const In : set set prop term iIn = In infix iIn 2000 2000 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term TransSet = \x:set.!y:set.y iIn x -> Subq y x const Pi : set (set set) set term setexp = \x:set.\y:set.Pi y \z:set.x const SNo : set prop const SNoLt : set set prop term < = SNoLt infix < 2020 2020 term SNoCutP = \x:set.\y:set.(!z:set.z iIn x -> SNo z) & (!z:set.z iIn y -> SNo z) & !z:set.z iIn x -> !w:set.w iIn y -> z < w term nIn = \x:set.\y:set.~ x iIn y const ordinal : set prop const omega : set axiom omega_ordinal: ordinal omega const SNoS_ : set set const SNoLev : set set const SNo_ : set set prop axiom SNoS_E2: !x:set.ordinal x -> !y:set.y iIn SNoS_ x -> !P:prop.(SNoLev y iIn x -> ordinal (SNoLev y) -> SNo y -> SNo_ (SNoLev y) y -> P) -> P const Sep : set (set prop) set axiom SepE: !x:set.!p:set prop.!y:set.y iIn Sep x p -> y iIn x & p y const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom SNoLeLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x <= y -> y < z -> x < z axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z const real : set const ap : set set set lemma !x:set.!y:set.x iIn setexp real omega -> y iIn setexp real omega -> ~ (?z:set.z iIn real & !w:set.w iIn omega -> ap x w <= z & z <= ap y w) -> (!z:set.z iIn omega -> SNo (ap x z)) -> (!z:set.z iIn omega -> SNo (ap y z)) -> (!z:set.z iIn omega -> !w:set.w iIn omega -> ap x z <= ap y w) -> Subq (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & z < ap x w) (SNoS_ omega) -> Subq (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & ap y w < z) (SNoS_ omega) -> ~ SNoCutP (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & z < ap x w) (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & ap y w < z) var x:set var y:set hyp x iIn setexp real omega hyp y iIn setexp real omega hyp ~ ?z:set.z iIn real & !w:set.w iIn omega -> ap x w <= z & z <= ap y w hyp !z:set.z iIn omega -> SNo (ap x z) hyp !z:set.z iIn omega -> SNo (ap y z) hyp !z:set.z iIn omega -> !w:set.w iIn omega -> ap x z <= ap y w hyp Subq (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & z < ap x w) (SNoS_ omega) claim ~ Subq (Sep (SNoS_ omega) \z:set.?w:set.w iIn omega & ap y w < z) (SNoS_ omega)