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 Subsets Transitive, $X \subseteq S$, and by the definition of a well-ordered set, $X$ has a minimal element.

Therefore, any subset of $T$ has a minimal element.

Thus $T$ is itself well-ordered.