Set has Rank

Theorem
If $S$ is a set, then $S$ has a rank.

Proof
Let $G$ be the smallest transitive set containing $S$ as a subset, which must exist by Set Contained in Smallest Transitive Set.

By Transitive Set Contained in Von Neumann Hierarchy Level, for some ordinal $i$, $G \subseteq V_i$.

Since the ordinals are well-ordered, Well-Ordering Determines Minimal Elements implies that there is a least such $i$, so $S$ has a rank.