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 SNo : set prop const SNoS_ : set set const ordsucc : set set const SNoLev : set set axiom SNoS_SNoLev: !x:set.SNo x -> x iIn SNoS_ (ordsucc (SNoLev x)) const SNoR : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoR_E: !x:set.SNo x -> !y:set.y iIn SNoR x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> x < y -> P) -> P const Repl : set (set set) set axiom ReplE_impred: !x:set.!f:set set.!y:set.y iIn Repl x f -> !P:prop.(!z:set.z iIn x -> y = f z -> P) -> P const famunion : set (set set) set axiom famunionE: !x:set.!f:set set.!y:set.y iIn famunion x f -> ?z:set.z iIn x & y iIn f z const SNoL : set set axiom SNoL_E: !x:set.SNo x -> !y:set.y iIn SNoL x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> y < x -> P) -> P const binunion : set set set axiom binunionE: !x:set.!y:set.!z:set.z iIn binunion x y -> z iIn x | z iIn y const minus_SNo : set set term - = minus_SNo lemma !x:set.!y:set.!z:set.!w:set.!u:set.TransSet x -> (!v:set.v iIn x -> !x2:set.x2 iIn SNoS_ v -> Subq (SNoLev - x2) (SNoLev x2)) -> SNoLev y iIn x -> ordinal (SNoLev y) -> z iIn ordsucc (SNoLev w) -> w = - u -> SNo u -> SNoLev u iIn SNoLev y -> u iIn SNoS_ (ordsucc (SNoLev u)) -> z iIn SNoLev y lemma !x:set.!y:set.!z:set.!w:set.!u:set.TransSet x -> (!v:set.v iIn x -> !x2:set.x2 iIn SNoS_ v -> Subq (SNoLev - x2) (SNoLev x2)) -> SNoLev y iIn x -> ordinal (SNoLev y) -> z iIn ordsucc (SNoLev w) -> w = - u -> SNo u -> SNoLev u iIn SNoLev y -> u iIn SNoS_ (ordsucc (SNoLev u)) -> z iIn SNoLev y const SNoCut : set set set var x:set var y:set var z:set hyp TransSet x hyp !w:set.w iIn x -> !u:set.u iIn SNoS_ w -> Subq (SNoLev - u) (SNoLev u) 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) \w:set.ordsucc (SNoLev w)) (famunion (Repl (SNoL y) minus_SNo) \w:set.ordsucc (SNoLev w))) hyp TransSet (binunion (famunion (Repl (SNoR y) minus_SNo) \w:set.ordsucc (SNoLev w)) (famunion (Repl (SNoL y) minus_SNo) \w:set.ordsucc (SNoLev w))) hyp z iIn SNoLev (SNoCut (Repl (SNoR y) minus_SNo) (Repl (SNoL y) minus_SNo)) claim z iIn binunion (famunion (Repl (SNoR y) minus_SNo) \w:set.ordsucc (SNoLev w)) (famunion (Repl (SNoL y) minus_SNo) \w:set.ordsucc (SNoLev w)) -> z iIn SNoLev y