const In : set set prop term iIn = In infix iIn 2000 2000 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 const Empty : set axiom SNo_0: SNo Empty const Sing : set set axiom SingE: !x:set.!y:set.y iIn Sing x -> y = x const nat_p : set prop axiom nat_p_trans: !x:set.nat_p x -> !y:set.y iIn x -> nat_p y const omega : set axiom nat_p_omega: !x:set.nat_p x -> x iIn omega const eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) const Repl : set (set set) set axiom ReplE_impred: !x:set.!f:set set.!y:set.y iIn Repl x f -> !P:prop.(!z:set.z iIn x -> y = f z -> P) -> P axiom SNo_eps_pos: !x:set.x iIn omega -> Empty < eps_ x var x:set hyp x iIn omega claim nat_p x -> (!y:set.y iIn Sing Empty -> SNo y) & (!y:set.y iIn Repl x eps_ -> SNo y) & !y:set.y iIn Sing Empty -> !z:set.z iIn Repl x eps_ -> y < z