Union of Union of Cartesian Product with Empty Factor

Theorem
Let $A$ and $B$ be sets such that either $A = \O$ or $B = \O$.

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

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

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

That is, if either $A$ or $B$ is empty:
 * $\ds \bigcup \bigcup \paren {A \times B} = A \cup B$

holds they are both empty

Proof
Let $A = \O$ or $B = \O$.

From Cartesian Product is Empty iff Factor is Empty:
 * $A \times B = \O$

Hence from Union of Empty Set:
 * $\ds \bigcup \bigcup \paren {A \times B} = \O$

However, from Union is Empty iff Sets are Empty:


 * $A \cup B = \O \iff A = \O \text { and } B = \O$

The result follows.

Also see

 * Union of Union of Cartesian Product