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