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Theorem clatlubcl2 15295
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 5713 . . . . . 6  |-  ( Base `  K )  e.  _V
31, 2eqeltri 2513 . . . . 5  |-  B  e. 
_V
43elpw2 4468 . . . 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 2443 . . . . 5  |-  ( glb `  K )  =  ( glb `  K )
91, 7, 8isclat 15291 . . . 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 2520 1  |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  S  e.  dom  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2984    C_ wss 3340   ~Pcpw 3872   dom cdm 4852   ` cfv 5430   Basecbs 14186   Posetcpo 15122   lubclub 15124   glbcglb 15125   CLatccla 15289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-dm 4862  df-iota 5393  df-fv 5438  df-clat 15290
This theorem is referenced by:  lublem  15300
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