Let $\forall X,P \ (P \text{ is a partition of X} \to |P| \leq |X|)$ be intuitive partitioning $``\sf IP"$.
In English it means that every set can only be partitioned into pieces the amount of which is less than or equal to the amount of elements in it.
Question: What's the consistency strength of $$\sf ZF+IP+\neg BT$$, where BT is Banach-Tarski paradox?
$\sf \neg BT$ is just a simple straightforward negation of Banach-Tarski paradox.