Existence of Greatest Common Divisor

Theorem
$\forall a, b \in \Z: a \ne 0 \lor b \ne 0$, there exists a largest $d \in \Z_{>0}$ such that $d \mathrel \backslash a$ and $d \mathrel \backslash b$.

The greatest common divisor of $a$ and $b$ always exists.

Proof
Proof of its existence:

$\forall a, b \in \Z: 1 \mathrel \backslash a \land 1 \mathrel \backslash b$ so $1$ is always a common divisor of any two integers.

Proof of there being a largest:

As the definition of $\gcd$ shows that it is symmetric, we can assume without loss of generality that $a \ne 0$.

First we note that:
 * $\forall c \in \Z: \forall a \in \Z_{>0}: c \mathrel \backslash a \implies c \le \left|{c}\right| \le \left|{a}\right|$

... from Integer Absolute Value Greater than Divisors.

The same applies for $c \mathrel \backslash b$.

Now we have three different results depending on $a$ and $b$:

So if $a$ and $b$ are both zero, then any $n \in \Z$ divides both, and there is no greatest common divisor. This is why the proviso that $a \ne 0 \lor b \ne 0$.

So we have proved that common divisors exist and are bounded above. Therefore, from Set of Integers Bounded Above by Integer has Greatest Element there is always a greatest common divisor.