Show Set Theory Subset Relationship: x, y $\in$ B

[Mr Davis 97]

Homework Statement

Assume that $x$ and $y$ are members of a set $B$. Show that $\{ \{x\}, \{x,y\} \} \in \mathcal{P} \mathcal{P} B$

Homework Equations

The Attempt at a Solution

I know that $\{ \{x\}, \{x,y\} \} \in \mathcal{P} \mathcal{P} B$ iff $\{ \{x\}, \{x,y\} \} \subseteq \mathcal{P} B$, but I don't see where this gets me. To me it's obviously true, but I don't see how to show it.

 

[FactChecker]

If something seems obvious, just see if you can use the basic definitions to state it. State the definition of power set and start there.
Since x and y ∈ B, {x} and {x,y} are subsets of B. By the definition of the power set ...

 

[Mr Davis 97]

If something seems obvious, just see if you can use the basic definitions to state it. State the definition of power set and start there.
Since x and y ∈ B, {x} and {x,y} are subsets of B. By the definition of the power set ...

Oh, right. That seems really obvious now. So the power set of B is the set of all subsets. Since $\{x\}$ and $\{x,y\}$ are subsets of B, the set $\{ \{x\}, \{x,y\} \}$ must be a subset of the power set.