const PNoLt : set (set prop) set (set prop) prop const PNoEq_ : set (set prop) (set prop) prop term PNoLe = \x:set.\p:set prop.\y:set.\q:set prop.PNoLt x p y q | x = y & PNoEq_ x p q const SNoLev : set set const In : set set prop term iIn = In infix iIn 2000 2000 term SNoLt = \x:set.\y:set.PNoLt (SNoLev x) (\z:set.z iIn x) (SNoLev y) \z:set.z iIn y term < = SNoLt infix < 2020 2020 term SNoEq_ = \x:set.\y:set.\z:set.PNoEq_ x (\w:set.w iIn y) \w:set.w iIn z term SNoLe = \x:set.\y:set.PNoLe (SNoLev x) (\z:set.z iIn x) (SNoLev y) \z:set.z iIn y term <= = SNoLe infix <= 2020 2020 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 claim !x:set.!y:set.SNo x -> SNo y -> x <= y -> x < y | x = y