Classes, Membership, Sets and Proper Classes

Class theory has two primitive notions, which are not defined: class, and membership between classes. For classes $X$ and $Y$, membership is written $X\in Y$ and read "$X$ is an element of $Y$"; $X\notin Y$ means that $X\in Y$ does not hold. Every object of the theory is a class.

¶ A class $X$ is a set if there is a class $Y$ with $X\in Y$.

¶ A class that is not a set is a proper class.