Smallest Element WRT Restricted Ordering

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Theorem

Let $S$ be a set or class.

Let $\preceq$ be an ordering on $S$.

Let $T$ be a subset or subclass of $S$.

Let $\preceq'$ be the restriction of $\preceq$ to $T$.

Let $m \in T$.


Then $m$ is the $\preceq$-smallest element of $T$ iff $m$ is the $\preceq'$-smallest element of $T$.


Proof