Intersection of Non-Empty Class is Set/Corollary

Theorem

Let $x$ be a non-empty set.

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

Then $\bigcap x$ is a set.

Proof

It is assumed that $x$ is an element of a basic universe.

Hence from the Axiom of Transitivity, every set is a class.

Hence Intersection of Non-Empty Class is Set applies directly.

$\blacksquare$

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 $5.1 \ (1)$