How to prove a subset relation?

Let A and Be be sets. Prove A $B"<(B B if and only if A $B"A(B B = A
I know that A is subset of B if every element in A is in B
And A=B if all element of A is B and All element of B is in A
But besides knowing that from just visualizing and drawing a graph, I
have no idea how to prove such a thing. Can anyone give a proof of
this besides drawing a venn diagram?