Definition df-dif 4047

Description: Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. For example, ({1, 3} ∖ {1, 8}) = {3} (ex-dif 31024). Contrast this operation with union (𝐴 ∪ 𝐵) (df-un 4151) and intersection (𝐴 ∩ 𝐵) (df-in 4091). 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.)

Assertion

Ref	Expression
df-dif	⊢ (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 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 4045	. 2 class (𝐴 ∖ 𝐵)
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1568	. . . . 5 class 𝑥
6	5, 1	wcel 2144	. . . 4 wff 𝑥 ∈ 𝐴
7	5, 2	wcel 2144	. . . . 5 wff 𝑥 ∈ 𝐵
8	7	wn 3	. . . 4 wff ¬ 𝑥 ∈ 𝐵
9	6, 8	wa 401	. . 3 wff (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)
10	9, 4	cab 2738	. 2 class {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}
11	3, 10	wceq 1569	1 wff (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)}

This definition is used by:  dfdif2  4048  eldif  4049  notabw  4258  dfnul4  4280