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 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 SNo : set prop const ordsucc : set set const SNoLev : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 lemma !x:set.!y:set.!z:set.ordinal x -> ordinal y -> 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) -> SNo (ordsucc x + z) -> ordinal (x + SNoLev z) -> (ordsucc x + z) < ordsucc (x + y) var x:set var y:set var z:set hyp ordinal x hyp ordinal y hyp SNo x hyp SNo y hyp ordinal (x + y) hyp ordinal (ordsucc x) hyp SNo (ordsucc x) hyp ordinal (ordsucc (x + y)) hyp SNo (ordsucc (x + y)) hyp SNoLev z iIn y hyp ordinal (SNoLev z) hyp SNo z hyp ordsucc x + SNoLev z = ordsucc (x + SNoLev z) claim SNo (ordsucc x + z) -> (ordsucc x + z) < ordsucc (x + y)