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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 ∪ 𝐴 = { 𝑥 ∣ ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) }

Detailed syntax breakdown

Step Hyp Ref Expression
0 cA ⊢ 𝐴
1 0 cuni ⊢ ∪ 𝐴
2 vx ⊢ 𝑥
3 vy ⊢ 𝑦
4 2 cv ⊢ 𝑥
5 3 cv ⊢ 𝑦
6 4 5 wcel ⊢ 𝑥 ∈ 𝑦
7 5 0 wcel ⊢ 𝑦 ∈ 𝐴
8 6 7 wa ⊢ ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 )
9 8 3 wex ⊢ ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 )
10 9 2 cab ⊢ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) }
11 1 10 wceq ⊢ ∪ 𝐴 = { 𝑥 ∣ ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) }