Theorem bnrel 29223

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 29222	. . 3 ⊢ (𝑥 ∈ CBan → 𝑥 ∈ NrmCVec)
2	1	ssriv 3930	. 2 ⊢ CBan ⊆ NrmCVec
3		nvrel 28958	. 2 ⊢ Rel NrmCVec
4		relss 5691	. 2 ⊢ (CBan ⊆ NrmCVec → (Rel NrmCVec → Rel CBan))
5	2, 3, 4	mp2 9	1 ⊢ Rel CBan

Syntax hints:   ⊆ wss 3892  Rel wrel 5594  NrmCVeccnv 28940  CBanccbn 29218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-11 2158  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-xp 5595  df-rel 5596  df-iota 6389  df-fv 6439  df-oprab 7273  df-nv 28948  df-cbn 29219

This theorem is referenced by:  hlrel  29246