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_notin_tagged_Repl: !x:set.!y:set.ordinal x -> nIn x (Repl y \z:set.SetAdjoin z (Sing (ordsucc Empty))) axiom FalseE: ~ False var x:set var y:set hyp ordinal y hyp y iIn binunion x (Repl x \z:set.SetAdjoin z (Sing (ordsucc Empty))) claim y iIn x | y iIn Repl x (\z:set.SetAdjoin z (Sing (ordsucc Empty))) -> y iIn x