My teacher said the statement about "the addition of irrational numbers is closed" is true, by showing a proof by contradiction, as it is in the image. I'm really confused about this because someone in the class said for example π - ( π ) = 0, therefore 0 is not irrational and the statement is false, but my teacher said that as 0 isn't in the irrational numbers we can't use that as proof, and as that is an example we can't use it to prove the statement. At the end I can't understand what this proof of contradiction means, the class was like 1 week ago and I'm trying to make sense of the proof she showed. I hope someone could get a decent proof of the sum of irrational aren't closed, yet trying to look at the internet only appears the classic number negative of that number = 0 and not a formal proof.

https://preview.redd.it/1d1yhmwofwsc1.jpg?width=1600&format=pjpg&auto=webp&s=68dd8fa8cb0951415a18feb5bd25b2cb9380e033