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Theorem elch0 22709
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 22708 . . 3  |-  0H  =  { 0h }
21eleq2i 2468 . 2  |-  ( A  e.  0H  <->  A  e.  { 0h } )
3 ax-hv0cl 22459 . . . 4  |-  0h  e.  ~H
43elexi 2925 . . 3  |-  0h  e.  _V
54elsnc2 3803 . 2  |-  ( A  e.  { 0h }  <->  A  =  0h )
62, 5bitri 241 1  |-  ( A  e.  0H  <->  A  =  0h )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649    e. wcel 1721   {csn 3774   ~Hchil 22375   0hc0v 22380   0Hc0h 22391
This theorem is referenced by:  ocin  22751  ocnel  22753  shuni  22755  choc0  22781  choc1  22782  omlsilem  22857  pjoc1i  22886  shne0i  22903  h1dn0  23007  spansnm0i  23105  nonbooli  23106  eleigvec  23413  cdjreui  23888  cdj3lem1  23890
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-hv0cl 22459
This theorem depends on definitions:  df-bi 178  df-an 361  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-v 2918  df-sn 3780  df-ch0 22708
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