Can one define $\langle x,y\rangle$ in $P(C)$?

Question

I study at course Foundations of Mathematics the below definitions and lemma:
$\langle x,y\rangle:=\{\{x\},\{x,y\}\}$ (from Kuratowski 1921)
$\langle x,y\rangle:=\{\{\{x\},\varnothing\}\{\{y\}\}\}$ (from Norbert Wiener 1914)

Lemma : If $x\in C$ and $y\in C$ then $\langle x,y\rangle\in P(P(C))$. (Lemma is true with Kuratowski definitions and $P(A)$ means power set of $A$)

1. Now is there any proof that explain there is no definition compatible with axioms of set theory and $\langle x,y\rangle\in P(C)$ instead of $P(P(C))$?
   

It seems answer is "No" for me, but I can't present any proof.
If you know any reference which can help for above question please point it.

A new approach:

Can we replace the question with below one:

Prove that there are no formulas $\varphi$ of two free variable, such that for all sets $C$, $\varphi$ describes an injective map from $C\times C$ into $P(C)$.

An other view to problem:

1. How can I check compatibility of 1. with axioms of set theory?
2. What informations I will need?

Answer 1

The fact that you can't do it for very small sets invariably means that you can't do it at all.

First of all, it is important to note that the way we encode the pairs shouldn't depend on the set we are considering at the moment. So $\langle 0,0\rangle$ should be encoded to the same set regardless to use considering it as an ordered pair in $\omega\times\omega$ or in $V_\omega\times V_\omega$.

This means that given an element $a$, $\langle a,a\rangle$ must be in every $A\times A$ whenever $a\in A$. If we want $A\times A\subseteq\mathcal P(A)$, this means that $\langle a,a\rangle\in\bigcap\{\mathcal P(A)\mid a\in A\}=\{\varnothing,\{a\}\}$. Clearly $\varnothing$ cannot encode all ordered pairs, so $\langle a,a\rangle=\{a\}$.

This means that $\langle a,b\rangle\in\mathcal P(\{a,b\})$ and that if $a\neq b$ then $\langle a,b\rangle\neq\{a\},\{b\}$. Therefore $\langle a,b\rangle=\{a,b\}$. But therefore $\langle a,b\rangle=\langle b,a\rangle$ even if $a\neq b$. And this is a contradiction to the axioms of the ordered pair.

Therefore there is no consistent definition of an ordered pair such that for every set $A$ we have $A\times A\subseteq\mathcal P(A)$.