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) axiom ordinal_Hered: !x:set.ordinal x -> !y:set.y iIn x -> ordinal y axiom SNoS_E: !x:set.ordinal x -> !y:set.y iIn SNoS_ x -> ?z:set.z iIn x & SNo_ z y axiom SepE: !x:set.!p:set prop.!y:set.y iIn Sep x p -> y iIn x & p y const SNoCut : set set set 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)) -> (!y:set.y iIn Sep (SNoS_ (SNoLev x)) (\z:set.z < x) -> SNo y & SNoLev y iIn SNoLev x & y < x) -> (!y:set.y iIn Sep (SNoS_ (SNoLev x)) (SNoLt x) -> SNo y & SNoLev y iIn SNoLev x & x < y) -> P lemma !x:set.!y:set.!z:set.ordinal (SNoLev x) -> x < y -> z iIn SNoLev x -> SNo_ z y -> ordinal z -> SNo y & SNoLev y iIn SNoLev x & x < y 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 hyp ordinal (SNoLev x) hyp x = SNoCut (Sep (SNoS_ (SNoLev x)) \y:set.y < x) (Sep (SNoS_ (SNoLev x)) (SNoLt x)) claim (!y:set.y iIn Sep (SNoS_ (SNoLev x)) (\z:set.z < x) -> SNo y & SNoLev y iIn SNoLev x & y < x) -> P