Metamath Proof Explorer
Description: 'Less than or equal to' in terms of 'less than'. (Contributed by Mario
Carneiro, 27-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
ltd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
|
|
ltd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
|
Assertion |
leloed |
⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ) ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
ltd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
ltd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
leloe |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ) ) ) |
4 |
1 2 3
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ) ) ) |