Suppose I have a family of sets $\mathcal{A}$.
Suppose that $x$ is contained in some set in $\mathcal{A}$.
Is there a common notation/convention for this relation? IE, $x \in_{\in} \mathcal{A}$?
I suppose the normal way to say that is something along the lines of "$\{x\}$ is covered by $\mathcal{A}$", but that's a little weird.
There is no single symbol that is commonly used to denote this. However, one slick way you could write it is $x\in\bigcup\mathcal{A}$. Here $\bigcup\mathcal{A}$ denotes the union of all the elements of $\mathcal{A}$, so $x\in\bigcup\mathcal{A}$ means exactly that $x$ is an element of some element of $\mathcal{A}$.