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Theorem pstr2 14629
Description: A poset is transitive. (Contributed by FL, 3-Aug-2009.)
Assertion
Ref Expression
pstr2  |-  ( R  e.  PosetRel  ->  ( R  o.  R )  C_  R
)

Proof of Theorem pstr2
StepHypRef Expression
1 isps 14626 . . 3  |-  ( R  e.  PosetRel  ->  ( R  e.  PosetRel  <->  ( Rel  R  /\  ( R  o.  R )  C_  R  /\  ( R  i^i  `' R )  =  (  _I  |`  U. U. R ) ) ) )
21ibi 233 . 2  |-  ( R  e.  PosetRel  ->  ( Rel  R  /\  ( R  o.  R
)  C_  R  /\  ( R  i^i  `' R
)  =  (  _I  |`  U. U. R ) ) )
32simp2d 970 1  |-  ( R  e.  PosetRel  ->  ( R  o.  R )  C_  R
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725    i^i cin 3311    C_ wss 3312   U.cuni 4007    _I cid 4485   `'ccnv 4869    |` cres 4872    o. ccom 4874   Rel wrel 4875   PosetRelcps 14616
This theorem is referenced by:  pslem  14630  cnvps  14636  psss  14638  tsrdir  14675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-v 2950  df-in 3319  df-ss 3326  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-res 4882  df-ps 14621
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