Class

A class is a Collection of different things with Propositional equality which may not be a set, though the exact meaning depends on the foundation being used. Thus it is a generalization of a small set. A Proper class is a class which is not a set. A Subclass generalizes a subset. A mapping between classes is a Class function.

Foundation-agnostic usage

We often use the word class in foundation-agnostic contexts. The appropriate interpretations are then:

In a theory like ZF one may treat a class indirectly as a predicate $Φ$ ranging over sets and possibly urelements and we say $𝑥 \in Φ$ iff $Φ (𝑥)$ .
In an extension such as NBG, a class becomes an object in its own right which largely supersedes the notion of a set, with a set becoming a class which is contained in some class. Classhood of $𝑥$ is denoted by $ℭ 𝔩 𝔰 (𝑥)$ . Again, classes are taken to satisfy the Axiom of Extensionality
In TG a $U$ -class is a subset of $U$ .
Similarly, in a Type theory with universes a class is the same as an h-set.