Theorem elch0 28111

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 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

Proof of Theorem elch0

Step	Hyp	Ref	Expression
1		df-ch0 28110	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2693	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 27860	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3213	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4211	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 264	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

Syntax hints:   ↔ wb 196   = wceq 1483   ∈ wcel 1990  {csn 4177   ℋchil 27776  0_ℎc0v 27781  0_ℋc0h 27792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-hv0cl 27860

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sn 4178  df-ch0 28110

This theorem is referenced by:  ocin  28155  ocnel  28157  shuni  28159  choc0  28185  choc1  28186  omlsilem  28261  pjoc1i  28290  shne0i  28307  h1dn0  28411  spansnm0i  28509  nonbooli  28510  eleigvec  28816  cdjreui  29291  cdj3lem1  29293