Subset of Well-Ordered Set is Well-Ordered

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

Proof
Let $S$ be a well-ordered set.

Let $T \subseteq S$. We want to show that $T$ is well-ordered.

So let $X \subseteq T$ be arbitrary.

Because the subset relation is transitive, it follows that $X \subseteq S$; if $X$ is non-empty, then $X$ has a minimal element because $S$ is well-ordered.

That is, $T$ is well-ordered.