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 axiom SNo_eq: !x:set.!y:set.SNo x -> SNo y -> SNoLev x = SNoLev y -> SNoEq_ (SNoLev x) x y -> x = y const SNoCut : set set set const ordinal : set prop var x:set hyp SNo x hyp ordinal (SNoLev x) hyp SNo (SNoCut (SNoL x) (SNoR x)) hyp !y:set.y iIn SNoL x -> y < SNoCut (SNoL x) (SNoR x) hyp !y:set.y iIn SNoR x -> SNoCut (SNoL x) (SNoR x) < y hyp ordinal (SNoLev (SNoCut (SNoL x) (SNoR x))) hyp Subq (SNoLev (SNoCut (SNoL x) (SNoR x))) (SNoLev x) hyp PNoEq_ (SNoLev (SNoCut (SNoL x) (SNoR x))) (\y:set.y iIn SNoCut (SNoL x) (SNoR x)) \y:set.y iIn x claim SNoLev (SNoCut (SNoL x) (SNoR x)) = SNoLev x -> x = SNoCut (SNoL x) (SNoR x)