const binunion : set set set const Sing : set set term SetAdjoin = \x:set.\y:set.binunion x (Sing y) const In : set set prop term iIn = In infix iIn 2000 2000 axiom SingI: !x:set.x iIn Sing x axiom binunionI2: !x:set.!y:set.!z:set.z iIn y -> z iIn binunion x y const ordinal : set prop const ordsucc : set set const Empty : set lemma !x:set.ordinal (SetAdjoin x (Sing (ordsucc Empty))) -> ~ Sing (ordsucc Empty) iIn SetAdjoin x (Sing (ordsucc Empty)) claim !x:set.~ ordinal (SetAdjoin x (Sing (ordsucc Empty)))