TITLE: How to prove that a set exists in ZFC?
QUESTION [9 upvotes]: This is probably something really trivial, but I don't have any helpful set theory books (nor any library where I could borrow them for that matter) and googling such things as "proving a set exists in ZFC" doesn't give me any helpful links.
At first I thought that checking if a set I constructed doesn't contradict any axioms would be enough, but it would be really tiresome to check them all for any set I build and I have been wondering if there is any simpler method than this. 
Besides, I'm not even sure if it would be a correct technique considering that although the empty set doesn't contradict any axiom, people still needed to axiomatise its existence (I know that technically it can be deduced from other axioms, but many list it as an axiom). If not contradicting axioms is not yet a proof of existence, then what is?
EDIT: I see a lot of people thinking that I need a proof of existence of at least one set, i.e. that our universe of discourse is not empty. If the original post wasn't clear enoguh, my question is: "I have constructed a set, e.g. ${\{{\{v_0}\},{\{v_0,v_1}\}}\}$. How to prove that it exists? Just checking the axioms doesn't seem to be enough, since the axioms of existence and infinity explicitly say that the sets they describe exist, and if not contradicting axioms would be enough to justify a set's existence, then we could simply build the sets described by these axioms, see that they don't contradict anything and there would be no need to list these two axioms"

REPLY [2 votes]: In the usual form of the axioms of ZFC, only the Axiom of Infinity and the Axiom of Existence ($\exists x\;(x=x)$) begin with " $\exists$ ". All the other axioms begin " $\forall$ ". We cannot deduce a statement beginning with "$\exists$ " from a  collection  of axioms that all begin " $\forall$ " because it is a sub-system of the  system $\neg\exists x\;( S(x)$ for every formula $S(x)$. And this system is consistent, for the simple reason that it is satisfied if the universe of our discourse has nothing in it. It is desirable to separate Infinity from the other axioms, and not desirable to explore the insides of nothingness, so we add Existence.