Poset Elements Equal iff Equal Weak Lower Closure

Theorem

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

Let $s, t \in S$.

Then $s = t$ if and only if:

$s^\preccurlyeq = t^\preccurlyeq$

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

That is, if and only if, for all $r \in S$:

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

Proof

Necessary Condition

If $s = t$, then trivially also:

$s^\preccurlyeq = t^\preccurlyeq$

$\Box$

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 $s = t$.

$\blacksquare$