Theorem chne0i 31482

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 31256	. 2 ⊢ 𝐴 ∈ Sℋ
3	2	shne0i 31477	1 ⊢ (𝐴 ≠ 0ℋ ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ)

Syntax hints:   ↔ wb 206   ∈ wcel 2106   ≠ wne 2938  ∃wrex 3068  0_ℎc0v 30953   C_ℋ cch 30958  0_ℋc0h 30964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-hilex 31028  ax-hv0cl 31032

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fv 6571  df-ov 7434  df-sh 31236  df-ch 31250  df-ch0 31282

This theorem is referenced by:  chne0  31523  hne0  31576  h1datomi  31610  riesz3i  32091  pjnmopi  32177