Intersection with Complement

Theorem
The intersection of a set and its complement is the empty set:


 * $S \cap \map \complement S = \O$

Proof
Substitute $\mathbb U$ for $S$ and $S$ for $T$ in $T \cap \relcomp S T = \O$ from Intersection with Relative Complement is Empty.

Also see

 * Union with Complement

Notice the similarity with the Principle of Non-Contradiction.

The complement of a set is similar to the negation of a proposition, and intersection is similar to conjunction.