We ought never to allow ourselves to be persuaded of the truth of anything unless on the evidence of our own reason – René Descartes

Show that P(A) is a subset of P(B) if A is a subset of B

Proof: Suppose A is a subset of B.
Let x be any element of $\mathcal{P}(A)$, the power set of A.
Then x is a subset of A by the definition of the power set.
But because A is a subset of B, x must also be a subset of B.
Hence x is an element of $\mathcal{P}(B)$ by the definition of power set.
Because this condition holds for any element of $\mathcal{P}(A)$,
$\mathcal{P}(A)$ is a subset of $\mathcal{P}(B)$.
Q.E.D.

Does this proof work for uncountably infinite sets A and B?
It is intuitively clear that this proof works for any finite set A.
But if A was countably infinite, or uncountably infinite,
then I’m not sure if this argument makes any sense at all.