The first law implicitly contains the claim that an inertial frame exists because it assumes that the motion of objects may only be described in the "right frame" and it is actually using the "right frame" to do so.
When an inertial frame exists, there are infinitely many other inertial reference frames that are in uniform motion relatively to each other (including the original one). But that's an additional insight. When you read Newton's first law as stated, you shouldn't be thinking about many frames at all. You should think that there's only one (right) frame to describe physics. In this right frame, bodies that are not subject to forces stay at rest or uniform motion.
Once you realize that you can use various frames, i.e. coordinate transformations you may find out that there are infinitely many frames in which the first law holds – you call them inertial - and infinitely many frames in which it doesn't hold – non-inertial frames. But this is a treatment that goes beyond the original formulation of the first law which assumes that there is one right frame and doesn't even want you to question that.