Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > df-un | Structured version Visualization version GIF version |
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 28774). Contrast this operation with difference (𝐴 ∖ 𝐵) (df-dif 3890) and intersection (𝐴 ∩ 𝐵) (df-in 3894). For an alternate definition in terms of class difference, requiring no dummy variables, see dfun2 4194. For union defined in terms of intersection, see dfun3 4200. (Contributed by NM, 23-Aug-1993.) |
Ref | Expression |
---|---|
df-un | ⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | cB | . . 3 class 𝐵 | |
3 | 1, 2 | cun 3885 | . 2 class (𝐴 ∪ 𝐵) |
4 | vx | . . . . . 6 setvar 𝑥 | |
5 | 4 | cv 1538 | . . . . 5 class 𝑥 |
6 | 5, 1 | wcel 2106 | . . . 4 wff 𝑥 ∈ 𝐴 |
7 | 5, 2 | wcel 2106 | . . . 4 wff 𝑥 ∈ 𝐵 |
8 | 6, 7 | wo 844 | . . 3 wff (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) |
9 | 8, 4 | cab 2715 | . 2 class {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)} |
10 | 3, 9 | wceq 1539 | 1 wff (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)} |
Colors of variables: wff setvar class |
This definition is referenced by: elun 4083 nfun 4099 unabw 4232 uniprg 4857 uniprOLD 4859 iinuni 5027 fvclss 7108 bnj98 32833 |
Copyright terms: Public domain | W3C validator |