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 SNo : set prop const SNoLev : set set axiom SNoLev_: !x:set.SNo x -> SNo_ (SNoLev x) x lemma !x:set.!y:set.!z:set.Subq (SNoLev x) (SNoLev y) -> (!w:set.w iIn SNoLev x -> (w iIn x <-> w iIn y)) -> Subq x (SNoElts_ (SNoLev x)) -> (!w:set.w iIn SNoLev x -> ~(SetAdjoin w (Sing (ordsucc Empty)) iIn x <-> w iIn x)) -> (!w:set.w iIn SNoLev y -> ~(SetAdjoin w (Sing (ordsucc Empty)) iIn y <-> w iIn y)) -> z iIn x -> z iIn binunion (SNoLev x) (Repl (SNoLev x) \w:set.SetAdjoin w (Sing (ordsucc Empty))) -> z iIn y claim !x:set.!y:set.SNo x -> SNo y -> Subq (SNoLev x) (SNoLev y) -> (!z:set.z iIn SNoLev x -> (z iIn x <-> z iIn y)) -> !z:set.z iIn x -> z iIn y