Metamath Proof Explorer


Theorem lttri3d

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

Ref Expression
Hypotheses ltd.1 φ A
ltd.2 φ B
Assertion lttri3d φ A = B ¬ A < B ¬ B < A

Proof

Step Hyp Ref Expression
1 ltd.1 φ A
2 ltd.2 φ B
3 lttri3 A B A = B ¬ A < B ¬ B < A
4 1 2 3 syl2anc φ A = B ¬ A < B ¬ B < A