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 add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoLev : set set const ordsucc : set set lemma !x:set.!y:set.!z:set.!w:set.SNo x -> ordinal (SNoLev x + SNoLev y) -> SNo w -> 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 -> ~ Subq (SNoLev x + SNoLev w) (SNoLev x + SNoLev y) 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 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 hyp Subq (SNoLev (x + w)) (SNoLev x + SNoLev w) claim ~ SNoLev x + SNoLev w iIn SNoLev x + SNoLev y