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 nIn = \x:set.\y:set.~ x iIn y 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 In_0_1: Empty iIn ordsucc Empty axiom SingE: !x:set.!y:set.y iIn Sing x -> y = x axiom EmptyE: !x:set.nIn x Empty axiom FalseE: ~ False 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 SingI: !x:set.x iIn Sing x axiom binunionI1: !x:set.!y:set.!z:set.z iIn x -> z iIn binunion x y axiom cases_1: !x:set.x iIn ordsucc Empty -> !p:set prop.p Empty -> p x axiom set_ext: !x:set.!y:set.Subq x y -> Subq y x -> x = y claim eps_ Empty = ordsucc Empty