Join Operation on Ordered Set such that Every Doubleton admits Supremum is Entropic

Theorem
Let $\struct {S, \preccurlyeq}$ be an ordered set.

Let $\struct {S, \preccurlyeq}$ be such that every doubleton subset of $S$ admits a supremum.

Let $\vee$ be the join operation on $S$, defined as:
 * $\forall a, b \in S: a \vee b = \sup_\preccurlyeq \set {a, b}$

Then $\vee$ is an entropic operation.

Proof
By definition, $\struct {S, \vee, \preccurlyeq}$ is a join semilattice.

From Join Semilattice is Semilattice, $\struct {S, \vee, \preccurlyeq}$ is indeed a semilattice.

Then: