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 Empty : set axiom EmptyE: !x:set.nIn x Empty axiom FalseE: ~ False const nat_p : set prop const omega : set axiom nat_p_omega: !x:set.nat_p x -> x iIn omega const ordsucc : set set axiom nat_ind: !p:set prop.p Empty -> (!x:set.nat_p x -> p x -> p (ordsucc x)) -> !x:set.nat_p x -> p x const real : set const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 const ap : set set set lemma !x:set.!y:set.x iIn setexp real omega -> y iIn setexp real omega -> (!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) -> ~ (?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.nat_p z -> !w:set.w iIn z -> ap x w <= ap x z lemma !x:set.!y:set.!z:set.!w:set.(!u:set.u iIn omega -> ap x u <= ap y u & ap x u <= ap x (ordsucc u) & ap y (ordsucc u) <= ap y u) -> (!u:set.u iIn omega -> SNo (ap x u)) -> nat_p z -> (!u:set.u iIn z -> ap x u <= ap x z) -> w iIn ordsucc z -> z iIn omega -> ap x w <= ap x (ordsucc z) 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) claim ~ !z:set.z iIn omega -> SNo (ap y z)