# Intersection of Non-Empty Class is Set/Proof 2

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## Theorem

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

Then $\bigcap A$ is a set.

## Proof

Since $A$ is a non-empty class, there exists $S \in A$.

Since $S$ is an element of a class, it is not a proper class, and is thus a set.

By definition of class intersection:

- $x \in \bigcap A \implies x \in S$

By the subclass definition:

- $\bigcap A \subseteq S$

By Subclass of Set is Set, $\bigcap A$ is a set.

$\blacksquare$

## Sources

- 2002: Thomas Jech:
*Set Theory*(3rd ed.) ... (previous) ... (next): Chapter $1$: Separation Schema