Independent Events are Independent of Complement/Corollary
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Corollary to Independent Events are Independent of Complement
Let $A$ and $B$ be events in a probability space $\struct {\Omega, \Sigma, \Pr}$.
$A$ and $B$ are independent if and only if $\Omega \setminus A$ 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.
$\blacksquare$