This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem lubub

Description: The LUB of a complete lattice subset is an upper bound. (Contributed by NM, 19-Oct-2011)

Ref Expression
Hypotheses lublem.b 
|- B = ( Base ` K )
lublem.l 
|- .<_ = ( le ` K )
lublem.u 
|- U = ( lub ` K )
Assertion lubub 
|- ( ( K e. CLat /\ S C_ B /\ X e. S ) -> X .<_ ( U ` S ) )

Proof

Step Hyp Ref Expression
1 lublem.b 
 |-  B = ( Base ` K )
2 lublem.l 
 |-  .<_ = ( le ` K )
3 lublem.u 
 |-  U = ( lub ` K )
4 1 2 3 lublem 
 |-  ( ( K e. CLat /\ S C_ B ) -> ( A. y e. S y .<_ ( U ` S ) /\ A. z e. B ( A. y e. S y .<_ z -> ( U ` S ) .<_ z ) ) )
5 4 simpld 
 |-  ( ( K e. CLat /\ S C_ B ) -> A. y e. S y .<_ ( U ` S ) )
6 breq1 
 |-  ( y = X -> ( y .<_ ( U ` S ) <-> X .<_ ( U ` S ) ) )
7 6 rspccva 
 |-  ( ( A. y e. S y .<_ ( U ` S ) /\ X e. S ) -> X .<_ ( U ` S ) )
8 5 7 stoic3 
 |-  ( ( K e. CLat /\ S C_ B /\ X e. S ) -> X .<_ ( U ` S ) )