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Metamath Proof Explorer

Theorem lenlti

Description: 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 24-May-1999)

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

Proof

Step Hyp Ref Expression
1 lt.1 ⊢ A ∈ ℝ
2 lt.2 ⊢ B ∈ ℝ
3 lenlt ⊢ A ∈ ℝ ∧ B ∈ ℝ → A ≤ B ↔ ¬ B < A
4 1 2 3 mp2an ⊢ A ≤ B ↔ ¬ B < A