 Clearly, \(d_1\) is a subset of \(d'\), and so \(M'\) is a submodel of \(M\). Let \(\kappa\) be the maximum of the size of \(K\), the size of \(d_1\), and denumerably infinite. A calculation reveals that the size of \(d'\) is at most \(\kappa\), based on the fact that there are at most \(\kappa\)-many formulas, and thus, at most \(\kappa\)-many witnesses at each stage. Notice, incidentally, that this calculation relies on the fact that a denumerable union of sets of size at most \(\kappa\) is itself at most \(\kappa\). This also relies on the axiom of choice.