Class Member of Class Builder

Theorem
Let $A$ be a class.

Let $x$ be a set.

Let $\map P x$ be a well-formed formula in the language of set theory.

Let $\map P A$ denote the formula $\map P x$ with all free instances of $x$ replaced with instances of $A$.

Let $\set {x: \map P x}$ be a class specified using class builder notation.

Then:
 * $A \in \set {x: \map P x} \iff \paren {\exists x: x = A \land \map P A}$

Also see

 * Set Definition by Predicate