Theorem suble0d

Description: Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	ltnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
Assertion	suble0d	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) ≤ 0 ↔ 𝐴 ≤ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		suble0	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐴 − 𝐵 ) ≤ 0 ↔ 𝐴 ≤ 𝐵 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) ≤ 0 ↔ 𝐴 ≤ 𝐵 ) )