Countable Set is Well-Orderable

Theorem
Let $S$ be a countable set.

Then $S$ is well-orderable.

Proof
By the Well-Ordering Principle, the set of natural numbers $\N$ under the usual ordering $\le$ forms a well-ordered set.

By definition of countable set, there exists an injection:
 * $f: S \to \N$

Let $V$ be a basic universe.

By definition of basic universe:
 * $S \in V$

and:
 * $\N \in V$

By the Axiom of Transitivity, both $S$ and $\N$ are classes.

From Class which has Injection to Subclass of Well-Orderable Class is Well-Orderable, it follows that $S$ is well-orderable.

Hence the result.