Definition df-in 3714

Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, ({1, 3} ∩ {1, 8}) = {1} (ex-in 27585). Contrast this operation with union (𝐴 ∪ 𝐵) (df-un 3712) and difference (𝐴 ∖ 𝐵) (df-dif 3710). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 3995 and dfin4 4002. For intersection defined in terms of union, see dfin3 4001. (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 3706	. 2 class (𝐴 ∩ 𝐵)
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1623	. . . . 5 class 𝑥
6	5, 1	wcel 2131	. . . 4 wff 𝑥 ∈ 𝐴
7	5, 2	wcel 2131	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wa 383	. . 3 wff (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)
9	8, 4	cab 2738	. 2 class {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}
10	3, 9	wceq 1624	1 wff (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}

This definition is referenced by:  dfin5  3715  dfss2  3724  elin  3931  disj  4152  iinxprg  4745  disjex  29704  disjexc  29705  eulerpartlemt  30734  iocinico  38291  csbingVD  39611