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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