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Theorem latref 14159
Description: A lattice ordering is reflexive. (ssid 3197 analog.) (Contributed by NM, 8-Oct-2011.)
Hypotheses
Ref Expression
latref.b  |-  B  =  ( Base `  K
)
latref.l  |-  .<_  =  ( le `  K )
Assertion
Ref Expression
latref  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X  .<_  X )

Proof of Theorem latref
StepHypRef Expression
1 latpos 14155 . 2  |-  ( K  e.  Lat  ->  K  e.  Poset )
2 latref.b . . 3  |-  B  =  ( Base `  K
)
3 latref.l . . 3  |-  .<_  =  ( le `  K )
42, 3posref 14085 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  X  .<_  X )
51, 4sylan 457 1  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X  .<_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   Posetcpo 14074   Latclat 14151
This theorem is referenced by:  latleeqj1  14169  latjidm  14180  latleeqm1  14185  latmidm  14192  olj01  28788  olm01  28799  cmtidN  28820  ps-1  29039  3at  29052  llnneat  29076  2atnelpln  29106  lplnneat  29107  lplnnelln  29108  3atnelvolN  29148  lvolneatN  29150  lvolnelln  29151  lvolnelpln  29152  4at  29175  lplncvrlvol  29178  lncmp  29345  lhpocnle  29578  ltrnel  29701  ltrncnvel  29704  ltrnmw  29713  tendoidcl  30331  cdlemk39u  30530  dia1eldmN  30604  dia1N  30616  dihwN  30852  dihglblem5apreN  30854  dihmeetbclemN  30867
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  ax-nul 4149
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-eu 2147  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-sbc 2992  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-poset 14080  df-lat 14152
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