Theorem elch0 31273

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 31272	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2833	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 31022	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3503	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4665	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 275	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2108  {csn 4626   ℋchba 30938  0_ℎc0v 30943  0_ℋc0h 30954

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 2708  ax-hv0cl 31022

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-sn 4627  df-ch0 31272

This theorem is referenced by:  ocin  31315  ocnel  31317  shuni  31319  choc0  31345  choc1  31346  omlsilem  31421  pjoc1i  31450  shne0i  31467  h1dn0  31571  spansnm0i  31669  nonbooli  31670  eleigvec  31976  cdjreui  32451  cdj3lem1  32453