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Mirrors > Home > ILE Home > Th. List > df-dif | GIF version |
Description: Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. Contrast this operation with union (A ∪ B) (df-un 2916) and intersection (A ∩ B) (df-in 2918). 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.) |
Ref | Expression |
---|---|
df-dif | ⊢ (A ∖ B) = {x ∣ (x ∈ A ∧ ¬ x ∈ B)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class A | |
2 | cB | . . 3 class B | |
3 | 1, 2 | cdif 2908 | . 2 class (A ∖ B) |
4 | vx | . . . . . 6 setvar x | |
5 | 4 | cv 1241 | . . . . 5 class x |
6 | 5, 1 | wcel 1390 | . . . 4 wff x ∈ A |
7 | 5, 2 | wcel 1390 | . . . . 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 2023 | . 2 class {x ∣ (x ∈ A ∧ ¬ x ∈ B)} |
11 | 3, 10 | wceq 1242 | 1 wff (A ∖ B) = {x ∣ (x ∈ A ∧ ¬ x ∈ B)} |
Colors of variables: wff set class |
This definition is referenced by: dfdif2 2920 eldif 2921 bdcdif 9316 |
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