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Theorem latref 14474
Description: A lattice ordering is reflexive. (ssid 3359 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 14470 . 2  |-  ( K  e.  Lat  ->  K  e.  Poset )
2 latref.b . . 3  |-  B  =  ( Base `  K
)
3 latref.l . . 3  |-  .<_  =  ( le `  K )
42, 3posref 14400 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  X  .<_  X )
51, 4sylan 458 1  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X  .<_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446   Basecbs 13461   lecple 13528   Posetcpo 14389   Latclat 14466
This theorem is referenced by:  latleeqj1  14484  latjidm  14495  latleeqm1  14500  latmidm  14507  olj01  29960  olm01  29971  cmtidN  29992  ps-1  30211  3at  30224  llnneat  30248  2atnelpln  30278  lplnneat  30279  lplnnelln  30280  3atnelvolN  30320  lvolneatN  30322  lvolnelln  30323  lvolnelpln  30324  4at  30347  lplncvrlvol  30350  lncmp  30517  lhpocnle  30750  ltrnel  30873  ltrncnvel  30876  ltrnmw  30885  tendoidcl  31503  cdlemk39u  31702  dia1eldmN  31776  dia1N  31788  dihwN  32024  dihglblem5apreN  32026  dihmeetbclemN  32039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-poset 14395  df-lat 14467
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