Meet with Complement is Bottom

Theorem
Let $\struct {S, \vee, \wedge, \neg}$ be a Boolean algebra, defined as in Definition 2.

Then:
 * $\exists \bot \in S: \forall a \in S: a \wedge \neg a = \bot$

where $\wedge$ denotes the meet operation in $S$.

This element $\bot$ is unique for any given $S$, and is named bottom.

Proof
Let $\exists r, s \in S: r \wedge \neg r = a, \ s \wedge \neg s = b$

Then:

Thus, whatever $r$ and $s$ may be:
 * $r \wedge \neg r = s \wedge \neg s$

This unique element can be assigned the symbol $\bot$ and named bottom as required.