const In : set set prop term iIn = In infix iIn 2000 2000 term nIn = \x:set.\y:set.~ x iIn y 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))) 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 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 SepI: !x:set.!p:set prop.!y:set.y iIn x -> p y -> y iIn Sep x p axiom binunionI1: !x:set.!y:set.!z:set.z iIn x -> z iIn binunion x y axiom exactly1of2_E: !P:prop.!Q:prop.~(P <-> Q) -> !R:prop.(P -> ~ Q -> R) -> (~ P -> Q -> R) -> R axiom binunionI2: !x:set.!y:set.!z:set.z iIn y -> z iIn binunion x y axiom ReplE_impred: !x:set.!f:set set.!y:set.y iIn Repl x f -> !P:prop.(!z:set.z iIn x -> y = f z -> P) -> P axiom binunionE: !x:set.!y:set.!z:set.z iIn binunion x y -> z iIn x | z iIn y axiom SepE: !x:set.!p:set prop.!y:set.y iIn Sep x p -> y iIn x & p y axiom FalseE: ~ False 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 set_ext: !x:set.!y:set.Subq x y -> Subq y x -> x = y lemma !x:set.!y:set.!z:set.!w:set.(!u:set.u iIn x -> ~(SetAdjoin u (Sing (ordsucc Empty)) iIn y <-> u iIn y)) -> z iIn y -> w iIn x -> z = SetAdjoin w (Sing (ordsucc Empty)) -> nIn w y -> SetAdjoin w (Sing (ordsucc Empty)) iIn ReplSep x (\u:set.nIn u y) \u:set.SetAdjoin u (Sing (ordsucc Empty)) const ordinal : set prop claim !x:set.!y:set.ordinal x -> SNo_ x y -> y = PSNo x \z:set.z iIn y