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Theorem relcat 25173
Description: A category is a relation. (Contributed by FL, 26-Oct-2007.)
Assertion
Ref Expression
relcat  |-  Rel  Cat OLD

Proof of Theorem relcat
StepHypRef Expression
1 strcat 25172 . . 3  |-  Cat OLD  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
2 xpss 4793 . . 3  |-  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )  C_  ( _V  X.  _V )
31, 2sstri 3188 . 2  |-  Cat OLD  C_  ( _V  X.  _V )
4 df-rel 4696 . 2  |-  ( Rel 
Cat OLD  <->  Cat OLD  C_  ( _V  X.  _V ) )
53, 4mpbir 200 1  |-  Rel  Cat OLD
Colors of variables: wff set class
Syntax hints:   _Vcvv 2788    C_ wss 3152    X. cxp 4687   Rel wrel 4694    Cat OLD ccatOLD 25164
This theorem is referenced by:  catded  25176
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695  df-rel 4696  df-catOLD 25165
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