Definition df-uni 4801

Description: Define the union of a class i.e. the collection of all members of the members of the class. Definition 5.5 of [TakeutiZaring] p. 16. For example, ∪ {{1, 3}, {1, 8}} = {1, 3, 8} (ex-uni 28211). This is similar to the union of two classes df-un 3886. (Contributed by NM, 23-Aug-1993.)

Assertion

Ref	Expression
df-uni	⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)}

Distinct variable group:   𝑥,𝑦,𝐴

Detailed syntax breakdown of Definition df-uni

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2	1	cuni 4800	. 2 class ∪ 𝐴
3		vx	. . . . . 6 setvar 𝑥
4		vy	. . . . . 6 setvar 𝑦
5	3, 4	wel 2112	. . . . 5 wff 𝑥 ∈ 𝑦
6	4	cv 1537	. . . . . 6 class 𝑦
7	6, 1	wcel 2111	. . . . 5 wff 𝑦 ∈ 𝐴
8	5, 7	wa 399	. . . 4 wff (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)
9	8, 4	wex 1781	. . 3 wff ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)
10	9, 3	cab 2776	. 2 class {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)}
11	2, 10	wceq 1538	1 wff ∪ 𝐴 = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)}

This definition is referenced by:  dfuni2  4802  eluni  4803  unipr  4817  csbuni  4829  uniuni  7464  csbunigVD  41604