Prime not Divisor implies Coprime/Proof 2

Proof
Let $p$ be a prime number.

Let $a \in \Z$ be such that $p$ is not a divisor of $a$.

$p$ and $a$ are not coprime.

Then:
 * $\exists c \in \Z_{>1}: c \mathrel \backslash p, c \mathrel \backslash a$

where $\backslash$ denotes divisibility.

But then by definition of prime:
 * $c = p$

Thus:
 * $p \mathrel \backslash a$

The result follows by Proof by Contradiction.