Order Generating iff Every Superset Closed on Infima is Whole Space

Theorem
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice.

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

Then
 * $X$ is order generating


 * $\forall Y \subseteq S: Y \supseteq X \land \left({\forall Z \subseteq Y: \inf Z \in Y}\right) \implies Y = S$

Sufficient Condition
Let $X$ be order generating.

Let $Y \subseteq S$ such that
 * $Y \supseteq X \land \left({\forall Z \subseteq Y: \inf Z \in Y}\right)$

We will prove that
 * $S \subseteq Y$

Let $s \in S$.

By Order Generating iff Every Element is Infimum:
 * $\exists Z \subseteq X: s = \inf Z$

By Subset Relation is Transitive:
 * $Z \subseteq Y$

Thus by assumption:
 * $s \in Y$

By definition of set equality:
 * $Y = S$

Necessary Condition
Assume that
 * $\forall Y \subseteq S: Y \supseteq X \land \left({\forall Z \subseteq Y: \inf Z \in Y}\right) \implies Y = S$

Let $x \in S$.

Define $Y := \left\{ {\inf Z: Z \subseteq X}\right\}$