Set equality

Proave A = B by showing A is a subset of B and B is a subset of A using a direct proof.

I have no fucking idea how to do this.

You'll have to specify which axiomatic system.
If you're just looking for a naive proof: if A is a subset of B, then A/B=C. If C is empty (∅), then A=B (can be seen as an alternate definition of equality), thus A=B ∪  C. Now you can write B/A=D (again if ∅, it's equal), thus you have B = A ∪  D, now substitute A: B = B ∪  D ∪  C. Now subtract B from them:
B/B = B∪ D∪ C/B => ∅ = D∪ C. Now given the axiom that the empty set is the only subset of itself, we can conclude that D=C=∅, and thus A/B=∅ => A=B.