Definition df-in 3909

Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 30483). Contrast this operation with union (𝐴 ∪ 𝐵) (df-un 3907) and difference (𝐴 ∖ 𝐵) (df-dif 3905). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 4224 and dfin4 4231. For intersection defined in terms of union, see dfin3 4230. (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 3901	. 2 class (𝐴 ∩ 𝐵)
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1541	. . . . 5 class 𝑥
6	5, 1	wcel 2114	. . . 4 wff 𝑥 ∈ 𝐴
7	5, 2	wcel 2114	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wa 395	. . 3 wff (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)
9	8, 4	cab 2715	. 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  3910  elin  3918  dfss2  3920  disj  4403  iinxprg  5045  disjex  32649  disjexc  32650  eulerpartlemt  34509  in-ax8  36399  iocinico  43490  csbingVD  45160