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_In_eq: !x:set.!y:set.ordinal x -> y iIn x -> ordsucc y iIn x | x = ordsucc y const SNoS_ : set set const SNoLev : set set const minus_SNo : set set term - = minus_SNo const SNo : set prop lemma !x:set.!y:set.!z:set.!w:set.!u:set.TransSet x -> (!v:set.v iIn x -> !x2:set.x2 iIn SNoS_ v -> Subq (SNoLev - x2) (SNoLev x2)) -> SNoLev y iIn x -> ordinal (SNoLev y) -> z iIn ordsucc (SNoLev w) -> w = - u -> SNo u -> SNoLev u iIn SNoLev y -> u iIn SNoS_ (ordsucc (SNoLev u)) -> ordsucc (SNoLev u) iIn x -> z iIn SNoLev y var x:set var y:set var z:set var w:set var u:set hyp TransSet x hyp !v:set.v iIn x -> !x2:set.x2 iIn SNoS_ v -> Subq (SNoLev - x2) (SNoLev x2) hyp SNoLev y iIn x hyp ordinal (SNoLev y) hyp z iIn ordsucc (SNoLev w) hyp w = - u hyp SNo u hyp SNoLev u iIn SNoLev y claim u iIn SNoS_ (ordsucc (SNoLev u)) -> z iIn SNoLev y