Smallest Element WRT Restricted Ordering

 It has been suggested that this article or section be renamed. One may discuss this suggestion on the talk page.

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 $T$.