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 Empty : set const eps_ : set set axiom SNo_eps_pos: !x:set.x iIn omega -> Empty < eps_ x const Sing : set set axiom SingE: !x:set.!y:set.y iIn Sing x -> y = x const Repl : set (set set) set const SNoCut : set set set const SNoLev : set set const ordsucc : set set const SNoEq_ : set set set prop 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) -> (!y:set.y iIn Sing Empty -> y < eps_ x) -> eps_ x = SNoCut (Sing Empty) (Repl x eps_) var x:set hyp x iIn omega hyp !y:set.y iIn Repl x eps_ -> SNo y hyp SNo (SNoCut (Sing Empty) (Repl x eps_)) hyp SNoLev (SNoCut (Sing Empty) (Repl x eps_)) iIn ordsucc (ordsucc x) hyp !y:set.y iIn Sing Empty -> y < SNoCut (Sing Empty) (Repl x eps_) hyp !y:set.y iIn Repl x eps_ -> SNoCut (Sing Empty) (Repl x eps_) < y hyp !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 claim SNo (eps_ x) -> eps_ x = SNoCut (Sing Empty) (Repl x eps_)