Theorem elch0 31340

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 31339	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2829	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 31089	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3453	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4610	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 275	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

Syntax hints:   ↔ wb 206   = wceq 1542   ∈ wcel 2114  {csn 4568   ℋchba 31005  0_ℎc0v 31010  0_ℋc0h 31021

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-hv0cl 31089

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-sn 4569  df-ch0 31339

This theorem is referenced by:  ocin  31382  ocnel  31384  shuni  31386  choc0  31412  choc1  31413  omlsilem  31488  pjoc1i  31517  shne0i  31534  h1dn0  31638  spansnm0i  31736  nonbooli  31737  eleigvec  32043  cdjreui  32518  cdj3lem1  32520