Quantifier: "For all sets" I've seen the following statement a few times:

"Let $A$ be a set, then $\emptyset\subseteq A$".
Or, written 'more formally':
  $$
\forall A\,\, \emptyset\subseteq A
$$

My doubt is: I've always seen the quantifier $\forall x$ to mean "for all $x$ elements of some set $S$". However, when talking about all the sets, how do we define this quantification? 

Without restriction $\forall A$ actually means "for all $A$", while with restriction like $\forall A\in B$ it means "for all $A$ that's a member of $B$".
The only counter intuitive thing here is that in formal set theory it's that everything is a set (ie $\forall A: A\mbox{ is a set}$).
Another thing to know is that the initial universal quantifier is normally omitted and only restrictions on the variables are mentioned. This is useful if you don't like formal set theory and don't accept that everything is a set. Then by this you can write only "if $A$ is a set, then $\emptyset\subseteq A$". If accepting that everything is a set you already accepted the premise and it could even formally be written as compact as $\emptyset\subseteq A$.
