Identities of Boolean Algebra are also Zeroes

Theorem
Let $\left({S, \vee, \wedge, \neg}\right)$ be a Boolean algebra whose identity for $\vee$ is $\bot$ and whose identity for $\wedge$ is $\top$.

Then:
 * $\forall x \in S: x \vee \top = \top$
 * $\forall x \in S: x \wedge \bot = \bot$

That is, $\bot$ is a zero element for $\wedge$, and $\bot$ is a zero element for $\vee$.

Proof
Let $x \in S$.

Then:

So $x \vee \top = \top$.

The result $x \wedge \bot = \bot$ follows from the Duality Principle.