const binunion : set set set const Sep : set (set prop) set const ReplSep : set (set prop) (set set) set const SetAdjoin : set set set const Sing : set set const ordsucc : set set const Empty : set term PSNo = \x:set.\p:set prop.binunion (Sep x p) (ReplSep x (\y:set.~ p y) \y:set.SetAdjoin y (Sing (ordsucc Empty))) 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 Repl : set (set set) set term SNoElts_ = \x:set.binunion x (Repl x \y:set.SetAdjoin y (Sing (ordsucc Empty))) 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 ordinal : set prop var p:set prop var x:set var y:set hyp ordinal x hyp p x hyp ~ p y hyp SetAdjoin x (Sing (ordsucc Empty)) = SetAdjoin y (Sing (ordsucc Empty)) hyp ordinal y claim x != y