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 ordsucc : set set axiom ordinal_ordsucc: !x:set.ordinal x -> ordinal (ordsucc x) axiom FalseE: ~ False axiom ordinal_ordsucc_In_eq: !x:set.!y:set.ordinal x -> y iIn x -> ordsucc y iIn x | x = ordsucc y const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNo : set prop const binunion : set set set const Repl : set (set set) set const SNoS_ : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const Empty : set const SNoLev : set set const SNoEq_ : set set set prop 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.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) -> ordsucc (x + y) iIn ordsucc x + y -> ~ ordinal (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) hyp ordinal (ordsucc x + y) hyp !z:set.z iIn binunion (Repl (SNoS_ (ordsucc x)) \w:set.w + y) (Repl (SNoS_ y) (add_SNo (ordsucc x))) -> z < ordsucc x + y hyp !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 claim x + y iIn ordsucc x + y -> ordsucc x + y = ordsucc (x + y)