Axiom ax-un 4410

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 4412 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4413. A version using class notation is uniex 4414.

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 4102), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 264).

The union of a class df-uni 3789 should not be confused with the union of two classes df-un 3119. Their relationship is shown in unipr 3802. (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 2137	. . . . . 6 wff 𝑧 ∈ 𝑤
4		vx	. . . . . . 7 setvar 𝑥
5	2, 4	wel 2137	. . . . . 6 wff 𝑤 ∈ 𝑥
6	3, 5	wa 103	. . . . 5 wff (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)
7	6, 2	wex 1480	. . . 4 wff ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)
8		vy	. . . . 5 setvar 𝑦
9	1, 8	wel 2137	. . . 4 wff 𝑧 ∈ 𝑦
10	7, 9	wi 4	. . 3 wff (∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
11	10, 1	wal 1341	. 2 wff ∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
12	11, 8	wex 1480	1 wff ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

This axiom is referenced by:  zfun  4411  axun2  4412  bj-axun2  13757