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Theorem ch0le 31460
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 31243 . 2 (𝐴C𝐴S )
2 sh0le 31459 . 2 (𝐴S → 0𝐴)
31, 2syl 17 1 (𝐴C → 0𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wss 3951   S csh 30947   C cch 30948  0c0h 30954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-hilex 31018
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fv 6569  df-ov 7434  df-sh 31226  df-ch 31240  df-ch0 31272
This theorem is referenced by:  chnlen0  31463  ch0pss  31464  ch0lei  31470  chssoc  31515  atcveq0  32367
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