Theorem xrltnled

Description: 'Less than' in terms of 'less than or equal to'. (Contributed by Glauco Siliprandi, 3-Mar-2021)

	Ref	Expression
Hypotheses	xrltnled.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
	xrltnled.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
Assertion	xrltnled	⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		xrltnled.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
2		xrltnled.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
3		xrltnle	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴 ) )