const In : set set prop term iIn = In infix iIn 2000 2000 const PNo_upc : (set (set prop) prop) set (set prop) prop const PNoLt : set (set prop) set (set prop) prop term PNo_rel_strict_lowerbd = \P:set (set prop) prop.\x:set.\p:set prop.!y:set.y iIn x -> !q:set prop.PNo_upc P y q -> PNoLt x p y q const PNo_downc : (set (set prop) 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_rel_strict_imv = \P:set (set prop) prop.\Q:set (set prop) prop.\x:set.\p:set prop.PNo_rel_strict_upperbd P x p & PNo_rel_strict_lowerbd Q x p const ordsucc : set set term PNo_rel_strict_split_imv = \P:set (set prop) prop.\Q:set (set prop) prop.\x:set.\p:set prop.PNo_rel_strict_imv P Q (ordsucc x) (\y:set.p y & y != x) & PNo_rel_strict_imv P Q (ordsucc x) \y:set.p y | y = x term PNoEq_ = \x:set.\p:set prop.\q:set prop.!y:set.y iIn x -> (p y <-> q y) term PNo_rel_strict_uniq_imv = \P:set (set prop) prop.\Q:set (set prop) prop.\x:set.\p:set prop.PNo_rel_strict_imv P Q x p & !q:set prop.PNo_rel_strict_imv P Q x q -> PNoEq_ x p q 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 PNoLt_pwise = \P:set (set prop) prop.\Q:set (set prop) prop.!x:set.ordinal x -> !p:set prop.P x p -> !y:set.ordinal y -> !q:set prop.Q y q -> PNoLt x p y q term nIn = \x:set.\y:set.~ x iIn y term PNoLt_ = \x:set.\p:set prop.\q:set prop.?y:set.y iIn x & (PNoEq_ y p q & ~ p y & q y) term PNoLe = \x:set.\p:set prop.\y:set.\q:set prop.PNoLt x p y q | x = y & PNoEq_ x p q axiom iffEL: !P:prop.!Q:prop.(P <-> Q) -> P -> Q axiom iffI: !P:prop.!Q:prop.(P -> Q) -> (Q -> P) -> (P <-> Q) axiom ordinal_ordsucc_In_eq: !x:set.!y:set.ordinal x -> y iIn x -> ordsucc y iIn x | x = ordsucc y 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 axiom ordinal_Hered: !x:set.ordinal x -> !y:set.y iIn x -> ordinal y axiom ordinal_lim_or_succ: !x:set.ordinal x -> (!y:set.y iIn x -> ordsucc y iIn x) | ?y:set.y iIn x & x = ordsucc y lemma !P:set (set prop) prop.!Q:set (set prop) prop.!x:set.ordinal x -> TransSet x -> ~ (?p:set prop.PNo_rel_strict_uniq_imv P Q x p) -> (!y:set.y iIn x -> ?p:set prop.PNo_rel_strict_uniq_imv P Q y p) -> (!y:set.y iIn x -> ordsucc y iIn x) -> ~ !y:set.ordinal y -> y iIn x -> PNo_rel_strict_uniq_imv P Q y \z:set.!p:set prop.PNo_rel_strict_imv P Q (ordsucc z) p -> p z lemma !P:set (set prop) prop.!Q:set (set prop) prop.!x:set.!y:set.!p:set prop.TransSet x -> ordinal y -> (!z:set.z iIn y -> z iIn x -> PNo_rel_strict_uniq_imv P Q z \w:set.!q:set prop.PNo_rel_strict_imv P Q (ordsucc w) q -> q w) -> y iIn x -> PNo_rel_strict_imv P Q y p -> PNo_rel_strict_upperbd P y p -> PNo_rel_strict_lowerbd Q y p -> (!q:set prop.PNo_rel_strict_imv P Q y q -> PNoEq_ y p q) -> PNoEq_ y (\z:set.!q:set prop.PNo_rel_strict_imv P Q (ordsucc z) q -> q z) p -> PNo_rel_strict_imv P Q y (\z:set.!q:set prop.PNo_rel_strict_imv P Q (ordsucc z) q -> q z) & !q:set prop.PNo_rel_strict_imv P Q y q -> PNoEq_ y (\z:set.!p2:set prop.PNo_rel_strict_imv P Q (ordsucc z) p2 -> p2 z) q lemma !P:set (set prop) prop.!Q:set (set prop) prop.!x:set.!y:set.!p:set prop.!z:set.TransSet x -> ordinal y -> (!w:set.w iIn y -> w iIn x -> PNo_rel_strict_uniq_imv P Q w \u:set.!q:set prop.PNo_rel_strict_imv P Q (ordsucc u) q -> q u) -> y iIn x -> PNo_rel_strict_imv P Q y p -> ordsucc z iIn y -> ordsucc z iIn x -> ((!q:set prop.PNo_rel_strict_imv P Q (ordsucc z) q -> q z) <-> p z) lemma !P:set (set prop) prop.!Q:set (set prop) prop.!x:set.!y:set.PNoLt_pwise (PNo_downc P) (PNo_upc Q) -> ordinal x -> ~ (?p:set prop.PNo_rel_strict_uniq_imv P Q x p) -> ~ (?z:set.z iIn x & ?p:set prop.PNo_rel_strict_split_imv P Q z p) -> (!z:set.z iIn x -> ?p:set prop.PNo_rel_strict_uniq_imv P Q z p) -> y iIn x -> x = ordsucc y -> ~ ordinal y var P:set (set prop) prop var Q:set (set prop) prop var x:set hyp PNoLt_pwise (PNo_downc P) (PNo_upc Q) hyp ordinal x hyp TransSet x hyp !y:set.y iIn x -> (?p:set prop.PNo_rel_strict_uniq_imv P Q y p) | ?z:set.z iIn y & ?p:set prop.PNo_rel_strict_split_imv P Q z p hyp ~ ?p:set prop.PNo_rel_strict_uniq_imv P Q x p hyp ~ ?y:set.y iIn x & ?p:set prop.PNo_rel_strict_split_imv P Q y p claim ~ !y:set.y iIn x -> ?p:set prop.PNo_rel_strict_uniq_imv P Q y p