I'm not sure if this is a soft question, or should be community wiki.

I was explaining to a student how to prove that two sets were equal using what I called the 'oldest trick in the book': to show that $A = B$, prove $A \subseteq B$ and $B \subseteq A$. This got me thinking: what are the other ways of showing that two sets are equal. There's of course the bijection method (establish a 1-1 onto correspondence), but I couldn't think of others off the top of my head.

Are there many more general-ish techniques for proving two sets equal ?