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Theorem psrel 14598
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 14597 . . 3  |-  ( A  e.  PosetRel  ->  ( A  e.  PosetRel  <->  ( Rel  A  /\  ( A  o.  A )  C_  A  /\  ( A  i^i  `' A )  =  (  _I  |`  U. U. A ) ) ) )
21ibi 233 . 2  |-  ( A  e.  PosetRel  ->  ( Rel  A  /\  ( A  o.  A
)  C_  A  /\  ( A  i^i  `' A
)  =  (  _I  |`  U. U. A ) ) )
32simp1d 969 1  |-  ( A  e.  PosetRel  ->  Rel  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1721    i^i cin 3287    C_ wss 3288   U.cuni 3983    _I cid 4461   `'ccnv 4844    |` cres 4847    o. ccom 4849   Rel wrel 4850   PosetRelcps 14587
This theorem is referenced by:  pslem  14601  cnvps  14607  psss  14609  cnvtsr  14617  spwpr4  14626  spwpr4c  14627  tsrdir  14646
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-rex 2680  df-v 2926  df-in 3295  df-ss 3302  df-uni 3984  df-br 4181  df-opab 4235  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-res 4857  df-ps 14592
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