# Upper Closure of Subset is Subset of Upper Closure

## Theorem

Let $\left({S, \preceq}\right)$ be an ordered set.

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

Then

$X \subseteq Y \implies X^\succeq \subseteq Y^\succeq$

where $X^\succeq$ denotes the upper closure of $X$.

## Proof

Let $X \subseteq Y$.

Let $x \in X^\succeq$.

By definition of upper closure of subset:

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

By definition of subset:

$y \in Y$

Thus by definition of upper closure of subset:

$x \in Y^\succeq$

$\blacksquare$