const In : set set prop term iIn = In infix iIn 2000 2000 term nIn = \x:set.\y:set.~ x iIn y term PNoEq_ = \x:set.\p:set prop.\q:set prop.!y:set.y iIn x -> (p y <-> q y) term SNoEq_ = \x:set.\y:set.\z:set.PNoEq_ x (\w:set.w iIn y) \w:set.w iIn z const binunion : set set set const Sing : set set const Empty : set const Repl : set (set set) set const SetAdjoin : set set set const ordsucc : set set term eps_ = \x:set.binunion (Sing Empty) (Repl x \y:set.SetAdjoin (ordsucc y) (Sing (ordsucc Empty))) axiom SingI: !x:set.x iIn Sing x axiom binunionI1: !x:set.!y:set.!z:set.z iIn x -> z iIn binunion x y const ordinal : set prop axiom eps_ordinal_In_eq_0: !x:set.!y:set.ordinal y -> y iIn eps_ x -> y = Empty axiom iffI: !P:prop.!Q:prop.(P -> Q) -> (Q -> P) -> (P <-> Q) const nat_p : set prop var x:set var y:set var z:set hyp nat_p y hyp z iIn ordsucc y claim ordinal z -> (z iIn eps_ x <-> z iIn eps_ y)