Definition df-int

Description: Define the intersection of a class. Definition 7.35 of TakeutiZaring p. 44. For example, |^| { { 1 , 3 } , { 1 , 8 } } = { 1 } . Compare this with the intersection of two classes, df-in . (Contributed by NM, 18-Aug-1993)

	Ref	Expression
Assertion	df-int	⊢ ∩ 𝐴 = { 𝑥 ∣ ∀ 𝑦 ( 𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦 ) }

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cA	⊢ 𝐴
1	0	cint	⊢ ∩ 𝐴
2		vx	⊢ 𝑥
3		vy	⊢ 𝑦
4	3	cv	⊢ 𝑦
5	4 0	wcel	⊢ 𝑦 ∈ 𝐴
6	2	cv	⊢ 𝑥
7	6 4	wcel	⊢ 𝑥 ∈ 𝑦
8	5 7	wi	⊢ ( 𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦 )
9	8 3	wal	⊢ ∀ 𝑦 ( 𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦 )
10	9 2	cab	⊢ { 𝑥 ∣ ∀ 𝑦 ( 𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦 ) }
11	1 10	wceq	⊢ ∩ 𝐴 = { 𝑥 ∣ ∀ 𝑦 ( 𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦 ) }