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 SepI: !x:set.!p:set prop.!y:set.y iIn x -> p y -> y iIn Sep x p axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z const minus_SNo : set set term - = minus_SNo const Repl : set (set set) set var x:set var y:set var z:set var w:set hyp !u:set.u iIn SNoS_ (SNoLev x) -> SNo - u & (!v:set.v iIn SNoL u -> - u < - v) & (!v:set.v iIn SNoR u -> - v < - u) & SNoCutP (Repl (SNoR u) minus_SNo) (Repl (SNoL u) minus_SNo) hyp SNo y hyp SNo z hyp SNo - z hyp !u:set.u iIn SNoR z -> - u < - z hyp SNo - y hyp !u:set.u iIn SNoL y -> - y < - u hyp SNo w hyp z < w hyp w < y hyp SNoLev w iIn SNoLev z hyp SNoLev w iIn SNoLev y hyp w iIn SNoS_ (SNoLev x) claim SNo - w -> - y < - z