Is a ceiling fan box brace required?
Brace Yourself!
The problem is not the simple strength of the ceiling joists - we'll presume that even a single joist is strong enough to hold up a fan. The problem is that a box next to a joist will not be stable enough to support a fan if it is only supported on one side. A typical brace solves that problem.
An alternative is to place the box under a joist. However, unless this is an open ceiling, that will (a) have the box dropping below the finished ceiling and (b) not look very nice.
So unless you really like a wobbling fan, use a brace:

You also get the nice advantage of more flexibility for placement of the fan, though if you are making use of an existing hole in the ceiling then that is a non-issue.