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 term TransSet = \x:set.!y:set.y iIn x -> Subq y x term ordinal = \x:set.TransSet x & !y:set.y iIn x -> TransSet y term nIn = \x:set.\y:set.~ x iIn y const SNo : set prop const SNoCutP : set set prop const Repl : set (set set) set const SNoR : set set const minus_SNo : set set term - = minus_SNo const SNoL : set set axiom minus_SNo_SNoCutP: !x:set.SNo x -> SNoCutP (Repl (SNoR x) minus_SNo) (Repl (SNoL x) minus_SNo) const SNoS_ : set set const SNoLev : set set const SNo_ : set set prop axiom SNoS_E2: !x:set.ordinal x -> !y:set.y iIn SNoS_ x -> !P:prop.(SNoLev y iIn x -> ordinal (SNoLev y) -> SNo y -> SNo_ (SNoLev y) y -> P) -> P axiom ordinal_ind: !p:set prop.(!x:set.ordinal x -> (!y:set.y iIn x -> p y) -> p x) -> !x:set.ordinal x -> p x lemma !x:set.!y:set.TransSet x -> (!z:set.z iIn x -> !w:set.w iIn SNoS_ z -> Subq (SNoLev - w) (SNoLev w)) -> SNoLev y iIn x -> ordinal (SNoLev y) -> SNo y -> SNoCutP (Repl (SNoR y) minus_SNo) (Repl (SNoL y) minus_SNo) -> Subq (SNoLev - y) (SNoLev y) claim !x:set.ordinal x -> !y:set.y iIn SNoS_ x -> Subq (SNoLev - y) (SNoLev y)