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Theorem phrel 21393
Description: The class of all complex inner product spaces is a relation. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
Assertion
Ref Expression
phrel  |-  Rel  CPreHil OLD

Proof of Theorem phrel
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 phnv 21392 . . 3  |-  ( x  e.  CPreHil OLD  ->  x  e.  NrmCVec )
21ssriv 3184 . 2  |-  CPreHil OLD  C_  NrmCVec
3 nvrel 21158 . 2  |-  Rel  NrmCVec
4 relss 4775 . 2  |-  ( CPreHil OLD  C_  NrmCVec  ->  ( Rel  NrmCVec  ->  Rel  CPreHil OLD ) )
52, 3, 4mp2 17 1  |-  Rel  CPreHil OLD
Colors of variables: wff set class
Syntax hints:    C_ wss 3152   Rel wrel 4694   NrmCVeccnv 21140   CPreHil OLDccphlo 21390
This theorem is referenced by:  phop  21396
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-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-oprab 5862  df-nv 21148  df-ph 21391
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