Ordering of Integers is Reversed by Negation

Theorem
Let $x, y \in \Z$ such that $x > y$.

Then:
 * $-x < -y$

Proof
From the formal definition of integers, $\eqclass {a, b} {}$ is an equivalence class of ordered pairs of natural numbers.

Let $x = \eqclass {a, b} {}$ and $y = \eqclass {c, d} {}$ for some $x, y \in \Z$.

We have