Theorem elat 10222

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 2619	. 2 |- (x = A -> (0H <o x <-> 0H <o A))
2		df-at 10221	. 2 |- Atoms = {x e. CH | 0H <o x}
3	1, 2	elrab2 1904	1 |- (A e. Atoms <-> (A e. CH /\ 0H <o A))

Syntax hints:   <-> wb 146   /\ wa 223   e. wcel 957   class class class wbr 2615  CHcch 8753  0Hc0h 8759  Atomscat 8788   <o ccv 8789

This theorem is referenced by:  elat2 10223  elatcv0 10224  atcv0 10225

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-rab 1650  df-v 1809  df-un 2047  df-sn 2409  df-pr 2410  df-op 2413  df-br 2616  df-at 10221

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