# Metamath Proof Explorer

## Definition df-un

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

Ref Expression
Assertion df-un ${⊢}{A}\cup {B}=\left\{{x}|\left({x}\in {A}\vee {x}\in {B}\right)\right\}$

### Detailed syntax breakdown

Step Hyp Ref Expression
0 cA ${class}{A}$
1 cB ${class}{B}$
2 0 1 cun ${class}\left({A}\cup {B}\right)$
3 vx ${setvar}{x}$
4 3 cv ${setvar}{x}$
5 4 0 wcel ${wff}{x}\in {A}$
6 4 1 wcel ${wff}{x}\in {B}$
7 5 6 wo ${wff}\left({x}\in {A}\vee {x}\in {B}\right)$
8 7 3 cab ${class}\left\{{x}|\left({x}\in {A}\vee {x}\in {B}\right)\right\}$
9 2 8 wceq ${wff}{A}\cup {B}=\left\{{x}|\left({x}\in {A}\vee {x}\in {B}\right)\right\}$