const PNoEq_ : set (set prop) (set prop) prop const In : set set prop term iIn = In infix iIn 2000 2000 term SNoEq_ = \x:set.\y:set.\z:set.PNoEq_ x (\w:set.w iIn y) \w:set.w iIn z const PNoLe : set (set prop) set (set prop) prop const SNoLev : set set 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 const ordinal : set prop axiom SNoLev_ordinal: !x:set.SNo x -> ordinal (SNoLev x) axiom SNo_eq: !x:set.!y:set.SNo x -> SNo y -> SNoLev x = SNoLev y -> SNoEq_ (SNoLev x) x y -> x = y axiom PNoLe_antisym: !x:set.!y:set.ordinal x -> ordinal y -> !p:set prop.!q:set prop.PNoLe x p y q -> PNoLe y q x p -> x = y & PNoEq_ x p q claim !x:set.!y:set.SNo x -> SNo y -> x <= y -> y <= x -> x = y