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 axiom In_irref: !x:set.nIn x x const SNoLev : set set axiom ordinal_SNoLev: !x:set.ordinal x -> SNoLev x = x const SNo : set prop const ordsucc : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 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 SNoEq_ : set set set prop 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)) hyp SNo (ordsucc (x + y)) claim ~ (Subq (SNoLev (ordsucc x + y)) (SNoLev (ordsucc (x + y))) & SNoEq_ (SNoLev (ordsucc x + y)) (ordsucc x + y) (ordsucc (x + y)))