TITLE: Proving that $\{a,\{b\}\}$ is a representation of an ordered pair in ZFC
QUESTION [5 upvotes]: I know that you can represent an ordered pair using the ZFC axioms like that : $\{\{x\},\{x,y\}\}$. 
The question is :  

Prove that it is possible to represent an ordered pair in the form of : $\{x,\{y\}\}$. 

I need to show that $\{x,\{y\}\} = \{a,\{b\}\}$ if and only if $x = a \wedge y = b$.
The first side of the proof is trivial because all you need to do is to place $x$ instead of $a$ and $y$ instead of $b$. 
I'm stuck in the other side of the proof. If dividing it into $2$ cases :   

If $x = a$ than $\{y\} = \{b\}$ and thus $y = b$.
If $x = \{b\}$ than $\{y\} = a$, and this is where I'm stuck at.

Thanks in advance !

REPLY [10 votes]: It isn't true. Consider $\{\{a\},\{b\}\}$. It can be viewed either as $\langle \{a\},b\rangle$ or as $\langle \{b\},a\rangle$, according to your new definition of ordered pair. Since these can be different, this definition doesn't have the ordered-pair property.