Can classes in ZFC be identified with logical predicates A(x)? I was wondering whether any class in ZFC can be identified with a logical predicate A(x) about sets.
Is that true?
I have read in Wikipedia that 'class' is an informal notion in ZFC. In that case, since category theory uses 'classes' from the very beginning, is category theory something informal?
 A: The fact that in ZFC everything is a set and hence it cannot "talk" about classes, dos not make the notion informal. For one there are alternative set theories that also formalize classes. And on the other hand, yes, you may identify classes with predicates. Thus the class of objects of Set is simply the class of all sets, i.e., to test if $x$ is an object of Set we test whether $x=x$ holds or the like.
A: Some, but not all classes are usually defined as definite conditions (the set theorist's talk for what you call 'predicates'). This definition yields an unrestricted comprehension schema for classes

For every definite condition $P(x)$ there is a class A s.t. $\forall y(y\in A \Leftrightarrow P(y))$ 

To see how this definition works let's assume we have a well-defined collection of definite conditions (for example, take $x=y, x \in y, set(x)$ as basic conditions and close the collection under the usual logical operations). As a matter of definition let  $x \in P \Leftrightarrow P(x)$, where $P$ is a definite condition. Let a condition $P$ be coextensive with a set $A$ ($P =_e A$), if $\forall x(P(x) \Leftrightarrow x \in A)$. By the axiom of extensionality it's obvious that each condition is coextensive with at most one set. 
Now, for some condition $P$ the class $\lbrack x : P(x) \rbrack$ can be defined as follows: If $P =_e A$, for some set $A$ we let $\lbrack x : P(x) \rbrack = A$; otherwise we let $ \lbrack x : P(x) \rbrack = P$  
