Theorem vcex 27433

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 >. e. CVecOLD -> ( G e. _V /\ S e. _V ) )

Proof of Theorem vcex

Step	Hyp	Ref	Expression
1		df-br 4654	. 2 |- ( G CVecOLD S <-> <. G , S >. e. CVecOLD )
2		vcrel 27415	. . . 4 |- Rel CVecOLD
3		df-rel 5121	. . . 4 |- ( Rel CVecOLD <-> CVecOLD C_ ( _V X. _V ) )
4	2, 3	mpbi 220	. . 3 |- CVecOLD C_ ( _V X. _V )
5	4	brel 5168	. 2 |- ( G CVecOLD S -> ( G e. _V /\ S e. _V ) )
6	1, 5	sylbir 225	1 |- ( <. G , S >. e. CVecOLD -> ( G e. _V /\ S e. _V ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   _Vcvv 3200   C_ wss 3574   <.cop 4183   class class class wbr 4653   X. cxp 5112   Rel wrel 5119   CVecOLDcvc 27413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-vc 27414

This theorem is referenced by:  isvcOLD  27434  nvex  27466  isnv  27467