Theorem leltned

Description: 'Less than or equal to' implies 'less than' is not 'equals'. (Contributed by Mario Carneiro, 27-May-2016)

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

Proof

Step	Hyp	Ref	Expression
1		ltd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		ltd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		leltned.3	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
4		leltne	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴 ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴 ) )