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 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term nIn = \x:set.\y:set.~ x iIn y const omega : set const Sing : set set const Empty : set const Repl : set (set set) set const eps_ : set set axiom eps_SNoCutP: !x:set.x iIn omega -> SNoCutP (Sing Empty) (Repl x eps_) const nat_p : set prop const SNoCut : set set set lemma !x:set.x iIn omega -> nat_p x -> SNoCutP (Sing Empty) (Repl x eps_) -> eps_ x = SNoCut (Sing Empty) (Repl x eps_) var x:set hyp x iIn omega claim nat_p x -> eps_ x = SNoCut (Sing Empty) (Repl x eps_)