Theorem bnrel 30956

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 30955	. . 3 ⊢ (𝑥 ∈ CBan → 𝑥 ∈ NrmCVec)
2	1	ssriv 3926	. 2 ⊢ CBan ⊆ NrmCVec
3		nvrel 30691	. 2 ⊢ Rel NrmCVec
4		relss 5732	. 2 ⊢ (CBan ⊆ NrmCVec → (Rel NrmCVec → Rel CBan))
5	2, 3, 4	mp2 9	1 ⊢ Rel CBan

Syntax hints:   ⊆ wss 3890  Rel wrel 5630  NrmCVeccnv 30673  CBanccbn 30951

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5631  df-rel 5632  df-iota 6449  df-fv 6501  df-oprab 7365  df-nv 30681  df-cbn 30952

This theorem is referenced by:  hlrel  30979