As the title suggest, I'm trying to express that there is no (e.g., $x$), or if there is any, there will be one and only one. Is the following a formal/valid writing?
$$\nexists \lor \exists! x \quad \text{bla bla}$$
If no, what would be the proper way of stating this?
If we want to say "There exists either exactly one, or no $x$ such that $P(x)$", then I think the most elegant way to say it is to rethink a little how we say what we want to say. I propose $$ \forall x, y\,(P(x)\land P(y)\to x = y) $$ The way to read this is that if you ever find yourself in a situation where you have two such elements, they are really one and the same.