Theorem elatcv0 10268

Description: A Hilbert lattice element is an atom iff it covers the zero subspace.

Assertion

Ref	Expression
elatcv0	|- (A e. CH -> (A e. Atoms <-> 0H <o A))

Proof of Theorem elatcv0

Step	Hyp	Ref	Expression
1		elat 10266	. 2 |- (A e. Atoms <-> (A e. CH /\ 0H <o A))
2	1	baib 685	1 |- (A e. CH -> (A e. Atoms <-> 0H <o A))

Syntax hints:   -> wi 3   <-> wb 146   e. wcel 958   class class class wbr 2619  CHcch 8798  0Hc0h 8804  Atomscat 8833   <o ccv 8834

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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rab 1652  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-at 10265

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