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?

The symbol $\exists$ is redundant. You could also state: 
$$\forall m,n\in 2\mathbb{Z} \ \ and \ \ k=m+n, \ \ k\in2\mathbb{Z}$$
