Theorem nn0ltp1ne

Description: Nonnegative integer ordering relation. (Contributed by BTernaryTau, 24-Sep-2023)

	Ref	Expression
Assertion	nn0ltp1ne	⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) → ( ( 𝐴 + 1 ) < 𝐵 ↔ ( 𝐴 < 𝐵 ∧ 𝐵 ≠ ( 𝐴 + 1 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		nn0z	⊢ ( 𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ )
2		nn0z	⊢ ( 𝐵 ∈ ℕ0 → 𝐵 ∈ ℤ )
3		zltp1ne	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( ( 𝐴 + 1 ) < 𝐵 ↔ ( 𝐴 < 𝐵 ∧ 𝐵 ≠ ( 𝐴 + 1 ) ) ) )
4	1 2 3	syl2an	⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) → ( ( 𝐴 + 1 ) < 𝐵 ↔ ( 𝐴 < 𝐵 ∧ 𝐵 ≠ ( 𝐴 + 1 ) ) ) )