Definition df-int 3582

Description: Define the intersection of a class. Definition 7.35 of [TakeutiZaring] p. 44. For example, ∩ { { 1 , 3 } , { 1 , 8 } } = { 1 } . Compare this with the intersection of two classes, df-in 2895. (Contributed by NM, 18-Aug-1993.)

Assertion

Ref	Expression
df-int	⊢ ∩ A = {x ∣ ∀y(y ∈ A → x ∈ y)}

Distinct variable group:   x,y,A

Detailed syntax breakdown of Definition df-int

Step	Hyp	Ref	Expression
1		cA	. . 3 class A
2	1	cint 3581	. 2 class ∩ A
3		vy	. . . . . . 7 setvar y
4	3	cv 1225	. . . . . 6 class y
5	4, 1	wcel 1369	. . . . 5 wff y ∈ A
6		vx	. . . . . 6 setvar x
7	6, 3	wel 1370	. . . . 5 wff x ∈ y
8	5, 7	wi 4	. . . 4 wff (y ∈ A → x ∈ y)
9	8, 3	wal 1224	. . 3 wff ∀y(y ∈ A → x ∈ y)
10	9, 6	cab 2002	. 2 class {x ∣ ∀y(y ∈ A → x ∈ y)}
11	2, 10	wceq 1226	1 wff ∩ A = {x ∣ ∀y(y ∈ A → x ∈ y)}

This definition is referenced by:  dfint2  3583  elint  3587  int0  3595  dfiin2g  3656  bdcint  8454  bdcriota  8460