const In : set set prop term iIn = In infix iIn 2000 2000 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 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 const Sep : set (set prop) set const SNoS_ : set set const SNoLev : set set term SNoL = \x:set.Sep (SNoS_ (SNoLev x)) \y:set.y < x term SNoR = \x:set.Sep (SNoS_ (SNoLev x)) (SNoLt x) term ordinal = \x:set.TransSet x & !y:set.y iIn x -> TransSet y axiom SNoS_I2: !x:set.!y:set.SNo x -> SNo y -> SNoLev x iIn SNoLev y -> x iIn SNoS_ (SNoLev y) 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 axiom SNoLev_ind: !p:set prop.(!x:set.SNo x -> (!y:set.y iIn SNoS_ (SNoLev x) -> p y) -> p x) -> !x:set.SNo x -> p x const minus_SNo : set set term - = minus_SNo const Repl : set (set set) set lemma !x:set.SNo x -> (!y:set.y iIn SNoS_ (SNoLev x) -> SNo - y & (!z:set.z iIn SNoL y -> - y < - z) & (!z:set.z iIn SNoR y -> - z < - y) & SNoCutP (Repl (SNoR y) minus_SNo) (Repl (SNoL y) minus_SNo)) -> (!y:set.y iIn SNoL x -> SNo - y & (!z:set.z iIn SNoL y -> - y < - z) & !z:set.z iIn SNoR y -> - z < - y) -> SNo - x & (!y:set.y iIn SNoL x -> - x < - y) & (!y:set.y iIn SNoR x -> - y < - x) & SNoCutP (Repl (SNoR x) minus_SNo) (Repl (SNoL x) minus_SNo) lemma !x:set.!y:set.SNo x -> (!z:set.z iIn SNoS_ (SNoLev x) -> SNo - z & (!w:set.w iIn SNoL z -> - z < - w) & (!w:set.w iIn SNoR z -> - w < - z) & SNoCutP (Repl (SNoR z) minus_SNo) (Repl (SNoL z) minus_SNo)) -> SNo y -> SNoLev y iIn SNoLev x -> y iIn SNoS_ (SNoLev x) -> SNo - y & (!z:set.z iIn SNoL y -> - y < - z) & !z:set.z iIn SNoR y -> - z < - y claim !x:set.SNo x -> SNo - x & (!y:set.y iIn SNoL x -> - x < - y) & (!y:set.y iIn SNoR x -> - y < - x) & SNoCutP (Repl (SNoR x) minus_SNo) (Repl (SNoL x) minus_SNo)