Independent Events are Independent of Complement/Corollary

Corollary to Independent Events are Independent of Complement
Let $A$ and $B$ be events in a probability space $\left({\Omega, \Sigma, \Pr}\right)$.

$A$ and $B$ are independent iff $\Omega \setminus A$ and $\Omega \setminus B$ are independent.

Proof
Let $A$ and $B$ be independent.

Then from Independent Events are Independent of Complement, $A$ and $\Omega \setminus B$ are 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.