Theorem elch0 31235

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 31234	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2826	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 30984	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3482	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4641	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 275	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2108  {csn 4601   ℋchba 30900  0_ℎc0v 30905  0_ℋc0h 30916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-hv0cl 30984

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-sn 4602  df-ch0 31234

This theorem is referenced by:  ocin  31277  ocnel  31279  shuni  31281  choc0  31307  choc1  31308  omlsilem  31383  pjoc1i  31412  shne0i  31429  h1dn0  31533  spansnm0i  31631  nonbooli  31632  eleigvec  31938  cdjreui  32413  cdj3lem1  32415