Theorem bnrel 31159

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 31158	. . 3 ⊢ (𝑥 ∈ CBan → 𝑥 ∈ NrmCVec)
2	1	ssriv 3949	. 2 ⊢ CBan ⊆ NrmCVec
3		nvrel 30894	. 2 ⊢ Rel NrmCVec
4		relss 5769	. 2 ⊢ (CBan ⊆ NrmCVec → (Rel NrmCVec → Rel CBan))
5	2, 3, 4	mp2 9	1 ⊢ Rel CBan

Syntax hints:   ⊆ wss 3913  Rel wrel 5667  NrmCVeccnv 30876  CBanccbn 31154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-11 2198  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-xp 5668  df-rel 5669  df-iota 6493  df-fv 6545  df-oprab 7415  df-nv 30884  df-cbn 31155

This theorem is referenced by:  hlrel  31182