Metamath Proof Explorer


Theorem xrltlen

Description: 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 6-Nov-2015)

Ref Expression
Assertion xrltlen ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( 𝐴 < 𝐵 ↔ ( 𝐴𝐵𝐵𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 xrlttri ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( 𝐴 < 𝐵 ↔ ¬ ( 𝐴 = 𝐵𝐵 < 𝐴 ) ) )
2 ioran ( ¬ ( 𝐴 = 𝐵𝐵 < 𝐴 ) ↔ ( ¬ 𝐴 = 𝐵 ∧ ¬ 𝐵 < 𝐴 ) )
3 2 biancomi ( ¬ ( 𝐴 = 𝐵𝐵 < 𝐴 ) ↔ ( ¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 = 𝐵 ) )
4 1 3 bitrdi ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( 𝐴 < 𝐵 ↔ ( ¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 = 𝐵 ) ) )
5 xrlenlt ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( 𝐴𝐵 ↔ ¬ 𝐵 < 𝐴 ) )
6 nesym ( 𝐵𝐴 ↔ ¬ 𝐴 = 𝐵 )
7 6 a1i ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( 𝐵𝐴 ↔ ¬ 𝐴 = 𝐵 ) )
8 5 7 anbi12d ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( 𝐴𝐵𝐵𝐴 ) ↔ ( ¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 = 𝐵 ) ) )
9 4 8 bitr4d ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( 𝐴 < 𝐵 ↔ ( 𝐴𝐵𝐵𝐴 ) ) )