Theorem nvrel 30574

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 30565	. 2 ⊢ NrmCVec ⊆ (CVecOLD × V)
2		relxp 5629	. 2 ⊢ Rel (CVecOLD × V)
3		relss 5717	. 2 ⊢ (NrmCVec ⊆ (CVecOLD × V) → (Rel (CVecOLD × V) → Rel NrmCVec))
4	1, 2, 3	mp2 9	1 ⊢ Rel NrmCVec

Syntax hints:  Vcvv 3436   ⊆ wss 3897   × cxp 5609  Rel wrel 5616  CVec_OLDcvc 30530  NrmCVeccnv 30556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-11 2160  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-opab 5149  df-xp 5617  df-rel 5618  df-oprab 7345  df-nv 30564

This theorem is referenced by:  nvop2  30580  nvop  30648  phrel  30787  bnrel  30839