Union Distributes over Intersection/General Result

Theorem
Set union is distributive over set intersection:

Let $S$ and $T$ be sets.

Let $\powerset T$ be the power set of $T$.

Let $\mathbb T$ be a subset of $\powerset T$.

Then:
 * $\ds S \cup \bigcap \mathbb T = \bigcap_{X \mathop \in \mathbb T} \paren {S \cup X}$

Also see

 * Intersection Distributes over Union/General Result