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Theorem shle0 22794
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 22792 . . 3  |-  ( A  e.  SH  ->  0H  C_  A )
21biantrud 494 . 2  |-  ( A  e.  SH  ->  ( A  C_  0H  <->  ( A  C_  0H  /\  0H  C_  A ) ) )
3 eqss 3308 . 2  |-  ( A  =  0H  <->  ( A  C_  0H  /\  0H  C_  A ) )
42, 3syl6bbr 255 1  |-  ( A  e.  SH  ->  ( A  C_  0H  <->  A  =  0H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3265   SHcsh 22281   0Hc0h 22288
This theorem is referenced by:  chle0  22795  shne0i  22800  shs00i  22802  cdj3lem1  23787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-hilex 22352
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-br 4156  df-opab 4210  df-xp 4826  df-cnv 4828  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-sh 22559  df-ch0 22605
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