Class is Subclass of Universal Class
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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