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 SNoS_ : set set const SNoLev : set set const minus_SNo : set set term - = minus_SNo const SNo : set prop const SNoCut : set set set const Repl : set (set set) set const SNoR : set set const SNoL : set set const ordsucc : set set const binunion : set set set const famunion : set (set set) set lemma !x:set.!y:set.TransSet x -> (!z:set.z iIn x -> !w:set.w iIn SNoS_ z -> Subq (SNoLev - w) (SNoLev w)) -> SNoLev y iIn x -> ordinal (SNoLev y) -> SNo y -> SNoLev (SNoCut (Repl (SNoR y) minus_SNo) (Repl (SNoL y) minus_SNo)) iIn ordsucc (binunion (famunion (Repl (SNoR y) minus_SNo) \z:set.ordsucc (SNoLev z)) (famunion (Repl (SNoL y) minus_SNo) \z:set.ordsucc (SNoLev z))) -> ordinal (binunion (famunion (Repl (SNoR y) minus_SNo) \z:set.ordsucc (SNoLev z)) (famunion (Repl (SNoL y) minus_SNo) \z:set.ordsucc (SNoLev z))) -> TransSet (binunion (famunion (Repl (SNoR y) minus_SNo) \z:set.ordsucc (SNoLev z)) (famunion (Repl (SNoL y) minus_SNo) \z:set.ordsucc (SNoLev z))) -> !z:set.z iIn SNoLev (SNoCut (Repl (SNoR y) minus_SNo) (Repl (SNoL y) minus_SNo)) -> z iIn SNoLev y var x:set var y:set hyp TransSet x hyp !z:set.z iIn x -> !w:set.w iIn SNoS_ z -> Subq (SNoLev - w) (SNoLev w) hyp SNoLev y iIn x hyp ordinal (SNoLev y) hyp SNo y hyp SNoLev (SNoCut (Repl (SNoR y) minus_SNo) (Repl (SNoL y) minus_SNo)) iIn ordsucc (binunion (famunion (Repl (SNoR y) minus_SNo) \z:set.ordsucc (SNoLev z)) (famunion (Repl (SNoL y) minus_SNo) \z:set.ordsucc (SNoLev z))) claim ordinal (binunion (famunion (Repl (SNoR y) minus_SNo) \z:set.ordsucc (SNoLev z)) (famunion (Repl (SNoL y) minus_SNo) \z:set.ordsucc (SNoLev z))) -> !z:set.z iIn SNoLev (SNoCut (Repl (SNoR y) minus_SNo) (Repl (SNoL y) minus_SNo)) -> z iIn SNoLev y