Vertical Section of Empty Set

Theorem
Let $X$ and $Y$ be sets.

Let $x \in X$.

Then:


 * $\O_x = \O$

where $\O$ is the empty set and $\O_x$ is the $x$-vertical section of $\O$.

Proof
suppose that:


 * $y \in \O_x$

Then from the definition of the $x$-vertical section, we have:


 * $\tuple {x, y} \in \O$

This is impossible from the definition of the empty set.

So:


 * there exists no $y \in \O_x$

giving:


 * $\O_x = \O$