Prime not Divisor implies Coprime

Theorem
Let $p, a \in \Z$.

If $p$ is a prime number then:
 * $p \nmid a \implies p \perp a$

where:
 * $p \nmid a$ denotes that $p$ does not divide $a$
 * $p \perp a$ denotes that $p$ and $a$ are coprime.

It follows directly that if $p$ and $q$ are primes, then:
 * $p \divides q \implies p = q$
 * $p \ne q \implies p \perp q$.