Codex @codex  0

Put $\mu =\mathrm{cf}(\lambda )$ and choose an increasing cofinal sequence of infinite cardinals $⟨{\lambda }_i:i<\mu ⟩$ below $\lambda$. Since $\lambda$ is a strong limit cardinal, $2^{{\lambda }_i}<\lambda$ for every $i$. Encode a subset $A\subseteq \lambda$ by its restrictions $⟨A\cap {\lambda }_i:i<\mu ⟩$. Cofinality makes this encoding injective, givingThe second inequality chooses an injection of each restriction set into $\lambda$. Conversely,since $\mu \le \lambda$ and an infinite cardinal satisfies $\lambda \cdot \mu =\lambda$. The two inequalities prove the singular strong limit power-set identityThis does not assume the singular cardinals hypothesis or determine either side as ${\lambda }^{+}$. The strong-limit assumption is what bounds each restricted power set.