Superset of Order Generating is Order Generating

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

Let $X, Y$ be subsets of $S$ such that
 * $X$ is order generating

and
 * $X \subseteq Y$

Then $Y$ is order generating.

Proof
Let $x \in S$.

Thus by definition of complete lattice:
 * $x^\succeq \cap Y$ admits an infimum.

By definition of complete lattice:
 * $x^\succeq$ admits an infimum

and
 * $x^\succeq \cap X$ admits an infimum.

By Intersection is Subset:
 * $x^\succeq \cap Y \subseteq x^\succeq$

By Infimum of Subset:
 * $\inf \left({x^\succeq}\right) \preceq \inf \left({x^\succeq \cap Y}\right)$

By Infimum of Upper Closure of Element:
 * $x \preceq \inf \left({x^\succeq \cap Y}\right)$

By Set Intersection Preserves Subsets/Corollary:
 * $x^\succeq \cap X \subseteq x^\succeq \cap Y$

By Infimum of Subset:
 * $\inf \left({x^\succeq \cap Y}\right) \preceq \inf \left({x^\succeq \cap X}\right)$

By definition of order generating:
 * $\inf \left({x^\succeq \cap Y}\right) \preceq x$

Thus by definition of antisymmetry:
 * $x = \inf \left({x^\succeq \cap Y}\right)$