Editing Axiom of pairing (section)

=== Stronger ===
Together with the [[axiom of empty set]] and the [[axiom of union]], the axiom of 
pairing can be generalised to the following schema:
:<math>\forall A_1 \, \ldots \, \forall A_n \, \exists C \, \forall D \, [D \in C \iff (D = A_1 \lor \cdots \lor D = A_n)]</math>
that is:
:Given any [[finite set|finite]] number of objects ''A''<sub>1</sub> through ''A''<sub>''n''</sub>, there is a set ''C'' whose members are precisely ''A''<sub>1</sub> through ''A''<sub>''n''</sub>.
This set ''C'' is again unique by the [[axiom of extensionality]], and is denoted {''A''<sub>1</sub>,...,''A''<sub>''n''</sub>}.

Of course, we can't refer to a ''finite'' number of objects rigorously without already having in our hands a (finite) set to which the objects in question belong. Thus, this is not a single statement but instead a [[schema (logic)|schema]], with a separate statement for each [[natural number]] ''n''.
*The case ''n'' = 1 is the axiom of pairing with ''A'' = ''A''<sub>1</sub> and ''B'' = ''A''<sub>1</sub>.
*The case ''n'' = 2 is the axiom of pairing with ''A'' = ''A''<sub>1</sub> and ''B'' = ''A''<sub>2</sub>.
*The cases ''n'' > 2 can be proved using the axiom of pairing and the [[axiom of union]] multiple times.
For example, to prove the case ''n'' = 3, use the axiom of pairing three times, to produce the pair {''A''<sub>1</sub>,''A''<sub>2</sub>}, the singleton {''A''<sub>3</sub>}, and then the pair {{''A''<sub>1</sub>,''A''<sub>2</sub>},{''A''<sub>3</sub>}}.
The [[axiom of union]] then produces the desired result, {''A''<sub>1</sub>,''A''<sub>2</sub>,''A''<sub>3</sub>}. We can extend this schema to include ''n''=0 if we interpret that case as the [[axiom of empty set]].

Thus, one may use this as an [[axiom schema]] in the place of the axioms of empty set and pairing. Normally, however, one uses the axioms of empty set and pairing separately, and then proves this as a [[theorem]] schema. Note that adopting this as an axiom schema will not replace the [[axiom of union]], which is still needed for other situations.