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Metamath Proof Explorer

Theorem xrletri3

Description: Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009)

Ref Expression
Assertion xrletri3 
|- ( ( A e. RR* /\ B e. RR* ) -> ( A = B <-> ( A <_ B /\ B <_ A ) ) )

Proof

Step Hyp Ref Expression
1 xrlttri3 
 |-  ( ( A e. RR* /\ B e. RR* ) -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
2 1 biancomd 
 |-  ( ( A e. RR* /\ B e. RR* ) -> ( A = B <-> ( -. B < A /\ -. A < B ) ) )
3 xrlenlt 
 |-  ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -. B < A ) )
4 xrlenlt 
 |-  ( ( B e. RR* /\ A e. RR* ) -> ( B <_ A <-> -. A < B ) )
5 4 ancoms 
 |-  ( ( A e. RR* /\ B e. RR* ) -> ( B <_ A <-> -. A < B ) )
6 3 5 anbi12d 
 |-  ( ( A e. RR* /\ B e. RR* ) -> ( ( A <_ B /\ B <_ A ) <-> ( -. B < A /\ -. A < B ) ) )
7 2 6 bitr4d 
 |-  ( ( A e. RR* /\ B e. RR* ) -> ( A = B <-> ( A <_ B /\ B <_ A ) ) )