Set of Integers can be Well-Ordered

Theorem
The set of integers $\Z$ can be well-ordered with an appropriately chosen ordering.

Proof
Consider the ordering $\preccurlyeq \subseteq \Z \times \Z$ defined as:
 * $x \preccurlyeq y \iff \left({\left\vert{x}\right\vert < \left\vert{y}\right\vert}\right) \lor \left({\left\vert{x}\right\vert = \left\vert{y}\right\vert \land x \le y}\right)$