satisfies . For example, the class function mapping each set to its powerset may be expressed as the formula . The fact that the ordered pair satisfies may be expressed with the shorthand notation .

Another approach is taken by the von Neumann-Bernays-Gödel axioms (NBG); classes are the basic objects in this theory, and a set is then defined to be a class that is an element of some other class. However, the class existence axioms of NBG are restricted so that they only