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*𝐴𝐵 ) → ( 𝐴 < 𝐵𝐵𝐴 ) )

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 syl6bi ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( 𝐴 = 𝐵 → ¬ 𝐴 < 𝐵 ) )
4 3 adantr ( ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) ∧ 𝐴𝐵 ) → ( 𝐴 = 𝐵 → ¬ 𝐴 < 𝐵 ) )
5 xrleloe ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( 𝐴𝐵 ↔ ( 𝐴 < 𝐵𝐴 = 𝐵 ) ) )
6 5 biimpa ( ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) ∧ 𝐴𝐵 ) → ( 𝐴 < 𝐵𝐴 = 𝐵 ) )
7 6 ord ( ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) ∧ 𝐴𝐵 ) → ( ¬ 𝐴 < 𝐵𝐴 = 𝐵 ) )
8 4 7 impbid ( ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) ∧ 𝐴𝐵 ) → ( 𝐴 = 𝐵 ↔ ¬ 𝐴 < 𝐵 ) )
9 8 necon2abid ( ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) ∧ 𝐴𝐵 ) → ( 𝐴 < 𝐵𝐴𝐵 ) )
10 necom ( 𝐵𝐴𝐴𝐵 )
11 9 10 bitr4di ( ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) ∧ 𝐴𝐵 ) → ( 𝐴 < 𝐵𝐵𝐴 ) )
12 11 3impa ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵 ) → ( 𝐴 < 𝐵𝐵𝐴 ) )