Finite Suprema Set and Lower Closure is Ideal

Theorem
Let $P = \struct {S, \vee, \preceq}$ be a join semilattice.

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

Then
 * $\map {\operatorname{finsups} } X^\preceq$ is ideal in $P$.

where
 * $\map {\operatorname{finsups} } X$ denotes the finite suprema set of $X$,
 * $X^\preceq$ denotes the lower closure of $X$.

Proof
By Finite Suprema Set and Lower Closure is Smallest Ideal:
 * $X \subseteq \map {\operatorname{finsups} } X^\preceq$

By definition of non-empty set:
 * $\map {\operatorname{finsups} } X^\preceq$ is a non-empty set.

We will prove that
 * $\map {\operatorname{finsups} } X$ is directed.

Let $x, y \in \map {\operatorname{fininfs} } X$

By definition of finite suprema set:
 * $\exists A \in \map {\operatorname {Fin} } X: x = \sup A \land A$ admits a supremum

and
 * $\exists B \in \map {\operatorname {Fin} } X: y = \sup B \land B$ admits an supremum

where $\map {\operatorname {Fin} } X$ denotes the set of all finite subsets of $X$.

Define $C = A \cup B$.

By Union of Subsets is Subset:
 * $C \subseteq X$

By Union of Finite Sets is Finite:
 * $C$ is finite.

Then
 * $C \in \map {\operatorname {Fin} } X$

By Existence of Non-Empty Finite Suprema in Join Semilattice:
 * $C \ne \O \implies C$ admits a supremum.

By Union is Empty iff Sets are Empty:
 * $C = \O \implies A = \O$

So
 * $C$ admits a supremum.

By definition of finite suprema set:
 * $\sup C \in \map {\operatorname{finsups} } X$

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

Thus by Supremum of Subset:
 * $x \preceq \sup C$ and $y \preceq \sup C$

Hence $\map {\operatorname{finsups} } X$ is directed.

By Directed iff Lower Closure Directed:
 * $\map {\operatorname{finsups} } X^\preceq$ is directed.

By Lower Closure is Lower Set:
 * $\map {\operatorname{finsups} } X^\preceq$ is lower.

Hence $\map {\operatorname{finsups}} X^\preceq$ is ideal in $P$.