Prove that: If C ⊆ A and C ⊆ B then C ⊆ A∩B

Question

I'm struggling with this proof for almost 2 hours and i just can't seem to reach a solution. So it would mean a lot if y'all could help me!! Again the question is Prove that, for all sets A, B and C, if C ⊆ A and C ⊆ B then C ⊆ A∩B

Thanks a lot :)

Answer 1

Here, we want to show that if $C \subseteq A$ and $C \subseteq B$, then $C \subseteq A\cap B$.

You can take an arbitrary element from $C$, call it $x$. So $x\in C$. Since $x$ is in the set $C$, and $C$ is contained in $A$, then we know that $x$ must be in $A$ also. So $x\in A$. Likewise, since $C$ is also contained in $B$, $x$ must also be in $B$. So $x\in B$.

Since $x\in A$ and $x\in B$, then... ????