Theorem elch0 31190

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 31189	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2821	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 30939	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3473	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4632	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 275	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2109  {csn 4592   ℋchba 30855  0_ℎc0v 30860  0_ℋc0h 30871

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-hv0cl 30939

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-sn 4593  df-ch0 31189

This theorem is referenced by:  ocin  31232  ocnel  31234  shuni  31236  choc0  31262  choc1  31263  omlsilem  31338  pjoc1i  31367  shne0i  31384  h1dn0  31488  spansnm0i  31586  nonbooli  31587  eleigvec  31893  cdjreui  32368  cdj3lem1  32370