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 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term TransSet = \x:set.!y:set.y iIn x -> Subq y x term ordinal = \x:set.TransSet x & !y:set.y iIn x -> TransSet y axiom ordinal_Hered: !x:set.ordinal x -> !y:set.y iIn x -> ordinal y lemma !P:set (set prop) prop.!x:set.!p:set prop.!y:set.ordinal x -> y iIn x -> TransSet x -> ordinal y -> (!z:set.z iIn x -> !q:set prop.PNo_downc P z q -> PNoLt z q x p) -> !z:set.z iIn y -> !q:set prop.PNo_downc P z q -> PNoLt z q y p claim !P:set (set prop) prop.!x:set.ordinal x -> !p:set prop.!y:set.y iIn x -> PNo_rel_strict_upperbd P x p -> PNo_rel_strict_upperbd P y p