const In : set set prop term iIn = In infix iIn 2000 2000 term nIn = \x:set.\y:set.~ x iIn y 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 Sing : set set term SetAdjoin = \x:set.\y:set.binunion x (Sing y) const ordsucc : set set const Empty : set term SNoElts_ = \x:set.binunion x (Repl x \y:set.SetAdjoin y (Sing (ordsucc Empty))) 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_Hered: !x:set.ordinal x -> !y:set.y iIn x -> ordinal y axiom not_ordinal_Sing2: ~ ordinal (Sing (ordsucc (ordsucc Empty))) axiom Sing2_notin_SingSing1: nIn (Sing (ordsucc (ordsucc Empty))) (Sing (Sing (ordsucc Empty))) axiom binunionE: !x:set.!y:set.!z:set.z iIn binunion x y -> z iIn x | z iIn y var x:set var y:set hyp Sing (ordsucc (ordsucc Empty)) iIn x hyp x = SetAdjoin y (Sing (ordsucc Empty)) hyp ordinal y claim ~ Sing (ordsucc (ordsucc Empty)) iIn SetAdjoin y (Sing (ordsucc Empty))