Intersection of Empty Set

Theorem
Consider the set of sets $$\mathbb S$$ such that $$\mathbb S$$ is the empty set $$\varnothing$$.

Then the intersection of $$\mathbb S$$ is $$\mathbb U$$:


 * $$\mathbb S = \left\{{\varnothing}\right\} \implies \bigcap \mathbb S = \mathbb U$$

where $$\mathbb U$$ is the universe.

A paradoxical result.

Proof
Let $$\mathbb S = \left\{{\varnothing}\right\}$$.

Then from the definition:
 * $$\bigcap \mathbb S = \left\{{x: \forall X \in \mathbb S: x \in X}\right\}$$

Consider any $$x \in \mathbb U$$.

Then as $$\mathbb S = \left\{{\varnothing}\right\}$$, it follows that:
 * $$\forall X \in \mathbb S: x \in X$$

from the definition of vacuous truth.

It follows directly that:
 * $$\bigcap \mathbb S = \left\{{x: x \in \mathbb U}\right\}$$

That is:
 * $$\bigcap \mathbb S = \mathbb U$$

Comment
Although it appears counter-intuitive, the reasoning is sound.

This result is therefore classed as a veridical paradox.