const binunion : set set set const Sep : set (set prop) set const ReplSep : set (set prop) (set set) set const SetAdjoin : set set set const Sing : set set const ordsucc : set set const Empty : set term PSNo = \x:set.\p:set prop.binunion (Sep x p) (ReplSep x (\y:set.~ p y) \y:set.SetAdjoin y (Sing (ordsucc Empty))) 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 const Repl : set (set set) set term SNoElts_ = \x:set.binunion x (Repl x \y:set.SetAdjoin y (Sing (ordsucc Empty))) term nIn = \x:set.\y:set.~ x iIn y term SNo_ = \x:set.\y:set.Subq y (SNoElts_ x) & !z:set.z iIn x -> ~(SetAdjoin z (Sing (ordsucc Empty)) iIn y <-> z iIn y) axiom binunionI1: !x:set.!y:set.!z:set.z iIn x -> z iIn binunion x y axiom SepE: !x:set.!p:set prop.!y:set.y iIn Sep x p -> y iIn x & p y axiom ReplI: !x:set.!f:set set.!y:set.y iIn x -> f y iIn Repl x f axiom binunionI2: !x:set.!y:set.!z:set.z iIn y -> z iIn binunion x y axiom ReplSepE_impred: !x:set.!p:set prop.!f:set set.!y:set.y iIn ReplSep x p f -> !P:prop.(!z:set.z iIn x -> p z -> y = f z -> P) -> P axiom binunionE: !x:set.!y:set.!z:set.z iIn binunion x y -> z iIn x | z iIn y const ordinal : set prop axiom ordinal_Hered: !x:set.ordinal x -> !y:set.y iIn x -> ordinal y lemma !x:set.!p:set prop.!y:set.ordinal x -> y iIn x -> ordinal y -> ~(SetAdjoin y (Sing (ordsucc Empty)) iIn PSNo x p <-> y iIn PSNo x p) claim !x:set.ordinal x -> !p:set prop.Subq (PSNo x p) (SNoElts_ x) & !y:set.y iIn x -> ~(SetAdjoin y (Sing (ordsucc Empty)) iIn PSNo x p <-> y iIn PSNo x p)