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 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 nIn = \x:set.\y:set.~ x iIn y axiom In_irref: !x:set.nIn x x axiom FalseE: ~ False 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 In_no2cycle: !x:set.!y:set.x iIn y -> ~ y iIn x 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 PNoLtI3: !x:set.!y:set.!p:set prop.!q:set prop.y iIn x -> PNoEq_ y p q -> ~ p y -> PNoLt x p y q 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_tra: !x:set.!y:set.!z:set.ordinal x -> ordinal y -> ordinal z -> !p:set prop.!q:set prop.!p2:set prop.PNoLt x p y q -> PNoLt y q z p2 -> PNoLt x p z p2 axiom PNoLt_irref: !x:set.!p:set prop.~ PNoLt x p x 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 !x:set.!p:set prop.!y:set.!q:set prop.!z:set.ordinal x -> TransSet x -> PNoEq_ x p (\w:set.p w & w != x) -> y iIn x -> z iIn y -> z iIn x -> PNoEq_ z (\w:set.p w & w != x) q & ~ (p z & z != x) & q z -> PNoLt x p y q const ordsucc : set set var P:set (set prop) prop var x:set var p:set prop var y:set var q:set prop hyp ordinal x hyp TransSet x hyp PNo_rel_strict_upperbd P x p hyp ordinal (ordsucc x) hyp PNoEq_ x p \z:set.p z & z != x hyp ordinal y hyp y iIn x hyp y iIn ordsucc x hyp PNo_downc P y q claim PNoLt y q x p -> PNoLt y q (ordsucc x) \z:set.p z & z != x