Intersection of Singleton

Theorem
Consider the set of sets $A$ such that $A$ consists of just one set $x$:
 * $A = \set x$

Then the intersection of $A$ is $x$:


 * $\ds \bigcap A = x$

Proof
Let $A = \set x$.

Then from the definition:
 * $\ds \bigcap \set x = \set {y: \forall z \in \set x: y \in z}$

from which it follows directly:
 * $\ds \bigcap \set x = \set {y: y \in x}$

as $x$ is the only set in $\set x$.

That is:
 * $\ds \bigcap A = x$

Also see

 * Union of Singleton