Metamath Proof Explorer

Theorem xrleltne

Description: 'Less than or equal to' implies 'less than' is not 'equals', for extended reals. (Contributed by NM, 9-Feb-2006)

Ref Expression
Assertion xrleltne ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 xrlttri3 ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 = 𝐵 ↔ ( ¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴 ) ) )
2 simpl ⊢ ( ( ¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴 ) → ¬ 𝐴 < 𝐵 )
3 1 2 biimtrdi ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 = 𝐵 → ¬ 𝐴 < 𝐵 ) )
4 3 adantr ⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 = 𝐵 → ¬ 𝐴 < 𝐵 ) )
5 xrleloe ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ) ) )
6 5 biimpa ⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ) )
7 6 ord ⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → ( ¬ 𝐴 < 𝐵 → 𝐴 = 𝐵 ) )
8 4 7 impbid ⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 = 𝐵 ↔ ¬ 𝐴 < 𝐵 ) )
9 8 necon2abid ⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 < 𝐵 ↔ 𝐴 ≠ 𝐵 ) )
10 necom ⊢ ( 𝐵 ≠ 𝐴 ↔ 𝐴 ≠ 𝐵 )
11 9 10 bitr4di ⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴 ) )
12 11 3impa ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 < 𝐵 ↔ 𝐵 ≠ 𝐴 ) )