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 nat_p : set prop const ordinal : set prop axiom nat_p_ordinal: !x:set.nat_p x -> ordinal x const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom SNoLe_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x <= y -> y <= z -> x <= z axiom ordinal_trichotomy_or_impred: !x:set.!y:set.ordinal x -> ordinal y -> !P:prop.(x iIn y -> P) -> (x = y -> P) -> (y iIn x -> P) -> P const omega : set const ap : set set set const ordsucc : set set var x:set var y:set var z:set var w:set hyp !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 hyp !u:set.u iIn omega -> SNo (ap x u) hyp !u:set.u iIn omega -> SNo (ap y u) hyp !u:set.nat_p u -> !v:set.v iIn u -> ap x v <= ap x u hyp !u:set.nat_p u -> !v:set.v iIn u -> ap y u <= ap y v hyp z iIn omega hyp w iIn omega hyp nat_p z hyp nat_p w claim ap x z <= ap y z -> ap x z <= ap y w