const In : set set prop term iIn = In infix iIn 2000 2000 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 const PNoLe : set (set prop) set (set prop) prop 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 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 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 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_lenbdd = \x:set.\P:set (set prop) prop.!y:set.!p:set prop.P y p -> y iIn x term nIn = \x:set.\y:set.~ x iIn y 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) const binintersect : set set set axiom binintersectI: !x:set.!y:set.!z:set.z iIn x -> z iIn y -> z iIn binintersect x y axiom PNoEq_sym_: !x:set.!p:set prop.!q:set prop.PNoEq_ x p q -> PNoEq_ x q p 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 In_irref: !x:set.nIn x x axiom FalseE: ~ False axiom PNoLtI1: !x:set.!y:set.!p:set prop.!q:set prop.PNoLt_ (binintersect x y) p q -> PNoLt x p y q var x:set var p:set prop var y:set var q:set prop var z:set hyp ordinal x hyp TransSet x hyp PNoEq_ x p \w:set.p w | w = x hyp y iIn x hyp z iIn y claim z iIn x -> PNoEq_ z q (\w:set.p w | w = x) & ~ q z & (p z | z = x) -> PNoLt y q x p