Isospectral finite simple groups

The spectrum of a finite group is the set of its element orders. Two groups are called isospectral if their spectra coincide. It is known that $PSp_6(2)$ is isospectral to $P\Omega^+_8(2)$ and $\Omega_7(3)$ is isospectral to $P\Omega^+_8(3)$. In the present paper we prove that there are no other pairs of non-isomorphic isospectral finite simple groups. In particular, we prove that there are no three finite simple groups with the same spectrum.