# Proving that $\{a,\{b\}\}$ is a representation of an ordered pair in ZFC

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 !

• To the person who voted to close this as "missing context or other details": I do not understand this. The question is perfectly clear. We often vote to close as "missing context" when OP pastes a homework question with no indication of their background or what their question is, but that did not happen here.
– MJD
Mar 8, 2015 at 15:56
• @MJD someone is rampaging through everything and voting to close with this reason. There are 50 close votes in the queue at a time, probably all from this person. Mar 8, 2015 at 16:29

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.