Definition df-in 3912

Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 30785). Contrast this operation with union (𝐴 ∪ 𝐵) (df-un 3910) and difference (𝐴 ∖ 𝐵) (df-dif 3908). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 4224 and dfin4 4231. For intersection defined in terms of union, see dfin3 4230. (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 3904	. 2 class (𝐴 ∩ 𝐵)
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1569	. . . . 5 class 𝑥
6	5, 1	wcel 2143	. . . 4 wff 𝑥 ∈ 𝐴
7	5, 2	wcel 2143	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wa 400	. . 3 wff (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)
9	8, 4	cab 2741	. 2 class {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
10	3, 9	wceq 1570	1 wff (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}

This definition is used by:  dfin5  3913  elin  3921  dfss2  3923  disj  4410  iinxprg  5055  disjex  32946  disjexc  32947  eulerpartlemt  34770  in-ax8  36764  iocinico  43967  csbingVD  45620