Poset Elements Equal iff Equal Weak Upper Closure

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

Let $s, t \in S$.

Then $s = t$ :


 * $s^\succcurlyeq = t^\succcurlyeq$

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

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


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

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


 * $s^\succcurlyeq = t^\succcurlyeq$

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 $s = t$.