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 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 axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z const SNo_ : set set prop axiom SNoLev_prop: !x:set.SNo x -> ordinal (SNoLev x) & SNo_ (SNoLev x) x 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 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) -> (!y:set.y iIn SNoR x -> SNo - y & (!z:set.z iIn SNoL y -> - y < - z) & !z:set.z iIn SNoR y -> - z < - y) -> SNoCutP (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) lemma !x:set.!y:set.!z:set.SNo x -> (!w:set.w iIn SNoS_ (SNoLev x) -> SNo - w & (!u:set.u iIn SNoL w -> - w < - u) & (!u:set.u iIn SNoR w -> - u < - w) & SNoCutP (Repl (SNoR w) minus_SNo) (Repl (SNoL w) minus_SNo)) -> SNo y -> SNoLev y iIn SNoLev x -> x < y -> SNo z -> z < x -> SNo - z -> (!w:set.w iIn SNoR z -> - w < - z) -> SNo - y -> (!w:set.w iIn SNoL y -> - y < - w) -> z < y -> - y < - z var x:set hyp SNo x hyp !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) hyp !y:set.y iIn SNoL x -> SNo - y & (!z:set.z iIn SNoL y -> - y < - z) & !z:set.z iIn SNoR y -> - z < - y claim (!y:set.y iIn SNoR 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)