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Theorem laps 14660
Description: A lattice is a poset. (Contributed by NM, 12-Jun-2008.)
Assertion
Ref Expression
laps  |-  ( R  e.  LatRel  ->  R  e.  PosetRel )

Proof of Theorem laps
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2435 . . 3  |-  dom  R  =  dom  R
21isla 14657 . 2  |-  ( R  e.  LatRel 
<->  ( R  e.  PosetRel  /\  A. x  e.  dom  R A. y  e.  dom  R ( ( R  sup w  { x ,  y } )  e.  dom  R  /\  ( R  inf w  { x ,  y } )  e.  dom  R ) ) )
32simplbi 447 1  |-  ( R  e.  LatRel  ->  R  e.  PosetRel )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   A.wral 2697   {cpr 3807   dom cdm 4870  (class class class)co 6073   PosetRelcps 14616    sup w cspw 14618    inf w cinf 14619   LatRelcla 14620
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
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-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  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-dm 4880  df-iota 5410  df-fv 5454  df-ov 6076  df-lar 14625
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