Step |
Hyp |
Ref |
Expression |
1 |
|
nnnn0 |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℕ0 ) |
2 |
|
nnge1 |
⊢ ( 𝑁 ∈ ℕ → 1 ≤ 𝑁 ) |
3 |
1 2
|
jca |
⊢ ( 𝑁 ∈ ℕ → ( 𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ) ) |
4 |
|
0lt1 |
⊢ 0 < 1 |
5 |
|
0re |
⊢ 0 ∈ ℝ |
6 |
|
1re |
⊢ 1 ∈ ℝ |
7 |
|
nn0re |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ ) |
8 |
|
ltletr |
⊢ ( ( 0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑁 ∈ ℝ ) → ( ( 0 < 1 ∧ 1 ≤ 𝑁 ) → 0 < 𝑁 ) ) |
9 |
5 6 7 8
|
mp3an12i |
⊢ ( 𝑁 ∈ ℕ0 → ( ( 0 < 1 ∧ 1 ≤ 𝑁 ) → 0 < 𝑁 ) ) |
10 |
4 9
|
mpani |
⊢ ( 𝑁 ∈ ℕ0 → ( 1 ≤ 𝑁 → 0 < 𝑁 ) ) |
11 |
10
|
imdistani |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ) → ( 𝑁 ∈ ℕ0 ∧ 0 < 𝑁 ) ) |
12 |
|
elnnnn0b |
⊢ ( 𝑁 ∈ ℕ ↔ ( 𝑁 ∈ ℕ0 ∧ 0 < 𝑁 ) ) |
13 |
11 12
|
sylibr |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ) → 𝑁 ∈ ℕ ) |
14 |
3 13
|
impbii |
⊢ ( 𝑁 ∈ ℕ ↔ ( 𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ) ) |