Does having unique normal forms imply weak normalization and confluence? Consider a term rewriting system $\mathcal{R} = (\Sigma, R)$ over a signature $\Sigma$ with basic rewrite rules $R$. If $\mathcal{R}$ is weakly normalizing and confluent, then we know that each $\Sigma$-term has a unique normal form (with respect to $\mathcal{R}$). Is the converse also true? I.e. if each term has a unique normal form with respect to $\mathcal{R}$, then does it follow that $\mathcal{R}$ is weakly normalizing and confluent? Obviously if each term has a unique normal form, then the system is weakly normalizing, so it remains to ask whether it will be confluent. 

In short, I am wondering whether the property of a term rewriting system to be weakly normalizing and confluent is strictly stronger than the property of each term having a unique normal form (with respect to the system).
It is a little bit sketchy, but here is my argument: Suppose that there are three terms t, u, v such that t reduces to u, t reduces to v and suppose that each term has a unique normal form. Since you are weakly normalizing as you said, there exists a normal form for v and u. But since the normal form is unique, then v and u are joinable. Hence, the system is confluent.