Axiom ax-un 4223

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 4225 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4226. A version using class notation is uniex 4227.

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 3925), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 262).

The union of a class df-uni 3628 should not be confused with the union of two classes df-un 2988. Their relationship is shown in unipr 3641. (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 1435	. . . . . 6 wff 𝑧 ∈ 𝑤
4		vx	. . . . . . 7 setvar 𝑥
5	2, 4	wel 1435	. . . . . 6 wff 𝑤 ∈ 𝑥
6	3, 5	wa 102	. . . . 5 wff (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)
7	6, 2	wex 1422	. . . 4 wff ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)
8		vy	. . . . 5 setvar 𝑦
9	1, 8	wel 1435	. . . 4 wff 𝑧 ∈ 𝑦
10	7, 9	wi 4	. . 3 wff (∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
11	10, 1	wal 1283	. 2 wff ∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
12	11, 8	wex 1422	1 wff ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

This axiom is referenced by:  zfun  4224  axun2  4225  bj-axun2  11147