Preceding iff Join equals Larger Operand

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

Let $x, y \in S$.

Then
 * $x \preceq y$ $x \vee y = y$

Sufficient Condition
Let
 * $x \preceq y$

By definition of join:
 * $x \vee y = \sup \left\{ {x, y}\right\}$

By definitions of upper bound and reflexivity:
 * $y$ is upper bound for $\left\{ {x, y}\right\}$

and
 * $\forall z \in S: z$ is upper bound for $\left\{ {x, y}\right\} \implies y \preceq z$

Thus by definition of supremum:
 * $y = \sup \left\{ {x, y}\right\} = x \vee y$

Necessary Condition
Let
 * $x \vee y = y$

Thus by Join Succeeds Operands:
 * $x \preceq y$