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Metamath Proof Explorer

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 ) )