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See Proposition 11.5 (and the discussion leading up to it) in Kanamori's book The Higher Infinite. Note that Kanamori states the result a bit more optimally: if $M\models\mathsf{ZF}$ + "$\omega_1$ is regular" + $\mathsf{PCP}$ then $\omega_1^M$ is inaccessible in $L^M$ (and in fact $M\models$ "$\omega_1$ is inaccessible to reals," that is, $\omega_1^M$ is inaccessible in $L[a]^M$ for every $a\in\mathbb{R}^M$).