Greatest Integer such that all Coprime and Less are Prime

Theorem
$30$ is the greatest integer such that all positive integers less than and coprime to it, excluding $1$, are prime.

Proof
First we demonstrate that $30$ has this property.

From Euler Phi Function of 30:
 * $\map \phi {30} = 8$

and by inspection, these numbers can be enumerated as:
 * $1, 7, 11, 13, 17, 19, 23, 29$

all of which, except $1$, can be seen to be prime.

It remains to be shown that this is the largest integer with this property.