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Theorem psrel 14312
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 14311 . . 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 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   U.cuni 3827    _I cid 4304   `'ccnv 4688    |` cres 4691    o. ccom 4693   Rel wrel 4694   PosetRelcps 14301
This theorem is referenced by:  pslem  14315  cnvps  14321  psss  14323  cnvtsr  14331  spwpr4  14340  spwpr4c  14341  tsrdir  14360  nfwpr4c  25285
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-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-res 4701  df-ps 14306
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