Theorem ltm1

Description: A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006)

	Ref	Expression
Assertion	ltm1	|- ( A e. RR -> ( A - 1 ) < A )

Proof

Step	Hyp	Ref	Expression
1		0lt1	|- 0 < 1
2		0re	|- 0 e. RR
3		1re	|- 1 e. RR
4		ltsub2	|- ( ( 0 e. RR /\ 1 e. RR /\ A e. RR ) -> ( 0 < 1 <-> ( A - 1 ) < ( A - 0 ) ) )
5	2 3 4	mp3an12	|- ( A e. RR -> ( 0 < 1 <-> ( A - 1 ) < ( A - 0 ) ) )
6	1 5	mpbii	|- ( A e. RR -> ( A - 1 ) < ( A - 0 ) )
7		recn	|- ( A e. RR -> A e. CC )
8	7	subid1d	|- ( A e. RR -> ( A - 0 ) = A )
9	6 8	breqtrd	|- ( A e. RR -> ( A - 1 ) < A )