[EDIT] the prior question was trivially false, however the intention is to *arrange* a possible world of such universes, in other words the question is about if it is possible to have a proper class $\mathcal {G}$ of Grothenderick universes that contains $V$ in them, such that every object that is an element of some class (i.e. a set) then it is in $\bigcup(\mathcal{G})$, here $\mathcal{G}$ is a primitive constant symbol. So the formulation would be:

if $\varphi(a_1,..,a_n)$ is a formula in $L(\in,=)$, in which all and only $a_1,...,a_n$ occur free, then:

$\forall W \big{(} W \in \mathcal {G} \to \\\forall a_1,..,a_n \in V [\text{if } \langle W,\in \rangle \models \varphi(a_1,..,a_n) \text{ then } \langle V,\in \rangle \models \varphi(a_1,..,a_n)] \big{)}$

Add a constant primitive symbol $V$ to the language of $ZF$, standing for some "sub-world" of sets.Now axiomatize that $V$ is a Grothendieck universe. Add Tarki's axiom of every object is an element of some Grothendieck universe. Now is it possible to arrange that all Grothendieck universes containing $V$ do not add additional theorems about elements of $V$ in the pure set theoretic language (i.e. in $L(\in,=)$)?

Formally: if $\varphi(a_1,..,a_n)$ is a formula in $L(\in,=)$, in which all and only $a_1,...,a_n$ occur free, then:

$\forall W \big{(}W \text { is a Grothendieck Universe } \wedge V \in W \to \\\forall a_1,..,a_n \in V [ \langle W,\in \rangle \models \varphi(a_1,..,a_n) \implies \\\langle V,\in \rangle \models \varphi(a_1,..,a_n)] \big{)}$

The second question can we upgrade that to be from free variables in $W$? i.e. can we have the following?

if $\varphi(a_1,..,a_n)$ is a formula in $L(\in,=)$, in which all and only $a_1,...,a_n$ occur free, then:

$\forall W \big{(}W \text { is a Grothendieck Universe } \wedge V \in W \to \\\forall a_1,..,a_n \in W [ \langle W,\in \rangle \models \varphi(a_1,..,a_n) \implies \\\langle V,\in \rangle \models \varphi(a_1,..,a_n)] \big{)}$