This is the theory that the EU tries to push on it's member states. They have accepted that international treaties carry greater weight than the local laws and even the constitution of the local countries :
https://en.wikipedia.org/wiki/Supremacy_(European_Union_law)
As to whether it applies to all treaties or just the EU ones, ... good question. In practice courts haven't really followed it.
Also, just as a data point, I've been working in Brussels for close to 20 years now and seen the evolution. I work as a consultant, so it has been a mixed bag for me. However, for just about everyone else, workers, middle class, business owners, ... it has been a consistently worsening disaster.
Having regularly worked for the EU commission, let me tell you this : any idea you may have that this organization has anything but their own economic interests at heart can be cured by chatting with one of them for 5 minutes.