Proposition 14

Let $(\mathfrak{a}_\iota)_{\iota \in I}$ and $(\mathfrak{b}_\iota)_{\iota \in I}$ be two families of cardinals, both indexed by the same set I, and such that $\mathfrak{a}_\iota \geqslant \mathfrak{b}_\iota$ for all $\iota \in I$. Then

The second inequality follows from the inclusion relations between products of sets (Chapter II, § 5, no. 4, Proposition 6, Corollary 3). As to the first inequality, if a set E is the union of a family $(A_\iota)_{\iota \in I}$ of mutually disjoint subsets and if $B_\iota \subset A_\iota$ for all $\iota \in I$, then the $B_\iota$ are also mutually disjoint and $\bigcup_\iota B_\iota \subset \bigcup_\iota A_\iota$ (Chapter II, § 4, no. 2).