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Theorem phrel 28188
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 28187 . . 3 (𝑥 ∈ CPreHilOLD𝑥 ∈ NrmCVec)
21ssriv 3801 . 2 CPreHilOLD ⊆ NrmCVec
3 nvrel 27975 . 2 Rel NrmCVec
4 relss 5410 . 2 (CPreHilOLD ⊆ NrmCVec → (Rel NrmCVec → Rel CPreHilOLD))
52, 3, 4mp2 9 1 Rel CPreHilOLD
Colors of variables: wff setvar class
Syntax hints:  wss 3768  Rel wrel 5316  NrmCVeccnv 27957  CPreHilOLDccphlo 28185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-sep 4974  ax-nul 4982  ax-pr 5096
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-rab 3097  df-v 3386  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-if 4277  df-sn 4368  df-pr 4370  df-op 4374  df-opab 4905  df-xp 5317  df-rel 5318  df-oprab 6881  df-nv 27965  df-ph 28186
This theorem is referenced by:  phop  28191
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