Theorem nvrel 31083

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 31074	. 2 ⊢ NrmCVec ⊆ (CVecOLD × V)
2		relxp 5673	. 2 ⊢ Rel (CVecOLD × V)
3		relss 5762	. 2 ⊢ (NrmCVec ⊆ (CVecOLD × V) → (Rel (CVecOLD × V) → Rel NrmCVec))
4	1, 2, 3	mp2 9	1 ⊢ Rel NrmCVec

This proof depends on syntax axioms:  Vcvv 3450   ⊆ wss 3899   × cxp 5653  Rel wrel 5660  CVec_OLDcvc 31039  NrmCVeccnv 31065

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-opab 5168  df-xp 5661  df-rel 5662  df-oprab 7417  df-nv 31073

This theorem is used by:  nvop2  31089  nvop  31157  phrel  31296  bnrel  31348