const In : set set prop term iIn = In infix iIn 2000 2000 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 const PNoEq_ : set (set prop) (set prop) prop term PNoLt_ = \x:set.\p:set prop.\q:set prop.?y:set.y iIn x & (PNoEq_ y p q & ~ p y & q y) term nIn = \x:set.\y:set.~ x iIn y const PNoLe : set (set prop) set (set prop) prop term PNo_downc = \P:set (set prop) prop.\x:set.\p:set prop.?y:set.ordinal y & ?q:set prop.P y q & PNoLe x p y q 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 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 ordsucc : set set axiom ordinal_ordsucc: !x:set.ordinal x -> ordinal (ordsucc x) lemma !P:set (set prop) prop.!x:set.!y:set.!p:set prop.!z:set.!q:set prop.ordinal x -> y iIn ordsucc x -> PNo_strict_upperbd P x p -> z iIn y -> PNo_downc P z q -> TransSet x -> ordinal (ordsucc x) -> PNoLt z q y p claim !P:set (set prop) prop.!x:set.ordinal x -> !y:set.y iIn ordsucc x -> !p:set prop.PNo_strict_upperbd P x p -> !z:set.z iIn y -> !q:set prop.PNo_downc P z q -> PNoLt z q y p