Definition df-dif 2899

Description: Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. Contrast this operation with union (A ∪ B) (df-un 2901) and intersection (A ∩ B) (df-in 2903). Several notations are used in the literature; we chose the ∖ convention used in Definition 5.3 of [Eisenberg] p. 67 instead of the more common minus sign to reserve the latter for later use in, e.g., arithmetic. We will use the terminology "A excludes B " to mean A ∖ B. We will use "B is removed from A " to mean A ∖ {B} i.e. the removal of an element or equivalently the exclusion of a singleton. (Contributed by NM, 29-Apr-1994.)

Assertion

Ref	Expression
df-dif	⊢ (A ∖ B) = {x ∣ (x ∈ A ∧ ¬ x ∈ B)}

Distinct variable groups:   x,A   x,B

Detailed syntax breakdown of Definition df-dif

Step	Hyp	Ref	Expression
1		cA	. . 3 class A
2		cB	. . 3 class B
3	1, 2	cdif 2893	. 2 class (A ∖ B)
4		vx	. . . . . 6 setvar x
5	4	cv 1373	. . . . 5 class x
6	5, 1	wcel 1376	. . . 4 wff x ∈ A
7	5, 2	wcel 1376	. . . . 5 wff x ∈ B
8	7	wn 3	. . . 4 wff ¬ x ∈ B
9	6, 8	wa 97	. . 3 wff (x ∈ A ∧ ¬ x ∈ B)
10	9, 4	cab 2009	. 2 class {x ∣ (x ∈ A ∧ ¬ x ∈ B)}
11	3, 10	wceq 1374	1 wff (A ∖ B) = {x ∣ (x ∈ A ∧ ¬ x ∈ B)}

This definition is referenced by:  dfdif2  2905  eldif  2906