Theorem elch0 31329

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 31328	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2828	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 31078	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3463	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4622	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 275	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

Syntax hints:   ↔ wb 206   = wceq 1541   ∈ wcel 2113  {csn 4580   ℋchba 30994  0_ℎc0v 30999  0_ℋc0h 31010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-hv0cl 31078

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3442  df-sn 4581  df-ch0 31328

This theorem is referenced by:  ocin  31371  ocnel  31373  shuni  31375  choc0  31401  choc1  31402  omlsilem  31477  pjoc1i  31506  shne0i  31523  h1dn0  31627  spansnm0i  31725  nonbooli  31726  eleigvec  32032  cdjreui  32507  cdj3lem1  32509