Exercise 1.2.8 (Part 2), p.8, from Categories for Types by Roy L. Crole.
Definition: Let $X$ be a preordered set and $A \subseteq X$. A join of $A$, if such exists, is a least element in the set of upper bounds for $A$. A meet of $A$, if such exists, is a greatest element in the set of lower bounds for $A$.
Exercise: Suppose that $X$ is a poset (and thus also a preorder). Show that meets and joins in a poset are unique if they exist.
Let $A \subseteq X$ and suppose $x$ and $y$ are both joins of $A$. Then $x$ and $y$ are both upper bounds of $A$. So $A \le x$ and $A \le y$. But since $x$ and $y$ are joins of $A$ and $x, y \in X$, $y \le x$ and $x \le y$. Therefore, $x = y$ because because $\le$ is anti-symmetric. Meets are unique by a similar argument.