More proofs of set relations

Let $A = \{ 1 \}$ and $B = \{ \{ 1 \}, 1 \}$ . Prove or disprove the following.

True.
Proof. Since 1 is the only element in $A$ , and $1 \in B$ , we have that every element of $A$ is contained in $B$ . Hence, $A \subseteq B$ . Further, since $\{ 1 \} \in B$ , but is not in $A$ , we see that $B \nsubseteq A$ . Therefore, $A \subset B$ . ∎
True.
Proof. Again, since $A \subset B \implies A \subseteq B$ , we have $A \subseteq B$ by part (a). ∎
True.
Proof. By the definition of $B$ we have that $\{ 1 \} \in B$ . Since $A = \{ 1 \}$ , we have $A \in B$ . ∎
True.
Proof. By definition $A = \{ 1 \}$ ; hence, $1 \in A$ .∎
False.
Since 1 is not a set, we cannot have $1 \subseteq A$ .
False.
Since 1 is still not a set, we cannot have $1 \subset A$ .

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