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)

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)