Theorem ela 22915

Description: Atoms in a Hilbert lattice are the elements that cover the zero subspace. Definition of atom in [Kalmbach] p. 15. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
ela	|- ( A e. HAtoms <-> ( A e. CH /\ 0H <oH A ) )

Proof of Theorem ela

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 4028	. 2 |- ( x = A -> ( 0H <oH x <-> 0H <oH A ) )
2		df-at 22914	. 2 |- HAtoms = { x e. CH | 0H <oH x }
3	1, 2	elrab2 2926	1 |- ( A e. HAtoms <-> ( A e. CH /\ 0H <oH A ) )

Syntax hints:   <-> wb 176   /\ wa 358   e. wcel 1685   class class class wbr 4024   CHcch 21505   0Hc0h 21511   <oH ccv 21540  HAtomscat 21541

This theorem is referenced by:  elat2  22916  elatcv0  22917  atcv0  22918

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-at 22914