Prove that A = B ⇒ A⋂B = A⋃B

Question

This one's a simple one, just asking for confirmation...

Here's my proof :

Since A=B, then A⋃B = A⋃A = A                             ...1
and for the same reason : A⋂B = A⋂A = A                   ...2

So, from 1 and 2, we get : A⋂B = A⋃B
So, A = B ⇒ A⋂B = A⋃B

The reason I'm asking for confirmation for this is because my textbook had a longer method for doing this and this method just seems too small compared to that one.

Let me know if this is right

Thanks