Theorem elch0 31183

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 31182	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2820	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 30932	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3470	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4629	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 275	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2109  {csn 4589   ℋchba 30848  0_ℎc0v 30853  0_ℋc0h 30864

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 2701  ax-hv0cl 30932

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3449  df-sn 4590  df-ch0 31182

This theorem is referenced by:  ocin  31225  ocnel  31227  shuni  31229  choc0  31255  choc1  31256  omlsilem  31331  pjoc1i  31360  shne0i  31377  h1dn0  31481  spansnm0i  31579  nonbooli  31580  eleigvec  31886  cdjreui  32361  cdj3lem1  32363