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Theorem shle0 26760
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 26758 . . 3  |-  ( A  e.  SH  ->  0H  C_  A )
21biantrud 505 . 2  |-  ( A  e.  SH  ->  ( A  C_  0H  <->  ( A  C_  0H  /\  0H  C_  A ) ) )
3 eqss 3456 . 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 367    = wceq 1405    e. wcel 1842    C_ wss 3413   SHcsh 26245   0Hc0h 26252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-hilex 26316
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-xp 4828  df-cnv 4830  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-sh 26524  df-ch0 26571
This theorem is referenced by:  chle0  26761  shne0i  26766  shs00i  26768  cdj3lem1  27752
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