Definition df-in 3924

Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 30361). Contrast this operation with union (𝐴 ∪ 𝐵) (df-un 3922) and difference (𝐴 ∖ 𝐵) (df-dif 3920). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 4237 and dfin4 4244. For intersection defined in terms of union, see dfin3 4243. (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 3916	. 2 class (𝐴 ∩ 𝐵)
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1539	. . . . 5 class 𝑥
6	5, 1	wcel 2109	. . . 4 wff 𝑥 ∈ 𝐴
7	5, 2	wcel 2109	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wa 395	. . 3 wff (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)
9	8, 4	cab 2708	. 2 class {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
10	3, 9	wceq 1540	1 wff (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}

This definition is referenced by:  dfin5  3925  elin  3933  dfss2  3935  disj  4416  iinxprg  5056  disjex  32528  disjexc  32529  eulerpartlemt  34369  in-ax8  36219  iocinico  43208  csbingVD  44880