Duality Principle for Sets

Theorem
Any identity in set theory which uses any or all of the operations: and none other, remains valid if:
 * intersection $$\cap$$
 * union $$\cup$$
 * Empty set $$\varnothing$$
 * Universal set $$\mathbb U$$
 * $$\cap$$ and $$\cup$$ are exchanged throughout;
 * $$\varnothing$$ and $$\mathbb U$$ are exchanged throughout.

Proof
Follows from:
 * Algebra of Sets is a Boolean Ring
 * Principle of Duality of Boolean Rings