Predecessor is Infimum

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Theorem

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

Let $x, y \in S$.

Then:

$x \preceq y$ if and only if $x = \map \inf {x, y}$

where $\map \inf {x, y}$ is the infimum of $x$ and $y$

Proof

This is the dual statement of Successor is Supremum by Dual Pairs (Order Theory).

The result follows from the Duality Principle.

$\blacksquare$