I would gladly use a notation for “$A$ is a finite subset of $B$”, like
$$A\sqsubset B \text{ or } A\underset{fin}{\subset} B,$$
but I have never seen a notation for that. Are there any?
EDIT: While waiting for a future standard, I will use Joffan’s $\ddot{\subset}$ coded as
$\newcommand{\finsub}[0]{\mathrel{\ddot{\subset}}}$
$A\finsub B$
$\newcommand{\finsub}[0]{\mathrel{\ddot{\subset}}}$
I will paste the new command in the first row, and then use \finsub, resulting in $A\finsub B$ which I will explain after first use in each text. I guess that is satisfying enough.
And really, as you define sets and functions in a text, you could as well define relations without standard notations.
Best Answer
The usual way is to use two different notations, one of which means that $A$ is finite, and the other means it's a subset of $B$. $$A\subset B,\qquad|A|<\infty.$$