Theorem elch0 31084

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 31083	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2821	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 30833	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3493	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4672	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 274	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

Syntax hints:   ↔ wb 205   = wceq 1533   ∈ wcel 2098  {csn 4632   ℋchba 30749  0_ℎc0v 30754  0_ℋc0h 30765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-hv0cl 30833

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-sn 4633  df-ch0 31083

This theorem is referenced by:  ocin  31126  ocnel  31128  shuni  31130  choc0  31156  choc1  31157  omlsilem  31232  pjoc1i  31261  shne0i  31278  h1dn0  31382  spansnm0i  31480  nonbooli  31481  eleigvec  31787  cdjreui  32262  cdj3lem1  32264