How do I prove the existence of infinite union in ZFC?

Given an infinite set of sets A - how can I prove in ZFC that the union of all the elements of A exists?

-
See Wikipedia: Axiom of union. – Martin Sleziak Aug 6 '12 at 9:57

The axiom states that if $A$ is a set, then there exists a set $B$ such that $B=\bigcup A$, that is to say
For every $x$, $x\in B$ if and only if there exists $y$ such that $y\in A$ and $x\in y$.