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 term PNoEq_ = \x:set.\p:set prop.\q:set prop.!y:set.y iIn x -> (p y <-> q y) term SNoEq_ = \x:set.\y:set.\z:set.PNoEq_ x (\w:set.w iIn y) \w:set.w iIn z axiom ordinal_Hered: !x:set.ordinal x -> !y:set.y iIn x -> ordinal y const SNo : set prop const SNoLev : set set const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 lemma !x:set.!y:set.ordinal x -> SNo y -> SNo x -> SNoLev x = x -> SNoLev y = x -> ~ y <= x -> (!z:set.ordinal z -> z iIn x -> z iIn y) -> ~ Subq x y var x:set var y:set hyp ordinal x hyp TransSet x hyp SNo y hyp SNo x hyp SNoLev x = x hyp SNoLev y = x hyp ~ y <= x claim ~ !z:set.ordinal z -> z iIn x -> z iIn y