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Mirrors > Home > MPE Home > Th. List > ltle | Structured version Visualization version GIF version |
Description: 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.) |
Ref | Expression |
---|---|
ltle | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orc 863 | . 2 ⊢ (𝐴 < 𝐵 → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)) | |
2 | leloe 11045 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) | |
3 | 1, 2 | syl5ibr 245 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) |
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