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 SNoCut : set set set axiom SNoCutP_SNo_SNoCut: !x:set.!y:set.SNoCutP x y -> SNo (SNoCut x y) axiom SNo_ind: !p:set prop.(!x:set.!y:set.SNoCutP x y -> (!z:set.z iIn x -> p z) -> (!z:set.z iIn y -> p z) -> p (SNoCut x y)) -> !x:set.SNo x -> p x const minus_SNo : set set term - = minus_SNo lemma !x:set.!y:set.SNoCutP x y -> (!z:set.z iIn x -> - - z = z) -> (!z:set.z iIn y -> - - z = z) -> (!z:set.z iIn x -> SNo z) -> (!z:set.z iIn y -> SNo z) -> SNo (SNoCut x y) -> - - SNoCut x y = SNoCut x y claim !x:set.SNo x -> - - x = x