Step |
Hyp |
Ref |
Expression |
1 |
|
elnn0 |
⊢ ( 𝑁 ∈ ℕ0 ↔ ( 𝑁 ∈ ℕ ∨ 𝑁 = 0 ) ) |
2 |
|
elnnz |
⊢ ( 𝑁 ∈ ℕ ↔ ( 𝑁 ∈ ℤ ∧ 0 < 𝑁 ) ) |
3 |
|
eqcom |
⊢ ( 𝑁 = 0 ↔ 0 = 𝑁 ) |
4 |
2 3
|
orbi12i |
⊢ ( ( 𝑁 ∈ ℕ ∨ 𝑁 = 0 ) ↔ ( ( 𝑁 ∈ ℤ ∧ 0 < 𝑁 ) ∨ 0 = 𝑁 ) ) |
5 |
|
id |
⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ ℤ ) |
6 |
|
0z |
⊢ 0 ∈ ℤ |
7 |
|
eleq1 |
⊢ ( 0 = 𝑁 → ( 0 ∈ ℤ ↔ 𝑁 ∈ ℤ ) ) |
8 |
6 7
|
mpbii |
⊢ ( 0 = 𝑁 → 𝑁 ∈ ℤ ) |
9 |
5 8
|
jaoi |
⊢ ( ( 𝑁 ∈ ℤ ∨ 0 = 𝑁 ) → 𝑁 ∈ ℤ ) |
10 |
|
orc |
⊢ ( 𝑁 ∈ ℤ → ( 𝑁 ∈ ℤ ∨ 0 = 𝑁 ) ) |
11 |
9 10
|
impbii |
⊢ ( ( 𝑁 ∈ ℤ ∨ 0 = 𝑁 ) ↔ 𝑁 ∈ ℤ ) |
12 |
11
|
anbi1i |
⊢ ( ( ( 𝑁 ∈ ℤ ∨ 0 = 𝑁 ) ∧ ( 0 < 𝑁 ∨ 0 = 𝑁 ) ) ↔ ( 𝑁 ∈ ℤ ∧ ( 0 < 𝑁 ∨ 0 = 𝑁 ) ) ) |
13 |
|
ordir |
⊢ ( ( ( 𝑁 ∈ ℤ ∧ 0 < 𝑁 ) ∨ 0 = 𝑁 ) ↔ ( ( 𝑁 ∈ ℤ ∨ 0 = 𝑁 ) ∧ ( 0 < 𝑁 ∨ 0 = 𝑁 ) ) ) |
14 |
|
0re |
⊢ 0 ∈ ℝ |
15 |
|
zre |
⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ ℝ ) |
16 |
|
leloe |
⊢ ( ( 0 ∈ ℝ ∧ 𝑁 ∈ ℝ ) → ( 0 ≤ 𝑁 ↔ ( 0 < 𝑁 ∨ 0 = 𝑁 ) ) ) |
17 |
14 15 16
|
sylancr |
⊢ ( 𝑁 ∈ ℤ → ( 0 ≤ 𝑁 ↔ ( 0 < 𝑁 ∨ 0 = 𝑁 ) ) ) |
18 |
17
|
pm5.32i |
⊢ ( ( 𝑁 ∈ ℤ ∧ 0 ≤ 𝑁 ) ↔ ( 𝑁 ∈ ℤ ∧ ( 0 < 𝑁 ∨ 0 = 𝑁 ) ) ) |
19 |
12 13 18
|
3bitr4i |
⊢ ( ( ( 𝑁 ∈ ℤ ∧ 0 < 𝑁 ) ∨ 0 = 𝑁 ) ↔ ( 𝑁 ∈ ℤ ∧ 0 ≤ 𝑁 ) ) |
20 |
1 4 19
|
3bitri |
⊢ ( 𝑁 ∈ ℕ0 ↔ ( 𝑁 ∈ ℤ ∧ 0 ≤ 𝑁 ) ) |