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Metamath Proof Explorer

Theorem nnleltp1

Description: Positive integer ordering relation. (Contributed by NM, 13-Aug-2001) (Proof shortened by Mario Carneiro, 16-May-2014)

Ref Expression
Assertion nnleltp1 ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( 𝐴 ≤ 𝐵 ↔ 𝐴 < ( 𝐵 + 1 ) ) )

Proof

Step Hyp Ref Expression
1 nnz ⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℤ )
2 nnz ⊢ ( 𝐵 ∈ ℕ → 𝐵 ∈ ℤ )
3 zleltp1 ⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( 𝐴 ≤ 𝐵 ↔ 𝐴 < ( 𝐵 + 1 ) ) )
4 1 2 3 syl2an ⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( 𝐴 ≤ 𝐵 ↔ 𝐴 < ( 𝐵 + 1 ) ) )