const SNo : set prop const omega : set axiom SNo_omega: SNo omega const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const minus_SNo : set set term - = minus_SNo axiom minus_SNo_Lt_contra1: !x:set.!y:set.SNo x -> SNo y -> - x < y -> - y < x const In : set set prop term iIn = In infix iIn 2000 2000 const SNoS_ : set set const ordsucc : set set lemma !x:set.- omega < x -> SNo x -> - x iIn SNoS_ (ordsucc omega) -> - x < omega -> ?y:set.y iIn omega & - y < x var x:set hyp x iIn SNoS_ (ordsucc omega) hyp - omega < x hyp SNo x claim - x iIn SNoS_ (ordsucc omega) -> ?y:set.y iIn omega & - y < x