I need to prove «If P(A) is a subset of P(B) => A is a subset of B», generally, I understand the main way I should prove it, but the problem is in the formal, pedantic language I have to use to prove such statement.
General proof is:
1. Let suppose «a» is any element of A
2. If «a» is in A, thus «a» is a subset of P(A)…
The problem is in the formal and pedantic way to right it, can I just write, that element «a» is a subset of P(A) or should I define additional set, which will be a subset of P(A) and element «a» will be a part of this new set.
Please, help me to write such prove in a formal way with the correct syntax.