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 SNoCutP : set set prop const SNoCut : set set set const ordsucc : set set const binunion : set set set const famunion : set (set set) set axiom SNoCutP_SNoCut: !x:set.!y:set.SNoCutP x y -> SNo (SNoCut x y) & SNoLev (SNoCut x y) iIn ordsucc (binunion (famunion x \z:set.ordsucc (SNoLev z)) (famunion y \z:set.ordsucc (SNoLev z))) & (!z:set.z iIn x -> z < SNoCut x y) & (!z:set.z iIn y -> SNoCut x y < z) & !z:set.SNo z -> (!w:set.w iIn x -> w < z) -> (!w:set.w iIn y -> z < w) -> Subq (SNoLev (SNoCut x y)) (SNoLev z) & SNoEq_ (SNoLev (SNoCut x y)) (SNoCut x y) z lemma !x:set.SNo x -> ordinal (SNoLev x) -> SNo (SNoCut (SNoL x) (SNoR x)) -> (!y:set.y iIn SNoL x -> y < SNoCut (SNoL x) (SNoR x)) -> (!y:set.y iIn SNoR x -> SNoCut (SNoL x) (SNoR x) < y) -> (!y:set.SNo y -> (!z:set.z iIn SNoL x -> z < y) -> (!z:set.z iIn SNoR x -> y < z) -> Subq (SNoLev (SNoCut (SNoL x) (SNoR x))) (SNoLev y) & SNoEq_ (SNoLev (SNoCut (SNoL x) (SNoR x))) (SNoCut (SNoL x) (SNoR x)) y) -> ordinal (SNoLev (SNoCut (SNoL x) (SNoR x))) -> x = SNoCut (SNoL x) (SNoR x) var x:set hyp SNo x hyp ordinal (SNoLev x) claim SNoCutP (SNoL x) (SNoR x) -> x = SNoCut (SNoL x) (SNoR x)