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Theorem elch0 28662
 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 28661 . . 3 0 = {0}
21eleq2i 2898 . 2 (𝐴 ∈ 0𝐴 ∈ {0})
3 ax-hv0cl 28411 . . . 4 0 ∈ ℋ
43elexi 3430 . . 3 0 ∈ V
54elsn2 4434 . 2 (𝐴 ∈ {0} ↔ 𝐴 = 0)
62, 5bitri 267 1 (𝐴 ∈ 0𝐴 = 0)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   = wceq 1656   ∈ wcel 2164  {csn 4399   ℋchba 28327  0ℎc0v 28332  0ℋc0h 28343 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803  ax-hv0cl 28411 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-v 3416  df-sn 4400  df-ch0 28661 This theorem is referenced by:  ocin  28706  ocnel  28708  shuni  28710  choc0  28736  choc1  28737  omlsilem  28812  pjoc1i  28841  shne0i  28858  h1dn0  28962  spansnm0i  29060  nonbooli  29061  eleigvec  29367  cdjreui  29842  cdj3lem1  29844
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