Shoenfield absoluteness is well known for $\Pi\_2^1$-statements, but it does not hold between a countable transitive model of ZFC and the universe.
 But it is also known that $\Pi\_1^1$ statements are absolute between a countable transitive model of set theory and the universe.


My question is: 


Where can I find a good reference for the last fact?



Absolutness of $\Pi_1^1$ statements
This is an immediate consequence of the fact that every $\Pi^1\_1$ statement is equivalent to the assertion that a certain relation is well-founded, and well-foundedness is absolute between transitive models. If a larger model thinks a relation is well-founded, then the smaller model must agree since it can have no infinite descending sequence. If a smaller model thinks a relation is well-founded, then it will have an ordinal ranking function, which will witness well-foundedness in the larger model. 


This theorem is contained in every descriptive set theory book, such as the excellent one of Moschovakis's. 


