Definition df-in 3897

Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 30520). Contrast this operation with union (𝐴 ∪ 𝐵) (df-un 3895) and difference (𝐴 ∖ 𝐵) (df-dif 3893). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 4206 and dfin4 4213. For intersection defined in terms of union, see dfin3 4212. (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 3889	. 2 class (𝐴 ∩ 𝐵)
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1546	. . . . 5 class 𝑥
6	5, 1	wcel 2119	. . . 4 wff 𝑥 ∈ 𝐴
7	5, 2	wcel 2119	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wa 396	. . 3 wff (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)
9	8, 4	cab 2718	. 2 class {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
10	3, 9	wceq 1547	1 wff (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}

This definition is referenced by:  dfin5  3898  elin  3906  dfss2  3908  disj  4385  iinxprg  5025  disjex  32688  disjexc  32689  eulerpartlemt  34562  in-ax8  36453  iocinico  43658  csbingVD  45328