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 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 var P:set (set prop) prop var x:set var p:set prop var q:set prop var y:set var p2:set prop hyp ordinal x hyp PNoEq_ x p q hyp PNo_rel_strict_upperbd P x p hyp y iIn x hyp PNo_downc P y p2 claim ordinal y -> PNoLt y p2 x q