> > how to prove ?
> >
> > (a,b)=(c,d) <==> a=c, b=d
>
> It's obvious that if a=c and b=d then (a,b)=(c,d). For the
> other direction, you can start by proving that if
> {x,y}={x,z} then y=z. Now suppose (a,b)=(c,d), that is,
> {{a},{a,b}}={{c},{c,d}}. Either {a}={c} or {a}={c,d}, both
> of which imply that c=a. So we have {{c},{c,b}}={{c},{c,d}},
> and therefore {c,b}={c,d}, and therefore b=d.
I assume you mean that, out of {c} or {c,d}, {a} = {c} is the only valid choice, as {a} = {c,d} leads to the contradiction {c,d} = {c}.
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