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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 ) ) )