Definition df-in 3911

Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 30571). Contrast this operation with union (𝐴 ∪ 𝐵) (df-un 3909) and difference (𝐴 ∖ 𝐵) (df-dif 3907). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 4223 and dfin4 4230. For intersection defined in terms of union, see dfin3 4229. (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 3903	. 2 class (𝐴 ∩ 𝐵)
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1558	. . . . 5 class 𝑥
6	5, 1	wcel 2141	. . . 4 wff 𝑥 ∈ 𝐴
7	5, 2	wcel 2141	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wa 399	. . 3 wff (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)
9	8, 4	cab 2739	. 2 class {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
10	3, 9	wceq 1559	1 wff (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}

This definition is referenced by:  dfin5  3912  elin  3920  dfss2  3922  disj  4403  iinxprg  5045  disjex  32739  disjexc  32740  eulerpartlemt  34627  in-ax8  36537  iocinico  43742  csbingVD  45412