sP221)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(239,plain,(sP221 | ~(sP44)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(240,plain,(sP44 | ~(sP326)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(241,plain,(sP326 | ~(sP80)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(242,plain,(sP80 | ~(sP54)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(243,plain,(sP54 | ~(sP141)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(244,plain,(sP141 | ~(sP64)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(245,plain,(sP64 | ~(sP289)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(246,plain,(sP289 | ~(sP129)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(247,plain,(sP129 | ~(sP305)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(248,plain,(sP305 | ~(sP210)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(249,plain,(sP210 | ~(sP199)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(250,plain,(sP199 | ~(sP276)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(251,plain,(sP276 | ~(sP22)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(252,plain,(sP22 | ~(sP314)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(253,plain,(sP314 | ~(sP220)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(254,plain,(sP220 | ~(sP166)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(255,plain,(sP166 | ~(sP48)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(256,plain,(sP48 | ~(sP321)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(257,plain,(sP321 | ~(sP172)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(258,plain,(sP172 | ~(sP95)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(259,plain,(sP95 | ~(sP297)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(260,plain,(sP297 | ~(sP293)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(261,plain,(sP293 | ~(sP275)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(262,plain,(sP275 | ~(sP309)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(263,plain,(sP309 | ~(sP31)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(264,plain,(sP31 | ~(sP320)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(265,plain,(sP320 | ~(sP262)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(266,plain,(sP262 | ~(sP255)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(267,plain,(sP255 | ~(sP97)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(268,plain,(sP97 | ~(sP265)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(269,plain,(sP265 | ~(sP273)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(270,plain,(sP273 | ~(sP303)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(271,plain,(sP303 | ~(sP239)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(272,plain,(sP239 | ~(sP82)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(273,plain,(sP82 | ~(sP110)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(274,plain,(sP110 | ~(sP59)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(275,plain,(sP59 | ~(sP15)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(276,plain,(sP15 | ~(sP106)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(277,plain,(sP106 | ~(sP315)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(278,plain,(sP315 | ~(sP191)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(279,plain,(sP191 | ~(sP278)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(280,plain,(sP278 | ~(sP254)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(281,plain,(sP254 | ~(sP308)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(282,plain,(sP308 | ~(sP244)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(283,plain,(sP244 | ~(sP238)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(284,plain,(sP238 | ~(sP24)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(285,plain,(sP24 | ~(sP69)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(286,plain,(sP69 | ~(sP147)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(287,plain,(sP147 | ~(sP292)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(288,plain,(sP292 | ~(sP259)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(289,plain,(sP259 | ~(sP130)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(290,plain,(sP130 | ~(sP47)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(291,plain,(sP47 | ~(sP294)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(292,plain,(sP294 | ~(sP209)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(293,plain,(sP209 | ~(sP188)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(294,plain,(sP188 | ~(sP111)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(295,plain,(sP111 | ~(sP319)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(296,plain,(sP319 | ~(sP1)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(297,plain,(sP1 | ~(sP122)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(298,plain,(sP122 | ~(sP136)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(299,plain,(sP136 | ~(sP201)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(300,plain,(sP201 | ~(sP144)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(301,plain,(sP144 | ~(sP202)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(302,plain,(sP202 | ~(sP90)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(303,plain,(sP90 | ~(sP58)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(304,plain,(sP58 | ~(sP37)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(305,plain,(sP37 | ~(sP325)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(306,plain,(sP325 | ~(sP135)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(307,plain,(sP135 | ~(sP71)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(308,plain,(sP71 | ~(sP30)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(309,plain,(sP30 | ~(sP194)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(310,plain,(sP194 | ~(sP84)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(311,plain,(sP84 | ~(sP197)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(312,plain,(sP197 | sP87),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(313,plain,(sP197 | ~(sP274)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(314,plain,(sP274 | ~(sP6)),inference(eigen_choice_rule,[status(thm),assumptions([h1])],[h1,eigendef_eigen__0])). 173.10/173.05 thf(315,plain,(sP6 | ~(sP307)),inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__1])). 173.10/173.05 thf(316,plain,(sP307 | ~(sP173)),inference(eigen_choice_rule,[status(thm),assumptions([h1])],[h1,eigendef_eigen__2])). 173.10/173.05 thf(317,plain,(sP173 | sP243),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(318,plain,(sP173 | sP234),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(319,plain,sP207,inference(@eq_ind,[status(thm)],[])). 173.10/173.05 thf(320,plain,(~(sP207) | sP283),inference(all_rule,[status(thm)],[])). 173.10/173.05 thf(321,plain,(~(sP283) | sP93),inference(all_rule,[status(thm)],[])). 173.10/173.05 thf(322,plain,((~(sP93) | ~(sP243)) | sP92),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(323,plain,(~(sP92) | sP114),inference(all_rule,[status(thm)],[])). 173.10/173.05 thf(324,plain,((~(sP114) | ~(sP73)) | ~(sP234)),inference(prop_rule,[status(thm)],[])). 173.10/173.05 thf(325,plain,(~(sP261) | sP73),inference(all_rule,[status(thm)],[])). 173.10/173.05 thf(326,plain,(~(sP128) | sP261),inference(all_rule,[status(thm)],[])). 173.10/173.05 thf(327,plain,(~(sP87) | sP128),inference(all_rule,[status(thm)],[])). 173.10/173.05 thf(328,plain,$false,inference(prop_unsat,[status(thm),assumptions([h2,h1,h0])],[h2,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327])). 173.10/173.05 thf(329,plain,$false,inference(eigenvar_choice,[status(thm),assumptions([h2,h0]),eigenvar_choice(discharge,[h1])],[328,h1])). 173.10/173.05 thf(330,plain,$false,inference(eigenvar_choice,[status(thm),assumptions([h2]),eigenvar_choice(discharge,[h0])],[329,h0])). 173.10/173.05 thf(0,theorem,(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => sP274)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),inference(contra,[status(thm),contra(discharge,[h2])],[328,h2])). 173.10/173.05 % SZS output end Proof 173.10/173.05 EOF