Theorem leneltd

Description: 'Less than or equal to' and 'not equals' implies 'less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	ltd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	ltd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	leltned.3	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
	leneltd.4	⊢ ( 𝜑 → 𝐵 ≠ 𝐴 )
Assertion	leneltd	⊢ ( 𝜑 → 𝐴 < 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ltd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		leltned.3	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
4		leneltd.4	⊢ ( 𝜑 → 𝐵 ≠ 𝐴 )
5	1 2 3	leltned	⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴 ) )
6	4 5	mpbird	⊢ ( 𝜑 → 𝐴 < 𝐵 )