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Definition df-dif 3542
Description: Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. For example, ({1, 3} ∖ {1, 8}) = {3} (ex-dif 26465). Contrast this operation with union (𝐴𝐵) (df-un 3544) and intersection (𝐴𝐵) (df-in 3546). 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 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)}
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 3536 . 2 class (𝐴𝐵)
4 vx . . . . . 6 setvar 𝑥
54cv 1473 . . . . 5 class 𝑥
65, 1wcel 1976 . . . 4 wff 𝑥𝐴
75, 2wcel 1976 . . . . 5 wff 𝑥𝐵
87wn 3 . . . 4 wff ¬ 𝑥𝐵
96, 8wa 382 . . 3 wff (𝑥𝐴 ∧ ¬ 𝑥𝐵)
109, 4cab 2595 . 2 class {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)}
113, 10wceq 1474 1 wff (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝑥𝐵)}
Colors of variables: wff setvar class
This definition is referenced by:  dfdif2  3548  eldif  3549
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