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Theorem sh0le 31187
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 31000 . 2 0 = {0}
2 sh0 30963 . . 3 (𝐴S → 0𝐴)
32snssd 4805 . 2 (𝐴S → {0} ⊆ 𝐴)
41, 3eqsstrid 4023 1 (𝐴S → 0𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  wss 3941  {csn 4621  0c0v 30671   S csh 30675  0c0h 30682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-hilex 30746
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-sh 30954  df-ch0 31000
This theorem is referenced by:  ch0le  31188  shle0  31189  orthin  31193  ssjo  31194  shs0i  31196  span0  31289
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