Definition df-in 3890

Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 28690). Contrast this operation with union (𝐴 ∪ 𝐵) (df-un 3888) and difference (𝐴 ∖ 𝐵) (df-dif 3886). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 4191 and dfin4 4198. For intersection defined in terms of union, see dfin3 4197. (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 3882	. 2 class (𝐴 ∩ 𝐵)
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1538	. . . . 5 class 𝑥
6	5, 1	wcel 2108	. . . 4 wff 𝑥 ∈ 𝐴
7	5, 2	wcel 2108	. . . 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 1539	1 wff (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}

This definition is referenced by:  dfin5  3891  elin  3899  dfss2OLD  3904  ss2abdv  3993  disj  4378  disjOLD  4379  iinxprg  5014  disjex  30832  disjexc  30833  eulerpartlemt  32238  iocinico  40959  csbingVD  42393