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

Aiming for a contradiction, suppose 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:

$\O_x = \O$

$\blacksquare$