Poset Elements Equal iff Equal Weak Lower Closure

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

Let $s, t \in S$.

Then $s = t$ iff:


 * $\mathop{\bar \downarrow} \left({s}\right) = \mathop{\bar \downarrow} \left({t}\right)$

where $\bar \downarrow$ denotes weak lower closure.

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


 * $r \preceq s \iff r \preceq t$

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


 * $\mathop{\bar \downarrow} \left({s}\right) = \mathop{\bar \downarrow} \left({t}\right)$

Sufficient Condition
Suppose that:


 * $\mathop{\bar \downarrow} \left({s}\right) = \mathop{\bar \downarrow} \left({t}\right)$

By definition of weak lower closure, we have:


 * $s \in \mathop{\bar \downarrow} \left({s}\right)$
 * $t \in \mathop{\bar \downarrow} \left({t}\right)$

and hence:


 * $s \in \mathop{\bar \downarrow} \left({t}\right)$
 * $t \in \mathop{\bar \downarrow} \left({s}\right)$

which by definition of weak lower closure means:


 * $s \preceq t$ and $t \preceq s$

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