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Theorem sh0le 28427
Description: The zero subspace is the smallest subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
sh0le (𝐴S → 0𝐴)

Proof of Theorem sh0le
StepHypRef Expression
1 df-ch0 28238 . 2 0 = {0}
2 sh0 28201 . . 3 (𝐴S → 0𝐴)
32snssd 4372 . 2 (𝐴S → {0} ⊆ 𝐴)
41, 3syl5eqss 3682 1 (𝐴S → 0𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2030  wss 3607  {csn 4210  0c0v 27909   S csh 27913  0c0h 27920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-hilex 27984
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-sh 28192  df-ch0 28238
This theorem is referenced by:  ch0le  28428  shle0  28429  orthin  28433  ssjo  28434  shs0i  28436  span0  28529
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