There is such a thing as a self, but it is not independent and standalone.
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The Buddha used the analogy of a lute. There is such a thing as music, but if you break down the lute into its constituent components, you will not be able to find music. Music only arises out of the inter-working of the different parts of the lute, when it is assembled, constructed and tuned correctly.
There is music, but it does not exist standalone or independently of other things. The self similarly arises virtually from the interplay of other phenomena. To say that the music or the self do not exist at all, is incorrect. "Sabbe dhamma anatta" means in all phenomena, there is no self, just as in all constituent parts of the lute, you cannot find something called music.
And there is also the mind.
"Sabbe sankhara anicca" means all conditioned things are not permanent. The mind and the self are not permanent. They are always changing, ceasing and arising, dependent on other things. Anicca does not mean that they do not exist.