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 \forally( 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!