Can ZF set theory prove this cardinal inequality?

Question

I know that ZF set theory without the axiom of choice can prove the cardinal inequality $2^{\aleph_0} \geq \aleph_1$. That raises the question, can ZF set theory without the axiom of choice prove the generalized cardinal inequality $2^{\aleph_\alpha} \geq \aleph_{\alpha + 1}$, for all ordinals $\alpha$?