TITLE: Notation showing a set is non-empty and finite
QUESTION [7 upvotes]: Does there exist any mathematical notation that would indicate that a set $V$ is non-empty and finite? Or would I have to write this out in words?

REPLY [8 votes]: To say that $V$ is not empty you can either say so or write $V\neq\emptyset$
or $|V|>0$. 
To say that $V$ is finite you can either say so or write $|V|<\aleph_{0}$. 
So you can write something like $0<|V|<\aleph_{0}$ to say that
$V$ is a non-empty finite set.
Added: in many context (mainly non set theory wise where just writing infinity is not common) you can replace $\aleph_{0}$ with $\infty$