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 $S$:


 * $\displaystyle \mathbb S = \set S \implies \bigcap \mathbb S = S$

Proof
Let $\mathbb S = \set S$.

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

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

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

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

Also see

 * Union of Singleton