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Theorem clatpos 15880
Description: A complete lattice is a poset. (Contributed by NM, 8-Sep-2018.)
Assertion
Ref Expression
clatpos  |-  ( K  e.  CLat  ->  K  e. 
Poset )

Proof of Theorem clatpos
StepHypRef Expression
1 eqid 2396 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2396 . . 3  |-  ( lub `  K )  =  ( lub `  K )
3 eqid 2396 . . 3  |-  ( glb `  K )  =  ( glb `  K )
41, 2, 3isclat 15879 . 2  |-  ( K  e.  CLat  <->  ( K  e. 
Poset  /\  ( dom  ( lub `  K )  =  ~P ( Base `  K
)  /\  dom  ( glb `  K )  =  ~P ( Base `  K )
) ) )
54simplbi 458 1  |-  ( K  e.  CLat  ->  K  e. 
Poset )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1836   ~Pcpw 3944   dom cdm 4930   ` cfv 5513   Basecbs 14657   Posetcpo 15709   lubclub 15711   glbcglb 15712   CLatccla 15877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-rex 2752  df-rab 2755  df-v 3053  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-br 4385  df-dm 4940  df-iota 5477  df-fv 5521  df-clat 15878
This theorem is referenced by: (None)
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