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.