Cardinal inequalities in set theory without choice

Question

Assume that we work in $\mathrm{ZF}$ without choice. Let $\kappa, \lambda, \mu, \nu$ be cardinals such that $\kappa < \lambda$ and $\mu < \nu$. Is the following true: $$\kappa + \mu < \lambda + \nu.$$ Is there a reference that contains the properties that cardinals in set theory without choice (Jech's book "The axiom of choice" does not cover such properties).

Answer 1

No, it is certainly possible that this doesn't hold.

For example, it is consistent that $\Bbb R$ can be split into two strictly smaller sets, $A$ and $B$.

Monro, G. P., Decomposable cardinals, Fundam. Math. 80, 101-104 (1973). ZBL0272.02085.

In that case, take $\lambda=\nu=2^{\aleph_0}$ and $\kappa=|A|$, $\mu=|B|$.