# Metamath Proof Explorer

## Definition df-uni

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, U. { { 1 , 3 } , { 1 , 8 } } = { 1 , 3 , 8 } ( ex-uni ). This is similar to the union of two classes df-un . (Contributed by NM, 23-Aug-1993)

Ref Expression
Assertion df-uni ${⊢}\bigcup {A}=\left\{{x}|\exists {y}\phantom{\rule{.4em}{0ex}}\left({x}\in {y}\wedge {y}\in {A}\right)\right\}$

### Detailed syntax breakdown

Step Hyp Ref Expression
0 cA ${class}{A}$
1 0 cuni ${class}\bigcup {A}$
2 vx ${setvar}{x}$
3 vy ${setvar}{y}$
4 2 cv ${setvar}{x}$
5 3 cv ${setvar}{y}$
6 4 5 wcel ${wff}{x}\in {y}$
7 5 0 wcel ${wff}{y}\in {A}$
8 6 7 wa ${wff}\left({x}\in {y}\wedge {y}\in {A}\right)$
9 8 3 wex ${wff}\exists {y}\phantom{\rule{.4em}{0ex}}\left({x}\in {y}\wedge {y}\in {A}\right)$
10 9 2 cab ${class}\left\{{x}|\exists {y}\phantom{\rule{.4em}{0ex}}\left({x}\in {y}\wedge {y}\in {A}\right)\right\}$
11 1 10 wceq ${wff}\bigcup {A}=\left\{{x}|\exists {y}\phantom{\rule{.4em}{0ex}}\left({x}\in {y}\wedge {y}\in {A}\right)\right\}$