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 SNo_ : set set prop axiom SNoLev_: !x:set.SNo x -> SNo_ (SNoLev x) x const ordinal : set prop axiom SNoS_I: !x:set.ordinal x -> !y:set.!z:set.z iIn x -> SNo_ z y -> y iIn SNoS_ x axiom SepI: !x:set.!p:set prop.!y:set.y iIn x -> p y -> y iIn Sep x p axiom SNoLt_irref: !x:set.~ x < x axiom In_irref: !x:set.nIn x x axiom SNoLt_trichotomy_or: !x:set.!y:set.SNo x -> SNo y -> x < y | x = y | y < x axiom FalseE: ~ False axiom ordinal_trichotomy_or: !x:set.!y:set.ordinal x -> ordinal y -> x iIn y | x = y | y iIn x const SNoCut : set set set 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) -> ordinal (SNoLev (SNoCut (SNoL x) (SNoR x))) -> Subq (SNoLev (SNoCut (SNoL x) (SNoR x))) (SNoLev x) -> PNoEq_ (SNoLev (SNoCut (SNoL x) (SNoR x))) (\y:set.y iIn SNoCut (SNoL x) (SNoR x)) (\y:set.y iIn x) -> SNoLev (SNoCut (SNoL x) (SNoR x)) = SNoLev x -> x = SNoCut (SNoL x) (SNoR x) 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 !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 hyp ordinal (SNoLev (SNoCut (SNoL x) (SNoR x))) hyp !y:set.y iIn SNoL x -> y < x claim (!y:set.y iIn SNoR x -> x < y) -> x = SNoCut (SNoL x) (SNoR x)