const In : set set prop term iIn = In infix iIn 2000 2000 term nIn = \x:set.\y:set.~ x iIn y const SNoCutP : set set prop const Empty : set axiom SNoCutP_0_0: SNoCutP Empty Empty const ordsucc : set set axiom cases_1: !x:set.x iIn ordsucc Empty -> !p:set prop.p Empty -> p x const binunion : set set set axiom binunion_idl: !x:set.binunion Empty x = x const famunion : set (set set) set axiom famunion_Empty: !f:set set.famunion Empty f = Empty const SNo : set prop const SNoCut : set set set const SNoLev : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const Subq : set set prop const SNoEq_ : set set set prop axiom SNoCutP_SNoCut_impred: !x:set.!y:set.SNoCutP x y -> !P:prop.(SNo (SNoCut x y) -> SNoLev (SNoCut x y) iIn ordsucc (binunion (famunion x \z:set.ordsucc (SNoLev z)) (famunion y \z:set.ordsucc (SNoLev z))) -> (!z:set.z iIn x -> z < SNoCut x y) -> (!z:set.z iIn y -> SNoCut x y < z) -> (!z:set.SNo z -> (!w:set.w iIn x -> w < z) -> (!w:set.w iIn y -> z < w) -> Subq (SNoLev (SNoCut x y)) (SNoLev z) & SNoEq_ (SNoLev (SNoCut x y)) (SNoCut x y) z) -> P) -> P lemma SNo (SNoCut Empty Empty) -> SNoLev (SNoCut Empty Empty) iIn ordsucc Empty -> SNoLev (SNoCut Empty Empty) = Empty -> SNoCut Empty Empty = Empty claim SNoCut Empty Empty = Empty