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 SNoR = \x:set.Sep (SNoS_ (SNoLev x)) (SNoLt x) term SNoL = \x:set.Sep (SNoS_ (SNoLev x)) \y:set.y < x const SNo : set prop const minus_SNo : set set term - = minus_SNo axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x const In : set set prop term iIn = In infix iIn 2000 2000 axiom SNoS_I2: !x:set.!y:set.SNo x -> SNo y -> SNoLev x iIn SNoLev y -> x iIn SNoS_ (SNoLev y) axiom SepI: !x:set.!p:set prop.!y:set.y iIn x -> p y -> y iIn Sep x p axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z 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 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 SNoCutP : set set prop const Repl : set (set set) set axiom minus_SNo_prop1: !x:set.SNo x -> SNo - x & (!y:set.y iIn SNoL x -> - x < - y) & (!y:set.y iIn SNoR x -> - y < - x) & SNoCutP (Repl (SNoR x) minus_SNo) (Repl (SNoL x) minus_SNo) claim !x:set.!y:set.SNo x -> SNo y -> x < y -> - y < - x