Proposition 9.1.7.8. Let $\kappa < \lambda $ be a regular cardinals, where $\lambda $ has exponential cofinality $\geq \kappa $. Then, for every $\lambda $-small set $S$, the collection of $\kappa $-small subsets $\{ S_ i \subseteq S \} _{i \in I}$ is also $\lambda $-small. In particular, $\kappa \triangleleft \lambda $.
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Proof. For every ordinal $\alpha $, let $\mathrm{Ord}_{<\alpha }$ denote the set of ordinals smaller than $\alpha $. Then a subset $S_ i \subseteq S$ is $\kappa $-small if and only if it is the image of a function $F: \mathrm{Ord}_{< \alpha } \rightarrow S$, for some ordinal $\alpha < \kappa $. It will therefore suffice to show that the set of all such pairs $(\alpha ,F)$ is $\lambda $-small. The set $\mathrm{Ord}_{< \kappa }$ has cardinality $\kappa $, and is therefore $\lambda $-small by virtue of our assumption that $\kappa < \lambda $. Since $\lambda $ is regular, it will suffice to show that for every fixed ordinal $\alpha < \kappa $, the set of functions $F: \mathrm{Ord}_{< \alpha } \rightarrow S$ is $\lambda $-small. This follows from our assumption that $\lambda $ has exponential cofinality $\geq \kappa $. $\square$