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 SNoL : set set const SNoS_ : set set const SNoLev : set set axiom SNoL_SNoS_: !x:set.Subq (SNoL x) (SNoS_ (SNoLev x)) const SNo : set prop const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoL_E: !x:set.SNo x -> !y:set.y iIn SNoL x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> y < x -> P) -> P const Repl : set (set set) set axiom ReplE_impred: !x:set.!f:set set.!y:set.y iIn Repl x f -> !P:prop.(!z:set.z iIn x -> y = f z -> P) -> P const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoCut : set set set const binunion : set set set const SNoR : set set const ordsucc : set set const famunion : set (set 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)) -> 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)))) iIn ordsucc (binunion (famunion (binunion (Repl (SNoL x) \z:set.z + y) (Repl (SNoL y) (add_SNo x))) \z:set.ordsucc (SNoLev z)) (famunion (binunion (Repl (SNoR x) \z:set.z + y) (Repl (SNoR y) (add_SNo x))) \z:set.ordsucc (SNoLev z))) -> (!z:set.z iIn Repl (SNoL x) (\w:set.w + y) -> !p:set prop.(!w:set.w iIn SNoS_ (SNoLev x) -> z = w + y -> SNo w -> SNoLev w iIn SNoLev x -> w < x -> p (w + y)) -> p z) -> (!z:set.z iIn Repl (SNoL y) (add_SNo x) -> !p:set prop.(!w:set.w iIn SNoS_ (SNoLev y) -> z = x + w -> SNo w -> SNoLev w iIn SNoLev y -> w < y -> p (x + w)) -> p z) -> 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) lemma !x:set.!y:set.!z:set.!p:set prop.!w:set.(!u:set.u iIn SNoS_ (SNoLev y) -> z = x + u -> SNo u -> SNoLev u iIn SNoLev y -> u < y -> p (x + u)) -> w iIn SNoL y -> z = x + w -> w iIn SNoS_ (SNoLev y) -> SNo w -> SNoLev w iIn SNoLev y -> w < y -> p (x + w) var x:set var y:set hyp SNo x hyp SNo y hyp ordinal (SNoLev x + SNoLev y) hyp !z:set.z iIn SNoS_ (SNoLev x) -> Subq (SNoLev (z + y)) (SNoLev z + SNoLev y) hyp !z:set.z iIn SNoS_ (SNoLev y) -> Subq (SNoLev (x + z)) (SNoLev x + SNoLev z) hyp 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)))) iIn ordsucc (binunion (famunion (binunion (Repl (SNoL x) \z:set.z + y) (Repl (SNoL y) (add_SNo x))) \z:set.ordsucc (SNoLev z)) (famunion (binunion (Repl (SNoR x) \z:set.z + y) (Repl (SNoR y) (add_SNo x))) \z:set.ordsucc (SNoLev z))) claim (!z:set.z iIn Repl (SNoL x) (\w:set.w + y) -> !p:set prop.(!w:set.w iIn SNoS_ (SNoLev x) -> z = w + y -> SNo w -> SNoLev w iIn SNoLev x -> w < x -> p (w + y)) -> p z) -> 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)