Metamath Proof Explorer


Theorem lttri3d

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

Ref Expression
Hypotheses ltd.1
|- ( ph -> A e. RR )
ltd.2
|- ( ph -> B e. RR )
Assertion lttri3d
|- ( ph -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )

Proof

Step Hyp Ref Expression
1 ltd.1
 |-  ( ph -> A e. RR )
2 ltd.2
 |-  ( ph -> B e. RR )
3 lttri3
 |-  ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
4 1 2 3 syl2anc
 |-  ( ph -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )