Use a technique similar to that used to derive Russell’s paradox to prove that for any set ???

Use a technique similar to that used to derive Russell’s paradox to prove that for any set A.

P(A)⊆A.

Given information:

Russell’s Paradox

Concept used:

Russell’s paradox says that,

If S={A|A is a set and A∉A} is the set of all sets that are not elements of themselves, then neither S∈S nor S∉S.

Calculation:

Russell’s paradox says that,

If S={A|A is a set and A∉A} is the set of all sets that are not elements of themselves, then neither S∈S nor S∉S.

Use the above technique to prove that P(A)⊆A

Suppose there exists a set A that contains its power set.

That is, P(A)⊆A

Consider the set B={x∈A|x∉x}. Then, B is contained in A, and so B∈P(A)