I had as a question on an assignment the question in the subject. I'll write it properly here: $A\subset B\leftrightarrow 2^A\subset 2^B$
How can I go about proving this? I had a few ideas how to start, but I don't know if I'm anywhere on the right track.
- $A\subset B\leftrightarrow \forall w\in A\{w\in B\}\wedge A\ne B$
- $2^A\leftrightarrow \forall x\{x\subseteq A\}$
- $2^B\leftrightarrow \forall y\{y\subseteq B\}$
- $2^A\subset 2^B\leftrightarrow \exists z\in 2^B\{z\notin 2^A\}$
I feel like I have a few good points, but it just doesn't feel complete. I wonder if anybody can help me along.