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Theorem chle0i 22802
Description: No Hilbert closed subspace is smaller than zero. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)
Hypothesis
Ref Expression
ch0le.1  |-  A  e. 
CH
Assertion
Ref Expression
chle0i  |-  ( A 
C_  0H  <->  A  =  0H )

Proof of Theorem chle0i
StepHypRef Expression
1 ch0le.1 . 2  |-  A  e. 
CH
2 chle0 22793 . 2  |-  ( A  e.  CH  ->  ( A  C_  0H  <->  A  =  0H ) )
31, 2ax-mp 8 1  |-  ( A 
C_  0H  <->  A  =  0H )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649    e. wcel 1717    C_ wss 3263   CHcch 22280   0Hc0h 22286
This theorem is referenced by:  chj00i  22837  chsup0  22898  spansnm0i  23000  largei  23618
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-hilex 22350
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 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-xp 4824  df-cnv 4826  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fv 5402  df-ov 6023  df-sh 22557  df-ch 22572  df-ch0 22603
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