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 axiom FalseE: ~ False axiom ordinal_In_Or_Subq: !x:set.!y:set.ordinal x -> ordinal y -> x iIn y | Subq y x const ordsucc : set set const SNoLev : set set const minus_SNo : set set term - = minus_SNo lemma !x:set.!y:set.!z:set.!w:set.y iIn ordsucc (SNoLev z) -> z = - w -> SNoLev w iIn SNoLev x -> Subq (SNoLev - w) (SNoLev w) -> Subq (SNoLev x) y -> ~ SNoLev w iIn y var x:set var y:set var z:set var w:set hyp ordinal (SNoLev x) hyp y iIn ordsucc (SNoLev z) hyp z = - w hyp SNoLev w iIn SNoLev x hyp Subq (SNoLev - w) (SNoLev w) hyp ordinal (ordsucc (SNoLev z)) claim ordinal y -> y iIn SNoLev x