As an addendum to Joel Hamkins answer: the weaker assertion (`Transitive Containment')
that every set x is contained in a transitive set 
(not necessarily its transitive closure) is not provable
in ZF - Replacement (sometimes known as Z Zermelo-set theory).

As Joel says in his answer we need to collect together the results of taking successive
$\bigcup$. For this $\Sigma_1$-Replacement is more than enough (if AxFoundation is formulated
in the right way for $\Pi_1$ classes).