Definition df-in 3865

Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 28309). Contrast this operation with union (𝐴 ∪ 𝐵) (df-un 3863) and difference (𝐴 ∖ 𝐵) (df-dif 3861). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 4165 and dfin4 4172. For intersection defined in terms of union, see dfin3 4171. (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 3857	. 2 class (𝐴 ∩ 𝐵)
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1537	. . . . 5 class 𝑥
6	5, 1	wcel 2111	. . . 4 wff 𝑥 ∈ 𝐴
7	5, 2	wcel 2111	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wa 399	. . 3 wff (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)
9	8, 4	cab 2735	. 2 class {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
10	3, 9	wceq 1538	1 wff (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}

This definition is referenced by:  dfin5  3866  elin  3874  dfss2OLD  3879  ss2abdv  3968  disj  4344  disjOLD  4345  iinxprg  4976  disjex  30453  disjexc  30454  eulerpartlemt  31857  iocinico  40557  csbingVD  41985