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 const SNoLev : set set axiom ordinal_SNoLev: !x:set.ordinal x -> SNoLev x = x const SNo : set prop const ordsucc : set set const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 lemma !x:set.!y:set.ordinal x -> TransSet x -> SNo y -> SNoLev y iIn ordsucc x -> SNo x -> SNoLev x = x -> y <= x var x:set var y:set hyp ordinal x hyp TransSet x hyp SNo y hyp SNoLev y iIn ordsucc x claim SNo x -> y <= x