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 TransSet = \x:set.!y:set.y iIn x -> Subq y x term ordinal = \x:set.TransSet x & !y:set.y iIn x -> TransSet y const famunion : set (set set) set axiom famunionE: !x:set.!f:set set.!y:set.y iIn famunion x f -> ?z:set.z iIn x & y iIn f z lemma !x:set.!f:set set.!y:set.!z:set.(!w:set.w iIn x -> ordinal (f w)) -> z iIn x -> y iIn f z -> ordinal (f z) -> !w:set.w iIn y -> w iIn famunion x f lemma !x:set.!f:set set.!y:set.!z:set.(!w:set.w iIn x -> ordinal (f w)) -> z iIn x -> y iIn f z -> ordinal (f z) -> TransSet y claim !x:set.!f:set set.(!y:set.y iIn x -> ordinal (f y)) -> TransSet (famunion x f) & !y:set.y iIn famunion x f -> TransSet y