A set S is called a hereditary set if every x ∈ S is also a hereditary set.
If H is a set, then ℘(H) is the power set of H.
Theorem: If H is a hereditary set, then ℘(H) is also a hereditary set.
Proof: Let H be a hereditary set. We will prove that ℘(H) is also a hereditary set. To do so, let F ∈ ℘(H) and let x ∈ F. Since F ∈ ℘(H), we know that F ⊆ H. This means x ∈ H. Since H is a hereditary set, it follows that x is also a hereditary set. Therefore, F must be a hereditary set which means ℘(H) is also a hereditary set. ∎
Question: Is my proof on the right track? Any critique on style is also welcome. Thank you.