Theorem elch0 30507

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 30506	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2826	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 30256	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3494	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4668	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 275	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

Syntax hints:   ↔ wb 205   = wceq 1542   ∈ wcel 2107  {csn 4629   ℋchba 30172  0_ℎc0v 30177  0_ℋc0h 30188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-hv0cl 30256

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-sn 4630  df-ch0 30506

This theorem is referenced by:  ocin  30549  ocnel  30551  shuni  30553  choc0  30579  choc1  30580  omlsilem  30655  pjoc1i  30684  shne0i  30701  h1dn0  30805  spansnm0i  30903  nonbooli  30904  eleigvec  31210  cdjreui  31685  cdj3lem1  31687