On the cardinality of quotient set

Question

I have read here that a quotient set may have a cardinality strictly greater of the starting set in ZF. I would to know if we add to ZF the axiom "for every set $X$ and equivalence relation $\thicksim$ on $X$, we have $|X/\thicksim|\leq|X|$". I was wondering what theory we obtain, certainly it is strictly stronger than ZF and weaker than ZFC, but is strictly weaker than ZFC or is equivalent? I was looking to proof that is equivalent considering a nonempty family $\{X_i|i\in I\}$ of nonempty set and i considered the relation on the product $X=\prod\{X_i|i\in I\}$ fixed a factor $X_j$, and let $p_j$ the projection, $(x_i)\thicksim(y_i)$ iff $x_j=y_j$, my idea was that the quotient $X/\thicksim$ is substantially $X_j$ (in particular they are in bijection) but now i think this doesn't work.