Definition:First-Order Property of Sets

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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.