Independent Events are Independent of Complement/Corollary
Corollary to Independent Events are Independent of Complement
Let $A$ and $B$ be independent.
Setting $A' = \Omega \setminus B$ and $B' = A$, we see clearly that $A'$ and $B'$ are independent.
So from the main result, $A'$ and $\Omega \setminus B'$ are independent.
That is, $\Omega \setminus B$ and $\Omega \setminus A$ are independent.
The "only if" part of the result follows directly from Relative Complement of Relative Complement and another application of this result.