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Metamath Proof Explorer

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. )