Intersection of Non-Empty Class is Set/Proof 1

Theorem

Let $A$ be a non-empty class.

Let $\bigcap A$ denote the intersection of $A$.

Then $\bigcap A$ is a set.

Proof

Let $V$ denote the basic universe such that $A \subseteq V$.

We are given that $A$ is non-empty.

Then $\exists x \in A$ where $x$ is a set.

By definition of intersection of class, every element of $\bigcap A$ is an element of all elements of $A$.

Thus:

$\bigcap A \subseteq x$

We are given that $A$ is a subclass of the basic universe $V$.

Thus $x \in V$ by definition of basic universe.

By the Axiom of Swelledness, $V$ is a swelled class.

By definition of swelled class, every subclass of a set $x \in V$ is a set.

It follows $\bigcap A$ is a set.

$\blacksquare$

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: $\S 5$ The union axiom: Theorem $5.1 \ (1)$