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 $i$, $G \subseteq V_i$, so $G \in V_{i+1}$.

Since the ordinals are well-ordered, Well-Ordering Determines Minimal Elements implies that there is a smallest ordinal $k$ such that $G \in V_{k+1}$, so $G$ has rank $k$.