Metamath Proof Explorer


Theorem elnnnn0c

Description: The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006)

Ref Expression
Assertion elnnnn0c ( 𝑁 ∈ ℕ ↔ ( 𝑁 ∈ ℕ0 ∧ 1 ≤ 𝑁 ) )

Proof

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 ∈ ℕ0𝑁 ∈ 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 ) )