Axiom ax-un 4188

Description: Axiom of Union. An axiom of Intuitionistic Zermelo-Fraenkel set theory. It states that a set 𝑦 exists that includes the union of a given set 𝑥 i.e. the collection of all members of the members of 𝑥. The variant axun2 4190 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4191. A version using class notation is uniex 4192.

This is Axiom 3 of [Crosilla] p. "Axioms of CZF and IZF", except (a) unnecessary quantifiers are removed, (b) Crosilla has a biconditional rather than an implication (but the two are equivalent by bm1.3ii 3899), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 262).

The union of a class df-uni 3602 should not be confused with the union of two classes df-un 2977. Their relationship is shown in unipr 3615. (Contributed by NM, 23-Dec-1993.)

Assertion

Ref	Expression
ax-un	⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

Distinct variable group:   𝑥,𝑤,𝑦,𝑧

Detailed syntax breakdown of Axiom ax-un

Step	Hyp	Ref	Expression
1		vz	. . . . . . 7 setvar 𝑧
2		vw	. . . . . . 7 setvar 𝑤
3	1, 2	wel 1434	. . . . . 6 wff 𝑧 ∈ 𝑤
4		vx	. . . . . . 7 setvar 𝑥
5	2, 4	wel 1434	. . . . . 6 wff 𝑤 ∈ 𝑥
6	3, 5	wa 102	. . . . 5 wff (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)
7	6, 2	wex 1421	. . . 4 wff ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)
8		vy	. . . . 5 setvar 𝑦
9	1, 8	wel 1434	. . . 4 wff 𝑧 ∈ 𝑦
10	7, 9	wi 4	. . 3 wff (∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
11	10, 1	wal 1282	. 2 wff ∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
12	11, 8	wex 1421	1 wff ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

This axiom is referenced by:  zfun  4189  axun2  4190  bj-axun2  10706