const ordinal : set prop const In : set set prop term iIn = In infix iIn 2000 2000 const SNoS_ : set set const SNoLev : set set const SNo : set prop const SNo_ : set set prop axiom SNoS_E2: !x:set.ordinal x -> !y:set.y iIn SNoS_ x -> !P:prop.(SNoLev y iIn x -> ordinal (SNoLev y) -> SNo y -> SNo_ (SNoLev y) y -> P) -> P 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 lemma !r:set set prop.(!x:set.!y:set.SNo x -> SNo y -> (!z:set.z iIn SNoS_ (SNoLev x) -> r z y) -> (!z:set.z iIn SNoS_ (SNoLev y) -> r x z) -> (!z:set.z iIn SNoS_ (SNoLev x) -> !w:set.w iIn SNoS_ (SNoLev y) -> r z w) -> r x y) -> (!x:set.ordinal x -> !y:set.ordinal y -> !z:set.z iIn SNoS_ x -> !w:set.w iIn SNoS_ y -> r z w) -> !x:set.!y:set.SNo x -> SNo y -> r x y claim !r:set set prop.(!x:set.!y:set.SNo x -> SNo y -> (!z:set.z iIn SNoS_ (SNoLev x) -> r z y) -> (!z:set.z iIn SNoS_ (SNoLev y) -> r x z) -> (!z:set.z iIn SNoS_ (SNoLev x) -> !w:set.w iIn SNoS_ (SNoLev y) -> r z w) -> r x y) -> !x:set.!y:set.SNo x -> SNo y -> r x y