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Theorem phrel 28010
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 28009 . . 3 (𝑥 ∈ CPreHilOLD𝑥 ∈ NrmCVec)
21ssriv 3756 . 2 CPreHilOLD ⊆ NrmCVec
3 nvrel 27797 . 2 Rel NrmCVec
4 relss 5345 . 2 (CPreHilOLD ⊆ NrmCVec → (Rel NrmCVec → Rel CPreHilOLD))
52, 3, 4mp2 9 1 Rel CPreHilOLD
Colors of variables: wff setvar class
Syntax hints:  wss 3723  Rel wrel 5255  NrmCVeccnv 27779  CPreHilOLDccphlo 28007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-opab 4848  df-xp 5256  df-rel 5257  df-oprab 6800  df-nv 27787  df-ph 28008
This theorem is referenced by:  phop  28013
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