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 SNoLev : set set const Empty : set axiom SNoLev_0: SNoLev Empty = Empty const Sing : set set axiom SingE: !x:set.!y:set.y iIn Sing x -> y = x const famunion : set (set set) set axiom famunionE_impred: !x:set.!f:set set.!y:set.y iIn famunion x f -> !P:prop.(!z:set.z iIn x -> y iIn f z -> P) -> P axiom SingI: !x:set.x iIn Sing x const ordsucc : set set axiom In_0_1: Empty iIn ordsucc Empty axiom famunionI: !x:set.!f:set set.!y:set.!z:set.y iIn x -> z iIn f y -> z iIn famunion x f axiom cases_1: !x:set.x iIn ordsucc Empty -> !p:set prop.p Empty -> p x axiom set_ext: !x:set.!y:set.Subq x y -> Subq y x -> x = y const omega : set const nat_p : set prop const Repl : set (set set) set const eps_ : set set const SNoCut : set set set lemma !x:set.x iIn omega -> nat_p x -> SNoCutP (Sing Empty) (Repl x eps_) -> (!y:set.y iIn Repl x eps_ -> SNo y) -> famunion (Sing Empty) (\y:set.ordsucc (SNoLev y)) = ordsucc Empty -> eps_ x = SNoCut (Sing Empty) (Repl x eps_) var x:set hyp x iIn omega hyp nat_p x hyp SNoCutP (Sing Empty) (Repl x eps_) claim (!y:set.y iIn Repl x eps_ -> SNo y) -> eps_ x = SNoCut (Sing Empty) (Repl x eps_)