Lower Closure of Subset is Subset of Lower Closure

Theorem
Let $\struct {S, \preceq}$ be an ordered set.

Let $X, Y$ be subsets of $S$.

Then
 * $X \subseteq Y \implies X^\preceq \subseteq Y^\preceq$

where $X^\preceq$ is the lower closure of $X$.

Proof
Let $X \subseteq Y$.

Let $x \in X^\preceq$.

By definition of lower closure of subset:


 * $\exists y \in X: x \preceq y$

By definition of subset:


 * $y \in Y$

Thus by definition of lower closure of subset:


 * $x \in Y^\preceq$