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Metamath Proof Explorer

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 ) )