Theorem chne0 9371

Description: A non-zero closed subspace has a non-zero vector.

Hypothesis

Ref	Expression
ch0le.1	|- A e. CH

Assertion

Ref	Expression
chne0	|- (A =/= 0H <-> E.x e. A x =/= 0h)

Distinct variable group:   x,A

Proof of Theorem chne0

Step	Hyp	Ref	Expression
1		ch0le.1	. . 3 |- A e. CH
2	1	chshi 9092	. 2 |- A e. SH
3	2	shne0 9366	1 |- (A =/= 0H <-> E.x e. A x =/= 0h)

Syntax hints:   <-> wb 146   e. wcel 960   =/= wne 1588  E.wrex 1649  0hc0v 8786  CHcch 8793  0Hc0h 8799

This theorem is referenced by:  chne0t 9412  hne0 9465  h1datom 9499  riesz3 9990  pjnmop 10070

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-hilex 8864  ax-hv0cl 8868

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-un 2053  df-in 2054  df-ss 2056  df-sn 2416  df-pr 2417  df-sh 9071  df-ch 9087  df-ch0 9120

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