Definition df-in 2897

Description: Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. Contrast this operation with union (A ∪ B) (df-un 2895) and difference (A ∖ B) (df-dif 2893). (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 2889	. 2 class (A ∩ B)
4		vx	. . . . . 6 setvar x
5	4	cv 1225	. . . . 5 class x
6	5, 1	wcel 1370	. . . 4 wff x ∈ A
7	5, 2	wcel 1370	. . . 4 wff x ∈ B
8	6, 7	wa 97	. . 3 wff (x ∈ A ∧ x ∈ B)
9	8, 4	cab 2004	. 2 class {x ∣ (x ∈ A ∧ x ∈ B)}
10	3, 9	wceq 1226	1 wff (A ∩ B) = {x ∣ (x ∈ A ∧ x ∈ B)}

This definition is referenced by:  dfin5  2898  dfss2  2907  elin  3099  disj  3241  iinxprg  3701  bdcin  7229