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Theorem preotr1 25234
Description: A preset is transitive. (Contributed by FL, 22-May-2011.)
Assertion
Ref Expression
preotr1  |-  ( R  e. PresetRel  ->  ( R  o.  R )  C_  R
)

Proof of Theorem preotr1
StepHypRef Expression
1 isprsr 25224 . . 3  |-  ( R  e. PresetRel  ->  ( R  e. PresetRel  <->  ( Rel  R  /\  ( R  o.  R )  C_  R  /\  (  _I  |`  U. U. R )  C_  R
) ) )
21ibi 232 . 2  |-  ( R  e. PresetRel  ->  ( Rel  R  /\  ( R  o.  R
)  C_  R  /\  (  _I  |`  U. U. R )  C_  R
) )
32simp2d 968 1  |-  ( R  e. PresetRel  ->  ( R  o.  R )  C_  R
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    e. wcel 1684    C_ wss 3152   U.cuni 3827    _I cid 4304    |` cres 4691    o. ccom 4693   Rel wrel 4694  PresetRelcpresetrel 25215
This theorem is referenced by:  preotr2  25235
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-co 4698  df-res 4701  df-prs 25223
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