= Power-set equality is not upward absolute
{title2=$L\models x=\mathcal P(y),\quad V\not\models x=\mathcal P(y)$}
Assume $V\ne L$. At the first cumulative stage containing a nonconstructible <set>, the index is a successor $\beta+1$. <set-theoretic absoluteness>[Rank absoluteness] gives $y=V_\beta=(V_\beta)^L\in L$, not just $V_\beta\subseteq L$. Let $x$ be the <power set> of $y$ computed in $L$. It omits a nonconstructible <subset> of $y$ from the next external rank stage. Thus $L$ satisfies <power-set equality is not upward absolute>[power-set equality] for the parameters $x,y\in L$, while $V$ does not. Both are transitive models of <ZF>. <Subset> inclusion itself is bounded and absolute; quantification over all possible <subsets> causes the failure.
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