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Theorem hlpos 29555
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 29553 . 2  |-  ( K  e.  HL  ->  K  e.  Lat )
2 latpos 14155 . 2  |-  ( K  e.  Lat  ->  K  e.  Poset )
31, 2syl 15 1  |-  ( K  e.  HL  ->  K  e.  Poset )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   Posetcpo 14074   Latclat 14151   HLchlt 29540
This theorem is referenced by:  hlhgt2  29578  hl0lt1N  29579  cvrval3  29602  cvrexchlem  29608  cvratlem  29610  cvrat  29611  atlelt  29627  2atlt  29628  athgt  29645  1cvratex  29662  ps-2  29667  llnnleat  29702  llncmp  29711  2llnmat  29713  lplnnle2at  29730  llncvrlpln  29747  lplncmp  29751  lvolnle3at  29771  lplncvrlvol  29805  lvolcmp  29806  pmaple  29950  2lnat  29973  2atm2atN  29974  lhp2lt  30190  lhp0lt  30192  dia2dimlem2  31255  dia2dimlem3  31256  dih1  31476
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-lat 14152  df-atl 29488  df-cvlat 29512  df-hlat 29541
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