MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-z Structured version   Visualization version   GIF version

Definition df-z 11213
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
StepHypRef Expression
1 cz 11212 . 2 class
2 vn . . . . . 6 setvar 𝑛
32cv 1473 . . . . 5 class 𝑛
4 cc0 9792 . . . . 5 class 0
53, 4wceq 1474 . . . 4 wff 𝑛 = 0
6 cn 10869 . . . . 5 class
73, 6wcel 1976 . . . 4 wff 𝑛 ∈ ℕ
83cneg 10118 . . . . 5 class -𝑛
98, 6wcel 1976 . . . 4 wff -𝑛 ∈ ℕ
105, 7, 9w3o 1029 . . 3 wff (𝑛 = 0 ∨ 𝑛 ∈ ℕ ∨ -𝑛 ∈ ℕ)
11 cr 9791 . . 3 class
1210, 2, 11crab 2899 . 2 class {𝑛 ∈ ℝ ∣ (𝑛 = 0 ∨ 𝑛 ∈ ℕ ∨ -𝑛 ∈ ℕ)}
131, 12wceq 1474 1 wff ℤ = {𝑛 ∈ ℝ ∣ (𝑛 = 0 ∨ 𝑛 ∈ ℕ ∨ -𝑛 ∈ ℕ)}
Colors of variables: wff setvar class
This definition is referenced by:  elz  11214
  Copyright terms: Public domain W3C validator