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Theorem shle0 22021
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 22019 . . 3  |-  ( A  e.  SH  ->  0H  C_  A )
21biantrud 493 . 2  |-  ( A  e.  SH  ->  ( A  C_  0H  <->  ( A  C_  0H  /\  0H  C_  A ) ) )
3 eqss 3194 . 2  |-  ( A  =  0H  <->  ( A  C_  0H  /\  0H  C_  A ) )
42, 3syl6bbr 254 1  |-  ( A  e.  SH  ->  ( A  C_  0H  <->  A  =  0H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   SHcsh 21508   0Hc0h 21515
This theorem is referenced by:  chle0  22022  shne0i  22027  shs00i  22029  cdj3lem1  23014
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-sh 21786  df-ch0 21832
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