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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