Order Generating iff Every Superset Closed on Infima is Whole Space

Theorem
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice.

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

Then
 * $X$ is order generating


 * $\forall Y \subseteq S: Y \supseteq X \land \paren {\forall Z \subseteq Y: \inf Z \in Y} \implies Y = S$

Sufficient Condition
Let $X$ be order generating.

Let $Y \subseteq S$ such that
 * $Y \supseteq X \land \paren {\forall Z \subseteq Y: \inf Z \in Y}$

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:
 * $\forall Y \subseteq S: Y \supseteq X \land \paren {\forall Z \subseteq Y: \inf Z \in Y} \implies Y = S$

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

By definition of subset:
 * $Y \subseteq S$

We will prove that
 * $X \subseteq Y$

Let $x \in X$.

By definitions of singleton and subset:
 * $\set x \subseteq X$

By Infimum of Singleton:
 * $\inf \set x = x$

Thus by definition of $Y$:
 * $x \in Y$

We will prove that:
 * $\forall x \in S: \exists Z \subseteq X: x = \inf Z$

Let $x \in S$.

We will prove that:
 * $\forall Z \subseteq Y: \inf Z \in Y$

Let $Z \subseteq Y$.

Define:
 * $T := \bigcup \set {A \subseteq X: \inf A \in Z}$

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

Define:
 * $N := \set {\inf A: A \subseteq X \land \inf A \in Z}$

We will prove that:
 * $Z \subseteq N$

Let $y \in Z$.

By definition of subset:
 * $y \in Y$

By definition of $Y$:
 * $\exists Z \subseteq X: y = \inf Z$

Thus by definition of $N$:
 * $y \in N$

By definition of subset:
 * $N \subseteq Z$

By definition of set equality:
 * $Z = N$

By Infimum of Infima:
 * $\inf Z = \inf T$

Thus
 * $\inf Z \in Y$

By assumption:
 * $Y = S$

Then
 * $x \in Y$

Thus by definition of $Y$:
 * $\exists Z \subseteq X: x = \inf Z$

Hence $X$ is order generating by Order Generating iff Every Element is Infimum.