Theorem elat 10174

Description: Atoms in a Hilbert lattice are the elements that cover the zero subspace. Definition of atom in [Kalmbach] p. 15.

Assertion

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

Proof of Theorem elat

Step	Hyp	Ref	Expression
1		breq2 2613	. 2 |- (x = A -> (0H <o x <-> 0H <o A))
2		df-at 10173	. 2 |- Atoms = {x e. CH | 0H <o x}
3	1, 2	elrab2 1898	1 |- (A e. Atoms <-> (A e. CH /\ 0H <o A))

Syntax hints:   <-> wb 146   /\ wa 223   e. wcel 955   class class class wbr 2609  CHcch 8737  0Hc0h 8743  Atomscat 8772   <o ccv 8773

This theorem is referenced by:  elat2 10175  elatcv0 10176  atcv0 10177

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-rab 1644  df-v 1803  df-un 2040  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-at 10173

Copyright terms: Public domain