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Theorem clatlubcl2 15616
Description: Any subset of the base set has an LUB in a complete lattice. (Contributed by NM, 13-Sep-2018.)
Hypotheses
Ref Expression
clatlubcl.b  |-  B  =  ( Base `  K
)
clatlubcl.u  |-  U  =  ( lub `  K
)
Assertion
Ref Expression
clatlubcl2  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  S  e.  dom  U )

Proof of Theorem clatlubcl2
StepHypRef Expression
1 clatlubcl.b . . . . . 6  |-  B  =  ( Base `  K
)
2 fvex 5882 . . . . . 6  |-  ( Base `  K )  e.  _V
31, 2eqeltri 2551 . . . . 5  |-  B  e. 
_V
43elpw2 4617 . . . 4  |-  ( S  e.  ~P B  <->  S  C_  B
)
54biimpri 206 . . 3  |-  ( S 
C_  B  ->  S  e.  ~P B )
65adantl 466 . 2  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  S  e.  ~P B )
7 clatlubcl.u . . . . 5  |-  U  =  ( lub `  K
)
8 eqid 2467 . . . . 5  |-  ( glb `  K )  =  ( glb `  K )
91, 7, 8isclat 15612 . . . 4  |-  ( K  e.  CLat  <->  ( K  e. 
Poset  /\  ( dom  U  =  ~P B  /\  dom  ( glb `  K )  =  ~P B ) ) )
10 simprl 755 . . . 4  |-  ( ( K  e.  Poset  /\  ( dom  U  =  ~P B  /\  dom  ( glb `  K
)  =  ~P B
) )  ->  dom  U  =  ~P B )
119, 10sylbi 195 . . 3  |-  ( K  e.  CLat  ->  dom  U  =  ~P B )
1211adantr 465 . 2  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  dom  U  =  ~P B )
136, 12eleqtrrd 2558 1  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  S  e.  dom  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    C_ wss 3481   ~Pcpw 4016   dom cdm 5005   ` cfv 5594   Basecbs 14506   Posetcpo 15443   lubclub 15445   glbcglb 15446   CLatccla 15610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-dm 5015  df-iota 5557  df-fv 5602  df-clat 15611
This theorem is referenced by:  lublem  15621
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