Set is Subset of Finite Infima Set
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Theorem
Let $\left({S, \preceq}\right)$ be an ordered set.
Let $X$ be a subset of $S$.
Then $X \subseteq \operatorname{fininfs}\left({X}\right)$
where $\operatorname{fininfs}\left({X}\right)$ denotes the finite infima set of $X$.
Proof
Let $x \in X$.
- $\left\{ {x}\right\}$ admits an infimum and $\inf \left\{ {x}\right\} = x$
By definitions of subset and singleton:
- $\left\{ {x}\right\} \subseteq X$
- $\left\{ {x}\right\}$ is a finite set.
Thus by definition of finite infima set:
- $x \in \operatorname{fininfs}\left({X}\right)$
$\blacksquare$
Sources
- Mizar article WAYBEL_0:50