Intersection of Empty Set/Class Theory

Theorem
Let $V$ be a basic universe.

Let $\O$ denote the empty class.

Then the intersection of $\O$ is $V$:


 * $\displaystyle \bigcap \O = V$

Proof
By definition of $V$ we have that $\displaystyle \bigcap \O \subseteq V$.

By definition of empty class, there exists no set $x \in V$ which is an elements of $\O$.

Hence it is vacuously true that every element of $V$ is an element of every element of $\O$.

Therefore every $x \in V$ is an element of $\displaystyle \bigcap \O$.

Therefore $V \subseteq \displaystyle \bigcap \O$.

It follows by definition of class equality that $\displaystyle \bigcap \O = V$.