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 )