Strictly Minimal Element is Minimal Element
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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:
- $\forall x \in T: \tuple {x, m} \notin \RR$
Aiming for a contradiction, suppose $m$ 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 $\RR$.
$\blacksquare$