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Metamath Proof Explorer

Theorem nn0p1nn

Description: A nonnegative integer plus 1 is a positive integer. Strengthening of peano2nn . (Contributed by Raph Levien, 30-Jun-2006) (Revised by Mario Carneiro, 16-May-2014)

Ref Expression
Assertion nn0p1nn ( 𝑁 ∈ ℕ0 → ( 𝑁 + 1 ) ∈ ℕ )

Proof

Step Hyp Ref Expression
1 1nn ⊢ 1 ∈ ℕ
2 nn0nnaddcl ⊢ ( ( 𝑁 ∈ ℕ0 ∧ 1 ∈ ℕ ) → ( 𝑁 + 1 ) ∈ ℕ )
3 1 2 mpan2 ⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 + 1 ) ∈ ℕ )