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Theorem pstr2 14565
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 14562 . . 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 1649    e. wcel 1717    i^i cin 3263    C_ wss 3264   U.cuni 3958    _I cid 4435   `'ccnv 4818    |` cres 4821    o. ccom 4823   Rel wrel 4824   PosetRelcps 14552
This theorem is referenced by:  pslem  14566  cnvps  14572  psss  14574  tsrdir  14611
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-rex 2656  df-v 2902  df-in 3271  df-ss 3278  df-uni 3959  df-br 4155  df-opab 4209  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-res 4831  df-ps 14557
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