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 ) }