Smallest Element is Minimal

Theorem
Let $\struct {S, \preceq}$ be an ordered set which has a smallest element.

Let $m$ be the smallest element of $\struct {S, \preceq}$.

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