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 const SNo : set prop const SNoLt : set set prop term < = SNoLt infix < 2020 2020 term SNoCutP = \x:set.\y:set.(!z:set.z iIn x -> SNo z) & (!z:set.z iIn y -> SNo z) & !z:set.z iIn x -> !w:set.w iIn y -> z < w const TransSet : set prop const ordsucc : set set axiom TransSet_In_ordsucc_Subq: !x:set.!y:set.TransSet y -> x iIn ordsucc y -> Subq x y const Sep : set (set prop) set const SNoS_ : set set const omega : set const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const Empty : set const SNoLev : set set const SNoCut : set set set const binunion : set set set const famunion : set (set set) set lemma !x:set.(!y:set.y iIn Sep (SNoS_ omega) (\z:set.x * z < ordsucc Empty) -> !P:prop.(SNo y -> SNoLev y iIn omega -> x * y < ordsucc Empty -> P) -> P) -> (!y:set.y iIn Sep (SNoS_ omega) (\z:set.ordsucc Empty < x * z) -> !P:prop.(SNo y -> SNoLev y iIn omega -> ordsucc Empty < x * y -> P) -> P) -> SNoLev (SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y)) iIn ordsucc (binunion (famunion (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) \y:set.ordsucc (SNoLev y)) (famunion (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y) \y:set.ordsucc (SNoLev y))) -> omega iIn SNoLev (SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y)) -> TransSet (binunion (famunion (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) \y:set.ordsucc (SNoLev y)) (famunion (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y) \y:set.ordsucc (SNoLev y))) -> ~ omega iIn binunion (famunion (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) \y:set.ordsucc (SNoLev y)) (famunion (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y) \y:set.ordsucc (SNoLev y)) var x:set hyp !y:set.y iIn Sep (SNoS_ omega) (\z:set.x * z < ordsucc Empty) -> !P:prop.(SNo y -> SNoLev y iIn omega -> x * y < ordsucc Empty -> P) -> P hyp !y:set.y iIn Sep (SNoS_ omega) (\z:set.ordsucc Empty < x * z) -> !P:prop.(SNo y -> SNoLev y iIn omega -> ordsucc Empty < x * y -> P) -> P hyp SNoLev (SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y)) iIn ordsucc (binunion (famunion (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) \y:set.ordsucc (SNoLev y)) (famunion (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y) \y:set.ordsucc (SNoLev y))) hyp omega iIn SNoLev (SNoCut (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y)) claim ~ TransSet (binunion (famunion (Sep (SNoS_ omega) \y:set.x * y < ordsucc Empty) \y:set.ordsucc (SNoLev y)) (famunion (Sep (SNoS_ omega) \y:set.ordsucc Empty < x * y) \y:set.ordsucc (SNoLev y)))