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 SNoLev : set set const minus_SNo : set set term - = minus_SNo axiom minus_SNo_Lev: !x:set.SNo x -> SNoLev - x = SNoLev x const SNoS_ : set set axiom SNoS_I2: !x:set.!y:set.SNo x -> SNo y -> SNoLev x iIn SNoLev y -> x iIn SNoS_ (SNoLev y) axiom FalseE: ~ False axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x const Subq : set set prop 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 ordinal : set prop axiom ordinal_In_Or_Subq: !x:set.!y:set.ordinal x -> ordinal y -> x iIn y | Subq y x const SNoCut : set set set const Repl : set (set set) set lemma !x:set.!y:set.!z:set.SNo x -> (!w:set.w iIn SNoS_ (SNoLev x) -> !u:set.!v:set.SNoCutP u v -> w = SNoCut u v -> - w = SNoCut (Repl v minus_SNo) (Repl u minus_SNo)) -> (!w:set.w iIn y -> SNo w) -> (!w:set.w iIn z -> SNo w) -> x = SNoCut y z -> SNoCutP (Repl z minus_SNo) (Repl y minus_SNo) -> 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) iIn SNoS_ (SNoLev x) var x:set var y:set var z:set hyp SNo x hyp !w:set.w iIn SNoS_ (SNoLev x) -> !u:set.!v:set.SNoCutP u v -> w = SNoCut u v -> - w = SNoCut (Repl v minus_SNo) (Repl u minus_SNo) hyp !w:set.w iIn y -> SNo w hyp !w:set.w iIn z -> SNo w hyp x = SNoCut y z hyp SNoCutP (Repl z minus_SNo) (Repl y minus_SNo) hyp SNo (SNoCut (Repl z minus_SNo) (Repl y minus_SNo)) hyp Subq (SNoLev (SNoCut (Repl z \w:set.- w) (Repl y \w:set.- w))) (SNoLev - x) hyp SNoEq_ (SNoLev (SNoCut (Repl z \w:set.- w) (Repl y \w:set.- w))) (SNoCut (Repl z \w:set.- w) (Repl y \w:set.- w)) - x hyp ordinal (SNoLev (SNoCut (Repl z minus_SNo) (Repl y minus_SNo))) claim ordinal (SNoLev - x) -> - x = SNoCut (Repl z minus_SNo) (Repl y minus_SNo)