Definition df-in

Description: Define the intersection of two classes. Definition 5.6 of TakeutiZaring p. 16. For example, ( { 1 , 3 } i^i { 1 , 8 } ) = { 1 } ( ex-in ). Contrast this operation with union ( A u. B ) ( df-un ) and difference ( A \ B ) ( df-dif ). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 and dfin4 . For intersection defined in terms of union, see dfin3 . (Contributed by NM, 29-Apr-1994)

	Ref	Expression
Assertion	df-in	⊢ ( 𝐴 ∩ 𝐵 ) = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ) }

Detailed syntax breakdown

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