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Theorem phrel 21827
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 21826 . . 3  |-  ( x  e.  CPreHil OLD  ->  x  e.  NrmCVec )
21ssriv 3270 . 2  |-  CPreHil OLD  C_  NrmCVec
3 nvrel 21592 . 2  |-  Rel  NrmCVec
4 relss 4878 . 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 3238   Rel wrel 4797   NrmCVeccnv 21574   CPreHil OLDccphlo 21824
This theorem is referenced by:  phop  21830
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-opab 4180  df-xp 4798  df-rel 4799  df-oprab 5985  df-nv 21582  df-ph 21825
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