Superset of Order Generating is Order Generating

Theorem
Let $L = \struct {S, \vee, \wedge, \preceq}$ 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:
 * $\map \inf {x^\succeq} \preceq \map \inf {x^\succeq \cap Y}$

By Infimum of Upper Closure of Element:
 * $x \preceq \map \inf {x^\succeq \cap Y}$

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

By Infimum of Subset:
 * $\map \inf {x^\succeq \cap Y} \preceq \map \inf {x^\succeq \cap X}$

By definition of order generating:
 * $\map \inf {x^\succeq \cap Y} \preceq x$

Thus by definition of antisymmetry:
 * $x = \map \inf {x^\succeq \cap Y}$