\begin{definition}[Definition:First-Order Property of Sets]
A '''first-order property of sets''' is a property defined by a logical formula whose domain is entirely over sets.
Thus the quantifiers $\forall x$ and $\exists x$ are valid when $x$ is a set, but $\forall A$ and $\exists A$ are ''not'' valid when $A$ is a class.
Hence we allow, for example:
:$\map \phi {A_1, A_2, \ldots, A_n, x}$
to be a logical formula whose variables $A_1, A_2, \ldots, A_n$ represent classes, and are all free, and $x$ is the only free variable representing a set.
\end{definition}