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Theorem bnrel 28639
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
StepHypRef Expression
1 bnnv 28638 . . 3 (𝑥 ∈ CBan → 𝑥 ∈ NrmCVec)
21ssriv 3955 . 2 CBan ⊆ NrmCVec
3 nvrel 28374 . 2 Rel NrmCVec
4 relss 5637 . 2 (CBan ⊆ NrmCVec → (Rel NrmCVec → Rel CBan))
52, 3, 4mp2 9 1 Rel CBan
Colors of variables: wff setvar class
Syntax hints:  wss 3918  Rel wrel 5541  NrmCVeccnv 28356  CBanccbn 28634
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pr 5311
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-xp 5542  df-rel 5543  df-iota 6295  df-fv 6344  df-oprab 7142  df-nv 28364  df-cbn 28635
This theorem is referenced by:  hlrel  28662
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