Definition df-in 3914

Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 30685). Contrast this operation with union (𝐴 ∪ 𝐵) (df-un 3912) and difference (𝐴 ∖ 𝐵) (df-dif 3910). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 4226 and dfin4 4233. For intersection defined in terms of union, see dfin3 4232. (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 3906	. 2 class (𝐴 ∩ 𝐵)
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1562	. . . . 5 class 𝑥
6	5, 1	wcel 2145	. . . 4 wff 𝑥 ∈ 𝐴
7	5, 2	wcel 2145	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wa 400	. . 3 wff (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)
9	8, 4	cab 2743	. 2 class {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
10	3, 9	wceq 1563	1 wff (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}

This definition is referenced by:  dfin5  3915  elin  3923  dfss2  3925  disj  4407  iinxprg  5051  disjex  32847  disjexc  32848  eulerpartlemt  34678  in-ax8  36597  iocinico  43801  csbingVD  45457