This follows from a stronger theorem of Henry Mann according to which the Schnirelmann density of a sum of two sequences is at least the sum of their Schnirelmann densities, unless their sum consists of all natural numbers.
The current statement is based on a more robust theorem by Henry Mann, stating that the Schnirelmann density of the sum of two sequences is at least the sum of their individual Schnirelmann densities, except when the sum comprises all natural numbers.