Theorem elch0 31511

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 31510	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2857	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 31260	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3479	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4627	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 278	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

Syntax hints:   ↔ wb 209   = wceq 1563   ∈ wcel 2145  {csn 4585   ℋchba 31176  0_ℎc0v 31181  0_ℋc0h 31192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-hv0cl 31260

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-sn 4586  df-ch0 31510

This theorem is referenced by:  ocin  31553  ocnel  31555  shuni  31557  choc0  31583  choc1  31584  omlsilem  31659  pjoc1i  31688  shne0i  31705  h1dn0  31809  spansnm0i  31907  nonbooli  31908  eleigvec  32214  cdjreui  32689  cdj3lem1  32691