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 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 axiom add_SNo_ordinal_InR: !x:set.ordinal x -> !y:set.ordinal y -> !z:set.z iIn y -> x + z iIn x + y const ordsucc : set set lemma !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> ordinal (SNoLev x + SNoLev y) -> SNo w -> SNoLev w iIn SNoLev y -> z iIn ordsucc (SNoLev (x + w)) -> ordinal z -> Subq (SNoLev x + SNoLev y) z -> Subq (SNoLev (x + w)) (SNoLev x + SNoLev w) -> ~ SNoLev x + SNoLev w iIn SNoLev x + SNoLev y const SNoS_ : set set var x:set var y:set var z:set var w:set hyp SNo x hyp SNo y hyp ordinal (SNoLev x + SNoLev y) hyp !u:set.u iIn SNoS_ (SNoLev y) -> Subq (SNoLev (x + u)) (SNoLev x + SNoLev u) hyp w iIn SNoS_ (SNoLev y) hyp SNo w hyp SNoLev w iIn SNoLev y hyp z iIn ordsucc (SNoLev (x + w)) hyp ordinal z hyp Subq (SNoLev x + SNoLev y) z claim ~ Subq (SNoLev (x + w)) (SNoLev x + SNoLev w)