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Theorem psrel 14405
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 14404 . . 3  |-  ( A  e.  PosetRel  ->  ( A  e.  PosetRel  <->  ( Rel  A  /\  ( A  o.  A )  C_  A  /\  ( A  i^i  `' A )  =  (  _I  |`  U. U. A ) ) ) )
21ibi 232 . 2  |-  ( A  e.  PosetRel  ->  ( Rel  A  /\  ( A  o.  A
)  C_  A  /\  ( A  i^i  `' A
)  =  (  _I  |`  U. U. A ) ) )
32simp1d 967 1  |-  ( A  e.  PosetRel  ->  Rel  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1642    e. wcel 1710    i^i cin 3227    C_ wss 3228   U.cuni 3906    _I cid 4383   `'ccnv 4767    |` cres 4770    o. ccom 4772   Rel wrel 4773   PosetRelcps 14394
This theorem is referenced by:  pslem  14408  cnvps  14414  psss  14416  cnvtsr  14424  spwpr4  14433  spwpr4c  14434  tsrdir  14453
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-rex 2625  df-v 2866  df-in 3235  df-ss 3242  df-uni 3907  df-br 4103  df-opab 4157  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-res 4780  df-ps 14399
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