Metamath Proof Explorer


Theorem latnlej1l

Description: An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012)

Ref Expression
Hypotheses latlej.b
|- B = ( Base ` K )
latlej.l
|- .<_ = ( le ` K )
latlej.j
|- .\/ = ( join ` K )
Assertion latnlej1l
|- ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ -. X .<_ ( Y .\/ Z ) ) -> X =/= Y )

Proof

Step Hyp Ref Expression
1 latlej.b
 |-  B = ( Base ` K )
2 latlej.l
 |-  .<_ = ( le ` K )
3 latlej.j
 |-  .\/ = ( join ` K )
4 1 2 3 latnlej
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ -. X .<_ ( Y .\/ Z ) ) -> ( X =/= Y /\ X =/= Z ) )
5 4 simpld
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ -. X .<_ ( Y .\/ Z ) ) -> X =/= Y )