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