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Metamath Proof Explorer

Theorem leloe

Description: 'Less than or equal to' expressed in terms of 'less than' or 'equals'. (Contributed by NM, 13-May-1999)

Ref Expression
Assertion leloe ⊢ A ∈ ℝ ∧ B ∈ ℝ → A ≤ B ↔ A < B ∨ A = B

Proof

Step Hyp Ref Expression
1 lenlt ⊢ A ∈ ℝ ∧ B ∈ ℝ → A ≤ B ↔ ¬ B < A
2 axlttri ⊢ B ∈ ℝ ∧ A ∈ ℝ → B < A ↔ ¬ B = A ∨ A < B
3 2 ancoms ⊢ A ∈ ℝ ∧ B ∈ ℝ → B < A ↔ ¬ B = A ∨ A < B
4 3 con2bid ⊢ A ∈ ℝ ∧ B ∈ ℝ → B = A ∨ A < B ↔ ¬ B < A
5 eqcom ⊢ B = A ↔ A = B
6 5 orbi1i ⊢ B = A ∨ A < B ↔ A = B ∨ A < B
7 orcom ⊢ A = B ∨ A < B ↔ A < B ∨ A = B
8 6 7 bitri ⊢ B = A ∨ A < B ↔ A < B ∨ A = B
9 4 8 bitr3di ⊢ A ∈ ℝ ∧ B ∈ ℝ → ¬ B < A ↔ A < B ∨ A = B
10 1 9 bitrd ⊢ A ∈ ℝ ∧ B ∈ ℝ → A ≤ B ↔ A < B ∨ A = B