Complement of Subset with Property (S) is Closed under Directed Suprema

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Theorem

Let $L = \left({S, \preceq}\right)$ be an up-complete ordered set.

Let $X$ be a subset of $S$ with property (S).


Then $\complement_S\left({X}\right)$ is closed under directed suprema.


Proof

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

$D \subseteq \complement_S\left({X}\right)$

Aiming for a contradiction suppose that

$\sup D \notin \complement_S\left({X}\right)$

By definition of relative complement:

$\sup D \in X$

By definition of property (S):

$\exists y \in D: \forall x \in D: y \preceq x \implies x \in X$

By definition of reflexivity:

$y \in X$

By definitions of intersection and non-empty:

$D \cap X \ne \varnothing$

Thus this by Empty Intersection iff Subset of Complement contradicts

$D \subseteq \complement_S\left({X}\right)$

$\blacksquare$


Sources