What is the difference between a believer (mu'min) and a Muslim, and is every Muslim a believer?

The difference between a believer (mu'min) and a Muslim depends on the difference between Islam and Iman (faith). The rule is: if they are mentioned together, they differ; if they are mentioned separately, they encompass each other. So, if they appear together, Islam refers to outward actions, and Iman refers to inner beliefs, as in the Almighty's saying: "The Bedouins say, 'We have believed.' Say, 'You have not believed; but say, "We have submitted," for faith has not yet entered your hearts.'" And if Islam is mentioned alone, it includes Iman, as in the Almighty's saying: "Indeed, the religion in the sight of Allah is Islam." And if Iman is mentioned alone, it includes Islam, as in the Almighty's saying: "And whoever disbelieves in faith—then his work has become worthless." Accordingly, every believer is a Muslim, but not every Muslim is a believer. For the hypocrite is a Muslim in worldly rulings but without faith in his heart.