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 axiom ordinal_Hered: !x:set.ordinal x -> !y:set.y iIn x -> ordinal y lemma !f:set set.!x:set.!y:set.x iIn f y -> ordinal (f y) -> ordinal x -> TransSet x 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) -> TransSet y