Axiom ax-un 4393

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 4395 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4396. A version using class notation is uniex 4397.

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 4085), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 264).

The union of a class df-uni 3773 should not be confused with the union of two classes df-un 3106. Their relationship is shown in unipr 3786. (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 2129	. . . . . 6 wff 𝑧 ∈ 𝑤
4		vx	. . . . . . 7 setvar 𝑥
5	2, 4	wel 2129	. . . . . 6 wff 𝑤 ∈ 𝑥
6	3, 5	wa 103	. . . . 5 wff (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)
7	6, 2	wex 1472	. . . 4 wff ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)
8		vy	. . . . 5 setvar 𝑦
9	1, 8	wel 2129	. . . 4 wff 𝑧 ∈ 𝑦
10	7, 9	wi 4	. . 3 wff (∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
11	10, 1	wal 1333	. 2 wff ∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
12	11, 8	wex 1472	1 wff ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

This axiom is referenced by:  zfun  4394  axun2  4395  bj-axun2  13501