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Theorem shle0 24845
Description: No subspace is smaller than the zero subspace. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)
Assertion
Ref Expression
shle0  |-  ( A  e.  SH  ->  ( A  C_  0H  <->  A  =  0H ) )

Proof of Theorem shle0
StepHypRef Expression
1 sh0le 24843 . . 3  |-  ( A  e.  SH  ->  0H  C_  A )
21biantrud 507 . 2  |-  ( A  e.  SH  ->  ( A  C_  0H  <->  ( A  C_  0H  /\  0H  C_  A ) ) )
3 eqss 3371 . 2  |-  ( A  =  0H  <->  ( A  C_  0H  /\  0H  C_  A ) )
42, 3syl6bbr 263 1  |-  ( A  e.  SH  ->  ( A  C_  0H  <->  A  =  0H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3328   SHcsh 24330   0Hc0h 24337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-hilex 24401
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-opab 4351  df-xp 4846  df-cnv 4848  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-sh 24609  df-ch0 24656
This theorem is referenced by:  chle0  24846  shne0i  24851  shs00i  24853  cdj3lem1  25838
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