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Metamath Proof Explorer

Theorem ltlei

Description: 'Less than' implies 'less than or equal to'. (Contributed by NM, 14-May-1999)

Ref Expression
Hypotheses lt.1 ⊢ A ∈ ℝ
lt.2 ⊢ B ∈ ℝ
Assertion ltlei ⊢ A < B → A ≤ B

Proof

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