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Metamath Proof Explorer

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 )