Theorem posdifi

Description: Comparison of two numbers whose difference is positive. (Contributed by NM, 19-Aug-2001)

	Ref	Expression
Hypotheses	lt2.1	⊢ 𝐴 ∈ ℝ
	lt2.2	⊢ 𝐵 ∈ ℝ
Assertion	posdifi	⊢ ( 𝐴 < 𝐵 ↔ 0 < ( 𝐵 − 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		lt2.1	⊢ 𝐴 ∈ ℝ
2		lt2.2	⊢ 𝐵 ∈ ℝ
3		posdif	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ 0 < ( 𝐵 − 𝐴 ) ) )
4	1 2 3	mp2an	⊢ ( 𝐴 < 𝐵 ↔ 0 < ( 𝐵 − 𝐴 ) )