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Theorem psrel 14666
Description: A poset is a relation. (Contributed by NM, 12-May-2008.)
Assertion
Ref Expression
psrel  |-  ( A  e.  PosetRel  ->  Rel  A )

Proof of Theorem psrel
StepHypRef Expression
1 isps 14665 . . 3  |-  ( A  e.  PosetRel  ->  ( A  e.  PosetRel  <->  ( Rel  A  /\  ( A  o.  A )  C_  A  /\  ( A  i^i  `' A )  =  (  _I  |`  U. U. A ) ) ) )
21ibi 234 . 2  |-  ( A  e.  PosetRel  ->  ( Rel  A  /\  ( A  o.  A
)  C_  A  /\  ( A  i^i  `' A
)  =  (  _I  |`  U. U. A ) ) )
32simp1d 970 1  |-  ( A  e.  PosetRel  ->  Rel  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1727    i^i cin 3305    C_ wss 3306   U.cuni 4039    _I cid 4522   `'ccnv 4906    |` cres 4909    o. ccom 4911   Rel wrel 4912   PosetRelcps 14655
This theorem is referenced by:  pslem  14669  cnvps  14675  psss  14677  cnvtsr  14685  spwpr4  14694  spwpr4c  14695  tsrdir  14714
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 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-rex 2717  df-v 2964  df-in 3313  df-ss 3320  df-uni 4040  df-br 4238  df-opab 4292  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-res 4919  df-ps 14660
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