# Math Help - Simple Set Theory Theorem

1. ## Simple Set Theory Theorem

Hello,

I am trying to prove that

A is a subset of B
if and only if
the intersection of A and B is the set A.

It it seems obvious from the definitons but I am having trouble formalizing the proof.

In the forward direction, we know an element x in A implies x in B, and that the intersection of A and B is just all the elements y such that y is in A AND y is in B...

So it seems obvious to me that A intersection B is just A, but how can I state it formally? Similarily for the backwards direction.

Thanks!

2. I do it one way. Suppose that $A\cap B=A$.
If $x\in A$ then it follows that $x\in A\cap B$ implying that $x\in B$.
So $A\subseteq B$.