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Theorem elch0 31084
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 31083 . . 3 0 = {0}
21eleq2i 2821 . 2 (𝐴 ∈ 0𝐴 ∈ {0})
3 ax-hv0cl 30833 . . . 4 0 ∈ ℋ
43elexi 3493 . . 3 0 ∈ V
54elsn2 4672 . 2 (𝐴 ∈ {0} ↔ 𝐴 = 0)
62, 5bitri 274 1 (𝐴 ∈ 0𝐴 = 0)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wcel 2098  {csn 4632  chba 30749  0c0v 30754  0c0h 30765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-hv0cl 30833
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-sn 4633  df-ch0 31083
This theorem is referenced by:  ocin  31126  ocnel  31128  shuni  31130  choc0  31156  choc1  31157  omlsilem  31232  pjoc1i  31261  shne0i  31278  h1dn0  31382  spansnm0i  31480  nonbooli  31481  eleigvec  31787  cdjreui  32262  cdj3lem1  32264
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