fundamental property of ordered pairs

Question

Can someone explain me how to show that $(x,y):=$ {{$x,∅$}, {$y$, {$y,∅$}}}

satisfies the fundamental property of ordered pair...

I.e. if $(x,y)=(x',y')$, then $x=x'$ and $y=y'$.

Thanks in advance.

Answer 1

Suppose that $\{\{x,\varnothing\},\{y,\{y,\varnothing\}\}=\{\{x',\varnothing\},\{y',\{y',\varnothing\}\}$.

Then either $\{x,\varnothing\}=\{x',\varnothing\}$, in which case it has to be that $x=x'$ and from a similar reasoning, $y=y'$; or $\{x,\varnothing\}=\{y',\{y',\varnothing\}\}$.

In the latter case, it has to be that $y'=\varnothing$ and therefore $x=\{\varnothing\}$. But now it also follows that $\{x',\varnothing\}=\{y,\{y,\varnothing\}\}$, and here again we get that $x'=\{\varnothing\}$ and $y=\varnothing$. So in this case also $x=x'$ and $y=y'$.