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 SNo : set prop const SNoLev : set set axiom SNoLev_ordinal: !x:set.SNo x -> ordinal (SNoLev x) const ordsucc : 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 -> SNo w -> SNoLev w iIn SNoLev x -> Subq (SNoLev - w) (SNoLev w) -> ordinal (SNoLev w) -> y iIn SNoLev x const SNoS_ : set set var x:set var y:set var z:set var w:set var u:set hyp !v:set.v iIn x -> !x2:set.x2 iIn SNoS_ v -> Subq (SNoLev - x2) (SNoLev x2) hyp ordinal (SNoLev y) hyp z iIn ordsucc (SNoLev w) hyp w = - u hyp SNo u hyp SNoLev u iIn SNoLev y hyp u iIn SNoS_ (ordsucc (SNoLev u)) hyp ordsucc (SNoLev u) iIn x claim Subq (SNoLev - u) (SNoLev u) -> z iIn SNoLev y