Theorem 1.7: (a, b ) = (c, d ) if and only if a = c and b = d.

Proof: If a = c and b = d then

so

For the converse, assume

then

by Definition 1.8

case 1: a is not equal to b.

so

The elements of {{a }, {a , b }} are {a } and {a, b }. so either

or

If {c } = {a } then since

then

in which case

The only thing that could go wrong, then would be if the other eventuality were the case i.e. that

But in that case,

so

in which case

but the same argument would show that

This would give us

and

which is not the case here. Conclude that

and

Next,

so

Thus either

or

We can't have {a , b } = {c } because we know that a = c, but b is not equal to a so b cannot be equal to c. Thus

and so

The only other possibility would be that

Thus

we know that

so

case 2: a = b

In this case

So

So if

then

so

and hence

so

so

so

We then have that

so

and

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