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 ) ) )