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Metamath Proof Explorer

Theorem lesub0i

Description: Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999) (Proof shortened by Andrew Salmon, 19-Nov-2011)

Ref Expression
Hypotheses lt2.1 ⊢ 𝐴 ∈ ℝ
lt2.2 ⊢ 𝐵 ∈ ℝ
Assertion lesub0i ( ( 0 ≤ 𝐴 ∧ 𝐵 ≤ ( 𝐵 − 𝐴 ) ) ↔ 𝐴 = 0 )

Proof

Step Hyp Ref Expression
1 lt2.1 ⊢ 𝐴 ∈ ℝ
2 lt2.2 ⊢ 𝐵 ∈ ℝ
3 lesub0 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 0 ≤ 𝐴 ∧ 𝐵 ≤ ( 𝐵 − 𝐴 ) ) ↔ 𝐴 = 0 ) )
4 1 2 3 mp2an ⊢ ( ( 0 ≤ 𝐴 ∧ 𝐵 ≤ ( 𝐵 − 𝐴 ) ) ↔ 𝐴 = 0 )