Prove or disprove some set relations

If $A = \{ 1 \}$ and $B = \{ 1,2 \}$ prove or disprove the following:

True Proof. To show $A \subset B$ we must show that every element in $A$ is also in $B$ . Further (since Apostol seems to distinguish between $\subset$ and $\subseteq$ ) we want to show that $A \neq B$ .
First, since $1$ is the only element of $A$ , and $1 \in B$ , we see that every element of $A$ is indeed contained in $B$ . Hence, $A \subseteq B$ .

Then, since $2 \in B$ , but $2 \notin A$ , we see that $B \nsubseteq A$ . Hence $A \neq B$ , so $A$ is a proper subset of $B$ , or $A \subset B$ . ∎
True
Proof. This follows immediately from part (a) since $A \subset B \implies A \subseteq B$ . ∎
Not True
This is not true since $A = \{ 1 \}$ and the set $\{ 1 \}$ is not an element of $B$ (since $1 \neq \{ 1 \}$ ).
True
Proof. There is really nothing to prove here. By definition, $A$ is the set containing 1, so 1 is in $A$ . ∎
Not true
This is not true since 1 is not a set, so is not a subset of $A$ .
Not true
Again, 1 is not a set, so cannot be a subset of $B$ .

Stumbling Robot