Definition df-un 3913

Description: Define the union of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∪ {1, 8}) = {1, 3, 8} (ex-un 30805). Contrast this operation with difference (𝐴 ∖ 𝐵) (df-dif 3911) and intersection (𝐴 ∩ 𝐵) (df-in 3915). For an alternate definition in terms of class difference, requiring no dummy variables, see dfun2 4226. For union defined in terms of intersection, see dfun3 4232. (Contributed by NM, 23-Aug-1993.)

Assertion

Ref	Expression
df-un	⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Detailed syntax breakdown of Definition df-un

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cB	. . 3 class 𝐵
3	1, 2	cun 3906	. 2 class (𝐴 ∪ 𝐵)
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1569	. . . . 5 class 𝑥
6	5, 1	wcel 2146	. . . 4 wff 𝑥 ∈ 𝐴
7	5, 2	wcel 2146	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wo 861	. . 3 wff (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)
9	8, 4	cab 2744	. 2 class {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}
10	3, 9	wceq 1570	1 wff (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}

This definition is used by:  elun  4110  unabw  4263  uniprg  4893  iinuni  5069  fvclss  7246  bnj98  35279