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Metamath Proof Explorer

Theorem ltnle

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

Ref Expression
Assertion ltnle ⊢ A ∈ ℝ ∧ B ∈ ℝ → A < B ↔ ¬ B ≤ A

Proof

Step Hyp Ref Expression
1 lenlt ⊢ B ∈ ℝ ∧ A ∈ ℝ → B ≤ A ↔ ¬ A < B
2 1 ancoms ⊢ A ∈ ℝ ∧ B ∈ ℝ → B ≤ A ↔ ¬ A < B
3 2 con2bid ⊢ A ∈ ℝ ∧ B ∈ ℝ → A < B ↔ ¬ B ≤ A