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 SNoCutP : set set prop const binunion : set set set const Repl : set (set set) set const SNoL : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoR : set set axiom add_SNo_SNoCutP: !x:set.!y:set.SNo x -> SNo y -> SNoCutP (binunion (Repl (SNoL x) \z:set.z + y) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) \z:set.z + y) (Repl (SNoR y) (add_SNo x))) const SNoCut : set set set axiom add_SNo_eq: !x:set.SNo x -> !y:set.SNo y -> x + y = SNoCut (binunion (Repl (SNoL x) \z:set.z + y) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) \z:set.z + y) (Repl (SNoR y) (add_SNo x))) const SNoLev : set set const SNoS_ : set set lemma !x:set.!y:set.SNo x -> SNo y -> ordinal (SNoLev x + SNoLev 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)) -> SNoCutP (binunion (Repl (SNoL x) \z:set.z + y) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) \z:set.z + y) (Repl (SNoR y) (add_SNo x))) -> Subq (SNoLev (SNoCut (binunion (Repl (SNoL x) \z:set.z + y) (Repl (SNoL y) (add_SNo x))) (binunion (Repl (SNoR x) \z:set.z + y) (Repl (SNoR y) (add_SNo x))))) (SNoLev x + SNoLev y) var x:set var y:set hyp SNo x hyp SNo y claim ordinal (SNoLev x + SNoLev 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)