TITLE: first-order definability transitive closure operator
QUESTION [8 upvotes]: I know this sounds dumb, but I can't for the life of me remember how to expand "TC(x)" into a first-order term in the language of set theory (ZFC, not NBG) where epsilon is the only nonlogical symbol.
The obvious definition is an $\omega$-long sentence $x\cup (\bigcup x)\cup (\bigcup\bigcup x)...$, but that isn't in $L_{\omega\omega}$.
The definition given in Jech, p64 appeals to "the intersection of any class with a set is a set" (p8), which is really expressible only in NBG, right?  I'm at a loss to figure out how to turn this into simple ZFC using separation and replacement.
I also don't have much trouble proving that for every set there exists some other set which is its transitive closure; I just can't seem to turn this proof of $(\exists y)\phi$ (for $\phi$ being "y is the transitive closure of x") into an explicit description of the $y$.
I'm starting to suspect that TC(x) isn't definable in ZFC, but that it can be defined as a class-function in NBG (which is a conservative extension of ZFC, so being able to define TC(x) doesn't actually get you any new theorems about sets).
Thanks!

REPLY [6 votes]: 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).
As a second addendum (as the discussion continues) one should remark that we have in full ZF the $\in$-recursion theorem: thus we may define the class function $x\rightarrow TC(x)$  within ZFC (no need for NBG) via the recursion scheme
                  $$TC(x)= \bigcup [TC(y) : y\in x] \cup x $$.
The function defined has just the same status as $\alpha \rightarrow V_\alpha$ (which is also defined by such a recursion scheme). The former guarantees the existence of $TC(x)$ for any $x$, the latter of $V_\alpha$ for any $\alpha$. Neither function is problematic for the full ZF-set theorist. 
The original question seems to be spurred on by the fact that we do not have an obvious {\em explicit  definition} for $TC(x)$ in the way that we do for say ordered pairs. However that is often the case: turning recursive definitions into explicit definitions
may not be possible.