Singleton of Bottom is Ideal
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Theorem
Let $\struct {S, \preceq}$ be a bounded below ordered set.
Then
- $\set \bot$ is an ideal in $\struct {S, \preceq}$
where $\bot$ denotes the smallest element in $S$.
Proof
Non-empty
By definition of singleton:
- $\bot \in \set \bot$
By definition:
- $\set \bot$ is a non-empty set.
$\Box$
Directed
Thus by Singleton is Directed and Filtered Subset:
- $\set \bot$ is directed.
$\Box$
Lower
Let $x \in \set \bot, y \in S$ such that
- $y \preceq x$
By definition of singleton:
- $x = \bot$
By definition of smallest element:
- $\bot \preceq y$
By definition of antisymmetry:
- $y = \bot$
Thus by definition of singleton:
- $y \in \set \bot$
Thus by definition:
- $\set \bot$ is a lower section.
$\Box$
Thus by definition:
- $\set \bot$ is an ideal in $\struct {S, \preceq}$
$\blacksquare$
Sources
- Mizar article WAYBEL_4:21