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 SNoCutP : set set prop const binunion : set set set const Repl : set (set set) set const SNoS_ : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const Empty : set axiom add_SNo_ordinal_SNoCutP: !x:set.ordinal x -> !y:set.ordinal y -> SNoCutP (binunion (Repl (SNoS_ x) \z:set.z + y) (Repl (SNoS_ y) (add_SNo x))) Empty const ordsucc : set set axiom ordsuccI2: !x:set.x iIn ordsucc x const SNo_ : set set prop axiom ordinal_SNo_: !x:set.ordinal x -> SNo_ x x axiom SNoS_I: !x:set.ordinal x -> !y:set.!z:set.z iIn x -> SNo_ z y -> y iIn SNoS_ x axiom ReplI: !x:set.!f:set set.!y:set.y iIn x -> f y iIn Repl x f axiom binunionI1: !x:set.!y:set.!z:set.z iIn x -> z iIn binunion x y const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom ordinal_SNoLt_In: !x:set.!y:set.ordinal x -> ordinal y -> x < y -> x iIn y const SNoCut : set set set axiom add_SNo_ordinal_eq: !x:set.ordinal x -> !y:set.ordinal y -> x + y = SNoCut (binunion (Repl (SNoS_ x) \z:set.z + y) (Repl (SNoS_ y) (add_SNo x))) Empty const SNo : set prop const SNoLev : set set const famunion : set (set set) set const SNoEq_ : set set set prop axiom SNoCutP_SNoCut: !x:set.!y:set.SNoCutP x y -> SNo (SNoCut x y) & SNoLev (SNoCut x y) iIn ordsucc (binunion (famunion x \z:set.ordsucc (SNoLev z)) (famunion y \z:set.ordsucc (SNoLev z))) & (!z:set.z iIn x -> z < SNoCut x y) & (!z:set.z iIn y -> SNoCut x y < z) & !z:set.SNo z -> (!w:set.w iIn x -> w < z) -> (!w:set.w iIn y -> z < w) -> Subq (SNoLev (SNoCut x y)) (SNoLev z) & SNoEq_ (SNoLev (SNoCut x y)) (SNoCut x y) z lemma !x:set.!y:set.ordinal x -> ordinal y -> (!z:set.z iIn y -> ordsucc x + z = ordsucc (x + z)) -> SNo x -> SNo y -> ordinal (x + y) -> ordinal (ordsucc x) -> SNo (ordsucc x) -> ordinal (ordsucc x + y) -> (!z:set.z iIn binunion (Repl (SNoS_ (ordsucc x)) \w:set.w + y) (Repl (SNoS_ y) (add_SNo (ordsucc x))) -> z < ordsucc x + y) -> (!z:set.SNo z -> (!w:set.w iIn binunion (Repl (SNoS_ (ordsucc x)) \u:set.u + y) (Repl (SNoS_ y) (add_SNo (ordsucc x))) -> w < z) -> (!w:set.w iIn Empty -> z < w) -> Subq (SNoLev (ordsucc x + y)) (SNoLev z) & SNoEq_ (SNoLev (ordsucc x + y)) (ordsucc x + y) z) -> x + y iIn ordsucc x + y -> ordsucc x + y = ordsucc (x + y) var x:set var y:set hyp ordinal x hyp ordinal y hyp !z:set.z iIn y -> ordsucc x + z = ordsucc (x + z) hyp SNo x hyp SNo y hyp ordinal (x + y) hyp ordinal (ordsucc x) hyp SNo (ordsucc x) claim ordinal (ordsucc x + y) -> ordsucc x + y = ordsucc (x + y)