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Theorem sh0le 29226
 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 29039 . 2 0 = {0}
2 sh0 29002 . . 3 (𝐴S → 0𝐴)
32snssd 4726 . 2 (𝐴S → {0} ⊆ 𝐴)
41, 3eqsstrid 4001 1 (𝐴S → 0𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2115   ⊆ wss 3919  {csn 4550  0ℎc0v 28710   Sℋ csh 28714  0ℋc0h 28721 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-hilex 28785 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-xp 5548  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-sh 28993  df-ch0 29039 This theorem is referenced by:  ch0le  29227  shle0  29228  orthin  29232  ssjo  29233  shs0i  29235  span0  29328
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