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 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) term TransSet = \x:set.!y:set.y iIn x -> Subq y x term nIn = \x:set.\y:set.~ x iIn y 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 const binintersect : set set set lemma !x:set.!y:set.SNo x -> y iIn SNoLev x -> SNo_ (SNoLev x) x -> Subq (binintersect x (SNoElts_ y)) (SNoElts_ y) & !z:set.z iIn y -> ~(SetAdjoin z (Sing (ordsucc Empty)) iIn binintersect x (SNoElts_ y) <-> z iIn binintersect x (SNoElts_ y)) claim !x:set.SNo x -> !y:set.y iIn SNoLev x -> SNo_ y (binintersect x (SNoElts_ y))