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Theorem ch0le 31189
Description: The zero subspace is the smallest member of C. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
Assertion
Ref Expression
ch0le (𝐴C → 0𝐴)

Proof of Theorem ch0le
StepHypRef Expression
1 chsh 30972 . 2 (𝐴C𝐴S )
2 sh0le 31188 . 2 (𝐴S → 0𝐴)
31, 2syl 17 1 (𝐴C → 0𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  wss 3941   S csh 30676   C cch 30677  0c0h 30683
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 30747
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-uni 4901  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-iota 6486  df-fv 6542  df-ov 7405  df-sh 30955  df-ch 30969  df-ch0 31001
This theorem is referenced by:  chnlen0  31192  ch0pss  31193  ch0lei  31199  chssoc  31244  atcveq0  32096
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