Reflexive and Transitive Relation is not necessarily Symmetric/Proof 2

Proof
Proof by Counterexample:

Let $S = \Z$ be the set of integers.

Let $\alpha$ be the relation on $S$ defined as:
 * $\forall x, y \in S: x \mathrel \alpha y \iff x \le y$

It is seen that:
 * $\forall x \in \Z: x \le x$

and so:
 * $\forall x \in \Z: x \mathrel \alpha x$

Thus $\alpha$ is reflexive.

Then it is seen that:


 * $\forall x, y, z \in \Z: x \le y, y \le z \implies x \le z$

Thus $\alpha$ is transitive.

Now let $x = 1$ and $y = 2$.

Then:
 * $x \le y$ but it is not the case that $y \le x$

and so $\alpha$ is not symmetric.

Hence $\alpha$ is both reflexive and transitive but not symmetric.