Sophie Germain Prime cannot be 6n+1

Theorem
Let $p$ be a Sophie Germain prime.

Then $p$ cannot be of the form $6 n + 1$, where $n$ is a positive integer.

Proof
Let $p$ be a Sophie Germain prime.

Then, by definition, $2 p + 1$ is prime.

$p = 6 n + 1$ for some $n \in \Z_{>0}$.

Then:

and so $2 p + 1$ is not prime.

The result follows by Proof by Contradiction.