Definition df-in 3954

Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 30234). Contrast this operation with union (𝐴 ∪ 𝐵) (df-un 3952) and difference (𝐴 ∖ 𝐵) (df-dif 3950). 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 3946	. 2 class (𝐴 ∩ 𝐵)
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1533	. . . . 5 class 𝑥
6	5, 1	wcel 2099	. . . 4 wff 𝑥 ∈ 𝐴
7	5, 2	wcel 2099	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wa 395	. . 3 wff (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)
9	8, 4	cab 2705	. 2 class {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
10	3, 9	wceq 1534	1 wff (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}

This definition is referenced by:  dfin5  3955  elin  3963  ss2abdv  4058  disj  4448  disjOLD  4449  iinxprg  5092  disjex  32381  disjexc  32382  eulerpartlemt  33991  iocinico  42640  csbingVD  44323