Set Union expressed as Intersection Complement

Theorem
Let $A$ and $B$ be subsets of a universal set $\Bbb U$.

Let $\uparrow$ denote the operation on $A$ and $B$ defined as:
 * $\paren {A \uparrow B} \iff \paren {\relcomp {\Bbb U} {A \cap B} }$

where $\relcomp {\Bbb U} A$ denotes the complement of $A$ in $\Bbb U$.

Then:
 * $A \cup B = \paren {A \uparrow A} \uparrow \paren {B \uparrow B}$