Subclasses and Subsets

Let $X$ and $Y$ be classes.

¶ $X$ is a subclass of $Y$, written $X\subseteq Y$, if every element of $X$ is an element of $Y$; this is the defined predicate abbreviating $\forall u\,(u\in X\Rightarrow u\in Y)$, with $u$ a set variable, which expresses the same thing because 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. When $X$ is a set, $X$ is also called a subset of $Y$.