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 PNoEq_antimon_: !p:set prop.!q:set prop.!x:set.ordinal x -> !y:set.y iIn x -> PNoEq_ x p q -> PNoEq_ y p q axiom PNoEq_tra_: !x:set.!p:set prop.!q:set prop.!p2:set prop.PNoEq_ x p q -> PNoEq_ x q p2 -> PNoEq_ x p p2 axiom PNoLtI2: !x:set.!y:set.!p:set prop.!q:set prop.x iIn y -> PNoEq_ x p q -> q x -> PNoLt x p y q lemma !x:set.!p:set prop.!q:set prop.!y:set.ordinal x -> PNoEq_ x (\z:set.p z & z != x) p -> y iIn x -> PNoEq_ y q (\z:set.p z & z != x) -> p y & y != x -> ordinal y -> PNoLt x p y q -> ~ PNoLt y q x p var P:set (set prop) prop var x:set var p:set prop var y:set var q:set prop var z:set hyp ordinal x hyp !w:set.w iIn x -> !p2:set prop.PNo_upc P w p2 -> PNoLt x p w p2 hyp PNoEq_ x (\w:set.p w & w != x) p hyp PNo_upc P y q hyp ordinal y hyp y = x hyp z iIn x hyp PNoEq_ z q \w:set.p w & w != x hyp ~ q z hyp p z & z != x hyp ordinal z claim ~ PNoLt x p z q