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Metamath Proof Explorer

Theorem nleltd

Description: 'Not less than or equal to' implies 'grater than'. (Contributed by Glauco Siliprandi, 2-Jan-2022)

Ref Expression
Hypotheses nleltd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
nleltd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
nleltd.3 ⊢ ( 𝜑 → ¬ 𝐵 ≤ 𝐴 )
Assertion nleltd ( 𝜑 → 𝐴 < 𝐵 )

Proof

Step Hyp Ref Expression
1 nleltd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 nleltd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3 nleltd.3 ⊢ ( 𝜑 → ¬ 𝐵 ≤ 𝐴 )
4 1 2 ltnled ⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴 ) )
5 3 4 mpbird ⊢ ( 𝜑 → 𝐴 < 𝐵 )