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 SetAdjoin : set set set const Sing : set set const ordsucc : set set const Empty : set term SNo_pair = \x:set.\y:set.binunion x (Repl y \z:set.SetAdjoin z (Sing (ordsucc (ordsucc Empty)))) axiom binunionI1: !x:set.!y:set.!z:set.z iIn x -> z iIn binunion x y const SNo : set prop lemma !x:set.!y:set.!z:set.!w:set.!u:set.SNo x -> SNo_pair x y = SNo_pair z w -> u iIn x -> u iIn SNo_pair z w -> u iIn z claim !x:set.!y:set.!z:set.!w:set.SNo x -> SNo_pair x y = SNo_pair z w -> !u:set.u iIn x -> u iIn z