Way Below implies Preceding
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Theorem
Let $\left({S, \preceq}\right)$ be an ordered set.
Let $x, y \in S$ such that
- $x \ll y$
where $\ll$ denotes element is way below second element.
Then
- $x \preceq y$
Proof
By Singleton is Directed and Filtered Subset:
- $\left\{ {y}\right\}$ is directed.
- $\left\{ {y}\right\}$ admits a supremum and $\sup \left\{ {y}\right\} = y$
By definition of reflexivity:
- $y \preceq \sup \left\{ {y}\right\}$
By definition of way below:
- $\exists d \in \left\{ {y}\right\}: x \preceq d$
Thus by definition of singleton:
- $x \preceq y$
$\blacksquare$
Sources
- 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices
- Mizar article WAYBEL_3:1