# The intersection of sets is a subset of each set

## Theorem

That $A\cap B\subset A$

Of course by commutivity of $\cap$ we have $A\cap B\subset B$ (as $A\cap B=B\cap A$ and $B\cap A\subset B$ by the statement above)[1]

## Proof

We will show $x\in A\cap B\implies x\in A$ then use the implies and subset relation to conclude $A\cap B\subset A$

Suppose $x\in A\cap B$

Then $x\in A$ and $x\in B$

QED (we have shown that if $x\in A\cap B$ then $x\in A$, this is what $\implies$ means)