const ordinal : set prop const SNo_ : set set prop term SNo = \x:set.?y:set.ordinal y & SNo_ y x const In : set set prop term iIn = In infix iIn 2000 2000 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 Sep : set (set prop) set const SNoS_ : set set const SNoLev : set set term SNoL = \x:set.Sep (SNoS_ (SNoLev x)) \y:set.y < x term SNoR = \x:set.Sep (SNoS_ (SNoLev x)) (SNoLt x) const SNoCut : set set set axiom SNo_eta: !x:set.SNo x -> x = SNoCut (SNoL x) (SNoR x) lemma !x:set.!P:prop.SNo x -> (!y:set.!z:set.SNoCutP y z -> (!w:set.w iIn y -> SNoLev w iIn SNoLev x) -> (!w:set.w iIn z -> SNoLev w iIn SNoLev x) -> x = SNoCut y z -> P) -> ordinal (SNoLev x) -> x = SNoCut (Sep (SNoS_ (SNoLev x)) \y:set.y < x) (Sep (SNoS_ (SNoLev x)) (SNoLt x)) -> P var x:set var P:prop hyp SNo x hyp !y:set.!z:set.SNoCutP y z -> (!w:set.w iIn y -> SNoLev w iIn SNoLev x) -> (!w:set.w iIn z -> SNoLev w iIn SNoLev x) -> x = SNoCut y z -> P claim ordinal (SNoLev x) -> P