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 iffI: !P:prop.!Q:prop.(P -> Q) -> (Q -> P) -> (P <-> Q) axiom ordsuccE: !x:set.!y:set.y iIn ordsucc x -> y iIn x | y = x axiom PNoEq_rel_strict_imv: !P:set (set prop) prop.!Q:set (set prop) prop.!x:set.ordinal x -> !p:set prop.!q:set prop.PNoEq_ x p q -> PNo_rel_strict_imv P Q x p -> PNo_rel_strict_imv P Q x q axiom FalseE: ~ False axiom xm: !P:prop.P | ~ P lemma !x:set.!p:set prop.!q:set prop.!y:set.PNoEq_ x p q -> y iIn x -> (p y <-> q y) -> (q y <-> p y | y = x) lemma !x:set.!p:set prop.!q:set prop.!y:set.PNoEq_ x p q -> y iIn x -> (p y <-> q y) -> (p y & y != x <-> q y) var P:set (set prop) prop var Q:set (set prop) prop var x:set var y:set var p:set prop var q:set prop hyp x = ordsucc y hyp ordinal (ordsucc y) hyp !p2:set prop.PNo_rel_strict_imv P Q y p2 -> PNoEq_ y p p2 hyp ~ (PNo_rel_strict_imv P Q x (\z:set.p z & z != y) & PNo_rel_strict_imv P Q x \z:set.p z | z = y) hyp PNo_rel_strict_imv P Q (ordsucc y) \z:set.p z & z != y hyp (!z:set.z iIn ordsucc y -> !p2:set prop.PNo_downc P z p2 -> PNoLt z p2 (ordsucc y) q) & !z:set.z iIn ordsucc y -> !p2:set prop.PNo_upc Q z p2 -> PNoLt (ordsucc y) q z p2 hyp PNo_rel_strict_imv P Q y q claim PNoEq_ y p q -> PNoEq_ (ordsucc y) (\z:set.p z & z != y) q