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 ordinal : set prop axiom PNoEqLt_tra: !x:set.!y:set.ordinal x -> ordinal y -> !p:set prop.!q:set prop.!p2:set prop.PNoEq_ x p q -> PNoLt x q y p2 -> PNoLt x p y p2 axiom PNoEq_tra_: !x:set.!p:set prop.!q:set prop.!p2:set prop.PNoEq_ x p q -> PNoEq_ x q p2 -> PNoEq_ x p p2 claim !x:set.!y:set.ordinal x -> ordinal y -> !p:set prop.!q:set prop.!p2:set prop.PNoEq_ x p q -> PNoLe x q y p2 -> PNoLt x p y p2 | x = y & PNoEq_ x p p2