Definition df-in 2893

Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. Contrast this operation with union (A ∪ B) (df-un 2891) and difference (A ∖ B) (df-dif 2889). (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 2885	. 2 class (A ∩ B)
4		vx	. . . . . 6 setvar x
5	4	cv 1223	. . . . 5 class x
6	5, 1	wcel 1367	. . . 4 wff x ∈ A
7	5, 2	wcel 1367	. . . 4 wff x ∈ B
8	6, 7	wa 97	. . 3 wff (x ∈ A ∧ x ∈ B)
9	8, 4	cab 2000	. 2 class {x ∣ (x ∈ A ∧ x ∈ B)}
10	3, 9	wceq 1224	1 wff (A ∩ B) = {x ∣ (x ∈ A ∧ x ∈ B)}

This definition is referenced by:  dfin5  2894  dfss2  2903  elin  3095  disj  3237  iinxprg  3695  bdcin  8320