Preliminaries (see e.g. Jech, Set Theory, p. 5): To every formula $\varphi(x)$ of ZF set theory corresponds a class $C = \lbrace x : \varphi(x)\rbrace$, but only to some formulas corresponds a set.
Every class is - by definition - definable, but not so every set.
Consider the following handwaving argument:
If the collection of definable sets (which is well-defined but not in first-order language) were a class $\Delta$, there would be a first-order formula $\delta(x)$ such that $\Delta = \lbrace x : \delta(x)\rbrace$.
If the collection of definable sets were a class, it would be a set.
So if it were a class, it would be a set that contains itself (because it is definable). But since there is no set that contains itself (by the axiom of regularity) the collection of definable sets cannot be a class, ergo: there is no formula expressing first-order definability.
Question: Can the emphasized claim - if the collection of definable sets were a class, it would be a set - be proved? E.g. in the line of "it is not too big to be a set (because there are only countably many formulas), so it is a set".