Bottom Relation is Auxiliary Relation

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

Let $B = \left\{ {\left({\bot, x}\right): x \in S}\right\}$

where $\bot$ denotes the smallest element in $L$.

Then
 * $B$ is auxiliary relation.

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

Thus by definition of $B$:
 * $\forall x, y \in S: \left({x, y}\right) \in B \implies x \preceq y$

We will prove that
 * $\forall x, y, z, u \in S: x \preceq y \land \left({y, z}\right) \in B \land z \preceq u \implies \left({x, u}\right) \in B$

Let $x, y, z, u \in S$ such that
 * $x \preceq y \land \left({y, z}\right) \in B \land z \preceq u$

By definition of $B$:
 * $y = \bot$

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

By definition of antisymmetry:
 * $x = \bot$

Thus by definition of $B$:
 * $\left({x, u}\right) \in B$

We will prove that
 * $\forall x, y, z \in S: \left({x, z}\right) \in B \land \left({y, z}\right) \in B \implies \left({x \vee y, z}\right) \in B$

Let $x, y, z \in S$ such that
 * $\left({x, z}\right) \in B \land \left({y, z}\right) \in B$

By definition of $B$:
 * $x = y = \bot$

By Join is Idempotent:
 * $x \vee y = \bot$

Thus by definition of $B$:
 * $\left({x \vee y, z}\right) \in B$

Thus by definition of $B$:
 * $\forall x \in S: \left({\bot, x}\right) \in B$

Thus by definition:
 * $B$ is auxiliary relation.