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 ) ) )