 In particular, one can identify the "class of all sets" with the set of all predicates equivalent to formula_10

Because classes do not have any formal status in the theory of ZF, the axioms of ZF do not immediately apply to classes.However, if an inaccessible cardinal formula_11 is assumed, then the sets of smaller rank form a model of ZF (a Grothendieck universe), and its subsets can be thought of as "classes".In ZF, the concept of a function can also be generalised to classes.