Definition df-uni 3893

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 in set.mm). This is similar to the union of two classes df-un 3215. (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 3892	. 2 class ∪A
3		vx	. . . . . 6 setvar x
4		vy	. . . . . 6 setvar y
5	3, 4	wel 1711	. . . . 5 wff x ∈ y
6	4	cv 1641	. . . . . 6 class y
7	6, 1	wcel 1710	. . . . 5 wff y ∈ A
8	5, 7	wa 358	. . . 4 wff (x ∈ y ∧ y ∈ A)
9	8, 4	wex 1541	. . 3 wff ∃y(x ∈ y ∧ y ∈ A)
10	9, 3	cab 2339	. 2 class {x ∣ ∃y(x ∈ y ∧ y ∈ A)}
11	2, 10	wceq 1642	1 wff ∪A = {x ∣ ∃y(x ∈ y ∧ y ∈ A)}

This definition is referenced by:  dfuni2  3894  eluni  3895  csbunig  3900