Theorem nvrel 30349

Description: The class of all normed complex vectors spaces is a relation. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
nvrel	⊢ Rel NrmCVec

Proof of Theorem nvrel

Step	Hyp	Ref	Expression
1		nvss 30340	. 2 ⊢ NrmCVec ⊆ (CVecOLD × V)
2		relxp 5685	. 2 ⊢ Rel (CVecOLD × V)
3		relss 5772	. 2 ⊢ (NrmCVec ⊆ (CVecOLD × V) → (Rel (CVecOLD × V) → Rel NrmCVec))
4	1, 2, 3	mp2 9	1 ⊢ Rel NrmCVec

Syntax hints:  Vcvv 3466   ⊆ wss 3941   × cxp 5665  Rel wrel 5672  CVec_OLDcvc 30305  NrmCVeccnv 30331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-opab 5202  df-xp 5673  df-rel 5674  df-oprab 7406  df-nv 30339

This theorem is referenced by:  nvop2  30355  nvop  30423  phrel  30562  bnrel  30614