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Theorem ch0le 31264
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 31047 . 2 (𝐴C𝐴S )
2 sh0le 31263 . 2 (𝐴S → 0𝐴)
31, 2syl 17 1 (𝐴C → 0𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  wss 3947   S csh 30751   C cch 30752  0c0h 30758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-hilex 30822
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fv 6556  df-ov 7423  df-sh 31030  df-ch 31044  df-ch0 31076
This theorem is referenced by:  chnlen0  31267  ch0pss  31268  ch0lei  31274  chssoc  31319  atcveq0  32171
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