Definition df-dif 1489

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.

Assertion

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

Distinct variable group(s):   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 1484	. 2 class (A ∖ B)
4		vx	. . . . . 6 set x
5	4	cv 1089	. . . . 5 class x
6	5, 1	wcel 1092	. . . 4 wff x ∈ A
7	5, 2	wcel 1092	. . . . 5 wff x ∈ B
8	7	wn 1	. . . 4 wff ¬ x ∈ B
9	6, 8	wa 196	. . 3 wff (x ∈ A ∧ ¬ x ∈ B)
10	9, 4	cab 1090	. 2 class {x∣(x ∈ A ∧ ¬ x ∈ B)}
11	3, 10	wceq 1091	1 wff (A ∖ B) = {x∣(x ∈ A ∧ ¬ x ∈ B)}

This definition is referenced by:  dfdif2 1495  eldif 1496  difeq1 1582  difeq2 1583  difeqri 1589

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