Definition df-un 3579

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 27281). Contrast this operation with difference (𝐴 ∖ 𝐵) (df-dif 3577) and intersection (𝐴 ∩ 𝐵) (df-in 3581). For an alternate definition in terms of class difference, requiring no dummy variables, see dfun2 3859. For union defined in terms of intersection, see dfun3 3865. (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 3572	. 2 class (𝐴 ∪ 𝐵)
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1482	. . . . 5 class 𝑥
6	5, 1	wcel 1990	. . . 4 wff 𝑥 ∈ 𝐴
7	5, 2	wcel 1990	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wo 383	. . 3 wff (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)
9	8, 4	cab 2608	. 2 class {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}
10	3, 9	wceq 1483	1 wff (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)}

This definition is referenced by:  elun  3753  nfun  3769  unipr  4449  iinuni  4609  fvclss  6500  bnj98  30937