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 omega : set const nat_p : set prop axiom omega_nat_p: !x:set.x iIn omega -> nat_p x const Sing : set set const Empty : set const Repl : set (set set) set const eps_ : set set lemma !x:set.x iIn omega -> 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 claim !x:set.x iIn omega -> SNoCutP (Sing Empty) (Repl x eps_)