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 const Subq : set set prop const SNoLev : set set const minus_SNo : set set term - = minus_SNo axiom minus_SNo_Lev_lem2: !x:set.SNo x -> Subq (SNoLev - x) (SNoLev x) axiom Subq_tra: !x:set.!y:set.!z:set.Subq x y -> Subq y z -> Subq x z axiom set_ext: !x:set.!y:set.Subq x y -> Subq y x -> x = y const SNoEq_ : set set set prop axiom SNo_eq: !x:set.!y:set.SNo x -> SNo y -> SNoLev x = SNoLev y -> SNoEq_ (SNoLev x) x y -> x = y const SNoCut : set set set var x:set var y:set hyp SNoCutP x y hyp !z:set.z iIn x -> - - z = z hyp !z:set.z iIn y -> - - z = z hyp !z:set.z iIn x -> SNo z hyp !z:set.z iIn y -> SNo z hyp SNo (SNoCut x y) hyp SNo - SNoCut x y hyp SNo - - SNoCut x y claim Subq (SNoLev (SNoCut x y)) (SNoLev - - SNoCut x y) & SNoEq_ (SNoLev (SNoCut x y)) (SNoCut x y) - - SNoCut x y -> - - SNoCut x y = SNoCut x y