Then $X = \{ X_j \}_{j \in J}$ is a

**disjoint set collection**if and only if \begin{equation} \forall \, N \in 2, 3, 4, \ldots : \forall \, j_1, \ldots, j_N \in J \left( \forall \, n, m \in 1, \ldots, N \left( n \neq m \quad \implies \quad j_n \neq j_m \right) \quad \implies \quad \bigcap_{n = 1}^N X_{j_n} = \emptyset \right) \end{equation}