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Theorem hlpos 29480
Description: A Hilbert lattice is a poset. (Contributed by NM, 20-Oct-2011.)
Assertion
Ref Expression
hlpos  |-  ( K  e.  HL  ->  K  e.  Poset )

Proof of Theorem hlpos
StepHypRef Expression
1 hllat 29478 . 2  |-  ( K  e.  HL  ->  K  e.  Lat )
2 latpos 14405 . 2  |-  ( K  e.  Lat  ->  K  e.  Poset )
31, 2syl 16 1  |-  ( K  e.  HL  ->  K  e.  Poset )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717   Posetcpo 14324   Latclat 14401   HLchlt 29465
This theorem is referenced by:  hlhgt2  29503  hl0lt1N  29504  cvrval3  29527  cvrexchlem  29533  cvratlem  29535  cvrat  29536  atlelt  29552  2atlt  29553  athgt  29570  1cvratex  29587  ps-2  29592  llnnleat  29627  llncmp  29636  2llnmat  29638  lplnnle2at  29655  llncvrlpln  29672  lplncmp  29676  lvolnle3at  29696  lplncvrlvol  29730  lvolcmp  29731  pmaple  29875  2lnat  29898  2atm2atN  29899  lhp2lt  30115  lhp0lt  30117  dia2dimlem2  31180  dia2dimlem3  31181  dih1  31401
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-iota 5358  df-fv 5402  df-ov 6023  df-lat 14402  df-atl 29413  df-cvlat 29437  df-hlat 29466
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