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 binintersect : set set set axiom binintersect_Subq_2: !x:set.!y:set.Subq (binintersect x y) y const SNo : set prop const ordinal : set prop const SNoLev : set set axiom SNoLev_ordinal: !x:set.SNo x -> ordinal (SNoLev x) axiom ordinal_TransSet: !x:set.ordinal x -> TransSet x lemma !x:set.!y:set.!z:set.SNo x -> y iIn SNoLev x -> (!w:set.w iIn SNoLev x -> ~(SetAdjoin w (Sing (ordsucc Empty)) iIn x <-> w iIn x)) -> z iIn y -> z iIn SNoLev x -> ~(SetAdjoin z (Sing (ordsucc Empty)) iIn binintersect x (SNoElts_ y) <-> z iIn binintersect x (SNoElts_ y)) var x:set var y:set hyp SNo x hyp y iIn SNoLev x claim 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))