Definition df-dif 3418

Description: Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. For example, ( { 1 , 3 } \ { 1 , 8 } ) = { 3 } (ex-dif 25921). Contrast this operation with union ( A u. B ) (df-un 3420) and intersection ( A i^i B ) (df-in 3422). 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 e. A /\ -. x e. 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 3412	. 2 class ( A \ B )
4		vx	. . . . . 6 setvar x
5	4	cv 1453	. . . . 5 class x
6	5, 1	wcel 1897	. . . 4 wff x e. A
7	5, 2	wcel 1897	. . . . 5 wff x e. B
8	7	wn 3	. . . 4 wff -. x e. B
9	6, 8	wa 375	. . 3 wff ( x e. A /\ -. x e. B )
10	9, 4	cab 2447	. 2 class { x | ( x e. A /\ -. x e. B ) }
11	3, 10	wceq 1454	1 wff ( A \ B ) = { x | ( x e. A /\ -. x e. B ) }

This definition is referenced by:  dfdif2  3424  eldif  3425