## Dimension of intersection of two sets

I am stuck with the following problem:
Let A, B be subsets of finite-dimensional vector space V. Prove that dim(A_intersection_B)=dim(A) then A is a subset of B.
I started proving by contradiction, that is, I want to show that if A is not a subset of B, then dim(A_intersection_B) is not dim(A). Assume A is not a subset of B,then there exists an x in A s.t. x is not in B. But then I don't really see how to relate that to a dimensions of the sets individually or their intersection.
(PS: my apologies for typing style, I just really have not enough time to type it up properly!)
Any help will be appreciated!
Thanks!