There is a standard notation $\mathrm{ZF}[n]$ for Zermelo Fraenkel set theory with the power set axiom restricted to saying the set of natural numbers has $n$ successive power sets $\beth\_0\dots\beth\_n$. 


Is there a similarly standard notation for the extension of $\mathrm{ZF}[n]$ by an axiom saying every set has an hereditary embedding in $\beth\_n$? 



Notation for upperbound power sets.
In modern set theory $H(\kappa)$ denotes the collection of sets $X$ such that $X$ is hereditarily of cardinality strictly less than $\kappa$. 



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