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Theorem pstr2 14330
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 14327 . . 3  |-  ( R  e.  PosetRel  ->  ( R  e.  PosetRel  <->  ( Rel  R  /\  ( R  o.  R )  C_  R  /\  ( R  i^i  `' R )  =  (  _I  |`  U. U. R ) ) ) )
21ibi 232 . 2  |-  ( R  e.  PosetRel  ->  ( Rel  R  /\  ( R  o.  R
)  C_  R  /\  ( R  i^i  `' R
)  =  (  _I  |`  U. U. R ) ) )
32simp2d 968 1  |-  ( R  e.  PosetRel  ->  ( R  o.  R )  C_  R
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   U.cuni 3843    _I cid 4320   `'ccnv 4704    |` cres 4707    o. ccom 4709   Rel wrel 4710   PosetRelcps 14317
This theorem is referenced by:  pslem  14331  cnvps  14337  psss  14339  tsrdir  14376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-v 2803  df-in 3172  df-ss 3179  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-res 4717  df-ps 14322
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