Integer Addition is Commutative/Proof 1

Theorem

The operation of addition on the set of integers $\Z$ is commutative:

$\forall x, y \in \Z: x + y = y + x$

Proof

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

From Integers under Addition form Abelian Group, the integers under addition form an abelian group, from which commutativity follows a fortiori.

$\blacksquare$