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 In_irref: !x:set.nIn x x axiom FalseE: ~ False axiom ordinal_Hered: !x:set.ordinal x -> !y:set.y iIn x -> ordinal y const binintersect : set set set axiom binintersectE: !x:set.!y:set.!z:set.z iIn binintersect x y -> z iIn x & z iIn y axiom ordsuccI2: !x:set.x iIn ordsucc x axiom In_no2cycle: !x:set.!y:set.x iIn y -> ~ y iIn x axiom PNoLtE: !x:set.!y:set.!p:set prop.!q:set prop.PNoLt x p y q -> !P:prop.(PNoLt_ (binintersect x y) p q -> P) -> (x iIn y -> PNoEq_ x p q -> q x -> P) -> (y iIn x -> PNoEq_ y p q -> ~ p y -> P) -> P axiom PNoLt_trichotomy_or: !x:set.!y:set.!p:set prop.!q:set prop.ordinal x -> ordinal y -> PNoLt x p y q | x = y & PNoEq_ x p q | PNoLt y q x p lemma !P:set (set prop) prop.!Q:set (set prop) prop.!x:set.!y:set.!p:set prop.!z:set.(!w:set.w iIn x -> ordsucc w iIn x) -> (!w:set.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) -> PNo_upc Q y p -> ordinal y -> z iIn y -> z iIn x -> PNoEq_ z p (\w:set.!q:set prop.PNo_rel_strict_imv P Q (ordsucc w) q -> q w) -> ~ p z -> (!q:set prop.PNo_rel_strict_imv P Q (ordsucc z) q -> q z) -> ordinal z -> PNoLt x (\w:set.!q:set prop.PNo_rel_strict_imv P Q (ordsucc w) q -> q w) y p lemma !P:set (set prop) prop.!Q:set (set prop) prop.!x:set.!y:set.!p:set prop.PNo_upc Q y p -> ordinal y -> ordinal (ordsucc y) -> PNoEq_ y p (\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 (ordsucc y) q -> q y) -> PNo_rel_strict_lowerbd Q (ordsucc y) (\z:set.!q:set prop.PNo_rel_strict_imv P Q (ordsucc z) q -> q z) -> PNoLt (ordsucc y) (\z:set.!q:set prop.PNo_rel_strict_imv P Q (ordsucc z) q -> q z) y p -> PNoLt x (\z:set.!q:set prop.PNo_rel_strict_imv P Q (ordsucc z) q -> q z) y p var P:set (set prop) prop var Q:set (set prop) prop var x:set var y:set var p:set prop hyp ordinal x hyp !z:set.z iIn x -> ordsucc z iIn x hyp !z:set.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 hyp y iIn x hyp PNo_upc Q y p hyp ordinal y hyp ordinal (ordsucc y) claim ordsucc y iIn x -> PNoLt x (\z:set.!q:set prop.PNo_rel_strict_imv P Q (ordsucc z) q -> q z) y p