Subset of Well-Ordered Set is Well-Ordered

Theorem
Every non-empty subset of a well-ordered set is itself well-ordered.

Proof
Let $\left({S, \preceq}\right)$ be a well-ordered set.

Let $T \subseteq S$.

Let $X \subseteq T$.

By Subset Relation is Transitive, $X \subseteq S$.

By the definition of a well-ordered set, $X$ has a smallest element.

It follows by definition that $T$ is well-ordered.

Hence the result.