TITLE: Is there a notation for being "a finite subset of"?
QUESTION [13 upvotes]: 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.

REPLY [12 votes]: In set theory there are two standard notations for the set of finite subsets of $X$:

$[X]^{<\omega}$,
$\mathcal P_\omega(X)$ or $\mathcal P_{\aleph_0}(X)$.

In naive set theory, you can also find $\operatorname{Fin}(X)$ quite often.
So if you'd want to write that $A$ is a finite subset of $B$, you could say that $A\in[B]^{<\omega}$ or $A\in\mathcal P_\omega(B)$ or $A\in\operatorname{Fin}(B)$.
Outside of set theory, I believe writing $|A|<\infty$ is probably one of the most accepted ways of writing that $A$ is finite. But I'm sure that in some fields of mathematics there are more or less common notations, and you should probably align yourself to the crowd which will read your work. 
If you are writing for yourself, then it really doesn't matter what you're using, right?
In any case, do remember the standard advice about notation:

Notation is used to reduce clutter, not to abbreviate. Define your notations and be consistent with them. Don't force the reader to keep track of your symbols, if there's no need to do so.