const ordinal : set prop const PNoLt : set (set prop) set (set prop) prop term PNo_strict_upperbd = \P:set (set prop) prop.\x:set.\p:set prop.!y:set.ordinal y -> !q:set prop.P y q -> PNoLt y q x p const PNoEq_ : set (set prop) (set prop) prop axiom PNoLtEq_tra: !x:set.!y:set.ordinal x -> ordinal y -> !p:set prop.!q:set prop.!p2:set prop.PNoLt x p y q -> PNoEq_ y q p2 -> PNoLt x p y p2 claim !P:set (set prop) prop.!x:set.ordinal x -> !p:set prop.!q:set prop.PNoEq_ x p q -> PNo_strict_upperbd P x p -> !y:set.ordinal y -> !p2:set prop.P y p2 -> PNoLt y p2 x q