"proves that if A and B are sets with A equivalent to a subset of B and B equivalent to a subset of A then A and B are equivalent. This theorem was also proved by Felix Bernstein and independently by E. Schroeder."
As you all know, this famous result is almost universally known as the "Schroeder-Bernstein theorem". But if Cantor did prove it, as stated above, why his name is not mentioned? I do not remember now where I saw "Cantor-Schroeder-Bernstein theorem", but I know of no other text (written before the 1970s) where it is said that Cantor did really <<prove>> it.
Let me examine with you some authoritative sources with which I am very well acquainted. According to Abraham Fraenkel in his book "Abstract Set Theory" (1952), "the first complete proof" of the theorem was given by Bernstein in 1897 and published in 1898 in Borel's "Lecons sur la theorie des fonctions". We are also informed that in 1897 Schroeder gave a "similar" proof which, however, was "defective" (although Fraenkel is not explicit on this point). On the other hand, I know from other sources that Cantor had already guessed the theorem earlier. Thus, to do historical justice, the theorem would deserve to be called "Cantor-Bernstein theorem". Now, Cantor in fact tried to prove the result in his final double treatise. So the next thing to ask is: Is Cantor's argument incomplete in some respect? To get an answer we will have to examine his proof itself.
Cantor's argument is as follows (according to the well-known Jourdain's translation):
"We have seen that, of the three relations a = b, a < b, b < a, each one exclude the two others. (...) Not until later, when we shall have gained a survey over the ascending sequence of the transfinite cardinal numbers and an insight into their connexion, will result the truth of the theorem: A. If a and b are any two cardinal numbers, then either a = b or a < b or a > b. From this theorem the following theorems (...) can be very simply derived: B. If two aggregates M and N are such that M is equivalent to a part N1 of N and N to a part M1 of M, them M and N are equivalent."
Note that B is precisely the result under discussion. But as we can now see, here Cantor is simply saying that A implies B and that he will prove A later. However, he never did it! That is why his proof was considered unsatisfactory. Indeed, theorem A (that any two sets are comparable) was proved (modulo the axiom of choice) by Zermelo in 1904. On the other hand, Bernstein's proof is "complete" in that it is entirely independent of the existence of a well-ordering between arbitrary cardinals. In this sense, therefore, Cantor did not prove the Schroeder-Bernstein theorem. But even accepting the comparability hypothesis, his chain of reasoning is not fully cogent.
We can get some more exciting information in Zermelo's papers "A new proof of the possibility of a well-ordering" and "Investigations in the foundations of set theory I" both published in 1908 (and reproduced in Heijenoort's anthology). Referring to the " Schroeder-Bernstein theorem", Zermelo says that in 1906 he discovered a proof that "rests solely upon Dedekind's chain theory" and which "unlike the older proofs by Schroeder and F. Bernstein as well as the latest proof by Konig (1906), avoids any reference to ordered sequences of order omega or to the principle of mathematical induction". He also mentions a "quite similar" proof published by Peano in that same year. What nobody knew was that they had discovered essentially a proof that had already been obtained by Dedekind in 1887 and 1899, but which would be published only in 1932. So Bernstein's proof was not really the "first complete" one. In many disguised forms, Dedekind's proof was rediscovered and published in many numbers of the American Mathematical Monthly.