Theorem elch0 9047

Description: Membership in zero for closed subspaces of Hilbert space.

Assertion

Ref	Expression
elch0	|- (A e. 0H <-> A = 0h)

Proof of Theorem elch0

Step	Hyp	Ref	Expression
1		df-ch0 9046	. . 3 |- 0H = {0h}
2	1	eleq2i 1530	. 2 |- (A e. 0H <-> A e. {0h})
3		ax-hv0cl 8794	. . . 4 |- 0h e. H~
4	3	elisseti 1809	. . 3 |- 0h e. V
5	4	elsnc2 2427	. 2 |- (A e. {0h} <-> A = 0h)
6	2, 5	bitr 173	1 |- (A e. 0H <-> A = 0h)

Syntax hints:   <-> wb 146   = wceq 953   e. wcel 955  {csn 2399  H~chil 8727  0hc0v 8730  0Hc0h 8743

This theorem is referenced by:  ocin 9085  ocnelt 9086  chocuni 9088  omlsilem 9159  pjoc1 9179  choc0 9205  choc1 9206  shne0 9286  h1dn0 9390  spansnm0 9512  nonbool 9513  cdjreu 10264  cdj3lem1 10266

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-hv0cl 8794

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-un 2040  df-sn 2402  df-pr 2403  df-ch0 9046

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