Definition df-in 3956

Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 29678). Contrast this operation with union (𝐴 ∪ 𝐵) (df-un 3954) and difference (𝐴 ∖ 𝐵) (df-dif 3952). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 4261 and dfin4 4268. For intersection defined in terms of union, see dfin3 4267. (Contributed by NM, 29-Apr-1994.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Detailed syntax breakdown of Definition df-in

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cB	. . 3 class 𝐵
3	1, 2	cin 3948	. 2 class (𝐴 ∩ 𝐵)
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1541	. . . . 5 class 𝑥
6	5, 1	wcel 2107	. . . 4 wff 𝑥 ∈ 𝐴
7	5, 2	wcel 2107	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wa 397	. . 3 wff (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)
9	8, 4	cab 2710	. 2 class {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
10	3, 9	wceq 1542	1 wff (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}

This definition is referenced by:  dfin5  3957  elin  3965  dfss2OLD  3970  ss2abdv  4061  disj  4448  disjOLD  4449  iinxprg  5093  disjex  31823  disjexc  31824  eulerpartlemt  33370  iocinico  41961  csbingVD  43645