const In : set set prop term iIn = In infix iIn 2000 2000 term PNoEq_ = \x:set.\p:set prop.\q:set prop.!y:set.y iIn x -> (p y <-> q y) term SNoEq_ = \x:set.\y:set.\z:set.PNoEq_ x (\w:set.w iIn y) \w:set.w iIn z term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term nIn = \x:set.\y:set.~ x iIn y const Sep : set (set prop) set const SNoS_ : set set const SNoLev : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 term SNoL = \x:set.Sep (SNoS_ (SNoLev x)) \y:set.y < x term SNoR = \x:set.Sep (SNoS_ (SNoLev x)) (SNoLt x) const SNo : set prop const ordinal : set prop axiom SNoLev_ordinal: !x:set.SNo x -> ordinal (SNoLev x) const SNoCut : set set set lemma !x:set.SNo x -> ordinal (SNoLev x) -> x = SNoCut (SNoL x) (SNoR x) claim !x:set.SNo x -> x = SNoCut (SNoL x) (SNoR x)