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))) const nat_p : set prop axiom nat_p_trans: !x:set.nat_p x -> !y:set.y iIn x -> nat_p y const omega : set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 lemma !x:set.!y:set.x iIn omega -> y iIn x -> nat_p x -> nat_p y -> eps_ x < eps_ y var x:set var y:set hyp x iIn omega hyp y iIn x claim nat_p x -> eps_ x < eps_ y