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Theorem elch0 31273
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 31272 . . 3 0 = {0}
21eleq2i 2833 . 2 (𝐴 ∈ 0𝐴 ∈ {0})
3 ax-hv0cl 31022 . . . 4 0 ∈ ℋ
43elexi 3503 . . 3 0 ∈ V
54elsn2 4665 . 2 (𝐴 ∈ {0} ↔ 𝐴 = 0)
62, 5bitri 275 1 (𝐴 ∈ 0𝐴 = 0)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wcel 2108  {csn 4626  chba 30938  0c0v 30943  0c0h 30954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-hv0cl 31022
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-sn 4627  df-ch0 31272
This theorem is referenced by:  ocin  31315  ocnel  31317  shuni  31319  choc0  31345  choc1  31346  omlsilem  31421  pjoc1i  31450  shne0i  31467  h1dn0  31571  spansnm0i  31669  nonbooli  31670  eleigvec  31976  cdjreui  32451  cdj3lem1  32453
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