const In : set set prop term iIn = In infix iIn 2000 2000 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y 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 ordinal : set prop const SNoLev : set set axiom SNoLev_ordinal: !x:set.SNo x -> ordinal (SNoLev x) const ordsucc : set set axiom ordinal_ordsucc: !x:set.ordinal x -> ordinal (ordsucc x) const famunion : set (set set) set axiom ordinal_famunion: !x:set.!f:set set.(!y:set.y iIn x -> ordinal (f y)) -> ordinal (famunion x f) const binunion : set set set axiom ordinal_binunion: !x:set.!y:set.ordinal x -> ordinal y -> ordinal (binunion x y) const TransSet : set prop axiom ordinal_TransSet: !x:set.ordinal x -> TransSet x const Sep : set (set prop) set const SNoS_ : set set const omega : set const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const Empty : set const SNoCut : set set set lemma !x:set.(!y:set.y iIn Sep (SNoS_ omega) (\z:set.x * z < ordsucc Empty) -> !P:prop.(SNo y -> SNoLev y iIn omega -> x * y < ordsucc Empty -> P) -> P) -> (!y:set.y iIn Sep (SNoS_ omega) (\z:set.ordsucc Empty < x * z) -> !P:prop.(SNo y -> SNoLev y iIn omega -> ordsucc Empty < x * y -> P) -> P) -> SNoLev (SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y)) iIn ordsucc (binunion (famunion (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) \y:set.ordsucc (SNoLev y)) (famunion (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y) \y:set.ordsucc (SNoLev y))) -> omega iIn SNoLev (SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y)) -> ~ TransSet (binunion (famunion (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) \y:set.ordsucc (SNoLev y)) (famunion (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y) \y:set.ordsucc (SNoLev y))) var x:set hyp !y:set.y iIn Sep (SNoS_ omega) (\z:set.x * z < ordsucc Empty) -> !P:prop.(SNo y -> SNoLev y iIn omega -> x * y < ordsucc Empty -> P) -> P hyp !y:set.y iIn Sep (SNoS_ omega) (\z:set.ordsucc Empty < x * z) -> !P:prop.(SNo y -> SNoLev y iIn omega -> ordsucc Empty < x * y -> P) -> P hyp SNoLev (SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y)) iIn ordsucc (binunion (famunion (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) \y:set.ordsucc (SNoLev y)) (famunion (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y) \y:set.ordsucc (SNoLev y))) hyp Subq (ordsucc omega) (SNoLev (SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y))) claim ~ omega iIn SNoLev (SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y))