Theorem xrleloe

Description: 'Less than or equal' expressed in terms of 'less than' or 'equals', for extended reals. (Contributed by NM, 19-Jan-2006)

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

Proof

Step	Hyp	Ref	Expression
1		xrlenlt	|- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -. B < A ) )
2		xrlttri	|- ( ( B e. RR* /\ A e. RR* ) -> ( B < A <-> -. ( B = A \/ A < B ) ) )
3	2	ancoms	|- ( ( A e. RR* /\ B e. RR* ) -> ( B < A <-> -. ( B = A \/ A < B ) ) )
4	3	con2bid	|- ( ( A e. RR* /\ B e. RR* ) -> ( ( B = A \/ A < B ) <-> -. B < A ) )
5		eqcom	|- ( B = A <-> A = B )
6	5	orbi1i	|- ( ( B = A \/ A < B ) <-> ( A = B \/ A < B ) )
7		orcom	|- ( ( A = B \/ A < B ) <-> ( A < B \/ A = B ) )
8	6 7	bitri	|- ( ( B = A \/ A < B ) <-> ( A < B \/ A = B ) )
9	4 8	bitr3di	|- ( ( A e. RR* /\ B e. RR* ) -> ( -. B < A <-> ( A < B \/ A = B ) ) )
10	1 9	bitrd	|- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> ( A < B \/ A = B ) ) )