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Theorem phrel 25503
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 25502 . . 3  |-  ( x  e.  CPreHil OLD  ->  x  e.  NrmCVec )
21ssriv 3508 . 2  |-  CPreHil OLD  C_  NrmCVec
3 nvrel 25268 . 2  |-  Rel  NrmCVec
4 relss 5090 . 2  |-  ( CPreHil OLD  C_  NrmCVec  ->  ( Rel  NrmCVec  ->  Rel  CPreHil OLD ) )
52, 3, 4mp2 9 1  |-  Rel  CPreHil OLD
Colors of variables: wff setvar class
Syntax hints:    C_ wss 3476   Rel wrel 5004   NrmCVeccnv 25250   CPreHil OLDccphlo 25500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-opab 4506  df-xp 5005  df-rel 5006  df-oprab 6289  df-nv 25258  df-ph 25501
This theorem is referenced by:  phop  25506
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