Theorem elch0 31325

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 31324	. . 3 ⊢ 0ℋ = {0ℎ}
2	1	eleq2i 2828	. 2 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 ∈ {0ℎ})
3		ax-hv0cl 31074	. . . 4 ⊢ 0ℎ ∈ ℋ
4	3	elexi 3452	. . 3 ⊢ 0ℎ ∈ V
5	4	elsn2 4609	. 2 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ)
6	2, 5	bitri 275	1 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ)

Syntax hints:   ↔ wb 206   = wceq 1542   ∈ wcel 2114  {csn 4567   ℋchba 30990  0_ℎc0v 30995  0_ℋc0h 31006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-hv0cl 31074

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-sn 4568  df-ch0 31324

This theorem is referenced by:  ocin  31367  ocnel  31369  shuni  31371  choc0  31397  choc1  31398  omlsilem  31473  pjoc1i  31502  shne0i  31519  h1dn0  31623  spansnm0i  31721  nonbooli  31722  eleigvec  32028  cdjreui  32503  cdj3lem1  32505