We prove the following conjecture of Furstenberg (1969): if $A,B\subset [0,1]$ are closed and invariant under $\times p\ \mathrm{mod}\ 1$ and $\times_q\ \mathrm{mod}\ 1$, respectively, and if $\mathrm{log}\ p/\mathrm{log}\ q\ne \mathbb{Q}$, then for all real numbers $u$ and $v$,$$\mathrm{dim}_{\mathrm{H}}(uA+v) \cap B \le \mathrm{max}\{0,\mathrm{dim}_{\mathrm{H}} A + \mathrm{dim}_{\mathrm{H}} B -1\}.$$.We obtain this result as a consequence of our study on the intersections of incommensurable self-similar sets on $\mathbb{R}$. Our methods also allow us to give upper bounds for dimensions of arbitrary slices of planar self-similar sets satisfying SSC and certain natural irreducible conditions.