Theorem opnlen0

Description: An element not less than another is nonzero. TODO: Look for uses of necon3bd and op0le to see if this is useful elsewhere. (Contributed by NM, 5-May-2013)

	Ref	Expression
Hypotheses	op0le.b	|- B = ( Base ` K )
	op0le.l	|- .<_ = ( le ` K )
	op0le.z	|- .0. = ( 0. ` K )
Assertion	opnlen0	|- ( ( ( K e. OP /\ X e. B /\ Y e. B ) /\ -. X .<_ Y ) -> X =/= .0. )

Proof

Step	Hyp	Ref	Expression
1		op0le.b	|- B = ( Base ` K )
2		op0le.l	|- .<_ = ( le ` K )
3		op0le.z	|- .0. = ( 0. ` K )
4	1 2 3	op0le	|- ( ( K e. OP /\ Y e. B ) -> .0. .<_ Y )
5	4	3adant2	|- ( ( K e. OP /\ X e. B /\ Y e. B ) -> .0. .<_ Y )
6		breq1	|- ( X = .0. -> ( X .<_ Y <-> .0. .<_ Y ) )
7	5 6	syl5ibrcom	|- ( ( K e. OP /\ X e. B /\ Y e. B ) -> ( X = .0. -> X .<_ Y ) )
8	7	necon3bd	|- ( ( K e. OP /\ X e. B /\ Y e. B ) -> ( -. X .<_ Y -> X =/= .0. ) )
9	8	imp	|- ( ( ( K e. OP /\ X e. B /\ Y e. B ) /\ -. X .<_ Y ) -> X =/= .0. )