User:Dfeuer/Intersection of Non-Empty Class is Set

Theorem

Let $A$ be a non-empty class.

Then $\bigcap A$ is a set.

Proof

As $A$ is non-empty, it has an element $y$.

Let $x \in \bigcap A$.

Then $x \in y$.

As this holds for all $x \in \bigcap A$, $\bigcap A \subseteq y$.

Thus by the subset axiom, $\bigcap A$ is a set.

$\blacksquare$