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Theorem elch0 22267
Description: Membership in zero for closed subspaces of Hilbert space. (Contributed by NM, 6-Apr-2001.) (New usage is discouraged.)
Assertion
Ref Expression
elch0  |-  ( A  e.  0H  <->  A  =  0h )

Proof of Theorem elch0
StepHypRef Expression
1 df-ch0 22266 . . 3  |-  0H  =  { 0h }
21eleq2i 2430 . 2  |-  ( A  e.  0H  <->  A  e.  { 0h } )
3 ax-hv0cl 22017 . . . 4  |-  0h  e.  ~H
43elexi 2882 . . 3  |-  0h  e.  _V
54elsnc2 3758 . 2  |-  ( A  e.  { 0h }  <->  A  =  0h )
62, 5bitri 240 1  |-  ( A  e.  0H  <->  A  =  0h )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1647    e. wcel 1715   {csn 3729   ~Hchil 21933   0hc0v 21938   0Hc0h 21949
This theorem is referenced by:  ocin  22309  ocnel  22311  shuni  22313  choc0  22339  choc1  22340  omlsilem  22415  pjoc1i  22444  shne0i  22461  h1dn0  22565  spansnm0i  22663  nonbooli  22664  eleigvec  22971  cdjreui  23446  cdj3lem1  23448
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-hv0cl 22017
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-v 2875  df-sn 3735  df-ch0 22266
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