Proving $\forall n \in N, \exists p \in N$, $p>n$ $\wedge$ $(prime(p) \wedge p \equiv 2\pmod3)$

Question

I am trying to prove that $\forall n \in N, \exists p \in N$, $p>n$ $\wedge$ $(prime(p) \wedge p = 2mod3)$ using the definitions below:

I genuinely have no idea where to start and how to go about this proof.

I would really appreciate it if someone could explain to how prove the statement.