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Theorem laps 14345
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 2283 . . 3  |-  dom  R  =  dom  R
21isla 14342 . 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 446 1  |-  ( R  e.  LatRel  ->  R  e.  PosetRel )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   A.wral 2543   {cpr 3641   dom cdm 4689  (class class class)co 5858   PosetRelcps 14301    sup w cspw 14303    inf w cinf 14304   LatRelcla 14305
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
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  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-dm 4699  df-iota 5219  df-fv 5263  df-ov 5861  df-lar 14310
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