A worldly cardinal $\kappa $ is defined by $V_\kappa \vDash ZFC $ . An inaccessible cardinal $\iota $ is defined in such a way that $V_\iota \ $ is a Grothendieck universe and so provides a model of ZFC. Therefore inaccessible cardinals are worldly. But if they exist the smallest worldly cardinal is singular so not inaccessible.
My question is how can it be that $V_\kappa \vDash ZFC $ and yet not be Grothendieck? For example $V_\kappa \vDash \forall x(set x \implies set \wp x) $ so $V_\kappa $ is closed under $\wp $, and similarly for all the other Grothendieck properties (transitive, infinity, pairs, unions, powers, substitutions). If $V_\kappa $ is worldly shouldn't it therefore be Grothendieck, what am i missing?