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Metamath Proof Explorer

Theorem xrletri3

Description: Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009)

Ref Expression
Assertion xrletri3 ( ( 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 1 biancomd ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 = 𝐵 ↔ ( ¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 < 𝐵 ) ) )
3 xrlenlt ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴 ) )
4 xrlenlt ⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( 𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵 ) )
5 4 ancoms ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵 ) )
6 3 5 anbi12d ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴 ) ↔ ( ¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 < 𝐵 ) ) )
7 2 6 bitr4d ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 = 𝐵 ↔ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴 ) ) )