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.

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.

$\blacksquare$

1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations