Greatest Element is Maximal

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

Let $M$ be the greatest element of $\struct {S, \preceq}$.

Then $M$ is a maximal element.


By definition of greatest element:

$\forall y \in S: y \preceq M$

Suppose $M \preceq y$.

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

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

That is:

$M \preceq y \implies M = y$

which is precisely the definition of a maximal element.


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