Set of Finite Suprema is Directed

Theorem
Let $\left({S, \vee, \preceq}\right)$ be a join semilattice.

Let $X$ be a non-empty subset of $S$.

Then
 * $\left\{ {\sup A: A \in {\it Fin}\left({X}\right) \land A \ne \varnothing}\right\}$ is directed.

where ${\it Fin}\left({X}\right)$ denotes the set of all finite subsets of $X$.

Proof
By Existence of Non-Empty Finite Suprema in Join Semilattice:
 * for every $A \in {\it Fin}\left({S}\right)$ if $A \ne \varnothing$, then $A$ admits a supremum.

By definition of non-empty set:
 * $\exists a: a \in X$

By definitions of subset and singleton:
 * $\left\{ {x}\right\} \subseteq X$

By Singleton is Finite:
 * $\left\{ {x}\right\}$ is finite

By definitions of non-empty set and singleton:
 * $\left\{ {x}\right\} \ne \varnothing$

By definition of ${\it Fin}$:
 * $\sup \left\{ {x}\right\} \in \left\{ {\sup A: A \in {\it Fin}\left({X}\right) \land A \ne \varnothing}\right\}$

Thus by definition:
 * $\left\{ {\sup A: A \in {\it Fin}\left({X}\right) \land A \ne \varnothing}\right\}$ is a non-empty set.

Let $x, y \in \left\{ {\sup A: A \in {\it Fin}\left({X}\right) \land A \ne \varnothing}\right\}$

Then
 * $\exists A \in {\it Fin}\left({X}\right): x = \sup A$ and $A \ne \varnothing$

and
 * $\exists B \in {\it Fin}\left({X}\right): y = \sup B$ and $B \ne \varnothing$

By Finite Union of Finite Sets is Finite:
 * $A \cup B$ is finite

By Union of Subsets is Subset:
 * $A \cup B \subseteq X$

By definitions of non-empty set and union:
 * $A \cup B \ne \varnothing$

By definition of $\it Fin$:
 * $\sup \left({A \cup B}\right) \in \left\{ {\sup A: A \in {\it Fin}\left({X}\right) \land A \ne \varnothing}\right\}$

By Set is Subset of Union:
 * $A \subseteq A \cup B$ and $B \subseteq A \cup B$

Thus by Supremum of Subset:
 * $x \preceq \sup \left({A \cup B}\right)$ and $y \preceq \sup \left({A \cup B}\right)$

Thus by definition:
 * $\left\{ {\sup A: A \in {\it Fin}\left({X}\right) \land A \ne \varnothing}\right\}$ is directed.