Finite Suprema Set and Lower Closure is Smallest Ideal

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

Let $X$ be a subset of $S$.

Then $X \subseteq \operatorname{finsups}\left({X}\right)^\preceq$ and
 * for every filter $F$ on $L$: $X \subseteq F \implies \operatorname{finsups}\left({X}\right)^\preceq \subseteq F$

Proof
By Set is Subset of Finite Suprema Set:
 * $X \subseteq \operatorname{finsups}\left({X}\right)$

By Lower Closure of Subset is Subset of Lower Closure:
 * $X^\preceq \subseteq \operatorname{finsups}\left({X}\right)^\preceq$

By Set is Subset of Lower Closure:
 * $X \subseteq X^\preceq$

Thus by Subset Relation is Transitive:
 * $X \subseteq \operatorname{finsups}\left({X}\right)^\preceq$