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Theorem laps 14673
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 2438 . . 3  |-  dom  R  =  dom  R
21isla 14670 . 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 448 1  |-  ( R  e.  LatRel  ->  R  e.  PosetRel )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726   A.wral 2707   {cpr 3817   dom cdm 4881  (class class class)co 6084   PosetRelcps 14629    sup w cspw 14631    inf w cinf 14632   LatRelcla 14633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4216  df-dm 4891  df-iota 5421  df-fv 5465  df-ov 6087  df-lar 14638
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