On pg 190 of Jech's Set Theory, he proves V = L implies GCH. I understand it all except the following:

Thus let X ⊂ $ω_α$. There exists a limit ordinal δ>$ω_α$ such that X ∈ $L_δ$. Let M be an elementary submodel of $L_δ$ such that $ω_α$ ⊂ M and X ∈ M, and $|M| = \aleph_{\alpha}$.

I think this uses the Lowenheim Skolem theorem to get the model of size $\aleph_{\alpha}$, but why should $\omega_{\alpha} \subset M$ and why should $X \in M$, or how do we construct such an M?

I know there are other proofs of this theorem which uses a version of the reflection principle which gives you this directly, but the version of the reflection principle I know is the more basic one which doesn't deal with the cardinality of the sets for which the formulas are absolute (the beginning reflection theorems in Kunen).

Basically, I would prefer an explanation that uses the Lowenheim Skolem theorem instead. How is this done?

in$L$. $\endgroup$ – Not Mike Jun 3 '19 at 11:11