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 nIn = \x:set.\y:set.~ x iIn y const minus_SNo : set set term - = minus_SNo axiom minus_SNo_invol: !x:set.SNo x -> - - x = x const Repl : set (set set) set axiom Repl_invol_eq: !p:set prop.!f:set set.(!x:set.p x -> f (f x) = x) -> !x:set.(!y:set.y iIn x -> p y) -> Repl (Repl x f) f = x const SNoCut : set set set const SNoLev : set set lemma !x:set.!y:set.!z:set.SNo x -> (!w:set.w iIn z -> SNo w) -> x = SNoCut y z -> SNo (SNoCut (Repl z minus_SNo) (Repl y minus_SNo)) -> SNoLev (SNoCut (Repl z \w:set.- w) (Repl y \w:set.- w)) iIn SNoLev - x -> - SNoCut (Repl z minus_SNo) (Repl y minus_SNo) = SNoCut (Repl (Repl y minus_SNo) minus_SNo) (Repl (Repl z minus_SNo) minus_SNo) -> Repl (Repl y minus_SNo) minus_SNo = y -> Repl (Repl z minus_SNo) minus_SNo != z var x:set var y:set var z:set hyp SNo x hyp !w:set.w iIn y -> SNo w hyp !w:set.w iIn z -> SNo w hyp x = SNoCut y z hyp SNo (SNoCut (Repl z minus_SNo) (Repl y minus_SNo)) hyp SNoLev (SNoCut (Repl z \w:set.- w) (Repl y \w:set.- w)) iIn SNoLev - x hyp - SNoCut (Repl z minus_SNo) (Repl y minus_SNo) = SNoCut (Repl (Repl y minus_SNo) minus_SNo) (Repl (Repl z minus_SNo) minus_SNo) claim Repl (Repl y minus_SNo) minus_SNo != y