The first math course as math major in university involved proving that a+b=b+a and that a+(b+c)=(a+b)+c. It's quite fun to go from 'advanced' calculus during the final classes of high school, to "OK, let's consider the expression 1+1=2. What does it actually mean and why is it true?"
It's been 20+ years, and I don't remember the exact steps the course went thought. But basically we started with defining N using the Peano axioms (although I don't recall the name 'Peano' being mentioned) and proving some basic rules of addition and equality. Then we defined subtraction, multiplication and inverse elements, and constructed Z and Q. From there you get to algebraic groups, and from there you can hand wave a lot of details.
As I said it was literally one of the first math courses we did, so it wasn't super rigorous with all the technical and logical details.
Then in university, there’s Elementary Functional Analysis!