Metamath Proof Explorer


Theorem lttri3d

Description: Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses ltd.1 ( 𝜑𝐴 ∈ ℝ )
ltd.2 ( 𝜑𝐵 ∈ ℝ )
Assertion lttri3d ( 𝜑 → ( 𝐴 = 𝐵 ↔ ( ¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 ltd.1 ( 𝜑𝐴 ∈ ℝ )
2 ltd.2 ( 𝜑𝐵 ∈ ℝ )
3 lttri3 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 = 𝐵 ↔ ( ¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴 ) ) )
4 1 2 3 syl2anc ( 𝜑 → ( 𝐴 = 𝐵 ↔ ( ¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴 ) ) )