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Theorem phrel 30063
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 30062 . . 3 (𝑥 ∈ CPreHilOLD𝑥 ∈ NrmCVec)
21ssriv 3986 . 2 CPreHilOLD ⊆ NrmCVec
3 nvrel 29850 . 2 Rel NrmCVec
4 relss 5781 . 2 (CPreHilOLD ⊆ NrmCVec → (Rel NrmCVec → Rel CPreHilOLD))
52, 3, 4mp2 9 1 Rel CPreHilOLD
Colors of variables: wff setvar class
Syntax hints:  wss 3948  Rel wrel 5681  NrmCVeccnv 29832  CPreHilOLDccphlo 30060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-opab 5211  df-xp 5682  df-rel 5683  df-oprab 7412  df-nv 29840  df-ph 30061
This theorem is referenced by:  phop  30066
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