Intersection of Empty Set

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

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


 * $\mathbb S = \O \implies \displaystyle \bigcap \mathbb S = \mathbb U$

where $\mathbb U$ is the universe.

A paradoxical result.

Intersection of Empty Class
In the context of class theory the result is in the same form:

Proof
Let $\mathbb S = \O$.

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

Consider any $x \in \mathbb U$.

Then as $\mathbb S = \O$, it follows that:
 * $\forall X \in \mathbb S: x \in X$

from the definition of vacuous truth.

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

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

Also presented as
Using the terminology of indexed families, this can also be written as:
 * $\displaystyle \bigcap_{i \mathop \in \O} S_i = \set {x: \forall i \in \O: x \in S_i} = \mathbb U$

where it is important to specify the nature of $\mathbb U$.