This question is almost more of an economics question, but I suppose it could be a philosophy or politics question too. It is indeed a false dichotomy. It would not be, if the economy were a closed environment and thus a zero sum game.
However, the sun gives the planet energy which humans can then use to expand the technosphere: the total system of all things created, used, and shared by humans, and there is a lot of energy available to us to use, for millions, if not hundreds of billions of years. Therefore one human does not have to diminish another person's wealth in order to gain wealth. People can work together to collectively increase wealth, through various methods, including reciprocity, consumerism, capitalism, and so on.
The connection between my answer and the question seems to have been misunderstood, so let me try to explain it in better detail. Because economics is not a zero sum game, everyone can become "rich." The existence of a "poor" group is not necessary in order for there to be a rich group and vise versa. Furthermore, neither "poor" or "rich" have to be defined in terms of the other.
What does it mean to be poor? Economically, it means not having enough resources to reasonably survive, long term, without the immediate and constant aid of another. What does it mean to be rich? It means having enough resources in order to be able to do what you wish, without the need to expend additional energy in order to maintain that status.
Under these two definitions, it is clear that a society could be made up entirely of rich people. In this scenario, no one would need to work in order to "make a living." In order for such a society to exist, I think we would need to have a significant advancement in energy availability however. Such advancement could include efficient nuclear fusion, access to the currently theoretical and certainly debated "zero point energy," or some other form of yet to be discovered energy reservoir, as well as technology which would be able to efficiently translate that energy into production, such as a replicator.
On the specific issue of dichotomy, while "poor" and "rich" do not need to be defined in terms of the other, and while we do not need poor people in order for there to be rich people, I would say that one cannot be both poor and rich. They are mutually exclusive. You can however be neither poor or rich. Both of these points follow from the definitions that I provided.