Anarchism and democracy are not diametrically opposed.
The current democratic system, maybe - but democracy in it's purest sense? The rule of the people?
To quote an essay I quite like "Actually, the record is more mixed than that. Of course all anarchists, from the start, have opposed bourgeois-democratic representative capitalist governments, often referring to them simply as “democracies.” And anarchists have supported the rights of individuals and minorities to freedom in many areas (religion, sex, speech, art, life styles, etc.)—areas which do not require collective decision-making. At the same time, almost all anarchists have advocated group decision-making, at work or in communes, using terms such as “self-governing, self-managing, autogestion, self-rule, self-determination.” What they have meant in practice has been usually indistinguishable from radical, participatory, democracy."