Intersection of Strict Upper Closures in Toset

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

Let $a,b \in S$.

Then:


 * $\mathop{\uparrow} \left({a}\right) \cap \mathop{\uparrow} \left({b}\right) = \mathop{\uparrow} \left({\max \left({a, b}\right)}\right)$

where $\uparrow$ denotes strict upper closure, and $\max$ denotes the max operation.

Proof
As $\left({S, \preceq}\right)$ is a totally ordered set, have either $a \preceq b$ or $b \preceq a$.

Since both sides are seen to be invariant upon interchanging $a$ and $b$, let WLOG $a \preceq b$.

Then it follows by definition of $\max$ that $\max \left({a, b}\right) = b$.

Thus, from Intersection with Subset is Subset, it suffices to show that $\mathop{\uparrow} \left({b}\right) \subseteq \mathop{\uparrow} \left({a}\right)$.

By definition of $\uparrow$, this comes down to showing that:


 * $\forall c \in S: b \prec c \implies a \prec c$

So let $c \in S$ with $b \prec c$, and recall that $a \preceq b$.

Now by Strictly Precedes is a Strict Ordering, $a \prec c$.