Given cardinal numbers $\kappa < \nu$ with $\nu$ an aleph fixed-point, do we necessarily have $\aleph_\kappa < \nu$?

Question

Given cardinal numbers $\kappa < \nu$ with $\nu$ an aleph fixed-point, its clear that $\aleph_\kappa \leq \nu$. Is this inequality in fact strict?

Answer 1

If $\nu$ is a fixed point then $\nu=\aleph_\nu$, since $\kappa<\nu$ and the $\aleph$ function is strictly increasing (and so injective), you automatically have $\aleph_\kappa<\aleph_\nu=\nu$.