Poset Elements Equal iff Equal Weak Upper Closure

Theorem

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

Let $s, t \in S$.

Then $s = t$ if and only if:

$s^\succcurlyeq = t^\succcurlyeq$

where $s^\succcurlyeq$ denotes weak upper closure of $s$.

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

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

Proof

Necessary Condition

If $s = t$, then trivially also:

$s^\succcurlyeq = t^\succcurlyeq$

$\Box$

Sufficient Condition

Suppose that:

$s^\succcurlyeq = t^\succcurlyeq$

By definition of weak upper closure, we have:

$s \in s^\succcurlyeq$

$t \in t^\succcurlyeq$

and hence:

$s \in s^\succcurlyeq$

$t \in s^\succcurlyeq$

which by definition of weak upper closure means:

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

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

$\blacksquare$