Theorem nnltp1le

Description: Positive integer ordering relation. (Contributed by NM, 19-Aug-2001)

	Ref	Expression
Assertion	nnltp1le	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 + 1 ) ≤ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		nnz	⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℤ )
2		nnz	⊢ ( 𝐵 ∈ ℕ → 𝐵 ∈ ℤ )
3		zltp1le	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 + 1 ) ≤ 𝐵 ) )
4	1 2 3	syl2an	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 + 1 ) ≤ 𝐵 ) )