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Theorem ch0le 22896
Description: The zero subspace is the smallest member of  CH. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
Assertion
Ref Expression
ch0le  |-  ( A  e.  CH  ->  0H  C_  A )

Proof of Theorem ch0le
StepHypRef Expression
1 chsh 22680 . 2  |-  ( A  e.  CH  ->  A  e.  SH )
2 sh0le 22895 . 2  |-  ( A  e.  SH  ->  0H  C_  A )
31, 2syl 16 1  |-  ( A  e.  CH  ->  0H  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1721    C_ wss 3280   SHcsh 22384   CHcch 22385   0Hc0h 22391
This theorem is referenced by:  chnlen0  22899  ch0pss  22900  ch0lei  22906  chssoc  22951  atcveq0  23804
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-hilex 22455
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-xp 4843  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fv 5421  df-ov 6043  df-sh 22662  df-ch 22677  df-ch0 22708
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