Upper Closure is Decreasing

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

Let $x, y$ be elements of $S$ such that
 * $x \preceq y$

then $y^\succeq \subseteq x^\succeq$

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

Proof
Let $z \in y^\succeq$.

By definition of upper closure of element:
 * $y \preceq z$

By definition of ordering, $\preceq$ is transitive.

From $x \preceq y$ and $y \preceq z$:


 * $x \preceq z$

Thus again by definition of upper closure of element:
 * $z \in x^\succeq$