const In : set set prop term iIn = In infix iIn 2000 2000 const PNo_downc : (set (set prop) prop) set (set prop) prop const PNoLt : set (set prop) set (set prop) prop term PNo_rel_strict_upperbd = \P:set (set prop) prop.\x:set.\p:set prop.!y:set.y iIn x -> !q:set prop.PNo_downc P y q -> PNoLt y q x p const ordinal : set prop axiom ordinal_Hered: !x:set.ordinal x -> !y:set.y iIn x -> ordinal y const PNoEq_ : set (set prop) (set prop) prop lemma !P:set (set prop) prop.!x:set.!p:set prop.!q:set prop.!y:set.!p2:set prop.ordinal x -> PNoEq_ x p q -> PNo_rel_strict_upperbd P x p -> y iIn x -> PNo_downc P y p2 -> ordinal y -> PNoLt y p2 x q claim !P:set (set prop) prop.!x:set.ordinal x -> !p:set prop.!q:set prop.PNoEq_ x p q -> PNo_rel_strict_upperbd P x p -> !y:set.y iIn x -> !p2:set prop.PNo_downc P y p2 -> PNoLt y p2 x q