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Theorem sh0le 31127
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 30940 . 2 0 = {0}
2 sh0 30903 . . 3 (𝐴S → 0𝐴)
32snssd 4812 . 2 (𝐴S → {0} ⊆ 𝐴)
41, 3eqsstrid 4030 1 (𝐴S → 0𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  wss 3948  {csn 4628  0c0v 30611   S csh 30615  0c0h 30622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-hilex 30686
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-sh 30894  df-ch0 30940
This theorem is referenced by:  ch0le  31128  shle0  31129  orthin  31133  ssjo  31134  shs0i  31136  span0  31229
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