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Theorem elch0 28239
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 ∈ 0𝐴 = 0)

Proof of Theorem elch0
StepHypRef Expression
1 df-ch0 28238 . . 3 0 = {0}
21eleq2i 2722 . 2 (𝐴 ∈ 0𝐴 ∈ {0})
3 ax-hv0cl 27988 . . . 4 0 ∈ ℋ
43elexi 3244 . . 3 0 ∈ V
54elsn2 4244 . 2 (𝐴 ∈ {0} ↔ 𝐴 = 0)
62, 5bitri 264 1 (𝐴 ∈ 0𝐴 = 0)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1523  wcel 2030  {csn 4210  chil 27904  0c0v 27909  0c0h 27920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-hv0cl 27988
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-sn 4211  df-ch0 28238
This theorem is referenced by:  ocin  28283  ocnel  28285  shuni  28287  choc0  28313  choc1  28314  omlsilem  28389  pjoc1i  28418  shne0i  28435  h1dn0  28539  spansnm0i  28637  nonbooli  28638  eleigvec  28944  cdjreui  29419  cdj3lem1  29421
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