Exponential tail of subcritical cluster size

For independent bond percolation on ${\mathbb{Z}}^d$ below the percolation critical probability, there are $K,c>0$ such that ${\mathbb{P}}_p(|C(x)|\ge n)\le K{e}^{-cn}$, uniformly in $x$. This concerns the number of vertices in a percolation cluster, a stronger conclusion than decay of its radius alone.