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 term nIn = \x:set.\y:set.~ x iIn y const ordsucc : set set axiom ordinal_ordsucc: !x:set.ordinal x -> ordinal (ordsucc x) const SNoLev : set set const minus_SNo : set set term - = minus_SNo lemma !x:set.!y:set.!z:set.!w:set.ordinal (SNoLev x) -> y iIn ordsucc (SNoLev z) -> z = - w -> SNoLev w iIn SNoLev x -> Subq (SNoLev - w) (SNoLev w) -> ordinal (SNoLev - w) -> ordinal (ordsucc (SNoLev z)) -> y iIn SNoLev x const SNo : set prop var x:set var y:set var z:set var w:set hyp ordinal (SNoLev x) hyp y iIn ordsucc (SNoLev z) hyp z = - w hyp SNoLev w iIn SNoLev x hyp Subq (SNoLev - w) (SNoLev w) hyp SNo - w claim ordinal (SNoLev - w) -> y iIn SNoLev x