Absolute Value Function on Integers induces Equivalence Relation

Theorem
Let $\Z$ be the set of integers.

Let $\RR$ be the relation on $\Z$ defined as:
 * $\forall x, y \in \Z: \tuple {x, y} \in \RR \iff \size x = \size y$

where $\size x$ denotes the absolute value of $x$.

Then $\RR$ is an equivalence relation.

Proof
$\RR$ is shown to be an equivalence relation thus:

Reflexivity

 * $\forall x \in \Z: \size x = \size x$

Thus $\RR$ is reflexive.

Symmetry

 * $\forall x, y \in \Z: \size x = \size y \implies \size y = \size x$

Thus $\RR$ is symmetric.

Transitive

 * $\forall x, y, z \in \Z: \size x = \size y \land \size y = \size z \implies \size x = \size z$

Thus $\RR$ is transitive.

Thus, by definition, $\RR$ is an equivalence relation.