Complement of Closed under Directed Suprema Subset is Inaccessible by Directed Suprema

Theorem

Let $L = \struct {S, \preceq}$ be an up-complete ordered set.

Let $X$ be a closed under directed suprema subset of $S$.

Then $\relcomp S X$ is inaccessible by directed suprema.

Proof

Let $D$ be a directed subset of $S$ such that

$\sup D \in \relcomp S X$

By definition of relative complement:

$\sup D \notin X$

By definition of closed under directed suprema:

$D \nsubseteq X$

By Complement of Complement:

$D \nsubseteq \relcomp S {\relcomp S X}$

Thus by Empty Intersection iff Subset of Relative Complement:

$D \cap \relcomp S X \ne \O$

$\blacksquare$

Also See

• Complement of Inaccessible by Directed Suprema Subset is Closed under Directed Suprema

Sources

• Mizar article WAYBEL11:funcreg 4