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:

$\gcd \set {a, b} = 1$

Aiming for a contradiction, suppose $a = b = 0$.

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 $a = b = 0$.

$\blacksquare$