This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem xreqnltd

Description: A consequence of trichotomy. (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypotheses xreqnltd.1 ⊢ φ → A ∈ ℝ *
xreqnltd.2 ⊢ φ → A = B
Assertion xreqnltd ⊢ φ → ¬ A < B

Proof

Step Hyp Ref Expression
1 xreqnltd.1 ⊢ φ → A ∈ ℝ *
2 xreqnltd.2 ⊢ φ → A = B
3 2 1 eqeltrrd ⊢ φ → B ∈ ℝ *
4 xrlttri3 ⊢ A ∈ ℝ * ∧ B ∈ ℝ * → A = B ↔ ¬ A < B ∧ ¬ B < A
5 1 3 4 syl2anc ⊢ φ → A = B ↔ ¬ A < B ∧ ¬ B < A
6 2 5 mpbid ⊢ φ → ¬ A < B ∧ ¬ B < A
7 6 simpld ⊢ φ → ¬ A < B