Preceding is Auxiliary Relation

Theorem
Let $\left({S, \vee, \preceq}\right)$ be a bounded below join semilattice.

Then
 * $\preceq$ is auxiliary relation.

Proof

 * $\forall x, y \in S: x \preceq y \implies x \preceq y$

Then condition $(i)$ of auxiliary relation is satisfied.

By definition of transitivity:
 * $\forall x, y, z, u \in S: x \preceq y \preceq z \preceq u \implies x \preceq u$

Then the condition $(ii)$ of auxiliary relation is satisfied.

By definition of supremum:
 * $\forall x, y, z \in S: x \preceq z \land y \preceq z \implies x \vee y \preceq z$

Then the condition $(iii)$ of auxiliary relation is satisfied.

By definition of smallest element:
 * $\forall x \in S: \bot \preceq x$

Then the condition $(iv)$ of auxiliary relation is satisfied.

Thus the result by definition auxiliary relation.