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Theorem vcex 28940
Description: The components of a complex vector space are sets. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
Assertion
Ref Expression
vcex (⟨𝐺, 𝑆⟩ ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V))

Proof of Theorem vcex
StepHypRef Expression
1 df-br 5075 . 2 (𝐺CVecOLD𝑆 ↔ ⟨𝐺, 𝑆⟩ ∈ CVecOLD)
2 vcrel 28922 . . 3 Rel CVecOLD
32brrelex12i 5642 . 2 (𝐺CVecOLD𝑆 → (𝐺 ∈ V ∧ 𝑆 ∈ V))
41, 3sylbir 234 1 (⟨𝐺, 𝑆⟩ ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  Vcvv 3432  cop 4567   class class class wbr 5074  CVecOLDcvc 28920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-vc 28921
This theorem is referenced by:  isvcOLD  28941  nvex  28973  isnv  28974
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