Is "constructible from" a transitive relation?

Question

In Jech's Set Theory, exercise 13.27, it is hinted that $X \in L[Y]$ and $Y \in L[X]$ together imply $L[X]=L[Y]$. I tried to prove this fact without success, although I suspect the proof is simple.

My approach has been to try to show that if $Y \in L[X]$, then $L[Y] \subset L[X]$; i.e. that "constructible from" is a transitive relation. Is this true? If so, can someone suggest an approach to proving it (without giving away the proof)? I have mostly been trying induction arguments.

Answer 1

You need to use/prove the following theorem (13.22 in Jech):

$L[X]$ is the smallest transitive model $M$ which includes all the ordinals and $M\cap X\in M$.

Then this is more or less immediate by minimality.