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Metamath Proof Explorer

Theorem ltleii

Description: 'Less than' implies 'less than or equal to' (inference). (Contributed by NM, 22-Aug-1999)

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

Proof

Step Hyp Ref Expression
1 lt.1 ⊢ A ∈ ℝ
2 lt.2 ⊢ B ∈ ℝ
3 ltlei.1 ⊢ A < B
4 1 2 ltlei ⊢ A < B → A ≤ B
5 3 4 ax-mp ⊢ A ≤ B