Theorem elch0 9121

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 9120	. . 3 |- 0H = {0h}
2	1	eleq2i 1541	. 2 |- (A e. 0H <-> A e. {0h})
3		ax-hv0cl 8868	. . . 4 |- 0h e. H~
4	3	elisseti 1821	. . 3 |- 0h e. V
5	4	elsnc2 2441	. 2 |- (A e. {0h} <-> A = 0h)
6	2, 5	bitr 173	1 |- (A e. 0H <-> A = 0h)

Syntax hints:   <-> wb 146   = wceq 958   e. wcel 960  {csn 2413  H~chil 8783  0hc0v 8786  0Hc0h 8799

This theorem is referenced by:  ocin 9164  ocnelt 9165  chocuni 9167  omlsilem 9239  pjoc1 9259  choc0 9285  choc1 9286  shne0 9366  h1dn0 9470  spansnm0 9590  nonbool 9591  cdjreu 10354  cdj3lem1 10356

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-hv0cl 8868

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417  df-ch0 9120

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