Subset of Well-Ordered Set is Well-Ordered

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Theorem

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


Proof

Let $\struct {S, \preceq}$ 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.

$\blacksquare$


Sources