Theorem vcex 31237

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 5109	. 2 ⊢ (𝐺CVecOLD𝑆 ↔ ⟨𝐺, 𝑆⟩ ∈ CVecOLD)
2		vcrel 31219	. . 3 ⊢ Rel CVecOLD
3	2	brrelex12i 5713	. 2 ⊢ (𝐺CVecOLD𝑆 → (𝐺 ∈ V ∧ 𝑆 ∈ V))
4	1, 3	sylbir 238	1 ⊢ (⟨𝐺, 𝑆⟩ ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2147  Vcvv 3454  ⟨cop 4840   class class class wbr 5108  CVec_OLDcvc 31217

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-vc 31218

This theorem is used by:  isvcOLD  31238  nvex  31270  isnv  31271