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If I understand correctly, the following enforces your desired behavior: \begin{align} y_1 &= d_1 \\ y_2 &= d_2 \\ y_3 &= d_3 \\ y_4 &\ge d_1 + d_2 - 1\\ y_5 &\ge d_1 + d_2 + d_3 - 2\\ \end{align} If you also want to enforce $y_4 \implies (d_1 \land d_2)$ and $y_5 \implies (d_1 \land d_2 \land d_3)$, then include these additional ...
Here's one possible formulation, where $a_1,\dots, a_n$ are the values of the $n$ integers. Let binary decision variable $x_{i,j}$ indicate whether integer $i$ is assigned to subset $j\in\{1,\dots,k\}$. The problem is to maximize $z$ subject to \begin{align} \sum_j x_{i,j} &= 1 &&\text{for all $i$} \tag1\\ \sum_i a_i x_{i,j} &\ge z &&...