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Definition df-dif 2897
Description: Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. Contrast this operation with union (AB) (df-un 2899) and intersection (AB) (df-in 2901). 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 AB. 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 (AB) = {x ∣ (x A ¬ x B)}
Distinct variable groups:   x,A   x,B

Detailed syntax breakdown of Definition df-dif
StepHypRef Expression
1 cA . . 3 class A
2 cB . . 3 class B
31, 2cdif 2891 . 2 class (AB)
4 vx . . . . . 6 setvar x
54cv 1227 . . . . 5 class x
65, 1wcel 1374 . . . 4 wff x A
75, 2wcel 1374 . . . . 5 wff x B
87wn 3 . . . 4 wff ¬ x B
96, 8wa 97 . . 3 wff (x A ¬ x B)
109, 4cab 2008 . 2 class {x ∣ (x A ¬ x B)}
113, 10wceq 1228 1 wff (AB) = {x ∣ (x A ¬ x B)}
Colors of variables: wff set class
This definition is referenced by:  dfdif2  2903  eldif  2904  bdcdif  7088
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