On sums of finite subsets of the primes

Let $A\subset [1,x]$ be a non-empty set of primes with $\mathrm{\mid }A\mathrm{\mid }=\alpha x(\log x{)}^{-1}$. We prove that there exist absolute constants $c_1,c_2>0$ such that, as $x$ gets sufficiently large, we have $\mathrm{\mid }A+A\mathrm{\mid }\ge c_1(\log x)(\log \log 3{\alpha }^{-1}{)}^{-1}\mathrm{\mid }A\mathrm{\mid }$ if $\alpha \ge c_2(\log x{)}^{-1\mathrm{/}2}\log \log x$ and otherwise $\mathrm{\mid }A+A\mathrm{\mid }\ge c_1(\log x)(\log 2{\alpha }^{-1}{)}^{-1}\mathrm{\mid }A\mathrm{\mid }$.