axiom of union

The axiom of union then produces the desired result, .

And this could be used to generate all hereditarily finite sets without using the axiom of union.

This union exists regardless of the set's size, by the axiom of union.

The axiom of union is generally considered uncontroversial, and it or an equivalent appears in just about any alternative axiomatization of set theory.

So one chooses the least element of each member of x to form y using the axioms of union and separation in L.

Note that adopting this as an axiom schema will not replace the axiom of union, which is still needed for other situations.

This is the set consisting of all objects which are elements of A or of B or of both (see axiom of union).

The usual axiom of extensionality for sets, as well as one for functions, and the usual axiom of union.

Axiom of union: For any set x there is a set y whose elements are precisely the elements of the elements of x.

In ZFC, one proves that these notions all generate or apply to sets via the ZFC axioms of union, separation, and power set.

If this subtheory is augmented with the axiom of infinity, each of the axioms of union, choice, and infinity is independent of the five remaining axioms.

That this union of M is a set no matter how large a set M itself might be, is the content of the axiom of union in axiomatic set theory.

It implies the axiom schema of specification, axiom schema of replacement, axiom of global choice, and even, as noticed later by Azriel Levy, axiom of union at one stroke.