Projection of Complement Contains Complement of Projection

Theorem
Let $S$ and $T$ be non-empty sets.

Let $X \subseteq S \times T$ be a subset of the Cartesian product $S \times T$.

Denote with $\operatorname{pr}_1, \operatorname{pr}_2$ and $\complement$ the first and second projections, and the complement operation, respectively.

Then:

Proof
Let $s \in S$.

Then:

In conclusion:


 * $s \in \complement \left({\operatorname{pr}_1 \left({X}\right)}\right) \implies s \in \operatorname{pr}_1 \left({\complement \left({X}\right)}\right)$

and by definition of subset, the first relation follows.

Mutatis mutandis the other relation can be established.