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 axiom ordinal_SNo: !x:set.ordinal x -> SNo x const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom add_SNo_ordinal_ordinal: !x:set.ordinal x -> !y:set.ordinal y -> ordinal (x + y) const SNoS_ : set set const SNoLev : set set const SNo_ : set set prop axiom SNoS_E2: !x:set.ordinal x -> !y:set.y iIn SNoS_ x -> !P:prop.(SNoLev y iIn x -> ordinal (SNoLev y) -> SNo y -> SNo_ (SNoLev y) y -> 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 binunion : set set set axiom binunionE: !x:set.!y:set.!z:set.z iIn binunion x y -> z iIn x | z iIn y const Empty : set axiom EmptyE: !x:set.nIn x Empty axiom FalseE: ~ False const ordsucc : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 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)) -> SNo (ordsucc (x + y)) -> ~ (Subq (SNoLev (ordsucc x + y)) (SNoLev (ordsucc (x + y))) & SNoEq_ (SNoLev (ordsucc x + y)) (ordsucc x + y) (ordsucc (x + y))) lemma !x:set.!y:set.!z:set.ordinal x -> ordinal y -> SNo x -> SNo y -> ordinal (x + y) -> ordinal (ordsucc (x + y)) -> SNo (ordsucc (x + y)) -> SNoLev z iIn ordsucc x -> ordinal (SNoLev z) -> SNo z -> ordinal (SNoLev z + y) -> (z + y) < ordsucc (x + y) lemma !x:set.!y:set.!z:set.ordinal x -> ordinal y -> (!w:set.w iIn y -> ordsucc x + w = ordsucc (x + w)) -> SNo x -> SNo y -> ordinal (x + y) -> ordinal (ordsucc x) -> SNo (ordsucc x) -> ordinal (ordsucc (x + y)) -> SNo (ordsucc (x + y)) -> SNoLev z iIn y -> ordinal (SNoLev z) -> SNo z -> ordsucc x + SNoLev z = ordsucc (x + SNoLev z) -> (ordsucc x + z) < 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.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 hyp ordsucc (x + y) iIn ordsucc x + y hyp ordinal (ordsucc (x + y)) claim ~ SNo (ordsucc (x + y))