Strictly Precede and Step Condition and not Precede implies Joins are equal

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

Let $p, q, u \in S$ such that
 * $p \prec q$ and $\left({\forall s \in S: p \prec s \implies q \preceq s}\right)$ and $u \npreceq p$

Then $p \vee u = q \vee u$

Proof
We will prove that
 * $\forall s \in S: p \preceq s \land u \preceq s \implies q \vee u \preceq s$