What is the best way to denote a set of solutions in a professional setting?
It needs to be (5) because (3), (4) imply $(x,\,y)$ is equal to its set of possible values rather than an element of that set.
I would probably write
$n,m\in\mathbb N:n,m\ge2$ (equivalently $\mathbb Z$),
$(n,m)=(5,4),(5,5),(1,9)$.
If you want a truly "professional" way, such as in a math review, you need to adhere to the conventions of that particular publication.