Theorem bnrel 30891

Description: The class of all complex Banach spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Assertion

Ref	Expression
bnrel	⊢ Rel CBan

Proof of Theorem bnrel

Step	Hyp	Ref	Expression
1		bnnv 30890	. . 3 ⊢ (𝑥 ∈ CBan → 𝑥 ∈ NrmCVec)
2	1	ssriv 3935	. 2 ⊢ CBan ⊆ NrmCVec
3		nvrel 30626	. 2 ⊢ Rel NrmCVec
4		relss 5729	. 2 ⊢ (CBan ⊆ NrmCVec → (Rel NrmCVec → Rel CBan))
5	2, 3, 4	mp2 9	1 ⊢ Rel CBan

Syntax hints:   ⊆ wss 3899  Rel wrel 5627  NrmCVeccnv 30608  CBanccbn 30886

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 2115  ax-9 2123  ax-11 2162  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

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 2713  df-cleq 2726  df-clel 2809  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-xp 5628  df-rel 5629  df-iota 6446  df-fv 6498  df-oprab 7360  df-nv 30616  df-cbn 30887

This theorem is referenced by:  hlrel  30914