Theorem nvrel 30954

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 30945	. 2 ⊢ NrmCVec ⊆ (CVecOLD × V)
2		relxp 5679	. 2 ⊢ Rel (CVecOLD × V)
3		relss 5768	. 2 ⊢ (NrmCVec ⊆ (CVecOLD × V) → (Rel (CVecOLD × V) → Rel NrmCVec))
4	1, 2, 3	mp2 9	1 ⊢ Rel NrmCVec

Syntax hints:  Vcvv 3455   ⊆ wss 3905   × cxp 5659  Rel wrel 5666  CVec_OLDcvc 30910  NrmCVeccnv 30936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-11 2192  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-opab 5174  df-xp 5667  df-rel 5668  df-oprab 7414  df-nv 30944

This theorem is referenced by:  nvop2  30960  nvop  31028  phrel  31167  bnrel  31219