Proving Set Inclusion: P(A) ⊆ P(B) Implies A ⊆ B

[LaMantequilla]

Homework Statement

Prove that if P(A) $\subseteq$ P(B) then A $\subseteq$ B,
where A and B are two sets and P symbolizes the power set (set of all subsets) of a particular set.

Homework Equations

The Attempt at a Solution

Okay, so here goes.

Because it's a conditional, we suppose P(A)$\subseteq$ P(B), and make it a "given."

From there, we look at the goal ( A$\in$ B ), and let x be arbitrary such that x $\in$ A $\rightarrow$ x $\in$ B. Because x is arbitrary, we suppose x $\in$ A.

So far, we have:

Givens:
P(A) is a subset of P(B), or $\forall$y( y $\in$ P(A) $\rightarrow$ y $\in$ P(B)
x $\in$ A

Goals:
x $\in$ B

So this is where it falls apart. Looking at the given above, I see the opportunity for universal instantiation. However, in order to do that I need to know some variable that y $\in$ P(A), or that y $\subseteq$ A. I see neither. Can you help me?

 

[Dick]

If x is an element of A then the set {x} is in P(A). Does that help?

 

[LaMantequilla]

Thanks!