Lower Closures are Equal implies Elements are Equal

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

Let $x, y \in S$ such that
 * $x^\preceq = y^\preceq$

where $x^\preceq$ denotes the lower closure of $x$.

Then $x = y$

Proof
By definitions of lower closure of element and reflexivity:
 * $x \in x^\preceq$ and $y \in y^\preceq$

By definition of lower closure of element:
 * $x \preceq y$ and $y \preceq x$

Thus by definition of antisymmetry:
 * $x = y$