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Theorem clatpos 15400
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 2454 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2454 . . 3  |-  ( lub `  K )  =  ( lub `  K )
3 eqid 2454 . . 3  |-  ( glb `  K )  =  ( glb `  K )
41, 2, 3isclat 15399 . 2  |-  ( K  e.  CLat  <->  ( K  e. 
Poset  /\  ( dom  ( lub `  K )  =  ~P ( Base `  K
)  /\  dom  ( glb `  K )  =  ~P ( Base `  K )
) ) )
54simplbi 460 1  |-  ( K  e.  CLat  ->  K  e. 
Poset )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   ~Pcpw 3969   dom cdm 4949   ` cfv 5527   Basecbs 14293   Posetcpo 15230   lubclub 15232   glbcglb 15233   CLatccla 15397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-dm 4959  df-iota 5490  df-fv 5535  df-clat 15398
This theorem is referenced by: (None)
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