I don`t think this statement is correct. Certainly, co-operation between humans is very important and has benefits, but it is not unique to humans and so to say it is this which makes us human is too simplistic.
We can find levels of co-operation in animals, not only when food is involved. Elephants have been observed in the wild working together to pull another free from mudflaps where it had become stuck, they worked together for same end, the release of another. This could be likened to the helping the man who fell off subway platform. Birds will work together to swarm at a predator to chase it away when a nest is under attack, co-operation. Also the arthritic chimp that was hoisted up to safety in the tree and then the other chimps co-operated to fetch and bring water to it. These are all levels of co-operation..working to the same end, in this case, to feed the arthritic chimp.
As for toddlers opening doors without prompting, some will, others wont. Some see the problem and reason the solution...they then may or may not co-operate, but first they use human mechanisms to see the problem before co-operation is produced. Take away the ability to reason and problem solve, you will not get much co-operation without prompting.
As co-operation can be seen across nature, it therefore follows this cannot be what makes us human, and whilst it can be seen to be done at times altruistically, it often serves us on a personal level. If we co-operate within the group, we can share the benefits. It is, as I said, a very important part of being human, it has many benefits..but I feel it is a layer of what makes us human.
Just for interest and not for debate...here is a really nice video presented by Frans de Waal regarding traits animals may [or may not] have. There is a piece at beginning on co-operation.
http://www.ted.com/talks/frans_de_waal_do_animals_have_morals?language=en