const In : set set prop term iIn = In infix iIn 2000 2000 const SNo : set prop 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 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term TransSet = \x:set.!y:set.y iIn x -> Subq y x 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) term ordinal = \x:set.TransSet x & !y:set.y iIn x -> TransSet y axiom SNoLev_ordinal: !x:set.SNo x -> ordinal (SNoLev x) axiom SNoS_I2: !x:set.!y:set.SNo x -> SNo y -> SNoLev x iIn SNoLev y -> x iIn SNoS_ (SNoLev y) const binintersect : set set set axiom binintersectE: !x:set.!y:set.!z:set.z iIn binintersect x y -> z iIn x & z iIn y const SNo_ : set set prop axiom SNoLev_prop: !x:set.SNo x -> ordinal (SNoLev x) & SNo_ (SNoLev x) x axiom SepI: !x:set.!p:set prop.!y:set.y iIn x -> p y -> y iIn Sep x p const SNoEq_ : set set set prop const nIn : set set prop axiom SNoLtE: !x:set.!y:set.SNo x -> SNo y -> x < y -> !P:prop.(!z:set.SNo z -> SNoLev z iIn binintersect (SNoLev x) (SNoLev y) -> SNoEq_ (SNoLev z) z x -> SNoEq_ (SNoLev z) z y -> x < z -> z < y -> nIn (SNoLev z) x -> SNoLev z iIn y -> P) -> (SNoLev x iIn SNoLev y -> SNoEq_ (SNoLev x) x y -> SNoLev x iIn y -> P) -> (SNoLev y iIn SNoLev x -> SNoEq_ (SNoLev y) x y -> nIn (SNoLev y) x -> P) -> P const minus_SNo : set set term - = minus_SNo const Repl : set (set set) set lemma !x:set.!y:set.!z:set.!w:set.SNo x -> (!u:set.u iIn SNoS_ (SNoLev x) -> SNo - u & (!v:set.v iIn SNoL u -> - u < - v) & (!v:set.v iIn SNoR u -> - v < - u) & SNoCutP (Repl (SNoR u) minus_SNo) (Repl (SNoL u) minus_SNo)) -> SNo y -> SNoLev y iIn SNoLev x -> SNo z -> SNo - z -> (!u:set.u iIn SNoR z -> - u < - z) -> SNo - y -> (!u:set.u iIn SNoL y -> - y < - u) -> SNo w -> z < w -> w < y -> SNoLev w iIn SNoLev z -> SNoLev w iIn SNoLev y -> w iIn SNoS_ (SNoLev x) -> - y < - z var x:set var y:set var z:set hyp SNo x hyp !w:set.w iIn SNoS_ (SNoLev x) -> SNo - w & (!u:set.u iIn SNoL w -> - w < - u) & (!u:set.u iIn SNoR w -> - u < - w) & SNoCutP (Repl (SNoR w) minus_SNo) (Repl (SNoL w) minus_SNo) hyp SNo y hyp SNoLev y iIn SNoLev x hyp x < y hyp SNo z hyp z < x hyp SNo - z hyp !w:set.w iIn SNoR z -> - w < - z hyp SNo - y hyp !w:set.w iIn SNoL y -> - y < - w claim z < y -> - y < - z