  In a previous letter (arXiv:1306.2287) we determined the isospin mass
splittings of the baryon octet from a lattice calculation based on quenched QED
and $N_f{=}2{+}1$ QCD simulations with 5 lattice spacings down to
$0.054~\mathrm{fm}$, lattice sizes up to $6~\mathrm{fm}$ and average up-down
quark masses all the way down to their physical value. Using the same data we
determine here the corrections to Dashen's theorem and the individual up and
down quark masses. For the parameter which quantifies violations to Dashens's
theorem, we obtain $\epsilon=0.73(2)(5)(17)$, where the first error is
statistical, the second is systematic, and the third is an estimate of the QED
quenching error. For the light quark masses we obtain,
$m_u=2.27(6)(5)(4)~\mathrm{MeV}$ and $m_d=4.67(6)(5)(4)~\mathrm{MeV}$ in the
$\bar{\mathrm{MS}}$ scheme at $2~\mathrm{GeV}$ and the isospin breaking ratios
$m_u/m_d=0.485(11)(8)(14)$, $R=38.2(1.1)(0.8)(1.4)$ and
$Q=23.4(0.4)(0.3)(0.4)$. Our results exclude the $m_u=0$ solution to the strong
CP problem by more than $24$ standard deviations.
