Writing a simple expression in set notation I am trying to state that for every pair of even integers, the sum of the two integers is even.
$\forall m, n \in 2 \mathbb Z, \exists a = m + n \ni a \in 2 \mathbb Z$ 
Does this make sense?

While your expression is logically true, according to your statement, it's better to say:
$$\forall m, n \in 2 \mathbb Z, m + n \in 2 \mathbb Z$$
