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 Sep : set (set prop) set axiom Sep_Subq: !x:set.!p:set prop.Subq (Sep x p) x const real : set const omega : set const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 const ap : set set set const SNoS_ : 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) const nat_p : set prop const ordsucc : set set var x:set var y:set hyp x iIn setexp real omega hyp y iIn setexp real omega hyp !z:set.z iIn omega -> ap x z <= ap y z & ap x z <= ap x (ordsucc z) & ap y (ordsucc z) <= ap y z 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.nat_p z -> !w:set.w iIn z -> ap x w <= ap x z hyp !z:set.nat_p z -> !w:set.w iIn z -> ap y z <= ap y w claim ~ !z:set.z iIn omega -> !w:set.w iIn omega -> ap x z <= ap y w