The Boolean Operations on Classes, Disjointness, and the Universal Class

Let $X$ and $Y$ be classes.

¶ Using class abstraction, the intersection, union and difference of $X$ and $Y$ are the class terms

$$X\cap Y=\{u:u\in X\wedge u\in Y\},\qquad X\cup Y=\{u:u\in X\vee u\in Y\},\qquad X\setminus Y=\{u:u\in X\wedge u\notin Y\}.$$

¶ $X$ and $Y$ are disjoint if $X\cap Y$ has no element; this is the defined predicate abbreviating $\neg\exists u\,(u\in X\wedge u\in Y)$, with $u$ a set variable.

¶ The universal class is $V=\{u:u=u\}$. Its elements are exactly the sets, since every element of a class is a set by Elements Are Sets, and the Empty Set and the Pair of Two Sets Exist and Are Unique §elements and $u=u$ holds for every set $u$.