const ordinal : set prop 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 const In : set set prop term iIn = In infix iIn 2000 2000 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 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 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 Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term PNo_upc = \P:set (set prop) prop.\x:set.\p:set prop.?y:set.ordinal y & ?q:set prop.P y q & PNoLe y q x p 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 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 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 axiom ordinal_Hered: !x:set.ordinal x -> !y:set.y iIn x -> ordinal y axiom dneg: !P:prop.~ ~ P -> P lemma !P:set (set prop) prop.!x:set.!p:set prop.!y:set.!q:set prop.ordinal x -> ordinal (ordsucc x) -> PNo_rel_strict_lowerbd P (ordsucc x) (\z:set.p z | z = x) -> p x | x = x -> ordinal y -> P y q -> (y iIn ordsucc x -> PNoLt x p y q) -> (!z:set.z iIn ordsucc x -> z iIn y -> PNoEq_ z q p -> p z | z = x -> q z) -> PNoLt x p y q lemma !P:set (set prop) prop.!x:set.!p:set prop.!y:set.!q:set prop.!z:set.ordinal x -> ordinal (ordsucc x) -> PNo_rel_strict_lowerbd P (ordsucc x) (\w:set.p w | w = x) -> ordinal y -> P y q -> z iIn ordsucc x -> z iIn y -> PNoEq_ z q p -> p z | z = x -> ~ q z -> ~ ordinal z var P:set (set prop) prop var x:set var p:set prop var y:set var q:set prop hyp ordinal x hyp ordinal (ordsucc x) hyp PNo_rel_strict_lowerbd P (ordsucc x) \z:set.p z | z = x hyp p x | x = x hyp PNoLt x p (ordsucc x) \z:set.p z | z = x hyp ordinal y hyp P y q hyp PNo_upc P y q claim (y iIn ordsucc x -> PNoLt x p y q) -> PNoLt x p y q