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Theorem phrel 22277
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 22276 . . 3  |-  ( x  e.  CPreHil OLD  ->  x  e.  NrmCVec )
21ssriv 3320 . 2  |-  CPreHil OLD  C_  NrmCVec
3 nvrel 22042 . 2  |-  Rel  NrmCVec
4 relss 4930 . 2  |-  ( CPreHil OLD  C_  NrmCVec  ->  ( Rel  NrmCVec  ->  Rel  CPreHil OLD ) )
52, 3, 4mp2 9 1  |-  Rel  CPreHil OLD
Colors of variables: wff set class
Syntax hints:    C_ wss 3288   Rel wrel 4850   NrmCVeccnv 22024   CPreHil OLDccphlo 22274
This theorem is referenced by:  phop  22280
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-opab 4235  df-xp 4851  df-rel 4852  df-oprab 6052  df-nv 22032  df-ph 22275
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