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.