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 Repl : set (set set) set const SetAdjoin : set set set const Sing : set set const ordsucc : set set const Empty : 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) const ordinal : set prop axiom ordinal_Hered: !x:set.ordinal x -> !y:set.y iIn x -> ordinal y lemma !x:set.!y:set.!z:set.!w:set.ordinal y -> ordinal z -> w iIn y -> (!u:set.u iIn y -> ~(SetAdjoin u (Sing (ordsucc Empty)) iIn x <-> u iIn x)) -> Subq x (SNoElts_ z) -> ordinal w -> w iIn z claim !x:set.!y:set.!z:set.ordinal y -> ordinal z -> SNo_ y x -> SNo_ z x -> !w:set.w iIn y -> w iIn z