Theorem elch0 28662

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 28661	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2898	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 28411	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3430	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4434	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 267	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

Syntax hints:   ↔ wb 198   = wceq 1656   ∈ wcel 2164  {csn 4399   ℋchba 28327  0_ℎc0v 28332  0_ℋc0h 28343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803  ax-hv0cl 28411

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-v 3416  df-sn 4400  df-ch0 28661

This theorem is referenced by:  ocin  28706  ocnel  28708  shuni  28710  choc0  28736  choc1  28737  omlsilem  28812  pjoc1i  28841  shne0i  28858  h1dn0  28962  spansnm0i  29060  nonbooli  29061  eleigvec  29367  cdjreui  29842  cdj3lem1  29844