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Theorem sh0le 27471
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 27282 . 2 0 = {0}
2 sh0 27245 . . 3 (𝐴S → 0𝐴)
32snssd 4184 . 2 (𝐴S → {0} ⊆ 𝐴)
41, 3syl5eqss 3516 1 (𝐴S → 0𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1938  wss 3444  {csn 4028  0c0v 26953   S csh 26957  0c0h 26964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-hilex 27028
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-rab 2809  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-op 4035  df-br 4482  df-opab 4542  df-xp 4938  df-cnv 4940  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-sh 27236  df-ch0 27282
This theorem is referenced by:  ch0le  27472  shle0  27473  orthin  27477  ssjo  27478  shs0i  27480  span0  27573
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