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 SNoLev : set set axiom SNoLev_ordinal: !x:set.SNo x -> ordinal (SNoLev x) const ordsucc : set set axiom ordinal_ordsucc: !x:set.ordinal x -> ordinal (ordsucc x) axiom ordinal_Hered: !x:set.ordinal x -> !y:set.y iIn x -> ordinal y const binunion : set set set axiom binunionE: !x:set.!y:set.!z:set.z iIn binunion x y -> z iIn x | z iIn y const famunion : set (set set) set axiom famunionE_impred: !x:set.!f:set set.!y:set.y iIn famunion x f -> !P:prop.(!z:set.z iIn x -> y iIn f z -> P) -> P axiom binunion_Subq_min: !x:set.!y:set.!z:set.Subq x z -> Subq y z -> Subq (binunion x y) z const SNoS_ : set set const Repl : set (set set) set const SNoL : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const SNoR : set set const SNoCut : 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)) -> (!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) -> (!z:set.z iIn Repl (SNoR 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 -> x < w -> p (w + y)) -> p z) -> (!z:set.z iIn Repl (SNoR y) (add_SNo x) -> !p:set prop.(!w:set.w iIn SNoS_ (SNoLev y) -> z = x + w -> SNo w -> SNoLev w iIn SNoLev y -> y < w -> 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))))) (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))) -> Subq (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))) (SNoLev x + SNoLev y) -> 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.!w:set.SNo x -> SNo y -> ordinal (SNoLev x + SNoLev y) -> (!u:set.u iIn SNoS_ (SNoLev x) -> Subq (SNoLev (u + y)) (SNoLev u + SNoLev y)) -> w iIn SNoS_ (SNoLev x) -> SNo w -> SNoLev w iIn SNoLev x -> z iIn ordsucc (SNoLev (w + y)) -> ordinal z -> z iIn SNoLev x + SNoLev y lemma !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> ordinal (SNoLev x + SNoLev y) -> (!u:set.u iIn SNoS_ (SNoLev y) -> Subq (SNoLev (x + u)) (SNoLev x + SNoLev u)) -> w iIn SNoS_ (SNoLev y) -> SNo w -> SNoLev w iIn SNoLev y -> z iIn ordsucc (SNoLev (x + w)) -> ordinal z -> z iIn SNoLev x + SNoLev y lemma !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> ordinal (SNoLev x + SNoLev y) -> (!u:set.u iIn SNoS_ (SNoLev x) -> Subq (SNoLev (u + y)) (SNoLev u + SNoLev y)) -> w iIn SNoS_ (SNoLev x) -> SNo w -> SNoLev w iIn SNoLev x -> z iIn ordsucc (SNoLev (w + y)) -> ordinal z -> z iIn SNoLev x + SNoLev y lemma !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> ordinal (SNoLev x + SNoLev y) -> (!u:set.u iIn SNoS_ (SNoLev y) -> Subq (SNoLev (x + u)) (SNoLev x + SNoLev u)) -> w iIn SNoS_ (SNoLev y) -> SNo w -> SNoLev w iIn SNoLev y -> z iIn ordsucc (SNoLev (x + w)) -> ordinal z -> z iIn SNoLev x + SNoLev y 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))) hyp !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 hyp !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 hyp !z:set.z iIn Repl (SNoR 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 -> x < w -> p (w + y)) -> p z hyp !z:set.z iIn Repl (SNoR y) (add_SNo x) -> !p:set prop.(!w:set.w iIn SNoS_ (SNoLev y) -> z = x + w -> SNo w -> SNoLev w iIn SNoLev y -> y < w -> p (x + w)) -> p z hyp ordinal (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 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))))) (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))) -> 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)