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 term PNoEq_ = \x:set.\p:set prop.\q:set prop.!y:set.y iIn x -> (p y <-> q y) 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 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 term PNo_strict_imv = \P:set (set prop) prop.\Q:set (set prop) prop.\x:set.\p:set prop.PNo_strict_upperbd P x p & PNo_strict_lowerbd Q x p term PNo_least_rep = \P:set (set prop) prop.\Q:set (set prop) prop.\x:set.\p:set prop.ordinal x & PNo_strict_imv P Q x p & !y:set.y iIn x -> !q:set prop.~ PNo_strict_imv P Q y q axiom ordinal_ind: !p:set prop.(!x:set.ordinal x -> (!y:set.y iIn x -> p y) -> p x) -> !x:set.ordinal x -> p x lemma !P:set (set prop) prop.!Q:set (set prop) prop.!x:set.!p:set prop.!q:set prop.ordinal x -> TransSet x -> (!y:set.y iIn x -> !p2:set prop.~ PNo_strict_imv P Q y p2) -> PNo_strict_upperbd P x p -> PNo_strict_lowerbd Q x p -> PNo_strict_upperbd P x q -> PNo_strict_lowerbd Q x q -> (!y:set.ordinal y -> y iIn x -> (p y <-> q y)) -> !y:set.y iIn x -> (p y <-> q y) lemma !P:set (set prop) prop.!Q:set (set prop) prop.!x:set.!p:set prop.!q:set prop.!y:set.ordinal x -> TransSet x -> (!z:set.z iIn x -> !p2:set prop.~ PNo_strict_imv P Q z p2) -> PNo_strict_upperbd P x p -> PNo_strict_lowerbd Q x p -> PNo_strict_upperbd P x q -> PNo_strict_lowerbd Q x q -> ordinal y -> (!z:set.z iIn y -> z iIn x -> (p z <-> q z)) -> y iIn x -> PNoEq_ y p q -> (p y <-> q y) const PNoLt_pwise : (set (set prop) prop) (set (set prop) prop) prop claim !P:set (set prop) prop.!Q:set (set prop) prop.PNoLt_pwise P Q -> !x:set.ordinal x -> !p:set prop.!q:set prop.PNo_least_rep P Q x p -> PNo_strict_imv P Q x q -> !y:set.y iIn x -> (p y <-> q y)