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 add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) const SNoS_ : set set const SNoLev : set set axiom SNoLev_ind2: !r:set set prop.(!x:set.!y:set.SNo x -> SNo y -> (!z:set.z iIn SNoS_ (SNoLev x) -> r z y) -> (!z:set.z iIn SNoS_ (SNoLev y) -> r x z) -> (!z:set.z iIn SNoS_ (SNoLev x) -> !w:set.w iIn SNoS_ (SNoLev y) -> r z w) -> r x y) -> !x:set.!y:set.SNo x -> SNo y -> r x y lemma !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) -> (!z:set.z iIn SNoS_ (SNoLev x) -> Subq (SNoLev (z + y)) (SNoLev z + SNoLev y)) -> (!z:set.z iIn SNoS_ (SNoLev y) -> Subq (SNoLev (x + z)) (SNoLev x + SNoLev z)) -> (!z:set.z iIn SNoS_ (SNoLev x) -> !w:set.w iIn SNoS_ (SNoLev y) -> Subq (SNoLev (z + w)) (SNoLev z + SNoLev w)) -> Subq (SNoLev (x + y)) (SNoLev x + SNoLev y) claim !x:set.!y:set.SNo x -> SNo y -> Subq (SNoLev (x + y)) (SNoLev x + SNoLev y)