For every set $X$ there cannot be a partition on $X$ of a larger size than the set $\iota``X$ of all singleton subsets of $X$. Formally: $$\begin{align} \forall X \forall P: P \text { is a partition on } X \to \neg [&\exists f (f: \iota``X \rightarrowtail P ) \land \\ & \not \exists g (g: P \rightarrowtail \iota ``X)] \end{align} $$

Where:

$$\begin{align} P \text { is a partition on } X \iff &\forall k \in P (k \neq \emptyset) \land \bigcup P = X \ \land \\ & \forall a ,b \in P (a \neq b \to a \cap b = \emptyset) \end{align}$$

Where $`` \rightarrowtail "$ stand for injections.

What forms of Choice this principle with the other axioms of $\sf ZF$ can prove?

Same question but with regards Stratified $\sf ZF$?

Stratified $\sf ZF$ is the theory axiomatized by the stratified axioms of $\sf ZF$, where stratified is defined after Quine as in $\sf NF$.