Class is Subclass of Universal Class

Theorem

Let $V$ denote the universal class.

Let $A$ be a class.

Then $A$ is a subclass of $V$.

Proof

By definition of class, $A$ is a collection of sets.

Let $x \in A$ be a set.

By definition of universal class, $V$ contains all sets as elements.

Hence $x \in V$.

So we have that:

$x \in A \implies x \in V$

and the result follows by definition of subclass.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 1$ Extensionality and separation