Strictly Minimal Element is Minimal 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 minimal element of $T$ under $\RR$.

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

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

Then by definition:
 * $\forall x \in T: \tuple {x, m} \notin \RR$

$m$ is not a minimal element of $T$ under $\RR$.

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

such that $y \ne m$.

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

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

That is:
 * $m$ is a minimal element of $T$ under $\RR$.