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 SNo_minus_SNo: !x:set.SNo x -> SNo - x const ordinal : set prop const SNoLev : set set axiom SNoLev_ordinal: !x:set.SNo x -> ordinal (SNoLev x) const SNoS_ : set set const SNoCut : set set set const Repl : set (set set) set const Subq : set set prop const SNoEq_ : set set set prop 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)) -> Subq (SNoLev (SNoCut (Repl z \w:set.- w) (Repl y \w:set.- w))) (SNoLev - x) -> 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 -> ordinal (SNoLev (SNoCut (Repl z minus_SNo) (Repl y minus_SNo))) -> - x = SNoCut (Repl z minus_SNo) (Repl y minus_SNo) 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 SNoCutP y z 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 !w:set.SNo w -> (!u:set.u iIn Repl z minus_SNo -> u < w) -> (!u:set.u iIn Repl y minus_SNo -> w < u) -> Subq (SNoLev (SNoCut (Repl z minus_SNo) (Repl y minus_SNo))) (SNoLev w) & SNoEq_ (SNoLev (SNoCut (Repl z minus_SNo) (Repl y minus_SNo))) (SNoCut (Repl z minus_SNo) (Repl y minus_SNo)) w hyp !w:set.w iIn Repl z minus_SNo -> w < - x claim (!w:set.w iIn Repl y minus_SNo -> - x < w) -> - x = SNoCut (Repl z minus_SNo) (Repl y minus_SNo)