Is there some compact notation for saying "all the elements of a set"?
Just trying to save time and be concise really when:
$\forall x\in S:P(x)$ is a bit long-winded to some potential alternative:
Sometimes people define $P(S)$ to mean $\forall x\in S, P(x)$. If you explicitly define it this way, then you can use it this way.