Definition df-z 12583

Description: Define the set of integers, which are the positive and negative integers together with zero. Definition of integers in [Apostol] p. 22. The letter Z abbreviates the German word Zahlen meaning "numbers." (Contributed by NM, 8-Jan-2002.)

Assertion

Ref	Expression
df-z	⊢ ℤ = {𝑛 ∈ ℝ ∣ (𝑛 = 0 ∨ 𝑛 ∈ ℕ ∨ -𝑛 ∈ ℕ)}

Detailed syntax breakdown of Definition df-z

Step	Hyp	Ref	Expression
1		cz 12582	. 2 class ℤ
2		vn	. . . . . 6 setvar 𝑛
3	2	cv 1533	. . . . 5 class 𝑛
4		cc0 11132	. . . . 5 class 0
5	3, 4	wceq 1534	. . . 4 wff 𝑛 = 0
6		cn 12236	. . . . 5 class ℕ
7	3, 6	wcel 2099	. . . 4 wff 𝑛 ∈ ℕ
8	3	cneg 11469	. . . . 5 class -𝑛
9	8, 6	wcel 2099	. . . 4 wff -𝑛 ∈ ℕ
10	5, 7, 9	w3o 1084	. . 3 wff (𝑛 = 0 ∨ 𝑛 ∈ ℕ ∨ -𝑛 ∈ ℕ)
11		cr 11131	. . 3 class ℝ
12	10, 2, 11	crab 3428	. 2 class {𝑛 ∈ ℝ ∣ (𝑛 = 0 ∨ 𝑛 ∈ ℕ ∨ -𝑛 ∈ ℕ)}
13	1, 12	wceq 1534	1 wff ℤ = {𝑛 ∈ ℝ ∣ (𝑛 = 0 ∨ 𝑛 ∈ ℕ ∨ -𝑛 ∈ ℕ)}

This definition is referenced by:  elz  12584