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Theorem laps 14596
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 2388 . . 3  |-  dom  R  =  dom  R
21isla 14593 . 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 1717   A.wral 2650   {cpr 3759   dom cdm 4819  (class class class)co 6021   PosetRelcps 14552    sup w cspw 14554    inf w cinf 14555   LatRelcla 14556
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-dm 4829  df-iota 5359  df-fv 5403  df-ov 6024  df-lar 14561
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