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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 ∪ 𝐵 ) = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) }

Detailed syntax breakdown

Step Hyp Ref Expression
0 cA ⊢ 𝐴
1 cB ⊢ 𝐵
2 0 1 cun ⊢ ( 𝐴 ∪ 𝐵 )
3 vx ⊢ 𝑥
4 3 cv ⊢ 𝑥
5 4 0 wcel ⊢ 𝑥 ∈ 𝐴
6 4 1 wcel ⊢ 𝑥 ∈ 𝐵
7 5 6 wo ⊢ ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 )
8 7 3 cab ⊢ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) }
9 2 8 wceq ⊢ ( 𝐴 ∪ 𝐵 ) = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) }