Metamath Proof Explorer


Theorem xrlenlt

Description: "Less than or equal to" expressed in terms of "less than", for extended reals. (Contributed by NM, 14-Oct-2005)

Ref Expression
Assertion xrlenlt ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( 𝐴𝐵 ↔ ¬ 𝐵 < 𝐴 ) )

Proof

Step Hyp Ref Expression
1 df-br ( 𝐴𝐵 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ ≤ )
2 opelxpi ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ⟨ 𝐴 , 𝐵 ⟩ ∈ ( ℝ* × ℝ* ) )
3 df-le ≤ = ( ( ℝ* × ℝ* ) ∖ < )
4 3 eleq2i ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ≤ ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ ( ( ℝ* × ℝ* ) ∖ < ) )
5 eldif ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( ( ℝ* × ℝ* ) ∖ < ) ↔ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( ℝ* × ℝ* ) ∧ ¬ ⟨ 𝐴 , 𝐵 ⟩ ∈ < ) )
6 4 5 bitri ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ≤ ↔ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( ℝ* × ℝ* ) ∧ ¬ ⟨ 𝐴 , 𝐵 ⟩ ∈ < ) )
7 6 baib ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ( ℝ* × ℝ* ) → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ≤ ↔ ¬ ⟨ 𝐴 , 𝐵 ⟩ ∈ < ) )
8 2 7 syl ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ ≤ ↔ ¬ ⟨ 𝐴 , 𝐵 ⟩ ∈ < ) )
9 1 8 syl5bb ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( 𝐴𝐵 ↔ ¬ ⟨ 𝐴 , 𝐵 ⟩ ∈ < ) )
10 df-br ( 𝐵 < 𝐴 ↔ ⟨ 𝐵 , 𝐴 ⟩ ∈ < )
11 opelcnvg ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ < ↔ ⟨ 𝐵 , 𝐴 ⟩ ∈ < ) )
12 10 11 bitr4id ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( 𝐵 < 𝐴 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ < ) )
13 12 notbid ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ¬ 𝐵 < 𝐴 ↔ ¬ ⟨ 𝐴 , 𝐵 ⟩ ∈ < ) )
14 9 13 bitr4d ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( 𝐴𝐵 ↔ ¬ 𝐵 < 𝐴 ) )