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Theorem ch0le 29234
 Description: The zero subspace is the smallest member of Cℋ. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
Assertion
Ref Expression
ch0le (𝐴C → 0𝐴)

Proof of Theorem ch0le
StepHypRef Expression
1 chsh 29017 . 2 (𝐴C𝐴S )
2 sh0le 29233 . 2 (𝐴S → 0𝐴)
31, 2syl 17 1 (𝐴C → 0𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111   ⊆ wss 3881   Sℋ csh 28721   Cℋ cch 28722  0ℋc0h 28728 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-hilex 28792 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-xp 5526  df-cnv 5528  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fv 6333  df-ov 7139  df-sh 29000  df-ch 29014  df-ch0 29046 This theorem is referenced by:  chnlen0  29237  ch0pss  29238  ch0lei  29244  chssoc  29289  atcveq0  30141
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