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Theorem latpos 14480
Description: A lattice is a poset. (Contributed by NM, 17-Sep-2011.)
Assertion
Ref Expression
latpos  |-  ( K  e.  Lat  ->  K  e.  Poset )

Proof of Theorem latpos
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2438 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2438 . . 3  |-  ( join `  K )  =  (
join `  K )
3 eqid 2438 . . 3  |-  ( meet `  K )  =  (
meet `  K )
41, 2, 3islat 14478 . 2  |-  ( K  e.  Lat  <->  ( K  e.  Poset  /\  A. x  e.  ( Base `  K
) A. y  e.  ( Base `  K
) ( ( x ( join `  K
) y )  e.  ( Base `  K
)  /\  ( x
( meet `  K )
y )  e.  (
Base `  K )
) ) )
54simplbi 448 1  |-  ( K  e.  Lat  ->  K  e.  Poset )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726   A.wral 2707   ` cfv 5456  (class class class)co 6083   Basecbs 13471   Posetcpo 14399   joincjn 14403   meetcmee 14404   Latclat 14476
This theorem is referenced by:  latref  14484  latasymb  14485  lattr  14487  latjcom  14490  latjle12  14493  latleeqj1  14494  latmcom  14506  latlem12  14509  latleeqm1  14510  atlpos  30161  cvlposN  30187  hlpos  30225
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086  df-lat 14477
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