Metamath Proof Explorer


Theorem xrlelttric

Description: Trichotomy law for extended reals. (Contributed by Thierry Arnoux, 12-Sep-2017)

Ref Expression
Assertion xrlelttric ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( 𝐴𝐵𝐵 < 𝐴 ) )

Proof

Step Hyp Ref Expression
1 pm2.1 ( ¬ 𝐵 < 𝐴𝐵 < 𝐴 )
2 xrlenlt ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( 𝐴𝐵 ↔ ¬ 𝐵 < 𝐴 ) )
3 2 orbi1d ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( 𝐴𝐵𝐵 < 𝐴 ) ↔ ( ¬ 𝐵 < 𝐴𝐵 < 𝐴 ) ) )
4 1 3 mpbiri ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( 𝐴𝐵𝐵 < 𝐴 ) )