Theorem elch0 5158

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

Assertion

Ref	Expression
elch0	⊢ (A ∈ 0ℋ ↔ A = 0v)

Proof of Theorem elch0

Step	Hyp	Ref	Expression
1		df-ch0 5157	. . 3 ⊢ 0ℋ = {0v}
2	1	eleq2i 1153	. 2 ⊢ (A ∈ 0ℋ ↔ A ∈ {0v})
3		ax-hvzercl 4987	. . . 4 ⊢ 0v ∈ ℋ
4	3	elisseti 1355	. . 3 ⊢ 0v ∈ V
5	4	elsnc2 1832	. 2 ⊢ (A ∈ {0v} ↔ A = 0v)
6	2, 5	bitr 151	1 ⊢ (A ∈ 0ℋ ↔ A = 0v)

Syntax hints:   ↔ wb 127   = wceq 1091   ∈ wcel 1092  {csn 1808   ℋ chil 4958  0_vc0v 4961  0_ℋc0h 4974

This theorem is referenced by:  ocin 5177  chocuni 5179  omlsilem 5249  pjoc1 5268  pjoml 5271  choc0 5291  choc1 5292  shne0 5372  h1dn0 5457

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074  ax-hvzercl 4987

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-ch0 5157

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