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 lemma !x:set.!f:set set.!y:set.!z:set.!w:set.z iIn x -> y iIn f z -> TransSet (f z) -> w iIn y -> w iIn f z -> w iIn famunion x f var x:set var f:set set var y:set var z:set hyp !w:set.w iIn x -> ordinal (f w) hyp z iIn x hyp y iIn f z claim ordinal (f z) -> !w:set.w iIn y -> w iIn famunion x f