Smallest Element is Minimal

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

Let $m$ be the smallest element of $\left({S, \preceq}\right)$.

Then $m$ is a minimal element.

Proof
By definition of smallest element:
 * $\forall y \in S: m \preceq y$

Suppose $y \preceq m$.

As $\preceq$ is an ordering, $\preceq$ is by definition antisymmetric.

Thus it follows by definition of antisymmetry that $m = y$.

That is:
 * $y \preceq m \implies m = y$

which is precisely the definition of a minimal element.

Also see

 * Greatest Element is Maximal