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 eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) const SNoCut : set set set const SNoLev : set set const ordsucc : set set const binunion : set set set const famunion : set (set set) set const SNoEq_ : set set set prop axiom SNoCutP_SNoCut_impred: !x:set.!y:set.SNoCutP x y -> !P:prop.(SNo (SNoCut x y) -> SNoLev (SNoCut x y) iIn ordsucc (binunion (famunion x \z:set.ordsucc (SNoLev z)) (famunion y \z:set.ordsucc (SNoLev z))) -> (!z:set.z iIn x -> z < SNoCut x y) -> (!z:set.z iIn y -> SNoCut x y < z) -> (!z:set.SNo z -> (!w:set.w iIn x -> w < z) -> (!w:set.w iIn y -> z < w) -> Subq (SNoLev (SNoCut x y)) (SNoLev z) & SNoEq_ (SNoLev (SNoCut x y)) (SNoCut x y) z) -> P) -> P const Repl : set (set set) set const Sing : set set const Empty : set lemma !x:set.x iIn omega -> (!y:set.y iIn Repl x eps_ -> SNo y) -> SNo (SNoCut (Sing Empty) (Repl x eps_)) -> SNoLev (SNoCut (Sing Empty) (Repl x eps_)) iIn ordsucc (ordsucc x) -> (!y:set.y iIn Sing Empty -> y < SNoCut (Sing Empty) (Repl x eps_)) -> (!y:set.y iIn Repl x eps_ -> SNoCut (Sing Empty) (Repl x eps_) < y) -> (!y:set.SNo y -> (!z:set.z iIn Sing Empty -> z < y) -> (!z:set.z iIn Repl x eps_ -> y < z) -> Subq (SNoLev (SNoCut (Sing Empty) (Repl x eps_))) (SNoLev y) & SNoEq_ (SNoLev (SNoCut (Sing Empty) (Repl x eps_))) (SNoCut (Sing Empty) (Repl x eps_)) y) -> SNo (eps_ x) -> eps_ x = SNoCut (Sing Empty) (Repl x eps_) const nat_p : set prop var x:set hyp x iIn omega hyp nat_p x hyp SNoCutP (Sing Empty) (Repl x eps_) hyp !y:set.y iIn Repl x eps_ -> SNo y hyp famunion (Sing Empty) (\y:set.ordsucc (SNoLev y)) = ordsucc Empty hyp x != Empty -> famunion (Repl x eps_) (\y:set.ordsucc (SNoLev y)) = ordsucc x claim binunion (famunion (Sing Empty) \y:set.ordsucc (SNoLev y)) (famunion (Repl x eps_) \y:set.ordsucc (SNoLev y)) = ordsucc x -> eps_ x = SNoCut (Sing Empty) (Repl x eps_)