Is it true that: $A\subseteq B\implies A^\circ\subseteq B^\circ ?$

Question

Let $A,B$ be algebraic groups. Denote the identity component of an algebraic group $C$ by $C^\circ$.

Is it true that: $$A\subseteq B\implies A^\circ\subseteq B^\circ ?$$

Does this follow from:

1. $A^\circ$ is irreducible in $A$, and hence irreducible in $B$, containing the identity of $B$.
2. $B^\circ$ is the unique maximal irreducible subset of $B$, containing the identity.
3. Hence $A^\circ \subseteq B^\circ$

I am fairly sure this was that straight forward, but just want to be careful due to my misunderstanding leading to my previous question. Better to be a fool now, than to be a fool forever!

Answer 1

Yes, your reasoning is correct indeed.