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 TransSet = \x:set.!y:set.y iIn x -> Subq y x term ordinal = \x:set.TransSet x & !y:set.y iIn x -> TransSet y const SNo : set prop const SNoLev : set set axiom SNoLev_ordinal: !x:set.SNo x -> ordinal (SNoLev x) const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoS_ : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const SNoR : set set const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 lemma !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo (x + y) -> SNo z -> (!u:set.u iIn SNoS_ (SNoLev z) -> SNoLev u iIn SNoLev (x + y) -> (x + y) < u -> (?v:set.v iIn SNoR x & (v + y) <= u) | ?v:set.v iIn SNoR y & (x + v) <= u) -> SNoLev z iIn SNoLev (x + y) -> ~ ((?u:set.u iIn SNoR x & (u + y) <= z) | ?u:set.u iIn SNoR y & (x + u) <= z) -> SNo w -> SNoLev w iIn SNoLev z -> w < z -> (x + y) < w -> w iIn SNoS_ (SNoLev z) -> ~ SNoLev w iIn SNoLev (x + y) const SNoL : set set var x:set var y:set var z:set var w:set hyp SNo x hyp SNo y hyp SNo (x + y) hyp SNo z hyp !u:set.u iIn SNoS_ (SNoLev z) -> SNoLev u iIn SNoLev (x + y) -> (x + y) < u -> (?v:set.v iIn SNoR x & (v + y) <= u) | ?v:set.v iIn SNoR y & (x + v) <= u hyp SNoLev z iIn SNoLev (x + y) hyp ~ ((?u:set.u iIn SNoR x & (u + y) <= z) | ?u:set.u iIn SNoR y & (x + u) <= z) hyp w iIn SNoL z hyp SNo w hyp SNoLev w iIn SNoLev z hyp w < z hyp (x + y) < w claim ~ w iIn SNoS_ (SNoLev z)