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 const Repl : set (set set) set axiom ReplI: !x:set.!f:set set.!y:set.y iIn x -> f y iIn Repl x f 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 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 minus_SNo : set set term - = minus_SNo const SNoCut : set set set axiom minus_SNo_eq: !x:set.SNo x -> - x = SNoCut (Repl (SNoR x) minus_SNo) (Repl (SNoL x) minus_SNo) lemma !x:set.!y:set.SNoCutP (Repl (SNoR x) minus_SNo) (Repl (SNoL x) minus_SNo) -> y iIn Sep (SNoS_ (SNoLev x)) (\z:set.z < x) -> - y iIn Repl (SNoL x) minus_SNo -> SNoCut (Repl (SNoR x) minus_SNo) (Repl (SNoL x) minus_SNo) < - y lemma !x:set.!y:set.SNoCutP (Repl (SNoR x) minus_SNo) (Repl (SNoL x) minus_SNo) -> y iIn Sep (SNoS_ (SNoLev x)) (SNoLt x) -> - y iIn Repl (SNoR x) minus_SNo -> - y < SNoCut (Repl (SNoR x) minus_SNo) (Repl (SNoL x) minus_SNo) var x:set hyp SNo x hyp SNoCutP (Repl (SNoR x) minus_SNo) (Repl (SNoL x) minus_SNo) claim SNo (SNoCut (Repl (SNoR x) minus_SNo) (Repl (SNoL x) minus_SNo)) -> 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)