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Theorem ch0le 31369
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 31152 . 2 (𝐴C𝐴S )
2 sh0le 31368 . 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 30856   C cch 30857  0c0h 30863
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 2697  ax-sep 5295  ax-hilex 30927
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3421  df-v 3465  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-br 5145  df-opab 5207  df-xp 5679  df-cnv 5681  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6496  df-fv 6552  df-ov 7417  df-sh 31135  df-ch 31149  df-ch0 31181
This theorem is referenced by:  chnlen0  31372  ch0pss  31373  ch0lei  31379  chssoc  31424  atcveq0  32276
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