[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?