[EDIT] This posting had been edited to assert that we are speaking about regular mutual stages. 


Let $H\_{\kappa}$ be the set of all sets that are hereditarily strictly smaller in cardinality than cardinality $\kappa$. Formally


$H\_\kappa = \{x: |TC(x)|<\kappa\}$


Where $TC(x)$ is the transitive closure of $x$ defined as the intersectional set of all transitive supersets of $x$.


Now a stage of the cumulative hierarchy $V\_{\kappa}$ is either identical to $H\_{\kappa}$ or not. For example $V\_\omega = H\_\omega$, and of course $V\_\emptyset = H\_\emptyset$. Lets call such stages as "mutual" stages.


Define: $mutual(x) \iff \exists \kappa (x=V\_\kappa =H\_\kappa) $


**Axiom:** The class of all regular mutual stages is proper.


Formally: $\forall x \exists \kappa (x \in V\_\kappa \land mutual(V\_\kappa) \land regular(\kappa) )$



> 
> What is the strength of adding this axiom to ZFC?
> 
> 
> 



What is the strength of claiming that the class of all $V_\kappa$ stages that are $H_\kappa$ when $\kappa$ is regular, is inaccessible?
The axiom in your question adds no strength; it's provable in ZFC. The claim in your title, involving inaccessibility, has the same strength as the existence of an inaccessible cardinal.


