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 nIn = \x:set.\y:set.~ x iIn y const SNo : set prop const SNoLt : set set prop term < = SNoLt infix < 2020 2020 term SNoCutP = \x:set.\y:set.(!z:set.z iIn x -> SNo z) & (!z:set.z iIn y -> SNo z) & !z:set.z iIn x -> !w:set.w iIn y -> z < w axiom In_irref: !x:set.nIn x x 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 const SNoLev : set set const SNoCut : set set set const SNoEq_ : set set set prop const ordinal : set prop var x:set var y:set var z:set hyp ordinal x hyp ordinal y hyp SNo (x + y) hyp SNoLev z iIn SNoLev (x + y) hyp SNo z hyp (x + y) < z hyp !w:set.w iIn binunion (Repl (SNoS_ x) \u:set.u + y) (Repl (SNoS_ y) \u:set.x + u) -> SNo w hyp !w:set.w iIn binunion (Repl (SNoS_ x) \u:set.u + y) (Repl (SNoS_ y) \u:set.x + u) -> w < SNoCut (binunion (Repl (SNoS_ x) \u:set.u + y) (Repl (SNoS_ y) \u:set.x + u)) Empty hyp !w:set.SNo w -> (!u:set.u iIn binunion (Repl (SNoS_ x) \v:set.v + y) (Repl (SNoS_ y) (add_SNo x)) -> u < w) -> (!u:set.u iIn Empty -> w < u) -> Subq (SNoLev (SNoCut (binunion (Repl (SNoS_ x) \u:set.u + y) (Repl (SNoS_ y) (add_SNo x))) Empty)) (SNoLev w) & SNoEq_ (SNoLev (SNoCut (binunion (Repl (SNoS_ x) \u:set.u + y) (Repl (SNoS_ y) (add_SNo x))) Empty)) (SNoCut (binunion (Repl (SNoS_ x) \u:set.u + y) (Repl (SNoS_ y) (add_SNo x))) Empty) w claim ~ (Subq (SNoLev (x + y)) (SNoLev z) & SNoEq_ (SNoLev (x + y)) (x + y) z)