Poset Elements Equal iff Equal Weak Lower Closure

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

Let $s, t \in S$.

Then $s = t$ :


 * $s^\preccurlyeq = t^\preccurlyeq$

where $s^\preccurlyeq$ denotes weak lower closure of $s$.

That is,, for all $r \in S$:


 * $r \preccurlyeq s \iff r \preccurlyeq t$

Necessary Condition
If $s = t$, then trivially also:


 * $s^\preccurlyeq = t^\preccurlyeq$

Sufficient Condition
Suppose that:


 * $s^\preccurlyeq = t^\preccurlyeq$

By definition of weak lower closure, we have:


 * $s \in s^\preccurlyeq$
 * $t \in t^\preccurlyeq$

and hence:


 * $s \in t^\preccurlyeq$
 * $t \in s^\preccurlyeq$

which by definition of weak lower closure means:


 * $s \preccurlyeq t$ and $t \preccurlyeq s$

Since $\preccurlyeq$ is antisymmetric it follows that $s = t$.