Theorem vcex 30930

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

Step	Hyp	Ref	Expression
1		df-br 5110	. 2 ⊢ (𝐺CVecOLD𝑆 ↔ ⟨𝐺, 𝑆⟩ ∈ CVecOLD)
2		vcrel 30912	. . 3 ⊢ Rel CVecOLD
3	2	brrelex12i 5716	. 2 ⊢ (𝐺CVecOLD𝑆 → (𝐺 ∈ V ∧ 𝑆 ∈ V))
4	1, 3	sylbir 238	1 ⊢ (⟨𝐺, 𝑆⟩ ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V))

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2143  Vcvv 3455  ⟨cop 4595   class class class wbr 5109  CVec_OLDcvc 30910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-vc 30911

This theorem is referenced by:  isvcOLD  30931  nvex  30963  isnv  30964