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 !P:set set set prop.(!x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> (!w:set.w iIn SNoS_ (SNoLev x) -> P w y z) -> (!w:set.w iIn SNoS_ (SNoLev y) -> P x w z) -> (!w:set.w iIn SNoS_ (SNoLev z) -> P x y w) -> (!w:set.w iIn SNoS_ (SNoLev x) -> !u:set.u iIn SNoS_ (SNoLev y) -> P w u z) -> (!w:set.w iIn SNoS_ (SNoLev x) -> !u:set.u iIn SNoS_ (SNoLev z) -> P w y u) -> (!w:set.w iIn SNoS_ (SNoLev y) -> !u:set.u iIn SNoS_ (SNoLev z) -> P x w u) -> (!w:set.w iIn SNoS_ (SNoLev x) -> !u:set.u iIn SNoS_ (SNoLev y) -> !v:set.v iIn SNoS_ (SNoLev z) -> P w u v) -> P x y z) -> (!x:set.ordinal x -> !y:set.ordinal y -> !z:set.ordinal z -> !w:set.w iIn SNoS_ x -> !u:set.u iIn SNoS_ y -> !v:set.v iIn SNoS_ z -> P w u v) -> !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> P x y z claim !P:set set set prop.(!x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> (!w:set.w iIn SNoS_ (SNoLev x) -> P w y z) -> (!w:set.w iIn SNoS_ (SNoLev y) -> P x w z) -> (!w:set.w iIn SNoS_ (SNoLev z) -> P x y w) -> (!w:set.w iIn SNoS_ (SNoLev x) -> !u:set.u iIn SNoS_ (SNoLev y) -> P w u z) -> (!w:set.w iIn SNoS_ (SNoLev x) -> !u:set.u iIn SNoS_ (SNoLev z) -> P w y u) -> (!w:set.w iIn SNoS_ (SNoLev y) -> !u:set.u iIn SNoS_ (SNoLev z) -> P x w u) -> (!w:set.w iIn SNoS_ (SNoLev x) -> !u:set.u iIn SNoS_ (SNoLev y) -> !v:set.v iIn SNoS_ (SNoLev z) -> P w u v) -> P x y z) -> !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> P x y z