For a set $X$, let $|X|$ denote its cardinality. A block of a partition is a non-empty element of the partition.
Let $P$ and $Q$ be two partitions of a set $X$. If $|P| < |Q|$ then $P$ contains a block $B$ which intersects two distinct blocks of $Q$.
What is this principle called?
The proof is immediate for finite $X$, and is an easy consequence of the Axiom of Choice for infinite $X$. I am interested in using this result, and would like to refer to its proper name and to cite it correctly. Jech's Set Theory and several online sources don't seem to mention this result. In particular the Consequences of the Axiom of Choice project doesn't seem to list it (at least not in this specific form).
I would also be interested to know whether this principle is equivalent to some known consequence of the Axiom of Choice.