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Metamath Proof Explorer

Theorem lttri2i

Description: Consequence of trichotomy. (Contributed by NM, 19-Jan-1997)

Ref Expression
Hypotheses lt.1 ⊢ A ∈ ℝ
lt.2 ⊢ B ∈ ℝ
Assertion lttri2i ⊢ A ≠ B ↔ A < B ∨ B < A

Proof

Step Hyp Ref Expression
1 lt.1 ⊢ A ∈ ℝ
2 lt.2 ⊢ B ∈ ℝ
3 lttri2 ⊢ A ∈ ℝ ∧ B ∈ ℝ → A ≠ B ↔ A < B ∨ B < A
4 1 2 3 mp2an ⊢ A ≠ B ↔ A < B ∨ B < A