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Theorem phrel 27654
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 CPreHilOLD

Proof of Theorem phrel
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 phnv 27653 . . 3 (𝑥 ∈ CPreHilOLD𝑥 ∈ NrmCVec)
21ssriv 3605 . 2 CPreHilOLD ⊆ NrmCVec
3 nvrel 27441 . 2 Rel NrmCVec
4 relss 5204 . 2 (CPreHilOLD ⊆ NrmCVec → (Rel NrmCVec → Rel CPreHilOLD))
52, 3, 4mp2 9 1 Rel CPreHilOLD
Colors of variables: wff setvar class
Syntax hints:  wss 3572  Rel wrel 5117  NrmCVeccnv 27423  CPreHilOLDccphlo 27651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-opab 4711  df-xp 5118  df-rel 5119  df-oprab 6651  df-nv 27431  df-ph 27652
This theorem is referenced by:  phop  27657
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