Theorem nvrel 29842

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 29833	. 2 ⊢ NrmCVec ⊆ (CVecOLD × V)
2		relxp 5693	. 2 ⊢ Rel (CVecOLD × V)
3		relss 5779	. 2 ⊢ (NrmCVec ⊆ (CVecOLD × V) → (Rel (CVecOLD × V) → Rel NrmCVec))
4	1, 2, 3	mp2 9	1 ⊢ Rel NrmCVec

Syntax hints:  Vcvv 3474   ⊆ wss 3947   × cxp 5673  Rel wrel 5680  CVec_OLDcvc 29798  NrmCVeccnv 29824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-opab 5210  df-xp 5681  df-rel 5682  df-oprab 7409  df-nv 29832

This theorem is referenced by:  nvop2  29848  nvop  29916  phrel  30055  bnrel  30107