Union of Union of Cartesian Product

Theorem
Let $A$ and $B$ be sets such that $A \ne \O$ and $B \ne \O$.

Let the ordered pair $\tuple {a, b}$ be defined using the Kuratowski formalization:
 * $\tuple {a, b} := \set {\set a, \set {a, b} }$

Then:
 * $\ds \bigcup \bigcup \paren {A \times B} = A \cup B$

where:
 * $\cup$ denotes union
 * $\times$ denotes Cartesian product.

Also see

 * Union of Union of Cartesian Product with Empty Factor