Intersection of Singleton

Theorem
Consider the set of sets $$\mathbb S$$ such that $$\mathbb S$$ consists of just one set $$S$$.

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


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

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

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

from which it follows directly:
 * $$\bigcap \mathbb S = \left\{{x: x \in S}\right\}$$

as $$S$$ is the only set in $$\mathbb S$$.

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