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Metamath Proof Explorer

Theorem ltnlei

Description: 'Less than' in terms of 'less than or equal to'. (Contributed by NM, 11-Jul-2005)

Ref Expression
Hypotheses lt.1 ⊢ A ∈ ℝ
lt.2 ⊢ B ∈ ℝ
Assertion ltnlei ⊢ A < B ↔ ¬ B ≤ A

Proof

Step Hyp Ref Expression
1 lt.1 ⊢ A ∈ ℝ
2 lt.2 ⊢ B ∈ ℝ
3 2 1 lenlti ⊢ B ≤ A ↔ ¬ A < B
4 3 con2bii ⊢ A < B ↔ ¬ B ≤ A