Strictly Maximal Element is Maximal Element

Theorem

Let $\struct {S, \RR}$ be a relational structure.

Let $T \subseteq S$ be a subset of $S$.

Let $m \in T$ be a strictly maximal element of $T$ under $\RR$.

Then $m$ is a maximal element of $T$ under $\RR$.

Proof

Let $m \in T$ be a strictly maximal element of $T$ under $\RR$.

Then by definition:

$\forall x \in T: \tuple {m, x} \notin \RR$

Aiming for a contradiction, suppose $m$ is not a maximal element of $T$ under $\RR$.

Then:

$\exists y \in T: \tuple {m, y} \in \RR$

such that $y \ne m$.

But this contradicts the assertion that $\tuple {m, y} \notin \RR$.

Hence it cannot be the case that $m$ is not a maximal element of $T$ under $\RR$.

That is:

$m$ is a maximal element of $T$ under $\RR$.

$\blacksquare$