Ordered pairs

We wish to demonstrate how ordered pairs may be defined in terms of ordinary sets, without introducing new axioms. We begin with a LEMMA: For any objects x, y, z if {x , y} = {x , z} then y = z
Proof: y is a member of {x , y}, so y is a member of {x , z}. Therefore either y = z, and we are done, or else y = x. In this case {x , z} = {x , y} = {x} and so z = x = y.

We now make the
DEFINITION: The ordered pair (x , y) is (x , y) = { {x} , {x , y} }
and prove the fundamental
THEOREM: For every a, b, x, y, (a , b) = (x , y) if and only if x = a and y = b
Proof: It is obvious that if x = a and y = b, then (x , y) = (a , b).
To prove the converse, assume that { {a} , {a , b} } = { {x} , {x , y} }. Then {a} is an element of { {x} , {x , y} } so either {a} = {x} or {a} = {x , y}. In the first case x = a and in the second x = y = a. In either case {a} = {x}, so that { {a} , {a , b} } = { {a} , {a , y} }. By our lemma, this implies that {a , b} = {a , y} and so, by the lemma again, b = y, as required.

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