Is $\{A\}\subseteq B$ ⟺ $A\in B$ in Set Theory? I am learning Set Theory using the book "Axioms and Set Theory" by Robert André, and it asks me to prove the following exercise:

If $A$ and $B$ are sets, show that $P(A) \in P(B)$ implies $P(A) \subseteq B$, and so $A \in B$.

And for it, I thought it could help me to know if my assumption is true.
$$\{A\} \subseteq B⟺A \in B.$$
I thought so because by the book's definition $A\subseteq B$ means "All elements of $A$ are elements of $B$" and $A\in B$ means "$A$ is an element of $B$". And I thought the element of $\{A\}$ is $A$, so it could be right.
I am not a mathematician or in University. Thank you for reading it.
 A: Yes: saying that $\{A\}\subseteq B$ is the same as $A\in B$ is correct.
A: Great question!  You wrote "Is $\{A\} \subseteq B  = A \in B$?"
A better (at least, more standard) way to write your question is:
Is $\{A \} \subseteq B \iff A \in B$ true?
In math, we tend to save the $=$ symbol for algebraic expressions (like, $x = 2$, or $2x + 1 = y$, etc.).  Your question is asking if the statement "$\{A \} \subseteq B$" is equivalent to the statement "$A \in B$".  So instead of the middle $=$ symbol, you should use the $\iff$ symbol.
The answer is yes.  Let's look at the definition of the symbol $\subseteq$.
Suppose $C$ and $D$ are two sets.  We say $C \subseteq D$ if every element of $C$ is in $D$, i.e., for each $c \in C$, we have $c \in D$.
Your first statement says $\{A \} \subseteq B$.  By the definition above, every element of the set $\{A \}$ must be in $B$.  But the set $\{A\}$ has only one element: $A$.  So if $\{A\} \subseteq B$, we have $A \in B$.
Let me know if you have any questions!
