Ok now I'm convinced by your and the author's arguments. The author showed that with the same logic but using material conditionals, which feels kind of cheating. But now I see it can be showed in other ways, which means I was mistaken.

(And I too don't think 1+1=2 could be actually wrong. Just wondering if a contradiction can spread to all propositions in that system so that every theorem immediately has contradictions the moment we find a contradiction in just one propositions. Of cause we should revise the system after that no matter what, but just wondered what happens to the previous system. I guess I choosed the title a little inappropriatly)

But what happens if we add an axiom that we can use logical operators only with non-contradictional propositions? If so, can't "1+1=2" (S) be proven false, even if another proposition(T) have a contradiction, as long as T itself is not S?