Theorem chne0i 29222

Description: A nonzero closed subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ch0le.1	⊢ 𝐴 ∈ Cℋ

Assertion

Ref	Expression
chne0i	⊢ (𝐴 ≠ 0ℋ ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ)

Distinct variable group:   𝑥,𝐴

Proof of Theorem chne0i

Step	Hyp	Ref	Expression
1		ch0le.1	. . 3 ⊢ 𝐴 ∈ Cℋ
2	1	chshii 28996	. 2 ⊢ 𝐴 ∈ Sℋ
3	2	shne0i 29217	1 ⊢ (𝐴 ≠ 0ℋ ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ)

Syntax hints:   ↔ wb 208   ∈ wcel 2108   ≠ wne 3014  ∃wrex 3137  0_ℎc0v 28693   C_ℋ cch 28698  0_ℋc0h 28704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-hilex 28768  ax-hv0cl 28772

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fv 6356  df-ov 7151  df-sh 28976  df-ch 28990  df-ch0 29022

This theorem is referenced by:  chne0  29263  hne0  29316  h1datomi  29350  riesz3i  29831  pjnmopi  29917