= Singular strong limit power-set identity
{title2=$2^\lambda=\lambda^{\operatorname{cf}(\lambda)}$}
For a singular <strong limit cardinal> $\lambda$, let $\mu=\operatorname{cf}(\lambda)$ and choose cofinal cardinals $\lambda_i<\lambda$. Subset restrictions inject $\mathcal P(\lambda)$ into $\prod_{i<\mu}\mathcal P(\lambda_i)$, whose cardinal is at most $\lambda^\mu$ by the strong-limit property. Conversely $\lambda^\mu\le(2^\lambda)^\mu=2^\lambda$. This equality alone does not imply the <singular cardinals hypothesis>.
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