When does the cardinality of a set equal the cardinality of an element of $V_\lambda$ for $\lambda$ being a limit ordinal? Consider the following proposition.
Proposition: let $\lambda$ be a limit ordinal and $V$ be the cumulative hierarchy starting with the null  set, and $S$ be a set with
$\vert S\vert<\vert V_\lambda\vert$. Then there exists an $x\in V_\lambda$ with $\vert x\vert=\vert S\vert$.
The proposition is trivially true for $\lambda=\omega$, and is trivially true for all limit ordinals assuming the axiom of choice,  but is it true for any other limit ordinal without  assuming the axiom of choice? My uneducated guess is no.
I find this question interesting because what appears to be a gch condition on successor ordinals in the cumulative hierarchy is merely an axiom of choice condition on limit ordinals. If the above proposition holds then what appears to be a gch condition on successor ordinals is unconditionally true on all limit ordinals. This could be used as evidence (not decisive evidence but evidence none the less) for accepting gch.
I had to answer the below questions only by editing the above text because the software won’t let me comment. In my original post I forgot to indicate that I was asking about the case where we don’t have the axiom of choice. I edited the above appropriately. Sorry!
 A: The proposition holds for the limit ordinal $\lambda$ iff $V_\lambda$ is wellorderable.
For if $V_\lambda$ is wellorderable where $\lambda$ is a limit, then it proposition easily follows at $\lambda$. Conversely, suppose the proposition holds at limit ordinal $\lambda$. (Without using AC) we can specify a certain ordinal $\eta$ and an injection $\pi:\eta\to V_\lambda$, such that $\eta$ cannot be injected into any $V_\alpha$ with $\alpha<\lambda$, but then applying the hypothesis with $S=\eta$, it follows that $|V_\lambda|=\eta$.
To define $\eta,\pi$, first for each $\alpha<\lambda$, define $\eta_\alpha$ and an injection $\pi_\alpha:\eta_\alpha\to V_{\alpha+1}$, as follows. For ordinals $\beta$, let $W^\alpha_\beta$ be the set of all binary relations $R\in V_\alpha$ which are wellorders in ordertype $\beta$, and let $\eta_\alpha$ be the least $\beta$ such that $W^\alpha_\beta=\emptyset$. Then define $\pi_\alpha:\eta_\alpha\to V_{\alpha+1}$ by $\pi_\alpha(\beta)=(\alpha,W^\alpha_\beta)$. Now let $D=\{(\alpha,\beta)\bigm|\alpha<\lambda\wedge\beta<\eta_\alpha\}$, and define $\pi':D\to V_\lambda$ as $\pi'(\alpha,\beta)=\pi_\alpha(\beta)$. Note that $D$ is wellorderable, and let $\eta$ be an ordinal and $\sigma:\eta\to D$ a bijection. Then define $\pi:\eta\to V_\lambda$ by $\pi(\xi)=\pi'(\sigma(\xi))$. Then $\pi$ is an injection, so $|\eta|\leq|V_\lambda|$. I claim that $|\eta|=|V_\lambda|$. For otherwise we have $|\eta|<|V_\lambda|$. So applying the proposition at $\lambda$ with $S=\eta$, we get that $|\eta|<|V_\alpha|$ for some $\alpha<\lambda$. But this easily contradicts the construction.
