Ordered pair
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An ordered pair [math](a,b)=\{\{a\},\{a,b\}\}[/math], this way [math](a,b)\ne(b,a)[/math].
Ordered pairs are vital in the study of relations which leads to functions
Proof of existence
It is easy to prove ordered pairs exist
Suppose we are given [math]a,b[/math] (so we can be sure they exist).
By the axiom of a pair we may create [math]\{a,b\}[/math] and [math]\{a,a\}=\{a\}[/math], then we simply have a pair of these, thus [math]\{\{a\},\{a,b\}\}[/math] exists.
The axioms may be found here
Set Theory
