How to translate set propositions involving power sets and cartesian products, into first-order logic statements? As seen from an earlier question of mine one can translate between set algebra and logic, as long as they speak about elements (a named set A is the same as {x ∣ x ∈ A}).
However I've stumbled upon propositions that involve cartesian products and power sets and I'm not sure how to translate those into logic statements. For instance:
(A  × B)    =   (B  × A)    if and only if  A   =   B
or


*

*if   A   =   ℘(A)    then    ℘(A)    =   ∅

*if   A   ⊆ ℘(A)  then    ℘(A)    =   ∅

*if A ∈ ℘(A)  then    ℘(A)    =   ∅


and even a combination of the two:
℘(A ×   B)  ⊆       ℘(A)    ×   ℘(B)
Note that "×" is the cartesian product symbol, and ℘ the power-set.
Can someone provide any insight on this?

There will be differences of opinion about notation, but I like to use:
The Cartesian Product of sets $A$ and $B$ is given by $ A\times B\$ such that:

$\forall x,y:[(x,y)\in A\times B\iff x\in A\land y\in B]$

The power set of set A is given by $\mathcal P(A)$ such that:

$\forall x: [x\in \mathcal P(A)\iff \forall y:[y\in x\implies y\in A]] $

