Theorem bnrel 31067

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 31066	. . 3 ⊢ (𝑥 ∈ CBan → 𝑥 ∈ NrmCVec)
2	1	ssriv 3940	. 2 ⊢ CBan ⊆ NrmCVec
3		nvrel 30802	. 2 ⊢ Rel NrmCVec
4		relss 5754	. 2 ⊢ (CBan ⊆ NrmCVec → (Rel NrmCVec → Rel CBan))
5	2, 3, 4	mp2 9	1 ⊢ Rel CBan

Syntax hints:   ⊆ wss 3904  Rel wrel 5652  NrmCVeccnv 30784  CBanccbn 31062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-11 2191  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-iota 6477  df-fv 6529  df-oprab 7400  df-nv 30792  df-cbn 31063

This theorem is referenced by:  hlrel  31090