Smallest Element is Unique

Theorem
Let $\left({S, \preceq}\right)$ be a poset.

If $S$ has a smallest element, then it can have only one.

That is, if $a$ and $b$ are both smallest elements of $S$, then $a = b$.

Proof
Let $a$ and $b$ both be smallest elements of $S$.

Then by definition:
 * $\forall y \in S: a \preceq y$
 * $\forall y \in S: b \preceq y$

Thus it follows that:
 * $a \preceq b$
 * $b \preceq a$

But as $\preceq$ is an ordering, it is antisymmetric.

Hence by definition of antisymmetric, $a = b$.