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 ordinal : set prop const SNoLev : set set axiom SNoLev_ordinal: !x:set.SNo x -> ordinal (SNoLev x) axiom ordinal_ind: !p:set prop.(!x:set.ordinal x -> (!y:set.y iIn x -> p y) -> p x) -> !x:set.ordinal x -> p x const SNoCut : set set set lemma !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.ordinal x -> !y:set.SNo y -> SNoLev y iIn x -> p y) -> !x:set.SNo x -> p x lemma !p:set prop.!x:set.!y:set.(!z:set.!w:set.SNoCutP z w -> (!u:set.u iIn z -> p u) -> (!u:set.u iIn w -> p u) -> p (SNoCut z w)) -> (!z:set.z iIn x -> !w:set.SNo w -> SNoLev w iIn z -> p w) -> SNo y -> SNoLev y iIn x -> ordinal (SNoLev y) -> p y claim !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