Ordered Set has Lower Bound iff has Smallest Element

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Theorem

Let $\struct{S,\preceq}$ be an ordered set.

Then:

$S$ has a lower bound in $\struct{S,\preceq}$ if and only if $\struct{S,\preceq}$ has a smallest element

Proof

This follows immediately from:

• Definition:Lower Bound
• Definition:Smallest Element

$\blacksquare$