Strict Well-Ordering Isomorphic to Unique Ordinal under Unique Mapping

Theorem
Let $S$ be a set.

Let $\left({S, \prec}\right)$ be a strict well-ordering.

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

Proof
The existence of $x$ and $f$ follows from Woset Isomorphic to Unique Ordinal.

The uniqueness of $x$ follows from Woset Isomorphic to Unique Ordinal.

The uniqueness of $f$ follows from Order Isomorphism Between Wosets is Unique.