Unique Isomorphism between Ordinal Subset and Unique Ordinal

Theorem
Let $\On$ be the class of all ordinals.

Let $S \subset \On$ where $S$ is a set.

Then there exists a unique mapping $\phi$ and a unique ordinal $x$ such that $\phi : x \to S$ is an order isomorphism.

Proof
Since $S \subset \On$, $\struct {S, \in}$ is a strict well-ordering.

The result follows directly from Strict Well-Ordering Isomorphic to Unique Ordinal under Unique Mapping.