Finite Subset Bounds Element of Finite Suprema Set and Lower Closure

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

Let $I$ be ideal in $L$.

Let $X$ be non empty finite subset of $S$.

Let $x \in S$ such that
 * $x \in \left({\operatorname{finsups}\left({F \cup X}\right)}\right)^\preceq$

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

Then there exists $a \in S$: $a \in I \land x \preceq a \vee \sup X$

Proof
This follows by mutatis mutandis of the proof of Finite Subset Bounds Element of Finite Infima Set and Upper Closure.