Rank is Ordinal

Theorem
Let $S$ be a small class

The rank of $S$ is an ordinal.

Proof
The rank of $S$ is an intersection of a set of ordinals $B$.

$B$ is non-empty by the fact that Set has Rank.

Thus, $B$ has a minimal element, which is the rank of $S$ plus $1$.

Therefore, the rank is itself an ordinal.