Definition df-dif 2039

Description: Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. 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.

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 2034	. 2 class (A \ B)
4		vx	. . . . . 6 set x
5	4	cv 952	. . . . 5 class x
6	5, 1	wcel 955	. . . 4 wff x e. A
7	5, 2	wcel 955	. . . . 5 wff x e. B
8	7	wn 2	. . . 4 wff -. x e. B
9	6, 8	wa 223	. . 3 wff (x e. A /\ -. x e. B)
10	9, 4	cab 1456	. 2 class {x | (x e. A /\ -. x e. B)}
11	3, 10	wceq 953	1 wff (A \ B) = {x | (x e. A /\ -. x e. B)}

This definition is referenced by:  dfdif2 2046  eldif 2047  difeq1 2143  difeq2 2144  difeqri 2150  difeqri2 10344

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