The Wikipedia article on the algebra of sets briefly mentions the following:
These are examples of an extremely important and powerful property of set algebra, namely, the principle of duality for sets, which asserts that for any true statement about sets, the dual statement obtained by interchanging unions and intersections, interchanging U and Ø and reversing inclusions is also true. A statement is said to be self-dual if it is equal to its own dual.
The article doesn't talk about how this is proven and the linked article wasn't particularly enlightening to me. This is a surprising theorem to me and I am interested in finding out how to prove it. What kind of math would be involved in proving it? How would we prove this?