Coprime Integers cannot Both be Zero

Theorem
Let $a$ and $b$ be integers.

Let $a$ and $b$ be coprime.

Then it cannot be the case that $a = b = 0$.

Proof
Let $a$ and $b$ be coprime.

Then by definition:
 * $\gcd \set {a, b} = 1$

$a = b = 0$.

Then $\gcd \set {a, b}$ is undefined.

But it is not possible both:
 * for $\gcd \set {a, b}$ to be undefined
 * for $\gcd \set {a, b} = 1$.

Hence by Proof by Contradiction it follows that it cannot be the case that $a = b = 0$.