Theorem chle0 9375

Description: No Hilbert closed subspace is smaller than zero.

Hypothesis

Ref	Expression
ch0le.1	|- A e. CH

Assertion

Ref	Expression
chle0	|- (A (_ 0H <-> A = 0H)

Proof of Theorem chle0

Step	Hyp	Ref	Expression
1		ch0le.1	. 2 |- A e. CH
2		chle0t 9367	. 2 |- (A e. CH -> (A (_ 0H <-> A = 0H))
3	1, 2	ax-mp 7	1 |- (A (_ 0H <-> A = 0H)

Syntax hints:   <-> wb 146   = wceq 956   e. wcel 958   (_ wss 2047  CHcch 8798  0Hc0h 8804

This theorem is referenced by:  chj00 9410  chsup0 9471  spansnm0 9595  large 10194

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-hilex 8869

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-in 2051  df-ss 2053  df-sn 2412  df-sh 9076  df-ch 9092  df-ch0 9125

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