Smallest Element is Minimal

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Theorem

Let $\left({S, \preceq}\right)$ be an ordered set 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.

$\blacksquare$


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