Definition df-in 3895

Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 28798). Contrast this operation with union (𝐴 ∪ 𝐵) (df-un 3893) and difference (𝐴 ∖ 𝐵) (df-dif 3891). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 4195 and dfin4 4202. For intersection defined in terms of union, see dfin3 4201. (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 3887	. 2 class (𝐴 ∩ 𝐵)
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1538	. . . . 5 class 𝑥
6	5, 1	wcel 2107	. . . 4 wff 𝑥 ∈ 𝐴
7	5, 2	wcel 2107	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wa 396	. . 3 wff (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)
9	8, 4	cab 2716	. 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  3896  elin  3904  dfss2OLD  3909  ss2abdv  3998  disj  4382  disjOLD  4383  iinxprg  5019  disjex  30940  disjexc  30941  eulerpartlemt  32347  iocinico  41050  csbingVD  42511